Essays on and Time Series Analysis

Dissertation

Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

By

Jongwook Park, M.A.

Graduate Program in

The Ohio State University

2016

Dissertation Committee:

Robert de Jong, Advisor Pok-Sang Lam Jason Blevins c Copyright by

Jongwook Park

2016 Abstract

My dissertation explores the effect of monetary policy and weak dependence con- ditions for nonlinear time series models. The first chapter analyzes the asymmetric effects of contractionary and expansionary monetary policy shocks on income in- equality. Using the Consumer Expenditure Survey we calculate the adjusted Gini coefficients suggested by Chen et al. (1982) and Berrebi and Silber (1985) for five types of incomes such as total, , business, financial, and other incomes. We then adopt the Kilian and Vigfusson (2011) model to examine the existence of asymmetry in the responses of inequality for each type of income to monetary policy shocks.

The main finding is that monetary policy shocks have asymmetric effects on total, wage, and financial income inequalities during about two years after a shock. The responses of total and wage income inequalities to contractionary shocks are much larger than those to expansionary shocks. The degree of asymmetric response of fi- nancial income inequality is modest even though it is significant. We also find that these considerable asymmetric responses of total and wage income inequalities are due to the asymmetric responses of the bottom income bracket. The responses of total and wage income inequalities in the bottom to contractionary shocks are larger than those to expansionary shocks while the responses in the top to contractionary and expansionary shocks are not much different.

ii The second chapter examines the conditions for the near epoch dependence in nonlinear dynamic time series models and applies them to the Kilian and Vigfusson

(2011) model.

The last chapter investigates the validity of sign restrictions for identifying the monetary policy shocks in a VAR model. A simple version of new Keynesian DSGE model is estimated using Bayesian techniques. Then the artificial data set is generated from the estimated model and are then used in a sign-restriction VAR model to estimate the effect of monetary policy. The result shows that the sign restrictions does not well identify the monetary policy shocks: after a contractionary monetary policy shock, the sign-restriction VAR estimation shows that output increases even though it decreases in the estimated model where the artificial data set is generated.

The sign restrictions seems to well identify the monetary policy shocks only when the shocks are extremely volatile. On the other hand, the recursive assumption captures the negative response of output after a contractionary monetary policy shock.

iii To my wife Gui Jeong and our son Minjoon

iv Acknowledgments

I would like to express my deep gratitude to Robert de Jong. It would be hardly possible for me to complete this dissertation without his continuous support and guid- ance. Through countless conversations he not only was willing to share his knowledge about economics and with me but also gave me invaluable advice when my research was stuck. Moreover, his enthusiasm for learning and research as a scholar inspired me. I hope to have and maintain the passion and attitude like him in the future. I am also grateful to Pok-Sang Lam and Jason Blevins for their valuable comments on my dissertation.

In addition, I appreciate Paul Evans who was my former advisor. He helped me in developing the last chapter of this dissertation with his insights. I also thank Hajime

Miyazaki and the department staffs. Due to their support, I could adapt to new surroundings smoothly.

Finally, I am thankful to my wife, Gui Jeong who encouraged me whenever I suffered hardship and discussed ideas, and our son, Minjoon who gave us happiness of parenting. I always appreciate my parents and family for their unconditional love and support.

v Vita

2007 ...... B.A. Economics, Yonsei University, Re- public of Korea 2007 - 2011 ...... Junior , Bank of Korea, Re- public of Korea 2012 ...... M.A. Economics, The Ohio State Uni- versity 2012 - 2016 ...... Graduate Teaching Associate, The Ohio State University

Fields of Study

Major Field: Economics

vi Table of Contents

Page

Abstract ...... ii

Dedication ...... iv

Acknowledgments ...... v

Vita ...... vi

List of Tables ...... x

List of Figures ...... xi

1. Asymmetric Effects of Monetary Policy Shocks on Income Inequality . . 1

1.1 Introduction ...... 1 1.1.1 Literature review ...... 5 1.2 Effects of Monetary Policy on Inequality ...... 9 1.2.1 Channels through which monetary policy affects inequality . 9 1.2.2 Discussion ...... 10 1.3 Data ...... 11 1.3.1 Consumer Expenditure Survey ...... 11 1.3.2 Gini Coefficient with Negative Incomes and Weighted Data 13 1.3.3 Unit Root Tests ...... 17 1.4 The Response of Income Inequality to Monetary Shocks ...... 18 1.4.1 Econometric Specifications ...... 18 1.4.2 Impulse Responses ...... 22 1.4.3 Symmetry Tests for Impulse Responses ...... 27 1.4.4 Size Asymmetry ...... 28 1.5 The Responses of Total and Wage Income Inequalities in Top and Bottom Income Brackets ...... 29 1.6 Conclusion ...... 32

vii 2. Near Epoch Dependence in Nonlinear Dynamic Time Series Models . . . 34

2.1 Introduction ...... 34 2.2 Near Epoch Dependence ...... 35 2.3 Nonlinear AR Model ...... 36 2.3.1 Scalar Case ...... 36 2.3.2 Multivariate Case ...... 40 2.4 Application to the Kilian and Vigfusson Model ...... 46 2.4.1 The Kilian and Vigfusson Model ...... 46 2.4.2 Near Epoch Dependence Conditions with One Lag . . . . . 47 2.4.3 Near Epoch Dependence Conditions with Arbitrary Lags . . 51 2.5 Conclusion ...... 53

3. Can Sign Restrictions Identify Monetary Policy Shocks in a VAR? . . . . 54

3.1 Introduction ...... 54 3.2 Motivation ...... 57 3.2.1 Uhlig’s Sign-restriction VAR Estimation ...... 57 3.2.2 Motivation ...... 59 3.3 Model ...... 60 3.3.1 Household ...... 61 3.3.2 Firms ...... 62 3.3.3 and Resource Constraint ...... 63 3.3.4 Log-linearization ...... 63 3.4 Parameter Estimation ...... 65 3.4.1 Data ...... 65 3.4.2 Priors ...... 66 3.4.3 Estimation Result ...... 67 3.4.4 DSGE Impulse Responses ...... 68 3.5 Monte Carlo Simulation ...... 69 3.6 Conclusion ...... 72

Bibliography ...... 75

A. Appendix to Chapter 1 ...... 82

A.1 Categories of Income ...... 82 A.2 Gini Coefficient with Negative Incomes and Weighted Data . . . . 83 A.3 Monte Carlo Integration for Constructing Impulse Responses . . . . 86

viii B. Appendix to Chapter 2 ...... 87

B.1 Weak Dependence Conditions with One Lag ...... 87 B.2 Weak Dependence Conditions with Arbitrary Lags ...... 92

C. Appendix to Chapter 3 ...... 100

ix List of Tables

Table Page

1.1 Standard Deviation and Correlation among Income Gini Coefficients . 17

1.2 Unit Root Tests ...... 18

1.3 Symmetry Tests for Impulse Responses (1 s.d.) ...... 28

1.4 Symmetry Tests for Impulse Responses (2 s.d.) ...... 29

3.1 Sign Restrictions in Uhlig (2005) ...... 58

3.2 Prior and Posterior Mean of Structural Parameters . . . 67

3.3 Sign Restrictions for the Simulation ...... 70

A.1 Four Categories of Income ...... 82

A.1 Four Categories of Income (continued) ...... 83

x List of Figures

Figure Page

1.1 Income Gini Coefficients ...... 16

1.2 Federal Funds Rate and Wu and Xia’s Shadow Rate ...... 21

1.3 Impulse Responses of Income Gini Coefficients to Monetary Policy Shocks in a Two-variable Model ...... 24

1.4 Impulse Responses of Income Gini Coefficients to Romer and Romer Monetary Policy Shocks ...... 25

1.5 Impulse Responses of Income Gini Coefficients in Top and Bottom Brackets to Monetary Policy Shocks in a Two-variable VAR (1 s.d.) . 30

1.6 Impulse Responses of Income Gini Coefficients in Top and Bottom Brackets to Romer and Romer Monetary Policy Shocks (1 s.d.) . . . . 31

3.1 Impulse Responses from the Sign-restriction VAR ...... 60

3.2 Impulse Responses from the Sign-restriction VAR during Pre-Volcker Period ...... 61

3.3 Impulse Responses from the Estimated DSGE Model ...... 69

3.4 Impulse Responses to a Contractionary Monetary Shock ...... 70

0 3.5 Impulse Responses to a Contractionary Monetary Shock (σm = 20σm) 71

0 0 3.6 Impulse Responses to a Contractionary Monetary Shock (σz = σd = 0) 72

3.7 Impulse Responses to a contractionary Monetary Shock from a Recur- sive VAR ...... 73

xi A.1 Lorenz Curve with Negative Incomes and Weighted Data ...... 85

xii Chapter 1: Asymmetric Effects of Monetary Policy Shocks on Income Inequality

1.1 Introduction

Inequality in U.S. has increased during the past three decades. As an influential early work, Piketty and Saez (2003) construct series on top shares of income for a long period covering 1913 to 1998 and show that top income shares have dropped in the first part of the 20th century but increased since 1980s. More recently, Piketty and Saez (2013) provide the updated top income shares through 2010 and show that the top income shares rebounded from a sharp fall during 2008-09 financial crisis.

A large body of literature also shows similar results using different data sets and diverse measures for inequality in income, consumption, and wealth. For example,

Heathcote et al. (2010) integrate data from the Current Population Survey, the Panel

Study of Income Dynamics, the Consumer Expenditure Survey, and the Survey of

Consumer Finances and document that the inequalities for income, consumption, and wealth have increased from 1967 to 2006. More recently, Fisher et al. (2013), using the Consumer Expenditure Survey, show that the Gini coefficients for income and consumption have also increased during the recent financial crisis.

1 Changes in some deep structural factors have been explored as the main culprits of rising inequality. Bound and Johnson (1992) argue that skill-biased technological progress increases the demand for the highly educated workers, which leads to a huge increase in the relative of them. Feenstra and Hanson (2001) find that the main reason for a relative increase in the demand for the skilled workers is the expanding international instead of technological progress. Card (2001) shows that the decline in union membership can account for up to one-quarter of the rise in male wage inequality. On the other hand, monetary policy has been ignored as a source of rising inequality since the effects of monetary policy are believed to be neutral in the long run while the trend of rising inequality is a long-run phenomenon. Central bankers also have doubts about the role of conventional monetary policy in widening inequality (Bernanke, 2015; Mersch, 2014).

However, monetary policy has recently gained attention as a factor affecting cycli- cal inequality patterns. There is an argument that unconventional monetary policy during and after the recent financial crisis increased financial asset and so it widened the degree of inequality. But there is still considerable disagreement among about whether and how much the unconventional monetary policy affects the degree of inequality. Even central bankers have different views on whether un- conventional monetary policy worsens inequality. Fisher (2014), the former president of FRB Dallas, and Mersch (2014), a member of ECB’s executive board, argue that quantitative easing program had an impact on inequalities by putting upward pres- sure on financial asset prices while it did not help stimulating job creation. On the other hand, Bernanke (2015), the former chairman of FRB, and Bullard (2014), the

2 president of FRB St.Louis, state that the program did not worsen inequality even

though they agree on the fact that the program led to increases in asset prices.

Since monetary policy has been ignored as a source of rising inequality empirical

studies about the effect of monetary policy on inequality are not much available.

Coibion et al. (2012) are, as far as we know, the first empirical study estimating

the effect of monetary policy shocks on inequality. Using the Consumer Expenditure

Survey from 1980 to 2008, they find that a contractionary (expansionary) monetary

policy shock raises (reduces) the income and consumption inequalities. However,

assuming a linear relationship between monetary policy shocks and the degree of

inequality Coibion et al. do not examine any nonlinear relationships between them.

As documented by Kilian and Vigfusson (2011), the estimation results from a linear

model would be invalid if the true data generating process is asymmetric. Since

literature1 suggests that monetary policy has an asymmetric effect on macro variables, it is possible that the results of Coibion et al. would be misleading.

The main contribution of this paper is to examine the existence of the sign- and size-asymmetric effect of monetary policy shocks on income inequality. If the expan- sionary and contractionary monetary policy shocks have different effects on macro variables, the response of income inequality could be asymmetric. For example, if a contractionary monetary policy is more effective in affecting GDP and employment than an expansionary monetary policy, then the responses of wage income inequality to contractionary and expansionary monetary policies would be different.

The other contributions are as follows. First, we consider the inequality in various sources of income. Not only total and wage income inequalities but also business,

1For example, see Cover (1992), Morgan (1993), and Karras (1996) among others. See the literature review in the next subsection for details.

3 financial, and other income inequalities are examined. Second, we calculate the ad- justed Gini coefficients suggested by Chen et al. (1982) and Berrebi and Silber (1985) for each type of incomes. The conventional Gini coefficient is not appropriate since total, business, and financial incomes are negative for some households and so the coefficient can be larger than one. Third, we adopt and estimate the model recently suggested by Kilian and Vigfusson (2011) to examine the existence of asymmetric responses of income inequality to monetary policy shocks. Finally, we estimate the effects on income inequality of monetary policy shocks identified in two alternative specifications for robustness.

The main finding is that monetary policy shocks have sign-asymmetric effects on total, wage, and financial income inequalities during about two years after a shock.

The responses of total and wage income inequalities to contractionary shocks are much larger than those to expansionary shocks. The degree of asymmetric response of financial income inequality is modest even though it is significant.

We also find that these considerable asymmetric responses of total and wage in- come inequalities are due to the asymmetric responses of the bottom income bracket.

The responses of inequality in the bottom income bracket to contractionary shocks are larger than those to expansionary shocks while the responses of inequality in the top income bracket to contractionary and expansionary shocks are not much different.

In addition, monetary policy shocks have different effects on each type of income inequality depending on whether they are contractionary or expansionary and on how they are identified. Generally speaking, total, wage, business, and other income inequalities worsens (reduces) following a contractionary (expansionary) monetary

4 policy shock. On the other hand, financial income inequality reduces (worsens) fol- lowing a contractionary (expansionary) shock.

Finally, we cannot find any evidences on the size-asymmetric effects of monetary policy on income inequality.

1.1.1 Literature review

Monetary policy has been ignored as a source of rising inequality and so empirical studies about the effect of monetary policy on inequality are not much available. Es- timating a univariate model with the international data, Romer and

Romer (1998) show that in the short run a low rate is associated with improved conditions for the poor by creating more jobs, but this cyclical movement in unemployment has little impact on income distribution. Also they find that the potential redistributive effects of unanticipated inflation are very small. Thus, they state that expansionary monetary policy aimed at rapid output growth improves the well-being of the poor in the short run. On the other hand, they argue that in the long run prudent monetary policy aimed at low inflation and steady output growth is associated with enhanced well-being of the poor and greater equality in income.

Coibion et al. (2012) are, as far as we know, the first empirical study estimating the effect of monetary policy shocks on inequality. Using the Consumer Expenditure

Survey from 1980 to 2008, they find that a contractionary (expansionary) monetary policy shock raises (reduces) the income and consumption inequalities.2 Also they show that monetary policy shocks have played a non-trivial role in accounting for

2They argue that this is driven by the earnings heterogeneity channel and the redis- tribution channel. For the description about the channels through which monetary policy affects inequality, see Section 1.2.1.

5 cyclical fluctuations in inequality and monetary policy shocks can account for a sur- prising amount of the historical cyclical changes in inequality. Mumtaz et al. (2016) obtain very similar results for U.K from 1969 to 2012.

Empirical analysis about the effect of unconventional monetary policy on inequal- ity during the Great is much less available. Watkins (2014) presents some illustrations from Survey of Consumer Finances that income and wealth inequalities have increased with the quantitative easing program of the Fed. Saiki and Frost (2014) show that unconventional monetary policy increased income inequality in Japan be- tween 2008 and 2013 by estimating a linear VAR model.3 They suggest that it is possible that the portfolio channel will be even larger in the U.S., the U.K., and many European countries, where households hold a larger portion of their savings in equities and bonds. Mumtaz et al. (2016) capture the impact of quantitative easing program in U.K assuming that the program affects the economy by reducing the yield on long-term government bonds. Their result implies that the program contributed to the increase in inequality over the Great Recession.

Other studies build calibrated models to examine the effect of monetary policy on inequality. Gornemann et al. (2012) construct a new Keynesian model with heterogeneous agents and show how the effect of monetary policies differs by wealth distributions. The result is that a 1 percentage point (yearly) increase in the policy rate increases the income, consumption, and wealth Gini coefficients increase by 0.003,

0.0012, and 0.0008, respectively. Airaudo and Bossi (2014) incorporate limited asset participation and consumption into a new Keynesian model and

3Saiki and Frost (2014) argue that this was driven by the portfolio channel: an increase in the monetary base through purchases of both safe and risky assets tends to increase asset prices.

6 show that a 1 percentage point increases in the policy rate raises the Gini coefficient for consumption inequality by about 0.008.

This paper is also related to the asymmetric effects of monetary policy. The literature examines the asymmetric effects of monetary policy on macro variables, usually output, depending on the sign, size, and state-dependent asymmetry.4 The seminal paper by Cover (1992) employing the two-step procedure finds that positive supply shocks do not have an effect on output, while negative shocks do. Cover’s two-step procedure5 is adopted by other researchers who extend

Cover’s analysis. For instance, Morgan (1993) yields results in line with Cover using the federal funds rate and Boschen-Mill index instead of monetary aggregates as the monetary policy stance. Morgan shows that an increase in the federal funds rate has significant negative effect on output, while a decrease in the funds rate has no effect.

Karras (1996) reports similar results to Cover and Morgan not only for output but also for consumption and investment in 18 European countries. Karras also investigates whether monetary policy shocks have the state-dependent asymmetric effects but fails to provide any evidence for it. However, Ravn and Sola (1996) indicate that the evidence on sign asymmetry is not robust for the sample period. Instead, Rvan and

Sola provide evidence for size asymmetry that monetary policy shocks of large size are neutral but small shocks have real effects.

More recent papers consider the state-dependent asymmetric effects of monetary policy. Thoma (1994) generalizes a VAR model to allow positive and negative changes in money growth to have different effects on output, and allows the effects to vary

4For theoretical considerations on asymmetric effects of monetary policy on output, see Ravn and Sola (2004). 5The first step is to estimate the money equation. In the second step the residuals obtained from the first step is fed into the output equation.

7 over the . Thoma shows that positive changes in money growth have very little impact on the level of output regardless of the initial state of the economy.

However, negative changes in money growth have a much larger impact on the level of output especially when industrial production is at its potential level. Garcia and

Schaller (2002) and Lo and Piger (2005) use the Markov switching model to examine whether monetary policy affects output differently in different phases of the business cycle. They find that rate changes have a stronger effect on output growth during than during expansions and that changes in interest rates have a substantial effect on the probability of state switches.6 In addition, Lo and Piger find less evidence for sign or size asymmetry. Weise (1999) employs the logistic smooth transition VAR technique and finds that shocks to the money supply have stronger output effects and weaker effects when output growth is initially low, which is similar to Garcia and Schaller (2002) and Lo and Piger (2005). Weise also finds asymmetric effects in the response to monetary shocks of different sizes but finds near symmetric effects in the response to positive and negative monetary shocks.7

The remainder of this paper is organized as follows. In Section 1.2, we summarize the channels through which monetary policy affects inequality. Section 1.3 describes the data and calculates the adjusted Gini coefficients for each type of incomes. Section

1.4 presents the responses of income inequality to monetary policy shocks and provides evidences on asymmetric responses of total, wage, and financial income inequalities.

6Chen (2007) applies the Markov switching model to the stock market and finds similar results: the monetary policy has larger effects on returns in the bear-market regime and a contractionary monetary policy leads to a higher probability of switching to a bear-market regime. 7Castillo and Montoro (2008) incorporate nonhomothetic preferences into a standard new Keye- sian DSGE model and show that monetary policy shocks have a larger effect on output in booms than in recessions.

8 Section 1.5 examines the responses of income inequality by income brackets and shows that considerable asymmetric responses of total and wage income inequalities are due to asymmetric responses of the bottom income bracket. Finally, Section 1.6 concludes.

1.2 Effects of Monetary Policy on Inequality

This section briefly restate the channels through which monetary policy affects inequality and discuss the possibility of asymmetric effects of monetary policy on income inequality.

1.2.1 Channels through which monetary policy affects in- equality

One reason why there is a disagreement on whether monetary policy increases inequality is that there are various channels through which monetary policy affects inequality as well summarized by Coibion et al. (2012).

1. Channels through which expansionary monetary policy increases inequality:

(a) Income composition channel: The primary source of income for each house-

hold is different. If expansionary monetary policy shocks raise profits more

than wages, then income inequality would grow because the households

with higher income are likely to rely on business or financial income.

(b) Financial segmentation channel: A central bank injects money supply into

the economy through financial markets. Since expansionary monetary pol-

icy shocks redistribute wealth toward those households that are most con-

nected to financial markets (Williamson, 2008) and are likely to be rich,

the policy widen the wealth inequality.

9 (c) Portfolio channel: Poor households tend to hold a large fraction of their

wealth as (Erosa and Ventura, 2002) whose real is vulnera-

ble to inflation. Thus expansionary monetary policy shocks which lead to

high inflation increase wealth inequality.

2. Channels through which expansionary monetary policy decreases inequality:

(a) Savings redistribution channel: An unexpected increase in inflation lowers

the ex-post real interest rates will hurt savers and benefit borrowers. Since

the main losers are rich and old households (Doepke and Schneider, 2006)

expansionary monetary policy shocks which lead to high inflation decrease

wealth inequality.

