ABSTRACT

LI, HUACHEN. Essays in Macroeconometrics. (Under the direction of James M. Nason).

This dissertation consists of two essays in macroeconomics and macroeconometrics. The first paper studies macroeconomic labor market fluctuations and effects in a structural vector autoregres- sion (SVAR) with time-varying parameter (TVP) and stochastic volatility (SV). The second chapter extends the first chapter to study the macroeconomic effects of immigration. There is no consensus among macroeconomists about the response of hours worked to a TFP shock. The first chapter of this dissertation offers new evidence on the TFP - hours worked rela- tionship. I extend [Gal99]. The extension estimates Galí’s bi-variate regression of average labor productivity (ALP) and hours worked in a SVAR that includes TVP and SV to test the labor market predictions of the New-Keynesian (NK) models against Real Business Cycle (RBC) models. I update Galí’s SVAR by including time-varying parameter (TVP) and stochastic volatility (SV). The TVPs and SV are included in the SVAR to address structural change and large but short-run variation in the data. My estimates show that the TVPs and SV are important for resolving the lack of consensus surrounding Galí’s hypothesis, and present a new challenge that supports RBC and NK theories. The second essay explores the impact of immigration on the macroeconomy. Shocks to immigra- tion are identified within a SVAR with short- and long-run restrictions. The key short-run restriction is that immigration does not respond to other shocks at impact. The long-run restrictions are Galí’s long-run neutrality of labor productivity to labor market shocks, a permanent income shock, and long-run neutrality of immigration with respect to transitory shocks. I estimate TVP-SV-SVARs to account for structural change in the U.S. economy and changes in U.S. immigration policy over the sample. The estimates indicate that over the business cycle, productivity decreases with respect to an immigration shock. Over the same horizon, hours worked falls. © Copyright 2019 by Huachen Li

All Rights Reserved Essays in Macroeconometrics

by Huachen Li

A dissertation submitted to the Graduate Faculty of North Carolina State University in partial fulfillment of the requirements for the Degree of Doctor of Philosophy

Economics

Raleigh, North Carolina

2019

APPROVED BY:

Giuseppe Fiori Douglas Pearce

Xiaoyong Zheng James M. Nason Chair of Advisory Committee DEDICATION

This is for my grandparents, mom, dad, and Rachel.

ii BIOGRAPHY

Huachen Li was born and raised in Shenyang, China. After completing high school in Shenyang, he attended the College of Wooster as a member of Class of 2012 and received a bachelor’s degree in business economics. In 2013, he received his master’s degree in management from Wake Forest

University and worked as a search engine marketing analyst. Huachen began his doctoral study in economics at North Carolina State University in 2014. His graduate research is in macroeconomics and macroeconometrics. During his time in Raleigh, he worked as an instructor in the Poole College of Management and as an adjunct professor at Meredith College. Huachen will join the Department of Economics at Kenyon College as an assistant professor starting fall 2019.

iii ACKNOWLEDGEMENTS

This dissertation and the completion of my degree would not have been possible without the effort, advice, and encouragement of my advisor, James M. Nason. He has been a wonderful mentor pro- viding the ideal balance between guidance and autonomy, which allowed me to grow substantially to become an economist. I am deeply grateful for his patience, enthusiasm, and flexibility, especially on the days that I needed it the most.

I would like to thank my dissertation committee, Giuseppe Fiori, Douglas Pearce, and Xiaoyong

Zheng for their contribution towards this dissertation and enhancement of my education. I am very lucky to have had committee members who are experts in economics and incredible proofreaders.

I owe a tremendous amount of thanks to Lee Craig for his continuous help in preparing me to become a junior faculty. I would also like to thank Xiaoyong Zheng and Tamah Morant, without whom I would not have been where I am today.

Additionally, I thank Robin Carpenter, Ayse Kabukcuoglu Dur, Ilze Kalnina, Ivan Kandilov,

Soumendra Lahiri, Thayer Morrill, Denis Pelletier, Bobby Puryear, Allison Lowe Reed, Nora Traum,

Sherley Teel, Walter Thurman, Mark Walker, Jingjing Yang, Anne York, Curtis Youngblood, and many others who helped to shape me into the economist I am today.

The journey of my graduate studies was amazing, but also challenging. Thank you Andrew

Hessler, Sasha Naumenko, Kelly Nelson, Kenny Rich, Bila Zhaladai, and many other friends and colleagues for making it more of the former. Special thanks to Carmichael Gymnasium for keeping me sane.

