Economic Studies 88 Peter Welz Quantitative New

ECONOMIC STUDIES 88

PETER WELZ QUANTITATIVE NEW KEYNESIAN MACROECONOMICS AND MONETARY POLICY

PETER WELZ

QUANTITATIVE NEW KEYNESIAN MACROECONOMICS AND MONETARY POLICY Department of Economics, Uppsala University

Visiting address: Kyrkogårdsgatan 10, Uppsala, Sweden Postal address: Box 513, SE-751 20 Uppsala, Sweden Telephone: +46 18 471 11 06 Telefax: +46 18 471 14 78 Internet: http://www.nek.uu.se/

______

ECONOMICS AT UPPSALA UNIVERSITY

The Department of Economics at Uppsala University has a long history. The first chair in Economics in the Nordic countries was instituted at Uppsala University in 1741.

The main focus of research at the department has varied over the years but has typically been oriented towards policy-relevant applied economics, including both theoretical and empirical studies. The currently most active areas of research can be grouped into six categories: x Labour economics x Public economics x Macroeconomics x Microeconometrics x Environmental economics x Housing and urban economics

______

Additional information about research in progress and published reports is given in our project catalogue. The catalogue can be ordered directly from the Department of Economics.

Abstract Doctoral dissertation to be publicly examined in Horsal¨ 1, Ekonomikum, Uppsala University, 28 October, 2005, at 10.15 a.m. for the degree of Doctor of Philosophy. The examination will be held in English.

WELZ, Peter, 2005, Quantitative New Keynesian Macroeconomics and Monetary Policy; Department of Economics, Uppsala University, Economic Studies 88, xviii, 128 pp., ISBN 91-87268-95-7.

This thesis consists of four self-contained essays. Essay 1 compares the dynamic behaviour of an estimated New Keynesian sticky- price model with one-period delayed effects of monetary policy shocks to the dynamics of a structural vector autoregression model. The model is estimated with Bayesian techniques on German pre-EMU data. The dynamics of the sticky-price model following either a demand shock or monetary policy shock are qualitatively and quantitatively comparable to those of the estimated structural VAR. When compared to the delayed-effects model, an alternative model with contemporaneous effects of monetary policy is rejected according to the posterior-odds ratio criterion. Essay 2 addresses the transmission of exchange-rate variations in an estimated, small open-economy model. In contrast to the standard New Open Economy Macroeco- nomics framework, imported goods are treated here as material inputs to production. The resulting model structure is transparent and tractable while also able to account for imperfect pass through of exchange-rate shocks. The model is estimated with Bayesian methods on German data and the key finding is that a substantial depreciation of the nominal exchange rate leads to only modest effects on CPI inflation. An extended version of the model reveals that relatively small weight is placed on foreign consumption. Essay 3 (with Annika Alexius) analyses the strong responses of long-term interest rates to shocks that are difficult to explain with standard macroeconomic models. Augmenting the standard model to include a time-varying equilibrium real interest rate generates forward rates that exhibit considerable movement at long horizons in response to movements of the policy-controlled short rate. In terms of coefficients from regressions of long-rate changes on short-rate movements, incorporating a time- varying natural rate explains a significant fraction of the excess sensitivity puzzle. Essay 4 (with Par¨ Osterholm)¨ argues that the common finding of a large and significant coefficient on the lagged interest rate in Taylor rules may be the consequence of misspecification, specifically an omitted variables problem. Our Monte Carlo study shows that omitting relevant variables from the estimated Taylor rule can generate significant partial-adjustment coefficients, despite the data generating process containing no interest-rate smoothing. We further show that misspecification leads to considerable size distortions in two recently proposed tests to distinguish between interest-rate smoothing and serially-correlated disturbances. Peter Welz, Department of Economics, Uppsala University, P.O. Box 513, SE-75120 Uppsala, Sweden ISSN 0283-7668 ISBN 91-87268-95-7

Acknowledgements

Writing one’s thesis is as much a product of intellectual discourse and collaboration as of lonely hours at the writing desk. As a result, I am grateful to many people. I would like to thank my supervisor, Nils Gottfries, for many insightful discussions, encouragement and support throughout my graduate studies. His valuable suggestions, comments and careful reading of initial drafts have helped transform these into essays that form this thesis. I am indebted to Annika Alexius for her ideas, collaboration and comments and to Anders Klevmarken for guiding me through the first year of the graduate programme. Over the course of my graduate studies I have also benefited from stimulating research environments at the Monetary Policy and the Research Departments at Sveriges Riksbank, the Department of Economics at Universitat Pompeu Fabra, the Econometric Modelling Unit of the Research Department at the European Central Bank, the Ifo Institute and the Department of Economics at the University of Munich. I would like to thank the staff at these institutions for their kind hospitality. I am particularly grateful to Ulrich Woitek for his invitations to Munich. I have appreciated and benefited from insightful discussions with Peter McAdam, Malin Adolfson, Ramon´ Adalid-Lozano, Mikael Carlsson, Kirsten Hubrich, Keith Kuster,¨ Frank Smets, Ulf Soderstr¨ om,¨ Roland Straub, Mathias Trabandt, Mattias Villani, Ulrich Woitek and my co-author, Par¨ Osterholm.¨ Sune Karlsson and Jesper Linde´ were kind enough to offer thorough comments on my work, leading to considerable improvements. In Fabio Canova, Sune Karlsson and Mark Steel I have found three excellent teachers who introduced me into the exciting field of Bayesian Statistics and Econometrics. Volker Clausen provided encouragement throughout my studies. Working for him as a research assistant while undergraduate student gave me the opportunity to share his enthusiasm for economic research and laid the basis for my decision to embark on a doctoral programme. The administrative staff at the department, especially Eva Holst, have shown utmost efficiency and professionalism. Many thanks also to Åke Qvarfort for providing excellent computing services, and beyond that, entertaining conversations. A thank you goes to my felIow students for lightening life as a graduate student: especially to Qian Liu, for inspiring and entertaining lunches, Mikael Nordberg for being a discussion partner during the night shifts and Hanna Ågren, Ranjula Bali Swain and Jovan Zamac for our talks

ix x Acknowledgements beyond economics. In Alexandre Dmitriev I found an enthusiastic office mate, friend and team worker who keenly took on any intellectual discussion and contributed to finalising most mathematical proofs ‘q.e.d.’. Meredith Beechey carefully proofread the manuscript and provided valuable comments on Chapter 3. Financial support from the Jan Wallander and Tom Hedelius foundation, C. Borgstrom¨ and C. Berch’s Foundation, the Ifo Institute Munich and Centro de Estudios Monetarios y Financieros (CEMFI) Madrid, is gratefully acknowledged. Finally I am most grateful to Shanti, Uwe and my mother for their encouragement and support and their patient attempts to stabilise mood cycles. Without their help this project would not have reached this final stage.

Peter Welz Frankfurt am Main, September 2005 Table of Contents

Acknowledgements ix

Introduction 1

1 Assessing Predetermined Expectations in the Standard Sticky Price Model: A Bayesian Approach 9 1.1 Introduction ...... 9 1.2 The Sticky Price Model ...... 11 1.2.1 Households ...... 11 1.2.2 Firms ...... 13 1.2.3 Central Bank ...... 16 1.2.4 Solution of the Model ...... 17 1.3 Estimation ...... 18 1.3.1 Data ...... 18 1.3.2 Estimation Methodology ...... 19 1.3.3 Speciﬁcation of Priors ...... 21 1.4 Results ...... 22 1.4.1 Parameter Estimates ...... 22 1.4.2 Empirical Performance of the Model ...... 25 1.4.3 Impulse-Response Analysis ...... 28 1.4.4 Comparison to VAR ...... 30 1.4.5 Comparison to a Model with Contemporaneous Eﬀects ...... 31 1.4.6 Estimation Diagnostics ...... 33 1.5 Summary and Conclusions ...... 34 Appendices ...... 36 A Model with Delays ...... 36 A.1 Matrix Representation ...... 36 B Bayesian Concepts ...... 40 B.1 Metropolis-Hastings Algorithm ...... 40 B.2 Marginal Likelihood Computation...... 40

xi xii TABLE OF CONTENTS

C Model with Contemporaneous Eﬀects ...... 42 C.1 Estimation Results ...... 42 C.2 Prior and Posterior Kernels ...... 43 C.3 Impulse Responses ...... 44

2 Transmission of Exchange-Rate Variations in an Estimated, Small-Open Economy Model 51 2.1 An Alternative Open-Economy Model ...... 53 2.1.1 Aggregate Supply ...... 54 2.1.2 Aggregate Demand and Wage Setting ...... 56 2.1.3 Monetary Policy ...... 59 2.1.4 Foreign Economy ...... 59 2.2 Including Foreign Consumption Goods ...... 60 2.3 Solution and Estimation ...... 62 2.3.1 Model Solution ...... 62 2.3.2 Methodology ...... 62 2.4 Data and Prior Speciﬁcation ...... 63 2.4.1 Data ...... 63 2.4.2 Prior Speciﬁcation ...... 64 2.5 Results ...... 65 2.5.1 The Benchmark Model ...... 65 2.5.2 Transmission of Shocks in the Benchmark Model ...... 67 2.5.3 The Extended Model ...... 68 2.5.4 Evaluation ...... 69 2.6 Conclusions ...... 70 Appendices ...... 72 A Data and Sources ...... 72 B Derivation of Model Dynamics in the Benchmark Model ...... 74 B.1 Firms ...... 74 B.2 Aggregate Demand ...... 75 B.3 Net Foreign Assets ...... 76 C The Extended Model ...... 77 D Figures and Results ...... 79 D.1 Benchmark Model ...... 79 D.2 Exchange-Rate Pass Through and Price Stickiness ...... 83 D.3 Prior- and Posterior Density - Extended Model ...... 84 TABLE OF CONTENTS xiii

3 Can a Time-Varying Equilibrium Real Interest Rate Explain the Excess Sensitivity Puzzle? 91 3.1 Introduction ...... 91 3.2 A Stylised Model ...... 93 3.3 Estimating a Time-Varying Equilibrium Real Rate ...... 95 3.3.1 Data ...... 96 3.3.2 Empirical Speciﬁcation ...... 96 3.3.3 Results ...... 98 3.4 The Puzzle Explained? ...... 100 3.4.1 Impulse Responses of Nominal Interest Rates ...... 101 3.4.2 Regression Evidence from Interest Rate Changes ...... 103 3.4.3 Sensitivity Analysis ...... 104 3.5 Conclusions ...... 105 Appendices ...... 107 A Derivation of the Equilibrium Real Rate ...... 107 B State Space Representation ...... 108 C Conditional Likelihoods ...... 109 D Output Gap and Real Rate Gap ...... 110

4 Interest-Rate Smoothing versus Serially Correlated Errors in Taylor Rules: Testing the Tests 113 4.1 Introduction ...... 113 4.2 The Taylor Rule ...... 115 4.2.1 Basic Speciﬁcation and Empirical Evidence ...... 115 4.2.2 Related Literature ...... 117 4.3 Two Tests for Interest-Rate Smoothing ...... 118 4.4 Model and Data Generating Process ...... 119 4.5 Simulations and Results ...... 120 4.5.1 Results ...... 120 4.6 Discussion and Conclusion ...... 124

List of Figures

1.1 Delayed Eﬀects Model: Prior- and Posterior Density ...... 24 1.2 Autocorrelation Functions ...... 26 1.3 Demand and Monetary Policy Shock ...... 28 1.4 Cost- and Technology Shock ...... 29 1.5 Dynamics of DSGE- and VAR Model ...... 31 1.6 CUSUM-Test ...... 34 1.7 Contemporaneous Eﬀects Model: Prior- and Posterior Density ...... 43 1.8 Demand- and Monetary Policy Shock ...... 44 1.9 Cost- and Technology Shock ...... 45

2.1 Uncovered Interest-Parity Shock - Benchmark Model ...... 67 2.2 Monetary Policy Shock - Benchmark Model ...... 68 2.3 Uncovered Interest-Parity Shock - Extended Model ...... 69 2.4 Prior- and Posterior Density - Benchmark Model ...... 79 2.5 Prior- and Posterior Density (continued) - Benchmark Model ...... 80 2.6 CUSUM Diagnostic - Benchmark Model ...... 81 2.7 CUSUM Diagnostic (continued) - Benchmark Model ...... 82 2.8 Exchange-Rate Pass Through and Price Stickiness ...... 83 2.9 Prior- and Posterior Density - Extended Model ...... 85 2.10 Prior- and Posterior Density (continued) - Extended Model ...... 86

3.1 Estimated Equilibrium Real Rate ...... 98 3.2 Model With Constant Equilibrium Real Rate ...... 102 3.3 Model with Time-Varying Equilibrium Real Rate ...... 103 3.4 Optimisation Diagnostic ...... 109 3.5 Output Gap and Real Rate Gap ...... 110

4.1 First Test under the Assumption of Interest-Rate Smoothing ...... 121 4.2 First Test under the Assumption of Serially Correlated Errors ...... 122 4.3 First test for the Reduced Form ...... 123 4.4 Second Test: Allowing for Smoothing and Serial Correlation ...... 124

List of Tables

1.1 Prior Speciﬁcation and Posterior Estimates ...... 23 1.2 Standard Deviations of Simulated and Actual Data ...... 25 1.3 Acceleration Phenomenon and Output-Interest-Rate Dynamics ...... 27 1.4 Model Comparison by Bayes Factors ...... 33 1.5 Prior Speciﬁcation and Posterior Estimates ...... 42

2.1 Prior Specification and Estimation results - Benchmark model ...... 66 2.2 Correlations between Inflation and Nominal Depreciation ...... 70 2.3 Prior Specification and Estimation Results - Extended Model ...... 84

3.1 Estimation Results ...... 99 3.2 Model Calibration ...... 101 3.3 Regression Results ...... 104

4.1 First Test under the Assumption of Interest-Rate Smoothing ...... 121 4.2 First Test under the Assumption of Serially Correlated Errors ...... 122 4.3 First Test for the Reduced Form ...... 123 4.4 Second Test: Allowing for Smoothing and Serial Correlation ...... 124 4.5 Documenting the Distortion of the Test Size ...... 125

xvii

Introduction

This thesis consists of four self-contained essays. The first two essays investigate the empirical properties of dynamic stochastic general equilibrium (DSGE) models by means of Bayesian estimation. The third essay offers an explanation for the sensitivity of long-term interest rates and the last essay examines the econometric properties of the well-known Taylor rule. A common theme linking the four essays is the presence of endogenous monetary policy in which the central bank pursues the twin aims of restoring inflation to target and stabilising economic activity. The term ‘New Keynesian Macroeconomics’ arises throughout the thesis and war- rants explanation. The first and second essays analyse a class of models that has become prominent over the last decade, labelled collectively as the ‘New Neoclassical Synthesis’ by Goodfriend and King (1997) or as ‘New Keynesian Models’ by Gal´ı (2003). Well over a decade ago, Mankiw and Romer (1991) edited two volumes under the title ‘New Keynesian Economics’. What do these labels represent? As Goodfriend and King’s label suggests, such models are the result of a synthesis of real business cycle (RBC) theory and New Keynesian theory. Following in the same vein as RBC models, a central assumption in modern DSGE models is that rationally optimising agents make forward-looking decisions based on the solution to an intertemporal optimisation problem. Keynesian elements, such as nominal rigidities and imperfect competition, are introduced to make such models more suitable for the analysis of monetary policy. The neoclassical (RBC theory) aspect yields a dynamic equilibrium, a path that is reached when all prices and wages are perfectly flexible. Deviations around this equilibrium are crucial in influencing wage- and price-setting decisions. Prices are assumed to be sticky, yielding the Keynesian charac- teristic that monetary policy has real effects in the short-run but is neutral in the long-run, making it the ideal tool for stabilisation policy. This class of models is attractive from a theoretical perspective because the structural relations are based on microfoundations, rendering the parameters invariant to policy and thus robust to the Lucas critique. The utility-based framework allows derivation of welfare measures for normative policy analysis and has triggered a rich literature on the optimality of monetary policy actions (Woodford, 2003). Another important ingredient of the first two essays is the econometric approach taken

1 2 Introduction to estimate DSGE models. The standard approach in RBC theory, pioneered by Kydland and Prescott (1982) and Long and Plosser (1983), has been to calibrate parameters and compare moments generated from the model with those of actual data. This method, however, lacks formal statistical foundations (Kim and Pagan, 1994) and hinders testing the results. Sargent (1989) suggested as an alternative maximum likelihood estimation of DSGE models, but potential misspecification due to omitted non-linearities, incorrect assumptions about preferences and technology or incorrectly-specified exogenous shocks easily lead to computational difficulties (Lubik and Schorfheide, 2005).

The Bayesian estimation methodology chosen here follows developments by DeJong et al. (2000a,b), Otrok (2001) and Smets and Wouters (2003). Bayesian analysis formally incorporates uncertainty and prior information regarding the parametrisation of the model. It combines the likelihood with a priori information on the parameters of interest that may have come from earlier microeconometric or macroeconometric studies. By introducing prior information about the structural parameters in the form of probability densities, the likelihood function is re-weighted by the prior density. The degree of uncertainty about the prior information can thereby be expressed in terms of the standard deviation of the prior density. Hence, the common practice of fixing some parameters in maximum likelihood estimation has the Bayesian interpretation that no uncertainty exists about the chosen values. The Bayesian approach also allows a formal comparison of different, and not necessarily nested, models through the marginal likelihood of the model. This approach is employed in both the first and second essays to compare alternative model specifications.

In the following, I will summarise each chapter and discuss the main findings in turn. Chapter 1, Assessing Predetermined Expectations in the Standard Sticky-Price Model: A Bayesian Approach, estimates a small scale sticky-price model with delayed effects of monetary policy. Motivated by stylised facts rather than strict microfoundations, monetary impulses are assumed to have a one-period delayed effect on output and inflation. This is built into the model by having agents form their expectations conditional on information from the previous period. A fraction of consumers and firms use a simple rule-of-thumb when deciding on consumption expenditure and prices, respectively. Rule- of-thumb consumers simply choose the average aggregate consumption level from the previous period and rule-of-thumb price setters update last period’s average price using last period’s inflation. The approach permits comparison of the dynamics of the DSGE model to those of a recursively estimated VAR on the same data, even when the two identification schemes are not identical. The setup is particularly interesting because a rich literature on monetary VAR models has documented that a recursive identification scheme fits the data quite well. Introduction 3

Rather than use synthetic data for the Euro area as is often done, the model is estimated with German data prior to the advent of the European Monetary Union (EMU). The German economy represents the largest share in the EMU and its stable monetary policy regime over almost 25 years makes this an attractive choice of data set. The main findings of the chapter are the following. The inflation adjustment equation exhibits considerable weight on expected inflation while the demand equation is almost purely backward-looking, with prices estimated to be fixed for about 6.5 quarters. The dynamics of the DSGE model following a demand shock and monetary policy shock, respectively, are qualitatively and quantitatively comparable to those of the estimated structural VAR. When compared to the delayed-effects model, an alternative DSGE model that allows for contemporaneous effects of monetary policy is rejected according to the posterior-odds ratio criterion. Chapter 2, Transmission of Exchange Rate Variations in an Estimated, Small-Open Economy Model, extends the closed economy model with contemporaneous effects of monetary policy to the open economy. The focus is upon how exchange-rate shocks, modelled as exogenous impulses to the uncovered interest parity (UIP) condition, influence domestic consumer prices. As opposed to the standard assumption in the New Open Economy Macroeconomics (NOEM) literature but in line with McCallum and Nelson (1999, 2000), foreign goods are assumed to be intermediate inputs to production. Hence, no distinction is made between foreign and domestic consumption goods and their associated price indices. Domestic producers use labour and intermediate inputs to produce final consumption goods. As in Chapter 1, domestic firms are assumed to set their prices in a staggered fashion. In this international setting, it implies that exchange-rate shocks which influence the import price of intermediate goods (that is, the real exchange rate) do not completely pass through onto consumer price inflation in the initial period. How- ever, in the long-run pass-through is complete, in line with the empirical exchange-rate literature (Campa and Goldberg, 2002). The motivation for this approach is based on an observation of Burstein et al. (2005), amongst others, that final consumption goods account for a small fraction of all imported goods paired with the fact that most products contain services with non-tradeable goods characteristics. The model is again estimated on data from Germany, where according to the input-output tables, imported consumption goods account only for 9 percent of total final private consumption expenditure. The benchmark model is compared to one that explicitly allows for imported consumption goods. In order to make this channel comparable to the transmission channel on the production side, retail firms are assumed to import consumption goods and set prices in an analogous manner to domestic producers. The results are as follows: a shock to 4 Introduction the UIP condition that causes a 1.5 percent nominal depreciation raises marginal cost by 0.36 percent and prices by less than 0.1 percent (or 5.5 percent of the nominal depreciation). In the extended model, the estimated fraction of imported consumption is small (5 percent) but a low degree of price stickiness in the import sector results in a much larger pass through of exchange rate shocks to domestic consumer price inflation. On pure statistical grounds, that is in terms of the posterior odds ratio, the two models cannot be distinguished. However, judged by correlations between contemporaneous inflation and contemporaneous and past nominal depreciations, the benchmark model without foreign consumption goods comes closer to correlations in the actual data. Chapter 3, Can a Time-Varying Equilibrium Real Interest Rate Explain the Excess Sensitivity Puzzle?, is co-authored with Annika Alexius. We address the empirical regularity that long-term interest rates display considerable movements in response to short- term interest rate shocks, in contrast to the predictions of standard macroeconomic models. While the literature has largely focused on learning mechanisms (Beechey, 2004; Orphanides and Williams, 2003) and varying central bank preferences (Ellingsen and Soderstr¨ om,¨ 2005), our hypothesis is that a time-varying equilibrium real interest rate can contribute to solving the puzzle. A constant equilibrium real interest rate is a common assumption, for example, the mean of the real rate over the sample period. Our suggested mechanism instead assumes that the equilibrium real rate is time-varying and persistent because it is a linear combination of highly persistent shocks. We assume that monetary policy can be described by a Taylor rule in which the equilibrium real rate is embedded in the intercept. Because of the central bank’s response to movements in this real interest rate, we get the result that forward rates fluctuate more than in the model with a constant equilibrium real rate. We estimate the time-varying equilibrium real rate to be close to a random walk in an unobserved components model and this measure is then used to augment a semi-structural New Keynesian model. We find that forward rates move between 29 and 117 basis points at the ten year horizon in response to a 100 basis points increase in the short rate triggered by structural shocks. Simulated data from the model is used to show that in terms of the coefficients in regressions of long-rate movements on short-rate movements, incorporating a time-varying equilibrium real rate contributes significantly to explaining the sensitivity puzzle. Chapter 4, Interest-Rate Smoothing versus Serially Correlated Errors in Taylor Rules: Testing the Tests, is written with Par¨ Osterholm.¨ In the paper we examine the size properties of two tests recently designed by English et al. (2003) to distinguish between interest- rate smoothing and autocorrelated errors in estimated Taylor rules. A common empirical finding when estimating Taylor rules is a high degree of interest-rate smoothing, implying Introduction 5 an implausibly slow partial-adjustment mechanism. Even with a large coefficient on the lagged interest rate, estimated equations typically continue to exhibit serially correlated errors. Our working hypothesis is that a high degree of interest-rate smoothing may indicate omitted variable bias. If in fact the central bank responds to more variables than just deviations of inflation from target and the output gap, the omission of relevant explanatory variables that are not orthogonal to inflation and the output gap renders the estimated coefficients inconsistent. Consequently, this leads to the failure of the English et al. (2003) tests since they are based on autocorrelation correction to account for omitted variables. In this chapter we set up a Monte Carlo analysis that employs an estimated structural VAR for the U.S. economy as the basis for the data generating process. The results demonstrate that omitted variable bias has the potential to generate a falsely significant lagged interest-rate term in a Taylor rule. In addition, we are able to show that the tests have a much larger empirical size than the nominal size of 5 percent. Whilst we do not identify which variables are most likely to cause the omitted variable bias, our results do point to problems with empirical Taylor rules that feature a high degree of interest-rate smoothing. 6 Introduction

References

Beechey, M. (2004). Excess sensitivity and volatility of long interest rates: The role of limited information in bond markets. Sveriges Riksbank Working Paper No 173, Sveriges Riksbank.

Burstein, A., Eichenbaum, M., and Rebelo, S. (2005). Large devaluations and the real exchange rate. Journal of Political Economy (forthcoming).

Campa, J. M. and Goldberg, L. S. (2002). Exchange rate pass-through into import prices. Review of Economics and Statistics (forthcoming).

DeJong, D., Ingram, B., and Whiteman, C. (2000a). A Bayesian approach to dynamic macroeconomics. Journal of Econometrics, 98(2):203–223.

DeJong, D., Ingram, B., and Whiteman, C. (2000b). Keynesian impulses versus Solow residuals: Identifying sources of business cycle ﬂuctuations. Journal of Applied Econo- metrics, 15(3):311–329.

Ellingsen, T. and Soderstr¨ om,¨ U. (2005). Why are long rates sensitive to monetary policy? Working Paper, Bocconi University.

English, W. B., Nelson, W. R., and Sack, B. P. (2003). Interpreting the signiﬁcance of the lagged interest rate in estimated monetary policy rules. Contributions to Macroeco- nomics, 3(1):Article 5.

Gal´ı, J. (2003). New perspectives on monetary policy, inﬂation and the business cycle. In Dewatripont, M., Hansen, L., and Turnovsky, S., editors, Advances in Economics and Econometrics: Theory and Applications. Eighth World Congress, volume III, pages 151–197. Cambridge University Press.

Goodfriend, M. and King, R. G. (1997). The New Neoclassical Synthesis and the role of monetary policy. NBER Macroeconomics Annual, 12:231–283.

Kim, K. and Pagan, A. (1994). The econometric analysis of calibrated macroeconomic models. In Pesaran, H. and Wickens, M., editors, Handbook of Applied Econometrics, volume 1. Blackwell Press, London.

Kydland, F. E. and Prescott, E. C. (1982). Time to build and aggregate ﬂuctuations. Econometrica, 50(6):1345–1370.

Long, J. B. and Plosser, C. (1983). Real business cycles. Journal of Political Economy, 91(1):39–69.

Lubik, T. and Schorfheide, F. (2005). A Bayesian look at New Open Economy Macro- economics. NBER Macroeconomics Annual (forthcoming).

Mankiw, N. G. and Romer, D., editors (1991). New Keynesian Economics, volume 1+2. MIT Press, Cambridge. REFERENCES 7

McCallum, B. T. and Nelson, E. (1999). Nominal income targeting in an open-economy optimizing model. Journal of Monetary Economics, 43:553–578.

McCallum, B. T. and Nelson, E. (2000). Monetary policy for an open economy: An alternative framework with optimizing agents and sticky prices. Oxford Review of Economic Policy, 16(4):74–91.

Orphanides, A. and Williams, J. C. (2003). Imperfect knowledge, inﬂation expectations, and monetary policy. NBER Working Paper No. 9884.

Otrok, C. (2001). On measuring the welfare cost of business cycles. Journal of Monetary Economics, 47(1):61–92.

Sargent, T. (1989). Two models of measurements and the investment accelerator. Journal of Political Economy, 97(2):251–287.

Smets, F. and Wouters, R. (2003). An estimated dynamic stochastic general equilibrium model of the Euro area. Journal of the European Economic Association, 1(5):1123– 1175.

Woodford, M. (2003). Interest and Prices: Foundations of a Theory of Monetary Policy. Princeton University Press, Princeton, New Jersey.

Chapter 1

Assessing Predetermined Expectations in the Standard Sticky Price Model: A Bayesian Approach

1.1 Introduction

In recent years a new paradigm has arisen in macroeconomics that combines elements of real business cycle theory (RBC) and New Keynesian Macroeconomics (NKM). The standard model involves a dynamic stochastic general equilibrium (DSGE) structure with intertemporally optimising agents who are assumed to make decisions based on rational expectations, an assumption that reflects the RBC origins of the paradigm. As a result, equilibrium conditions for aggregate variables can be computed from the optimal individual behaviour of consumers and firms. NKM features are introduced by explicitly allowing for monopolistic competition as well as costly - and therefore gradual - price and/or wage adjustment. In this environment, monetary policy takes on a stabilisation role because actions taken by the monetary authority have significant effects on real economic activity in the short- to medium run. Furthermore, due to the rigorous microfoundations on which such models are based, it is possible to evaluate the welfare implications of alternative policy regimes. Ideally, such evaluations should serve as the basis for economic policy advice.

The purpose of the present study is to estimate a small-scale DSGE model with sticky prices for the German economy prior to European Monetary Union (EMU). A number of authors, including Smets and Wouters (2003, 2004), Adolfson et al. (2005), Levin et al. (2005), have recently estimated medium- to large scale models and found that these models ﬁt the data fairly well. This paper deviates from the existing literature in two respects. The model is kept much simpler and thus closer to the types of models commonly used for

9 10 1. Assessing Predetermined Expectations normative monetary policy analysis.1 In addition, I introduce the assumption that the consumption and price-setting decisions of optimising agents are determined one period in advance. In this way decisions are based on information up to and including the previous period which introduces a delayed effect of monetary impulses on output and inflation in the model.2 While not dictated by microfoundations, this assumption makes it possible to compare the impulse responses of the estimated DSGE model to those of a (recursively) identified VAR model, even when the two identification schemes differ. Many VAR studies of monetary policy have found that an identification scheme that leads to one-period delayed effects of monetary impulses on output and inflation fit the data quite well, at least in closed economies.3 It is therefore interesting to understand the effect of a similar identification strategy within a DSGE model. The model in this paper also deviates from its simplest counterpart through the assumption of endogenous persistence on both the demand and supply side. I introduce endogenous persistence by assuming that the population can be divided into two types. One group solves an optimisation problem according to the rational expectations hypothesis whereas the other group deviates from fully rational behaviour and follows a rule- of-thumb. Specifically, rule-of-thumb individuals make decisions based on information from the previous period rather than optimising over an infinite horizon. This assumption may be justified because such forward-looking optimisation is complicated and costly and requires acquiring large amounts of information. At a more general level, another motivation for introducing endogenous persistence is that the purely forward-looking sticky-price model cannot account for observed persistence in inflation and consumption (Fuhrer and Moore, 1995). Because the sticky price model is designed to analyse the short-run effects of monetary policy and to study optimal monetary policy, it is important that the model can account for the empirical regularities. Other studies have employed habit formation to introduce endogenous persistence on the demand side (McCallum and Nelson, 1999) or indexation of a fraction of prices to past inflation to generate persistence on the supply side (Christiano et al., 2005). In contrast, this paper provides a consistent modelling perspective by assuming rule-of-thumb behaviour in both consumption and price-setting, thus treating both sides of the economy symmetrically. Several of the above-mentioned studies use a synthetic Euro area data set rather than data for individual countries within the EMU. However, analysis at the country level is important and the focus of this paper will be on the German economy. Germany deserves special attention not only because of its relative importance in the aggregate EMU

1But see Levin et al. (2005) for an exception. 2Woodford (2003, Chapter 4) also discusses delayed effects of monetary policy. 3Favero (2001) provides an overview of this literature. 1.2. The Sticky Price Model 11 economy but because of its unique monetary regime for the two decades prior to EMU. The model is estimated with techniques developed by DeJong et al. (2000a,b) and Otrok (2001). The approach takes a Bayesian view that formally incorporates prior information about the parameters of the model. Smets and Wouters (2003, 2004) apply this technique to estimate the 34 parameters of a New Keynesian model with capital investment, sticky prices and wages using Euro area data. Adolfson et al. (2005) apply the same method to estimate an open-economy version of the model with a larger number of parameters. These larger models clearly have a greater chance of empirical success but deviate from the parsimonious sticky-price model commonly used for optimal monetary policy analysis. They also require a number of additional assumptions about the exact investment technology and the use of capital. The approach here is more modest in at- tempting to fit a small-scale New Keynesian model with 17 parameters. Thus, the model resembles more closely the standard class of models used in theory. The remainder of the paper is structured as follows. Section 1.2 presents the theoretical model and discusses the solution method while Section 1.3 covers the estimation methodology and specification of priors. In Section 1.4, the results are presented and the DSGE impulse response functions are compared to those from an identified VAR. Section 1.5 summarises and draws some conclusions.