(b) Earnings heterogeneity channel: Employment and labor earnings at the

bottom of the distribution are most affected by business cycle fluctuations

(Romer and Romer, 1998; Heathcote et al., 2010). Thus expansionary

monetary policy shocks decrease income inequality if they can boost the

economy.

1.2.2 Discussion

This paper focuses on income inequality and categorizes total income into wage, business, financial, and other income, which are summarized in Table A.1 in Appendix

A. If we know how monetary policy affects each source of income, we can predict how inequality of each income would respond to monetary policy.

First, earnings heterogeneity channel implies that a contractionary monetary pol- icy would worsen wage income inequality. Second, business income consists of non- farm business and farm incomes. While it is uncertain how monetary policy affects

10 farm income, Gertler and Gilchrist (1994) find that monetary policy affects the sales of small firms more than that of large firms. This implies that a contractionary monetary policy would worsen nonfarm business income inequality. Third, financial income covers various types of incomes on which monetary policy has different ef- fects. For example, a contractionary monetary policy would increase interest income by increasing the while it would decrease rental income by slowing down the economy. Thus how financial income inequality responds to monetary policy de- pends on the composition of financial income. Fourth, other income mostly consists of transfers. Since the poor more depend on transfers and transfers are countercycli- cal, a contractionary monetary policy would worsen other income inequality. Finally, as implied by income composition channel the response of total income inequality depends on the composition of total income.

1.3 Data

In this section, we describe the data and calculates the adjusted Gini coefficients for each type of incomes. Then we report the results of unit root tests for the adjusted

Gini coefficients.

1.3.1 Consumer Expenditure Survey

The Consumer Expenditure Survey (CEX) is a Bureau of Labor Statistics (BLS) survey that provides information on the expenditures and income by consumer units

(CU)8.9 It started at 1960, but the survey has been conducted continuously since

8A CU is the measurement unit in CEX. For the definition of a CU, see the CEX users’ docu- mentation available on BLS website. Generally speaking, a household consists of one CU though it can include more than one CU. 9CEX public-use microdata can be obtained from ICPSR (Inter-university Consortium for Polit- ical and Social Research) at the University of Michigan.

11 1980. CEX data is primarily used as weights for the Consumer Price Index (CPI) and collects data through two surveys: Interview Survey and Diary Survey. This paper uses the income data from Interview Survey which is conducted quarterly and covers 95 percent of the total expenditure and income.

The Interview Survey is a rotating panel of CUs: each CU is interviewed once per quarter for at most five consecutive quarters. CEX reports only the last four interviews in the published data. Income data are collected in the second and fifth interviews for all CU members over 13 years of age and in the third and fourth in- terviews for members over 13 who are new to the CU or who previously reported not working and are now working. Income data collected in the second interview are copied to the third and fourth interviews subject to the above conditions. In- come variables contain annual values for the 12 months prior to the interview month.

CEX covers various sources of income and we categorize them into four types: wage, business, financial, and other income.

Even though there are other data sources such as the Panel Study of Income Dy- namics (PSID), the Current Population Survey (CPS), and the Survey of Consumer

Finances (SCF) from which we can obtain the income data, the CEX is more appro- priate than others for examining the asymmetric effect of monetary policy on income inequality. First, CEX provides high frequency data which are necessary to analyze the effect of monetary policy because the monetary policy is believed to affect the economy in the short run. Second, CEX provides a long sample which is essential in capturing possible asymmetries. Finally, CEX provides various sources of income.

Despite these advantages, the CEX data also has some drawbacks. First, the cred- ibility of answers can be always questioned in survey data including CEX. Especially,

12 many respondents would not like to report the true amounts of their income. To

reduce this problem of invalid missing values for income, BLS has started to impute

them since 2004. So researchers have two options: using the reported income data

of complete income reporters or using the imputed income data of all CUs applying

the same imputation method used by BLS for the periods before 2004. Because the

details in procedures for imputations used by BLS are not available, we use the re-

ported values of income for consistency.10 Another potential problem in using the

CEX data is that there may be a break due to changes in topcoding. Before 1996

the CEX has replaced the income values that fall outside the predetermined critical

value with that critical value.11 But since 1996 the CEX has replaced them with a topcoded value that represents the mean of the subset of all outlying observations.

The topcoded values vary across income sources and across time. It is not yet known whether and how much this change in topcoding affects the statistics for income in- equality. However, we believe that this change does not affect much the statistics that capture the whole distribution such as Gini coefficient on which this paper focuses.

1.3.2 Gini Coefficient with Negative Incomes and Weighted Data

This paper focuses on Gini coefficient as a measurement for the degree of inequal- ity. We do not consider other popular measurements such as the standard deviation of logs because business, financial, and so total incomes can be negative if loss occurs and those measurements cannot handle negative values.

10Since 2006 CEX data set has included both reported and imputed values of income, but for 2004 and 2005 CEX data set included only imputed values. For 2004 and 2005 during which the reported income data are not available, we tease out the reported values from imputed ones following the procedures suggested by BLS. 11The critical values in absolute terms are $75,000 before 1983 and $100,000 since 1984.

13 The Gini coefficient is defined as a half of relative mean absolute difference where the relative mean absolute difference is the mean absolute difference divided by the mean. Thus, in the simplest form the income Gini coefficient can be written as

Pn Pn |Yi − Yj| G = i=1 j=1 (1.1) 2n2Y¯ where n and Yj represents the number of total population and the income of jth

Pn Y ¯ i=1 i person, respectively, and Y = n is the average income. One limitation of the standard Gini coefficient is that it can be greater than one when negative values are included in the distribution. Chen et al. (1982) propose the adjusted Gini coefficient which normalizes the standard Gini coefficient. This adjusted Gini coefficient always has an upper bound of one and so comparability between the distribution without negative incomes and the distribution with negative incomes can be attained. The adjusted Gini coefficient, which is corrected by Berrebi and Silber (1985) later, is

Pn n+1 (2/n) jyj − G∗ = j=1 n (1.2) h Pk y i Pk Pk j=1 j 1 + (2/n) jyj + (1/n) yj − (1 + 2k) j=1 j=1 yk+1

Yj where yj = nY¯ are the income share of the jth person, Yj are in increasing order Pk Pk+1 (Yj ≤ Yj+1), and k is defined such that j=1 yj ≤ 0 and j=1 yj > 0. If there is no negative values (k = 0), the adjusted Gini coefficient in Equation (1.2) collapses to

Equation (1.1).

We extend the adjusted Gini coefficient in Equation (1.2) to take the weight of each income values into account, which can be written as12

Pn Pj Pn 2 yj f(Yi) − 1 − yjf(Yj) G∗∗ = j=1 i=1 j=1 h Pk y i Pk Pj Pk j=1 j Pk 1 + 2 yj f(Yi) + yj f(Yk+1) − (f(Yj) + 2 f(Yi)) j=1 i=1 j=1 yk+1 j=1 (1.3)

12See Appendix A.2 for proof.

14 f(Yj )Yj where f(Yj) represents the fraction of population with Yj of income and yj = Pn j=1 f(Yj )Yj 1 are the share of the income Yj. If each income value has the same weight (f(Yj) = n ), the adjusted Gini coefficient with weighted data in Equation (1.3) collapses to the standard one in Equation (1.2).

The Gini coefficients are constructed for each income source by applying Equation

1.3 to CEX data as follows.13 First, the reported annual income for each CU is divided by 12 to be converted into monthly value. The monthly incomes are then deflated by

CPI and equivalized using the square root of CU size to take in consumption into account. This value is used as Yj in Equation (1.3). The weights for each CU provided by CEX are adjusted to reflect person weight and then are used as f(Yi) in Equation (1.3). Since CEX data are available from 1980:I to 2015:I and income data contain annual values for the 12 months prior to the date of interview,

Equation (1.3) produces the series of income Gini coefficients from 1979:01 to 2015:02.

Finally, to minimize the influences due to the change in the sample size we drop the

first and last 12 coefficients which are calculated based on much smaller numbers of

CUs than other coefficients in the middle.

Figure 1.1 describes monthly Gini coefficients for total income and four sources of income from 1980:01 to 2014:02. Table 1.1 shows the standard deviation and cor- relation among those Gini coefficients. The Gini coefficients for all types of income have an upward trend during the sample period. Since wage is the primary source

13We do not restrict the sample by age, residence, or income reporting status as previous studies such as Heathcote et al. (2010) and Aguiar and Bils (2015) did. Especially, even though Fisher (2006) finds that incomplete income reporters have lower consumption (and so maybe lower income) than complete income reporters, this does not affect much Gini coefficient considering the whole distribution as shown by Fisher (2006).

15 Figure 1.1: Income Gini Coefficients 1

0.9

0.8

0.7

0.6

0.5

0.4

0.3 1980 1985 1990 1995 2000 2005 2010 2015

Note: The solid line, dotted line, dashed line, dotted line with circle markers, and dashed line with asterisk represent total, wage, business, financial, and other income Gini coefficients, respectively.

of income for most households the coefficients for total and wage income have sim- ilar patterns and they are highly correlated having 0.921 correlation coefficient as expected. The Gini coefficients for business and financial incomes are higher and less volatile than others. This reflects the fact that these incomes are highly concentrated.

16 Table 1.1: Standard Deviation and Correlation among Income Gini Coefficients Total Wage Business Financial Other Total (0.031) Wage 0.921 (0.024) Business 0.484 0.189 (0.009) Financial 0.920 0.769 0.579 (0.014) Other 0.750 0.483 0.536 0.778 (0.019) Note: The numbers in parenthesis represent the standard deviations.

1.3.3 Unit Root Tests

We examine whether the series of income Gini coefficients from 1980:01 to 2014:02 calculated above have unit roots by applying several unit root tests for univariate pro- cesses.14 Even though no economic theory suggests that the income Gini coefficients should exhibit deterministic time trends, Figure 1.1 shows that these series have an obvious trend over the sample period. Thus we include the time trend term in the test regressions. Since the standard ADF test for the unit root are found to have difficulties in distinguishing between a unit root and a near unit root process in small samples, we also execute the DF-GLS test developed by Elliott et al. (1996), which is more likely to reject the null hypothesis of a unit root against the stationary alter- native when the alternative is true. As a complement to the ADF and DF-GLS tests, we also apply the KPSS test by Kwiatkowski et al. (1992) which assumes that the given process is stationary under the null hypothesis.

Table 1.2 shows the test results. The results generally favor the hypothesis that the series of each income Gini coefficient has a unit root against the hypothesis that

14Several studies suggest that there are unit roots in various indicators for income inequality across countries and across states in US. For example, see Malinen (2012), Lin and Huang (2012), and Ho (2015).

17 Table 1.2: Unit Root Tests ADF DF-GLS KPSS Detrend Difference Detrend Difference Detrend Difference Gini coefficients Total income -2.50 -8.80∗∗ -2.07 -2.00∗ 0.13 0.08 Wage income -2.75 -17.46∗∗ -1.72 -4.54∗∗ 0.15∗ 0.08 Business income -2.07 -11.07∗∗ -1.55 -1.58 0.44∗∗ 0.10 Financial income 0.08 -18.32∗∗ -1.61 -4.15∗∗ 0.12 0.21 Other income -2.88 -9.91∗∗ -1.95 -5.40∗∗ 0.16∗ 0.18 Shadow rate -3.32 -5.69∗∗ -3.14∗ -0.88 0.15∗ 0.05 Note: ∗∗ and ∗ imply rejection of the null hypothesis at 1% and 5%, respectively. For the ADF and DF-GLS tests, the lag length is chosen based on the SIC. For the KPSS test, the long-run variance is estimated using the Bartlett kernel with Newey-West bandwidth parameter.

each series is trend stationary. Given these results, we will carry out the empirical

analysis under the assumptions that all series have unit roots.

1.4 The Response of Income Inequality to Monetary Shocks

In this section, we present two alternative econometric specifications to estimate

the effects of monetary policy shocks on income inequality. Then the evidences on

asymmetric responses of total, wage, and financial income inequalities are provided.

1.4.1 Econometric Specifications

Since Mork (1989) finds that only oil price increases have a significant impact on

GDP, the censored oil price VAR models become widely used.15 A simple bivariate case is as follows (ignoring constant terms):

p p + X + X xt = aixt−i + biyt−i + t i=1 i=1

15For example, see Hamilton (1996), Bernanke et al. (1997), Lee and Ni (2002), and Hamilton and Herrera (2004).

18 p p X + X yt = cixt−i + diyt−i + vt (1.4) i=0 i=1 + where xt ≡ max(0, xt), xt is the change rate in oil price, and yt is a variable of interest such as real GDP growth rate. This specification has been widely used because it produces ”better-looking” impulse responses: an oil price increase causes a decrease in GDP and an increase in . On the other hand, as discussed in Bernanke et al. (1997), conventional linear VAR models show that an oil price increase causes an increase in GDP or a decrease in price level, which is not consistent with the conventional wisdom. However, Kilian and Vigfusson (2011) cast doubt on the validity of censored oil price VAR models. First, the right hand side of the first equation in model (1.4) can be negative if t is a very large negative number. But this

+ is a contradiction because the left hand side, xt , is positive by definition. Second, there would be an omitted variable problem. If oil price decreases also affect yt,

+ then xt−1 would be captured by the error term vt. This means that xt−1 should be correlated with vt in the second equation and so estimates will be biased.

Kilian and Vigfusson (2011) propose and estimate an alternative model which encompasses both linear and nonlinear specifications as following: p p X X xt = αixt−i + βiyt−i + t i=1 i=1 p p p X X + X yt = γixt−i + δixt−i + ρiyt−i + vt (1.5) i=0 i=0 i=1

Using the change rate in oil price as xt and real GDP growth rate, unemployment rate,

or the change rate in gas consumption as yt, they conclude that there is no compelling

evidence against the null hypothesis of symmetric response of U.S. economy to the

unanticipated oil price change. Herrera et al. (2015) show similar results for a set of

OECD countries.

19 In this paper we consider two alternative specifications to investigate an asym- metric effect of monetary policy on inequality. In the first specification, we estimate model (1.5) with the monetary policy instrument for x and one of income Gini coef-

ficients for y. In this case, the lag specification of model (1.5) implies that the degree of inequality does not have a contemporaneous effect on monetary policy, which is reasonable given the fact that most central banks do not respond to the changes in inequality. Many studies16 provide the evidence that the federal funds rate well rep- resents monetary policy instrument. One problem with using the federal funds rate as the monetary policy instrument is that since December 2008, the funds rate has been near zero and so has not been thought of as the primary instrument of monetary policy as seen in Figure 1.2. So we use the ‘shadow rate’ recently proposed by Wu and Xia (2016) as the monetary policy instrument17 Since the series of each income

Gini coefficient and the shadow rate18 are found to have unit roots as seen in Table

1.2 they are differenced before estimation. In addition, following literature estimat- ing the effect of monetary policy shocks we assume that the direct effect of monetary policy lasts for one year, p = 12. We apply equation-by-equation OLS estimation which yields consistent estimator for the sample periods of 1980:01-2014:02 to the

16See Bernanke and Mihov (1998) among others. 17For the robustness check, we use the federal funds rate as the monetary policy instrument while excluding the periods of recent financial crisis from the sample. In general, the results are virtually same. 18Early studies employing the standard methods such as ADF test and find that the nominal interest rate has a unit root. See Nelson and Plosser (1982), Engle and Granger (1987), and Stock and Watson (1988) among others. Recent studies apply panel-based unit root tests to increase statistical power and they reach different conclusions. For example, Wu and Zhang (1996) find that short-term nominal interest rates from 12 OECD countries are stationary, but Moon and Perron (2007) find that 11 nominal interest rates from U.S. and 14 rates from Canada contain a single I(1) factor.

20 Figure 1.2: Federal Funds Rate and Wu and Xia’s Shadow Rate 20

15

10

5

0

-5 1980 1985 1990 1995 2000 2005 2010 2015

Note: The solid and dotted lines describe the effective federal funds rate and Wu and Xia’s shadow rate, respectively.

first specification as follows:

12 12 X X i ∆Shadow ratet = αi∆Shadow ratet−i + βi∆Ginit−i + t i=1 i=1 12 12 12 X X + X ∆Ginit = γi∆Shadow ratet−i + δi∆Shadow ratet−i + ρi∆Ginit−i + vt i=0 i=0 i=1 (1.6)

i where t represents a shock to the shadow rate. However, this specification has the endogeneity problem since it is well known that the Fed adjusts the federal funds rate according to economic conditions such as

GDP and inflation rate. In this case, if GDP and inflation rate affect the degree of inequality, which seems true, then model (1.6) would have the endogeneity problem

21 making the estimation of the model invalid. To reduce this endogeneity problem, we

consider an alternative specification.

The second specification uses monetary policy shocks identified by Romer and

Romer (2004). Romer and Romer derive the innovations to the federal funds rate

from 1969 to 1996 that is free of endogenous and anticipatory movements. They first

derive the intended funds rate changes around meeting of the Federal Open Market

Committee by examining the narrative record of the meetings. The intended funds

rate changes are considered to be free of endogenous movements. Then Romer and

Romer regress the intended funds rate changes on the Federal Reserve’s forecasts

to eliminate the anticipatory movement in the intended funds rate changes. The

residuals are considered monetary policy shocks. Barakchian an Crowe (2013) extend

the series of Romer and Romer’s monetary policy shocks until June 2008. In the

second specification we estimate the effect of monetary policy shocks identified by

Romer and Romer on income inequality:

12 12 12 X RR X RR,+ X ∆Ginit = γit−i + δit−i + ρi∆Ginit−i + vt (1.7) i=0 i=0 i=1

RR where t represents monetary policy shocks identified by Romer and Romer (2004) and extended by Barakchian an Crowe (2013).

1.4.2 Impulse Responses

Figures 1.3 and 1.4 show the impulse responses of income Gini coefficients to mon- etary policy shocks in two specifications. For easy comparison, the impulse responses to the expansionary shock are shown as mirror images. As noted by Koop et al.

(1996), the impulse responses in nonlinear model depend on the history of the obser- vations and on the magnitude of the shock. The impulse responses are constructed

22 by Monte Carlo integration over all possible paths of the data following Kilian and

Vigfusson (2011) and Herrera et al. (2015).19 Also in this subsection we estimate

the effects of monetary policy shocks with a size of one standard deviation shown on

the left-hand side. Below we consider the responses to two standard deviation shocks

shown on the right-hand side to examine the existence of size asymmetry.

Figure 1.3 shows the impulse responses of each income Gini coefficient to monetary

policy rate shocks in a two-variable model (1.6). Panel (a) shows that increases

(decreases) in the policy rate raise (reduce) the degree of income inequality. This

result is consistent with Coibion et al. (2012) and Mumtaz et al. (2016) finding that

a contractionary (expansionary) monetary policy worsens (reduces) income inequality.

Panel (b) implies that increases (decreases) in the policy rate worsen (reduce) the wage

income inequality. This finding supports earnings heterogeneity channel which argues

that contractionary monetary policy decreases the employment and labor earning at

the bottom rather than at the top and so widens inequality. Note that the response

of the wage income inequality is very similar to the response of the total income

inequality in Panel (a). This reflects the fact that wage income takes most part of the

total income for the most people. Panel (c) shows that increases (decreases) in the

policy rate also worsen (reduce) the business income inequality.20 Panel (d) shows that increases (decreases) in the policy rate reduce (worsen) the financial income inequality. Financial income covers various types of incomes on which monetary policy has different effects. For example, following a contractionary monetary policy

19See Appendix A.3 for details. 20This is consistent with Gertler and Gilchrist (1994) who find that monetary policy affects the sales of small firms more than that of large firms. Thus, when the interest rate increases the sales of small firms decline more than that of large firms, which leads to increases in business income inequality.

23 Figure 1.3: Impulse Responses of Income Gini Coefficients to Monetary Policy Shocks in a Two-variable Model (a) Total income ×10-3 <1 s.d. shock> ×10-3 <2 s.d. shock> 3 3 2 2 1 1

10 20 30 40 50 10 20 30 40 50 (b) Wage income ×10-3 ×10-3 2 2 1 1 0 0 10 20 30 40 50 10 20 30 40 50 (c) Business income ×10-4 ×10-4 6 6 4 4 2 2 0 0 10 20 30 40 50 10 20 30 40 50 (d) Financial income ×10-4 ×10-4 0 0 -5 -5 -10 -10 10 20 30 40 50 10 20 30 40 50 (e) Other income ×10-3 ×10-3

2 2 1 1

10 20 30 40 50 10 20 30 40 50

Note: The solid and dotted lines describe the impulse responses to contractionary and expansionary monetary policy shocks in model (1.6). The responses to an expansionary shock are multiplied by −1. The impulse responses are calculated by Monte Carlo integration over 100 histories with 1,000 paths each.

24 Figure 1.4: Impulse Responses of Income Gini Coefficients to Romer and Romer Monetary Policy Shocks

(a) Total income ×10-3 <1 s.d. shock> ×10-3 <2 s.d. shock> 2 2 1 1 0 0

10 20 30 40 50 10 20 30 40 50 (b) Wage income ×10-4 ×10-4 20 20 10 10 0 0

10 20 30 40 50 10 20 30 40 50 (c) Business income ×10-4 ×10-4 5 5 0 0 -5 -5 10 20 30 40 50 10 20 30 40 50 (d) Financial income ×10-4 ×10-4 2 2 0 0 -2 -2 10 20 30 40 50 10 20 30 40 50 (e) Other income ×10-4 ×10-4 15 15 10 10 5 5 0 0 10 20 30 40 50 10 20 30 40 50

Note: The solid and dotted lines describe the impulse responses to the Romer and Romer contractionary and expansionary monetary policy shocks in model (1.7). See the note for Figure 1.3.