Finally, I wish to express my gratitude to my family who has always given me unconditional love.

Thank you, Rachel, for making all this work worthwhile.

iv TABLE OF CONTENTS

LIST OF TABLES ...... viii

LIST OF FIGURES ...... ix

Chapter 1 The Time-Varying Response of Hours Worked to a Productivity Shock ...... 1 1.1 Introduction ...... 1 1.2 Data and Instabilities...... 5 1.2.1 Data...... 5 1.2.2 Structural Breaks in a Reduced-form VAR...... 5 1.3 A TVP-SV-SVAR and Bayesian Estimation ...... 7 1.3.1 TVP-SV-SVAR...... 7 1.3.2 Priors...... 9 1.3.3 Sampling Algorithm ...... 10 1.4 The Effect of TFP Shocks with Time Variation ...... 11 1.4.1 IRFs of Hours Worked to a TFP Shock...... 11 1.4.2 IRFs of ALP to a Demand Shock...... 12 1.4.3 IRFs of Output to a TFP Shock...... 13 1.4.4 Evidence from FEVDs...... 14 1.5 Source of Structural Change...... 16 1.5.1 Estimates of Stochastic Volatility...... 16 1.5.2 Estimates of Structural Intercepts and Slope Parameters...... 17 1.5.3 Discussion...... 17 1.6 Conclusion ...... 18

Chapter 2 The Time-Varying and Volatile Macroeconomic Effects of Immigration ...... 32 2.1 Introduction ...... 32 2.2 U.S. Immigration Policy and Data...... 34 2.2.1 Background on Recent Changes to Immigration Policy...... 35 2.2.2 Data...... 35 2.3 TVP-SV-SVARs...... 36 2.3.1 TVP-SV-SVAR Methodology...... 37 2.3.2 Identification...... 38 2.3.3 Bayesian Estimation...... 43 2.4 Results ...... 43 2.4.1 Bayesian Model Selection Results ...... 44 2.4.2 Structural Dynamic Responses ...... 45 2.4.3 FEVD of the Structural Shocks...... 48 2.4.4 Evidence from Time-Varying Parameters and Stochastic Volatility...... 48 2.5 Conclusion ...... 49

BIBLIOGRAPHY ...... 63

APPENDICES ...... 70

v Appendix A Appendix for Chapter 1 ...... 71 A.1 Data Source and Construction ...... 71 A.1.1 Hours Worked ...... 71 A.1.2 ALP Data ...... 72 A.2 A Fixed-Coefficient SVAR and Identification ...... 73 A.2.1 A Fixed-Coefficient SVAR ...... 73 A.2.2 Identification Assumptions...... 75 A.3 Details about the Andrews Test...... 76 A.4 Details in Sampling Algorithms...... 77 A.4.1 Alternative Prior...... 77 A.4.2 Draw Bħ...... 77 A.4.3 Draw H ...... 78 A.4.4 Draw ...... 79 A.5 V Convergence Diagnosis ...... 80 A.6 Bayesian Model Selection...... 82 A.7 Additional IRFs ...... 83 Appendix B Appendix for Chapter 2 ...... 85 B.1 Relevant Recent Changes in U.S. Immigration Policy...... 85 B.2 Details in Data Construction ...... 87 B.2.1 Interpolation of Quarterly Immigration...... 87 B.2.2 Results of Alternative Interpolation Methods...... 89 B.2.3 Unit Root Test Results...... 90 B.2.4 Data Construction of Consumption...... 90 B.3 Alternative Assumption in Stationarity of Hours Worked...... 91 B.4 Necessary and Sufficient Conditions of Identification Schemes...... 92 B.5 Priors and Estimation Setup...... 96 B.6 Details in the Sampling Algorithm ...... 97 B.6.1 Draw Time-Varying Coefficients ...... 97 B.6.2 Draw A0 ...... 98 B.6.3 Draw Σ ...... 98 B.6.4 Draw ...... 99 B.7 ConvergenceV Diagnosis ...... 100 B.8 Computation of Log Marginal Likelihood ...... 102 B.8.1 Estimating p(z θ ) ...... 102 B.8.2 Estimating p(θ|) ...... 103

vi B.8.3 Estimating p(z)...... 103

vii LIST OF TABLES

Table 1.1 Priors ...... 9 Table 1.2 FEVD at NBER Recession Dates, Hours Worked in Growth Rates...... 15 Table 1.3 FEVD at NBER Recession Dates, Hours Worked in Log Levels...... 15