1.2 The Sticky Price Model

The theoretical model used for estimation purposes here is an extension of the standard sticky-price model with ﬁxed capital commonly used for the analysis of optimal monetary policy (Gal´ı, 2003; Woodford, 2003). Only a brief description is given in this section. No explicit reference is made to money balances because the central bank is assumed to follow an interest rate rule. Introducing money balances for instance into an additively separable utility function, would only add a money demand equation which endogenously determines the magnitude of money balances without aﬀecting the general results.

1.2.1 Households

The economy consists of a continuum of inﬁnitely-lived consumers of measure one where each individual is indexed by j ∈ [0, 1]. It is assumed that expenditure decisions are made one period in advance and subsequently altered only due to disturbances to preferences. Following Amato and Laubach (2003) I assume that re-optimisation is costly due to information-gathering or information-processing constraints. Hence, every period a randomly chosen fraction of households 1 − αy decides to base its consumption decision on optimising behaviour, whereas the remaining fraction αy follows a rule-of-thumb that 12 1. Assessing Predetermined Expectations simply implies choosing the optimal consumption level from the previous period, i.e.

r = . Ct Ct−1 (1.1) Assuming that the individual household is too small to affect the level of consumption ∞ Cr, the re-optimisation problem is to find a sequence Co that maximises the present t jt t=1 discounted value of expected life-time utility   ∞  o 1−σ +ϕ  C N 1  s−t gs js us js Et−1 β e  − e  , (1.2)  1 − σ 1 + ϕ  s=t where β is a discount factor, gt is a preference shock affecting the individual’s time discount factor, and an individual’s disutility derives from supplying work hours, N jt, perturbed by ut (to be explained below). The intertemporal elasticity of substitution is defined by σ−1 and labour supply elasticity is denoted by ϕ−1. Note that the expectation in (1.2) is conditional upon information up to and including time period t − 1, reflecting the predetermined nature of the expenditure decision. Aggregate consumption in the economy is given by the standard Dixit-Stiglitz aggregate 1 −1 −1 Ct = Cit di , (1.3) 0 where >1 denotes the elasticity of substitution among the varieties of goods. The associated aggregate price index that gives the minimum expenditure PitCit for which the amount Ct(i) of the composite consumption basket can be purchased is given by

1 1 1− 1− Pt = Pit di . (1.4) 0 This speciﬁcation leads to the familiar isoelastic demand function for each variety of the consumption good − Pit Cit = Ct. (1.5) Pt Financial markets are assumed to be complete in this economy, that is, each household can insure against any type of idiosyncratic risk through purchase of the appropriate portfolio of securities. Since by assumption households are identical ex ante they are willing to enter such insurance contracts. The advantage of this assumption is that the represen- tative agent framework can be preserved, avoiding the need to keep track of an additional state variable of households’ wealth distribution. As a result of the homogeneity assump- o, tion, all optimising households choose the same level of consumption Ct and per capita ≡ α o + − α r. consumption in period t is given by Ct yCt (1 y)Ct Each household then faces the same ﬂow budget constraint

−1 PtCt + (1 + Rt) Bt ≤ Bt−1 + WtNt + Tt, (1.6) 1.2. The Sticky Price Model 13

i.e. households’ income consists of security holdings from the previous period, Bt−1, labour income, Wt, and a transfer, Tt, that they receive in order to balance the wealth eﬀects of choosing consumption according to the optimality condition instead of the rule- of-thumb (Amato and Laubach, 2003). Since the model also abstracts from government expenditure, goods-market clearing requires that Ct = Yt in each time period. Thus the rule-of-thumb for consumption in (1.1) r = becomes Ct Yt−1 and output in period t is given by

= − α o + α . Yt (1 y)Ct yYt−1 (1.7)

Maximising (1.2) subject to the budget constraint (1.6) and substituting the market clearing condition and the output relation (1.7) yields an Euler equation whose log-linearised form leads to the following intertemporal IS equation: (1 − α )δ (1 − α )δ = δ { } + − δ − y { − π } + y − { } , yt Et−1 yt+1 (1 )yt−1 σ Et−1 it t+1 σ (gt Et−1 gt+1 ) (1.8) −1 where δ ≡ 1 + αy . The equation is log-linearised around a zero inflation steady state, so πt ≡ log(Pt/Pt−1) is the inflation rate and it is the percent deviation from its steady- state level associated with zero inflation. Furthermore, yt denotes the percent deviation of output from its steady state level. For the case in which all households base their consumption decisions on optimisation, i.e. αy = 0, and there are no implementation delays, the standard intertemporal IS equation is obtained:

−1 −1 yt = Et {yt+1} − σ (it − Et {πt+1}) − σ Et {∆gt+1} . (1.9)

Comparing (1.8) with (1.9) we notice that introducing rule-of-thumb behaviour in consumption generates a backward-looking term in the IS equation. This is appealing from an empirical point of view as will become clear below.

1.2.2 Firms

Firms indexed by i ∈ [0, 1] produce a continuum of goods in a monopolistically competitive market with a decreasing returns-to-scale technology perturbed by an exogenous labour productivity shock at that is common to all ﬁrms:

at α Yit = (e Nit) , (1.10)

Since α<1, firms with different production levels face different real marginal cost given by 1 W N W MC = t = it t , it α α α−1 α At Nit Pt Yit Pt 14 1. Assessing Predetermined Expectations which can be related to average marginal cost by

Nt Wt MCt = . αYt Pt

Using the production function, the demand function (2.9) and Yit = Cit, the following relationship can be derived in log-linearised form (1 − α) = − − .4 mcit mct α (pit pt)

Then, real marginal cost can be shown to be given by − α + ασ + ϕ = 1 − + ϕ + = + mct α yt (1 )at ut mct ut, (1.11) where the first-order condition with respect to the labour decision has been substituted in.5 Turning to price setting, I make the same assumption as in Gal´ı and Gertler (1999) and Amato and Laubach (2003) that a fraction of firms re-optimise their prices and another fraction sets prices following a rule-of-thumb. Those firms who are assumed to optimise follow the setup suggested by Calvo (1983); every period a random fraction 1 − θ of firms resets prices to the new optimal price whereas the remaining fraction of firms leaves prices unchanged from the period before. In addition, I assume that a fraction απ does not act according to Calvo’s price-setting mechanism but uses a backward-looking rule-of-thumb for setting their prices.6 Analogous to the motivation for the rule-of-thumb behaviour on the demand side, this could be justified by the fact that it is time-consuming to gather information about the stance of the economy, costly to obtain this information and that firms possess limited information-processing capacity. In addition, in order to match the commonly-made assumption in identified VAR models that monetary disturbances do not have contemporaneous effects on inflation, I assume that newly chosen prices take effect one period later (Woodford, 2003, Chapter 4). With these assumptions, the log-linearised aggregate price level evolves according to

= θ + − θ ∗, pt pt−1 (1 )pt (1.12)

∗ where pt is the (log-linearised) price index of prices set in period t, ∗ = − α f + α b. pt (1 π)pt π pt (1.13)

f The latter is a convex combination of the price pt set by the forward-looking firms follow- b ing the Calvo (1983) rule and the price pt set by the remaining backward-looking firms 4See for instance Sbordone (2002) or Walsh (2003), Chapter 5. 5 This condition (in log-linearised form) is given by wt − pt = ut + ϕnt + σct. 6This is the argument of Gal´ı and Gertler (1999). Amato and Laubach (2003) use a slightly different motivation that leads to the same specification of the Phillips curve below. 1.2. The Sticky Price Model 15 that follow the rule-of-thumb. The forward-looking price can be derived from firms’ profit maximisation and is given by7

f = − βθ + + βθ f . pt (1 )Et−1(mct pt) Et−1 pt+1 (1.14)

The backward-looking price setters are assumed to set their price equal to the average price in the previous period corrected for past inﬂation, i.e.

b = ∗ + π , pt pt−1 t−1 (1.15) where, importantly, past inﬂation serves as the forecast for actual inﬂation. Equations (1.12)-(1.15) can be combined to yield the following ‘hybrid’-Phillips curve

b f πt = γ πt−1 + γ Et−1{πt+1} + λ(Et−1{mct} + ut), (1.16) where the parameters are deﬁned as follows

−1 λ ≡ Φ (1 − απ)(1 − θ)(1 − βθ)µ f −1 b −1 γ ≡ Φ βθ, γ ≡ Φ απ α µ = + − α − 1 (1 )( 1) Φ ≡ θ + απ 1 − θ(1 − β) .

Thus, as first suggested by Fuhrer and Moore (1995), inflation is both forward- and backward-looking and depends on the forecastable component of real marginal cost. As in Clarida et al. (2001), the ‘cost-push’ shock ut derives from the random disturbance per- turbing the labour supply decision in the utility function in (1.2). In effect, it introduces a wedge between the marginal rate of substitution between leisure and consumption and the real wage and can be interpreted as a stochastic wage markup. Analogous, to the discussion of the IS equation, the purely forward-looking New Key- nesian Phillips curve results when all firms follow the Calvo pricing rule, i.e. απ = 0, and prices are not preset one period in advance,

πt = βEtπt+1 + λmct + λut, (1.17)

λ = (1−θ)(1−βθ) µ. where θ The lagged inflation term in (1.16) is again important to account for the empirically observed inflation persistence. In the purely forward-looking specification, inflation would become a jump variable and the price level a state variable. Estrella and Fuhrer (2002) have shown that purely forward-looking specifications like (1.17) and the IS relation (1.9) imply counterfactual

7The assumption is as in Gal´ı and Gertler (1999) that all consumers choose consumption optimally so that the marginal utility of consumption is identical across consumers. 16 1. Assessing Predetermined Expectations

− 8 relationships. The former implies that inflation and the output gap (yt yt) are positively correlated while the correlation between the change in inflation and output gap is negative. This is at odds with the ‘acceleration phenomenon’ according to which high economic activity should move hand-in-hand with positive movements in inflation. The argument is similar for the IS equation, that is equation (1.9) stipulates a negative correlation between the consumption level and the expected real interest rate and a positive correlation between consumption growth and the expected real interest rate. This implies that when the expected real interest rate rises above its steady state value, the level of consumption must decline but its growth rate remain positive. This is only possible when consumption ‘jumps’ down initially and approaches its lower level from below. To assess these predictions, in section 1.4 I compare the characteristics of the actual data with those of simulated data from the estimated model.

1.2.3 Central Bank

The model is closed by assuming that the central bank follows a Taylor-type interest-rate rule. That is, it adjusts its instrument in response to deviations of inﬂation and output from their respective target levels of price stability and potential output. In addition, I include a lagged interest rate term to account for the fact that central banks generally do not move their instrument in large steps (Goodhart, 1997), = φ + − φ φ − + φ π + εi. it iit−1 (1 i) y(yt yt) π t t (1.18)

εi Here t is a white-noise, exogenous shock to the interest rate that can be interpreted as the unsystematic component of monetary policy. All coefficients are assumed to be positive and the smoothing or partial-adjustment coefficient is assumed to obey the restriction φ ∈ , . i [0 1) Existence of a stable solution of the model requires certain restrictions on the policy coefficients (Clarida et al., 1999). Namely, in response to an increase in expected inflation, the central bank must increase the nominal interest rate sufficiently to achieve a rise in the real interest rate that dampens economic activity. I confine the analysis to stable unique solutions of the model in the estimation procedure. Specifically, stability and uniqueness of the model solution will be checked by the numerical solution algorithm. Against the background of the Bundesbank’s official money growth-targeting strategy, it may be surprising that in this model central bank behaviour is modelled in terms of the interest rate. However, the instrument of the Bundesbank when conducting monetary policy has always been a short term interest rate. Clarida and Gertler (1996) argue that the behaviour of the Bundesbank in the post Bretton-Woods era can be described

8Under the assumptions made in this model, there is a proportional relationship between marginal cost and the output gap. 1.2. The Sticky Price Model 17 well by a Taylor-type rule that also incorporates the output gap. Furthermore, between 1975 and 1985 the Bundesbank announced a rate of ‘unavoidable inflation’ that ranged between 4.5% and 3%. From 1986 onwards the Bundesbank went a step further, announc- ing that an inflation rate of 2% was consistent with price stability (Deutsche Bundesbank, 1995). Also supporting the interest-rate rule formulation, it has been observed that the Bundesbank allowed deviations of money growth from target more often than deviations of inflation from its prescribed values; by analysing the effects of changes in forecasted money growth and forecasted inflation on the interest rate instrument, Bernanke and Mi- hov (1997) find that money growth plays a quantitatively unimportant role in explaining variations in the interest rate. This leads them to conclude that implementation of Bun- desbank’s monetary policy is described well with an interest-rate rule. However a recent study by Gerberding et al. (2004) using real time data shows that a broad monetary aggregate enters significantly into a Taylor-type rule.

1.2.4 Solution of the Model

The three endogenous variables, yt,πt, it, are determined by three equations: the IS- equation (1.8), the Phillips curve (1.16) and the monetary-policy rule (1.18). The stochas- tics of this system of rational-expectations equations are assumed to be driven by four independent exogenous shocks: the preference shock gt, the productivity shock at, the i cost-push shock ut, and the monetary policy shock ε . The first three are assumed to follow stationary AR(1)-processes, while the monetary policy shock is assumed to be white noise. Because data for three series is employed, at least three shocks need to be specified in order to avoid a singular covariance matrix in the likelihood computation. However, Smets and Wouters (2003) note that allowing for richer stochastic specifications than dictated by the number of time series may be helpful in the estimation procedure. The system has the following matrix representation9

Γ0(ξ)st =Γ1(ξ)st−1 +Ψzt +Πϑt, (1.19) and is solved using the method developed by Sims (2002). ξ is a (17×1)-vector containing ﬃ ρ ,ρ,ρ, the parameters of the model including the autoregressive coe cients, g u a and the standard deviations of the shock processes σd, d ∈ {g, u, a, i}

ξ = β, α, , σ, ϕ, θ, φ ,φ ,φ ,α ,α ,ρ ,ρ ,σ ,σ ,σ ,σ . i π y π y g a g u a i

The matrices Γ0(ξ), Γ1(ξ), Ψ and Π are the (12 × 12), (12 × 12), (12 × 4) and (12 × 6) coeﬃcient matrices respectively, zt is the (4 × 1)-vector of exogenous disturbances, ϑt =

9See Appendix A for full details. 18 1. Assessing Predetermined Expectations

Xt − Et−1Xt isa(6× 1)-vector of expectational errors, i.e. Et(ϑt+1) = 0(6×1) and

= ,π, , , , ,1,0,π1,π0,0, 0 , st yt t it mct gt at yt yt t t it mct ≡ ∈ , 1,π,π1, , where I have deﬁned xt Et xt+1 for xt yt yt t t it mct and added the six equations = + ϑx 10 xt xt−1 t to the system. The general solution to (2.36) has a VAR(1)-representation

= ξ + ξ η . st T( )st−1 R( ) t (1.20)

Note that the system is stochastically singular since st has dimension 12 but there are only four stochastic shocks, rendering the covariance matrix of the disturbances singular. Hence the series for output, inﬂation and the interest rate are selected via the measurement equation

Yt = Zst, (1.21) where Yt is a (3 × 1)-vector and Z a(3× 12)-matrix. In the model the natural level of output - the level of output obtained when prices are ﬂexible and no cost shocks are present - is driven by the unobservable stochastic technology process. Hence, it is treated as unobservable in the estimation procedure as well.

1.3 Estimation

1.3.1 Data

The data ranges from the first quarter of 1975 to the fourth quarter of 1998, covering the post Bretton-Woods era up until the launch of European Monetary Union. Real Gross Domestic Product (GDP) and the Consumer Price Index (CPI) are taken from the OECD Main Economic Indicators Database, and the interest-rate series is constructed as the quarterly average of the monthly average of the bank call rate published in Deutsche Bun- desbank’s time-series database11. The raw data is transformed so that it is conformable with the theoretical model. GDP data for Western Germany is employed until 1991Q3, after which the GDP series is for unified Germany. I account for the level shift and the possible trend break by regressing each series on an individual constant and individual linear trend. An alternative method to treat the statistical effect of reunification on the output series would be to link the series for Western Germany, for which observations are available until 1994Q4, with the series for unified Germany for which data are available from 1991Q1 onwards. This strategy may understate the initial economic boom related to

10See Sims (2002) for a thorough discussion of this method and again Appendix A for a brief description. 11http://www.bundesbank.de/stat/zeitreihen/index.htm, series code SU0101. 1.3. Estimation 19 reunification that began shortly after the inner border was opened in the end of 1989 and would also assume that the Eastern and Western German economies had equal growth rates prior to reunification which seems implausible. The inflation series is calculated as the difference between CPI-inflation and a quasi inflation-target series. This series, published in Gerberding et al. (2004), is comprised of announcements made by the Bundesbank about what they first called ‘unavoidable inflation’ and later termed inflation consistent with price-stability. A series for the nominal interest rate is obtained by regressing the interest rate on this inflation-target series and removing the mean of the resulting series.

1.3.2 Estimation Methodology

Traditionally, DSGE models are calibrated such that certain theoretical moments given by the model match as closely as possible their empirical counterparts.12 However, this method lacks formal statistical foundations (Kim and Pagan, 1994) and makes testing the results difficult.13 One approach used recently in the monetary-economics literature that has improved on this shortcoming is to minimise the distance between the theoretical impulse response functions of the model and the empirical impulse responses estimated from a VAR (Christiano et al., 2005; Rotemberg and Woodford, 1997, for example). Since DSGE models provide by construction only an abstraction of reality, one advantage of this method is that it allows the researcher to focus on that dimension of the model for which it was designed, for example, the effects of a monetary policy shock. Following Sargent (1989), it has become more common to estimate monetary DSGE models with maximum likelihood (ML) (Bergin, 2003; Kim, 2000). Well known problems that arise with this method are that parameters take on corner solutions or implausible values, and that the likelihood function may be flat in some dimensions. GMM estimation is a popular alternative for estimating intertemporal models (Gal´ı and Gertler, 1999, and others). However, Christiano and Haan (1996) show by estimating a business cycle model on U.S. data that GMM estimators often do not have the distributions implied by asymptotic theory. In addition, Linde´ (2005) finds that parameters in a simple New Keynesian model are likely to be estimated imprecisely and with bias. Parameters sometimes need to be fixed beforehand, implying that results are valid conditional only on these a priori ‘calibrated’ parameters. This aspect often remains undiscussed in the final assessment of the model, despite the fact that calibration calls for a careful sensitivity analysis.

12For an overview see Favero (2001). 13See, however, Canova and Ortega (2000) for a discussion on how testing in calibrated DSGE models could be conducted. 20 1. Assessing Predetermined Expectations

The Bayesian approach taken in this paper follows work by DeJong et al. (2000a,b), Otrok (2001), Smets and Wouters (2003, 2004) 14 and can be seen as a combination of likelihood methods and the calibration methodology. Bayesian analysis allows uncertainty and prior information regarding the parametrisation of the model to be formally incorporated by combining the likelihood with prior information on the parameters of interest from earlier microeconometric or macroeconometric studies. In the Bayesian approach such values could be employed as the means or modes of the prior densities to be specified, while a priori uncertainty can be expressed by choosing the appropriate prior variance. For example, the restriction that AR(1)-coefficients lie within the unit interval can be implemented by choosing a prior density that covers only that interval, such as a truncated normal or a beta density. This strategy may help to mitigate such problems as a potentially flat likelihood as estimates of the maximum likelihood are pulled towards values that the researcher would consider sensible a priori. This effect will be stronger when the data carry little information about a certain parameter, that is the likelihood is relatively flat whereas the effect will only be moderate when the likelihood is very peaked. Uncertainty about the specification of the structural model can also be accommodated by the Bayesian approach. I do so in the robustness analysis in Section 1.4 when the model is compared to a model without delayed effects. By Bayes’ theorem, the posterior density ϕ(ξ | Y) is related to prior and likelihood as follows f (Y | ξ)π(ξ) ϕ(ξ | Y) = ∝ f (Y | ξ)π(ξ) = L(ξ | Y)π(ξ), (1.22) f (Y) π ξ ξ, ξ | ≡ | ξ where ( ) denotes the prior density of the parameter vector L( Y) f (Y ) is the likelihood of the sample Y and f (Y) = f (Y | ξ)π(ξ)dξ is the unconditional sample density. The unconditional sample density does not depend on the unknown parameters and consequently serves only as a proportionality factor that can be neglected for estimation purposes. In this context it becomes clear that the main difference between ‘classical’ and Bayesian statistics is a matter of conditioning. Likelihood-based non-Bayesian methods condition on the unknown parameters ξ and compare f (Y | ξ) with the observed data. Bayesian methods condition on the observed data and use the full distribution f (ξ, Y) = f (Y | ξ)π(ξ) and require specification of a prior density π(ξ). The likelihood function can be computed with the Kalman filter using the state-space representation of the above model, where (2.37) is the transition equation and (2.38) is the measurement equation. Denoting st as the optimal estimator of st based on observations up to Yt−1 and Pt = E (st − st)(st − st) as the covariance matrix of the estimation error,

14There are by now numerous applications of the approach, for example Adolfson et al. (2005), Justiniano and Preston (2004), Lubik and Schorfheide (2003), Rabanal and Rubio-Ram´ırez (2005). 1.3. Estimation 21 the prediction equations are given by

st|t−1 = Tst−1 (1.23) Pt|t−1 = TPt−1T + RQR (1.24) and the updating equations are

= + −1 − st st|t−1 Pt|t−1Z Ft (Yt Zst|t−1) (1.25) = − −1 , Pt Pt|t−1 Pt|t−1Z Ft ZPt|t−1 (1.26)

where Ft = ZPt|t−1Z (Harvey, 1989, p. 106). The updating equations describe the solution to the signal extraction problem based on information up to and including time t − 1, the prediction equations are one-step ahead = η η predictions and Q E t t . The recursions are then initialised with the values of the −1 15 unconditional distribution s1|0 = 0 and vec(P1|0) = (I−T⊗T) vec(RQR ) (Harvey, 1989, p. 121). Finally, the likelihood can be computed conditional upon the initial observation

Y0 using a prediction-error decomposition (Harvey, 1989, p. 125). The prediction error is deﬁned as νt = Yt − Zst|t−1, and assuming that st is Gaussian, st|t−1 is also Gaussian with covariance matrix Pt|t−1. It follows that the log-likelihood can be written as

T T NT 1 1 log L(Y | ξ) = − log 2π − log |F | − νF−1ν . (1.27) 2 2 t 2 t t t t=1 t=1 Computation of the posterior distribution ϕ(ξ | Y) requires calculating the likelihood and then multiplying by the prior density. The likelihood itself is computed by applying the Kalman ﬁlter to the state space system in (2.37) and (2.38), after solving the model given values of the elements in the parameter vector ξ.

1.3.3 Speciﬁcation of Priors

In specifying the prior density for the parameter vector I assume that all parameters are independently distributed of each other, i.e.

17 π ξ = π ξ , ( ) i( i) (1.28) i=1 ξ , = , .., ξ where i i 1 17 denotes elements in . However, the solution set of the DSGE model is restricted to unique and stable solutions which may imply prior dependence. Table 1.1 provides an overview of the priors used in the estimation. However, a number of parameters are diﬃcult to estimate given the available data and are ﬁxed a priori.

15This is possible because the transition equation is stationary. 22 1. Assessing Predetermined Expectations

Because the discount factor in the model, β, is related to the steady state interest rate by −logβ = i and the estimations are performed with demeaned data, an estimate for β cannot be pinned down. Hence I fix the discount factor to 0.99, implying an annual steady state interest rate of 4 percent. From a Bayesian perspective this is equivalent to imposing a strict prior on β with zero variance. The intertemporal elasticity of substitution is assumed to equal one, guaranteeing a balanced growth path, as is the elasticity of labour supply. Labour’s share in production is set to 0.67 and the markup is assumed to be 10 percent which implies = 11. The price stickiness parameter is assumed to be characterised by a beta distribution with a mean that implies an average duration of fixed prices of about half a year. The interest-rate smoothing parameter should lie between zero and one and, like all other autoregressive parameters in the model, is also assumed to follow a beta distribution. Its mean is chosen to be 0.8, whereas the prior densities of the shock processes are specified with a mean of 0.5 and fairly wide variance to account for the uncertainty about their persistence. Concerning the degree of rule-of-thumb behaviour in consumption, I take account of the findings in Campbell and Mankiw (1989) that the population can be divided into roughly equal shares of forward- and backward-looking agents. Thus a beta prior with mean of 0.5 and a relatively large standard deviation of 0.25 is specified to account for the a priori uncertainty of this value. The same prior is chosen for the fraction of rule- of-thumb price-setters. Finally, little is known about the standard deviations of the shock processes. I specify inverted gamma densities with infinite standard deviations to account for the lack of knowledge. The modes are based on simple AR(1)-regressions with data prior to the sample period.

1.4 Results

In this section the estimation results from the DSGE model are discussed and its empirical performance evaluated. Impulse response functions are compared to those of a VAR estimation. The results are also compared to a model without the one-period delay imposed on optimising consumers and price setters. Finally, one means of estimation diagnostic is discussed.

1.4.1 Parameter Estimates

Combining the joint prior with the likelihood leads to an analytically-intractable posterior density. In order to sample from the posterior, I employ a random-walk chain Metropolis- Hastings algorithm with a multivariate normal proposal density and generate 150 000 1.4. Results 23

Table 1.1: Prior Speciﬁcation and Posterior Estimates Prior Posterior Estimates Parameter Density Mean Std Dev Mode 5% Mean 95%

rule of thumb cons αy Beta 0.50 0.25 0.97 0.91 0.96 0.99 rule of thumb infl απ Beta 0.50 0.25 0.40 0.30 0.44 0.58 price stickiness θ Beta 0.50 0.20 0.83 0.76 0.84 0.91 Monetary policy rule interest rate fi Beta 0.80 0.15 0.87 0.84 0.89 0.94 inflation fπ Normal 1.50 0.25 1.14 0.93 1.25 1.64 output gap fy Normal 0.50 0.25 0.25 0.19 0.33 0.55 Shock persistence γ preference g Beta 0.50 0.25 0.03 0.01 0.05 0.11 γ productivity a Beta 0.50 0.25 0.93 0.72 0.87 0.96 Shock variances Mode Dof* preference σg Inv Gamma 0.80 2.00 1.11 0.99 1.12 1.26 cost push σu Inv Gamma 1.60 2.00 0.38 0.35 0.40 0.45 productivity σa Inv Gamma 0.80 2.00 0.78 0.42 1.21 2.39 monetary policy σi Inv Gamma 0.50 2.00 0.12 0.11 0.12 0.14 *Note: Dof = degrees of freedom draws from the posterior.16 A complete set of estimation results is reported in Table 1.1 while Figure 1.1 displays kernel estimates of the priors and the posteriors of each parameter. As the marginal posterior densities are reasonably symmetric I refer in the following discussion of the results to the means of the marginal posteriors. Turning first to the consumption decision, on average nearly all agents employ the backward-looking rule-of-thumb. Forward-looking optimisation plays only a minor role. The preference shock is found to approximate white noise, with a persistence coefficient insignificantly different from zero. In their larger model containing nominal and real rigidities, Smets and Wouters (2003) also report a large fraction of backward-looking individuals in out- put17 although not as high as is found here. In contrast, they find that preference shocks are highly persistent. Estimation of the posterior mode turned out to be sensitive to the starting values; sometimes the mode was estimated with a low rule-of-thumb fraction and a high persistence coefficient of the preference shock process. However, the marginal likelihood in these cases was lower than for those cases where rule-of-thumb behaviour is important and the preference shock process is close to white noise. This is, however, one example of potential identification problems in DSGE models that have recently been noted by Beyer and Farmer (2004), Canova and Sala (2005) and Lubik and Schorfheide (2005). The supply side exhibits a considerable degree of forward-looking behaviour in infla-

16Appendix B.1 reports details about this algorithm. 17They rationalise backward-looking behaviour with habit persistence in consumption which leads to a mathematical identical equation. 24 1. Assessing Predetermined Expectations

Figure 1.1: Delayed Eﬀects Model: Prior- and Posterior Density consumption backward inflation backward price stickiness 15 4 8 6 10 2 4 5 2 0 0 0 0 0.5 1 0 0.5 1 0 0.5 1 Taylor rule − lagged interest rate Taylor rule − inflation Taylor rule − output gap

10 1 2 5

0 0 0 0 0.5 1 0 1 2 0 1 2 AR(1) preference shock AR(1) technology shock stddev − preference shock

10 5 4

5 2

0 0 0 0 0.5 1 0 0.5 1 0 1 2 stddev − cost shock stddev − technology shock stddev − monetary policy shock

10 1 20 5 0.5

0 0 0 0 0.5 1 1.5 0 1 2 3 0 0.5 1

Notes: Prior (dashed lines) and posterior densities (solid lines) for the DSGE model with delayed effects. tion and stickier prices than a priori assumed. The estimated value of 0.84 implies that prices are fixed for 6.5 quarters on average.18 This is consistent with Gal´ı et al. (2001, 2003), who find a relatively low fraction of backward-looking price-setters using Euro area data. These studies estimate θ in the interval 0.77 to 0.87, varying according to the instruments used in their GMM approach. Similarly, Smets and Wouters (2003) estimate the Calvo price-stickiness parameter at 0.91 using Euro Area data. In order to clarify how backward-looking behaviour and price stickiness influence inflation dynamics, assume for a moment a purely forward-looking specification and consider a positive shock to marginal cost (or equivalently the output gap). The inflation rate jumps up instanta- neously19 to the maximum response and then reverts back to equilibrium. The degree of price stickiness governs the maximum response of inflation to cost shocks and the speed of convergence as it returns to equilibrium. The stickier prices are, that is, the fewer price

18 1 = 1 ≈ The Calvo specification implies that the average duration of fixed prices is calculated as 1−θ 1−0.84 6.5. 19With one-period delayed effects this happens after one period, the first period in which inflation is allowed to move. 1.4. Results 25 setters who change price in a given period, the smaller is the inflation response and the more prolonged its convergence back to equilibrium. Allowing for a lagged inflation term heightens inflation persistence and produces a ‘hump-shaped’ inflation response so that the maximum impact on inflation is delayed somewhat. However, the reduced-form coefficient on marginal cost is very small (0.0018), indicating a relatively weak transmission from marginal-cost changes onto prices with respect to other shocks in the model. The parameters of the Taylor rule display familiar values. The mean for the inflation coefficient is 1.25 and for the output gap coefficient 0.33 close to the values suggested by Taylor (1993) of 1.5 and 0.5 on inflation and the output gap, respectively. The partial- adjustment coefficient in the Taylor rule (mean of 0.89) is in line with results commonly found in most empirical studies irrespective of the method used. For example, Smets and Wouters (2003) estimate the mean of the lagged interest-rate term to be 0.93 using Euro area data. Finally, the technology shock is highly persistent, as has been found in other empirical work and commonly assumed in calibration studies. However, its standard deviation is not well identified.

1.4.2 Empirical Performance of the Model

Data Moments and Autocorrelation Functions

In this section I compare stylised facts from by the actual data to those of simulated data from the model. Altogether, 10 000 sets of parameter values are drawn from the posterior distribution and used to simulate 96 observations for each of the three variables, equivalent to the number of observations of the actual data. The mean of the distribution of standard deviations and their 10- and 90-percentile values are calculated for each set of time series and compared to the standard deviations of the actual data. The results reported in Table 1.2 indicate that the simulated data series are a good match to the actual data, with inﬂation and the interest rate slightly more and output slightly less volatile than the actual data.