25 interest income would increase while rental income would decrease. The responses in Panel (d) imply that the effect of the latter dominates that of the former. Panel

(e) shows that increases (decreases) in the policy rate raise (reduce) other income inequality. This is due to the fact that transfers which takes most part of other income are the main sources of income for low income households and they tend to be countercyclical. Thus increases in the policy rate slow down the economy and make other income more concentrated to low income households, which leads to increasing other income inequality.

Note that the effect of a contractionary shock on total, wage, business, and finan- cial income inequalities is larger than the effect of an expansionary shock, suggesting asymmetric effects on these income inequalities.

Figure 1.4 shows the impulse responses of each income Gini coefficient to the

Romer and Romer monetary policy shocks in model (1.7). Similar to Figures 1.3, the responses of total, wage, and business income inequalities to contractionary shocks is larger than the effect of expansionary shocks while the response of other income inequality shows no asymmetry. Contrast to Figures 1.3, the response of financial income inequality is almost symmetric. The response of other income inequality is almost symmetric as in Figure 1.3.

Two main findings robust to two alternative specifications are follows. First, contractionary monetary policy shocks are likely to worsen total, wage, business, and other income inequalities while contractionary shocks are likely to reduce financial income inequalities. Second, the responses of total and wage income inequalities to contractionary shocks are larger than those to expansionary shocks, suggesting that there exists asymmetry in the responses of those income inequalities.

26 1.4.3 Symmetry Tests for Impulse Responses

This subsection examines whether the asymmetric effects of monetary policy shocks described in Figures 1.3–1.4 are significant. Even though the estimates sug- gest some degrees of asymmetry in the responses of income inequalities, especially of total and wage income inequalities, it is necessary to conduct a formal test of the symmetry of the responses since the estimates are subject to sampling uncertainty.

Following Kilian and Vigfusson (2011) who suggest the impulse-responses-based tests, we execute the Wald test for the null hypothesis of symmetric impulse response functions as follows:21

Iy(h, δ) = −Iy(h, −δ), h = 1, 2, ··· ,H (1.8)

2 Table 1.3 reports the p-values from the χH+1 distribution for the test of symmetry in the responses to one standard deviation monetary shocks for horizons H=0, 1, ··· ,

12, 18, 24. Columns (1) and (2) represent two alternative specifications shown in

Equations (1.6) and (1.7), respectively. The result shows that the null of symmetries in the responses of total, wage, and financial income inequalities is rejected at 10% level across two specifications while the null of symmetry in the response of other income inequality cannot be rejected. The null of symmetry in the responses of business income inequality is rejected only in the last specification. We cannot find any evidence on asymmetry in the responses for the horizons longer than two years.

21Kilian and Vigfusson (2011) argue that the slope-based tests involving the null hypothesis of H0 : δ0 = δ1 = ··· = δ12 are not informative in assessing the symmetry of the impulse responses. Even though we agree with their argument, we conduct the slope-based tests and the results show that the joint null hypothesis is rejected at 5% significance level except for wage and other income inequalities.

27 Table 1.3: Symmetry Tests for Impulse Responses (1 s.d.) Total Wage Business Financial Other (1) (2) (1) (2) (1) (2) (1) (2) (1) (2) 0 0.96 0.48 0.27 0.56 0.61 0.00 0.94 0.54 0.56 0.09 1 0.73 0.62 0.50 0.77 0.26 0.00 0.17 0.54 0.18 0.23 2 0.27 0.78 0.08 0.89 0.38 0.00 0.18 0.70 0.19 0.35 3 0.41 0.18 0.13 0.00 0.49 0.00 0.05 0.83 0.28 0.47 4 0.09 0.24 0.16 0.00 0.40 0.00 0.07 0.09 0.12 0.61 5 0.10 0.34 0.18 0.01 0.45 0.00 0.02 0.02 0.19 0.73 6 0.13 0.37 0.12 0.01 0.37 0.00 0.03 0.02 0.27 0.72 7 0.08 0.36 0.03 0.01 0.48 0.00 0.03 0.02 0.36 0.69 8 0.12 0.44 0.04 0.02 0.44 0.00 0.05 0.03 0.29 0.76 9 0.07 0.10 0.05 0.01 0.53 0.00 0.04 0.03 0.15 0.80 10 0.07 0.02 0.08 0.00 0.59 0.00 0.03 0.03 0.09 0.82 11 0.07 0.03 0.05 0.00 0.59 0.00 0.02 0.05 0.12 0.86 12 0.09 0.06 0.08 0.01 0.67 0.00 0.01 0.07 0.08 0.83 18 0.38 0.15 0.33 0.06 0.94 0.04 0.09 0.32 0.35 0.98 24 0.70 0.43 0.68 0.23 0.99 0.17 0.31 0.66 0.67 1.00 2 Note: Tests are based on 500 simulations. p-values are based on the χH+1 distribution. Boldface indicates statistical significance at the 10% level.

1.4.4 Size Asymmetry

This subsection investigates whether there exists the size asymmetry. Since the sign and size asymmetries could be related to each other, it is meaningful to examine both of them. For instance, even if the sign asymmetry exists in a small sized shock, it would not in a large sized shock.

The right-hand sides in Figures 1.3 and 1.4 shows the responses of income in- equality when the size of a shock increases to two standard deviation. Compared to the responses to one standard deviation, the estimates are almost the same. Table

2 1.4 reports the p-values from the χH+1 distribution for the test of symmetry in the responses to two standard deviation monetary shocks. Again the results are similar

28 Table 1.4: Symmetry Tests for Impulse Responses (2 s.d.) Total Wage Business Financial Other (1) (2) (1) (2) (1) (2) (1) (2) (1) (2) 0 0.96 0.50 0.28 0.52 0.60 0.00 0.94 0.52 0.57 0.08 1 0.74 0.63 0.50 0.74 0.26 0.00 0.19 0.52 0.19 0.23 2 0.28 0.78 0.08 0.88 0.38 0.00 0.21 0.68 0.21 0.34 3 0.43 0.17 0.14 0.00 0.48 0.00 0.07 0.82 0.31 0.47 4 0.09 0.24 0.18 0.00 0.39 0.00 0.10 0.10 0.14 0.61 5 0.10 0.33 0.20 0.01 0.42 0.00 0.03 0.02 0.21 0.73 6 0.13 0.37 0.13 0.01 0.35 0.00 0.05 0.02 0.29 0.72 7 0.08 0.37 0.03 0.01 0.45 0.00 0.05 0.02 0.39 0.69 8 0.12 0.45 0.04 0.02 0.40 0.00 0.07 0.03 0.31 0.76 9 0.07 0.11 0.05 0.01 0.50 0.00 0.06 0.03 0.16 0.80 10 0.07 0.03 0.08 0.00 0.56 0.00 0.04 0.03 0.09 0.82 11 0.06 0.03 0.05 0.00 0.55 0.00 0.02 0.04 0.13 0.86 12 0.09 0.03 0.08 0.01 0.63 0.00 0.02 0.06 0.08 0.83 18 0.36 0.16 0.32 0.05 0.92 0.04 0.12 0.30 0.35 0.98 24 0.69 0.45 0.68 0.23 0.99 0.15 0.35 0.63 0.68 1.00 Note: See the note for Table 1.3.

to Table 1.3. That is, the null of symmetries in the responses of total, wage, and

financial income inequalities is rejected at 10% level across two specifications.

This implies that there is no size asymmetry and thus the results suggesting the existence of the sign asymmetry in a small sized shock are robust.

1.5 The Responses of Total and Wage Income Inequalities in Top and Bottom Income Brackets

In this section, we examine whether there exists asymmetry in the responses of income inequality in top and bottom income brackets. We focus on total and wage income inequalities since their responses show a large degree of asymmetry. Specifi- cally, we compute the total and wage income Gini coefficients in the top and bottom

29 Figure 1.5: Impulse Responses of Income Gini Coefficients in Top and Bottom Brack- ets to Monetary Policy Shocks in a Two-variable VAR (1 s.d.) (a) Total income ×10-3 ×10-3 6 6

4 4

2 2

0 0 10 20 30 40 50 10 20 30 40 50 (b) Wage income ×10-3 ×10-3 4 4

3 3

2 2

1 1

0 0

10 20 30 40 50 10 20 30 40 50

Note: The solid and dotted lines describe the impulse responses to con- tractionary and expansionary monetary policy shocks in model (1.6), re- spectively. The responses to an expansionary shock are multiplied by −1. The impulse responses are calculated by Monte Carlo integration over 100 histories with 1,000 paths each.

50% and estimate the responses to the alternative monetary policy shocks specified in Equations (1.6) and (1.7).

Figure 1.5 shows the impulse responses of total and wage income Gini coefficients in the top and bottom 50% to a one standard deviation monetary shock in a two- variable model (1.6). The responses of total income inequality in Panel (a) and wage

30 Figure 1.6: Impulse Responses of Income Gini Coefficients in Top and Bottom Brack- ets to Romer and Romer Monetary Policy Shocks (1 s.d.) (a) Total income ×10-3 ×10-3

4 4

2 2

0 0

10 20 30 40 50 10 20 30 40 50 (b) Wage income ×10-3 ×10-3 4 4

2 2

0 0

-2 -2 10 20 30 40 50 10 20 30 40 50

Note: The solid and dotted lines describe the impulse responses to the Romer and Romer contractionary and expansionary monetary policy shocks in model (1.7), respectively. See the note for Figure 1.5.

income inequality in Panel (b) show similar patterns. A contractionary shock worsens total and wage income inequalities in both of top and bottom 50%, which leads to increases in inequality for the whole population as seen in Panel (a) and (b) in Figure

1.3. Also note that the increases in inequality are more pronounced in the bottom

50% than in the top 50%, which suggests that a contractionary monetary policy affects the lower income bracket rather than the top bracket. An expansionary shock

31 reduces total and income inequalities in both brackets. Note that the responses in the bottom 50% to an expansionary shock are smaller than to a contractionary shock while the responses in the top 50% for both types of shocks are similar. This implies that the asymmetric responses of total and wage income inequalities found above are due to the asymmetric responses in the bottom 50%.

Figure 1.6 shows the impulse responses of total and wage income Gini coefficients in the top and bottom 50% to a one standard deviation shock to the Romer and

Romer monetary policy shocks in model (1.7). The results are similar to Figure 1.5.

First, a contractionary shock affects the bottom bracket more than the top bracket.

In addition, the responses in the bottom 50% to an expansionary shock are smaller than to a contractionary shock while the responses in the top 50% for both types of shocks are similar. This implies that the asymmetric responses of total and wage income inequalities are due to the asymmetric responses in the bottom 50%.

1.6 Conclusion

Unconventional monetary policy during and after the recent financial crisis makes economists and central bankers to reassess the relationship between monetary policy and income inequality. Recently Coibion et al. (2012) shed some light on this rela- tionship by finding that a contractionary monetary policy shock raises the income and that monetary policy shocks have played a non-trivial role in accounting for cyclical

fluctuations in inequality.

We shed more light on the relationship by examining the existence of the asym- metric effects of monetary policy shocks on inequality in various sources of income.

We find that monetary policy shocks have asymmetric effects on total, wage, and

32 financial income inequalities. The effects of contractionary shocks are larger than those of expansionary shocks, especially for total and wage income inequalities. The asymmetric responses of total and wage income inequalities are due to the asymmetric responses in the bottom 50%.

Our empirical results have important implications in conducting monetary policy.

Policymakers should consider the effects of monetary policy on income inequality especially when a contractionary monetary policy is needed.

The following topics will be valuable for future research. First, it will be of interest to understand the mechanism through which monetary policy shocks have asymmetric effects on income inequality and construct DSGE models including the mechanism, in the same spirit as Castillo and Montoro (2008) who incorporate nonhomothetic preferences into a standard new Keyesian DSGE model and show that monetary policy shocks have state-dependent asymmetric effects on output. Second, it will be also meaningful to examine the existence of state-dependent asymmetry in the responses of income inequality to monetary policy. This paper investigates the existence of asymmetric effects depending on sign and size but the recent literature finds that monetary policy has state-dependent asymmetric effects on output.22 Thus it is possible that monetary policy also has state-dependent asymmetric effects on income inequality.

22See Thoma (1994), Weise (1999), Garcia and Schaller (2002), and Lo and Piger (2005) among others.

33 Chapter 2: Near Epoch Dependence in Nonlinear Dynamic Time Series Models

2.1 Introduction

In static models zt = f(xt, vt), zt is represented by a function of a finite number of

23 exogenous variables xt and error term vt. This means that if the exogenous variables and error term are mixing processes then zt would also have mixing properties. Thus zt would satisfy the LLN and CLT because mixing processes satisfy those large sample theorems. However, in typical dynamic models zt = f(zt−1, xt, vt), zt depends on an infinite number of exogenous variables and error terms as shown by a recursive substitution zt = f(zt−1, xt, vt) = f(f(zt−2, xt−1, vt−1), xt, vt) = ··· . Therefore, the mixing property of the exogenous variables and error term does not necessarily carry over to zt.

This paper examines the conditions for the near epoch dependence in nonlinear dynamic time series models. The near epoch dependence allows us to approximate a stochastic process zt by a finite sequence of vt called the base process. The near epoch dependence of zt is especially useful when the basis process vt is a mixing process.

This is because under the appropriate moment conditions zt will be a mixing process

23Below bold lowercases will denote vectors of random variables.

34 and so zt would satisfy LLNs (see P¨otscher and Prucha, 1997, Theorem 6.2, 6.3, and

6.4) and CLTs (see P¨otscher and Prucha, 1997, Theorem 10.2).

The remainder of this paper is organized as follows. Section 2 defines the con- cept of near epoch dependence. Section 3 examines the conditions for the near epoch dependence in nonlinear AR and VAR models. In Section 4, we examines the con- ditions for the near epoch dependence in the model used by Kilian and Vigfusson

(2011). Section 5 concludes.

2.2 Near Epoch Dependence

r 1/r p Pk p 1/p Let kzkr ≡ [E(|z|p) ] where | · |p represents L norm: |z|p = ( i=1 |zi| )

where k denotes the dimension of z and zi is the i-th element of z. P¨otscher and

Prucha (1997) define the concept of near epoch dependence as follows.

Definition 1 The stochastic process zt is called Lr-near epoch dependent on vt if there exist sequences of constants dt and νm such that

kzt − E(zt|vt+m, vt+m−1, ..., vt−m)kr ≤ dtνm (2.1)

−1 Pn where supn n t=1 dt < ∞ and νm → 0 as m → ∞.

This paper focuses on the case of r = 2 which implies that the conditional mean minimizes the mean square error, the left-side of Equation (2.1), among all Borel measurable functions of vt+m, vt+m−1, ..., and vt−m.

35 2.3 Nonlinear AR Model

2.3.1 Scalar Case

Consider a univariate nonlinear AR(1) model, yt = f(yt−1, t), with supt∈Z kytk2 < ∞. Suppose that

|f(y, ) − f(y0, )| ≤ L|y − y0| (2.2) holds for L ≥ 0 where L is called a Lipschitz constant. If 0 ≤ L < 1, f is said to satisfy the contraction condition.

If the contraction condition holds, then yt is near epoch dependent on t because

vm ≡ sup kyt − E(yt|t, ..., t−m)k2 t∈Z

≤ sup kyt − E(yt|t−1, ..., t−m)k2 t∈Z

= sup kf(yt−1, t) − E(f(yt−1, t)|t−1, ..., t−m)k2 t∈Z

≤ sup kf(yt−1, t) − f(E(yt−1|t−1, ..., t−m), t)k2 t∈Z

≤ L sup kyt−1 − E(yt−1|t−1, ..., t−1−(m−1))k2 t∈Z

= Lvm−1 . .

m ≤ L v0

m ≤ 2L sup kytk2 t∈Z decays exponentially. The second inequality holds because the conditional mean minimizes the mean square error.

36 Example 2 For the dynamic Tobit model, yt = max(0, φyt−1 + t), the contraction condition holds as long as |φ| < 1 because

|f(y, ) − f(y0, ))| = | max(0, φy + ) − max(0, φy0 + )|

≤ |φy +  − φy0 − |

≤ |φ||y − y0|.

Example 3 For the model with a censored regressor, yt = ρyt−1 +α max(0, yt−1)+t,

the contraction condition holds as long as max(|ρ|, |ρ + α|) < 1 because

|f(y, ) − f(y0, )| = |ρy + α max(0, y) +  − ρy0 − α max(0, y0) − |

= |ρ(y − y0) + α max(0, y) − max(0, y0)|

≤ max(|ρ|, |ρ + α|)|y − y0|.

Next consider a univariate nonlinear AR(2) model, yt = f(yt−1, yt−2, t), with

supt∈Z kytk2 < ∞. Then yt is near epoch dependent on t if

0 0 0 0 |f(y1, y2, ) − f(y1 , y2 , )| ≤ L1|y1 − y1 | + L2|y2 − y2 |, 0 ≤ L1 + L2 < 1, (2.3)

holds because

vm ≡ sup kyt − E(yt|t, ..., t−m)k2 t∈Z

≤ sup kyt − E(yt|t−1, ..., t−m)k2 t∈Z

= sup kf(yt−1, yt−2, t) − E(f(yt−1, yt−2, t)|t−1, ..., t−m)k2 t∈Z

≤ sup kf(yt−1, yt−2, t) − f(E(yt−1|t−1, ..., t−m),E(yt−2|t−2, ..., t−m), t)k2 t∈Z

≤ sup kL1|yt−1 − E(yt−1|t−1, ..., t−1−(m−1))| t∈Z

+ L2|yt−2 − E(yt−2|t−2, ..., t−2−(m−2))|k2

37 ≤ L1 sup kyt−1 − E(yt−1|t−1, ..., t−1−(m−1))k2 t∈Z

+ L2 sup kyt−2 − E(yt−2|t−2, ..., t−2−(m−2))k2 t∈Z

= L1vm−1 + L2vm−2

≤ (L1 + L2)vm−2

converges to 0 as long as 0 ≤ L1 + L2 < 1.

Example 4 For a linear univariate AR(2) model, yt = ρ1yt−1 + ρ2yt−2 + t, the

following equation holds:

0 0 0 0 |f(y1, y2, ) − f(y1 , y2 , )| ≤ |ρ1||y1 − y1 | + |ρ2||y2 − y2 |.

Thus yt is near epoch dependent on t if 0 ≤ |ρ1| + |ρ2| < 1. This condition is

stricter than the stationary condition for AR(2) models, which is that the roots of lag

polynomial should be outside the unit circle.

Example 5 For the model with the censored regressors, yt = ρ1yt−1 + ρ2yt−2 +

α1 max(0, yt−1) + α2 max(0, yt−2) + t, the following equation holds:

0 0 0 |f(y1, y2, ) − f(y1 , y2 , )| ≤ max(|ρ1|, |ρ1 + α1|)|y1 − y1 |

0 + max(|ρ2|, |ρ2 + α2|)|y2 − y2 |.

Thus yt is near epoch dependent on t if 0 ≤ max(|ρ1|, |ρ1 +α1|)+max(|ρ2|, |ρ2 +α2|) <

1.

The next Claim shows the condition for the near epoch dependence for a univariate

nonlinear AR(p) model with an arbitrary p.

38 Claim 6 For a univariate nonlinear AR(p) model, yt = f(yt−1, yt−2, ..., yt−p, t), the sufficient condition for yt to be near epoch dependent on t is

p X 0 ≤ Li < 1 (2.4) i=1 for

p 0 0 0 X 0 |f(y1, yt−2, ··· , yp, ) − f(y1 , y2 , ..., yp , )| ≤ Li|yi − yi | (2.5) i=1

Proof.

vm ≡ sup kyt − E(yt|t, ..., t−m)k2 t∈Z

≤ sup kyt − E(yt|t−1, ..., t−m)k2 t∈Z

= sup kf(yt−1, yt−2, ..., yt−p, t) − E(f(yt−1, yt−2, ..., yt−p, t)|t−1, ..., t−m)k2 t∈Z

≤ sup kf(yt−1, yt−2, ..., yt−p, t) − f(E(yt−1|t−1, ..., t−m),E(yt−2|t−2, ..., t−m), ... t∈Z

,E(yt−p|t−p, ..., t−m), t)k2

≤ sup kL1|yt−1 − E(yt−1|t−1, ..., t−1−(m−1))| t∈Z

+ L2|yt−2 − E(yt−1|t−2, ..., t−2−(m−2))|

+ ··· + Lp|yt−p − E(yt−p|t−p, ..., t−p−(m−p))|k2

≤ L1 sup kyt−1 − E(yt−1|t−1, ..., t−1−(m−1))k2 t∈Z

+ L2 sup kyt−2 − E(yt−1|t−2, ..., t−2−(m−2))k2 + ··· t∈Z

+ Lp sup kyt−p − E(yt−p|t−p, ..., t−p−(m−p))k2 t∈Z

= L1vm−1 + L2vm−2 + ··· + Lpvm−p

≤ (L1 + L2 + ··· + Lp)vm−p

Pp converges to 0 as long as 0 ≤ i=1 Li < 1.

