Quantitative Macroeconomic Modeling with Structural Vector Autoregressions – an Eviews Implementation

Quantitative Macroeconomic Modeling with Structural Vector Autoregressions { An EViews Implementation S. Ouliaris1, A.R. Pagan2 and J. Restrepo3 August 2, 2018 [email protected] [email protected] [email protected] © 2016- S. Ouliaris, A.R. Pagan and J. Restrepo. All rights reserved. Preface This book began as a series of lectures given by the second author at the In- ternational Monetary Fund as part of the Internal Economics Training program conducted by the Institute for Capacity Development. They were delivered dur- ing 2011-2015 and were gradually adapted to describe the methods available for the analysis of quantitative macroeconomic systems with the Structural Vector Autoregression approach. A choice had to be made about the computer package that would be used to perform the quantitative work and EViews was eventually selected because of its popularity among IMF staff and central bankers more generally. Although the methodology developed in this book extends to other packages such as Stata, it was decided to illustrate the methods with EViews 9.5. The above was the preface to the book Ouliaris et al. (2016) (henceforth OPR). Many of the restrictions we needed to impose to estimate structural shocks could not be handled directly in EViews 9.5. We proposed workaround methods that enabled one to impose the identifying information upon the shocks in an indirect way. A key idea we used was that for exactly identified SVARs, i.e. an SVAR in which the number of moment restrictions for estimation equals the number of parameters, the Maximum Likelihood estimator (MLE) is iden- tical to an instrumental variable (IV) estimator. This was advantageous since the MLE requires non-linear optimization techniques, while the IV estimator uses linear Two Stage Least Squares (2SLS). The IV approach allowed us to estimate complex SVAR structures using EViews 9.5. Gareth Thomas and Glenn Sueyoshi at IHS Global Inc kindly helped us understand some of the functions of the EViews package. They also added new options to EViews 9.5 that were important for implementing and illustrating the methods described in OPR. The orientation of OPR was to take some classic studies from the literature and show how one could re-do them in EViews 9.5 using the IV approach. We covered a range of ways of identifying the shocks when variables were stationary or non-stationary or both. These methods involved both parametric and sign restrictions upon either the impulse responses or the structural equations. EViews 10 has many new features that deal with VARs and SVARs. For VARs it is now possible to exclude some lagged variables from particular equations using lagged matrices L1, L2, ... etc. For SVARs the specification of the A, B matrices used to define the model in EViews 9.5 is now augmented by two extra matrices S and F, which are used to impose short-and long-run restrictions on the model. Hence in this updated manuscript we first present how to estimate SVARs if one only has EViews 9.5 and then we re-do the same exercises using EViews 10. As we will see it is generally much easier to work with EViews 10, although thinking about the problem from an instrumental variables perspective can often be very valuable. There are still some issues that can arise in EViews 10. In particular, there may be convergence problems with the MLE, and the computation of standard errors for impulse responses is sometimes problematic. We provide an EViews add-in that may be able to deal with convergence problems by first estimating the system using the instrumental approach and using the resulting parameter estimates to initialize the new SVAR estimator. The standard error problem, which arises when there are long-run restrictions, is left to the EViews team to resolve, as it involves modification of the base code. Because the book developed out of a set of lectures we would wish to thank the many IMF staff and country officials who participated in the courses and whose reactions were important in allowing us to decide on what should be emphasised and what might be treated more lightly. The courses were excep- tionally well organized by Luz Minaya and Maria (Didi) Jones. Versions of the course were also given at the Bank of England and the Reserve Bank of Australia. Finally, on a personal level Adrian would like to dedicate this book to Janet who had to endure the many hours of its construction and execution. Contents Forward 3 1 An Overview of Macro-Econometric System Modeling 15 2 Vector Autoregressions: Basic Structure 21 2.1 Basic Structure . 