Table 1.2: Standard Deviations of Simulated and Actual Data Simulated Data Actual Data 10% mean 90% GDP 1.81 2.51 3.34 2.38 Inﬂation 1.69 2.07 2.50 1.83 Interest rate 1.61 2.74 4.25 2.35

Autocorrelation functions for both the actual and simulated data are then estimated 26 1. Assessing Predetermined Expectations from a VAR(1).20 Figure 1.2 summarises the results; the dashed lines indicate the 10- and 90-percentiles from the simulated data. The autocorrelations of the simulated data are typically in the vicinity of those of the actual data, but the DSGE model produces lower autocorrelations for inﬂation relative to the actual data. However, the wide error bands indicate that the autocorrelations from the DSGE model are estimated with greater uncertainty. This exercise demonstrates the advantage of the Bayesian approach since the full (small-sample) distributions are available for all statistics.

Figure 1.2: Autocorrelation Functions y ,y y ,π y ,r t t−h t t−h t t−h 1 1 1

0.5 0.5 0.5

0 0 0

−0.5 −0.5 −0.5

−1 −1 −1 0 5 10 0 5 10 0 5 10

π ,y π ,π π ,r t t−h t t−h t t−h 1 1 1

0.5 0.5 0.5

0 0 0

−0.5 −0.5 −0.5

−1 −1 −1 0 5 10 0 5 10 0 5 10

r ,y r ,π r ,r t t−h t t−h t t−h 1 1 1

0.5 0.5 0.5

0 0 0

−0.5 −0.5 −0.5 actual simulated −1 −1 −1 0 5 10 0 5 10 0 5 10

Notes: Autocorrelations for simulated data from the model DSGE model with delays (solid lines) and the actual data (crossed lines) with 10- and 90%-tiles from DSGE-model (dash- dotted lines).

Acceleration Phenomenon

An interesting question is whether the estimated DSGE model can account for the acceleration phenomenon discussed in Section 1.2. The dynamic relationship between output

20The Schwarz criterion selects one lag while the Akaike criterion selects a lag length of two for the VAR estimated on the actual data set. 1.4. Results 27

Table 1.3: Acceleration Phenomenon and Output-Interest-Rate Dynamics

Simulated data Actual data

= − n = Output Measure yt yt yt yt yt

Panel A: Phillips Curve corr(∆πt, yt) -0.001 0.10 0.32 [-0.15, 0.16] [-0.06, 0.25] corr(πt, yt) -0.05 0.30 0.35 [-0.28, 0.18] [0.01, 0.58] Panel B: IS Equation corr(∆yt, rt) 0.09 -0.26 -0.40 [-0.12, 0.29] [-0.44, -0.07] corr(yt, rt) -0.20 0.13 -0.59 [-0.52, 0.14] [-0.18, 0.45] − n Note: Mean of the model consistent output gap measure yt yt and the classical = n ( linearly detrended output) output gap measure yt. yt is the natural level of output (under flexible prices) implied by the DSGE model. 10 and 90 percentiles in brackets below. The real interest rate is calculated as rt = it − πt. and the expected real interest rate is also an open issue. In traditional empirical and theoretical analyses, the natural level of output is calculated as a deterministic trend, whereas New Keynesian models of the business cycle define natural output as the level of output obtained when all prices are flexible. In the simple model studied here, flexible-price output is simply proportional to the technology process, which may lead to a poor estimate of natural output. For this reason I use two measures of economic activity: (i) the theoretical output gap calculated from the DSGE model as the difference between output and output under flexible prices and (ii) the output variable measured as the deviation from steady-state output in the log-linearised model. These two measures are compared to those calculated from the data. Since mean and trend have already been removed from all variables, the empirical output measure can be interpreted as an output-gap measure in the classical sense. The same simulation procedure described above is used to generate data from the DSGE model. The real expected interest rate is approximated as the difference between the current nominal interest rate and current inflation and output growth and inflation growth are calculated as one year changes, that is ∆yt = yt+2 − yt−2 and

∆πt = πt+2 − πt−2. Panel A of Table 1.3 reports the results for the acceleration phenomenon and Panel B the correlation between output and the real expected interest rate. As can be seen in the first data column, correlations based on the first measure of the output gap are not significantly different from zero. The results for the second output 28 1. Assessing Predetermined Expectations gap measure (2nd data column) are closer to the actual data (the final column of the table). As Panel A demonstrates, the model can generate positive correlations between inflation growth and output as well as between the level of inflation and output, namely the acceleration phenomenon observed in the actual data. However, there is considerable uncertainty around these correlations and the correlation between inflation growth and the output gap is significantly smaller in the generated data than in the actual data. Looking at Panel B, the DSGE model is able to generate a plausible negative correlation between output growth and the real expected interest rate but fails to do so with the level of output. In this respect the model is lacking.

1.4.3 Impulse-Response Analysis

Impulse response functions to each of the four shocks, together with 10 and 90 percentile error bands, are calculated from 10 000 draws of the posterior distribution and shown in Figure 1.3 and 1.4. All shocks are one standard deviation shocks.

Figure 1.3: Demand and Monetary Policy Shock Demand Shock Interest rate shock 1 0.2

0.5 0.1 0

Inflation 0 −0.1 −0.2 −0.5 0 5 10 15 20 0 5 10 15 20

2 0.2

1 0.1 0 Output 0 −0.1 −0.2 −1 0 5 10 15 20 0 5 10 15 20

2 0.5

1 0 0 Nominal Interest −1 −0.5 0 5 10 15 20 0 5 10 15 20

Notes: Impulse responses (solid lines) from the DSGE model with delays with 10- and 90%-tiles (dash-dotted lines).

In response to a contractionary monetary policy shock (Figure 1.3, right column), the 1.4. Results 29 interest rate increases and output and inflation fall, consistent with the specification of the theoretical model. Both output and inflation show a hump-shaped and gradual re- version over time. The hump-shaped form of the impulse responses is due to significant backward-looking terms in both the Phillips-curve relation and the IS relation and is in line with stylised facts from VAR studies (Christiano et al., 2005). Note that in accordance with the specification of the theoretical model, output and inflation do not react in the period of the shock as is the case in recursively identified VAR models. A positive preference shock (Figure 1.3, left column) increases the discount factor in the intertemporal optimisation problem so that agents are willing to consume more, inducing a rise in output. In turn, excess demand triggers inflationary pressures due to increasing marginal cost. A positive output gap and inflation deviating from target consequently lead to an increase in the interest rate. Again, because expectations about marginal cost are predetermined, the rise in inflation begins with a one period delay.

Figure 1.4: Cost- and Technology Shock Cost−push shock Technology shock 1 0.2

0 0.1 0

Inflation −1 −0.1 −0.2 −2 0 5 10 15 20 0 5 10 15 20

0.5 0.2

0.1 0 0 Output −0.1 −0.5 gap −0.2 level −1 0 5 10 15 20 0 5 10 15 20

0.5 0.5

0 0 Nominal Interest −0.5 −0.5 0 5 10 15 20 0 5 10 15 20

Notes: Impulse responses (solid lines) from the DSGE model with delays with 10- and 90%-tiles (dash-dotted lines).

Following a positive technology shock (Figure 1.4, right column), output increases while inflation and the interest rate fall. Upon impact, marginal cost falls and natural output increases by more than the level of actual output, opening up a negative output 30 1. Assessing Predetermined Expectations gap. Since the monetary authority does not respond strongly enough to offset the shock, inflation falls. This result is in line with the New Keynesian literature on technology shocks (Gal´ı, 1999). Finally, a negative cost-push shock (Figure 1.4, left column) produces a qualitatively similar response to the technology shock. The fall in inflation causes the central bank to cut the interest rate which leads to a rise in output. However, the effect is quantitatively smaller.

1.4.4 Comparison to VAR

In order to gain insight into the quality of the results from the Bayesian estimation, I compare the monetary and preference shock impulse response functions of the DSGE model to those of an identified first order VAR estimated on the same data set. The VAR is ordered as inflation, output and nominal interest rate and takes the following form:

A0yt = A1yt−1 + εt,εt ∼ N(0, Ω). (1.29)

The identification is recursive with ones on the main diagonal and an additional zero entry at the (2,1) position,      100 =   . A0  010 (1.30) −a31 −a32 1 That is, an inflationary shock does not have an immediate impact on output which allows identification of an aggregate demand shock in addition to the monetary policy shock. Given the assumptions about technology and cost shocks, these cannot be sep- arately identified in this framework. In Figure 1.5, the impulse response functions for a contractionary one-standard-deviation monetary policy shock are shown (right column). Impulse responses from both the VAR and the DSGE model are shown, along with 10 and 90 percentile bands. For the VAR, the percentile bands are estimated with methods suggested by Sims and Zha (1999). Apart from the so called ‘price puzzle’, namely that inflation rises after a contractionary monetary policy shock, the estimated impulse response functions from the VAR are similar to those from the DSGE model. However, the high coefficient on the backward-looking term in the aggregate demand equation (1.8) implies weak transmission from real interest rate changes to consumption. This effect is mirrored in the weaker response of output in the DSGE model than in the VAR. The interest-rate dynamics match well between the DSGE model and the VAR. In both models the interest rate returns to zero after about eight quarters. Similar dynamics are also observed in both models in response to a preference shock (left column in figure 1.5). The response of inflation in the DSGE model is slightly less 1.4. Results 31 persistent than in the VAR model, which is in line with the evidence obtained from the autocorrelation functions, whereas there is almost a perfect fit with respect to the output response.

Figure 1.5: Dynamics of DSGE- and VAR Model Demand shock Interest rate shock

0.6 0.3 VAR 0.4 0.2 DSGE 0.1 0.2 Inflation 0 0 −0.1 0 5 10 15 20 0 5 10 15 20

1 0.1 0 0.5 −0.1 Output 0 −0.2

0 5 10 15 20 0 5 10 15 20

0.6 1 0.4

0.5 0.2 Interest

0 0 0 5 10 15 20 0 5 10 15 20

Notes: Impulse responses: DSGE model with delayed eﬀects (thick solid line) with VAR (thin solid line); 10- and 90%-tiles of DSGE-impulses (dash-dotted lines) and error bands of VAR (shaded area).

In conclusion, the estimated DSGE model qualitatively resembles the identiﬁed VAR. The models are also quantitatively similar with respect to the monetary and preference shocks.

1.4.5 Comparison to a Model with Contemporaneous Eﬀects

I have introduced delayed effects of monetary policy onto inflation and output in order to account for the assumptions often made in identified VAR studies. However, despite the fact that this recursive scheme has become the standard identification in the monetary- policy VAR literature, most DSGE models do not allow for such effects.21 Rather, DSGE

21Exceptions include Christiano et al. (2005) and Rotemberg and Woodford (1997). 32 1. Assessing Predetermined Expectations models postulate that monetary policy shocks have a contemporaneous impact on all variables. In this section I compare the results of the model in this paper to those of a baseline model in which all expectations are conditional on information up to and including period t. That is, optimising consumers’ and price setters’ decisions have immediate effects. Appendix C.1 presents the estimation results for such a model using the same prior specification as before; the estimation outcome is quite similar to the model with delayed effects (see Table 1.5). The prior and posterior density kernels are shown in Figure 1.7 in Appendix C.2 and the corresponding impulse responses are shown in Figures 1.9 and 1.8 in Appendix C.3. The coefficient estimates are similar to those from the model with delayed effects but in contrast, the contemporaneous responses of output and inflation to a monetary impulse are significantly different from zero. As discussed in Geweke (1999), for example, the Bayesian approach to estimation also allows a formal comparison of different models based on the marginal likelihood of the model. The marginal likelihood of a model Mi is defined by

f (YT |Mi) = ϕ(ξ|Mi) f (YT |ξ, Mi)dξ, (1.31) Ξ where ϕ(ξ|Mi) is the prior density for model Mi and f (YT |ξ, Mi) is the data density of model Mi given the parameter vector ξ. Integrating out the parameter vector, the marginal likelihood gives information about the overall likelihood of the model given the data.

Further, the posterior-odds ratio in favour of model Mi versus M j is deﬁned by

pi f (YT |Mi) POi = , (1.32) p j f (YT |M j) where pi and p j are the prior model probabilities for model i and j, respectively. Assuming that both models complete the model space and assigning equal prior probabilities of 1/2, the Bayes factor in favour of model i versus j can be calculated as

f (YT |Mi) Bij = . f (YT |Mj)

Assuming that falsely choosing a model incurs equal losses for both models, a Bayes factor greater than 1 indicates that model i is more likely than model j after having observed the data. The results in Table 1.4 show that the data favour the model with one-period delays over the model without delayed eﬀects, as can be seen by the higher (log) marginal likelihood for the former as well as the magnitude of the Bayes factor. According to Jeﬀreys a value of the log-Bayes factor greater than 2 is decisive evidence against the alternative model 1.4. Results 33

Table 1.4: Model Comparison by Bayes Factors

Model with Delayed Effects Model without Delayed Effects log( f (YT |Mi)) -421.8402 -427.2308 log(Bayes factor) 5.3906 Note: Modified harmonic mean estimation with p = 0.05 (Gelfand and Dey, 1994) See Appendix B.2 for more details.

1.4.6 Estimation Diagnostics

The Metropolis-Hastings sampler that is employed in order to generate random draws from the unknown posterior distribution falls in the class of Markov Chain Monte Carlo (MCMC) methods.22 Essentially, the sampler generates draws from a candidate generating density23 (a Markov chain) that is not identical to the posterior but one that ‘wanders’ over the posterior. The candidate draws are then accepted with a certain probability that is highest (lowest) in areas where the posterior probability is highest (lowest). The Markov chain is serially dependent but it can be shown that under some regularity conditions it converges asymptotically to the true posterior. Hence, convergence of the Markov chain becomes an important issue for validity of the results. One suggested diagnostic tool to analyse if the chain has converged is to look at the running means (CUSUM test) of the marginal posteriors.24 The standardised statistic used to calculate these means given N draws of the Markov chain is (Bauwens et al., 1999)    s  1  CS =  γ − µγ /σγ s = 1,...N, (1.33) s s i i=1 where µ and σγ are the mean and standard deviation of the N draws respectively, and γ 1 s γ s i=1 i is the running mean for a subset of s draws of the chain. If the chain converges, the graph of CS s should converge smoothly to zero. On the contrary, long and regular movements away from the zero line indicate that the chain has not converged. According to Bauwens et al. (1999), a CUSUM value of 0.05 after s draws means that the estimate of the posterior expectation deviates from the final estimate after N draws by 5 percent in units of the final estimate of the posterior standard deviation. The authors consider a value of 25 percent to be a good result. Figure 1.6 shows the CUSUM-paths along with 5 percent bands for each parameter for 150 000 draws (note that the overall interval captured by the figure corresponds to 25 percent bands). Overall, the figure points to a satisfactory degree of convergence.

22See Chib and Greenberg (1995) for an introduction to Metropolis-Hastings sampling. 23A random walk is often taken as the candidate generating density. 24See Koop (2003) for an overview of diagnostic methods for MCMC samplers. 34 1. Assessing Predetermined Expectations

Figure 1.6: CUSUM-Test consumption backward inflation backward price stickiness 0.2 0.2 0.2

0 0 0

−0.2 −0.2 −0.2 0 5 10 15 0 5 10 15 0 5 10 15 4 4 4 x 10 x 10 x 10 Taylor rule − lagged interest rate Taylor rule − inflation Taylor rule − output gap 0.2 0.2 0.2 0 0 0 −0.2 −0.2 −0.2 0 5 10 15 0 5 10 15 0 5 10 15 4 4 4 x 10 x 10 x 10 AR(1) preference shock AR(1) technology shock stddev − preference shock 0.2 0.2 0.2 0 0 0 −0.2 −0.2 −0.2 0 5 10 15 0 5 10 15 0 5 10 15 4 4 4 x 10 x 10 x 10 stddev − cost shock stddev − technology shock stddev − monetary policy shock 0.2 0.2 0.2

0 0 0

−0.2 −0.2 −0.2 0 5 10 15 0 5 10 15 0 5 10 15 4 4 4 x 10 x 10 x 10

Notes: The horizontal grey lines indicate 5% bands, the vertical line indicates the 40%- burn-in of the Markov chain with 150 000 simulations.

1.5 Summary and Conclusions

This paper has augmented a New Keynesian sticky-price model to include endogenous persistence in consumption and inflation as well as informational delays and then estimated using Bayesian methods for Germany pre-EMU. The estimated model features a high degree of persistence in consumption and output and sizeable backward-looking behaviour in inflation. Prices are estimated to be fixed for 6.5 quarters on average, quantitatively similar to those of Smets and Wouters (2003) based on Euro area data. Using a conventional output-gap measure, the model can account for the acceleration phenomenon. In contrast, the the output-gap measure suggested by the model, that is the deviation of output from its flexible price level, appears to be a poor estimate in this simple specification of the model. The data clearly favour the model with delayed effects on output and inflation when compared to a model that allows interest rate movements to have contemporaneous effects on these variables. This may justify the often used identification scheme in structural VAR 1.5. Summary and Conclusions 35 models, even though both models could of course be wrong. Moreover, the dynamics following monetary and preference shocks are comparable between the DSGE model and an identified VAR model. The VAR, however, displays more persistence in inflation. The estimated Taylor rule in the model confirms earlier studies by Clarida and Gertler (1996) and Bernanke and Mihov (1997) that in practice the Bundesbank’s behaviour can be described well as an inflation targeting strategy. On the other hand, this paper has not directly compared the Taylor rule to a monetary-policy strategy focussed exclusively on money balances as they do not play a meaningful role in the model. A potential way to introduce money balances into the model would be to specify a utility function that is non-additively separable in consumption and money balances as in Kim (2000). Kremer et al. (2003) find that estimating such a model with purely forward-looking agents leads to the conclusion that real money balances do play an important role for inflation and output dynamics. Finally, unlike VAR model does the DSGE model impose a restriction that produces the correct response of inflation to a contractionary monetary impulse, namely that inflation falls. The common empirical fix is to add a commodity prices index to the VAR as this may capture inflation expectations. However, it may be interesting to further investigate on theoretical grounds25 why the VAR generates a positive response.

25Giordani (2004) is one contribution. 36 1. Assessing Predetermined Expectations

Appendices A Model with Delays

A.1 Matrix Representation

Recall equation (2.36) from section 1.2 of the main text.

Γ0(ξ)st =Γ1(ξ)st−1 +Ψzt +Πϑt

= ,π, , , , , , , π , π , , state vector st (yt t it gt ut at Etyt+1 Etyt+2 Et t+1 Et t+1 Etit+1 Etyt+1)    y π i mc g a y1 y0 π1 π0 i0 mc0   t t t t t t t t t t t t   1000−α δ/σ 0000000  y   01000 0000000    00100 0000000  1−α+ασ+ϕ   − 00 1 0 1+ ϕ 00000 0  α   00000 0000000   Γ =  00000 1000000 0    10000 0000000    00000 0100000    01000 0000000    00000 0000000    00100 0000000 00010 0000000

Endogenous Errors = 1 + ηy1, 1 = , 1 = 0 + ηy0, 0 = 1 = yt yt−1 t where yt Etyt+1 yt yt−1 t where yt Etyt+1 ( Etyt+2) π = π1 + ηπ1, π1 = π ,π1 = π0 + ηπ, π0 = π1 = π t t−1 t where t Et t+1 t t−1 t where t Et t+1 ( Et t+2) = 0 + ηmc, 0 = mct mct−1 t where mct Etmct+1 = 0 + ηi, 0 = it it−1 t where it Etit+1 A. Model with Delays 37                                                               1 − 0 t 0 λ mc 1 δσ y − 0 0 t i α − 1 f − δσ 0 t y π α 1 βθ γ − b 1 t 0 γ π − 1 − δ 0 t y 1 − 00 0 0 0 0 00 0 0 0 0 1 t y a ψ y φ ) 1 i a − φ t 0000000 00 ρ a − (1 − δσ 1 y g − t α ρ 2 g g ρ − 1 − t 00 mc 1 i − 00 0 0 00 t φ i π φ ) i 1 b − φ t γ π − (1 y φ )000 ) δ i 1 φ − 0 00000 000000000000 0000 000000100000 000000010000 000000001000 000000000100 000000000010 000000000001 t − y − (1 (1                                                               = 1 Γ 38 1. Assessing Predetermined Expectations

     εi εg εu εa   ηy1 ηy0 ηπ1 ηπ0 ηi0 ηmc0   t t t t                     1             1        Ψ= 1  Π=       1   1         1         1         1         1  1 = ,π, , , , , , , π , π , , state vector st (yt t it gt ut at Etyt+1 Etyt+2 Et t+1 Et t+1 Etit+1 Etyt+1)

Aggregate Demand α δ = δ + − δ − y − π − + yt Et−1yt+1 (1 )yt−1 σ (Et−1it Et−1 t+1 gt Et−1gt+1) δ = 1 1 + αy

Phillips Curve

b f πt = γ πt−1 + γ Et−1(πt+1) + λEt−1mct + λut b −1 f −1 γ =Φαπ,γ=Φ βθ, λ =Φ−1 − βθ − θ − α , (1 )(1 )(1 π) Φ=θ + απ 1 − θ(1 − β)

Monetary Policy

= φ + − φ φ π + − φ φ − + εi it iit−1 (1 i) π t (1 i) x(yt yt) t

Marginal Cost (Without Cost Shock) − α + ασ + ϕ = 1 − + ϕ mct α yt (1 )at

Output Gap α y − y = ψ a ψ = t t a t a 1 − α + ασ + ϕ Demand Shock = ρ + εg gt ggt−1 t A. Model with Delays 39

Productivity Shock = ρ + εa at aat−1 t

Sims’ method turns out to be convenient. The solution algorithm is very fast, which is convenient here because the model needs to be solved many times, and it can handle singular Γ0 matrices. The algorithm uses the Schur decomposition to solve the generalised eigenvalue problem Γ0 s = λΓ1 s, i.e. matrices Q and Z can be found such that Q ΛZ =Γ0, Q ΩZ =Γ1 and Q Q = Z Z = I, where Q, Z, Λ and Ω are possibly conjugate complex and Λ and Ω are upper triangular.

To demonstrate this, consider the case where Γ0 has full rank. The dynamics of the Γ−1Γ system are governed by the eigenvalues of the 0 1-matrix. An eigenvalue-eigenvector Γ−1Γ = Λ −1 decomposition 0 1 C C is calculated in order to ﬁnd the stable subspace of the system. The matrix C contains the eigenvectors that are associated with the eigenvalues of the system that are collected on the diagonal of the matrix Λ. By imposing the restriction ci st = 0 for each eigenvector that is associated with an explosive eigenvalue (i.e. λ>1), a stationary solution can be found. The information that the algorithm reports about existence and uniqueness of the solution is then used in the estimation procedure to restrict the admissible parameter space to unique and stable solutions. 40 1. Assessing Predetermined Expectations

B Bayesian Concepts

B.1 Metropolis-Hastings Algorithm

As in the main text denote the data set as Y, the prior density as π(ξ) and the likelihood as L(Y|ξ). In order to obtain N random draws from the posterior density, the following algorithm is implemented:

ξ π ξ |ξ 1. Start with an initial value 0 and evaluate ( 0)L(Y 0)

2. For each draw s, ξ − α ξ ,ξ∗ ξ = s−1 with probability 1 ( s−1 s) , s ξ∗ α ξ ,ξ∗ s with probability ( s−1 s) ξ∗ = ξ + ν ν where s s−1 s, and s is called the increment random variable which is multivariate normally distributed as ν ∼ N(0, ΩM). The acceptance probability is calculated as (Gamerman, 1997, Chapter 6) π ξ∗ |ξ∗ α ξ ,ξ∗ = ( s)L(Y s) , . ( s−1 s) min π ξ |ξ 1 ( s−1)L(Y s−1) This deﬁnition ensures that the chain moves in the appropriate direction, that is it is more likely that a draw in an area of high probability is accepted. Prior to running the Markov chain the posterior mode is estimated. A possible starting vector is the ξ Ω mode M and M is taken to be the posterior covariance matrix.

B.2 Marginal Likelihood Computation

The presentation follows Koop (2003, Chapter 5). Given the posterior simulation output ξ N Θ s s=1 for model M j defined on the region , computation of the marginal likelihood makes use of the following relationship: for any p.d.f. f (ξ) with support in Θ f (ξ) 1 E | Y, M j = π(ξ|M j)L(Y|ξ, M j) L(Y|M j) Hence, using the posterior simulation output the empirical counterpart is f (ξ ) 1 = 1 N s . | s=1 π ξ | |ξ , L(Y M j) N ( s M j)L(Y s M j) Following Geweke (1999), f (ξ) is taken to be a truncated normal in order to ensure that f (ξ) ξ ξ Σ π ξ| |ξ, is finite. Next, the support of f ( ) is defined as follows: let and N be ( M j)L(Y M j) N estimates of E ξ|Y, M j and Var(ξ|Y, M j) from the posterior estimator. Then for some probability, p ∈ (0, 1), define the support, Θ,of f (ξ)as Θ= ξ ξ − ξ Σ−1 ξ − ξ ≤ χ2 , :( N ) N ( N ) 1−p(k) B. Bayesian Concepts 41

χ2 − where 1−p(k) is the (1 p)th percentile of the Chi-squared distribution with k degrees of freedom and k is the dimension of ξ. Then f (ξ)isgivenas 1 f (ξ) = p−1(2π)−k/2 Σ−1 exp − (ξ − ξ)Σ−1(ξ − ξ) 1(ξ ∈ Θ), N 2 N N N where 1(.) is the indicator function. The algorithm is as follows

1. Calculate ξ = 1 N ξ N N s=1 s 1 Σ = N (ξ − ξ )(ξ − ξ ) N N s=1 N s N s

2. Choose p

3. Calculate 1 using all ξ ∈ Θ. L(Y|M j) s 42 1. Assessing Predetermined Expectations

C Model with Contemporaneous Eﬀects

C.1 Estimation Results

Table 1.5: Prior Speciﬁcation and Posterior Estimates

Prior Posterior Estimates Parameter Density Mean Std Dev Mode 5% Mean 95%

rule of thumb cons αy Beta 0.50 0.25 0.95 0.87 0.93 0.98 rule of thumb infl απ Beta 0.50 0.25 0.39 0.26 0.41 0.54 price stickiness θ Beta 0.50 0.20 0.87 0.81 0.89 0.96 Monetary Policy rule interest rate fi Beta 0.80 0.15 0.89 0.85 0.90 0.94 inflation fπ Normal 1.50 0.25 1.23 0.93 1.29 1.69 output gap fy Normal 0.50 0.25 0.32 0.21 0.38 0.64 shock persistence γ preference g Beta 0.50 0.25 0.07 0.02 0.13 0.28 γ productivity a Beta 0.50 0.25 0.93 0.71 0.87 0.97 shock variances Mode Dof* preference σg Inv Gamma 0.80 2.00 0.57 0.47 0.56 0.65 cost push σu Inv Gamma 1.60 2.00 0.30 0.26 0.31 0.36 productivity σa Inv Gamma 0.80 2.00 0.71 0.43 1.25 2.60 monetary policy σi Inv Gamma 0.50 2.00 0.12 0.10 0.12 0.14 *Note: Dof = degrees of freedom C. Model with Contemporaneous Effects 43

C.2 Prior and Posterior Kernels

Figure 1.7: Contemporaneous Eﬀects Model: Prior- and Posterior Density consumption backward inflation backward price stickiness 8 10 4 6

5 2 4 2 0 0 0 0 0.5 1 0 0.5 1 0 0.5 1 Taylor rule − lagged interest rate Taylor rule − inflation Taylor rule − output gap

10 1 2 5

0 0 0 0 0.5 1 0 1 2 0 1 2 AR(1) preference shock AR(1) technology shock stddev − preference shock 5 4 5

0 0 0 0 0.5 1 0 0.5 1 0 1 2 stddev − cost shock stddev − technology shock stddev − monetary policy shock

10 1 20 5 0.5

0 0 0 0 0.5 1 1.5 0 1 2 3 0 0.5 1

Notes: Prior (dashed lines) and posterior densities (solid lines) for the DSGE model without delayed eﬀects. 44 1. Assessing Predetermined Expectations

C.3 Impulse Responses

Figure 1.8: Demand- and Monetary Policy Shock Demand shock Interest rate shock 0.5 0.1

0 0 Inflation −0.1 −0.5 0 5 10 15 20 0 5 10 15 20

2 0.2

1 0.1 0 Output 0 −0.1 −0.2 −1 0 5 10 15 20 0 5 10 15 20

1 0.5

0.5 0 0 Nominal Interest −0.5 −0.5 0 5 10 15 20 0 5 10 15 20

Notes: Impulse responses (solid lines) from the DSGE model without delays with 10- and 90%-tiles (dash-dotted lines). C. Model with Contemporaneous Eﬀects 45

Figure 1.9: Cost- and Technology Shock Cost−push shock Technology shock 1 0.1 0 0

Inflation −1 −0.1 −2 0 5 10 15 20 0 5 10 15 20

0.5 0.2 0.1 0 0 Output −0.1 gap −0.2 level −0.5 0 5 10 15 20 0 5 10 15 20

0.5 0.5

0 0 Nominal Interest −0.5 −0.5 0 5 10 15 20 0 5 10 15 20

Notes: Impulse responses (solid lines) from the DSGE model without delays with 10- and 90%-tiles (dash-dotted lines). 46 1. Assessing Predetermined Expectations

References

Adolfson, M., Lasseen,´ S., Linde,´ J., and Villani, M. (2005). Bayesian estimation of an open economy DSGE model with incomplete pass-through. Sveriges Riksbank Work- ing Paper No 179.

Amato, J. D. and Laubach, T. (2003). Rule-of-thumb behaviour and monetary policy. European Economic Review, 47(5):791–831.

Bauwens, L., Lubrano, M., and Richard, J.-F. (1999). Bayesian Inference in Dynamic Econometric Models. Oxford University Press.

Bergin, P. (2003). Putting the New Open Economy Macroeconomics to a test. Journal of International Economics, 60:3–34.

Bernanke, B. S. and Mihov, I. (1997). What does the Bundesbank target? European Economic Review, 41(6):1025–1053.

Beyer, A. and Farmer, R. (2004). On the indeterminacy of New Keynesian economics. European Central Bank Working Paper No 323.

Calvo, G. (1983). Staggered prices in a utility maximising framework. Journal of Mone- tary Economics, 12(3):383–398.

Campbell, J. Y. and Mankiw, N. G. (1989). Consumption, income, and interest rates: Reinterpreting the time series evidence. NBER Macroeconomics Annual, pages 185– 216.

Canova, F. and Ortega, E. (2000). Testing calibrated general equilibrium models. In Mariano, R., Schuerman, T., and Weeks, M., editors, Simulation-based Inference in Econometrics: Methods and Applications. Cambridge University Press.

Canova, F. and Sala, L. (2005). Back to square one: Indentiﬁcation issues in DSGE models. Manuscript, IGIER and Bocconi University.

Christiano, L., Eichenbaum, M., and Evans, C. (2005). Nominal rigidities and the dynamic eﬀects of a shock to monetary policy. Journal of Political Economy, 118(1):1– 45.

Christiano, L. and Haan, W. D. (1996). Small sample properties of GMM for business cycle analysis. Journal of Business and Economic Statistics, 14(3):309–327.

Clarida, R., Gal´ı, J., and Gertler, M. (1999). The science of monetary policy: A New Keynesian perspective. Journal of Economic Literature, 37(4):1661–1707.

Clarida, R., Gal´ı, J., and Gertler, M. (2001). Optimal monetary policy in open versus closed economies: An integrated approach. American Economic Review, 91(2):248– 252.