39 Pp Example 7 For a linear univariate AR(p) model, yt = i=1 ρiyt−i +t, the following equation holds:

p 0 0 X 0 |f(y1, y2, ) − f(y1 , y2 , )| ≤ |ρi||yi − yi |. i=1 Pp Thus yt is near epoch dependent on t if 0 ≤ i=1 |ρi| < 1.

Pp Pp Example 8 For the model with the censored regressors, yt = i=1 ρiyt−i + i=1 αi

max(0, yt−i) + t, the following equation holds:

p 0 0 X 0 |f(y1, ··· , yp, ) − f(y1 , ··· , yp , )| ≤ max(|ρi|, |ρi + αi|)|yi − yi |. i=1 Pp Thus yt is near epoch dependent on t if 0 ≤ i=1 max(|ρi|, |ρi + αi|) < 1.

2.3.2 Multivariate Case

Multivariate models are widely used to capture the correlation among economic

variables. Consider a nonlinear VAR(1) model, zt = f(zt−1, vt), with supt∈Z k|zt|p1 k2 < ∞ and let k denote the dimension of z. The contraction condition on f is

0 0 |f(z, v) − f(z , v)|p1 ≤ L|z − z |p2 , 0 ≤ L < 1, (2.6)

While p1 and p2 are same as 1 in the univariate models, they can take different

values in the multivariate models. Thus it would be useful to see the relationship

among the Lp norm for different p’s.

k (1/r−1/p) Lemma 9 For any z ∈ R , |z|p ≤ |z|r ≤ k |z|p for 0 < r < p.

Proof. The first inequality holds because r/p < 1 and

k k k X p 1/p X p r/p 1/r X p(r/p) 1/r |z|p = ( |zi| ) = (( |zi| ) ) ≤ ( |zi| ) = |z|r i=1 i=1 i=1

40 where the inequality holds because (x + y)n ≤ xn + yn for x, y ≥ 0 and 0 < n < 1.

Now let q ≡ p/r > 1 and then g(x) = |x|q is a convex function. Jensen’s inequality

implies

k k 1 X 1 X ( |z |r)q ≤ (|z |r)q. k i k i i=1 i=1

Raising both sides to the power of 1/qr and noting qr = p, we have

k k 1 X 1 X ( |z |r)1/r ≤ ( |z |p)1/p, k i k i i=1 i=1

implying

(1/r−1/p) |z|r ≤ k |z|p.

Now we examine the condition for the near epoch dependence for a nonlinear

VAR(1) model. Similar to the scalar cases, we have

vm ≡ sup k|zt − E(zt|vt, ..., vt−m)|p1 k2 t∈Z

≤ sup k|zt − E(zt|vt−1, ..., vt−m)|p1 k2 t∈Z

= sup k|f(zt−1, vt) − E(f(zt−1, vt)|vt−1, ..., vt−m)|p1 k2 t∈Z

≤ sup k|f(zt−1, vt) − f(E(zt−1|vt−1, ..., vt−m), vt)|p1 k2 t∈Z

≤ L sup k|zt−1 − E(zt−1|vt−1, ..., vt−m)|p2 k2. t∈Z

In the case of a univariate AR(1), we could replace the last term with vm−1 as shown in the previous section. This is not possible here because p2 can be different from p1.

There are two possible cases. In the case of p1 ≤ p2, the first inequality in Lemma

8 holds and so the contraction condition 0 ≤ L < 1 is sufficient for the near epoch

41 dependence of zt on vt because

vm ≤ L sup k|zt−1 − E(zt−1|vt−1, ..., vt−m)|p2 k2 t∈Z

≤ L sup k|zt−1 − E(zt−1|vt−1, ..., vt−m)|p1 k2 t∈Z

= Lvm−1 . .

m ≤ L v0

m ≤ 2L sup k|zt|p1 k2 t∈Z

decays exponentially. In the other case of p1 > p2, we have

vm ≤ L sup k|zt−1 − E(zt−1|vt−1, ..., vt−m)|p2 k2 t∈Z

(1/p2−1/p1) ≤ k L sup k|zt−1 − E(zt−1|vt−1, ..., vt−m)|p1 k2 t∈Z

(1/p2−1/p1) = k Lvm−1 . .

(1/p2−1/p1) m ≤ (k L) v0

(1/p2−1/p1) m ≤ 2(k L) sup k|zt|p1 k2. t∈Z

Therefore in order for the near epoch dependence of zt on vt, L should be less than

1/k(1/p2−1/p1) implying that the contraction condition L < 1 is not sufficient for the near epoch dependence.

Next consider a nonlinear VAR(2) model, zt = f(zt−1, zt−2, vt), with supt∈Z k|zt|p1 k2 < ∞ and suppose

0 0 0 0 |f(z1, z2, v) − f(z1, z2, v)|p1 ≤ L1|z1 − z1|p2 + L2|z2 − z2|p2 . (2.7)

42 Similar to the scalar case, we have

vm ≡ sup k|zt − E(zt|vt, ..., vt−m)|p1 k2 t∈Z

≤ sup k|zt − E(zt|vt−1, ..., vt−m)|p1 k2 t∈Z

= sup k|f(zt−1, zt−2, vt) − E(f(zt−1, zt−2, vt)|vt−1, ..., vt−m)|p1 k2 t∈Z

≤ sup k|f(zt−1, zt−2, vt) − f(E(zt−1|vt−1, ..., vt−m),E(zt−2|vt−2, ..., vt−m), vt)|p1 k2 t∈Z

≤ sup kL1|zt−1 − E(zt−1|vt−1, ..., vt−1−(m−1))|p2 t∈Z

+ L2|zt−2 − E(zt−2|vt−2, ..., vt−2−(m−2))|p2 k2

≤ L1 sup k|zt−1 − E(zt−1|vt−1, ..., vt−1−(m−1))|p2 k2 t∈Z

+ L2 sup k|zt−2 − E(zt−2|vt−2, ..., vt−2−(m−2))|p2 k2. t∈Z

In the case of p1 ≤ p2, zt is near epoch dependent on vt as long as 0 ≤ L1 + L2 < 1 because

vm ≤ L1 sup k|zt−1 − E(zt−1|vt−1, ..., vt−1−(m−1))|p2 k2 t∈Z

+ L2 sup k|zt−2 − E(zt−2|vt−2, ..., vt−2−(m−2))|p2 k2 t∈Z

≤ L1 sup k|zt−1 − E(zt−1|vt−1, ..., vt−1−(m−1))|p1 k2 t∈Z

+ L2 sup k|zt−2 − E(zt−2|vt−2, ..., vt−2−(m−2))|p1 k2 t∈Z

= L1vm−1 + L2vm−2

≤ (L1 + L2)vm−2

converges to 0. In the other case of p1 > p2,

vm ≤ L1 sup k|zt−1 − E(zt−1|vt−1, ..., vt−1−(m−1))|p2 k2 t∈Z

+ L2 sup k|zt−2 − E(zt−2|vt−1, ..., vt−2−(m−2))|p2 k2 t∈Z

(1/p2−1/p1) ≤ k L1 sup k|zt−1 − E(zt−1|vt−1, ..., vt−1−(m−1))|p1 k2 t∈Z 43 (1/p2−1/p1) + k L2 sup k|zt−2 − E(zt−2|vt−1, ..., vt−2−(m−2))|p1 k2 t∈Z

(1/p2−1/p1) ≤ k L1 sup k|zt−1 − E(zt−1|vt−1, ..., vt−1−(m−1))|p1 k2 t∈Z

(1/p2−1/p1) + k L2 sup k|zt−2 − E(zt−2|vt−2, ..., vt−2−(m−2))|p1 k2 t∈Z

(1/p2−1/p1) (1/p2−1/p1) = k L1vm−1 + k L2vm−2

(1/p2−1/p1) ≤ k (L1 + L2)vm−2

(1/p2−1/p1) converges to 0 as long as 0 ≤ L1 + L2 < 1/(k ).

The next Claim shows the condition for the near epoch dependence for a nonlinear

VAR(p) model with an arbitrary p.

Claim 10 For a nonlinear VAR(p) model, zt = f(zt−1, zt−2, ..., zt−p, vt), the suffi- cient condition for zt to be near epoch dependent on vt is

 Pp 0 ≤ i=1 Li < 1 if p1 ≤ p2 Pp (1/p2−1/p1) (2.8) 0 ≤ i=1 Li < 1/(k ) if p1 > p2 for

p 0 0 0 X 0 |f(z1, zt−2, ..., zp, v) − f(z1, zt−2, ..., zp, v)|p1 ≤ Li|zi − zi|p2 . (2.9) i=1

Proof. The proof is similar to VAR(2) model.

vm ≡ sup k|zt − E(zt|vt, ..., vt−m)|p1 k2 t∈Z

≤ sup k|zt − E(zt|vt−1, ..., vt−m)|p1 k2 t∈Z

= sup k|f(zt−1, zt−2, ..., zt−p, vt) − E(f(zt−1, zt−2, ..., zt−p, vt)|vt−1, ..., vt−m)|p1 k2 t∈Z

≤ sup k|f(zt−1, zt−2, ..., zt−p, vt) − f(E(zt−1|vt−1, ..., vt−m),E(zt−2|vt−2, ..., vt−m), t∈Z

··· ,E(zt−p|vt−p, ..., vt−m), vt)|p1 k2

≤ sup kL1|zt−1 − E(zt−1|vt−1, ..., vt−1−(m−1))|p2 t∈Z 44 + L2|zt−2 − E(zt−2|vt−2, ..., vt−2−(m−2))|p2 + ···

+ Lp|zt−p − E(zt−p|vt−p, ..., vt−p−(m−p))|p2 k2

≤ L1 sup k|zt−1 − E(zt−1|vt−1, ..., vt−1−(m−1))|p2 k2 t∈Z

+ L2 sup k|zt−2 − E(zt−2|vt−2, ..., vt−2−(m−2))|p2 k2 + ··· t∈Z

+ Lp sup k|zt−p − E(zt−p|vt−p, ..., vt−p−(m−p))|p2 k2. t∈Z

In the case of p1 ≤ p2,

vm ≤ L1 sup k|zt−1 − E(zt−1|vt−1, ..., vt−1−(m−1))|p2 k2 t∈Z

+ L2 sup k|zt−2 − E(zt−2|vt−2, ..., vt−2−(m−2))|p2 k2 + ··· t∈Z

+ Lp sup k|zt−p − E(zt−p|vt−p, ..., vt−p−(m−p))|p2 k2 t∈Z

≤ L1 sup k|zt−1 − E(zt−1|vt−1, ..., vt−1−(m−1))|p1 k2 t∈Z

+ L2 sup k|zt−2 − E(zt−2|vt−2, ..., vt−2−(m−2))|p1 k2 + ··· t∈Z

+ Lp sup k|zt−p − E(zt−p|vt−p, ..., vt−p−(m−p))|p1 k2 t∈Z

= L1vm−1 + L2vm−2 + ··· + Lpvm−p

≤ (L1 + L2 + ··· + Lp)vm−p

Pp converges to 0 as long as 0 ≤ i=1 Li < 1. In the other case of p1 > p2,

vm ≤ L1 sup k|zt−1 − E(zt−1|vt−1, ..., vt−1−(m−1))|p2 k2 t∈Z

+ L2 sup k|zt−2 − E(zt−2|vt−2, ..., vt−2−(m−2))|p2 k2 + ··· t∈Z

+ Lp sup k|zt−p − E(zt−p|vt−p, ..., vt−p−(m−p))|p2 k2 t∈Z

(1/p2−1/p1) ≤ k L1 sup k|zt−1 − E(zt−1|vt−1, ..., vt−1−(m−1))|p1 k2 t∈Z

(1/p2−1/p1) + k L2 sup k|zt−2 − E(zt−2|vt−2, ..., vt−2−(m−2))|p1 k2 + ··· t∈Z

(1/p2−1/p1) + k Lp sup k|zt−p − E(zt−p|vt−p, ..., vt−p−(m−p))|p1 k2 t∈Z 45 (1/p2−1/p1) (1/p2−1/p1) (1/p2−1/p1) = k L1vm−1 + k L2vm−2 + ··· + k Lpvm−p

(1/p2−1/p1) ≤ k (L1 + L2 + ··· + Lp)vm−p

Pp (1/p2−1/p1) converges to 0 as long as 0 ≤ i=1 Li < 1/k .

2.4 Application to the Kilian and Vigfusson Model

In this section we examine the condition for the near epoch dependence for the

Kilian and Vigfusson (2011) model.

2.4.1 The Kilian and Vigfusson Model

One common view among the economists is that the oil price is an important factor affecting the economy. This view was supported by the fact that the recessions were followed by two oil price shocks in 1970s and 1980s. But as discussed in Bernanke et al. (1997) this view was not supported by linear VAR models. The VAR estimation shows that oil price shocks do not affect GDP significantly. Even more for some sample periods an oil price increase causes an increase in GDP or a decrease in price level, which is not consistent with the conventional wisdom.

However, Mork (1989) found the empirical evidence that the effect of oil shock is not symmetric: only oil price increases have a significant impact on GDP. Following him, censored oil price VAR models become widely used24:

p p + X + X xt = ρixt−i + γiyt−i + ut (2.10) i=1 i=1 p p X + X yt = βxt−i + φiyt−i + t (2.11) i=1 i=1

24For example, see Hamilton (1996), Bernanke et al. (1997), Lee and Ni (2002), and Hamilton and Herrera (2004).

46 + where xt ≡ max(0, xt), xt is the change rate in oil price, and yt is a variable of interest

such as real GDP growth rate. One reason why this specification is popular is that it

produces ”better-looking” impulse responses: an oil price increase causes a decrease

in GDP and an increase in price level.

However, Kilian and Vigfusson (2011) point out two problems of this specification.

First, the right hand side of Equation (2.10) can be negative if ut is a very large

+ negative number. But this is a contradiction because the left hand side, xt , is positive by definition. Second, there would be an omitted variable problem. If oil price decreases also affect yt, then xt−1 would be captured by the error term t. This means

+ that xt−1 should be correlated with t in the second equation and so estimates will be biased.

Kilian and Vigfusson propose and estimate an alternative specification without these problems as following:

p p X X xt = ρixt−i + γiyt−i + ut (2.12) i=1 i=1 p p p X + X X yt = βixt−i + αixt−i + φiyt−i + t (2.13) i=1 i=1 i=1

Using the change rate in oil price as xt and real GDP growth rate, unemployment rate, or the change rate in gas consumption as yt, they concluded that there is no compelling evidence against the null hypothesis of symmetric response of U.S. economy to the unanticipated oil price change.

2.4.2 Near Epoch Dependence Conditions with One Lag

In this section, we derive the condition for the near epoch dependence for the

Kilian and Vigfusson model with one lag (p = 1 in Equations (2.12) and (2.13)).

47 We consider nine cases, pi = 1, 2, ∞ for i = 1, 2 (Equation (2.6)) and find the most

0 0 general one among them. Below let zt ≡ (xt yt) and vt ≡ (ut t) .

We start with the case of p1 = 1. For p2 = 1, 2, ∞,

0 |f(z, v) − f(z , v)|1   0 max |ρ| + max(|α|, |α + β|), |γ| + |φ| · |z − z |1  1/2 h 2 2i 0 ≤ max 2 |ρ| + max(|α|, |α + β|) , 2(|γ| + |φ|) · |z − z |2   0 |ρ| + max(|α|, |α + β|) + |γ| + |φ| · |z − z |∞

25 holds. Let L1,L2,L3 be Lipschitz constants for p2 = 1, 2, ∞, respectively. Since p1

is always less than or equal to p2, the contraction condition 0 ≤ Li < 1 is sufficient

for the near epoch dependence.

Claim 11 The condition 0 ≤ L1 < 1 is the most general condition for the near epoch

dependence in the case of p1 = 1. That is, the parameter set satisfying 0 ≤ L1 < 1

includes the set satisfying 0 ≤ L2 < 1 or 0 ≤ L3 < 1:

{α, β, γ, ρ, φ|0 ≤ L1 < 1} ⊇ {α, β, γ, ρ, φ|0 ≤ L2 < 1 or 0 ≤ L3 < 1}

Proof. Suppose 0 ≤ L2 < 1. Then,

√ √ 0 ≤ L2 < 1 ⇒ 0 ≤ |ρ| + max(|α|, |α + β|) < 1/ 2 and 0 ≤ |γ| + |φ| < 1/ 2 √ ⇒ 0 ≤ max |ρ| + max(|α|, |α + β|), |γ| + |φ| < 1/ 2

⇒ 0 ≤ L1 < 1

Similarly, if 0 ≤ L3 < 1, then

0 ≤ L3 < 1 ⇒ 0 ≤ |ρ| + max(|α|, |α + β|) < 1 and 0 ≤ |γ| + |φ| < 1

⇒ 0 ≤ max |ρ| + max(|α|, |α + β|), |γ| + |φ| < 1

25For the details of derivation, see Appendix B.1.

48 ⇒ 0 ≤ L1 < 1

.

Next consider the case of p1 = 2. If p2 = 1, p1 is greater than p2 and Lemma 9

0 1/2 0 implies |z − z |1 ≤ 2 |z − z |2. Thus

0 |f(z, v) − f(z , v)|2  1/2 h 2 2i 0  max |ρ|, |γ| + max max(|α|, |α + β|), |φ| · |z − z |1  | {z }  1/2 0  ≤2 |z−z |2 ≤ h i1/2 max 2(|ρ|2 + max(|α|, |α + β|)2), 2(|γ|2 + |φ|2) · |z − z0|  2  1/2  h 2 2i 0  |ρ| + |γ| + max(|α|, |α + β|) + |φ| · |z − z |∞ holds. Let L4,L5,L6 be Lipschitz constants for p2 = 1, 2, ∞, respectively and let

0 1/2 L4 = 2 L4.

Claim 12 The parameter set satisfying 0 ≤ L1 < 1 includes the set satisfying 0 ≤

0 L4 < 1 or 0 ≤ L5 < 1 and the parameter set satisfying 0 ≤ L9 < 1 includes the set satisfying 0 ≤ L6 < 1:

0 {α, β, γ, ρ, φ|0 ≤ L1 < 1} ⊇ {α, β, γ, ρ, φ|0 ≤ L4 < 1 or 0 ≤ L5 < 1}

{α, β, γ, ρ, φ|0 ≤ L9 < 1} ⊇ {α, β, γ, ρ, φ|0 ≤ L6 < 1}

 where L9 ≡ max |ρ| + |γ|, max(|α|, |α + β|) + |φ| is defined below as the Lipschitz constant for p1 = p2 = 3.

Proof. It is easy to prove these claims by contradiction. First, we show that L1 ≥ 1

0 implies L4 ≥ 1. If L1 ≥ 1, then |ρ| + max(|α|, |α + β|) ≥ 1 or |γ| + |φ| ≥ 1. Without loss of generality, suppose |ρ| + max(|α|, |α + β|) ≥ 1. Then note that

0  2 2 1/2 L4 ≥ 2(|ρ| + max(|α|, |α + β|) ) ≥ |ρ| + max(|α|, |α + β|) ≥ 1.

49 The other two claims can be proved as follows.

L1 ≥ 1 ⇒ |ρ| + max(|α|, |α + β|) ≥ 1 or |γ| + |φ| ≥ 1

⇒ 2(|ρ|2 + max(|α|, |α + β|)2)1/2 ≥ 1 or 2(|γ|2 + |φ|)21/2 ≥ 1

⇒ L5 ≥ 1

L9 ≥ 1 ⇒ |ρ| + |γ| ≥ 1 or max(|α|, |α + β|) + |φ| ≥ 1

⇒ |ρ| + |γ|2 + max(|α|, |α + β|) + |φ|2 ≥ 1

⇒ L6 ≥ 1

Finally consider the case of p1 = ∞. In the case of p2 = 1, 2, p1 is greater than p2

0 0 0 1/2 0 and Lemma 9 implies |z − z |1 ≤ 2|z − z |∞ and |z − z |2 ≤ 2 |z − z |∞. Thus

0 |f(z, v) − f(z , v)|∞     0  max max |ρ|, |γ| , max max(|α|, |α + β|), |φ| · |z − z |1   | {z 0 }  ≤2|z−z |∞    ≤  2 2 1/2  2 2 1/2 0 max max(2|ρ| , 2|γ| ) , max(2 max(|α|, |α + β|) , 2|φ| ) · |z − z |2  | {z }  1/2 0  ≤2 |z−z |∞   0  max |ρ| + |γ|, max(|α|, |α + β|) + |φ| · |z − z |∞

holds. Let L7,L8,L9 be Lipschitz constants for p2 = 1, 2, ∞, respectively and let

0 0 1/2 L7 = 2L7 and L8 = 2 L8.