21 2.1.1 Maximum Likelihood Estimation of Basic VARs . 22 2.1.2 A Small Macro Model Example . 23 2.2 Specification of VARs . 25 2.2.1 Choosing p .......................... 27 2.2.1.1 Theoretical models . 27 2.2.1.2 Rules of Thumb . 28 2.2.1.3 Statistical Criteria . 28 2.2.2 Choice of Variables . 29 2.2.2.1 Institutional Knowledge . 29 2.2.2.2 Theoretical models . 29 2.2.3 Restricted VAR's . 31 2.2.3.1 Setting Some Lag Coefficients to Zero . 31 2.2.3.2 Imposing Exogeneity- the VARX Model . 32 2.2.4 Augmented VARs . 38 2.2.4.1 Trends and Dummy Variables . 38 2.2.4.2 With Factors . 40 2.3 Vector Autoregressions - Handling Restricted VARs in EViews 10 41 2.4 Conclusion . 45 3 Using and Generalizing a VAR 47 3.1 Introduction . 47 3.2 Testing Granger Causality . 47 3.3 Forecasting using a VAR . 48 3.3.1 Forecast Evaluation . 49 3.3.2 Conditional Forecasts . 49 3.3.3 Forecasting Using EViews . 49 3.4 Bayesian VARs . 55 3.4.1 The Minnesota prior ................... 58 4 3.4.1.1 Implementing the Minnesota prior in EViews . 61 3.4.2 Normal-Wishart prior .................. 63 3.4.2.1 Implementing the Normal-Wishart prior in EViews 63 3.4.3 Additional priors using dummy observations or pseudo data 65 3.4.3.1 Sum of Coefficients Dummy Prior . 65 3.4.3.2 The Initial Observations Dummy Prior . 67 3.4.4 Forecasting with Bayesian VARs.............. 68 3.5 Computing Impulse Responses . 68 3.6 Standard Errors for Impulse Responses . 71 3.7 Issues when Using the VAR as a Summative Model . 74 3.7.1 Missing Variables . 74 3.7.2 Latent Variables . 77 3.7.3 Non-Linearities . 78 3.7.3.1 Threshold VARs . 78 3.7.3.2 Markov Switching process . 79 3.7.3.3 Time Varying VARs . 80 3.7.3.4 Categorical Variables for Recurrent States . 81 4 Structural Vector Autoregressions with I(0) Processes 83 4.1 Introduction . 83 4.2 Mathematical Approaches to Finding Uncorrelated Shocks . 83 4.3 Generalized Impulse Responses . 84 4.4 Structural VAR's and Uncorrelated Shocks: Representation and Estimation . 85 4.4.1 Representation . 85 4.4.2 Estimation . 87 4.4.2.1 Maximum Likelihood Estimation . 88 4.4.2.2 Instrumental Variable (IV) Estimation . 88 4.5 Impulse Responses for an SVAR: Their Construction and Use . 90 4.5.1 Construction . 90 4.5.2 Variance and Variable Decompositions . 90 4.6 Restrictions on a SVAR . 92 4.6.1 Recursive Systems . 92 4.6.1.1 A Recursive SVAR with the US Macro Data . 94 4.6.1.2 Estimating the Recursive Small Macro Model with EViews 9.5 . 94 4.6.1.3 Estimating the Recursive Small Macro Model with EViews 10 . 102 4.6.1.4 Impulse Response Anomalies (Puzzles) . 102 4.6.2 Imposing Restrictions on the Impact of Shocks . 108 4.6.2.1 A Zero Contemporaneous Restriction using EViews 9.5 . 108 4.6.2.2 Zero Contemporaneous Restrictions in EViews 10 113 4.6.2.3 A Two Periods Ahead Zero Restriction . 117 4.6.3 Imposing Restrictions on Parameters - The Blanchard- Perotti Fiscal Policy Model . 120 4.6.4 Incorporating Stocks and Flows Plus Identities into an SVAR - A US Fiscal-Debt Model . 123 4.6.4.1 The Cherif and Hasanov (2012) Model . 123 4.6.4.2 Estimation with EViews 9.5 . 125 4.6.4.3 Estimation with EViews 10 . 128 4.6.5 Treating Exogenous Variables in an SVAR - the SVARX Model . 128 4.6.5.1 Estimation with EViews 9.5 . 128 4.6.5.2 Estimation with EViews 10 . 131 4.6.6 Restrictions on Parameters and Partial Exogeneity: Ex- ternal Instruments . 138 4.6.7 Factor Augmented SVARs . 139 4.6.8 Global SVARs (SGVARs) . 144 4.6.9 DSGE Models and the Origins of SVARs . 145 4.7 Standard Errors for Structural Impulse Responses. 146 4.8 Other Estimation Methods for SVARs . 147 4.8.1 Bayesian . 147 4.8.2 Using Higher Order Moment Information . 149 5 SVARs with I(0) Variables and Sign Restrictions 152 5.1 Introduction . 152 5.2 The Simple Structural Models Again and Their Sign Restrictions 153 5.3 How Do we Use Sign Restriction Information? . 155 5.3.1 The SRR Method . 156 5.3.1.1 Finding the Orthogonal Matrices . 157 5.3.2 The SRC method . 159 5.3.3 The SRC and SRR Methods Applied to a Market Model . 159 5.3.3.1 The SRR Method Applied to the Market Model 160 5.3.3.2 The SRC Method Applied to the Market Model 161 5.3.3.3 Comparing the SRR and SRC Methodologies . 162 5.3.4 Comparing SRC and SRR With a Simulated Market Model ...............................

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