Clarida, R. and Gertler, M. (1996). How the Bundesbank conducts monetary policy. NBER Working Paper No. 5581. REFERENCES 47

DeJong, D., Ingram, B., and Whiteman, C. (2000a). A Bayesian approach to dynamic macroeconomics. Journal of Econometrics, 98(2):203–223.

DeJong, D., Ingram, B., and Whiteman, C. (2000b). Keynesian impulses versus Solow residuals: Identifying sources of business cycle ﬂuctuations. Journal of Applied Econo- metrics, 15(3):311–329.

Deutsche Bundesbank (1995). Die Geldpolitik der Bundesbank. Deutsche Bundesbank, Frankfurt am Main.

Estrella, A. and Fuhrer, J. C. (2002). Dynamic inconsistencies: Counterfactual implications of a class of rational expectations models. American Economic Review, 92(4):1013–1028.

Favero, C. (2001). Applied Macroeconometrics. Oxford University Press.

Fuhrer, J. C. and Moore, G. R. (1995). Inﬂation persistence. Quarterly Journal of Eco- nomics, 110(1):127–159.

Gal´ı, J. (1999). Technology, employment, and the business-cycle: Do technology shocks explain aggregate ﬂuctuations? American Economic Review, 89(1):249–271.

Gal´ı, J. and Gertler, M. (1999). Inﬂation dynamics: A structural econometric analysis. Journal of Monetary Economics, 44(2):195–222.

Gal´ı, J., Gertler, M., and Lopez-Salido,´ J. D. (2001). European inﬂation dynamics. Euro- pean Economic Review, 45(7):1237–1270.

Gal´ı, J., Gertler, M., and Lopez-Salido,´ J. D. (2003). Erratum: European inﬂation dynamics. European Economic Review, 47(4):759–760.

Gamerman, D. (1997). Markov Chain Monte Carlo. Stochastic Simulation for Bayesian Inference. Chapman & Hall.

Gelfand, A. E. and Dey, D. K. (1994). Bayesian model choice: Asymptotics and exact calculations. Journal of the Royal Statistical Society Series B, 56(3):501–514.

Gerberding, C., Worms, A., and Seitz, F. (2004). How the Bundesbank really conducted monetary policy: An analysis based on real-time data. Discussion Paper Series 1: Studies of the Economic Research Center No 25/2004, Deutsche Bundesbank.

Geweke, J. (1999). Using simulation methods for Bayesian econometric models: Infer- ence, development and communication. Econometric Reviews, 18(1):1–126.

Giordani, P. (2004). An alternative explanation of the price puzzle. Journal of Monetary Economics, 51(6):1271–1296. 48 1. Assessing Predetermined Expectations

Goodhart, C. A. E. (1997). Why do monetary authorities smooth interest rates? In Collignon, S., editor, European Monetary Policy. Pinter, London and Washington.

Harvey, A. C. (1989). Forecasting, Structural Time Series Models and the Kalman Filter. Cambridge University Press.

Justiniano, A. and Preston, B. (2004). Small open economy DSGE models: Speciﬁcation, estimation and model ﬁt. Manuscript IMF.

Kim, J. (2000). Constructing and estimating a realistic optimising model of monetary policy. Journal of Monetary Economics, 45(2):329–360.

Koop, G. (2003). Bayesian Econometrics. John Wiley.

Kremer, J., Lombardo, G., and Werner, T. (2003). Money in a new Keynesian model estimated with german data. Discussion Paper Series 1, Studies of the Economic Research Center No 15/2003, Deutsche Bundesbank.

Levin, A. T., Onatski, A., Williams, J. C., and Williams, N. (2005). Monetary policy under uncertainty in micro-founded macroeconometric models. mimeo.

Linde,´ J. (2005). Estimating new Keynesian Phillips curves: A full information maximum likelihood approach. Journal of Monetary Economics (forthcoming).

Lubik, T. and Schorfheide, F. (2003). Do central banks respond to exchange rate movements? A structural investigation. Manuscript.

Lubik, T. and Schorfheide, F. (2005). A bayesian look at new open economy macroeconomics. NBER Macroeconomics Annual (forthcoming).

McCallum, B. T. and Nelson, E. (1999). Nominal income targeting in an open-economy optimizing model. Journal of Monetary Economics, 43(3):553–578.

Otrok, C. (2001). On measuring the welfare cost of business cycles. Journal of Monetary Economics, 47(1):61–92.

Rabanal, P. and Rubio-Ram´ırez, J. F. (2005). Comparing new Keynesian models of the business cycle: A Bayesian approach. Journal of Monetary Economics (forthcoming).

Rotemberg, J. J. and Woodford, M. (1997). An optimization-based econometric framework for the evaluation of monetary policy. NBER Macroeconomics Annual, 12:297– 346.

Sargent, T. (1989). Two models of measurements and the investment accelerator. Journal of Political Economy, 97(2):251–287.

Sbordone, A. M. (2002). Prices and unit labor costs: A new test of price stickiness. Journal of Monetary Economics, 49(2):265–292. REFERENCES 49

Sims, C. A. (2002). Solving linear rational expectations models. Computational Eco- nomics, 20(1-2):1–20.

Sims, C. A. and Zha, T. (1999). Error bands for impulse responses. Econometrica, 67(5):1113–1155.

Smets, F. and Wouters, R. (2003). An estimated dynamic stochastic general equilibrium model of the euro area. Journal of the European Economic Association, 1(5):1123– 1175.

Smets, F. and Wouters, R. (2004). Forecasting with a Bayesian DSGE model: An application to the euro area. Journal of Common Market Studies, 42(4):841–867.

Taylor, J. B. (1993). Discretion versus policy rules in practice. Carnegie-Rochester Con- ference Series on Public Policy, 39:195–214.

Walsh, C. E. (2003). Monetary Theory and Policy. The MIT Press, Cambridge, Massa- chusetts; London, England, 2 edition.

Woodford, M. (2003). Interest and Prices: Foundations of a Theory of Monetary Policy. Princeton University Press, Princeton, New Jersey.

Chapter 2

Transmission of Exchange-Rate Variations in an Estimated, Small-Open Economy Model

The spectrum of models comprising the ‘New Open Economy Macroeconomics’ (NOEM)- literature has widened considerably in the decade following the seminal contributions by Obstfeld and Rogoff (1995, 1996).1 Compared to their closed-economy counterparts, these open-economy models employ a greater degree of complexity in modelling households’ preferences, imported inputs, international risk sharing and nominal rigidities (wages, producer prices and/or consumer prices). In the Mundell-Fleming model, as well as many more recent contributions to the NOEM literature, perfect exchange rate pass-through is assumed. However, empirical evidence, such as that of Engel (1999), Goldberg and Knetter (1997) and Campa and Goldberg (2002), shows that exchange-rate pass through onto prices is gradual. As a consequence, researchers have sought various ways to include partial pass through of exchange-rate changes in open economy models (Adolfson, 2002; Monacelli, 2005). The transmission of exchange-rate changes is of particular importance for the conduct of monetary policy as it can determine, for example, whether reaction of the central bank to a depreciation is desirable. This question is not only of general interest but also topical in light of recent large movements of the Euro-Dollar exchange rate. In this paper I study the effects of nominal exchange-rate shocks on domestic inflation in a small open-economy model that differs from the standard NOEM framework. Build- ing on McCallum and Nelson (1999, 2000), I assume that imported goods are treated as material inputs to production. Consequently, only domestic goods are consumed and there are no foreign (directly) imported consumption goods. This approximation can be motivated on the grounds that final foreign consumption goods account for only a small

1For surveys of the literature see Lane (2001) and Sarno (2001).

51 52 2. Exchange-Rate Pass Through fraction of all imported goods and services in many countries (Burstein et al., 2005). Most imported goods are not sold directly to consumers but pass through an imperfectly competitive distribution sector that adds services with non-tradeable goods characteristics (Obstfeld, 2002). For example, German imported consumption expenditure in 2000 accounted for approximately 9 percent of total private consumption expenditure (Statistis- ches Bundesamt, 2000). Furthermore, Burstein et al. (2005) report direct import-content shares of consumption in the U.S. and Swedish consumer price indices (CPI) of 4.7 and 13.6 percent respectively. The modelling strategy chosen here postulates that the CPI pertains only to the prices of domestic goods, just as in a closed economy. Price effects from foreign goods arise, however, due to the use of imported goods in the production technology. Since it is assumed that domestic firms have pricing power and set prices in a staggered fashion, the model implies indirect pass-through of exchange-rate shocks to CPI-inflation via marginal costs. In contrast, exchange-rate shocks have a direct impact on the CPI in the standard NOEM setup because the CPI is comprised of a basket of domestic and foreign consumer goods prices. This feature will be added as an extension to the benchmark model of this paper. A more complex model environment could be created in which importing firms refine imported products and sell them as intermediate inputs to final-goods production firms (Adolfson et al., 2005; Smets and Wouters, 2002, for example). However, this sacrifices tractability and requires additional assumptions about the markets in which intermediate and final goods are traded and thus the price-setting behaviour of firms operating in these markets. The approach taken here is simpler and preserves straightforward economic intuition. The theoretical literature on NOEM models has grown rapidly during recent years but the empirical literature has been late to develop. Since many results in the theoretical literature are sensitive to the precise model specification, more empirical evidence and evaluation is needed. This is particularly true if such models are to be used for policy recommendations. Some recent examples of empirical work with NOEM models include the following. Ghironi (2000) estimates single structural equations in a large open-economy model by means of nonlinear least squares and Smets and Wouters (2002) construct an open economy DGE model with sticky prices in both the domestic and imported-goods sector. Smets and Wouters calibrate their model by matching implied theoretical impulse responses with the empirical impulse responses estimated from a VAR model. Using a synthetic EMU data set, they report a considerable degree of stickiness in domestic and import prices. Bergin (2003) uses full-information maximum-likelihood to estimate a model with different specifications of price setting for three small open economies while 2.1. An Alternative Open-Economy Model 53

Lubik and Schorfheide (2003) analyse whether central banks respond to exchange rate depreciations. They do so for a number of countries and find that only the UK exhibits a significant response in a Taylor-type monetary-policy rule.2 Furthermore, Adolfson et al. (2005) find that their large-scale model for a small open economy with about 50 estimated parameters can match Euro area data well. Following the suggestions of DeJong et al. (2002a,b), the estimation technique applied here is Bayesian. Similar techniques are also used in Welz (2005), Smets and Wouters (2003), Adolfson et al. (2005), Lubik and Schorfheide (2003, 2005) and Justiniano and Preston (2004). The model below is estimated on German data. Germany is of special interest not only because of its relative importance in the aggregate EMU economy (30 percent of aggregate GDP) but because of its unique and stable monetary regime for the two decades prior to EMU. The paper is structured as follows. Section 2.1 lays out the benchmark model and Section 2.2 shows how the model can be extended to allow for imported consumption goods. Section 2.3 discusses the estimation methodology and the data. In Section 2.4, the estimation results for the benchmark specification are analysed and its dynamic properties are studied by means of impulse response analysis. Section 2.5 compares these results to the extended model and is followed by concluding comments.

2.1 An Alternative Open-Economy Model

The standard model employed in the NOEM literature assumes that households consume home-produced and foreign-produced consumption goods. In this framework, consumption is a CES-aggregated composite of domestic and foreign consumption goods and the consumer price index is accordingly a CES-aggregate of the respective consumer price indices. The approach in the benchmark model below introduces foreign-economy aspects through the production side rather than the demand side in a New Keynesian macro model with staggered price and wage setting a` la Calvo (1983). Speciﬁcally, ﬁrms are assumed to produce goods for consumption and export with two inputs to production: labour services provided by domestic households and imported goods whose real price is the real exchange rate. This modelling strategy results in a consumption basket and a CPI that is analogous to the closed-economy case. In line with much of the literature on small open economies, the framework below assumes that the economy operates in the ‘cashless limit’, that is, money balances are not

2More recently, they study the empirical properties of a two country model (Lubik and Schorfheide, 2005). 54 2. Exchange-Rate Pass Through explicitly introduced. The model also employs the standard device of abstracting from capital investment. It is worth pointing out two main diﬀerences of the model in this paper from that presented by McCallum and Nelson (1999, 2000). Here workers have market power and set their wages in a staggered fashion. There is also a well-deﬁned steady state for consumption-asset holdings through the inclusion of a debt-elastic risk premium for foreign bonds. In the following section I focus on the open-economy aspects of the model. The New Keynesian features of price and wage setting are by now well established and thus not every equation is carefully derived. Appendix B provides more details.

2.1.1 Aggregate Supply Production and Marginal Cost

All consumption goods, Yit, are produced by a continuum of monopolistically competitive

ﬁrms using labour, Nit, and imported goods, Mit, as production inputs. That is,

= α 1−α, Yit ZtNit Mit (2.1) where Zt is a stationary, exogenous productivity shock common to all ﬁrms. With this Cobb-Douglas technology, real marginal cost is common across ﬁrms and given by the ratio of the real wage to the marginal product of labour, i.e.

1 WtNt MCt = . (2.2) α PtYt

Foreign ﬁrms set the price of products exported to the home country in their own currency. That is, they engage in producer-currency pricing. Furthermore, the law of one price M = ∗ holds for imported goods in this model so that domestic ﬁrms pay the price Pt S tPt for the imported inputs and the nominal wage Wt to workers. Here, S t is the nominal ∗ exchange rate denominated in domestic-currency units per foreign-currency unit and Pt is the foreign price level. As shown in Appendix B, real marginal cost can also be expressed as dependent on the two factor prices

α = α−α − α −(1−α) 1 Wt 1−α, MCt (1 ) Qt (2.3) Zt Pt where Qt is the real exchange rate deﬁned as

∗ S tPt Qt = . (2.4) Pt 2.1. An Alternative Open-Economy Model 55

Hence the real exchange rate can be interpreted as the price of imports in terms of consumption goods. A real depreciation increases real marginal cost via a higher domestic- currency price of imported inputs.3

Price setting

As is now standard in the New Keynesian literature, firms are assumed to set prices according to the mechanism suggested by Calvo (1983). That is, in any period t a random − θ opt 4 fraction, 1 p, of firms resets prices to the new optimal price, Pt . The remaining fraction of firms update their price to past inflation, Πt−1, according to the partial indexation rule

log Pt = log Pt−1 + απΠt−1, (2.5) where απ ∈ [0, 1] is the degree of price indexation. These assumptions yield the familiar log-linearised ‘hybrid’ New Keynesian Phillips curve 5

β απ (1 − βθ )(1 − θ ) π = π + π + p p + επ, t Et t+1 t−1 mct t (2.6) 1 + απβ 1 + απβ θp(1 + απβ) where β is a discount factor. Real marginal cost in log-linearised form becomes

mct = α(wt − pt) − zt + (1 − α)qt. (2.7)

Without derivation I have also added a cost-push term, ut, that is assumed be a white- noise process. Such a shock may arise due to a time-varying substitution elasticity among goods, as argued by Christiano et al. (2005), Smets and Wouters (2003) and Steinsson (2003). This leads to variations in ﬁrms’ monopoly power over time and hence to time- varying desired mark-up. Clarida et al. (2001) alternatively suggest that introducing a stochastic wage markup to represent deviations between the marginal rate of substitution between leisure and consumption and the real wage will lead to the same cost-push term as in (2.6).6,7 However, in an open-economy setting there may be additional sources of price shocks that cannot be captured by the structure of the simple model. For this reason, the shock process is not given a microeconomic foundation in this model.

3 ∗ Pt is also the foreign-currency price of the foreign consumption good because of the small fraction of domestic export goods in the foreign price level. 4I neglect the firm-specific index because each firm that is allowed to change its price will set the same optimal price. 5 Note that for απ = 0 (no indexation) the standard forward looking Phillips curve is obtained. 6 This approach was also taken in Welz (2005). Note that the standard deviation of the shock term in ut needs to be re-scaled in order to obtain a unit coefficient on ut in (2.6). 7 επ Alternatively, t could be derived as an endogenous cost term as suggested by Ravenna and Walsh (2004). By linking marginal costs to the nominal rate of interest in a cash-in-advance model, a cost channel is introduced that generates endogenous cost-push shocks because firms must borrow money to pay their wage bills. 56 2. Exchange-Rate Pass Through

2.1.2 Aggregate Demand and Wage Setting Households’ Consumption Decision

There is a continuum of households with measure 1, indexed by j ∈ [0, 1], that maximise life-time utility given by   1+ϕ ∞   −  N js  βs t gs  − −  , Et e log C js Hs  (2.8) s=t 1 + ϕ where Ht = hCt−1 is the habit stock that depends on last period’s aggregate consumption, and is thus taken as exogenous by the household for the period-t consumption decision.

N jt is the amount of labour supplied and ϕ denotes the inverse labour supply elasticity.

The shock to the discount factor, gt, is a stationary process and can be interpreted as a demand or preference shock. As in the closed-economy setup, aggregate consumption is given by the standard Dixit-Stiglitz aggregate and demand for each variety of products is given by8

− Pit Cit = Ct. (2.9) Pt The period budget constraint can be formulated as ∗ B S t B + jt + jt = + ∗ + + , PtC jt + + ∗ χ ,ξ B jt−1 S t B jt−1 WtNt PtVt (2.10) 1 it (1 it ) (At t)

9 where Vt are real profits from production. It is assumed that domestic households hold bonds denominated in domestic and foreign currency, whereas foreign households only hold bonds denominated in their own currency. Let B jt denote end-of-period nominal holdings of the risk-free one-period nominal bond in domestic currency and it the nominal −1 domestic interest rate, implying that (1 + it) is the price of a domestic bond. Likewise ∗ B jt denotes the holding of the foreign risk-free bond by domestic households in foreign + ∗ χ ,ξ −1 ∗ currency at the price of [(1 it ) (At t)] , where it is the foreign nominal interest rate. χ ,ξ The function (At t) is a risk premium on these foreign bond holdings that depends on the real, aggregate, net foreign-asset position in the domestic economy, defined as ∗ S t Bt At ≡ , Pt ξ = γ ξ + εx and an exogenous shock process t x t−1 t . 1 −1 −1 8 = Neglecting time-subscripts, the consumption aggregate is C 0 Ci di . The associated aggregate price index that yields the minimum expenditure (PiCi) on the composite consumption basket is 1 − = 1 1− 1 P 0 Pi di . 9Assuming that households own the firms. 2.1. An Alternative Open-Economy Model 57

χ ,ξ The function (At t) is assumed to be of the form

−φ − +ξ χ ,ξ = a(At A) t , (At t) e (2.11)

φ > where a 0 and A denotes steady-state net foreign-asset holdings, so that in steady state χ(A, 0) = 1 and A = 0 (see the Appendix B.3). Households take this function as given when they decide optimal bond holdings. This approach follows Benigno (2001) and Linde´ et al. (2004) and captures imperfect integration in international ﬁnancial markets. ∗ < , < If the domestic country is a net borrower (Bt 0 At 0), domestic households must pay a premium on the foreign interest rate whereas if the domestic country is a net lender ∗ > , > (Bt 0 At 0), households receive a lower payment on their savings. This ensures a well-deﬁned steady state in this small open-economy model.10

Export demand

The domestic country is assumed to be small with respect to the rest of the world in that its exports contribute an insigniﬁcant fraction to foreigners’ aggregate demand. Follow- ing McCallum and Nelson (1999), I assume that foreign demand for each variety of the domestically-produced good is analogous to demand for consumption goods by domestic citizens and is given by − Pit Xit = Xt. (2.12) Pt The structure of the foreign economy is not explicitly modelled and so aggregate foreign demand for domestic products is postulated to depend positively on the real exchange rate

= η∗ ∗, Xt Qt Yt (2.13) where η∗ > 0.

Aggregate demand

In the benchmark model all imported goods are used as inputs in production so that the resource constraint (in log-linearised form) is given by

yt = (1 − ωx)ct + ωx xt, (2.14) where x is the log-deviation of export demand from its steady state and ω ≡ X is the t x Y steady state export share. This implies that yt represents output, not value added. Max- imising utility in equation (2.8) subject to the budget constraint in equation (2.10) yields the ﬁrst order conditions shown in Appendix B.2. Combining the static condition for

10See Schmitt-Grohe´ and Uribe (2003) and Ghironi (2000) for alternative approaches. 58 2. Exchange-Rate Pass Through consumption and the intertemporal ﬁrst-order condition yields a dynamic aggregate demand condition in terms of consumption. Inserting the resource constraint (2.14) into that equation yields the aggregate demand equation in log-linearised form as

yt − ωx xt = δEt{yt+1 − ωx xt+1} + (1 − δ)(yt−1 − ωx xt−1) − ω − δ −(1 x)(1 h) − {π } − − γ , σ it Et t+1 (1 g)gt (2.15)

δ ≡ 1 .11 where 1+h

Uncovered interest parity

The ﬁrst order conditions for domestic and foreign bond holdings yield an uncovered interest parity condition that takes account of the assumptions about international ﬁnancial markets, − ∗ = − φ + ξ , it it Et st+1 aat t (2.16) where I show in Appendix B.3 that the dynamics of net foreign assets are guided by = 1 + Y ω − + + 1 , at βat−1 β x( yt xt) θ mct (2.17) where at = dAt.

Wage setting

Several studies have pointed out the importance of nominal wage rigidities for improving the empirical ﬁt of New Keynesian DSGE models.12 In light of these, I assume that households act as price-setters in the labour market as in Erceg et al. (2000) and numerous other empirical papers. Households face a Calvo-type restriction similar to ﬁrms and can reset their wage only after receiving a random price-change signal which arrives with constant probability

1 − θw. Furthermore, analogous to the ﬁrms’ price-setting behaviour, it is assumed that those households that do not get the opportunity to reset their wage to the optimal wage choose to partially index the nominal wage to past inﬂation according to

log Wt = log Wt−1 + αwΠt−1, (2.18)

11Note that the closed-economy, forward-looking aggregate demand relationship obtains for h = 0 and ωx = 0. 12Rabanal and Rubio-Ram´ırez (2003, 2005) apply a Bayesian approach for a systematic comparison of diﬀerent model speciﬁcations to U.S. and EMU data. 2.1. An Alternative Open-Economy Model 59

where αw ∈ [0, 1] is the degree of wage indexation to past inﬂation. These considerations result in the following log-linearised equation for nominal wage growth:

∆wt − αwπt−1 = βEt∆wt+1 − αwβπt − αwβπt + (2.19) (1 − θw)(1 − βθw) mrst − (wt − pt) θw (1 + wϕ) where mrst is the marginal rate of substitution between consumption and labour given by 1 σ mrs = ϕ y − z − (1 − α)(w − p − q ) + (c − hc − ) (2.20) t t α t t t t 1 − h t t 1 and w > 1 is the labour demand elasticity. Using the identity

wt − pt = wt−1 − pt−1 +∆wt − ∆pt, (2.21) it is possible to re-express (2.19) in terms of the real wage.

2.1.3 Monetary Policy

Rather than derive the optimal monetary policy reaction resulting from minimisation of an objective function, I take an empirical approach and assume that the central bank chooses the nominal interest rate as its instrument and reacts to deviations of inﬂation and the output gap from steady state. Speciﬁcally, the central bank behaves according to

= + − π + + εi, it fiit−1 (1 fi)( fπ t fyyt) t (2.22) εi ﬃ where t is assumed be white noise. All coe cients are assumed to be positive and the 13 smoothing coeﬃcient is assumed to regard the restriction fi ∈ [0, 1).

2.1.4 Foreign Economy

The foreign economy is approximated by a closed economy. For simplicity I do not formulate a full structural model but rather summarise the dynamics of the three foreign variables by an estimated structural VAR(p)-process,

∗ = ∗ + ε∗,ε∗ ∼ , Ω . A0 xt A(L)xt−1 t t N(0 ) (2.23) ∗ = π∗, ∗, ∗ π∗ Here, xt [ t yt it ] is the vector of foreign variables comprised of inﬂation, t , nominal ∗ ∗ interest rate, it , and output, yt . Demand and monetary shocks are identiﬁed by imposing the following restrictions on A0:      100 =   . A0  010 (2.24) −a31 −a32 1

13See the discussion in Welz (2005) for a justiﬁcation of this approach for the Bundesbank. 60 2. Exchange-Rate Pass Through

The identifying assumption is that monetary shocks in the foreign country have a one- period delayed effect on inflation and the output gap due to predetermined expectations. Furthermore, demand shocks are identified by assuming that inflationary shocks do not have contemporaneous effects on output. Note that the VAR representation in (2.23) could be derived as the reduced form of a structural closed-economy model with one-period delayed effects of monetary policy on output and inflation. The VAR is estimated prior to estimation of the home-country model. In effect, the estimated equations (2.23) are added to the structural model.14

2.2 Including Foreign Consumption Goods

The above framework introduces foreign-economy features through the production side rather than the demand side. In this section I extend the benchmark model to allow for foreign consumption goods that are imported by retail ﬁrms who act in an imperfectly competitive market. This assumption implies that the law of one price for consumption goods does not hold.

Assume that the fraction of imported consumption goods is ωm ∈ [0, 1] and the elasticity of substitution between domestic and foreign consumption goods is a constant, η>0. Then the consumption aggregate and the CPI are, respectively, given by

η 1 η−1 1 η−1 η−1 = − ω η h η + ω η f η Ct (1 m) (Ct ) m(Ct ) (2.25) and −η = − ω h 1−η + ω f η−1 1 , Pt (1 m)(Pt ) m(Pt ) (2.26)

h f h where Ct (Ct ) denotes consumption of domestic (foreign) consumption goods and Pt f (Pt ) is the average price of the domestic (foreign) consumption goods. To shorten the presentation, all other equations of the extended model will be presented in log-linearised form (see Appendix B for full details). Demand for domestic and foreign consumption goods are respectively given by

h = + ω ητ ct ct m t (2.27) f = − − ω ητ , ct ct (1 m) t (2.28)

τ ≡ f − h where the terms-of-trade are deﬁned as t pt pt . Export demand can be expressed as

= η∗ + ω τ + ∗. xt (qt m t) yt (2.29)

14Justiniano and Preston (2004) estimate a standard NOEM framework using Bayesian methods with a similar modelling strategy for the foreign sector and compare the case where the foreign VAR is estimated simultaneously with the case where it is estimated beforehand. 2.2. Including Foreign Consumption Goods 61

An expression for aggregate demand can be found when (2.27) and (2.28) are inserted into the log-linearised consumption aggregate (2.25) to yield − ω − δ = δ { } + − δ − (1 x)(1 h) − {π } − − γ yt Et yt+1 (1 )yt−1 σ it Et t+1 (1 g)gt (2.30) where yt ≡ yt − ωx xt − (1 − ωx)ωmητt. Real marginal cost of domestic producers is given by

= − h + − mct wt pt nt zt

= α(wt − pt) − zt + ωmτt + (1 − α)qt (2.31) and the Phillips curve can be formulated in terms of domestic consumption good inﬂation,

β απ (1 − βθ )(1 − θ ) πh = πh + πh + p p + επ. t Et t+1 t−1 mct t (2.32) 1 + απβ 1 + απβ θp(1 + απβ) This is related to consumer price inﬂation by

π = πh + ω ∆τ . t t m t (2.33) f ∗ If the law of one price for foreign consumption goods does not hold, i.e. Pt S tPt , the real exchange rate can be written as = ∗ + − qt pt st pt (2.34) = ∗ + − f + f − h + ω τ pt st pt pt pt m t = ψ + − ω τ . t (1 m) t ψ ≡ ∗ + − f Monacelli (2005) refers to t pt st pt the ’law of one price-gap’. I assume that retail firms face a similar, Calvo-style pricing problem as domestic producers. As explained in Monacelli (2005), the assumption that importers purchase foreign f ∗ consumption goods at world market prices pt is equivalent to assuming that the law of one price holds ‘at the dock’. However, they charge a markup on marginal cost when selling the products to domestic consumers so that the consumer price for imported consumption goods, measured in domestic currency, differs from world market prices. In addition, those retailers who do not reset their price to the new optimal price use the same partial-indexation rule as domestic producers. These considerations lead to the following equation for foreign consumption-good inflation: β απ (1 − βθ )(1 − θ ) π f = π f + π f + f f ψ , t Et t+1 t−1 t (2.35) 1 + απβ 1 + απβ θ f (1 + απβ) where 1 − θ f is the fraction of firms that can re-optimise prices each period. I assume that domestic retailers and produces choose the same partial indexation value απ. The other equations of the model are identical to the benchmark model described in Section 2.1.

Note, that the benchmark model is obtained when ωm = 0. 62 2. Exchange-Rate Pass Through

2.3 Solution and Estimation

2.3.1 Model Solution

The benchmark model is comprised of ten domestic endogenous variables ιt = (πt, yt, mct, − , , , , , ∆ ∗ = π∗, ∗, ∗ wt pt mrst, xt at it qt st) , the foreign block xt ( t yt it ) and three stationary but (possibly) persistent exogenous shock processes that are assumed to be independent of each other a priori: = + ζ ,ζ∼ , Φ , t B t−1 t t N(0 ) = , ,ξ . where t (gt zt t) The model can then be written in matrix form as

Γ ξ ι =Γ ξ ι +Ψζ +Πϑ . 0( ) t 1( ) t−1 t t (2.36)

The matrices Γ0(ξ), Γ1(ξ), Ψ and Π are coeﬃcient matrices, ϑt is a vector of expectational errors (Et(ϑt+1) = 0) that are introduced to construct the forward-looking variables in the model such that ϑt = xt − Et−1 xt. The matrix B is subsumed in Γ0(ξ) and Γ1(ξ) and ιt is the ι , ∗ 15 state vector of endogenous variables in the model that stacks t xt and t. The general solution to (2.36) has a VAR(1)-representation

ι = ξ ι + ξ η . t T( ) t−1 R( ) t (2.37)

Note that the system is stochastically singular since ιt has a higher dimension than the number of stochastic shocks, rendering the covariance matrix of the disturbances singular. Hence the observable time series are selected via the measurement equation

Yt = Fιt, (2.38) where Yt is the vector of observable variables.

2.3.2 Methodology

The model is estimated using Bayesian techniques. Extensive surveys of the advantages and disadvantages of the Bayesian approach are readily available and not repeated here. Welz (2005) and the papers cited in the introduction provide an overview. Since the small open-economy model studied here imposes strong restrictions across coefficients, model misspecification may be a relevant problem. However, as discussed by Lubik and Schorfheide (2005), relaxing restrictions and building larger models may result in identification problems. Unlike Smets and Wouters (2003) and Adolfson et al.

15The model is estimated using Dynare but all post-estimation analysis, e.g. impulse-response analysis and calculation of their highest posterior densities, is conducted using the method of Sims (2002) to solve linear rational expectations models. 2.4. Data and Prior Speciﬁcation 63

(2005), the strategy in the present study is not to aim to fit all aspects of the data but to study the model’s properties via impulse-response analysis and the estimated parameters. These are interesting exercises in themselves as to date only calibrated versions of the non-standard model presented here have been studied by McCallum and Nelson (1999, 2000). The focus will be on the transmission of exchange rate variations to CPI-inflation and the effects of monetary-policy shocks.