Claim 13 The condition 0 ≤ L9 < 1 is the most general condition for the case of

p1 = 3. That is, the parameter set satisfying 0 ≤ L9 < 1 includes the set satisfying

0 0 0 ≤ L7 < 1 or 0 ≤ L8 < 1:

0 0 {α, β, γ, ρ, φ|0 ≤ L9 < 1} ⊇ {α, β, γ, ρ, φ|0 ≤ L7 < 1 or 0 ≤ L8 < 1}

0 0 0 Proof. First, note that L7 = L8. Suppose 0 ≤ L7 < 1. Then,

0 0 ≤ L7 < 1 ⇒ 0 ≤ |ρ| < 1/2, 0 ≤ |γ| < 1/2, 0 ≤ max(|α|, |α + β|) < 1/2, and

50 0 ≤ |φ| < 1/2

⇒ 0 ≤ max |ρ| + |γ|, max(|α|, |α + β|) + |φ| < 1

⇒ 0 ≤ L9 < 1

Therefore, the most general conditions for the near epoch dependence, or the broadest parameter sets leading to the weak dependence, for the Kilian and Vigfusson model with one lag are

 0 ≤ L1 = max |ρ| + |β| + |α|, |γ| + |φ| < 1 and  0 ≤ L9 = max |ρ| + |γ|, max(|α|, |α + β|) + |φ| < 1.

The first condition requires that the sum of absolute values of coefficients for lagged x and for lagged y across two Equations (2.12) and (2.13) should be less than one. The second condition requires that the sum of absolute values of coefficients for lagged x and y in each Equation (2.12) and (2.13) should be less than one.

2.4.3 Near Epoch Dependence Conditions with Arbitrary Lags

In the section, we extend the results earned in the previous section to the general case. We derive the condition for the near epoch dependence for the Kilian and

Vigfusson model with arbitrary lags. We consider nine cases, pi = 1, 2, ∞ for i = 1, 2 and find the most general one among them.

First, in the case of p1 = 1,

0 0 |f(z1, ··· , zp, v) − f(z1, ··· , zp, v)|1

51  Pp max |ρ | + max(|α |, |α + β |), |γ | + |φ | ·|z − z0 |  i=1 i i i i i i i i 1  | {z }  ≡L1i  h i1/2  Pp max 2|ρ | + max(|α |, |α + β |)2, 2(|γ | + |φ |)2 ·|z − z0 | ≤ i=1 i i i i i i i i 2  | {z }  ≡L2i  Pp |ρ | + max(|α |, |α + β |) + |γ | + |φ | ·|z − z0 |  i=1 i i i i i i i i ∞  | {z } ≡L3i

26 holds. Next in the case of p1 = 2,

0 0 |f(z1, ··· , zp, v) − f(z1, ··· , zp, v)|2  h 2 2i1/2  max max(|ρi|), max(|γi|) + max max(max(|αi|, |αi + βi|)), max(|φi|)   | {z }  ≡L4  Pp 0  · i=1 |zi − zi|1  | {z }  1/2 0  ≤2 |zi−zi|2  1/2 h 2 2 2 2i ≤ 2p max max(|ρi|) + max(max(|αi|, |αi + βi|)) , max(|γi|) + max(|φi|)   | {z }  ≡L5  Pp 0  · i=1 |zi − zi|2  h i1/2  2 2 Pp 0  max(|ρi| + |γi|) + max(max(|αi|, |αi + βi|) + |φi|) · i=1 |zi − zi|∞   | {z } ≡L6 holds. Finally in the case of p1 = ∞,

0 0 |f(z1, ··· , zp, v) − f(z1, ··· , zp, v)|∞      max max max(|ρi|), max(|γi|) , max max(max(|αi|, |αi + βi|)), max(|φi|)   | {z }  ≡L7  Pp 0  · i=1 |zi − zi|1  | {z }  ≤2|z −z0 |  i i ∞     max 21/2 max(max(|ρ |2, |γ |2)1/2), 21/2 max(max(max(|α |, |α + β |)2, |φ |2)1/2) ≤ i i i i i i | {z }  ≡L8  Pp 0  · i=1 |zi − zi|2  | {z }  ≤21/2|z −z0 |  i i ∞       Pp 0  max max |ρi| + |γi| , max max(|αi|, |αi + βi|) + |φi| · i=1 |zi − zi|∞   | {z } ≡L9

26For the details of derivation, see Appendix B.2.

52 holds.

Using the results for one lag in the previous section, we can conclude that the most general conditions for the near epoch dependence are

p p X X  0 ≤ L1i = max |ρi| + max(|αi|, |αi + βi|), |γi| + |φi| < 1 and i=1 i=1 p X  0 ≤ L9 = p max max(|ρi| + |γi|), max(max(|αi|, |αi + βi|) + |φi|) < 1. i=1

Broadly speaking, the first condition restricts the sum of absolute values of coefficients for lagged x and for lagged y across two equations (2.12) and (2.13). The second condition restricts the sum of absolute values of coefficients for lagged x and y in each Equation (2.12) and (2.13).

2.5 Conclusion

This paper examines the conditions for the near epoch dependence in nonlinear dynamic time series models. Then we derive the conditions for the near epoch de- pendence for the Kilian and Vigfusson (2011) model and find the most general one among them.

For future research it will be interesting topics to examine whether there exists a strictly stationary solution to the the Kilian and Vigfusson model and whether that solution is unique.

53 Chapter 3: Can Sign Restrictions Identify Monetary Policy Shocks in a VAR?

3.1 Introduction

The effect of monetary policy on the economy has been one of the most interesting and controversial topics among economists and central bankers. Theoretically, it is important to understand it in order to assess different competing models. The models that predict what is consistent with the empirical evidences would be more reliable than others that do not. Practically, understanding that effect is essential in conducting monetary policy. It can help policy makers make the right decisions. But it is not easy to analyze the effect of monetary policy since policy makers consider general economic conditions before making decisions and so monetary policy actions reflect all the shocks to the economy. Thus it is not appropriate to determine its effect simply by seeing the correlation between the monetary policy actions and macro variables. In order to try to avoid this endogeneity problem, the literatures have developed several methods for identifying the monetary policy shocks.

54 Broadly speaking, two different approaches have been employed to analyze the

effect of monetary shocks. The first approach is to apply vector autoregression (here-

after, VAR) models.27 Especially, VAR models with recursiveness assumption have

been widely used. The recursive assumption identifies the monetary shocks by assum-

ing that the monetary shocks do not have contemporaneous effects on the variables

in central bank’s information set. Sims (1992) provided the robust empirical evi-

dence on the effects of monetary shocks across several countries.28 Christiano et al.

(1999), focusing on results from the recursive VAR models, reviewed the literatures on the effect of monetary shocks. Christiano et al. (2005) and Boivin and Giannoni

(2006) used the impulse responses to monetary shocks in the recursive VAR models to estimate the parameters in the dynamic stochastic general equilibrium (hereafter,

DSGE) models and assess the validity of those models. There are also literatures using VAR models without recursiveness assumption. Bernanke and Mihov (1998) set up a model for bank reserve market and estimated structural VAR models to see which variable can be well represented as the policy instrument of the Fed. Bernanke et al. (2005) estimated factor-augmented VAR models to solve the limited infor- mation problem in conventional VAR models. Primiceri (2005) modeled monetary policy of U.S as a time-varying coefficient VAR model. The other approach is narra- tive approach. Romer and Romer (2004) analyzed the record of FOMC and derived the intended federal funds rate and internal forecasts of inflation and real activity to purge the endogeneity and anticipatory movements in the actual federal funds rate.

Even though these literatures adopted different methods, they provided the similar

27See Stock and Watson (2001) for the review of the VAR method. 28Especially, Sims found that the temporary price level increases are followed by contractionary monetary policies across countries. He conjectured that this price puzzle is due to the fact that monetary policy authorities have more information than those in information set in the model.

55 empirical evidences on the effect of monetary shocks, which are well summarized in

Christiano et al. (1999):

”...after a contractionary monetary shock, short term interest rates rise, aggregate

output, employment, profits and various monetary aggregate fall, the aggregate price

level responds very slowly, and various measures of wages fall, albeit by very modest

amounts...” (p69 in Christiano et al. (1999))

That is, monetary shocks are not neutral: after contraction monetary shocks, output

decreases or unemployment rate increases.

However, Uhlig (2005) identified the monetary policy shocks by adopting sign-

restrictions in a VAR model and showed the neutrality of monetary policy shocks:

”...”contractionary” monetary policy shocks do not necessarily seem to have con- tractionary effects on real GDP...” (p385 in Uhlig (2005))

In this paper, we investigate the validity of sign restrictions for identifying the monetary policy shocks in a VAR model. First, a simple version of New Keynesian

DSGE model is constructed and estimated using Bayesian techniques. The artificial data set is generated from this DSGE model and are then used in a sign-restriction

VAR model to estimate the effect of monetary shocks. The result shows that the sign restrictions do not well identify the monetary policy shocks: after a contractionary monetary policy shock, the sign-restriction VAR estimation shows that output in- creases even though it decreases in the estimated model from which the artificial data set is generated. The sign restrictions seem to well identify the monetary policy shocks only when the shocks are extremely volatile. On the other hand, the recursive

56 assumption captures the negative response of output after a contractionary monetary policy shock.

The remainder of this paper is organized as follows. Section 2 provides the mo- tivation for this paper. Section 3 describes the model economy. In Section 4, the model is estimated using Bayesian techniques. In Section 5, the simulation results are presented. Section 6 concludes.

3.2 Motivation

There are a large number of studies on various topics using sign restrictions.29 The main idea of this procedure is that it identifies certain structural shocks by imposing the signs of the responses to certain shocks based on prior beliefs and letting the data speak. In this way, this procedure recognizes explicitly the uncertainty in imposing the identification assumptions. In this section, we provide the motivation for this paper by replicating the sign-restriction VAR estimation in Uhlig (2005).

3.2.1 Uhlig’s Sign-restriction VAR Estimation

Uhlig (2005) assumed that a contractionary monetary policy shock does not cause increases in price level, increases in nonborrowed reserves, and decreases in the federal funds rate for a certain period following a shock. In other words, he identified any shocks causing decreases in price level and nonborrowed reserves and increases in federal funds rate as contractionary monetary shocks. Table 3.1 summarizes these sign restrictions. No restriction is imposed on the response of real GDP.

Consider a VAR model with p lags,

Yt = B1Yt−1 + ··· + BpYt−p + ut (3.1)

29See Blanchard and Perotti (2002), Canova and De Nicol´o(2002), Faust (1998), and Uhilg (2005).

57 Table 3.1: Sign Restrictions in Uhlig (2005) Real GDP GDP deflator Commodity price index –– Federal funds rate Nonborrowed reserves Total reserves + –

0 where the dimension of Yt is k. Denote the k fundamental shocks by vt with E(vtvt) =

I. Suppose ut = Avt implying that the jth column of A represents the immediate response of Yt to the jth fundamental shock. Uhlig (2005) assumes that the prior and

0 posterior for (B, Σ) belong to the Normal-Wishart family where B = (B1 B2 ... Bp)

0 30 and Σ = E(utut), and constructs the impulse responses as follows.

Construction for the impulse responses:

1. Estimate a VAR model with 12 lags using the monthly data on the variables

shown as Table 3.1 by OLS and denote the estimates by Bˆ and Σ.ˆ Let A˜A˜0 = Σˆ ˆ be the Cholesky decomposition of Σ and let rj(h) be the impulse response at

horizon h to the jth shock in a Cholesky decomposition of Σ.ˆ

2. Using the prior distribution and estimates (B,ˆ Σ),ˆ derive the posterior distribu-

tion and draw n1 number of (B, Σ) from it.

3. Given (B, Σ), draw n2 number of α from the k-dimensional unit sphere.

Pk 4. Given α, calculate the impulse response r(h) = j=1 αjrj(h). If r(h) satisfies the sign restrictions in Table 3.1 for h = 1, ...6, then keep the r(h).

30This is the pure-sign-restriction approach which Uhlig (2005) focused on. For the details, refer to Uhlig (2005).

58 3.2.2 Motivation

We replicate the sign-restriction VAR estimation similar to Uhlig (2005).31 To do

that, we first fit a VAR model with 4 lags,

Yt = B1Yt−1 + ··· + B4Yt−4 + ut (3.2)

where Yt=[log(Real GDPt) log(GDP deflatort) Federal funds ratet/4 log(Nonborrowed

0 reservest) log(Total reservest)] from 1960:1Q to 2013:4Q. Then, we adopt the pure- sign-restriction approach with the sign restrictions given in Table 3.1 and assume that the restrictions are satisfied for 2 quarters after a shock. Figure 3.1 shows the impulse responses of real GDP, GDP deflator, and federal funds rate to a contrac- tionary monetary shock. The size of the shock is normalized so that the federal funds rate increases by 0.25%p at the impact. Figure 3.1 implies that considering the error band a contractionary monetary shock is neutral as in Uhlig (2005).32

However, when the same sign restrictions are applied to the pre-Volcker period

(1960:1Q-1979:2Q), the impulse responses show a different result. Even though it is not significant, the median implies that real GDP decreases after a contractionary monetary shock as shown in Figure 3.2. The reason seems to be obvious: It is well known that during the pre-Volcker period, monetary policy shocks were relatively volatile and so they had a large effect on the economy. This suggests that the sign restrictions are inappropriate in identifying monetary shocks if the shocks are not too volatile. This guess is the motivation for this paper.

31The specification used here is a little bit different from what was used in Uhlig (2005). The quarterly data from 1960 to 2013 is used here while the monthly data from 1965 to 2003 was used in Uhlig (2005). Also the commodity price index is not included in Yt here. However, the results are similar. 32By construction, there is no price puzzle.

59 Figure 3.1: Impulse Responses from the Sign-restriction VAR Real GDP 0.8 0.6 0.4

% 0.2 0 −0.2 −0.4 2 4 6 8 10 12 14 16 18 20

GDP deflator 0

−0.5 %

−1

2 4 6 8 10 12 14 16 18 20

Fed Funds Rate

median 0.4 upper/lower 16%

% 0.2

0

2 4 6 8 10 12 14 16 18 20

3.3 Model

Over the past few years, the new Keynesian DSGE model has emerged as the

workhorse for the analysis of monetary policy and its implications.33 In this section, a simple version of new Keynesian DSGE model is presented which will be estimated in the next section. In order to allow the monetary shocks to have a contemporaneous impact on output as in the previous sign-restriction VAR model, it is assumed that the shocks occur at the start of each period.

33See Rotemberg and Woodford (1997, 1999), Christiano et al. (2005), Boivin and Giannoni (2006), Smets and Wouters (2003, 2007), Gali (2008), and Walsh (2010).

60 Figure 3.2: Impulse Responses from the Sign-restriction VAR during Pre-Volcker Period Real GDP 0.5

0 %

−0.5

2 4 6 8 10 12 14 16 18 20

GDP deflator 0.4 0.2 0

% −0.2 −0.4 −0.6

2 4 6 8 10 12 14 16 18 20

Fed Funds Rate

0.4 0.2

% 0 −0.2 −0.4 2 4 6 8 10 12 14 16 18 20

3.3.1 Household

The representative household maximizes the expected lifetime given by

∞ 1+φ X t Nt max E0 β [log(Ct − bCt−1) − ] (3.3) Ct,Nt,Bt 1 + φ t=0

s.t. PtCt + Bt ≤ (1 + Rt−1)Bt−1 + WtNt − Tt (3.4)

where β ∈ (0, 1) is the discount factor, Nt is hours of work, and φ is the inverse of

1 −1  R  −1 Frisch . Ct ≡ [ 0 Ct(j) dj] is the aggregate consumption index where

Ct(j) is the consumption for a differentiated good j ∈ (0, 1) and  ≥ 1 is the elasticity

1 1 R 1− 1− of demand for each differentiated good. Also Pt ≡ [ 0 Pt(j) dj] is the aggregate price index where Pt(j) is the price for a differentiated good j, Bt is the quantity of one-period riskless bonds, Rt is the nominal interest rate, Wt is the nominal wage,

61 and Tt is the lump-sum taxes. Finally, the habit parameter b ∈ (0, 1) is introduced to replicate the hump-shaped response of consumption and output to monetary policy shocks.

3.3.2 Firms

There is a unit measure of monopolistic firms indexed by j ∈ (0, 1). Each firm faces three constraints. First, each firm faces the demand function derived from the previous household problem. Second, the technology available to each firm is summarized by the production function. For simplicity, it is assumed that there is no capital stock in the economy.34 So output depends only on labor input. Finally, each monopolistic firm j set prices according to a variant of Calvo pricing. In each period, each firm can reoptimize its price with the probability of 1 − θ. Firms that cannot reoptimize their price index to the lagged inflation and steady-state inflation rate. Therefore, a firm that reoptimizes its price at period t will choose the price

∗ Pt that maximizes the current market value of the profits generated while that price remains effective:

∞ X Pt+k−1 max E θk[Q {P ∗( )χ(1 +π ¯)k(1−χ)Y (j) − TC(Y (j))}] (3.5) ∗ t t,t+k t t+k t+k Pt Pt−1 k=0 ∗ χ k(1−χ) Pt (Pt+k−1/Pt−1) (1 +π ¯) − s.t. Yt+k(j) = Yt+k[ ] (3.6) Pt+k t α Yt(j) = γ exp(zt)Nt(j) (3.7)

k UC,t+k/Pt+k where Qt,t+k = β is the nominal discount factor,π ¯ is the steady state UC,t/Pt inflation rate, χ ∈ (0, 1) is a degree of indexation to lagged inflation, and TC(·) is the

total production cost function. Also γ is the trend growth rate, α is the labor ratio,

34Cogley and Nason (1995) and McCallum and Nelson (1999) showed that there is little relation- ship between the capital stock and output at business cycle frequencies.

62 and zt is a productivity shock. Full indexation produces a vertical long-run Phillips

curve and so the level of output does not depend on the inflation rate in the steady

state.

3.3.3 Central Bank and Resource Constraint

Central bank follows a generalized Taylor rule:

1 + Rt 1 + Et(πt+1) φπ Yt φy (1−ρ) 1 + Rt−1 ρ ¯ = [{ } { ¯ } ] ( ¯ ) exp(mt) (3.8) 1 + R 1 +π ¯ Yt 1 + R

where π ≡ log Pt+1 is the inflation rate between t and t + 1, R¯ is the steady-state t+1 Pt ¯ nominal interest rate, and Yt is the output on the balanced-growth path. Also φπ is

the coefficient for the deviation of the expected inflation rate from the steady-state

level, φy is the coefficient for the output gap, and ρ is the interest rate smoothing

parameter. Finally, mt is a monetary policy shock. This specification implies that

central bank is assumed to respond to fluctuation of the expected inflation and current

output, which is consistent with Christiano et al. (2005) and Boivin and Giannoni

(2006).

The resource constraint of this economy is given by

Yt = Ct + Gt (3.9)

where Gt represents other demand components than consumption. The aggregate

1 −1  R  −1 Gt ≡ [ 0 Gt(j) dj] has the same form as Ct.

3.3.4 Log-linearization

Three equations describing the economy are obtained by log-linearizing first-order conditions shown above around a steady state.35 With the exception of interest rates

35See Appendix C for the details.

63 and inflation rates, let the lowercases denote the detrended values of uppercases and

a hat over a variable denote the percentage deviation from its steady-state value. For

interest rates and inflation rates, a hat over a variable denotes the deviation from its

steady-state value.

¯ ˆ yˆt = A1yˆt−1 + A1βγEt(ˆyt+1) − A2λt + dt(ˆgt, gˆt−1, gˆt+1) (3.10)

¯ πˆt − χπˆt−1 = βγEt(ˆπt+1 − χπˆt) + A3mcˆ t(ˆyt, yˆt+1, yˆt−1, dt, zt) (3.11)

ˆ ˆ Rt = (1 − ρ)[φπEt(ˆπt+1) + φyyˆt] + ρRt−1 + mt (3.12)

b (1−βb¯ )(1− b ) ¯ ¯ β γ γ c¯ (1−θ)(1−βγθ) α where β ≡ γ ,A1 ≡ ¯ b2 ,A2 ≡ ¯ b2 y¯, and A3 ≡ θ α+(1−α) . Equation 1+β γ 1+β γ ˆ P∞ ˆ (3.10) is a dynamic IS equation where λt = k=0 Et(Rt+k − πˆt+k+1) is the deviation ¯ of the long-term nominal interest rate. Also dt ≡ gˆt − A1gˆt−1 − A1βγgˆt+1 is a function

gt−g¯ of percentage deviation of demand components whereg ˆt = y¯ and so dt can be considered as a . Equation (3.11) is a new Keynesian Phillips curve

1 1+φ A1 A1 ¯ 1+φ 1 equation wheremc ˆ t = ( − 1 + )ˆyt − yˆt−1 − βγEt(ˆyt+1) − zt − dt is the A2 α A2 A2 α A2 percentage deviation of real . Finally, Equation (3.12) is a Taylor rule.

We assume that each of three shocks follows the AR(1) process. Also as in Smets

and Wouters (2007), we assume that the demand shock depends on the productivity

shock because net exports are likely to depend on the productivity shock.

m m 2 mt = ρmmt−1 + et , (et ∼ N(0, σm)) (3.13)

z z 2 zt = ρzzt−1 + et , (et ∼ N(0, σz )) (3.14)

z d d 2 dt = ρddt−1 + ρdzet + et , (et ∼ N(0, σd)) (3.15)

64 3.4 Parameter Estimation

Various econometric methods have been proposed to estimate the parameters

in DSGE models, ranging from calibration, generalized method of moments (GMM)

estimation, minimum distance estimation based on the discrepancy between VAR and

DSGE model impulse response functions, to maximum likelihood (ML) estimation.

In this section, the model presented in previous section is estimated using Bayesian

techniques which are widely used recently.36

3.4.1 Data

Three US time series from 1960:1Q to 2013:4Q are used as observable variables.