2.4 Data and Prior Speciﬁcation

2.4.1 Data

The small open-economy model is estimated on German data for the period 1980Q1 to 2004Q3. The data set includes output measured as gross domestic product, annualised CPI-inflation, the annualised nominal short-term interest rate and a nominal effective exchange rate that is based on the nominal exchange rates and constant trade weights of Germany’s fifteen most important trading partners (a more detailed description of the data set is provided in Appendix A).16 The foreign country variables are output, inflation and nominal short-term interest rates. The first two are constructed, respectively, as trade- weighted averages of GDP and annualised growth rates of CPI-indices. The nominal interest rate is calculated as the trade-weighted arithmetic mean of annualised short-term interest rates. All data is seasonally adjusted. Trends are not explicitly taken account of in this model, thus all variables are transformed into stationary time series prior to estimation. The foreign variables are detrended with the HP-filter whereas German inflation, interest-rate, output and nominal depreciation measures were constructed from the residuals of OLS regressions on a constant, dummies and a linear time trend.17 The constant and time trend are intended to account for changes in the trend growth rate. The effective nominal exchange rate is included in percentage depreciation rates from a constant which is close to zero. One caveat should be mentioned. A single monetary policy for the Bundesbank did not exist during the last six years of the sample. Whilst this gives cause for caution in interpreting the estimation results, the size of the German economy is the Euro area should ensure that any bias is not large.18

16Eastern European countries have recently become important trading partners for Germany but due to the short period of time this has been of relevance, these countries are not included in the sample. 17The linking method by Fagan and Henry (1998) was used to construct a time series for German GDP. 18German data is available from 1970 but the time span was restricted by availability of GDP series for some foreign countries. 64 2. Exchange-Rate Pass Through

2.4.2 Prior Speciﬁcation

Calibrated Parameters

The formal inclusion of prior information about the model specification lies at the heart of Bayesian analysis. Some parameters were fixed a priori, because they may be difficult to estimate as they are related to steady-state values (the data is demeaned) or because no data is used which could provide direct information, such as consumption expenditure on foreign goods. In terms of the Bayesian approach, this implies a degenerate prior density with a given mean and infinite precision. The discount factor β is set to 0.99 implying an annual steady-state interest rate of 4 percent. The steady-state value of domestic GDP is set to the mean of the GDP-series before detrending, y = 4.4. The inverse of the labour supply elasticity, ϕ, is set to unity and the labour demand elasticity, w is set to

6, implying a wage markup of 20 percent. The steady state trade share ωx = 0.3is calculated as the sample mean of the ratio of export demand to GDP. Initially, a rather φ , uninformative prior for a the parameter that links the net foreign asset relation to the uncovered interest parity (2.16) was used, but this parameter is diﬃcult to identify and introduced some instabilities in the estimation procedure. It was thus ﬁxed to 0.002, the φ . value that was obtained when the posterior mode was estimated for a In the extended model, the substitution elasticity between foreign and domestic consumption goods is set to unity.

Estimated Parameters

The prior density for the parameter vector of the estimated parameters is specified under the assumption that all parameters are independently distributed of each other. However, the solution set of the DSGE model is restricted to unique and stable solutions which may imply prior dependence. The following principle has been applied to select the prior shape: for all parameters that should lie in the (0,1)-interval according to theory, a beta density is chosen as it is defined on the unit interval. Inverted gamma densities are chosen for the standard deviations of exogenous shocks. The lack of a priori information about the precision of these parameters is accounted for by specifying two degrees of freedom for the inverted gamma densities, implying that the variance is infinite. For all remaining parameters, normal densities are specified. I now turn to the choice of prior density means. A detailed specification can be found in Table 2.1 which also depicts the estimation results from the benchmark model. The prior mean of the inflation indexation coefficient απ is set to 0.5. While Christiano et al. (2005) set this value to unity, Smets and Wouters (2003) find a value significantly smaller 2.5. Results 65 than 1, implying stronger forward-looking. It is assumed that there is endogenous persistence in demand so a value of 0.8 is chosen as the prior mean for the habit persistence parameter. The prior mean for the price and wage stickiness parameter is set to 0.5 which implies that prices and wages are changed about twice a year. Using German data, Coenen and Levin (2004) estimate this duration in a generalised Calvo-pricing model as opposed to many other empirical studies that find longer durations. However, the large standard deviation of 0.25 implies that these priors are relatively uninformative. The prior means for the monetary policy rule parameters are set to standard values and the export demand elasticity is assumed to follow a normal distribution with mean 1. The prior means for the persistence parameters of the exogenous shock processes are set at intermediate values of 0.5 with large standard deviations of 0.25. However, for the technology shock process a fairly high value of 0.8 with tight standard deviation of 0.10 is specified. This choice can partly be rationalised by the fact, that technology processes have often been estimated to be highly persistent. The cost shock and the monetary policy shock are assumed to be white noise processes. The prior modes of the standard deviations of the shock processes are based on simple first order autoregressions estimated on German data for the period 1973Q1 to 1979Q4.

2.5 Results

2.5.1 The Benchmark Model

It is not possible to solve for the posterior density in analytical form. Instead, I estimate it with the random-walk chain Metropolis-Hastings algorithm with a multivariate normal proposal density, centered at the posterior mode and with covariance matrix calculated from the hessian at the posterior mode. The algorithm is used to generate 150 000 draws, of which the last 60 percent are used for sampling.19 The estimation results are reported in Table 2.1 while Figures 2.4 and 2.5 in Appendix D display kernel estimates of the marginal priors and marginal posteriors of each parameter. Overall, the results are promising: the posterior densities appear to have a single mode and are reasonably symmetric around that value. In most cases the posterior mode and mean are close to one another and the posteriors have tighter variances than the priors. The following discussion of the results is in terms of the means of the marginal posteriors. Labour’s share in production, α, is estimated at 0.75, implying an import share in production of 25 percent. The estimates imply a relatively modest degree of price indexation (0.21) which translates into a forward-looking coeﬃcient in the Phillips curve of

19The convergence diagnostics displayed in Appendix D indicate that this number of simulations is suf- ﬁcient to achieve convergence of the Markov chain. 66 2. Exchange-Rate Pass Through

Table 2.1: Prior Specification and Estimation results - Benchmark model Prior Specification Posterior Estimates Parameter Density Mean Std Dev Mode 5% Mean 95% import input share α Beta 0.67 0.15 0.84 0.58 0.75 0.90 inflation indexation απ Beta 0.50 0.25 0.19 0.05 0.21 0.40 wage indexation αw Beta 0.50 0.25 0.36 0.06 0.31 0.66 habit persistence h Beta 0.80 0.15 0.94 0.82 0.90 0.96 price stickiness θπ Beta 0.50 0.25 0.88 0.83 0.87 0.91 wage stickiness θw Beta 0.50 0.25 0.83 0.76 0.88 0.98 Monetary Policy rule interest rate fi Beta 0.80 0.15 0.88 0.83 0.88 0.92 inflation fπ Normal 1.50 0.15 1.08 0.86 1.11 1.38 output gap fy Normal 0.50 0.15 0.39 0.28 0.41 0.58 relation to foreign economy export demand elast η∗ Normal 1.00 0.25 1.31 0.84 1.19 1.54 shock persistence γ preference g Beta 0.50 0.25 0.05 0.01 0.10 0.25 γ productivity z Beta 0.80 0.10 0.91 0.72 0.85 0.95 risk premium γχ Beta 0.50 0.25 0.92 0.86 0.90 0.94 shock variances Mode Dofa preference σg Inv Gamma 1.60 2.00 1.15 0.98 1.16 1.38 wage cost σg Inv Gamma 1.30 2.00 0.70 0.61 0.69 0.80 cost push σu Inv Gamma 0.80 2.00 0.77 0.63 0.75 0.87 productivity σz Inv Gamma 0.80 2.00 0.57 0.43 1.64 3.63 interest rate σi Inv Gamma 0.50 2.00 0.36 0.32 0.37 0.42 risk premium shock σξ Inv Gamma 1.50 2.00 0.29 0.25 0.32 0.40 Marginal likelihood: -795.799 aNote: Dof = degrees of freedom

0.82. The value of 0.87 for the price stickiness parameter implies that prices are fixed for about 7.5 quarters, which is consistent with other empirical DSGE studies using EMU data (Adolfson et al., 2005; Smets and Wouters, 2003). It is also in line with the results in Welz (2005) for a closed-economy model estimated with German data, but significantly higher than the result found by Coenen and Levin (2004). The results also point to a similar degree of wage stickiness and wage partial indexation. While habit persistence is estimated to be high, the preference shock process is not very persistent. The technology and the risk premium shock both are highly persistent. Finally, the estimated Taylor-rule coefficients for inflation, the output gap and the lagged interest rate confirm the findings of other empirical studies that the dynamics of the German interest rate are described well by such a rule. 2.5. Results 67

2.5.2 Transmission of Shocks in the Benchmark Model

In this section I study the dynamic properties of the benchmark model with respect to risk-premium shocks and monetary shocks. A shock to the risk-premium process can be interpreted as an exogenous shock to expectations about future depreciation rates. Quan- titatively, a one standard deviation shock results in a nominal depreciation of about 1.5 percent and a slightly smaller real depreciation. The real depreciation leads to an increase in marginal cost of about 0.36 percent (23 percent of the magnitude of the nominal depreciation) as shown in Figure 2.1. However, the contemporaneous pass through to annualised inflation is only moderate: only about 5.5 percent of the nominal depreciation pass through onto consumer prices in the first quarter. Despite the strong rise of marginal cost following the depreciation, inflation rises very little because of the low sensitivity (mean coefficient estimate 0.0172) to marginal cost. These results are entirely guided by the degree of price stickiness. As can be seen clearly in Figure 2.8 in Appendix D, the pass through of a nominal depreciation onto annualised inflation is almost linear. A price stickiness parameter θp of 0.01, for example, implies almost complete exchange rate pass-through.

Figure 2.1: Uncovered Interest-Parity Shock - Benchmark Model Inflation Output

0.2 0.8

0.1 0.6 0.4 0 0.2 −0.1 0 0 5 10 15 20 0 5 10 15 20

Nominal Interest Marginal Cost

0.4 0.8 0.6 0.2 0.4 0.2 0 0 0 5 10 15 20 0 5 10 15 20

Real Exchange Rate Nominal Depreciation

2 2

1 1

0 0 0 5 10 15 20 0 5 10 15 20

Notes: Impulse responses (solid lines) from the benchmark model with 10- and 90%- tiles (dash-dotted lines) to a one standard deviation risk premium shock. 68 2. Exchange-Rate Pass Through

For reference I report the effects of a one standard deviation contractionary monetary policy impulse. The impulse responses are standard and presented in Figure 2.2. The rise in the interest rate leads to an immediate nominal and real appreciation with relatively small effects on inflation and output.

Figure 2.2: Monetary Policy Shock - Benchmark Model Inflation Output 0.05 0

0 −0.1

−0.2 −0.05 0 5 10 15 20 0 5 10 15 20

Nominal Interest Marginal Cost 0.05

0.4 0 −0.05 0.2 −0.1 −0.15 0 0 5 10 15 20 0 5 10 15 20

Real Exchange Rate Nominal Depreciation

0 0

−0.2 −0.2

−0.4 −0.4

0 5 10 15 20 0 5 10 15 20

Notes: Impulse responses (solid lines) from the benchmark model with 10- and 90%- tiles (dash-dotted lines) to a one standard-deviation interest-rate shock.

2.5.3 The Extended Model

Estimation results for the extended model are shown in Table 2.3 in Appendix D.3. Most of the estimated means are close to those of the benchmark model although interestingly, the fraction of imported consumption goods is estimated at only 5 percent. The estimated mode of 0.14 of the price-stickiness parameter for foreign consumption goods implies rather ﬂexible prices and a fast transmission of exchange-rate variations through this channel. However, the parameter does not appear to be well identiﬁed in the data, its posterior mean is 0.43. This also evident from the kernel estimate shown in Figure 2.9 in Appendix D. In any event, pass through to foreign consumption prices is faster than onto domestic consumer prices. The results in Table 2.3 indicate that the 5th percentile is located at 0.05 and the 95th percentile at 0.87. 2.5. Results 69

Figure 2.3 illustrates the dynamics of the model in response to a one standard deviation risk-premium shock. As expected from the estimation results, the picture is qualitatively similar to the benchmark model. The major difference is the much larger contemporaneous exchange-rate pass through onto CPI inflation (23 percent) despite a quantitatively similar response of marginal cost to the nominal depreciation (22 percent). Hence, the small fraction of imported consumption goods together with a low degree of price stickiness leads to a much stronger effect on annualised inflation.

Figure 2.3: Uncovered Interest-Parity Shock - Extended Model Inflation Output 1 0.6

0.4 0.5 0.2

0 0 0 5 10 15 20 0 5 10 15 20

Nominal Interest Marginal Cost

0.6 0.4 0.4 0.2 0.2

0 0

0 5 10 15 20 0 5 10 15 20

Real Exchange Rate Nominal Depreciation

2 2 1.5 1 1 0.5 0 0 0 5 10 15 20 0 5 10 15 20

Notes: Impulse responses (solid lines) from the benchmark model with 10- and 90%- tiles (dash-dotted lines) to a one standard-deviation risk premium shock.

2.5.4 Evaluation

The two models deliver quantitatively different answers which leaves open the question of which better matches the empirical facts. One way to evaluate the models is to study the correlations between inflation and current and past depreciations. In the first column of Table 2.2 are correlations for the actual data over the sample period that was used for estimation. The next two columns show the same relation for 5 000 observations of simulated data from the estimated benchmark and the extended models. 70 2. Exchange-Rate Pass Through

Table 2.2: Correlations between Inﬂation and Nominal Depreciation Corr(πt, ∆st−k) actual data benchmark model extended model k=0 0.13 0.34 0.60 k=1 0.22 0.36 0.39 k=2 0.13 0.31 0.30 k=3 0.19 0.25 0.19 k=4 0.08 0.20 0.11

The values in Table 2.2 suggest that neither model perfectly replicates the properties of the data but that the benchmark model comes closer to the data in the ﬁrst two quarters, whereas the extended model matches the correlations better at larger lags. Relying on the marginal likelihoods, however, puts the extended model (marginal likelihood = -794.746) slightly in favour, but formally the two are not distinguishable. Assuming that both models complete the model space and assigning equal prior probabilities of 0.5, the Bayes factor in favour of model i versus j can be calculated as

f (YT |Mi) Bij = , (2.39) f (YT |Mj) where

f (YT |Mi) = ϕ(ξ|Mi) f (YT |ξ, Mi)dξ (2.40) Ξ is the marginal likelihood of model Mi, ϕ(ξ|Mi) the prior density and f (YT |ξ, Mi) the likelihood of model i. Assuming that falsely choosing a model incurs equal losses for both models, a Bayes factor greater than 1 indicates that model i is more likely than model j after having observed the data. The calculation for the two models above yields exp(−795.799) B , = = 0.3489, benchmark extended exp(−744.746) which is no decisive evidence against the benchmark model.

2.6 Conclusions

This paper has outlined and estimated a small open-economy model in which open- economy aspects are introduced on the supply side. Imported goods serve exclusively as inputs to production which implies that the CPI is comprised of prices for domestic goods only. As a result, inflationary pressures from a depreciating exchange rate are transmit- ted through the cost channel of domestic firms. In this environment, the degree to which exchange-rate fluctuations pass through onto prices depends on the degree of price stickiness in the economy. The results show that a nominal depreciation of the exchange rate leads to instantaneous pass through onto inflation which amounts to only 5.5 percent of 2.6. Conclusions 71 the magnitude of the initial depreciation in the benchmark model. Estimating an extended model that permits foreign consumption goods raises exchange-rate pass through considerably to 23 percent. Furthermore, in terms of the marginal likelihood, the extended model fairs slightly better than the benchmark model, but the benchmark model is not formally rejected. The benchmark model fares better at matching correlations between current inflation and current and past exchange-rate movements contemporaneously and at short lags, whereas the extended model matches these correlations better between current inflation and higher lagged depreciations. In conclusion, this study shows that within the class of relatively simple small open- economy DSGE models, adopting the modelling approach that all imported goods enter the production process has the potential to explain the low degree of exchange-rate pass through. Both models feature a number of restrictions, for example on the substitution elasticities between the factors of production or domestic and foreign consumption goods which were assumed to be one. One interesting extension would be to find estimates of these elasticities in a more general formulation of the model. But this will probably require additional data on consumption expenditure and the production process that can help identify these parameters. 72 2. Exchange-Rate Pass Through

Appendices A Data and Sources

• 16 OECD countries: Austria (AUT), Belgium (BEL), Canada (CAN), Denmark (DEN), Finland (FIN), France (FRA), Germany (GER), Italy (ITA), Japan (JAP), Netherlands (NED), Norway (NOR), Spain (SPA), Sweden (SWE), Switzerland (SWI), United Kingdom (UK), United States (US).

• variables and data source

– interest rates from IMF International Financial Statistics (mainly call money market rates, FIN = ‘Cost of CB Debt’) – CPI from OECD Main Economic Indicators – Trade Weights (TCW) from Sveriges Riksbank (based on IMF calculations) the trade weights for Germany are: AUT = 0.06, BEL = 0.08, CAN = 0.01, DEN = 0.02, FIN = 0.01, FRA = 0.17, ITA = 0.13, JAP = 0.07, NED = 0.08, NOR = 0.01, SPA = 0.05, SWE = 0.03, SWI = 0.07, UK = 0.11, US = 0.11. – Nominal exchange rates: National currency/US$ from IMF International Fi- nancial Statistics calculate together with CPI and TCW nominal and real eﬀective exchange rates (REER) – GDP volume index (base year 2000): from IMF International Financial Sta- tistics and OECD Main Economic Indicators Germany: OECD Quarterly National Accounts for Westen Germany and Ger- many. The series were linked according to the ratio of Germany/Western Ger- many on the start date 1991:1 as in Fagan et al. (2001) Denmark: OECD Quarterly National Accounts, OECD Main Economic Indi- cators and IMF International Financial Statistics – all series are seasonally adjusted series or parts of them that were not seasonally adjusted have been adjusted with TRAMO/Seats

• time span: data used for estimation 1980:1 - 2003:4, for some countries data is available from some year in 1970 and later the time span is mainly restricted by available GDP and interest rate data

• aggregation method ﬀ = ”Index-method” for GDP, inﬂation and e ective exchange rates: ln Xa i wi ln Xi A. Data and Sources 73

arithmetic mean for interest rates (Fagan and Henry, 1998, p. 504, footnote 22) which are measured in annual percentages 74 2. Exchange-Rate Pass Through

B Derivation of Model Dynamics in the Benchmark Model

B.1 Firms

Neglecting firm specific indices, the cost minimisation problem of the firm is

, , F = + M C(Yt Wt Pt ) WtNt Pt Mt . . = α 1−α. s t Yt ZtNt Mt

Labour demand is then − −α α 1−α 1 W (1 ) N = t Y t − α M t 1 Zt Pt − −α α 1−α (1 ) = 1 Wt P − α ∗ Yt 1 Zt Pt S tPt − −α α 1−α (1 ) = 1 Wt 1−α Qt Yt 1 − α Zt Pt M = ∗ where I have used the fact that the import price in home currency is given by Pt S tPt ∗ S t Pt and the real exchange rate is deﬁned as Qt = . Pt Import demand α α −α 1 W M = t Y t 1 − α Z M t t Pt −α α α 1 Wt 1 = Yt 1 − α Zt Pt Qt can be written in log-linearised form:

mt = α(wt − pt − zt − qt) + yt.

With Cobb-Douglas technology, real marginal cost is given by

real wage 1 WtNt MCt = = MPN α PtYt and substituting labour demand yields − −α α 1−α (1 ) = 1 Wt 1 Wt 1−α MCt Qt α Pt 1 − α Zt Pt α 1 W = α−α(1 − α)−(1−α) t Q1−α. Zt Pt Log-linearising this relationship yields

mct = wt − pt + nt − yt or

mct = α(wt − pt − zt) + (1 − α)qt B. Derivation of Model Dynamics in the Benchmark Model 75

Proﬁts

Real proﬁts are given by = 1 opt − nom Vt Pt MCt Yt Pt = 1 , MC = ε−1 = opt Log-linearisation using the steady-state relationships V ε Y P ε and P P

Popt MCnom Vv = Y(popt − p + y ) − Y mcnom − p + y t P t t t P t t t ε − 1 = opt − + − 1 + εvt pt pt yt ε (mct yt)

vt = yt + mct

B.2 Aggregate Demand

The optimality conditions of the household’s optimisation problem are given by

egt C jt : λt = Uc = C jt − hCt−1 Wt ϕ λ = = gt Nt : t UN e Nt Pt Pt B jt : λt = (1 + it)βEt λt+1 Pt+1 ∗ λ = + ∗ + χ β λ Pt S t+1 B jt : t (1 it )(1 t) Et t+1 Pt+1 S t

The consumption Euler equation can be derived from the ﬁrst order conditions of the consumer’s problem: gt gt+1 e = + β e Pt − (1 it) Et − Ct hCt−1 Ct+1 hCt Pt+1 C − hC − ∆gt+1 t t 1 1 = βRtEt e Ct+1 − hCt

Pt where Rt = (1 + it) . Pt+1 Log-linearisation yields

1 h 1 − h c = E c + + c − − (i − E π + +∆g + ) t 1 + h t t 1 1 + h t 1 σ(1 + h) t t t 1 t 1

Then inserting the log-linearised resource constraint

1 ωx ct = yt + xt 1 − ωx 1 − ωx gives equation (2.15). 76 2. Exchange-Rate Pass Through

B.3 Net Foreign Assets

∗ S t Bt Assume that domestic bonds are in zero net supply. Deﬁne At ≡ and write the Pt aggregate budget constraint as

+ At = Pt−1 S t + Wt + Ct ∗χ ,ξ At−1 Nt Vt Rt (At t) Pt S t−1 Pt The risk premium is deﬁned as

−φ − +ξ χ ,ξ = a(At A) t (At t) e

ξ = γ ξ + εx where t x t−1 t Log-linearisation of the budget constraint yields      χ  CdCt − A ∗ + 1  1 − A (A)  ... ∗ dRt ∗   dAt C χ(A, 0)(R )2 R χ(A, 0) χ(A, 0) 2  = 1 =0 WN dWt WN dP WN dNt dVt = dA − + − + + V t 1 P W P P P N V = WN + , = − ω , = 1 Use the steady state relationships: C P V C (1 x)Y V ε Y and use the notation at ≡ dAt

WN Cc + βa = a − + (w − p + n ) + Vv t t t 1 P t t t t = 1 − + − − + + at β at−1 Cct (C V)(wt pt nt) Vvt = 1 − + − + + + β at−1 Cct (C V)(yt mct) V(yt mct) = 1 + − ω + − β at−1 (1 x)Y(yt mct ct) = 1 + Y ω − + + − ω βat−1 β x( yt xt) (1 x)mct C. The Extended Model 77

C The Extended Model

Demand

Demand for the domestic and foreign consumption good by domestic agents is

−ε  −ε Ph P f  h = it h, f =  it  f Cit h Ct Cit  f  Ct. Pt Pt The optimal allocation of expenditure across domestic and foreign goods is given by

−η  −η Ph P f  h = − ω t f = ω  t  Ct (1 m) Ct, Ct m   Ct Pt Pt

Export demand is given as −η∗ Ph/S = t t ∗b Xt ∗ Yt Pt

Supply

1. Domestic producers: The Dixit-Stiglitz price index for domestic prices is given by

 απ 1−ε  Ph  h 1−ε = − θ h,opt 1−ε + θ  h t−1  (Pt ) (1 p)(Pt ) p Pt−1 h Pt−2 The ﬁrms’ objective is to maximise future discounted proﬁts

 α  ∞  h π  s−t h  h,opt Ps−1 h  θ ,  −  Et p Dt sYis Pt h Ps MCs s=t Ps−2 subject to the demand schedule     , απ −ε Ph Ph opt   h =  it  t−1   h + Yit  h  h   Ct Xt Pt Pt−2 The ﬁrst order condition is given by:

 α  ∞  h π θ  s−t h  h,opt Ps−1 p h  E θ D , y P − P MC  = 0 t p t s is t h θ − 1 s s s=t Ps−2 p

where Dt,s is a stochastic discount factor.

2. Domestic retailers face an analogous problem to the domestic producers. The price index follows the law of motion

  απ 1−ε  P f   f 1−ε = − θ f,opt 1−ε + θ  f  t−1   . (Pt ) (1 f )(Pt ) f Pt−1  f   Pt−2 78 2. Exchange-Rate Pass Through

Retailers maximise   α  ∞ f π    ∗  s−t f  f,opt Ps−1  f  θ ,    −  Et f Dt sCs (i) Pt (i)  f  S sPs  s=t Ps−2 subject to the demand schedule     , απ −ε P f opt P f   f =  t  t−1   f Cit  f  f   Ct Pt Pt−2 and the ﬁrst order condition is given by

 α  ∞  h π θ  s−t  f,opt Ps−1 f f ∗  E θ D , P − S P  = 0 t f t s t h θ − 1 s s s=t Ps−2 f Log-linearising the price index and this ﬁrst order condition and combining them yields (2.35). D. Figures and Results 79

D Figures and Results

D.1 Benchmark Model

Figure 2.4: Prior- and Posterior Density - Benchmark Model α α α π w 2 3 3 2 2 1 1 1 0 0 0 0 0.5 1 0 0.5 1 0 0.5 1 θ θ h π w 4 10 5 2 5 0 0 0 0 0.5 1 0 0.5 1 0 0.5 1 f f f i π y 4 10 2

5 1 2

0 0 0 0 0.5 1 0 1 2 0 1 2 s γ γ η g z 5 5 1

0 0 0 0 1 2 0 0.5 1 0 0.5 1

Notes: Prior (dashed lines) and posterior densities (solid lines) for the benchmark open economy model. 80 2. Exchange-Rate Pass Through

Figure 2.5: Prior- and Posterior Density (continued) - Benchmark Model γ σ σ χ π w 15 3 6

10 2 4

5 1 2

0 0 0 0 0.5 1 0 1 2 3 0 1 2 3 σ σ σ g z i

4 1 10

2 0.5 5

0 0 0 0 1 2 3 0 1 2 3 0 1 2 3 σ χ 8 6 4 2 0 0 1 2 3

Notes: Notes: Prior (dashed lines) and posterior densities (solid lines) for the benchmark open economy model. D. Figures and Results 81

Figure 2.6: CUSUM Diagnostic - Benchmark Model α α α π w 0.2 0.2 0.2

0 0 0

−0.2 −0.2 −0.2 0 5 10 15 0 5 10 15 0 5 10 15 4 4 4 x 10 x 10 x 10 θ θ h π w 0.2 0.2 0.2 0 0 0 −0.2 −0.2 −0.2 0 5 10 15 0 5 10 15 0 5 10 15 4 4 4 x 10 x 10 x 10 f f f i π y 0.2 0.2 0.2 0 0 0 −0.2 −0.2 −0.2 0 5 10 15 0 5 10 15 0 5 10 15 4 4 4 x 10 x 10 x 10 s γ γ η g z 0.2 0.2 0.2

0 0 0

−0.2 −0.2 −0.2 0 5 10 15 0 5 10 15 0 5 10 15 4 4 4 x 10 x 10 x 10

Notes: Notes: The horizontal grey lines indicate 5% bands, the vertical line indicates the 40%-burn-in of the Markov chain with 150 000 simulations. 82 2. Exchange-Rate Pass Through

Figure 2.7: CUSUM Diagnostic (continued) - Benchmark Model γ σ σ χ π w

0.2 0.2 0.2 0.1 0.1 0.1 0 0 0 −0.1 −0.1 −0.1 −0.2 −0.2 −0.2 0 5 10 15 0 5 10 15 0 5 10 15 4 4 4 x 10 x 10 x 10 σ σ σ g z i 0.2 0.2 0.2

0 0 0

−0.2 −0.2 −0.2 0 5 10 15 0 5 10 15 0 5 10 15 4 4 4 x 10 x 10 x 10 σ χ

0.2

−0.2 0 5 10 15 4 x 10

Notes: Notes: The horizontal grey lines indicate 5% bands, the vertical line indicates the 40%-burn-in of the Markov chain with 150 000 simulations.

Given N draws from the Markov chain, the statistic is calculated as in Bauwens et al. (1999)    s  1  CS =  γ − µγ /σγ, s s i i=1 where µ and σγ are the mean and standard deviation of the N draws respectively, and γ 1 s γ s i=1 i is the running mean for a subset of s draws of the chain. If the chain converges, the graph of CS s should converge smoothly to zero. On the contrary, long and regular movements away from the zero line indicate that the chain has not converged. According to Bauwens et al. (1999), a CUSUM value of 0.05 after s draws means that the estimate of the posterior expectation deviates from the ﬁnal estimate after N draws by 5 percent in units of the ﬁnal estimate of the posterior standard deviation. The authors consider a value of 25 percent to be a good result. Figures 2.6 and 2.7 show the CUSUM-paths along with 5 percent bands for each parameter for 150 000 draws (note that the overall interval corresponds to 25 percent bands). D. Figures and Results 83

D.2 Exchange-Rate Pass Through and Price Stickiness

Figure 2.8: Exchange-Rate Pass Through and Price Stickiness 100

pass−through % 40

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 θ p

Notes: Relationship between price stickiness and exchange-rate pass through in the benchmark model, holding all other parameters constant. The inﬂation response (y-axis) is measure in percentage points of the magnitude of the nominal depreciation in the ﬁrst quarter, for values of θp ∈ [0.01, 0.99]. 84 2. Exchange-Rate Pass Through

D.3 Prior- and Posterior Density - Extended Model

Table 2.3: Prior Specification and Estimation Results - Extended Model Prior specification Model estimates Parameter Density Mean Std Dev Mode 5% Mean 95% import input share α Beta 0.67 0.15 0.76 0.60 0.77 0.90 inflation indexation απ Beta 0.50 0.25 0.09 0.03 0.15 0.30 wage indexation αw Beta 0.50 0.25 0.32 0.07 0.35 0.69 habit persistence h Beta 0.70 0.15 0.95 0.82 0.91 0.96 price stickiness θπ Beta 0.50 0.25 0.89 0.85 0.88 0.91 wage stickiness θw Beta 0.50 0.25 0.96 0.82 0.92 0.99 price stickiness foreign θ f Beta 0.50 0.25 0.14 0.05 0.43 0.87 Monetary Policy rule interest rate fi Beta 0.80 0.15 0.87 0.82 0.87 0.92 inflation fπ Normal 1.50 0.15 1.05 0.85 1.09 1.36 output gap fy Normal 0.50 0.15 0.37 0.28 0.40 0.56 relation to foreign economy export demand elast η∗ Normal 1.00 0.25 1.24 0.90 1.20 1.52 foreign cons ωm Beta 0.10 0.05 0.06 0.02 0.05 0.09 shock persistence γ preference g Beta 0.50 0.25 0.04 0.01 0.09 0.22 γ productivity z Beta 0.80 0.10 0.90 0.71 0.85 0.94 risk premium γχ Beta 0.50 0.25 0.87 0.79 0.86 0.91 shock variances Mode Dofa cost push σπ Inv Gamma 1.60 2.00 1.18 0.98 1.16 1.37 wage cost σw Inv Gamma 1.30 2.00 0.66 0.60 0.68 0.78 preference σg Inv Gamma 0.80 2.00 0.79 0.67 0.79 0.92 productivity σz Inv Gamma 0.80 2.00 1.64 0.81 2.29 4.40 interest rate σi Inv Gamma 0.50 2.00 0.36 0.33 0.37 0.43 risk premium shock σξ Inv Gamma 1.50 2.00 0.32 0.27 0.34 0.43 Marginal likelihood: -794.746 aNote: Dof = degrees of freedom D. Figures and Results 85

Figure 2.9: Prior- and Posterior Density - Extended Model α α α π w 4 3 1.5 1 2 2 1 0.5 0 0 0 0 0.5 1 0 0.5 1 0 0.5 1 θ θ h π w

5 5 10

0 0 0 0 0.5 1 0 0.5 1 0 0.5 1 θ f f f i π

1 10 2

0.5 5 1

0 0 0 0 0.5 1 0 0.5 1 0 1 2 f s ω y η m 4 20 1 2 10

0 0 0 0 1 2 0 1 2 0 0.5 1

Notes: Prior (dashed lines) and posterior densities (solid lines) for the benchmark open economy model. 86 2. Exchange-Rate Pass Through

Figure 2.10: Prior- and Posterior Density (continued) - Extended Model γ γ γ g z χ 10

4 4

5 2 2

0 0 0 0 0.5 1 0 0.5 1 0 0.5 1 σ σ σ π w g 3 6 4 2 4 2 1 2

0 0 0 0 1 2 3 0 1 2 3 0 1 2 3 σ σ σ z i χ

1 10 6

4 0.5 5 2

0 0 0 0 1 2 3 0 1 2 3 0 1 2 3

Notes: Prior (dashed lines) and posterior densities (solid lines) for the benchmark open economy model. REFERENCES 87

References

Adolfson, M. (2002). Monetary policy with incomplete exchange rate pass-through. Sveriges Riksbank Working Paper No. 127.

Adolfson, M., Lasseen,´ S., Linde,´ J., and Villani, M. (2005). Bayesian estimation of an open economy dsge model with incomplete pass-through. Sveriges Riksbank Working Paper No 179.

Bauwens, L., Lubrano, M., and Richard, J.-F. (1999). Bayesian Inference in Dynamic Econometric Models. Oxford University Press.

Benigno, P. (2001). Price stability with imperfect ﬁnancial integration. Manuscript, New York University.

Bergin, P. (2003). Putting the New Open Economy Macroeconomics to a test. Journal of International Economics, 60:3–34.

Burstein, A., Eichenbaum, M., and Rebelo, S. (2005). Large devaluations and the real exchange rate. Journal of Political Economy (forthcoming).

Calvo, G. (1983). Staggered prices in a utility maximising framework. Journal of Mone- tary Economics, 12:383–398.

Campa, J. M. and Goldberg, L. S. (2002). Exchange rate pass-through into import prices. Review of Economics and Statistics (forthcoming).

Christiano, L., Eichenbaum, M., and Evans, C. (2005). Nominal rigidities and the dynamic eﬀects of a shock to monetary policy. Journal of Political Economy, 118(1):1– 45.

Clarida, R., Gal´ı, J., and Gertler, M. (2001). Optimal monetary policy in open versus closed economies: An integrated approach. American Economic Review, 91:248–252.

Coenen, G. and Levin, A. T. (2004). Identifying the inﬂuences of nominal and real rigidities in aggregate price-setting behaviour. European Central Bank, Working Paper N0. 418, European Central Bank.

DeJong, D., Ingram, B., and Whiteman, C. (2002a). A Bayesian approach to dynamic macroeconomics. Journal of Econometrics, 98:203–223.

DeJong, D., Ingram, B., and Whiteman, C. (2002b). Keynesian impulses versus solow residuals: Identifying sources of business cycle ﬂuctuations. Journal of Applied Econo- metrics, 15:311–329.

Engel, C. M. (1999). Accounting for u.s. real exchange rate changes. Journal of Political Economy, 107:507–538.

Erceg, C., Henderson, D., and Levin, A. (2000). Optimal monetary policy with staggered wage and price contracts. Journal of Monetary Economics, 46(2):281–313. 88 2. Exchange-Rate Pass Through

Fagan, G. and Henry, J. (1998). Long run money demand in the eu: Evidence for area- wide aggregates. Empirical Economics, 23:483–506.

Fagan, G., Henry, J., and Mestre, R. (2001). An area-wide model (awm) of the Euro area. Working Paper No. 42, European Central Bank.

Ghironi, F. (2000). Towards New Open Macroeconometrics. Boston College Working Paper 469.

Goldberg, P. K. and Knetter, M. M. (1997). Goods prices and exchange rates. what have we learned? Journal of Economic Literature, 35:1243–1272.

Justiniano, A. and Preston, B. (2004). Small open economy DSGE models: speciﬁcation, estimation and model ﬁt. Manuscript IMF.

Lane, P. (2001). The New Open Macroeconomics: A survey. Journal of International Economics, 54:235–266.

Linde,´ J., Nessen,´ M., and Soderstr¨ om,¨ U. (2004). Monetary policy in an estimated open- economy model with imperfect pass-through. Sveriges Riksbank Working Paper No 167, Sveriges Riksbank.

Lubik, T. and Schorfheide, F. (2003). Do central banks respond to exchange rate movements? A structural investigation. Manuscript.

Lubik, T. and Schorfheide, F. (2005). A Bayesian look at New Open Economy Macro- economics. NBER Macroeconomics Annual (forthcoming).

McCallum, B. T. and Nelson, E. (1999). Nominal income targeting in an open-economy optimizing model. Journal of Monetary Economics, 43:553–578.

McCallum, B. T. and Nelson, E. (2000). Monetary policy for an open economy: An alternative framework with optimizing agents and sticky prices. Oxford Review of Economic Policy, 16(4):74–91.

Monacelli, T. (2005). Monetary policy in a low pass-through environment. Journal of Money Credit and Banking (forthcoming).

Obstfeld, M. (2002). Exchange rates and adjustment: Perspectives from the New Open Economy Macroeconomics. Manuscript.

Obstfeld, M. and Rogoﬀ, K. (1995). Exchange rate dynamics redux. Journal of Political Economy, 103:624–660.

Obstfeld, M. and Rogoﬀ, K. (1996). Foundations of International Macroeconomics. MIT Press.

Rabanal, P. and Rubio-Ram´ırez, J. F. (2003). Comparing New Keynesian models in the Euro area: A Bayesian approach. Federal Reserve Bank of Atlanta Working Paper.

Rabanal, P. and Rubio-Ram´ırez, J. F. (2005). Comparing New Keynesian models of the business cycle: A Bayesian approach. Journal of Monetary Economics (forthcoming). REFERENCES 89

Ravenna, F. and Walsh, C. E. (2004). Optimal monetary policy with the cost channel. Journal of Monetary Economcis (forthcoming).

Sarno, L. (2001). Toward a new paradigm in open economy modeling: Where do we stand? Federal Reserve Bank of St. Louis Review, 83(3):21–36.

Schmitt-Grohe,´ S. and Uribe, M. (2003). Closing small open economy models. Journal of International Economics, 61:163–185.

Sims, C. A. (2002). Solving linear rational expectations models. Computational Eco- nomics, 20:1–20.

Smets, F. and Wouters, R. (2002). Openness, imperfect exchange rate pass-through and monetary policy. Journal of Monetary Economics, 49:947–981.

Smets, F. and Wouters, R. (2003). An estimated dynamic stochastic general equilibrium model of the Euro area. Journal of the European Economic Association, 1(5):1123– 1175.

Statistisches Bundesamt (2000). Volkswirtschaftliche Gesamtrechnungen. Input-Output- Rechnung 2000.

Steinsson, J. (2003). Optimal monetary policy in an economy with inﬂation persistence. Journal of Monetary Economics, 50:1425–1456. Journal of Monetary Economics.

Welz, P. (2005). Assessing predetermined expectations in the standard sticky price model: A Bayesian approach. Manuscript, Uppsala University.

Chapter 3

Can a Time-Varying Equilibrium Real Interest Rate Explain the Excess Sensitivity Puzzle?

3.1 Introduction

Standard models of monetary policy assume that long-term interest rates should remain stable when shocks arrive in the economy. In contrast, observed long-term interest rates often exhibit considerable movement in the same direction as short-term interest rates, that is, the yield curve shifts in a parallel manner. This behaviour has been labelled the excess sensitivity and/or excess volatility puzzle. The excess volatility puzzle denotes the finding that the variance of long-term interest rates is far larger than standard models predict,1 whereas the excess sensitivity puzzle concerns the large movements in long-term interest rates associated with changes of the policy-controlled short rate. Because we are primarily interested in explaining the prevalence of parallel shifts of forward rates and the yield curve, our focus here is on the excess sensitivity puzzle. The excess sensitivity puzzle has been studied by Gurkaynak¨ et al. (2003), Beechey (2004) and Ellingsen and Soderstr¨ om¨ (2005) amongst others. Gurkaynak¨ et al. (2003) conclude that long-run inflation expectations are a crucial component of the puzzle and that the assumption of a time-invariant steady state is incapable of generating such behaviour. More specifically, they suggest that the private sector adjusts its expectations of the long-run inflation target in response to macroeconomic shocks in a manner that causes the observed movements of nominal long-term interest rates. Ellingsen and Soderstr¨ om¨ (2005) show that a model with private central bank information about future inflation generates moderate parallel shifts in the yield curve when the economy is hit by supply and demand shocks. They estimate that the ten-year interest rate rises on average by 25

1See e.g. Shiller (1979) for an early discussion.

91 92 3. The Excess Sensitivity Puzzle basis points in response to an unexpected one percentage point increase in the policy- controlled short rate. Beechey (2004) shows that a non-stationary inflation target and adaptive learning about the target can also generate sensitivity of long-term interest rates. Overall, existing explanations focus on whether the central bank has a credible and known inflation target. This paper investigates an alternative solution to the excess sensitivity puzzle, namely that it may be due to persistent effects of shocks to the equilibrium real interest rate. Models which are typically used to illustrate the excess sensitivity puzzle assume that the equilibrium interest rate is (i) constant over time and (ii) independent of structural shocks. In contrast, many dynamic stochastic general equilibrium (DSGE) models share the features that shocks to technology or consumers’ rate of time preference cause persistent movements in the equilibrium real rate. As supporting evidence, several studies find that the equilibrium real rate can be modelled as non-stationary (Laubach and Williams, 2003) or near unit root process (Mesonnier´ and Renne, 2004), implying that shocks have permanent effects or at least highly persistent effects.2 Following the just cited contributions we use an unobserved components model estimated with the Kalman filter to extract a measure of the equilibrium real rate. We then show that a general equilibrium model augmented with the estimated process has the potential to explain the excess sensitivity puzzle. Relative to the case when the equilibrium real interest rate is assumed constant, forward-interest rates at long horizons respond more to short-rate movements when shocks have persistent effects on the equilibrium real rate. An alternative empirical formulation of the excess sensitivity puzzle is that long rates react by more than predicted to changes in the short rate. Subject to the monetary policy reaction function that we employ, are we able to show that the slope coefficients from regressions of changes in long-term interest rates on changes in short-term interest rates are about 10 basis points higher when the equilibrium real interest rate is time-varying than when it is constant. One motivation for focusing on the equilibrium real interest rate rather than informational aspects such as inflation expectations is the finding by Pennacchi (1991) that real interest rates are more volatile than expected inflation. An implication of this finding is that variation in long-term interest rates is primarily due to shocks to the real interest rate. As such, it deserves attention as an explanation for the excess sensitivity puzzle. Other authors have reached the opposite conclusion, that is that nominal volatility is more important than real volatility and we do not deny that inflation expectations and/or a time- varying inflation target may also play a role. However, in this study attention will be con- fined to real explanations. We find that equilibrium real interest rates display sufficient

2Other contributions in the same vein are Andres´ et al. (2004) and Manrique and Marques´ (2004). 3.2. A Stylised Model 93 variation and dependence on structural shocks to come close the observed co-movements of long-term interest rates and short-term interest rates. To generate persistent effects on interest rates, either the stochastic process of the shock itself or the effects of the shock on other variables in the model need to be highly persistent. Gurkaynak¨ et al. (2003) conclude that the degree of persistence of nominal shocks required to solve the puzzle is unreasonably high. Our contribution is to show that the effects of real shocks propagated through the equilibrium real rate is plausibly high enough to contribute to the model’s predictions with observed data. The paper is organised as follows. Section 2 presents a semi-structural general equilibrium model with a time-varying equilibrium real rate. In Section 3 we estimate this time-varying real rate. Section 4 analyses the implications for the excess sensitivity puzzle in two ways: first by examining the effect of shocks on long-horizon forward rates and second through the coefficients of regressions of long-rate changes on short rate movements. Section 5 concludes.

3.2 A Stylised Model

We employ the stylised model estimated by Rudebusch and Svensson (1999) and Rude- busch (2002), used in Gurkaynak¨ et al. (2003) and Ellingsen and Soderstr¨ om¨ (2005) for the analysis of excess sensitivity of long-term interest rates. As discussed in Rudebusch and Svensson (1999) and Rudebusch (2002), such a model can account for several broad characteristics of larger models commonly used by central banks. It also shares features of modern DSGE models, including both forward-looking and backward-looking terms in the Phillips curve and aggregate demand equation. However, the model used in this paper features richer dynamics than standard sticky-price models that are usually parsimonious in the lag structure but based on ﬁrmer microfoundations.

Let yt denote the output gap measured as the percent deviation of output from its potential, πt, the annualised quarterly inflation rate and rt ≡ it − Et−1πt+3 the real interest rate measured as the difference between the nominal interest rate and four-quarter ended 1 3 − π + ≡ − π + inflation expectations. Inflation expectations are calculated as Et 1 t 3 Et 1 4 j=0 t j. Aggregate demand and is affected by monetary policy via the real interest rate gap, the de- ∗ viation of the real rate from its equilibrium level, rt , and its backward-looking component is an AR(2) process, = φ + − φ 2 α − α − ∗ + εy. yt yEt−1yt+1 1 y s=1 ysyt−s r(rt−1 rt−1) t (3.1)

The equilibrium real rate is deﬁned as the real interest rate that has a neutral eﬀect on demand, that is the real interest rate consistent with a zero output gap given that the 94 3. The Excess Sensitivity Puzzle

φ εy economy is in equilibrium. y governs the degree of forward-looking and t is a zero mean iid-shock. The Phillips curve allows for both backward-looking and forward-looking behaviour as well, where φπ is the weight on expected inflation and 1 − φπ is the weight on lags one to four of inflation. π = φ π + − φ 4 β π + β + επ. t πEt−1 t+3 (1 π) s=1 πs t−s y1yt−1 t (3.2) φ φ There is a continuing debate about the magnitude of the parameters y and π; Estrella and Fuhrer (2002) amongst others have argued that purely forward-looking frameworks are empirically implausible whereas Rudebusch (2002) and Clarida et al. (2000) argue that expectations about future inflation are an important component in hybrid models. The results also depend on the type of model and on the assumed policy regime. In estimated DSGE models with a simple Taylor rule chosen to describe monetary policy, it is often found that φπ is relatively high, in the range of 0.7, whereas forward-looking in consumption is low.3 However, Soderlind¨ et al. (2005) estimate a model like ours assuming that the central bank implements a discretionary optimal monetary policy with the objective to minimise the weighted unconditional variances of inflation, the output gap and the change in the nominal interest rate. They find that inflation expectations play a minor role whereas expectations about the output gap are important. On the other hand, single equation estimations of equation (3.1) and (3.2) seem to confirm the results found in DSGE models.4 In Rudebusch’s specification and other models used to analyse the excess sensitivity puzzle, the equilibrium real rate is assumed to be constant over time and unaffected by ∗ = structural shocks, i.e. rt r for all t. However, as mentioned in the introduction, there is abundant empirical evidence that the equilibrium real rate shifts over time, in particular in response to shocks hitting the economy. Therefore, we model the equilibrium real εy rate as an autoregressive progress that is driven by the demand disturbance t and own ε∗ disturbances, t , ∗ = ρ ∗ + ρ εy + ε∗. rt rrt−1 e t t (3.3) Recalling that the aggregate demand equation is formulated in terms of the output gap, we show in Appendix A that this dynamic specification can be derived by assuming that two persistent processes affect the output gap: a persistent aggregate demand shock and, for example, a persistent productivity shock. As shown in Laubach and Williams (2003), a link between the equilibrium real rate and trend productivity can be derived from a stochastic growth model. The combination of two persistent processes leads generally to a

3See e.g. Smets and Wouters (2003) and Welz (2005). 4 φ φ Gal´ı and Gertler (1999) estimate π around 0.8 and Fuhrer and Rudebusch (2004) find values for y between 0 and 0.45. 3.3. Estimating a Time-Varying Equilibrium Real Rate 95 more complex dynamic process but can be simplified to the expression in equation (3.3) by assuming that both processes share the same degree of persistence.5 In addition, our structural model can be viewed as a stylised version of more elaborate models which account for an explicit link between the marginal product of capital and the real interest rate. Specifically in DSGE models that feature endogenous investment decisions, the equilibrium real interest rate depends on the capital stock, itself a persistent variable (Woodford, 2003). These features motivate our choice of equation (3.3). As is shown in Appendix ε∗ ff A, the shock t can be interpreted as capturing e ects on the equilibrium real rate that do not have a contemporaneous effect on the output gap or inflation but act with a one-period delay. As the model is not fully structural, we label it for the remainder of the paper equilibrium real rate shock. The timing assumption is our central identifying criteria between the two shocks. Allowing for a highly-persistent, time-varying equilibrium real rate in the aggregate demand relation will increase persistence in the nominal interest rate if the central bank responds to it. Hence, the model is closed by adding a Taylor rule which permits interest- rate smoothing, a common empirical finding; we will consider both fi = 0 and fi 0 in

= + − ∗ + π + + εi. it fiit−1 (1 fi)(rt fπ t fyyt) t (3.4)

Note that a constant equilibrium real rate (and inflation target) would appear as an intercept term in (3.4). We assume that the inflation target is constant and equal to zero, in contrast to Gurkaynak¨ et al. (2003) and Ellingsen and Soderstr¨ om¨ (2005) who suggest a time-varying inflation target to explain the excess sensitivity puzzle. Most parameter values for the model are taken from Rudebusch (2002). Parameters pertaining to the time-varying equilibrium real interest rate need to be estimated in order to calibrate the model and analyse the response of the short-term and long-term interest rates to shocks.

3.3 Estimating a Time-Varying Equilibrium Real Rate

The extent to which a time-varying equilibrium real rate can explain the excess sensitivity ρ ρ puzzle depends on the size of the parameters r and e in equation (3.3) and the standard deviations of the shocks. If the equilibrium real rate is highly autocorrelated or close to a random walk, only small shocks are required to create large co-movements between long- term interest rates and short-term interest rates. In this section we estimate a time-varying equilibrium real rate using an unobserved components model.

5We assume the two shock processes to be AR(1); the sum of two AR(1)-processes that have equal persistence ρ is an AR(1)-process with the same persistence ρ. 96 3. The Excess Sensitivity Puzzle

3.3.1 Data

All data is measured at a quarterly frequency and obtained from the Federal Reserve Bank of St. Louis database (FRED) covering the period 1959Q1-2004Q3. For the level of potential output (ypot) we use the series provided by the Congressional Budget Oﬃce

(CBO), where output (yt) is measured as chained real GDP in billions of year 2000 U.S. = / pot dollars. We construct the output gap as yt 100 log(yt yt ). The nominal interest rate is measured as the quarterly average of the monthly Federal Funds rate and the real interest rate is defined as the difference between the nominal interest rate and the year-on-year inflation rate measured by the percentage change in the GDP chained price index.

3.3.2 Empirical Speciﬁcation

We do not estimate the complete model from the previous section but rather focus on the aggregate demand equation and the dynamic speciﬁcation for the equilibrium real interest rate. However, the empirical speciﬁcation is similar to those of Laubach and Williams (2003) and Mesonnier´ and Renne (2004). These authors estimated the equilibrium real rate and potential output jointly. In our approach potential output is treated as an observable variable and the empirical model is formulated as follows:6

= α + α − α + εy yt y1yt−1 y2yt−2 rrt−1 t (3.5) ∗ = ρ ∗ + ρ εy + ε∗ rt rrt−1 e t t (3.6) = + + εr rt d1rt−1 d2rt−2 t (3.7) = − ∗ rt rt rt (3.8)

The first equation is the empirical counterpart of our aggregate demand equation (3.1), that is we do not attempt to estimate the forward-looking component of output, as this would be an additional unobservable variable. The second equation describes the dynamics of the equilibrium real interest rate. We postulate a stationary AR(2)-process for the real-rate gap in equation (3.7), assuming that it follows similar dynamics to the (backward component of the) output gap.7 Note that our model nests stationary and nonstationary ∗ dynamics for the equilibrium real rate, where nonstationarity implies that rt and rt are cointegrated as long as the real rate gap is stationary, which we restrict it to be. In addition to treating potential output as observable variable, differs our specification from those of Laubach and Williams (2003) and Mesonnier´ and Renne (2004) with respect ∗ to equations (3.6) and (3.7). The former authors postulate that rt is nonstationary because 6See Clark and Kozicki (2004) for a comparison of models that jointly estimate the equilibrium real rate of interest and potential output with models that only estimate the equilibrium real rate of interest given the CBO-measure of potential output. 7A similar approach has earlier been used to estimate potential output, for example by Clark (1987). 3.3. Estimating a Time-Varying Equilibrium Real Rate 97 in their model the equilibrium real rate depends on the trend growth rate of potential output, which is itself driven by a random walk. Mesonnier´ and Renne (2004) postulate the same joint link between the trend growth rate and potential output and the equilibrium real rate but assume that this growth rate is stationary. In our model we do not explicitly relate the equilibrium real rate to the trend growth rate of output but only approximate it to be autocorrelated and focus on its dependence on shocks. As stated before, demand shocks have contemporaneous effects on the output gap and the equilibrium real rate whereas real rate shocks influence the output gap with one-period delay. In the absence of shocks, equation (3.5) implies that in the long run a zero output gap corresponds with a zero interest rate gap, in line with our definition of the equilibrium real rate. It would obviously be desirable to estimate the entire model rather than focusing on the time-varying equilibrium real rate. A long sample period is essential for capturing movements in the equilibrium real rate but we have not succeeded in obtaining a satisfactory empirical specification of the whole theoretical model with constant parameters over such a long sample. In particular, the monetary policy reaction function appears unstable, possibly because the estimated time interval spans several different monetary policy regimes. Against this background it is not clear how to treat for instance the inflation target. Hence, we do not take a particular stance on monetary policy here but simply postulate in equation (3.7) that the Federal Reserve has conducted monetary policy in a way that the deviation of the real interest rate from its equilibrium is stationary over the estimation period. This is a general assumption that encompasses a dynamic Taylor rule of the type in our theoretical model. All shocks in the model are assumed independent of each other, implying a diagonal variance-covariance matrix of the transition equation    σ2   y  Σ= σ2  .  r  σ2 r∗

The model is written in state-space form (see Appendix B for a detailed representation) and the value of the likelihood function calculated with the Kalman ﬁlter. The likelihood function is maximised by standard procedures and the negative inverse Hessian is computed in order to ﬁnd the standard errors of the estimates.8 In the following section the estimation results are discussed.

8Since we assume that the equilibrium real rate is stationary, a proper prior distribution of the state −1 vector can be found. The initial covariance matrix is computed by vec(P1|0) = (I − T ⊗ T) vec(RQR ), the ∗ means of the real rate gap and the output gap are set to zero and the mean of rt is set equal to the mean of the real interest rate over the sample period (see Appendix B for details of the state space model and the notation). For calculation of the log-likelihood we use Paul Soderlind’s¨ Kalman ﬁlter Gauss code adapted to Matlab. 98 3. The Excess Sensitivity Puzzle

3.3.3 Results

For comparison, two models are estimated. Model 1 estimates all parameters freely whereas Model 2 fixes the standard deviation of the equilibrium real rate to σr∗ = 0.322; ∗ this value is estimated by Clark and Kozicki (2004) assuming that rt follows a random walk. The smaller standard deviation leads to a smoother estimate of the equilibrium real rate relative to Model 1. The equilibrium interest rate series is intrinsically difficult to measure; for instance, Orphanides and Williams (2002) compare six alternative methods of measuring the equilibrium real rate and find considerable differences depending on the method used. With regard to the smoothness of the equilibrium real rate, it is not imme- diately obvious that it should be smoother than the actual real interest rate. For example, Smets and Wouters (2003) find that the equilibrium real rate, defined as the real interest rate under flexible prices and absent nominal shocks, varies more than the real interest rate.

Figure 3.1: Estimated Equilibrium Real Rate 10 r* Model 1 r* Model 2 real rate 5

−5 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005

6 r−r* Model 1 r−r* Model 2 4

−2

−4 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005

Notes: Top panel: Estimated equilibrium real interest rates from Model 1 (dashed line) and Model 2 (solid line) and the real interest rate (dash-dotted line). Bottom panel: Real interest rate gaps from Model 1 (dashed line) and Model 2 (solid line).

Figure 3.1 graphs the equilibrium real rates generated from the two estimated models. The top panel plots the estimates from the one-sided Kalman ﬁlter together with the 3.3. Estimating a Time-Varying Equilibrium Real Rate 99 observed real interest rate. The bottom panel presents the interest rate gaps from the two models. The estimated interest rate series from Model 1 tracks the real rate closely but the lower panel reveals sizeable variation in the real rate gap up to 2 percentage points. As expected, Model 2 produces a ﬂatter estimate of the equilibrium real interest rate.

Table 3.1: Estimation Results

Parameter Model 1 Model 2

αy1 1.061 1.111 (11.26) (14.12) αy2 −0.118 −0.161 (−1.30) (−2.00) αr 0.366 0.168 (2.47) (2.46) d1 0.965 0.923 (5.65) (8.89) d2 −0.277 −0.116 (−2.03) (−1.14) ρ . . r 0 977 0 987 (54.25) (89.91) ρ . . e 0 257 0 316 (2.43) (2.82) σy 0.731 0.786 (14.26) (18.16) σr 0.595 0.827 (4.01) (15.86) σr∗ 0.662 0.322 (4.60) (−) Log-likelihood -440.58 -444.69 AIC / BIC -4.978 / -4.796 -5.037 / -4.873

Notes: In Model 2, σr∗ is ﬁxed to 0.322. AIC (BIC) = Akaike (Bayesian) information criterion

The main purpose of this study is not to find the most accurate measure of the equilibrium real rate but to document that it is time-variant, highly persistent and affected by real shocks. Both of our estimated models produce an estimate of the equilibrium real rate that fulfills these criteria. The coefficient estimates for the two models are shown in ρ Table 3.1. The equilibrium real rate appears to be highly persistent with r estimated as 0.98 and 0.99 respectively. In Appendix C we show plots of the conditional likelihoods ρ in an interval around the optimum for each parameter. These plots confirm that r should be below 1 and thus the equilibrium real rate should be stationary. The other coefficient estimates are similar across the two models, with the exception of αr (the sensitivity of the output gap with respect to the real rate gap) which has a smaller value in Model 2. The lower value is similar to that estimated in models assuming a constant equilibrium 100 3. The Excess Sensitivity Puzzle real interest rate (Rudebusch, 2002) but also falls closer to the range of estimates rates found by Laubach and Williams (2003).9

3.4 The Puzzle Explained?

We employ two types of output from the model to assess whether a time-varying equilibrium real interest rate can resolve the excess sensitivity puzzle. We begin by deriving the impulse response function of the short-term interest rate to various shocks. The desired empirical property is that the model is capable of generating significant movements at 20 to 40 quarter horizons. Next we generate time series data of the term structure from the model and regress changes in the artificial long-term interest rates on changes in the short- term rate. If the long-term (ten-year) interest rate attracts a coefficient of 0.2 or more, we conclude that our approach has the potential to explain the excess sensitivity puzzle. In both exercises, we compare results from the model with the time-varying equilibrium real rate to the model where the rate is constant. The model is calibrated using values from Rudebusch (2002) and our estimated parameters for the time-varying equilibrium real rate.10 Because the equilibrium real rate is constant in the standard Rudebusch specification, he does not provide parameter values ρ ρ . ffi for r and e All parameter values are shown in Table 3.2. For the empirical di culties discussed before, we also choose the standard parameter values for the Taylor rule suggested by Taylor (1993) and perform a sensitivity analysis with respect to interest-rate smoothing and the output gap coefficient. Two main caveats to the analysis need to be stated. First, we have made the strong assumption that the central bank has full information about the equilibrium real interest rate in the same period that it sets the nominal short-term interest rate. Assuming instead that the central bank observes the equilibrium real rate imperfectly would serve to strengthen our results but would conflict with the assumption of full information about the output gap. There is a rich empirical and theoretical literature that discusses the implications for monetary policy that arise from imperfect information about the stance of the economy in general and about natural rates in particular. Orphanides (2001) shows, for instance, that the use of real-time data instead of ex-post revised data changes the recommendations for monetary policy considerably. Furthermore, Orphanides and Williams (2002) discuss the performance of Taylor-type monetary policy rules in the presence of imperfect knowledge of the natural rate of unemployment and the equilibrium real rate of interest. In our

9 They estimate αr between 0.088 and 0.122. Appendix D shows graphs of the relationship between the lagged interest rate gap and the output gap. 10Because this model cannot be solved analytically we use algorithms provided in the Dynare-package for Matlab to obtain numerical simulated responses of the nominal interest rate it to shocks for speciﬁc sets of parameter values. 3.4. The Puzzle Explained? 101

Table 3.2: Model Calibration Equilibrium real Aggregate Supply Aggregate Demand Monetary Policy Interest rate φ = . α = . = . ρ = . π 0 29 y1 1 15 fπ 1 50 r 0 987 β = . α = − . = . ρ = . π1 0 67 y2 0 27 fy 0 50 e 0 316 β = − . α = . = . σ = . π2 0 14 r 0 09 fi 0(050) r∗ 0 322 β = . σ εy = . σ εi = π3 0 40 ( ) 0 833 ( ) 1 β = . π4 0 07 β = . y 0 13 σ(επ) = 1.012 Sources: Rudebusch (2002) (aggregate demand and supply), Taylor (1993) (monetary policy), own estimates (equilibrium real interest rate). empirical model we take account of imperfect knowledge about the equilibrium real rate of interest only to the extent that we employ a one-sided Kalman filter rather than the smoothed estimates. The second caveat is that we have assumed the coefficients in the Taylor rule are unchanged when the time-varying equilibrium real interest rate is introduced. Given that a long estimation horizon is needed to estimate the equilibrium real interest rate, it is difficult to estimate a single time-invariant monetary policy rule. Hence we have chosen to adopt the parameters suggested by Taylor (1993).

3.4.1 Impulse Responses of Nominal Interest Rates

Constant Equilibrium Real Rate Figure 3.2 presents impulse responses of the future short-term nominal interest rate one to 40 quarters into the future for the model with a constant equilibrium real rate. These are the predicted forward-rate responses to shocks to aggregate supply, aggregate demand and the nominal short-term interest rate when the equilibrium real rate is constant. The impulse responses reveal that nominal interest rates return quickly to zero. For instance, following a demand shock, forward rates move by only 3.5 basis points after 40 quarters in the version with no interest-rate smoothing (static Taylor rule). This implies that forward interest rates respond very little at long horizons to short rate movements and is at odds with the empirical facts. For instance Gurkaynak¨ et al. (2003) show that forward rates at the 10-year horizon move signiﬁcantly in response to news releases of a broad set of macroeconomic indicators.

Time-Varying Equilibrium Real Rate We now turn to the case of a persistent time-varying equilibrium real rate and investigate the impulse response of the nominal short rate when the equilibrium real rate is hit 102 3. The Excess Sensitivity Puzzle

Figure 3.2: Model With Constant Equilibrium Real Rate Supply shock 1

0.5

−0.5 0 5 10 15 20 25 30 35 40

Demand shock 1

0.5

−0.5 0 5 10 15 20 25 30 35 40

Monetary shock 1.5 f =0 i 1 f =0.5 i 0.5

−0.5 0 5 10 15 20 25 30 35 40 quarters

Notes: Impulse responses of the nominal short-term rate in the model with constant equilibrium real rate, with static (solid line, fi = 0) and dynamic (dash-dotted line, fi = 0.50) Taylor rule. by a demand shock or an equilibrium real rate shock. Figure 3.3 shows the response to an aggregate demand shock in the left panel and the response to an equilibrium real rate shock in the right panel. As in Figure 3.2, the abscissa denote the time horizon of the short-term interest rate response as well as the forward-rate horizon. Because the equilibrium real rate shock is a ‘new’ source of fluctuation, only εy the response to the demand shock t can be compared to Figure 3.2. Given the assumed parametrisation of the model, there is now a considerable effect on forward interest rates at the ten-year horizon from a demand shock (left panel): in absolute terms the forward rate moves 12 basis into the same direction as the short rate when the Taylor rule is dynamic. This corresponds to a shift of 29 basis points when the short rate moves 100 basis points. Similarly, if the Taylor rule is static, forward rates move 20 basis points in absolute terms, or 35 basis points relative to a 100 basis point short-rate rise. The effects are stronger in response to an equilibrium real rate shock (right panel), namely 19 basis points, or 61 points when the short rate moves 100 basis points, in the presence of interest-rate smoothing. In the case with no smoothing, the forward rate responds by more 3.4. The Puzzle Explained? 103

Figure 3.3: Model with Time-Varying Equilibrium Real Rate Demand shock r* shock 1.4 0.35 f =0 i f =0.5 i 1.2 0.3

1 0.25

0.8 0.2

0.6 0.15

0.4 0.1

0.2 0.05

0 0 0 10 20 30 40 0 10 20 30 40 quarters quarters

Notes: Responses of the nominal short-term rate to a demand shock (left panel) and an equilibrium real rate shock (right panel), with static (solid line fi = 0) and dynamic (dash-dotted line fi = 0.50) Taylor rule. than the short rate, namely by 117 basis points relative to a 100 basis point shift in the policy-controlled rate. The effects in the no-smoothing case appear to be longer-lasting. With interest-rate smoothing the economy experiences larger swings because the central bank first responds relatively little to the shocks but subsequently increases the nominal rate by more than in the no-smoothing case. As a result, the inflation and output gaps are closed more quickly. Beyond 40 quarters, when the inflation and output gaps have nearly returned to equilibrium, the dynamics of the nominal interest rate are dominated by the high persistence of the equilibrium real rate. From this horizon onward, the two cases produce quantitatively similar results.