      100d(log(Real GDPt)) 100(γ − 1) yˆt − yˆt−1 100d(log(GDP deflatort)) =  100¯π  +  πˆt  (3.16) 1 ˆ Federal funds ratet/4 100( β¯(1 +π ¯) − 1) Rt

where d stands for the difference operator. 100(γ−1) is the net quarterly trend growth

1 rate of real GDP in percentage terms. Similarly, 100¯π and 100( β¯(1 +π ¯) − 1) imply the net quarterly steady-state inflation rate and nominal interest rate in percentage

terms, respectively. So the first equation shows that the actual growth rate of real

GDP is equal to the sum of the net trend growth rate of real GDP and the difference

in percentage deviation of detrended real GDP from its trend. Similarly, the last two

equations show that the actual inflation rate and nominal interest rate are equal to

the sum of its steady-state level and the deviation from its steady-state level.

36An and Schorfheide (2007) reviewed Bayesian estimation for DSGE models. In this paper, the estimation is executed in Dynare.

65 3.4.2 Priors

The priors are important in Bayesian estimation and are summarized in the second column in Table 3.2. We use similar priors as in Smets and Wouters (2003, 2007).

The trend growth rate 100(γ − 1), trend inflation rate 100¯π, and trend real interest

1 rate 100( β − 1) are assumed to be around 0.7% per quarter, 2.5% per year, and 1% per year, respectively. These are based on the data during the sample period.

The parameters related to firm behavior are assumed as follows. The degree of price stickiness θ is assumed to be around 0.5 implying that firms reoptimize their

1 prices twice a year (= 1−0.5 ). The degree of indexation to the lagged inflation χ is also assumed to be around 0.5 implying that firms that cannot reoptimize their prices change them at a half of the rate of past inflation rate. The labor share α is assumed to follow a Normal distribution with mean 0.7 which is consistent with data.

The parameters of the utility function are assumed as follows. The habit parame- ter b is assumed to fluctuate around 0.7, the inverse of Frisch elasticity φ is assumed to be around the unity, and the elasticity of demand  is assumed to be around 10.

These are all quite consistent with literatures.

In the monetary policy rule, the coefficient for expected inflation deviation φπ and for output deviation φy are assumed to be around 1.5 and 0.125, respectively. The smoothing parameter ρ is assumed to fluctuate around 0.5. These are quite consistent with the actual behavior of Fed after Volcker periods.

The standard errors of all three shocks are assumed to follow an inverse-gamma distribution with a mean of 0.1 and a standard deviation of 2. Also the persistence of all three shocks is assumed to follow a beta distribution with mean 0.5 and standard deviation 0.2.

66 Table 3.2: Prior Distribution and Posterior Mean of Structural Parameters Prior distribution (mean, std) Posterior mean (5%, 95%) 100(γ − 1) Normal (0.70, 0.10) 0.7648 (0.7341, 0.7997) 100¯π Gamma (0.63, 0.10) 0.6842 (0.5284, 0.8310) 1 100( β¯ − 1) Gamma (0.25, 0.10) 0.2997 (0.1483, 0.4369) θ Beta (0.50, 0.10) 0.5964 (0.5304, 0.6701) χ Beta (0.50, 0.15) 0.0920 (0.0345, 0.1518) α Normal (0.70, 0.05) 0.6669 (0.5992, 0.7325) b Beta (0.70, 0.10) 0.5185 (0.4412, 0.5927) φ Normal (1.00, 0.20) 1.0105 (0.7271, 1.3006)  Normal (10.0, 0.20) 9.9334 (9.6330, 10.2072) φπ Normal (1.50, 0.25) 1.6934 (1.3709, 2.0255) φy Normal (0.13, 0.05) 0.0501 (0.0133, 0.0839) ρ Beta (0.50, 0.20) 0.2122 (0.0953, 0.3242) ρm Beta (0.50, 0.20) 0.6893 (0.6264, 0.7525) ρz Beta (0.50, 0.20) 0.9462 (0.9127, 0.9785) ρd Beta (0.50, 0.20) 0.9191 (0.8922, 0.9494) ρdz Beta (0.50, 0.20) 0.2449 (0.0580, 0.4234) σm Invgamma (0.10, 2.00) 0.3323 (0.2721, 0.3987) σz Invgamma (0.10, 2.00) 0.7533 (0.5335, 0.9826) σd Invgamma (0.10, 2.00) 0.4929 (0.3375, 0.6511) c¯ y¯ Calibration (0.55) -

Finally, the consumption ratio is calibrated as 0.55. If we estimate the consump- tion ratio, then it is estimated too low, about 10%, because consumption is not considered as an observable variable.

3.4.3 Estimation Result

The last column in Table 3.2 shows the estimation result. The trend growth rate

100(γ −1) is estimated as 0.76% per quarter. Also the annualized trend inflation rate

1 100¯π is estimated as 2.74% and annualized trend real rate 100( β¯ − 1) is estimated as 1.2%.

67 The degree of price stickiness θ is estimated as 0.60. This implies that firms

1 reoptimize their prices every two and a half quarters (= 1−0.60 ) on average. This estimate is consistent with the evidence by Nakamura and Steinsson (2008) where

the estimated median duration of prices is between 8 and 11 months. The degree of

indexation to the lagged inflation χ is estimated as 0.09 which is small. This estimate is also consistent with the evidence by Gali and Gertler (1999) where backward- looking price setting is shown to be not quantitatively important. And labor ratio α is estimated as 0.67.

The habit parameter b is estimated as 0.52, the inverse of Frisch elasticity φ is

estimated as 1.01, and the elasticity of demand  is estimated as 10.04 implying the

1 markup rate is about 10% (=100 9.9−1 ) when prices are flexible. These estimates are also broadly consistent with those in literatures.

In the monetary policy rule, the coefficient for expected inflation deviation φπ is

large but the coefficient for output deviation φy is small, which are consistent with

the literatures. However, the smoothing parameters ρ is estimated to be low which is

not consistent with literatures where the smoothing parameter is large, usually more

than 0.5.

Each shock is estimated to be persistent (high ρm, ρz, and ρd). Also it is found

that the demand shock does not depend on the productivity shock much (low ρdz).

3.4.4 DSGE Impulse Responses

Figure 3.3 shows the impulse responses to a contractionary monetary shock con-

ditional on the posterior mean parameters. The size of a shock is normalized so that

68 Figure 3.3: Impulse Responses from the Estimated DSGE Model Output

−0.1 −0.2

% −0.3 −0.4 −0.5

2 4 6 8 10 12 14 16 18 20

Price level 0

−0.2 % −0.4

−0.6 2 4 6 8 10 12 14 16 18 20

Interest rate

0.2 0.15 % 0.1 0.05 0 2 4 6 8 10 12 14 16 18 20

the increases in the interest rate is 0.25%p corresponding to 1%p annually. The re- sult shows that output and inflation decrease after a contractionary monetary shock.

Especially, the response of output shows a hump-shaped movement because of the habit formation.

3.5 Monte Carlo Simulation

In this section, we investigate how well the sign restrictions can identify the effect of monetary policy shocks. To do that, we generate an 5000 artificial data set con- sisting of 216 observations of output (Y), price level (P), and interest rate (R) from the previously estimated model conditional on the posterior mean parameters. Then we estimate a VAR model with the sign restrictions given in Table 3.3. These sign

69 Figure 3.4: Impulse Responses to a Contractionary Monetary Shock Output

0

−0.2 %

−0.4

2 4 6 8 10 12 14 16 18 20

Price level 0

−0.2 % −0.4

−0.6 2 4 6 8 10 12 14 16 18 20

Interest rate

0.2 DSGE SRVAR (median) 0.15

% 0.1 0.05 0

2 4 6 8 10 12 14 16 18 20

restrictions correspond to those in Table 3.1, which are used in Uhlig (2005). Then impulse responses from the sign-restriction VAR and estimated models are compared.

Table 3.3: Sign Restrictions for the Simulation YPR – +

Figure 3.4 shows the results. Note that after a contractionary monetary policy shock output increases in the sign-restriction VAR model even though it decreases in the estimated model from which the artificial data set is generated.37

37As before, the size of a shock is normalized so that the increases in the interest rate is 0.25%p corresponding to 1%p annually.

70 0 Figure 3.5: Impulse Responses to a Contractionary Monetary Shock (σm = 20σm) Output 0

−0.2 % −0.4

2 4 6 8 10 12 14 16 18 20

Price level 0

−0.2 % −0.4

−0.6 2 4 6 8 10 12 14 16 18 20

Interest rate

0.2 DSGE SRVAR (median) 0.15

% 0.1 0.05 0

2 4 6 8 10 12 14 16 18 20

Then when could the sign restrictions well identify the effect of monetary policy shocks? As mentioned in Section 3.2.2, the sign restrictions seem to well identify the effect of a shock only if the shocks are volatile. To see that, we increase the standard deviation of monetary policy shocks and then apply the previous procedure again: generate an artificial data set and estimate the impulse responses from a VAR model with the sign restrictions. We find that when the standard deviation of monetary policy shocks is increased by twenty times, the sign restrictions start to capture the negative response of output to a contractionay monetary shock as shown in Figure

3.5. Similar result can be obtained when the artificial data set is generated only by monetary shocks as shown in Figure 3.6.

71 0 0 Figure 3.6: Impulse Responses to a Contractionary Monetary Shock (σz = σd = 0) Output 0

−0.2 % −0.4

2 4 6 8 10 12 14 16 18 20

Price level 0

−0.2 % −0.4

−0.6 2 4 6 8 10 12 14 16 18 20

Interest rate

0.2 DSGE SRVAR (median) 0.15

% 0.1 0.05 0

2 4 6 8 10 12 14 16 18 20

How well could other approaches identify the effect of monetary policy shocks? As

mentioned in Section 3.1, a VAR model with recursiveness assumption is widely used

and so we see if it could well identify the effect of monetary policy shocks. Figure

3.7 shows the result of a recursive VAR ordered as Yt =[log(Output) log(Price level)

Interest rate]. It shows that a recursive VAR captures the negative response of output which is consistent with the estimated model.38

3.6 Conclusion

This paper investigates the validity of sign restriction for identifying the monetary policy shocks in a VAR model. A simple version of new Keynesian DSGE model is

38Also it shows the price puzzle. However, this problem could be eliminated by including a commodity price index as shown in literatures.

72 Figure 3.7: Impulse Responses to a contractionary Monetary Shock from a Recursive VAR Output 0

−0.2 % −0.4

2 4 6 8 10 12 14 16 18 20

Price level 0

−0.2 % −0.4

−0.6 2 4 6 8 10 12 14 16 18 20

Interest rate

0.2 DSGE Recursive VAR (median) 0.15

% 0.1 0.05 0

2 4 6 8 10 12 14 16 18 20

estimated using Bayesian techniques. Then the artificial data set is generated from the estimated model to be used in a sign-restriction VAR model. The results show that the sign restrictions do not well identify the monetary policy shocks: after the contractionary monetary policy shock, the sign-restriction VAR estimation shows that output increases even though it decreases in the estimated model where the artificial data set is generated. The sign restrictions seem to well identify the monetary policy shocks only when the shocks are extremely volatile. On the other hand, the recursive

VAR captures the negative response of output after a contractionary monetary policy shock.

Introducing more variables and imposing more sign restrictions could improve the validity of sign restrictions, which will be an interesting topic for future research.

73 Also the underlying mechanism by which the size of a shock affects the validity of sign restrictions should be analyzed in the future.

74 Bibliography

[1] Aguiar, M., and Bils, M. Has consumption inequality mirrored income inequality? American Economic Review 105, 9 (2015), 2725–56.

[2] Airaudo, M., and Bossi, L. Trickle-down consumption, monetary policy, and inequality, 2014.

[3] An, S., and Schorfheide, F. Bayesian analysis of dsge models. Econometric reviews 26, 2-4 (2007), 113–172.

[4] Barakchian, S. M., and Crowe, C. Monetary policy matters: Evidence from new shocks data. Journal of 60, 8 (2013), 950–966.

[5] Bernanke, B. S. Monetary policy and inequality. https://www.brookings. edu/blog/ben-bernanke/2015/06/01/monetary-policy-and-inequality/, 2015.

[6] Bernanke, B. S., Boivin, J., and Eliasz, P. Measuring the effects of monetary policy: A factor-augmented vector autoregressive (favar) approach. The Quarterly Journal of Economics 120, 1 (2005), 387–422.

[7] Bernanke, B. S., Gertler, M., and Watson, M. Systematic monetary policy and the effects of oil price shocks. Brookings papers on economic activity (1997), 91–142.

[8] Bernanke, B. S., Mihov, I., et al. Measuring monetary policy. The Quar- terly Journal of Economics 113, 3 (1998), 869–902.

[9] Berrebi, Z. M., and Silber, J. The gini coefficient and negative income: a comment. Oxford Economic Papers 37, 3 (1985), 525–526.

[10] Blanchard, O., and Perotti, R. An empirical characterization of the dy- namic effects of changes in government spending and taxes on output. The Quarterly Journal of Economics 117, 4 (2002), 1329–1368.

[11] Boivin, J., and Giannoni, M. P. Has monetary policy become more effective? The Review of Economics and Statistics 88, 3 (2006), 445–462.

75 [12] Bound, J., and Johnson, G. E. Changes in the structure of wages in the 1980’s: An evaluation of alternative explanations. American Economic Review 82, 3 (1992), 371–92.

[13] Bullard, J. Income inequality and monetary policy: A framework with an- swers to three questions. https://www.stlouisfed.org/~/media/Files/PDFs/ Bullard/remarks/Bullard_CFR_26June2014_Final.pdf, 2014.

[14] Canova, F., and De Nicolo, G. Monetary disturbances matter for business fluctuations in the g-7. Journal of Monetary Economics 49, 6 (2002), 1131–1159.

[15] Card, D. The effect of unions on wage inequality in the us labor market. Industrial and Labor Relations Review 54, 2 (2001), 296–315.

[16] Castillo, P., and Montoro, C. The asymmetric effects of monetary policy in general equilibrium. Journal of Centrum Cathedra 1, 2 (2008), 28–46.

[17] Chen, C.-N., Tsaur, T.-W., and Rhai, T.-S. The gini coefficient and negative income. Oxford Economic Papers 34, 3 (1982), 473–478.

[18] Chen, S.-S. Does monetary policy have asymmetric effects on stock returns? Journal of Money, Credit and Banking 39, 2-3 (2007), 667–688.

[19] Christiano, L. J., Eichenbaum, M., and Evans, C. L. Monetary policy shocks: What have we learned and to what end? Handbook of 1 (1999), 65–148.

[20] Christiano, L. J., Eichenbaum, M., and Evans, C. L. Nominal rigidities and the dynamic effects of a shock to monetary policy. Journal of 113, 1 (2005), 1–45.

[21] Cogley, T., and Nason, J. M. Output dynamics in real-business-cycle mod- els. The American Economic Review (1995), 492–511.

[22] Coibion, O., Gorodnichenko, Y., Kueng, L., and Silvia, J. Innocent bystanders? monetary policy and inequality in the us. Tech. rep., National Bureau of Economic Research, 2012.

[23] Cover, J. P. Asymmetric effects of positive and negative money-supply shocks. The Quarterly Journal of Economics (1992), 1261–1282.

[24] Doepke, M., and Schneider, M. Inflation and the redistribution of nominal wealth. Journal of Political Economy 114, 6 (2006), 1069–1097.

[25] Elliott, G., Rothenberg, T. J., and Stock, J. H. Efficient tests for an autoregressive unit root. (1996), 813–836.

76 [26] Engle, R. F., and Granger, C. W. Co-integration and error correction: representation, estimation, and testing. Econometrica: journal of the Economet- ric Society (1987), 251–276.

[27] Erosa, A., and Ventura, G. On inflation as a regressive consumption tax. Journal of Monetary Economics 49, 4 (2002), 761–795.

[28] Faust, J. The robustness of identified var conclusions about money. In Carnegie- Rochester Conference Series on Public Policy (1998), vol. 49, Elsevier, pp. 207– 244.

[29] Feenstra, R. C., and Hanson, G. H. Global production sharing and rising inequality: A survey of trade and wages. NBER Working Paper, w8372 (2001).

[30] Fisher, J., Johnson, D. S., and Smeeding, T. M. Measuring the trends in inequality of individuals and families: Income and consumption. American Economic Review 103, 3 (2013), 184–88.

[31] Fisher, J. D. Income imputation and the analysis of consumer expenditure data. Monthly Lab. Rev. 129 (2006), 11.

[32] Gal´ı, J. Monetary policy, inflation, and the business cycle: an introduction to the new Keynesian framework and its applications. Princeton University Press, 2008.

[33] Galı, J., and Gertler, M. Inflation dynamics: A structural econometric analysis. Journal of monetary Economics 44, 2 (1999), 195–222.

[34] Garcia, R., and Schaller, H. Are the effects of monetary policy asymmet- ric? Economic Inquiry 40, 1 (2002), 102–119.

[35] Gertler, M., and Gilchrist, S. Monetary policy, business cycles, and the behavior of small manufacturing firms. The Quarterly Journal of Economics (1994), 309–340.

[36] Gornemann, N., Kuester, K., and Nakajima, M. Monetary policy with heterogeneous agents. Tech. rep., Federal Reserve Bank of Philadelphia, 2012.

[37] Hamilton, J. D. This is what happened to the oil price-macroeconomy rela- tionship. Journal of Monetary Economics 38, 2 (1996), 215–220.

[38] Hamilton, J. D., and Herrera, A. M. Oil shocks and aggregate macroe- conomic behavior: The role of monetary policy. Journal of Money, Credit and Banking (2004), 265–286.

77 [39] Heathcote, J., Perri, F., and Violante, G. L. Unequal we stand: An empirical analysis of economic inequality in the united states, 1967–2006. Review of Economic dynamics 13, 1 (2010), 15–51.

[40] Herrera, A. M., Lagalo, L. G., and Wada, T. Asymmetries in the response of economic activity to oil price increases and decreases? Journal of International Money and Finance 50 (2015), 108–133.

[41] Ho, T.-w. Income inequality may not converge after all: Testing panel unit roots in the presence of cross-section cointegration. The Quarterly Review of Economics and Finance 56 (2015), 68–79.

[42] Karras, G. Are the output effects of monetary policy asymmetric? evidence from a sample of european countries. Oxford Bulletin of Economics and Statistics 58, 2 (1996), 267–278.

[43] Kilian, L., and Vigfusson, R. J. Are the responses of the us economy asymmetric in energy price increases and decreases? Quantitative Economics 2, 3 (2011), 419–453.

[44] Koop, G., Pesaran, M. H., and Potter, S. M. Impulse response analysis in nonlinear multivariate models. 74, 1 (1996), 119–147.

[45] Kwiatkowski, D., Phillips, P. C., Schmidt, P., and Shin, Y. Testing the null hypothesis of stationarity against the alternative of a unit root: How sure are we that economic time series have a unit root? Journal of econometrics 54, 1 (1992), 159–178.

[46] Lee, K., and Ni, S. On the dynamic effects of oil price shocks: A study using industry level data. Journal of Monetary Economics 49, 4 (2002), 823–852.

[47] Lin, P.-C., and Huang, H.-C. Convergence in income inequality? evidence from panel unit root tests with structural breaks. Empirical Economics 43, 1 (2012), 153–174.

[48] Lo, M. C., and Piger, J. Is the Response of Output to Monetary Policy Asymmetric? Evidence from a Regime-Switching Coefficients Model. Journal of Money, Credit and Banking 37, 5 (October 2005), 865–86.

[49] Malinen, T. Estimating the long-run relationship between income inequality and economic development. Empirical Economics 42, 1 (2012), 209–233.

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

78 [51] Mersch, Y. Monetary policy and economic inequality. https://www.ecb. europa.eu/press/key/date/2014/html/sp141017_1.en.html, 2014.

[52] Moon, H. R., and Perron, B. An empirical analysis of nonstationarity in a panel of interest rates with factors. Journal of Applied Econometrics 22, 2 (2007), 383–400.

[53] Morgan, D. P. Asymmetric effects of monetary policy. Economic Review- Federal Reserve Bank of Kansas City 78, 2 (1993), 21.

[54] Mork, K. A. Oil and the macroeconomy when prices go up and down: An extension of hamilton’s results. Journal of political Economy (1989), 740–744.

[55] Mumtaz, H., Theophilopoulou, A., et al. The impact of monetary policy on inequality in the uk. an empirical analysis. Tech. rep., 2016.

[56] Nakamura, E., and Steinsson, J. Five facts about prices: A reevaluation of menu cost models. The Quarterly Journal of Economics (2008), 1415–1464.

[57] Nelson, C. R., and Plosser, C. R. Trends and random walks in macroecon- mic time series: some evidence and implications. Journal of monetary economics 10, 2 (1982), 139–162.

[58] Piketty, T., and Saez, E. Income inequality in the united states, 1913–1998. The Quarterly Journal of Economics 118, 1 (2003), 1–41.

[59] Piketty, T., and Saez, E. Top incomes and the great recession: Recent evolutions and policy implications. IMF Economic Review 61, 3 (2013), 456– 478.

[60] Potscher,¨ B. M., and Prucha, I. R. Dynamic nonlinear econometric mod- els: Asymptotic theory. Springer Science & Business Media, 1997.

[61] Primiceri, G. E. Time varying structural vector autoregressions and monetary policy. The Review of Economic Studies 72, 3 (2005), 821–852.