3.4.2 Regression Evidence from Interest Rate Changes

The original formulation of the excess sensitivity puzzle concerns the large average response of long-term interest rates to changes in the policy-controlled short rate. Hence, in this section we investigate the overall impact of a time-varying equilibrium real rate on 104 3. The Excess Sensitivity Puzzle long-rate fluctuations relative to short rate movements. Several authors have distinguished between expected and unexpected changes in monetary policy, based on the reasoning that expected events have already been factored into long-term interest rates. Ellingsen and Soderstr¨ om¨ (2005) report that an unexpected one percentage point rise in the Federal Funds rate is associated with a 25 basis point rise in ten-year yields. From the model we generate time-series data (10 000 observations) on the policy- controlled short rate and the ten-year interest rate calculated as n 1 n−1 i = E {i + }. (3.9) t n s=0 t t s First differences of the long-term rate are then regressed on the short-term rate. Because we construct the long-term rate according to the expectations hypothesis, only unexpected changes in the short-term rate affect longer yields. Hence it is not necessary to distinguish between expected and unexpected monetary policy changes in this analysis. Four cases are examined to assess the effects of a time-varying equilibrium real rate and separate them from the effects of interest-rate smoothing. Specifically, we consider cases with and without interest-rate smoothing, a constant and time-varying equilibrium real interest rate and combinations thereof.

Table 3.3: Regression Results

Equilibrium Real Rate Taylor Rule time-varying constant static 0.20 0.10 dynamic 0.13 0.04 Note: Slope coeﬃcients from regressing changes in long-term interest rates on changes in short- ∆ 40 = α + β∆ 1 + term interest rates it it t

Table 3.3 reports the results from regressing changes in long-term interest rates on changes in the policy-controlled short rate. As can be seen from the table, relative to the model with constant equilibrium real rate the β-coeﬃcient is estimated to be about 10 basis points larger when the equilibrium rate is time-varying at the 10-year horizon. Inclusion of a time-varying equilibrium real interest rate explains about 25 percent of the excess sensitivity puzzle.

3.4.3 Sensitivity Analysis

In addition to distinguishing between a dynamic and static Taylor rule, we perform a small-scale sensitivity analysis to the results in Table 3.3 with respect to the output gap 3.5. Conclusions 105 coefficient. In crude single equation estimations of the Taylor rule (3.4) that assume a constant equilibrium real rate, this coefficient turns out to be rather unstable over the sample period, ranging from values close to zero to values above 1.The inflation coefficient is more stable around 1.5 by comparison. We solve the theoretical model for the additional cases fy ∈{0, 1} keeping all other coefficients fixed and simulate again data from the model. While fy = 0(= 1) shifts the β−coefficient down (up), the 10 basis point difference between the model with constant and time-varying equilibrium rate is preserved.

3.5 Conclusions

This paper explores one explanation for the excess sensitivity of long-term interest rates, namely the covariance between the equilibrium real rate and the policy-controlled short rate when both react to the same shocks. Models of the monetary transmission mechanism used to analyse the excess sensitivity puzzle assume that the equilibrium real rate is constant over time and unrelated to structural shocks. However, empirical evidence has shown that equilibrium real rates do vary considerably over time and this fact is embedded in state-of-the-art DSGE models. We incorporate this feature into a familiar semi-structural general equilibrium model frequently employed in the monetary policy literature. We estimate the equilibrium real rate for the U.S. over the last 45 years using an unobserved components model which yields parameter values for a dynamic equilibrium real rate equation. The results suggest that the equilibrium real rate displays time variation and dependence on structural shocks, and confirm the findings of previous studies that the equilibrium real rate is highly persistent. Specifically, it is driven by demand shocks and own shocks, where the latter could be interpreted as a proxy for shocks to productivity. Given the estimated parameters we show that a time-varying equilibrium real rate influenced by structural shocks can account for greater co-movement of short-term and long-term interest rates than a constant real rate benchmark. Slope coefficients from regressing changes in long-term rates on changes in short-term rates are approximately 10 basis point higher at the 10-year horizon when the equilibrium real rate is allowed to vary than when it is assumed to be constant. Assuming that the central bank responds to the equilibrium real rate, shocks have long lasting effects on forward rates and thereby onto long-term interest rates. The approach we take focuses on the real side of the economy and we are unable to address nominal shocks. Thus our claim that a time-varying equilibrium real interest rate may explain the sensitivity puzzle does not exclude other explanations, such as the informational asymmetries and imperfect knowledge discussed by Gurkaynak¨ et al. (2003) and Ellingsen and Soderstr¨ om¨ (2005). However, allowing real shocks to affect the equi- 106 3. The Excess Sensitivity Puzzle librium real interest rate creates an additional mechanism through which the effects of shocks become more persistent. Developing models that incorporate both real and nominal approaches would be valuable as they would allow conclusions to be drawn about the explanatory power of each approach. A. Derivation of the Equilibrium Real Rate 107

Appendices A Derivation of the Equilibrium Real Rate

This appendix shows how a specification of a time-varying equilibrium real interest rate such as equation (3.3) could be derived. Assume that there are two persistent shock processes influencing the aggregate demand relation; a ‘demand’ shock (et) and a ‘productivity’ or ‘equilibrium real rate’ shock (at), respectively and suppose that their dynamics can be modelled by stationary AR(1)-processes: = ρ + εy et eet−1 t (A1) = ρ + εa. at aat−1 t (A2) Next assume that the demand shock process has a contemporaneous effect and the equilibrium real rate shock process has a one-period delayed effect on the output gap, i.e. = φ + − φ + − + + yt yEt−1yt+1 1 y by1yt−1 by2yt−2 brrt−1 et at−1 (A3) so that = φ + − φ + − + ρ + + εy. yt yEt−1yt+1 1 y by1yt−1 by2yt−2 brrt−1 eet−1 at−1 t (A3’) ∗ { } = Define the equilibrium real rate rt as the real interest rate that implies Et−1 yt 0given yt−s = 0 for all s > 0. Then = { } = − ∗ + ρ + 0 Et−1 yt brrt−1 eet−1 at−1 (A4) ∗ = 1 ρ + . rt ( eet at) (A4’) br ρ ρ , ∗ With e a rt follows an ARMA(2,1) process. Both processes are likely to be highly ρ ≡ ρ = ρ . persistent and as an approximation we set r e a Then equation (3.3) can be derived as follows ∗ = 1 ρ2 + ρ + ρ εy + εa rt eet−1 aat−1 e t t (A5) br ρ = ρ ∗ + e εy + 1 εa rrt−1 t t br br = ρ ∗ + ρ εy + ε∗ rrt−1 e t t ρ εa where ρ ≡ e and ε∗ ≡ t . e br t br Finally, combining equation (A3’) and (A4’) yields aggregate demand with time- varying equilibrium real interest rate as in equation (3.1): y y = φ E − y + + 1 − φ b y − + b y − − b r − + ρ e − + a − + ε (A6) t y t 1 t 1 y y1 t 1 y2 t 2 r t 1 e t 1 t 1 t 1 y = φ E − y + + 1 − φ b y − + b y − − b r − − ρ e − + a − + ε y t 1 t 1 y y1 t 1 y2 t 2 r t 1 b e t 1 t 1 t r = φ + − φ + − − ∗ + εy. yEt−1yt+1 1 y by1yt−1 by2yt−2 br rt−1 rt−1 t 108 3. The Excess Sensitivity Puzzle

B State Space Representation

Our state space model has the following general representation:

= ξ Yt Z t (Measurement equations) ξ = ξ + ε t T t−1 R t (Transition equations)

ε = , ε ε = ε ε = where E( t) 0 E( t s) 0 for all t s and E( t t ) Q. The state vector has the ξ = , ,, , ∗ ε = following elements, t (yt yt−1 rt rt−1 rt ) and the residual vector is comprised of t εy,εr,εr∗ t t t . Measurement equations

yt = yt, = + ∗ rt rt rt Transition equations = α + α − α + εy yt y1yt−1 y2yt−2 rrt−1 t = + + εr rt d1rt−1 d2rt−2 t ∗ = ρ ∗ + ρ εy + εr∗ rt rrt−1 e t t    α α −α   y1 y2 r 00    10 000 10000   Z = , T =  00d1 d2 0  00101    00 100 ρ   00 00r  100      000  σ2     y  =  010 =  σ2  R   Q  r    σ2  000 r∗ ρ e 01 C. Conditional Likelihoods 109

C Conditional Likelihoods

Figure 3.4: Optimisation Diagnostic α α α y1 y2 r 454 454 454 ML−estimate 452 neg. maximum 452 452 neg. cond. log−likelihood log−likelihood 450 450 450 448 448 448 446 446 446 444 444 444 0.6 0.8 1 −0.24 −0.2 −0.16 −0.12 0.1 0.15 0.2 0.25

d d ρ 1 2 r 454 454 454 452 452 452 450 450 450 448 448 448 446 446 446 444 444 444 0.6 0.8 1 −0.15 −0.1 −0.05 0 0.05 0.94 0.96 0.98 1

ρ σ σ ~ e y r 454 454 454 452 452 452 450 450 450 448 448 448 446 446 446 444 444 444 0.2 0.3 0.4 0.4 0.6 0.8 1 0.6 0.8 1 1.2

Notes: Negative conditional log-likelihoods around the optimum for Model 2. The plots are calculated with a diﬀuse prior on the initial state vector in order to allow the ρ = likelihood computation for r 1 as well. 110 3. The Excess Sensitivity Puzzle

D Output Gap and Real Rate Gap

Figure 3.5: Output Gap and Real Rate Gap 8 (r−r*) Model 1 −1 (r−r*) Model 2 6 −1 output gap

−2

−4

−6

−8

−10 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005

Notes: One-period lagged real interest rate gap from Model 1 (dashed line) and Model 2 (solid line) and the output gap (dash-dotted line). REFERENCES 111

References

Andres,´ J., Lopez-Salido,´ J. D., and Nelson, E. (2004). Money and the natural rate of interest: Structural estimates for the uk, the us and the euro area. CEPR Discussion Paper Series 4337, Centre for Economic Policy Research.

Beechey, M. (2004). Excess sensitivity and volatility of long interest rates: The role of limited information in bond markets. Sveriges Riksbank Working Paper No 173, Sveriges Riksbank.

Clarida, R., Gal´ı, J., and Gertler, M. (2000). Monetary policy rules and macroeconomic stability: Evidence and some theory. Quarterly Journal of Economics, 115:147–180.

Clark, P. K. (1987). The cyclical component of U.S. economic activity. Quarterly Journal of Economics, 102:797–814.

Clark, T. E. and Kozicki, S. (2004). Estimating equilibrium real interest rates in real-time. Deutsche Bundesbank Discussion Paper Series No 32/2004.

Ellingsen, T. and Soderstr¨ om,¨ U. (2005). Why are long rates sensitive to monetary policy? Working Paper, Bocconi University.

Estrella, A. and Fuhrer, J. C. (2002). Dynamic inconsistencies: counterfactual implications of a class of rational-expectations models. American Economic Review, 92(4):1013–1028.

Fuhrer, J. C. and Rudebusch, G. D. (2004). Estimating the Euler equation for output. Journal of Monetary Economics, 51(6):1133–1153.

Gal´ı, J. and Gertler, M. (1999). Inﬂation dynamics: a structural econometric analysis. Journal of Monetary Economics, 44(2):195–222.

Gurkaynak,¨ R. S., Sack, B., and Swanson, E. T. (2003). The excess sensitivity of long- term interest rates to economic news: evidence and implications for macroeconomic models. Federal Reserve Board: Finance and Economics Discussion Series 2003-50.

Laubach, T. and Williams, J. C. (2003). Measuring the natural rate of interest. Review of Economics and Statistics, 85(4):1063–1070.

Manrique, M. and Marques,´ J. M. (2004). An empirical approximation of the natural rate of interest and potential growth. Documentos de Trabajo No. 0416, Banco de Espana, Madrid.

Mesonnier,´ J.-S. and Renne, J.-P. (2004). A time-varying ’natural’ rate of interest for the Euro area. Working Paper.

Orphanides, A. (2001). Monetary policy rules based on real-time data. American Eco- nomic Review, 91(4):964–985.

Orphanides, A. and Williams, J. C. (2002). Robust monetary policy rules with unknown natural rates. Brookings Papers on Economic Activity, 2002:2:63–145. 112 3. The Excess Sensitivity Puzzle

Pennacchi, G. G. (1991). Identifying the dynamics of real interest rates and inﬂation : evidence using survey data. The Review of Financial Studies, 4(1):53–86.

Rudebusch, G. D. (2002). Assessing nominal income rules for monetary policy with model and data uncertainty. Economic Journal, 112:402–432.

Rudebusch, G. D. and Svensson, L. E. O. (1999). Policy rules for inﬂation targeting. In Taylor, J. B., editor, Monetary Policy Rules. Chicago University Press, Chicago.

Shiller, R. J. (1979). The volatility of long-term interest rates and expectations models of the term structure. Journal of Political Economy, 87(6):1190–1219.

Smets, F. and Wouters, R. (2003). An estimated dynamic stochastic general equilibrium model of the Euro area. Journal of the European Economic Association, 1(5):1123– 1175.

Soderlind,¨ P., Soderstr¨ om,¨ U., and Vredin, A. (2005). New-Keynesian models and monetary policy: A reexamination of the stylized facts. Scandinavian Journal of Economics (forthcoming).

Taylor, J. B. (1993). Discretion versus policy rules in practice. Carnegie-Rochester Con- ference Series on Public Policy, 39:195–214.

Welz, P. (2005). Assessing predetermined expectations in the standard sticky price model: A Bayesian approach. Manuscript, Uppsala University.

Woodford, M. (2003). Interest and Prices: Foundations of a theory of monetary policy. Princeton University Press, Princeton, New Jersey. Chapter 4

Interest-Rate Smoothing versus Serially Correlated Errors in Taylor Rules: Testing the Tests

4.1 Introduction

The Taylor (1993) rule has, over the last decade, become a largely unquestioned tool for monetary policy evaluation. Notwithstanding this, a number of authors have recently presented serious criticisms of the Taylor rule, Rudebusch (2002) being among the more influential. Rudebusch uses term-structure evidence to show that a Taylor rule with partial adjustment, interpreted as interest-rate smoothing by the central bank, implies more interest-rate predictability than can be found in the data. In contrast to much of the existing literature, Rudebusch also points to the implausibility of the slow partial adjustment implied by most coefficient estimates. The degree of interest-rate smoothing found in quarterly data implies that the central bank closes the gap between the actual and the desired target interest rate slowly, roughly half of the gap in a year. His explanation for the high degree of interest-rate smoothing commonly found in the literature is the presence of serially-correlated errors caused by ‘appropriate response[s] to special circumstances’ which are not captured by the variables in the Taylor rule. Although almost observationally equivalent, interest-rate smoothing and serially-correlated errors have not only different economic interpretations but also different statistical implications. English et al. (2003), ENS henceforth, use this fact to develop two tests that distinguish between the two cases. Employing these tests and Castelnuovo (2003a,b) find support for interest-rate smoothing when the tests are applied to U.S. data. The purpose of this paper is to further investigate the reasons for the finding of strong interest-rate smoothing in estimated Taylor rules and our focus is on omitted variables as a likely cause. We employ a Monte Carlo study of a data generating process based on a structural VAR for the U.S. economy and in addition to the federal funds rate, inflation

113 114 4. Interest-Rate Smoothing and the output gap, capture dynamics by including four potentially important variables. We investigate the magnitude of the resulting bias in the coefficients of the Taylor rule and the size properties of the ENS tests when the estimated reaction function is misspecified. As expected, estimating a standard Taylor rule with inflation and output gap as only explanatory variables results in biased coefficients when the central bank’s true reaction function contains additional persistent variables.1 The ENS tests are also found to overre- ject the null hypothesis of no interest-rate smoothing at conventional significance levels. While our exercise does not reproduce the degree of interest-rate smoothing that is commonly found in empirical studies, the results in this paper suggest that the ENS tests must be interpreted with caution and that upward bias of the partial-adjustment coefficient can be sizeable. The methodological setup in this paper takes as it starting point critique of Svensson (2003, 2005) that the Taylor rule is unlikely to be the solution to a typical central bank optimisation problem of stabilising inflation and the output gap. If important state variables other than inflation and the output gap exist, the rule will not be optimal; instead, the number of response coefficients that need to be fixed in the central bank’s reaction function must be increased in accordance with the number of state variables the central bank takes into account.Put differently, inflation and output deviations are unlikely to be sufficient statistics for the state of the economy, nor for characterisation of central bank behaviour.2 Svensson’s arguments are also empirically relevant. For instance, Osterholm¨ (2005) points out that given the highly persistent nature of variables in the Taylor rule, cointegration is a necessary condition for both consistent estimation of parameters and compatibility between theoretical models and data. The lack of empirical evidence for a cointegrating relationship between nominal interest rates, inflation and the output gap thus offers support for the idea that the Taylor rule is misspecified. Furthermore, Good- hart and Hofmann (2002) show how the omission of various asset price variables leads to considerable changes in the remaining coefficients in an estimated Taylor-type rule. The remainder of this paper is organised as follows. Section 4.2 presents the Tay- lor rule and replicates some well-known empirical results. In Section 4.3, the ENS test equations are presented. Section 4.4 describes the data generating process (DGP) used in the Monte Carlo study, the results of which are discussed in Section 4.5. Section 4.6

1An early source is Grilliches (1961). 2This viewpoint finds support among practitioners. For instance Ben S. Bernanke (2004), member of the Board of Governors of the U.S. Federal Reserve System, remarked: ‘..., my forecast of controlled inflation is based on more than output gap arguments. Other factors likely to keep inflation at modest levels include continuing rapid gains in productivity, which have kept growth of unit labor costs at a very low level; unusually high price-cost margins in industry, which provide scope for firms to absorb future cost increases without raising prices; globalization and intensified competition in product markets; and the recent strengthening of the dollar.’ 4.2. The Taylor Rule 115 concludes with a brief discussion of our findings and some general remarks.

4.2 The Taylor Rule

To organise the discussion, we ﬁrst give an account of the Taylor rule and then provide a brief overview of the theoretical and empirical literature on interest-rate smoothing. A more thorough exposition can be found in Taylor (1999).

4.2.1 Basic Speciﬁcation and Empirical Evidence

The original formulation of the Taylor (1993) rule is given by

= ∗ + π + π − π∗ + φ , it r t fπ ( t ) yyt (4.1)

∗ where it is the central bank policy rate, r the equilibrium real interest rate, πt the twelve ∗ month inflation rate, π the inflation target of the central bank and yt the output gap. Based on calibration, Taylor found that a rule with the parameters set to r∗ = π∗ = 2 and = φ = . fπ y 0 5 tracked the actual federal funds rate fairly well between 1987 and 1992. Note that the equilibrium real interest rate as well as the inflation target are assumed to be constant here. Adding an error term and collecting constants in the intercept, equation (4.1) can be reformulated as = φ + φ π + φ + εi, it 0 π t yyt t (4.2) φ ≡ ∗ − φ − π∗ φ ≡ + where 0 r π 1 and π 1 fπ. The rule in equation (4.2), or versions thereof allowing for forward-looking behaviour, has been the standard starting point in the empirical literature.3 In the literature, it has also been shown that adding a lagged interest rate term as = − λ φ + φ π + φ + λ + εi, it (1 ) 0 π t yyt it−1 t (4.3) where 0 ≤ λ<1, improves the empirical fit considerably. The lagged interest-rate term is commonly interpreted as deliberate interest-rate smoothing on the part of the monetary authority. Note also that equation (4.3) has a partial adjustment structure in which the ≡ φ + φ π + φ term it 0 π t yyt is the target interest rate dependent on the state of the economy that the central bank seeks to achieve. 3An extensive discussion in the literature addresses the timing of explanatory variables in the Taylor rule. For example, McCallum and Nelson (1999) have argued that due to informational delays, the central bank likely reacts to lagged values of inflation and the output gap. Others have suggested using forecasts of the regressors in order to capture the potentially forward-looking behaviour of central banks. For a study which addresses several different approaches, see Orphanides (2001). The timing issue is less relevant for the exercise in this paper because the data are generated synthetically. We therefore adhere to Taylor’s original formulation regarding timing. 116 4. Interest-Rate Smoothing

As an example of a typical finding in the empirical literature, we present the results from least squares regressions of the static equation (4.2) and the dynamic equation (4.3) using quarterly U.S. data on the federal funds rate, CPI inflation and output gap from 1987Q1 to 2004Q3.4 Newey-West standard errors are reported in parentheses and results for the constants have been omitted here for brevity. The results for the estimated static Taylor rule are shown in (S). The estimated coefficients are not far from those suggested by Taylor (1993)5 but the diagnostic test statistics indicate the presence of autocorrelation as well as heteroskedasticity in the residuals.

it = 1.32 πt + 0.77 yt (S) (0.12) (0.13) 2 = . σ = . = . Rad j 0 78 ε 1 05 DW 0 38 AR(4) : χ2(4) = 42.05 ARCH(4) : χ2(4) = 28.15

The estimated dynamic Taylor rule (D) appears to fit the data better, with a higher adjusted R2 and standard error halved relative to the static equation. The coefficient estimates are ff φ = . markedly di erent, with the restriction y 0 5 rejected at the 5 percent level while the restriction φπ = 1.5 is not. Despite inclusion of the lagged dependent variable, the autocorrelation tests still indicate serial correlation in the residuals.

it = 0.17(1.21πt + 1.30yt ) + 0.83 it−1 (D) (0.34) (0.21) (0.04) 2 = . σ = . = . = . Rad j 0 97 ε 0 41 DW 0 75 Durbin H 5 62 AR(4) : χ2(4) = 27.03 ARCH(4) : χ2(4) = 14.42

It is surprising how many researchers have interpreted the estimated partial adjustment coefficient as evidence of intentional interest-rate smoothing by the central bank, despite the basic fact that all coefficients in this equation are inconsistently estimated because of the significant lagged dependent variable together with autocorrelated disturbances. Clarida et al. (1998, 2000), Gerlach and Schnabel (2000) and Domenech´ et al. (2002) study dynamic Taylor rules over different sample periods across different countries and consistently report large and significant smoothing parameters. Each interpret these estimates as evidence for the hypothesis that central banks adjust their policy interest rate very gradually towards the target interest rate. This conclusion is questionable, however, given the implied speed of adjustment of similar parameter estimates. The implication of our estimation of (D) with quarterly data is that almost a year elapses before the monetary authority closes half of the gap between the actual federal funds rate and the intended interest rate target. The value of 0.92 found by Clarida et al. (1998) using U.S. monthly

4All data are from the Federal Reserve Bank of St Louis data base (FRED). 5 φ = . φ = . The restrictions π 1 5 and y 0 5 are not rejected at the 5 percent level. 4.2. The Taylor Rule 117 data from 1979 to 1994 implies that approximately half the intended adjustment has taken place after about nine months. Even though there seems to be agreement in the profession that central banks dislike aggressive movements of their instruments because such behaviour is believed to unsettle ﬁnancial markets, the adjustment implied by these estimates seems implausibly slow. More generally, Hendry (1995, p. 259) remarks that ‘long lags in partial adjustment models may be an artefact of that type of model’, and further notes that the adjustment parameter often lies in the interval (0.8, 0.95) ‘regardless of application’. These econometric issues raise doubts over the appropriateness of the dynamic Taylor rule in equation (4.3).

4.2.2 Related Literature

A wealth of theoretical literature has investigated whether interest-rate smoothing is optimal from a monetary policy perspective. The results are not clear cut and are dependent on the structural macroeconomic model. Aoki (2003), for example, shows that interest-rate smoothing may be optimal in the presence of noisy indicator variables because it allows policy cautiousness in the presence of uncertainty. Woodford (2003) establishes that in a purely forward-looking model with commitment to optimal policy, a lagged interest- rate term in the monetary reaction function may be optimal because it induces history dependence that helps to stabilise inflation expectations. Using larger models Levin et al. (1999) suggest that interest-rate smoothing in the short-term interest rate may provide control over long-term rates, because expected sustained movements of the short-term interest rate have a greater impact on long-term interest rates. In an empirical study, Goodhart (1997) investigates the interest rate setting behaviour of several central banks and finds that they tend to move the interest rate in small steps in the same direction between reversals. This observation does not, however, provide direct justification for interest-rate smoothing. Slow, stepwise movements in the interest rate may reflect an explicit smoothing objective or be the appropriate reaction to the central bank’s perception of the slow-moving state of the economy. In a follow-up study to Rudebusch (2002), Soderlind¨ et al. (2005a) find that the two sides of the Taylor rule do not match in terms of predictability. The inflation rate and output gap are relatively easy to predict, which should imply predictability of nominal interest rate changes, but the authors find that survey evidence does not support this conclusion. In a companion paper, the same authors find a high preference for interest-rate smoothing and almost no preference for output gap stabilisation in a model with optimal, discretionary monetary policy (Soderlind¨ et al., 2005b). This may be a particular result of that type of model. 118 4. Interest-Rate Smoothing

4.3 Two Tests for Interest-Rate Smoothing

This section reviews the testing procedure suggested by English et al. (2003). In the εi standard empirical specification, the error term t in (4.2) is assumed to be serially uncorrelated. However, as pointed out by Rudebusch (2002), serial correlation may give rise to a significant coefficient on the lagged interest rate (Grilliches, 1961). Assuming that error autocorrelation is of first order, the interest rate equation may be specified as

= φ + φ π + φ + υ ,υ= ρυ + εi, it 0 π t yyt t t t−1 t (4.4)

εi where t is assumed to be i.i.d. with mean zero. Subtracting it−1 from both sides and again ≡ φ + φ π + φ letting it 0 π t yyt denote the target rate of the central bank, equations (4.3) and (4.4) can both be rewritten in the form ∆ = γ ∆ + γ − + εi. it 1 it 2 it−1 it−1 t (4.5)

γ The parameter 1 indicates if the lagged interest rate is significant; with interest-rate γ = γ = − λ, γ = γ = − ρ smoothing, 1 2 1 but with serially-correlated errors 1 1 and 2 1 . English et al. (2003) suggest a test for serial correlation versus interest-rate smoothing that can be based on a non-linear least squares estimation of equation (4.5). Note, that the γ = ffi hypotheses are not nested; rejecting the null hypothesis H0 : 1 1 is not su cient to con- γ = γ clude interest-rate smoothing but also requires maintaining the hypothesis that 1 2. From this point of view, a likelihood ratio test might be preferable to a simple coefficient test as conducted by English et al. (2003). In order to obtain a nested test equation, the authors extend the model to a more general form and allow for both interest-rate smoothing and serially-correlated errors in one specification, that is,

= − λ + λ + υ ,υ= ρυ + εi. it (1 ) it it−1 t t t−1 t (4.6)

They estimate equation (4.6) on U.S. data and find significant estimates of λ and ρ.Aswe will show below, this test procedure may be misleading if the coefficients in (4.6) are biased due to omitted variables that are not proxied by the autocorrelated error term. English et al. (2003) also address whether a significant smoothing parameter may be spuriously obtained due to omitted variable bias but consider measurement error, parameter instabil- ity and changed levels of the target variables as likely culprits. Castelnuovo (2003b) tests for omitted variables in the same framework and finds that the square of the output gap6 and the growth rate of M3 are both significant when individually included in the equation. However, he still finds that the interest-rate smoothing parameter is significant and

6This is taken to represent asymmetric preferences of the central bank; see for instance Surico (2002). 4.4. Model and Data Generating Process 119 draws the conclusion that its magnitude adequately describes the degree of interest-rate smoothing. In the following we will investigate the importance of omitted variable bias for the above equations. For this purpose we base our Monte Carlo study on equations (4.3) to (4.6) and study the inﬂuence of omitted variables on (i) the partial adjustment coeﬃcient and (ii) the empirical size of the tests.

4.4 Model and Data Generating Process

The model used for the simulations is based on a structural VAR for the U.S. economy incorporating short-run restrictions.7 Estimation is performed using quarterly data from

1987Q1 to 2004Q3 on inflation (πt), the output gap (yt), the log-difference of M2 (∆m2t), 10y ff the yield on the ten year treasury bond (it ), the log-di erence of the Standard and Poor

500 index (∆sp500t), the log of the real effective exchange rate (qt), and the Federal funds rate (it). All data are from the Federal Reserve Bank of St. Louis data base (FRED), apart from the S&P 500 index which was taken from EcoWin Pro. We define inflation as π = . 3 π , π = / t 0 25 s=0 t−s where t−s 400 ln (Pt Pt−1) and Pt is the consumer price index. The = /pot pot output gap is defined as yt 100 ln yt yt , where yt is GDP and yt potential GDP. The structural VAR is estimated in order to find a reasonable summary of the data dynamics and covariances and to identify a monetary policy reaction function. No attempt is made to build a structural model – such as a DSGE model – of the economy. We first determine the lag length of the unconstrained VAR to be unity according to the Schwarz information criterion. Having established the lag length, we next turn to estimation of the model. Let the general form of the model be given by

A0 xt = µ + A1 xt−1 + εt, (4.7) where xt is the 7 × 1 vector of our macroeconomic variables, εt ∼ nid (0, D) and D is a diagonal matrix. Specifically xt = [πt yt zt it] , where zt is the vector containing the additional four variables money growth, long term interest rate, change in S&P 500 and the real effective exchange rate (in that order). We assume a recursive identification scheme in which the interest rate is ordered last for two reasons: (i) it has been widely used in the literature (Christiano et al., 1999; Sims, 1980), and (ii) we are not concerned with shocks in the economy.8 The main advantage of this identification assumption is that ordering the policy interest rate last in the recursion allows the central bank to react to all the other variables in the system contemporaneously.

7A K-model in the terminology of Amisano and Giannini (1997). 8For a range of identiﬁcation approaches, see Bernanke (1986), Shapiro and Watson (1988), Blanchard and Quah (1989) and Uhlig (2005). 120 4. Interest-Rate Smoothing

This is in line with the fairly general opinion that the central bank takes a large number of variables into account in its decision making. Second, and more importantly, the recursive structure means that we can identify the monetary policy innovation without having to define seven structural equations. We fix the contemporaneous coefficients on inflation and the output gap in the interest rate equation to 1.5 and 0.5, respectively, in accordance with Taylor’s original article and the general findings in the literature. Furthermore, because we want to study the effects of omitted variables in a static Taylor rule, we set all lags in the interest rate equation to zero when we estimate the system (4.7). All other coefficients are estimated without restrictions. Given the recursive structure of the system, we can consistently estimate it equation-by-equation with OLS.9 Having estimated the above model, we use the coefficient estimates A0, A0, µ, D to generate synthetic data. These data meet all of our underlying assumptions, enabling us to investigate the properties of the suggested econometric tests for an economy with serially- uncorrelated shocks and a central bank that does not employ interest-rate smoothing but does react to four additional macroeconomic variables.