[62] Ravn, M., and Sola, M. A reconsideration of the empirical evidence on the asymmetric effects of money-supply shocks: Positive vs. negative or big vs. small. Tech. rep., Birkbeck, Department of Economics, Mathematics & Statistics, 1996.

[63] Ravn, M. O., and Sola, M. Asymmetric effects of monetary policy in the united states. Review-Federal Reserve Bank of Saint Louis 86 (2004), 41–58.

[64] Romer, C. D., and Romer, D. H. Monetary policy and the well-being of the poor. Tech. rep., National Bureau of Economic Research, 1998.

79 [65] Romer, C. D., and Romer, D. H. A new measure of monetary shocks: Derivation and implications. The American Economic Review 94, 4 (2004), 1055– 1084.

[66] Rotemberg, J., and Woodford, M. An optimization-based econometric framework for the evaluation of monetary policy. In NBER Macroeconomics Annual 1997, Volume 12. MIT Press, 1997, pp. 297–361.

[67] Rotemberg, J. J., and Woodford, M. Interest-rate rules in an estimated sticky price model. Tech. rep., National Bureau of Economic Research, 1998.

[68] Saiki, A., and Frost, J. Does unconventional monetary policy affect in- equality? Evidence from Japan. 46, 36 (December 2014), 4445–4454.

[69] Sims, C. A. Interpreting the macroeconomic time series facts: The effects of monetary policy. European Economic Review 36, 5 (1992), 975–1000.

[70] Smets, F., and Wouters, R. An estimated dynamic stochastic general equi- librium model of the euro area. Journal of the European economic association 1, 5 (2003), 1123–1175.

[71] Smets, F., and Wouters, R. Shocks and frictions in us business cycles: A bayesian dsge approach. The American Economic Review 97, 3 (2007), 586–606.

[72] Stock, J. H., and Watson, M. W. Testing for common trends. Journal of the American statistical Association 83, 404 (1988), 1097–1107.

[73] Stock, J. H., and Watson, M. W. Vector autoregressions. The Journal of Economic Perspectives 15, 4 (2001), 101–115.

[74] Thoma, M. A. Subsample instability and asymmetries in money-income causal- ity. Journal of Econometrics 64, 1 (1994), 279–306.

[75] Uhlig, H. What are the effects of monetary policy on output? results from an agnostic identification procedure. Journal of Monetary Economics 52, 2 (2005), 381–419.

[76] Walsh, C. E. Monetary theory and policy. MIT press, 2010.

[77] Watkins, J. P. Quantitative easing as a means of reducing unemployment: a new version of trickle-down economics. Journal of Economic Issues 48, 2 (2014), 431–440.

[78] Weise, C. L. The asymmetric effects of monetary policy: A nonlinear vector autoregression approach. Journal of Money, Credit and Banking (1999), 85–108.

80 [79] Williamson, S. D. Monetary policy and distribution. Journal of Monetary Economics 55, 6 (2008), 1038–1053.

[80] Wu, J. C., and Xia, F. D. Measuring the macroeconomic impact of monetary policy at the zero lower bound. Journal of Money, Credit and Banking 48, 2-3 (2016), 253–291.

[81] Wu, Y., and Zhang, H. Mean reversion in interest rates: New evidence from a panel of countries. Journal of Money, Credit and Banking 28, 4 (1996), 604–621.

81 Appendix A: Appendix to Chapter 1

A.1 Categories of Income

Table A.1: Four Categories of Income Category Description Variable Name in CEX A. Through 2013Q1 Wage Wage and salary income FSALARYX Business Net nonfarm business income or loss FNONFRMX Net farm income or loss FFRMINCX Financial Interest income INTEARNX Dividends, royalties, estates, trusts income FININCX Net roomer and boarder income or loss INCLOSSA Other net rental income or loss INCLOSSB Other Social Security and railroad retirement income FRRETIRX Pensions and annuities PENSIONX Supplemental security income FSSIX Public assistance WELFAREX Food stamps and electronic benefits JFDSTMPA (thr 2001Q1) FOODSMPX Unemployment compensation UNEMPLX Workers’ compensation and veterans’ benefits COMPENSX Child support payments INCCONTX (thr 1993Q2) CHDLMPX CHDOTHX Other regular contributions including alimony INCCONTX (thr 1993Q2) ALIOTHX Other income OTHRINCX

82 Table A.1: Four Categories of Income (continued) Category Description Variable Name in CEX B. From 2013Q2 Wage Wage and salary income FSALARYX Business Net self-employment income or loss FSMPFRMX Financial Interest and dividends income INTRDVX Net rental income or loss NETRENTX Royalty, estate, trust income ROYESTX Other Social Security and railroad retirement income FRRETIRX Retirement, survivors, disability pension RETSURVX Supplemental security income FSSIX Public assistance WELFAREX Food stamps JFS AMT Other regular income OTHREGX Other income OTHRINCX

A.2 Gini Coefficient with Negative Incomes and Weighted Data

In this Appendix, I derive Gini coefficient with negative incomes and weighted

data given as Equation (1.3). Figure A.1 shows a Lorenz curve of an economy where

Yj for j = 1, ··· , n denotes income value in increasing order (Y1 < ··· < Y3 < 0 <

Y4 < ··· < Y10), f(Yj) represents the fraction of population with income Yj, yj =

f(Yj )Yj P4 P5 Pn is the share of the income Yj. Note that yj ≤ 0 and yj > 0. j=1 f(Yj )Yj j=1 j=1 Pk Pk+1 In general, suppose that j=1 yj ≤ 0 and j=1 yj > 0. Chen et al. (1982) propose the adjusted Gini coefficient which normalizes the standard Gini as

1 + 2(A − B) G∗∗ = (A.1) 1 + 2A

83 Figure A.1 implies

1 1 A = −[ y f(Y ) + ( y f(Y ) + y f(Y )) + ··· 2 1 1 2 2 2 1 2 k P 2 1 1 h j=1 yj i + ( ykf(Yk) + yk−1f(Yk) + ··· + y1f(Yk))] + yk+1f(Yk+1) 2 2 yk+1 k k Pk 2 1 X X 1 ( j=1 yj) = − y [f(Y ) + 2 f(Y )] + f(Y ) (A.2) 2 j j i 2 k+1 y j=1 i=j+1 k+1

and

k+1 k+1 k+1 P 2 1 h j=1 yj i X X B = y f(Y ) + y (1 − f(Y )) 2 k+1 k+1 y j i k+1 j=1 i=1 1 + [( y f(Y ) + y f(Y ) + ··· + y f(Y )) + ··· 2 k+2 k+2 k+2 k+3 k+2 n 1 1 + ( y f(Y ) + y f(Y )) + y f(Y )] 2 n−1 n−1 n−1 n 2 n n k+1 k+1 k+1 P 2 1 h j=1 yj i X X = y f(Y ) + y (1 − f(Y )) 2 k+1 k+1 y j i k+1 j=1 i=1 n n 1 X X + y [f(Y ) + 2 f(Y )] (A.3) 2 j j i j=k+2 i=j+1

Plugging A in Equation (A.2) and B in Equation (A.3) into Equation (A.1) yields

Equation (1.3).

84 Figure A.1: Lorenz Curve with Negative Incomes and Weighted Data

y 10

y 9

B y 8

y 7

y 6

y 0 5 y f(Y ) f(Y ) f(Y ) f(Y ) f(Y ) f(Y ) f(Y ) f(Y ) f(Y ) f(Y ) 1 1 2 3 4 5 6 7 8 9 10 y A y y2 4 3

A.3 Monte Carlo Integration for Constructing Impulse Re- sponses

In this Appendix, I summarize the procedures of constructing the impulse re- sponses from Monte Carlo integration suggested by Kilian and Vigfusson (2011) and

Herrera et al. (2015). After estimating model (3.2),

1. Choose a block of p consecutive xt and yt from the sample. This is defined as

the i-th history Ωi.

i 2. Given Ω , generate two series for each xt+h and yt+h for h = 0, 1, ··· ,H. In

the first series, the initial value of t is changed by the shock denoted by δ.

t+h for h = 1, ··· ,H and vt+h for h = 0, 1, ··· ,H are randomly drawn from

each of their residuals. With this time path of t+h and vt+h, I construct the

85 first series denoted by x1,t+h and y1,t+h. In the second series, t+h and vt+h for

h = 0, 1, ··· ,H are randomly drawn from each of their residuals. With this

time path of t+h and vt+h, I construct the second series denoted by x2,t+h and

y2,t+h.

3. Compute the difference between y1,t+h and y2,t+h for h = 0, 1, ··· ,H.

4. Average this difference across 1000 repetitions of 2 and 3. This average defines

the impulse response of yt+h to a shock of size δ conditional on the i-th history

Ωi.

5. Repeat 1-4 by choosing 300 different histories. Average the conditional impulse

response of yt+h. This average defines the unconditional impulse response of

yt+h to a shock of size δ.

86 Appendix B: Appendix to Chapter 2

In this appendix, we derive the weak dependence conditions for Kilian and Vig- fusson (2011) model.

B.1 Weak Dependence Conditions with One Lag

Kilian and Vigfusson’s model with one lag:

xt = ρxt−1 + γyt−1 + ut

+ yt = βxt−1 + αxt−1 + φyt−1 + t

+ 0 0 where xt ≡ max(0, xt). Below let zt ≡ (xt yt) and vt ≡ (ut t) .

Case 1 (p1 = 1):

 ρx + γy + u − ρx0 − γy0 − u  |f(z, v) − f(z0, v)| = | | 1 β max(0, x) + αx + φy +  − β max(0, x0) − αx0 − φy0 −  1  ρ(x − x0) + γ(y − y0)  = | | β max(0, x) − max(0, x0) + α(x − x0) + φ(y − y0) 1 0 0 = ρ(x − x ) + γ(y − y )

0  0 0 + β max(0, x) − max(0, x ) + α(x − x ) + φ(y − y )

0 0 0  0 ≤ |ρ||x − x | + |γ||y − y | + β max(0, x) − max(0, x ) + α(x − x )

+ |φ||y − y0|

≤ |ρ||x − x0| + |γ||y − y0| + max(|α|, |α + β|)|x − x0| + |φ||y − y0|

87 = |ρ| + max(|α|, |α + β|)|x − x0| + |γ| + |φ||y − y0|

Case 1-A (p1 = 1, p2 = 1):

0  0  0 |f(z, v) − f(z , v)|1 ≤ |ρ| + max(|α|, |α + β|) |x − x | + |γ| + |φ| |y − y |

≤ max |ρ| + max(|α|, |α + β|), |γ| + |φ| · |x − x0| + |y − y0|

 0 = max |ρ| + max(|α|, |α + β|), |γ| + |φ| · |z − z |1

Case 1-B (p1 = 1, p2 = 2):

0  0  0 |f(z, v) − f(z , v)|1 ≤ |ρ| + max(|α|, |α + β|) |x − x | + |γ| + |φ| |y − y | h i1/2 ≤ 2|ρ| + max(|α|, |α + β|)2|x − x0|2 + 2|γ| + |φ|2|y − y0|2

2 2 1/2 (∵ a + b ≤ (2a + 2b ) ) h i1/2 ≤ max 2|ρ| + max(|α|, |α + β|)2, 2(|γ| + |φ|)2

· |x − x0|2 + |y − y0|21/2

1/2 h 2 2i 0 = max 2 |ρ| + max(|α|, |α + β|) , 2(|γ| + |φ|) · |z − z |2

Case 1-C (p1 = 1, p2 = ∞):

0  0  0 |f(z, v) − f(z , v)|1 ≤ |ρ| + max(|α|, |α + β|) |x − x | + |γ| + |φ| |y − y |

≤ |ρ| + max(|α|, |α + β|) + |γ| + |φ| · max |x − x0|, |y − y0|

 0 = |ρ| + max(|α|, |α + β|) + |γ| + |φ| · |z − z |∞

Case 2 (p1 = 2):

 ρx + γy + u − ρx0 − γy0 − u  |f(z, v) − f(z0, v)| = | | 2 β max(0, x) + αx + φy +  − β max(0, x0) − αx0 − φy0 −  2  ρ(x − x0) + γ(y − y0)  = | | β max(0, x) − max(0, x0) + α(x − x0) + φ(y − y0) 2

88 h 0 0 2 = ρ(x − x ) + γ(y − y ) 1/2 0  0 0 2i + β max(0, x) − max(0, x ) + α(x − x ) + φ(y − y ) h ≤ |ρ||x − x0| + |γ||y − y0|2

1/2 0  0 0 2i + β max(0, x) − max(0, x ) + α(x − x ) + |φ||y − y | h ≤ |ρ||x − x0| + |γ||y − y0|2 i1/2 + max(|α|, |α + β|)|x − x0| + |φ||y − y0|2

Case 2-A (p1 = 2, p2 = 1):

0 h 0 0 2 |f(z, v) − f(z , v)|2 ≤ |ρ||x − x | + |γ||y − y | i1/2 + max(|α|, |α + β|)|x − x0| + |φ||y − y0|2 h ≤ max |ρ|, |γ|2|x − x0| + |y − y0|2 i1/2 + max max(|α|, |α + β|), |φ|2|x − x0| + |y − y0|2 h i1/2 = max |ρ|, |γ|2 + max max(|α|, |α + β|), |φ|2

· |x − x0| + |y − y0|

1/2 h 2 2i 0 = max |ρ|, |γ| + max max(|α|, |α + β|), |φ| · |z − z |1

Case 2-B (p1 = 2, p2 = 2):

0 h 0 0 2 |f(z, v) − f(z , v)|2 ≤ |ρ||x − x | + |γ||y − y | i1/2 + max(|α|, |α + β|)|x − x0| + |φ||y − y0|2 h ≤ (2|ρ||x − x0|2 + 2|γ||y − y0|2 + 2(max(|α|, |α + β|)|)|x − x0|2 i1/2 + 2|φ||y − y0|2

2 2 2 (∵ (a + b) ≤ 2a + 2b ) h i1/2 = 2|ρ|2 + max(|α|, |α + β|)2|x − x0|2 + 2|γ|2 + |φ|2|y − y0|2

89 h i1/2 ≤ max 2(|ρ|2 + max(|α|, |α + β|)2), 2(|γ|2 + |φ|2)

· |x − x0|2 + |y − y0|21/2

1/2 h 2 2 2 2 i 0 = max 2(|ρ| + max(|α|, |α + β|) ), 2(|γ| + |φ| ) · |z − z |2

Case 2-C (p1 = 2, p2 = ∞):

0 h 0 0 2 |f(z, v) − f(z , v)|2 ≤ |ρ||x − x | + |γ||y − y | i1/2 + max(|α|, |α + β|)|x − x0| + |φ||y − y0|2 h ≤ |ρ| + |γ|2 max |x − x0|, |y − y0|2 i1/2 + max(|α|, |α + β|) + |φ|2 max |x − x0|, |y − y0|2 h i1/2 = |ρ| + |γ|2 + max(|α|, |α + β|) + |φ|2

· max |x − x0|, |y − y0|

1/2 h 2 2i 0 = |ρ| + |γ| + max(|α|, |α + β|) + |φ| · |z − z |∞

Case 3 (p1 = ∞):

 ρx + γy + u − ρx0 − γy0 − u  |f(z, v) − f(z0, v)| = | | ∞ β max(0, x) + αx + φy +  − β max(0, x0) − αx0 − φy0 −  ∞  ρ(x − x0) + γ(y − y0)  = | | β max(0, x) − max(0, x0) + α(x − x0) + φ(y − y0) ∞  0 0 = max ρ(x − x ) + γ(y − y )

0  0 0  , β max(0, x) − max(0, x ) + α(x − x ) + φ(y − y )  ≤ max |ρ||x − x0| + |γ||y − y0|

0  0 0  , β max(0, x) − max(0, x ) + α(x − x ) + |φ||y − y |   ≤ max |ρ||x − x0| + |γ||y − y0|, max(|α|, |α + β|)|x − x0| + |φ||y − y0|

Case 3-A (p1 = ∞, p2 = 1):

0  0 0 0 0  |f(z, v) − f(z , v)|∞ ≤ max |ρ||x − x | + |γ||y − y |, max(|α|, |α + β|)|x − x | + |φ||y − y |

90  ≤ max max |ρ|, |γ||x − x0| + |y − y0|  , max max(|α|, |α + β|), |φ||x − x0| + |y − y0|   = max max |ρ|, |γ|, max max(|α|, |α + β|), |φ|

· |x − x0| + |y − y0|

   0 = max max |ρ|, |γ| , max max(|α|, |α + β|), |φ| · |z − z |1

Case 3-B (p1 = ∞, p2 = 2):

0  0 0 0 0  |f(z, v) − f(z , v)|∞ ≤ max |ρ||x − x | + |γ||y − y |, max(|α|, |α + β|)|x − x | + |φ||y − y |  ≤ max 2|ρ|2|x − x0|2 + 2|γ|2|y − y0|21/2  , 2 max(|α|, |α + β|)2|x − x0|2 + 2|φ|2|y − y0|21/2

2 2 1/2 (∵ a + b ≤ (2a + 2b ) )  ≤ max  max(2|ρ|2, 2|γ|2)1/2|x − x0|2 + |y − y0|21/2,   max(2 max(|α|, |α + β|)2, 2|φ|2)1/2|x − x0|2 + |y − y0|21/2   = max  max(2|ρ|2, 2|γ|2)1/2,  max(2 max(|α|, |α + β|)2, 2|φ|2)1/2

· |x − x0|2 + |y − y0|21/2   = max  max(2|ρ|2, 2|γ|2)1/2,  max(2 max(|α|, |α + β|)2, 2|φ|2)1/2

0 · |z − z |2

Case 3-C (p1 = ∞, p2 = ∞):

0  0 0 0 0  |f(z, v) − f(z , v)|∞ ≤ max |ρ||x − x | + |γ||y − y |, max(|α|, |α + β|)|x − x | + |φ||y − y |  ≤ max |ρ| + |γ| max |x − x0|, |y − y0|  , max(|α|, |α + β|) + |φ| max |x − x0|, |y − y0|

= max |ρ| + |γ|, max(|α|, |α + β|) + |φ| · max |x − x0|, |y − y0|

91  0 = max |ρ| + |γ|, max(|α|, |α + β|) + |φ| · |z − z |∞

B.2 Weak Dependence Conditions with Arbitrary Lags

Kilian and Vigfusson (2011) model with arbitrary lags:

p p X X xt = ρixt−i + γiyt−i + ut i=1 i=1 p p p X + X X yt = βixt−i + αixt−i + φiyt−i + t i=1 i=1 i=1

Case 1 (p1 = 1):

0 0 |f(z1, ··· , zp, v) − f(z1, ··· , zp, v)|1  Pp Pp Pp 0 Pp 0  i=1 ρixi + i=1 γiyi + u − i=1 ρixi − i=1 γiyi − u =| Pp + Pp Pp Pp + Pp Pp |1 i=1 βixi + i=1 αixi + i=1 φiyi +  − i=1 βixi − i=1 αixi − i=1 φiyi −   Pp 0 Pp 0  i=1 ρi(xi − xi) + i=1 γi(yi − yi) =| Pp 0  Pp 0 Pp 0 |1 i=1 βi max(0, xi) − max(0, xi) + i=1 αi(xi − xi) + i=1 φi(yi − yi) p p X 0 X 0 =| ρi(xi − xi) + γi(yi − yi)| i=1 i=1 p p p X 0  X 0 X 0 + | βi max(0, xi) − max(0, xi) + αi(xi − xi) + φi(yi − yi)| i=1 i=1 i=1 p p p X 0 X 0 X 0  0 ≤ |ρi||xi − xi| + |γi||yi − yi| + |βi max(0, xi) − max(0, xi) + αi(xi − xi)| i=1 i=1 i=1 p X 0 + |φi||yi − yi| i=1 p p p p X 0 X 0 X 0 X 0 ≤ |ρi||xi − xi| + |γi||yi − yi| + max(|αi|, |αi + βi|)|xi − xi| + |φi||yi − yi| i=1 i=1 i=1 i=1 p p X  0 X  0 = |ρi| + max(|αi|, |αi + βi|) |xi − xi| + |γi| + |φi| |yi − yi| i=1 i=1

Case 1-A (p1 = 1, p2 = 1):

0 0 |f(z1, ··· , zp, v) − f(z1, ··· , zp, v)|1

92 p p X  0 X  0 ≤ |ρi| + max(|αi|, |αi + βi|) |xi − xi| + |γi| + |φi| |yi − yi| i=1 i=1 p X  0 0 ≤ max |ρi| + max(|αi|, |αi + βi|), |γi| + |φi| · (|xi − xi| + |yi − yi|) i=1 p X  0 = max |ρi| + max(|αi|, |αi + βi|), |γi| + |φi| · |zi − zi|1 i=1

Case 1-B (p1 = 1, p2 = 2):

0 0 |f(z1, ··· , zp, v) − f(z1, ··· , zp, v)|1 p p X  0 X  0 ≤ |ρi| + max(|αi|, |αi + βi|) |xi − xi| + |γi| + |φi| |yi − yi| i=1 i=1 p 1/2 X h 2 0 2 2 0 2i ≤ 2 |ρi| + max(|αi|, |αi + βi|) |xi − xi| + 2 |γi| + |φi| |yi − yi| i=1 2 2 1/2 (∵ a + b ≤ (2a + 2b ) ) p 1/2 X h 2 2i 0 2 0 21/2 ≤ max 2 |ρi| + max(|αi|, |αi + βi|) , 2(|γi| + |φi|) · |xi − xi| + |yi − yi| i=1 p 1/2 X h 2 2i 0 = max 2 |ρi| + max(|αi|, |αi + βi|) , 2(|γi| + |φi|) · |zi − zi|2 i=1

Case 1-C (p1 = 1, p2 = ∞):

0 0 |f(z1, ··· , zp, v) − f(z1, ··· , zp, v)|1 p p X  0 X  0 ≤ |ρi| + max(|αi|, |αi + βi|) |xi − xi| + |γi| + |φi| |yi − yi| i=1 i=1 p X  0 0  ≤ |ρi| + max(|αi|, |αi + βi|) + |γi| + |φi| · max |xi − xi|, |yi − yi| i=1 p X  0 = |ρi| + max(|αi|, |αi + βi|) + |γi| + |φi| · |zi − zi|∞ i=1

Case 2 (p1 = 2):

0 0 |f(z1, ··· , zp, v) − f(z1, ··· , zp, v)|2  Pp Pp Pp 0 Pp 0  i=1 ρixi + i=1 γiyi + u − i=1 ρixi − i=1 γiyi − u =| Pp + Pp Pp Pp + Pp Pp |2 i=1 βixi + i=1 αixi + i=1 φiyi +  − i=1 βixi − i=1 αixi − i=1 φiyi −  93  Pp 0 Pp 0  i=1 ρi(xi − xi) + i=1 γi(yi − yi) =| Pp 0  Pp 0 Pp 0 |2 i=1 βi max(0, xi) − max(0, xi) + i=1 αi(xi − xi) + i=1 φi(yi − yi) p p h X 0 X 0 2 = | ρi(xi − xi) + γi(yi − yi)| i=1 i=1 p p p 1/2 X 0  X 0 X 0 2i + | βi max(0, xi) − max(0, xi) + αi(xi − xi) + φi(yi − yi)| i=1 i=1 i=1 p p h X 0 X 0 2 ≤ |ρi||xi − xi| + |γi||yi − yi| i=1 i=1 p p 1/2 X 0  0 X 0 2i + |βi max(0, xi) − max(0, xi) + αi(xi − xi)| + |φi||yi − yi| i=1 i=1 p p h X 0 X 0 2 ≤ |ρi||xi − xi| + |γi||yi − yi| i=1 i=1 p p 1/2 X 0 X 0 2i + max(|αi|, |αi + βi|)|xi − xi| + |φi||yi − yi| i=1 i=1

Case 2-A (p1 = 2, p2 = 1):

0 0 |f(z1, ··· , zp, v) − f(z1, ··· , zp, v)|2 p p h X 0 X 0 2 ≤ |ρi||xi − xi| + |γi||yi − yi| i=1 i=1 p p 1/2 X 0 X 0 2i + max(|αi|, |αi + βi|)|xi − xi| + |φi||yi − yi| i=1 i=1 p p h X 0 X 0 2 ≤ max(|ρi|) |xi − xi| + max(|γi|) |yi − yi| i=1 i=1 p p 1/2 X 0 X 0 2i + max(max(|αi|, |αi + βi|)) |xi − xi| + max(|φi|) |yi − yi| i=1 i=1 p p h 2 X 0 X 0 2 ≤ max max(|ρi|), max(|γi|) |xi − xi| + |yi − yi| i=1 i=1 p p 1/2 2 X 0 X 0 2i + max max(max(|αi|, |αi + βi|)), max(|φi|) |xi − xi| + |yi − yi| i=1 i=1 h 2 2i1/2 = max max(|ρi|), max(|γi|) + max max(max(|αi|, |αi + βi|)), max(|φi|) p p X 0 X 0  · |xi − xi| + |yi − yi| i=1 i=1 94 h 2 2i1/2 = max max(|ρi|), max(|γi|) + max max(max(|αi|, |αi + βi|)), max(|φi|)

0 · |z − z |1

Case 2-B (p1 = 2, p2 = 2):

0 0 |f(z1, ··· , zp, v) − f(z1, ··· , zp, v)|2 p p h X 0 X 0 2 ≤ |ρi||xi − xi| + |γi||yi − yi| i=1 i=1 p p 1/2 X 0 X 0 2i + max(|αi|, |αi + βi|)|xi − xi| + |φi||yi − yi| i=1 i=1 p p p h X 0 2 X 0 2 X 0 2 ≤ 2 |ρi||xi − xi| + 2 |γi||yi − yi| + 2 max(|αi|, |αi + βi|)|xi − xi| i=1 i=1 i=1 p 1/2 X 0 2i + 2 |φi||yi − yi| i=1 p p h 2 X 0 2 2 X 0 2 ≤ 2 max(|ρi|) |xi − xi| + 2 max(|γi|) |yi − yi| i=1 i=1 p p 1/2 2 X 0 2 2 X 0 2i + 2 max(max(|αi|, |αi + βi|)) |xi − xi| + 2 max(|φi|) |yi − yi| i=1 i=1 p p h 2 X 0 2 2 X 0 2 ≤ 2p max(|ρi|) |xi − xi| + 2p max(|γi|) |yi − yi| i=1 i=1 p p 1/2 2 X 0 2 2 X 0 2i + 2p max(max(|αi|, |αi + βi|)) |xi − xi| + 2p max(|φi|) |yi − yi| i=1 i=1 n n n X 2 X 2 X 2 (∵ By Cauchy-Schwarz inequality, ( aibi) ≤ ( ai )( bi )) i=1 i=1 i=1 1/2 h 2 2 2 2i ≤ 2p max max(|ρi|) + max(max(|αi|, |αi + βi|)) , max(|γi|) + max(|φi|) p X 0 2 0 2 1/2 · (|xi − xi| + |yi − yi| ) i=1 1/2 h 2 2 2 2i ≤ 2p max max(|ρi|) + max(max(|αi|, |αi + βi|)) , max(|γi|) + max(|φi|) p X 0 · |zi − zi|2 i=1

95 Case 2-C (p1 = 2, p2 = ∞):

0 0 |f(z1, ··· , zp, v) − f(z1, ··· , zp, v)|2 p p h X 0 X 0 2 ≤ |ρi||xi − xi| + |γi||yi − yi| i=1 i=1 p p 1/2 X 0 X 0 2i + max(|αi|, |αi + βi|)|xi − xi| + |φi||yi − yi| i=1 i=1 p 2 h X 0 0  ≤ (|ρi| + |γi|) max |xi − xi|, |yi − yi| i=1 p 2 1/2  X 0 0  i + (max(|αi|, |αi + βi|) + |φi|) max |xi − xi|, |yi − yi| i=1 p 2 h 2 X 0 0  ≤ max(|ρi| + |γi|) max |xi − xi|, |yi − yi| i=1 p 2 1/2 2 X 0 0  i + max(max(|αi|, |αi + βi|) + |φi|) max |xi − xi|, |yi − yi| i=1 p 1/2 h 2 2i X 0 0  = max(|ρi| + |γi|) + max(max(|αi|, |αi + βi|) + |φi|) · max |xi − xi|, |yi − yi| i=1 p 1/2 h 2 2i X 0 = max(|ρi| + |γi|) + max(max(|αi|, |αi + βi|) + |φi|) · |zi − zi|∞ i=1

Case 3 (p1 = ∞):

0 0 |f(z1, ··· , zp, v) − f(z1, ··· , zp, v)|∞  Pp Pp Pp 0 Pp 0  i=1 ρixi + i=1 γiyi + u − i=1 ρixi − i=1 γiyi − u =| Pp + Pp Pp Pp + Pp Pp |∞ i=1 βixi + i=1 αixi + i=1 φiyi +  − i=1 βixi − i=1 αixi − i=1 φiyi −   Pp 0 Pp 0  i=1 ρi(xi − xi) + i=1 γi(yi − yi) =| Pp 0  Pp 0 Pp 0 |∞ i=1 βi max(0, xi) − max(0, xi) + i=1 αi(xi − xi) + i=1 φi(yi − yi) p p  X 0 X 0 = max | ρi(xi − xi) + γi(yi − yi)| i=1 i=1 p p p X 0  X 0 X 0  , | βi max(0, xi) − max(0, xi) + αi(xi − xi) + φi(yi − yi)| i=1 i=1 i=1 p p  X 0 X 0 ≤ max |ρi||xi − xi| + |γi||yi − yi| i=1 i=1

96 p p X 0  0 X 0  , |βi max(0, xi) − max(0, xi) + αi(xi − xi)| + |φi||yi − yi| i=1 i=1 p p p p  X 0 X 0 X 0 X 0  ≤ max |ρi||xi − xi| + |γi||yi − yi|, max(|αi|, |αi + βi|)|xi − xi| + |φi||yi − yi| i=1 i=1 i=1 i=1

Case 3-A (p1 = ∞, p2 = 1):

0 0 |f(z1, ··· , zp, v) − f(z1, ··· , zp, v)|∞ p p  X 0 X 0 ≤ max |ρi||xi − xi| + |γi||yi − yi| i=1 i=1 p p X 0 X 0  , max(|αi|, |αi + βi|)|xi − xi| + |φi||yi − yi| i=1 i=1 p p  X 0 X 0 ≤ max max(|ρi|) |xi − xi| + max(|γi|) |yi − yi| i=1 i=1 p p X 0 X 0  , max(max(|αi|, |αi + βi|)) |xi − xi| + max(|φi|) |yi − yi| i=1 i=1 p p   X 0 X 0  ≤ max max max(|ρi|), max(|γi|) |x − x | + |y − y | i=1 i=1 p p  X 0 X 0  , max max(max(|αi|, |αi + βi|)), max(|φi|) |x − x | + |y − y | i=1 i=1    = max max max(|ρi|), max(|γi|) , max max(max(|αi|, |αi + βi|)), max(|φi|) p p X X · |x − x0| + |y − y0| i=1 i=1    = max max max(|ρi|), max(|γi|) , max max(max(|αi|, |αi + βi|)), max(|φi|) p X 0 · |zi − zi|1 i=1

Case 3-B (p1 = ∞, p2 = 2):

0 0 |f(z1, ··· , zp, v) − f(z1, ··· , zp, v)|∞ p p  X 0 X 0 ≤ max |ρi||xi − xi| + |γi||yi − yi| i=1 i=1 p p X 0 X 0  , max(|αi|, |αi + βi|)|xi − xi| + |φi||yi − yi| i=1 i=1 97 p  1/2 X 2 0 2 2 0 2 1/2 ≤ max 2 (|ρi| |xi − xi| + |γi| |yi − yi| ) i=1 p 1/2 X 2 0 2 2 0 2 1/2 , 2 (max(|αi|, |αi + βi|) |xi − xi| + |φi| |yi − yi| ) i=1 2 2 1/2 (∵ a + b ≤ (2a + 2b ) ) p  1/2 X 2 2 1/2 0 2 0 2 1/2 ≤ max 2 max(|ρi| , |γi| ) (|xi − xi| + |yi − yi| ) i=1 p 1/2 X 2 2 1/2 0 2 0 2 1/2 , 2 max(max(|αi|, |αi + βi|) , |φi| ) (|xi − xi| + |yi − yi| ) i=1  1/2 2 2 1/2 1/2 2 2 1/2  ≤ max 2 max(max(|ρi| , |γi| ) ), 2 max(max(max(|αi|, |αi + βi|) , |φi| ) ) p X 0 2 0 2 1/2 · (|xi − xi| + |yi − yi| ) i=1  1/2 2 2 1/2 1/2 2 2 1/2  ≤ max 2 max(max(|ρi| , |γi| ) ), 2 max(max(max(|αi|, |αi + βi|) , |φi| ) ) p X 0 · |zi − zi|2 i=1

Case 3-C (p1 = ∞, p2 = ∞):

0 0 |f(z1, ··· , zp, v) − f(z1, ··· , zp, v)|∞ p p  X 0 X 0 ≤ max |ρi||xi − xi| + |γi||yi − yi| i=1 i=1 p p X 0 X 0  , max(|αi|, |αi + βi|)|xi − xi| + |φi||yi − yi| i=1 i=1 p  X  0 0  ≤ max |ρi| + |γi| max |xi − xi|, |yi − yi| i=1 p X  0 0  , max(|αi|, |αi + βi|) + |φi| max |xi − xi|, |yi − yi| i=1 p   X 0 0  ≤ max max |ρi| + |γi| max |xi − xi|, |yi − yi| i=1 p  X 0 0  , max max(|αi|, |αi + βi|) + |φi| max |xi − xi|, |yi − yi| i=1    = max max |ρi| + |γi| , max max(|αi|, |αi + βi|) + |φi|

98 p X 0 0  · max |xi − xi|, |yi − yi| i=1    = max max |ρi| + |γi| , max max(|αi|, |αi + βi|) + |φi| p X 0 · |zi − zi|∞ i=1

99 Appendix C: Appendix to Chapter 3

In this appendix, we derive Equations (3.10), (3.11), and (3.12) that jointly de- scribe the model economy. First, the representative household maximizes the ex- pected lifetime utility given by

∞ 1+φ X t Nt max E0 β [log(Ct − bCt−1) − ] (C.1) Ct,Nt,Bt 1 + φ t=0

s.t. PtCt + Bt ≤ (1 + Rt−1)Bt−1 + WtNt − Tt (C.2)

The first-order conditions are

−1 −1 ΛtPt = (Ct − bCt−1) − βbEt[(Ct+1 − bCt) ] (C.3)

φ ΛtWt = Nt (C.4)

Λt = β(1 + Rt)EtΛt+1 (C.5) where Λ is the Lagrange multiplier associated with the budget constraint. Using the

t −t fact that Ct = γ ct and ΛtPt = γ λt, Equation (C.3) can be rewritten as

b β b λ = (c − c )−1 − bE [(c − c )−1]. (C.6) t t γ t−1 γ t t+1 γ t

Equation (C.6) can be log-linearized as

β b b (1 − β b)(1 − b ) cˆ = γ E cˆ + γ cˆ − γ γ λˆ . (C.7) t β b2 t t+1 β b2 t−1 β b2 t 1 + γ γ 1 + γ γ 1 + γ γ

100 By the way, the resource constraint can be log-linearized as

c¯ yˆ = cˆ +g ˆ (C.8) t y¯ t t

gt−g¯ whereg ˆt = y¯ . Using Equation (C.8), Equation (C.7) can be rewritten as

β β c¯ b c¯ b c¯ (1 − b)(1 − b ) c¯ cˆ = γ E ( cˆ ) + γ cˆ − γ γ λˆ (C.9) y¯ t β b2 t y¯ t+1 β b2 y¯ t−1 β b2 y¯ t 1 + γ γ 1 + γ γ 1 + γ γ ¯ ˆ yˆt = A1yˆt−1 + A1βγEt(ˆyt+1) − A2λt + dt(ˆgt, gˆt−1, gˆt+1) (C.10)

b (1−βb¯ )(1− b ) ¯ β γ γ c¯ ¯ where β ≡ γ ,A1 ≡ ¯ b2 ,A2 ≡ ¯ b2 y¯, and dt ≡ gˆt − A1gˆt−1 − A1βγgˆt+1. 1+β γ 1+β γ Also Equation (C.5) can be rewritten as

β 1 λt = (1 + Rt)Et[λt+1 ] (C.11) γ 1 + πt+1

where π = log Pt+1 . Equation (C.11) can be log-linearized as t+1 Pt

ˆ ˆ ˆ λt = Rt + Et[λt+1 − πˆt+1] (C.12)

Next, each monopolistic firm j set prices according to a variant of Calvo pricing.

∞ X Pt+k−1 max E θk[Q {P ∗( )χ(1 +π ¯)k(1−χ)Y (j) − TC(Y (j))}] (C.13) ∗ t t,t+k t t+k t+k Pt Pt−1 k=0 ∗ χ k(1−χ) Pt (Pt+k−1/Pt−1) (1 +π ¯) − s.t. Yt+k(j) = Yt+k[ ] (C.14) Pt+k t α Yt(j) = γ exp(zt)Nt(j) (C.15)

k UC,t+k/Pt+k k Λt+k where Qt,t+k = β = β is the nominal discount factor. The first-order UC,t/Pt Λt condition is

∞ X k Λt+k Pt+k−1 χ k(1−χ) ∗ Pt+k−1 χ k(1−χ) dYt+k(j) Et (βθ) [ {( ) (1 +π ¯) Yt+k(j) + Pt ( ) (1 +π ¯) ∗ Λt Pt−1 Pt−1 dP k=0 t dYt+k(j) − MC(Yt+k(j)) ∗ }] = 0 (C.16) dPt 101 dYt+k(j) where MC(·) represents the nominal marginal cost. Using the fact that ∗ = dPt

Yt+k(j) − ∗ , Equation (C.16) can be rewritten as Pt ∞ ∗ χ k(1−χ) X k Pt (Pt+k−1/Pt−1) (1 +π ¯) − Et (βθ) [λt+kyt+k{ } Pt+k k=0 P ∗ P {(1 − ) t ( t+k−1 )χ(1 +π ¯)k(1−χ) + mc(j)}] = 0 (C.17) Pt+k Pt−1

∗ where mc(·) represents the real marginal cost. Denoting q = Pt , Equation (C.17) t Pt can be rewritten as

∞ ∗ χ k(1−χ) X ¯ k Pt (Pt+k−1/Pt−1) (1 +π ¯) 1− qtEt (βγθ) [λt+kyt+k( − 1){ } ] Pt+k k=0 ∞ X ¯ k = Et (βγθ) [λt+kyt+kmc(j)]. (C.18) k=0

∗ χ k(1−χ) Pt (Pt+k−1/Pt−1) (1+¯π) Using the fact that can be log-linearized as 1 + χ(ˆπt + ··· + Pt+k

πˆt+k−1) − (ˆπt+1 + ··· +π ˆt+k), Equation (C.18) can be log-linearized as ∞ 1 X qˆ =E (βγθ¯ )kmcˆ (j) 1 − βγθ¯ t t t+k k=0 βγθ¯ + E [(ˆπ − χπˆ ) + βγθ¯ (ˆπ − χπˆ ) + ··· ]. (C.19) 1 − βγθ¯ t t+1 t t+2 t+1

Aggregate price index is

1− ∗1− χ 1−χ 1− Pt = (1 − θ)Pt + θ[(1 + πt−1) (1 +π ¯) Pt−1] (C.20) which can be log-linearized as

θ(ˆπ − χπˆ ) qˆ = t t−1 . (C.21) t 1 − θ

Also the relationship between individual firm’s real marginal cost and the econ- omy’s average real marginal cost can be derived as below. First, individual firm’s real marginal cost is

Wt/Pt wt/pt wt/pt mct(j) = t α−1 = α−1 = (C.22) αγ exp(zt)Nt(j) α exp(zt)Nt(j) αyt(j)/Nt(j) 102 α where yt(j) = exp(zt)Nt(j) . Equation (C.22) can be log-linearized as

α − 1 1 mcˆ (j) = (w ˆ − pˆ ) − yˆ (j) − z (C.23) t t t α t α t

Similarly, the log-linearized economy’s average real marginal cost is

α − 1 1 mcˆ = (w ˆ − pˆ ) − yˆ − z (C.24) t t t α t α t

which means that

α − 1 mcˆ (j) =mc ˆ − (ˆy (j) − yˆ ) (C.25) t t α t t

∗ χ k(1−χ) Pt (Pt+k−1/Pt−1) (1+¯π) − Using the fact that yt+k(j) = yt+k[ ] and so Pt+k

yˆt+k(j) =y ˆt+k − qˆt + [(ˆπt+1 − χπˆt) + ··· + (ˆπt+k − χπˆt+k−1)], (C.26)

Equation (C.25) can be rewritten as

mcˆ t+k(j) =mc ˆ t+k − A[ˆqt − [(ˆπt+1 − χπˆt) + ··· + (ˆπt+k − χπˆt+k−1)]] (C.27)

(1−α) where A = α . Therefore, using Equation (C.18), (C.21), and (C.27),

(1 − θ)(1 − βγθ¯ ) πˆ − χπˆ = βγE¯ (ˆπ − χπˆ ) + mcˆ (C.28) t t−1 t t+1 t (1 + A)θ t

Nowmc ˆ t can be derived as below. First, from Equation (C.3) and (C.4),

w N φ t = t (C.29) p b −1 β b −1 t (ct − γ ct−1) − γ bEt[(ct+1 − γ ct) ] Equation (C.29) can be log-linearized as

ˆ 1 ¯ 1 wˆt − pˆt = φNt + [ˆyt − A1βγEt(ˆyt+1) − A1yt−1] − dt (C.30) A2 A2

Therefore,

ˆ mcˆ t = (w ˆt − pˆt) − (ˆyt − Nt)

103 1 φ + 1 A1 ¯ A1 1 φ + 1 = ( − 1 + )ˆyt − βγEt(ˆyt+1) − yt−1 − dt − zt (C.31) A2 α A2 A2 A2 α

Finally, central bank follows a generalized Taylor rule:

1 + R 1 + E (π ) Y 1 + R 1 + R t t t+1 φπ t φy (1−ρ1−ρ2) t−1 ρ1 t−2 ρ2 ¯ = [{ } { ¯ } ] ( ¯ ) ( ¯ ) exp(mt) 1 + R 1 +π ¯ Yt 1 + R 1 + R (C.32) which can be log-linearized as

ˆ ˆ ˆ Rt = (1 − ρ)[φπEtπˆt+1 + φyyˆt] + ρ1Rt−1 + ρ2Rt−2 + mt (C.33)

104