4.5 Simulations and Results

The Monte Carlo experiment consists of 10 000 replications of the economy simulated for 100 periods. Because our purpose is to study the eﬀects of omitted variables bias, we estimate equations (4.3) to (4.6) on the generated data, omitting the four additional γ ,γ ,λ variables that are part of the true DGP. We then perform hypothesis tests on 1 2 and ρ as suggested by English et al. (2003).

4.5.1 Results

Table 4.1 presents the direct estimates of the smoothing coefficient, λ, from equation (4.3) while Figure 4.1 shows the empirical distribution function.10 The results for the autocorrelation coefficient, ρ, estimated from equation (4.4) are summarised in Table 4.2 and Figure 4.2. We report the median and the 5th and 95th percentiles of the empirical distribution function of the estimated coefficients. The vertical line in the histograms indicates the true value in the DGP.

9The restricted VAR is a recursive simultaneous equation system with diagonal variance-covariance matrix. 10 = β + β π + Instead of estimating equation (4.3) with non-linear least squares, the equivalent model it 0 π t β + λ + εi ﬃ yyt it−1 t was estimated and the coe cients were thereafter transformed to long-run values in order to be comparable to the other estimated coeﬃcients in the paper. 4.5. Simulations and Results 121

Table 4.1: First Test under the Assumption of Interest-Rate Smoothing

Parameter Median 5,95 percentile true value λ 0.18 [0.05,0.31] 0 φπ 1.79 [1.53,2.05] 1.5 φ y 0.59 [0.43,0.75] 0.5

Figure 4.1: First Test under the Assumption of Interest-Rate Smoothing λ

−0.1 0 0.1 0.2 0.3 0.4

Notes: Empirical distribution function of the partial adjustment coeﬃ- cient λ, obtained from 10 000 OLS estimations of equation (4.3).

The results in Table 4.1 reveal that all coefficients are biased. Specifically, the distribution of λ has its median at 0.18, suggesting the presence of interest-rate smoothing. This is a standard econometric result: omitting the four additional variables causes the residuals to be autocorrelated and this autocorrelation is taken up by the lagged interest rate term in (4.3) and the autocorrelation term in equation (4.4). The autocorrelation correction only succeeds if the omitted variables are orthogonal to inflation and the output gap, which is neither the case in this experiment nor likely to be true in real data. These two cases demonstrate the difficulty in distinguishing between interest-rate smoothing and serially-correlated errors when tested one at a time and are the motivation for the reduced-form equation (4.5) suggested by English et al. (2003). As explained before, the assumption underlying this equation is that only one of the two alternatives is γ < true, even though the two are nested in the same equation. Hence, 1 1 would indicate 122 4. Interest-Rate Smoothing

Table 4.2: First Test under the Assumption of Serially Correlated Errors

Parameter Median 5,95 percentile true value ρ 0.19 [0.00,0.35] 0 βπ 1.76 [1.50,2.00] 1.5 β y 0.55 [0.39,0.70] 0.5

Figure 4.2: First Test under the Assumption of Serially Correlated Errors ρ

−0.1 0 0.1 0.2 0.3 0.4 0.5

Notes: Empirical distribution function of the serial correlation coefficient ρ, obtained from 10 000 NLLS estimations of equation (4.4). that the coefficient on the lagged interest rate is significant. We present the Monte Carlo results for equation (4.5) in Table 4.3 and Figure 4.3. These results once again show that all coefficients are biased. In particular, the reduced γ form parameter 1, which provides an indication of the degree of interest-rate smoothing, γ γ is biased downward. In fact, the medians of 1 and 2 are almost equal, pointing to interest-rate smoothing in the data rather than autocorrelated errors. This is confirmed by a Wald test with the null hypothesis that the two coefficients are equal; at the 5 percent level, this test rejects the null in only 7 percent of cases. Next we discuss the results from the model permitting both interest-rate smoothing and autocorrelated errors. Table 4.4 shows the estimated parameters and Figure 4.4 displays the empirical distribution function. The Monte Carlo results imply that there is interest-rate smoothing rather than serial 4.5. Simulations and Results 123

Table 4.3: First Test for the Reduced Form Parameter Median 5,95 percentile true value γ 1 0.81 [0.61,1.02] 1 γ 2 0.83 [0.66,1.00] 1 φπ 1.79 [1.53,2.06] 1.5 φ y 0.59 [0.43,0.75] 0.5

Figure 4.3: First test for the Reduced Form γ γ 1 2

0.4 0.6 0.8 1 1.2 1.4 0.6 0.8 1 1.2

Notes: Empirical distribution function of the reduced form coefficients γ γ 1 and 2, obtained from 10 000 NLLS estimations of equation (4.5). correlation in the errors, in line with our results from equation (4.5). Bias in the inflation and output-gap coefficients has been reduced, but the magnitude of the bias in λ is substantial, even though the distribution of λ does not cover the values that are commonly found in empirical studies. For reference we also report the empirical test sizes in Table 4.5. These are severely distorted, in line with our expectations based on the coefficient estimates. Interestingly, for our specific data generating process, the omission of relevant explanatory variables leads to the conclusion that partial adjustment explains the dynamics of the data better than a static or dynamic relation with autocorrelated errors. The exercise demonstrates that modelling autocorrelated errors with autoregression is the wrong cure in this case and leads to incorrect conclusions about the dynamic behaviour 124 4. Interest-Rate Smoothing

Table 4.4: Second Test: Allowing for Smoothing and Serial Correlation

Parameter Median 5,95 percentile true value λ 0.20 [-0.03,0.37] 0 ρ -0.04 [0.66,0.28] 0 φπ 1.44 [1.23,1.65] 1.5 φ y 0.47 [0.34,0.61] 0.5

Figure 4.4: Second Test: Allowing for Smoothing and Serial Correlation λ ρ

−0.2 0 0.2 0.4 −0.4 −0.2 0 0.2 0.4

Notes: Empirical distribution function of the reduced form coeﬃcients λ and ρ, obtained from 10 000 NLLS estimations of equation (4.6). of the interest rate.

4.6 Discussion and Conclusion

The present study investigates the extent to which omitted variables can generate spurious interest-rate smoothing in an otherwise standard Taylor rule. Our results indicate that the tests suggested by English et al. (2003) may not be able to distinguish between interest-rate smoothing and serially-correlated disturbances in Taylor-type rules when the rule is misspecified. The omission of relevant variables leads to biased and inconsistent coefficient estimates and is likely to induce a disturbance structure for which the tests are not designed. However, the smoothing coefficient that we obtain is not as high as is 4.6. Discussion and Conclusion 125

Table 4.5: Documenting the Distortion of the Test Size

Rejection rates - Nominal size 0.05 Equation H0 Empirical rejection rate (4.3) λ = 0 0.76 (4.4) ρ = 00.54 γ = (4.5) 1 1 0.36 γ = γ (4.5) 1 2 0.07 (4.6) λ = 0 0.57 (4.6) ρ = 00.06 γ = γ Notes: Wald test on H0 : 1 2. All other tests are (one-sided) t-tests. commonly found in empirical studies and we cannot draw further conclusions about the magnitude of the partial adjustment coefficient in real data. Nevertheless, our analysis demonstrates that autocorrelation does not entail autoregression and that the interest-rate smoothing term is likely to have a sizeable upward bias when important variables are omitted from the Taylor rule. It also suggests that inference regarding central banks’ preferences could be hazardous if there is reason to believe that the monetary policy reaction function is incorrectly specified. The results of this paper question the standard conclusion that the large and significantly estimated coefficient on the lagged interest rate should be interpreted as intentional interest-rate smoothing. The partial adjustment coefficient may not be informative about the true degree of interest-rate smoothing because it may hide omitted variable bias due to misspecification of the estimated equation and its disturbance structure. This finding not only raises doubts about conclusions from the Taylor-rule literature evaluating central bank performance and preferences but offers a credible explanation for the inconsistencies between the Taylor rule and the data that have recently been brought to researchers’ attention. Variable omission is a likely source of misspecification in estimated central bank reaction functions. The practical decision-making process in central banks depends on a wider range of economic indicators than inflation and the output gap, including such variables as monetary aggregates, the exchange rate, the current account and financial market variables. The information content of this broad set of indicators may not be sufficiently approximated by inflation and the output gap. It may be reasonable to include forecasts of the output gap and inflation rather than their current values or the determinants for these forecasts. Moreover, as pointed out by Svensson (2003, 2005) and evident from central bank communications, judgement plays an important role in the decision-making process. How this judgement can be accounted for in a statistically sensible way is beyond 126 4. Interest-Rate Smoothing the scope of this paper. At a more general level, we can relate our study to macroeconomic modelling methodology. English et al. (2003) point out that serially-correlated errors signal that something systematic has been left out of the estimated equation and name one important implication: as long as the omitted variables that generate the serially-correlated error term are orthogonal to the regressors in the equation, modelling them as autoregressive process is a valid approach. However, this orthogonality assumption is difficult to justify with macroeconomic data. We believe that it is extremely rare for a central bank to observe and react to variables that are orthogonal to inflation and the output gap. REFERENCES 127

References

Amisano, G. and Giannini, C. (1997). Topics in Structural VAR Econometrics. Springer Verlag, Heidelberg, 2nd edition. Aoki, K. (2003). On the optimal monetary policy response to noisy indicators. Journal of Monetary Economics, 50:501–523. Bernanke, B. S. (1986). Alternative explanations of the money-income correlation. Carnegie-Randester Conference Series on Public Policy, 25:49–100. Bernanke, B. S. (2004). Gradualism. Speech at an economics luncheon co-sponsored by the Federal Reserve Bank of San Francisco (Seattle Branch) and the University of Washington, Seattle, Washington. Blanchard, O. J. and Quah, D. (1989). The dynamic eﬀects of aggregate demand and supply disturbances. American Economic Review, 79:655–673. Castelnuovo, E. (2003a). Describing the Fed’s conduct with Taylor rules: Is interest rate smoothing important? European Central Bank Working Paper No. 232. Castelnuovo, E. (2003b). Taylor rules, omitted variables, and interest rate smoothing in the U.S. Economics Letters, 81:55–59. Christiano, L., Eichenbaum, M., and Evans, C. (1999). Monetary policy shocks: What have we learned and to what end? In Taylor, J. B. and Woodford, M., editors, Handbook of Macroeconomics, volume 15. Elsevier, Amsterdam. Clarida, R., Gal´ı, J., and Gertler, M. (1998). Monetary policy rules in practice. some international evidence. European Economic Review, 42:1033–1067. Clarida, R., Gal´ı, J., and Gertler, M. (2000). Monetary policy rules and macroeconomic stability: Evidence and some theory. Quarterly Journal of Economics, 115:147–180. Domenech,´ R., Ledo, M., and Taguas, D. (2002). Some new results on interest rate rules in EMU and in the U.S. Journal of Economics and Business, 54:431–446. English, W. B., Nelson, W. R., and Sack, B. P. (2003). Interpreting the signiﬁcance of the lagged interest rate in estimated monetary policy rules. Contributions to Macroeco- nomics, 3(1):Article 5. Gerlach, S. and Schnabel, G. (2000). The Taylor rule and interest rates in the EMU area. Economics Letters, 67:165–171. Goodhart, C. A. E. (1997). Why do monetary authorities smooth interest rates? In Collignon, S., editor, European Monetary Policy. Pinter, London and Washington. Goodhart, C. A. E. and Hofmann, B. (2002). Asset prices and the conduct of monetary policy. Paper presented at the Royal Economic Society Annual Conference. Grilliches, Z. (1961). A note on serial correlation bias in estimates of distributed lags. Econometrica, 29(1):65–73. 128 4. Interest-Rate Smoothing

Hendry, D. F. (1995). Dynamic Econometrics. Oxford University Press, Oxford.

Levin, A., Wieland, V., and Williams, J. (1999). The robustness of simple monetary policy rules under model uncertainty. In Taylor, J. B., editor, Monetary Policy Rules. Chicago University Press, Chicago.

McCallum, B. T. and Nelson, E. (1999). Nominal income targeting in an open-economy optimizing model. Journal of Monetary Economics, 43:553–578.

Orphanides, A. (2001). Monetary policy rules based on real-time data. American Eco- nomic Review, 91:964–985.

Osterholm,¨ P. (2005). The Taylor rule: A spurious regression? Bulletin of Economic Research, 57:217–247.

Rudebusch, G. D. (2002). Term structure evidence on interest rate smoothing and monetary policy inertia. Journal of Monetary Economics, 49:1161–1187.

Shapiro, M. D. and Watson, M. (1988). Sources of business cycle ﬂuctuations. NBER Macroeconomics Annual, 3:111–148.

Sims, C. A. (1980). Macroeconomics and reality. Econometrica, 48.:1–48.

Soderlind,¨ P., Soderstr¨ om,¨ U., and Vredin, A. (2005a). Dynamic Taylor rules and the predictability of interest rates. Macroeconomic Dynamics, 9:412–428.

Soderlind,¨ P., Soderstr¨ om,¨ U., and Vredin, A. (2005b). New-Keynesian models and monetary policy: A reexamination of the stylized facts. Scandinavian Journal of Economics (forthcoming).

Surico, P. (2002). U.S. monetary policy rules: The case for asymmetric preferences. FEEM Working Paper 64-02.

Svensson, L. E. O. (2003). What is wrong with Taylor rules? Using judgment in monetary policy through targeting rules. Journal of Economic Literature, 41:426–477.

Svensson, L. E. O. (2005). Monetary policy with judgement: Forecast targeting. Interna- tional Journal of Central Banking, 1(1):1–54.

Taylor, J. B. (1993). Discretion versus policy rules in practice. Carnegie-Rochester Con- ference Series on Public Policy, 39:195–214.

Taylor, J. B. (1999). A historical analysis of monetary policy rules. In Taylor, J. B., editor, Monetary Policy Rules. Chicago University Press, Chicago.

Uhlig, H. (2005). What are the eﬀects of monetary policy on output? Results from an agnostic identiﬁcation procecure. Journal of Monetary Economics, 52(2):381–419.

Woodford, M. (2003). Optimal interest rate smoothing. Review of Economic Studies, 70:861–886. Economic Studies

______

1987:1 Haraldson,ȱ Marty:ȱȱToȱ Careȱ andȱ Toȱ Cure.ȱ Aȱ linearȱ programmingȱ approachȱ to nationalȱhealthȱplanningȱinȱdevelopingȱcountries.ȱ98ȱpp.

1989:1 Chryssanthou,ȱ Nikos:ȱ Theȱ Portfolioȱ Demandȱ forȱ theȱ ECU:ȱȱAȱ Transactionȱ Cost Approach.ȱȱ42ȱpp.

1989:2 Hansson,ȱBengt:ȱȱConstructionȱofȱSwedishȱCapitalȱStocks,ȱ1963Ȭ87:ȱAnȱApplicationȱof theȱHultenȬWykoffȱStudies.ȱ37ȱpp.

1989:3 Choe,ȱ ByungȬTae:ȱȱSomeȱ Notesȱ onȱ Utilityȱ Functionsȱ Demandȱ andȱ Aggregation.ȱ 39 pp.

1989:4 Skedinger,ȱ Per:ȱ Studiesȱ ofȱ Wageȱ andȱ Employmentȱ Determinationȱ inȱ theȱ Swedish WoodȱIndustry.ȱȱ89ȱpp.

1990:1 Gustafson,ȱClaesȬHåkan:ȱȱInventoryȱInvestmentȱinȱManufacturingȱFirms.ȱTheoryȱand Evidence.ȱ98ȱpp.

1990:2 Bantekas,ȱ Apostolos:ȱȱTheȱ Demandȱ forȱ Maleȱ andȱ Femaleȱ Workersȱ inȱ Swedish Manufacturing.ȱ56ȱpp.

1991:1 Lundholm,ȱMichael:ȱȱCompulsoryȱSocialȱInsurance.ȱAȱCriticalȱReview.ȱ109ȱpp.

1992:1 Sundberg,ȱGun:ȱȱTheȱDemandȱforȱHealthȱandȱMedicalȱCareȱinȱSweden.ȱ58ȱpp.

1992:2 Gustavsson,ȱThomas:ȱNoȱArbitrageȱPricingȱandȱtheȱTermȱStructureȱofȱInterestȱRates. 47ȱpp.

1992:3 Elvander,ȱ Nils:ȱ Labourȱ Marketȱ Relationsȱ inȱ Swedenȱ andȱ Greatȱ Britain.ȱ Aȱ ComȬ parativeȱStudyȱofȱLocalȱWageȱFormationȱinȱtheȱPrivateȱSectorȱduringȱtheȱ1980s.ȱȱ43 pp.

12 Dillén,ȱMats:ȱStudiesȱinȱOptimalȱTaxation,ȱStabilization,ȱandȱImperfectȱCompetition. 1993.ȱȱ143ȱpp.

13 Banks,ȱFerdinandȱE.:ȱAȱModernȱ Introductionȱ toȱ Internationalȱ Money,ȱBankingȱ and Finance.ȱ1993.ȱȱ303ȱpp.

14 Mellander,ȱ Erik:ȱ Measuringȱ Productivityȱ andȱ Inefficiencyȱ Withoutȱ Quantitative OutputȱData.ȱ1993.ȱ140ȱpp.

15 AckumȱAgell:ȱȱSusanne:ȱEssaysȱonȱWorkȱandȱPay.ȱ1993.ȱ116ȱpp.

16 Eriksson,ȱClaes:ȱȱEssaysȱonȱGrowthȱandȱDistribution.ȱ1994.ȱ129ȱpp.

17 Banks,ȱFerdinandȱE.:ȱAȱModernȱ Introductionȱ toȱ Internationalȱ Money,ȱBankingȱ and Finance.ȱ2ndȱversion,ȱ1994.ȱȱ313ȱpp. 18 Apel,ȱMikael:ȱȱEssaysȱonȱTaxationȱandȱEconomicȱBehavior.ȱ1994.ȱ144ȱpp.

19 Dillén,ȱ Hans:ȱȱAssetȱ Pricesȱ inȱ Openȱ Monetaryȱ Economies.ȱ Aȱ Contingentȱ Claims Approach.ȱ1994.ȱȱ100ȱpp.

20 Jansson,ȱPer:ȱȱEssaysȱonȱEmpiricalȱMacroeconomics.ȱ1994.ȱȱ146ȱpp.

21 Banks,ȱ Ferdinandȱ E.:ȱ Aȱ Modernȱ Introductionȱ toȱ Internationalȱ Money,ȱ Banking,ȱ and Finance.ȱ3rdȱversion,ȱ1995.ȱ313ȱpp.

22 Dufwenberg,ȱMartin:ȱOnȱRationalityȱandȱBeliefȱFormationȱinȱGames.ȱ1995.ȱȱ93ȱpp.

23 Lindén,ȱJohan:ȱJobȱSearchȱandȱWageȱBargaining.ȱ1995.ȱȱ127ȱpp.

24 Shahnazarian,ȱHovick:ȱThreeȱEssaysȱonȱCorporateȱTaxation.ȱ1996.ȱȱ112ȱpp.

25 Svensson,ȱ Roger:ȱ Foreignȱ Activitiesȱ ofȱ Swedishȱ Multinationalȱ Corporations.ȱ 1996. 166ȱpp.

26 Sundberg,ȱGun:ȱEssaysȱonȱHealthȱEconomics.ȱ1996.ȱ174ȱpp.

27 Sacklén,ȱHans:ȱEssaysȱonȱEmpiricalȱModelsȱofȱLaborȱSupply.ȱ1996.ȱȱ168ȱpp.

28 Fredriksson,ȱPeter:ȱEducation,ȱMigrationȱandȱActiveȱLaborȱMarketȱPolicy.ȱ1997.ȱ106ȱpp.

29 Ekman,ȱErik:ȱHouseholdȱandȱCorporateȱBehaviourȱunderȱUncertainty.ȱȱ1997.ȱȱ160ȱpp.

30 Stoltz,ȱBo:ȱȱEssaysȱonȱPortfolioȱBehaviorȱandȱAssetȱPricing.ȱȱ1997.ȱȱ122ȱpp.

31 Dahlberg,ȱMatz:ȱȱEssaysȱonȱEstimationȱMethodsȱandȱLocalȱPublicȱEconomics.ȱȱ1997.ȱȱ179 pp.

32 Kolm,ȱ AnnȬSofie:ȱȱTaxation,ȱ Wageȱ Formation,ȱ Unemploymentȱ andȱ Welfare.ȱ 1997.ȱ 162 pp.

33 Boije,ȱRobert:ȱCapitalisation,ȱEfficiencyȱandȱtheȱDemandȱforȱLocalȱPublicȱServices.ȱ1997. 148ȱpp.

34 Hort,ȱ Katinka:ȱȱOnȱ Priceȱ Formationȱ andȱ Quantityȱ Adjustmentȱ inȱ Swedishȱ Housing Markets.ȱ1997.ȱȱ185ȱpp.

35 Lindström,ȱThomas:ȱȱStudiesȱinȱEmpiricalȱMacroeconomics.ȱȱ1998.ȱȱ113ȱpp.

36 Hemström,ȱMaria:ȱȱSalaryȱDeterminationȱinȱProfessionalȱLabourȱMarkets.ȱȱ1998.ȱ127ȱpp.

37 Forsling,ȱGunnar:ȱȱUtilizationȱofȱTaxȱAllowancesȱandȱCorporateȱBorrowing.ȱȱ1998.ȱȱ96 pp.

38 Nydahl,ȱStefan:ȱȱEssaysȱonȱStockȱPricesȱandȱExchangeȱRates.ȱȱ1998.ȱȱ133ȱpp.

39 Bergström,ȱPål:ȱȱEssaysȱonȱLabourȱEconomicsȱandȱEconometrics.ȱȱ1998.ȱȱ163ȱpp. 40 Heiborn,ȱMarie:ȱȱEssaysȱonȱDemographicȱFactorsȱandȱHousingȱMarkets.ȱȱ1998.ȱȱ138ȱpp.

41 Åsberg,ȱPer:ȱȱFourȱEssaysȱinȱHousingȱEconomics.ȱȱ1998.ȱȱ166ȱpp.

42 Hokkanen,ȱJyry:ȱȱInterpretingȱBudgetȱDeficitsȱandȱProductivityȱFluctuations.ȱȱ1998.ȱȱ146 pp.

43 Lunander,ȱAnders:ȱȱBidsȱandȱValues.ȱȱ1999.ȱȱ127ȱpp.

44 Eklöf,ȱMatias:ȱȱStudiesȱinȱEmpiricalȱMicroeconomics.ȱȱ1999.ȱȱ213ȱpp.

45 Johansson,ȱEva:ȱȱEssaysȱonȱLocalȱPublicȱFinanceȱandȱIntergovernmentalȱGrants.ȱȱ1999. 156ȱpp.

46 Lundin,ȱDouglas:ȱȱStudiesȱinȱEmpiricalȱPublicȱEconomics.ȱȱ1999.ȱȱ97ȱpp.

47 Hansen,ȱSten:ȱȱEssaysȱonȱFinance,ȱTaxationȱandȱCorporateȱInvestment.ȱȱ1999.ȱ140ȱpp.

48 Widmalm,ȱFrida:ȱȱStudiesȱinȱGrowthȱandȱHouseholdȱAllocation.ȱȱ2000.ȱȱ100ȱpp.

49 Arslanogullari,ȱSebastian:ȱȱHouseholdȱAdjustmentȱtoȱUnemployment.ȱȱ2000.ȱȱ153ȱpp.

50 Lindberg,ȱSara:ȱȱStudiesȱinȱCreditȱConstraintsȱandȱEconomicȱBehavior.ȱȱ2000.ȱȱ135ȱpp.

51 Nordblom,ȱKatarina:ȱȱEssaysȱonȱFiscalȱPolicy,ȱGrowth,ȱandȱtheȱImportanceȱofȱFamily Altruism.ȱ2000.ȱȱ105ȱpp.

52 Andersson,ȱBjörn:ȱȱGrowth,ȱSaving,ȱandȱDemography.ȱ2000.ȱȱ99ȱpp.

53 Åslund,ȱOlof:ȱȱHealth,ȱImmigration,ȱandȱSettlementȱPolicies.ȱ2000.ȱȱ224ȱpp.

54 BaliȱSwain,ȱRanjula:ȱȱDemand,ȱSegmentationȱandȱRationingȱinȱtheȱRuralȱCreditȱMarkets ofȱPuri.ȱȱ2001.ȱȱ160ȱpp.

55 Löfqvist,ȱRichard:ȱȱTaxȱAvoidance,ȱDividendȱSignalingȱandȱShareholderȱTaxationȱinȱan OpenȱEconomy.ȱȱ2001.ȱȱ145ȱpp.

56 Vejsiu,ȱAltin:ȱȱEssaysȱonȱLaborȱMarketȱDynamics.ȱȱ2001.ȱȱ209ȱpp.

57 Zetterström,ȱErik:ȱȱResidentialȱMobilityȱandȱTenureȱChoiceȱinȱtheȱSwedishȱHousing Market.ȱȱ2001.ȱȱȱ125ȱpp.

58 Grahn,ȱSofia:ȱȱTopicsȱinȱCooperativeȱGameȱTheory.ȱȱ2001.ȱȱ106ȱpp.

59 Laséen,ȱStefan:ȱȱMacroeconomicȱFluctuationsȱandȱMicroeconomicȱAdjustments:ȱȱWages, Capital,ȱandȱLaborȱMarketȱPolicy.ȱȱ2001.ȱȱ142ȱpp.

60 Arnek,ȱMagnus:ȱȱEmpiricalȱEssaysȱonȱProcurementȱandȱRegulation.ȱȱ2002.ȱȱ155ȱpp. 61 Jordahl,ȱHenrik:ȱEssaysȱonȱVotingȱBehavior,ȱLaborȱMarketȱPolicy,ȱandȱTaxtion.ȱȱ2002. 172ȱpp.

62 Lindhe,ȱTobias:ȱȱCorporateȱTaxȱIntegrationȱandȱtheȱCostȱofȱCapital.ȱȱ2002.ȱȱ102ȱpp.

63 Hallberg,ȱDaniel:ȱȱEssaysȱonȱHouseholdȱBehaviorȱandȱTimeȬUse.ȱȱ2002.ȱȱ170ȱpp.

64 Larsson,ȱLaura:ȱEvaluatingȱSocialȱPrograms:ȱActiveȱLaborȱMarketȱPoliciesȱandȱSocial Insurance.ȱȱ2002.ȱȱ126ȱpp.

65 Bergvall,ȱAnders:ȱȱEssaysȱonȱExchangeȱRatesȱandȱMacroeconomicȱStability.ȱȱ2002. 122ȱpp.

66 NordströmȱSkans,ȱOskar:ȱȱLabourȱMarketȱEffectsȱofȱWorkingȱTimeȱReductionsȱand DemographicȱChanges.ȱȱ2002.ȱȱ118ȱpp.

67 Jansson,ȱJoakim:ȱȱEmpiricalȱStudiesȱinȱCorporateȱFinance,ȱTaxationȱandȱInvestment. 2002.ȱȱ132ȱpp.

68 Carlsson,ȱMikael:ȱȱMacroeconomicȱFluctuationsȱandȱFirmȱDynamics:ȱTechnology, ProductionȱandȱCapitalȱFormation.ȱȱ2002.ȱȱ149ȱpp.

69 Eriksson,ȱStefan:ȱȱTheȱPersistenceȱofȱUnemployment:ȱDoesȱCompetitionȱbetween EmployedȱandȱUnemployedȱJobȱApplicantsȱMatter?ȱȱ2002.ȱȱ154ȱpp.

70 Huitfeldt,ȱHenrik:ȱȱLabourȱMarketȱBehaviourȱinȱaȱTransitionȱEconomy:ȱȱTheȱCzech Experience.ȱȱ2003.ȱȱ110ȱpp.

71 Johnsson,ȱRichard:ȱȱTransportȱTaxȱPolicyȱSimulationsȱandȱSatelliteȱAccountingȱwithinȱa CGEȱFramework.ȱȱ2003.ȱȱ84ȱpp.

72 Öberg,ȱAnn:ȱȱEssaysȱonȱCapitalȱIncomeȱTaxationȱinȱtheȱCorporateȱandȱHousingȱSectors. 2003.ȱȱ183ȱpp.

73 Andersson,ȱFredrik:ȱȱCausesȱandȱLaborȱMarketȱConsequencesȱofȱProducer Heterogeneity.ȱ2003.ȱȱ197ȱpp.

74 Engström,ȱPer:ȱȱOptimalȱTaxationȱinȱSearchȱEquilibrium.ȱȱ2003.ȱȱ127ȱpp.

75 Lundin,ȱMagnus:ȱȱTheȱDynamicȱBehaviorȱofȱPricesȱandȱInvestment:ȱFinancial ConstraintsȱandȱCustomerȱMarkets.ȱ2003.ȱȱ125ȱpp.

76 Ekström,ȱErika:ȱȱEssaysȱonȱInequalityȱandȱEducation.ȱȱ2003.ȱȱ166ȱpp.

77 Barot,ȱBharat:ȱȱEmpiricalȱStudiesȱinȱConsumption,ȱHouseȱPricesȱandȱtheȱAccuracyȱof EuropeanȱGrowthȱandȱInflationȱForecasts.ȱȱ2003.ȱȱ137ȱpp.

78 Österholm,ȱPär:ȱȱTimeȱSeriesȱandȱMacroeconomics:ȱStudiesȱinȱDemographyȱand MonetaryȱPolicy.ȱȱ2004.ȱȱ116ȱpp.

79 Bruér,ȱMattias:ȱȱEmpiricalȱStudiesȱinȱDemographyȱandȱMacroeconomics.ȱȱ2004.ȱȱ113ȱpp. 80 Gustavsson,ȱMagnus:ȱEmpiricalȱEssaysȱonȱEarningsȱInequality.ȱȱ2004.ȱȱ154ȱpp.

81 Toll,ȱStefan:ȱȱStudiesȱinȱMortgageȱPricingȱandȱFinanceȱTheory.ȱȱ2004.ȱȱ100ȱpp.

82 Hesselius,ȱPatrik:ȱȱSicknessȱAbsenceȱandȱLabourȱMarketȱOutcomes.ȱȱ2004.ȱȱ109ȱpp.

83 Häkkinen,ȱIida:ȱȱEssaysȱonȱSchoolȱResources,ȱAcademicȱAchievementȱandȱStudent Employment.ȱȱ2004.ȱȱȱ123ȱpp.

84 Armelius,ȱHanna:ȱȱDistributionalȱSideȱEffectsȱofȱTaxȱPolicies:ȱAnȱAnalysisȱofȱTax AvoidanceȱandȱCongestionȱTolls.ȱȱ2004.ȱȱ96ȱpp.

85 Ahlin,ȱÅsa:ȱȱCompulsoryȱSchoolingȱinȱaȱDecentralizedȱSetting:ȱStudiesȱofȱtheȱSwedish Case.ȱȱ2004.ȱȱ148ȱpp.

86 Heldt,ȱTobias:ȱȱSustainableȱNatureȱTourismȱandȱtheȱNatureȱofȱTouristsȇȱCooperative Behavior:ȱRecreationȱConflicts,ȱConditionalȱCooperationȱandȱtheȱPublicȱGoodȱProblem. 2005.ȱȱ148ȱpp.

87 Holmberg,ȱPär:ȱModellingȱBiddingȱBehaviourȱinȱElectricityȱAuctions:ȱSupplyȱFunction EquilibriaȱwithȱUncertainȱDemandȱandȱCapacityȱConstraints.ȱ2005.ȱȱ43ȱpp.