Table 2.1 List of Models ...... 42 Table 2.2 Log Marginal Data Densities and Bayes Factors...... 44

Table D.3 Parameters of Mixture Normal Distributions...... 79 Table E.1 Inefficiency Factors for Different Sets of Parameters...... 80 Table F.1 List of Model Comparison ...... 82 Table F.2 Log Marginal Likelihood ...... 82

Table A1 SSC Interpolation Statistics ...... 88 Table B.1 Augmented Dickey-Fuller Test for Unit Root ...... 90 Table B.3 List of Model Comparison, Hours Worked in Growth Rates ...... 91 Table B.4 Log Marginal Likelihood and Bayes Factor of All Estimated Models...... 92 Table B.5 Impact and Long-run Restrictions of Model 1 ...... 93 Table B.6 Priors ...... 96 Table B.7 Inefficiency Factors for Different Sets of Parameters...... 100

viii LIST OF FIGURES

Figure 1.1 Average Labor Productivity in Growth Rates...... 20 Figure 1.2 Log Weekly Hours per capita...... 21

Figure 2.3 IRFs on Selected Dates of Changes to Immigration Policy...... 53 Figure 2.4 IRFs at 4 Quarters Post Selected Dates of Changes to Immigration Policy . . . . 54

T T Figure A.1 Recursive means for Bħ and H ...... 81 Figure A.2 IRFs of hours worked with respect to a TFP shock on selected NBER recession (trough) dates...... 83 Figure A.3 IRFs of ALP with respect to a demand shock on selected NBER recession (trough) dates...... 83 Figure A.4 IRFs of output with respect to a TFP shock on selected NBER recession (trough) dates...... 84

Figure B.1 Comparison across interpolation methods for immigration data. The results from Fernandez, Litterman, and SSC are shifted vertically of equal distance for better viewability...... 89 Figure B.2 Sequence of Retained Draws...... 101

ix CHAPTER

1

THE TIME-VARYING RESPONSE OF HOURS WORKED TO A PRODUCTIVITY SHOCK

1.1 Introduction

The impact of a total factor productivity (TFP) shock on hours worked has been debated by macroe- conomists since at least [Luc69]. One reason for the controversy is the disparate predictions made by neo-classical and new Keynesian models. Neo-classical models call for a positive response of hours worked to a TFP shock. Traditional and new Keynesian models promise that hours worked fall in reaction to a TFP shock. The lack of consensus among macroeconomists is given empirical content by studying the dynamic relationship between average labor productivity (ALP) and hours worked. [Gal99] stands out as important for changing the debate about TFP and hours worked. He finds evidence that a positive TFP shock leads to a decline in hours worked. This rejects a key prediction of real business cycle (RBC) theory. This is evidence for Galí to argue that output fluctuations are dominated by demand shocks. Hence, the result favors new Keynesian models with imperfect competition and sticky prices, according to Galí. I investigate the TFP-hours worked relationship following [Gam05] and [Gal09] to include time-

1 varying parameters (TVPs) in a structural vector autoregression (SVAR) that also has stochastic volatility (SV) in the errors. Advances in econometric methods allow me to estimate the TVP-SV- SVAR that is consistent with the non-linear identification induced by Galí’s identifying restriction. The TVP-SV-SVARs are estimated to ask: Can robust implications about the effect of hours worked to a TFP shock be drawn by allowing for structural change in the post-war U.S. data? Galí casts doubt on the RBC paradigm using a fixed-coefficient SVAR with ALP and hours worked. Motivated by the results of Dickey-Fuller tests, he assumes ALP and hours worked are integrated of order one, which means both variables are first-differenced (in logs) to achieve stationarity. Galí uses a long-run restriction to identify the orthogonalized structural shocks. The restriction imposes long-run neutrality on ALP.This restricts ALP to respond only to TFP shocks in the long-run.1 The identification scheme relies on the existence of a unit root in ALP,but can accommodate either I (0) or I (1) behavior in hours worked. Galí constructs the impulse response function (IRF) of hours worked with respect to a TFP shock in several steps. First, estimate a finite-order reduced-form VAR on ALP and hours worked. Next, create the associated reduced-form infinite-order vector moving average (VMA). The structural impact matrix is computed by Galí using the [Bla89] decomposition. The Blanchard and Quah (BQ) decomposition equates the reduced-form VMA( ) to a structural VMA( ) by a Cholesky factorization of the reduced-form error covariance matrix.∞ Given the esti- mated∞ structural impact matrix, Galí calculates the structural IRF of hours worked with respect to a TFP shock. Keeping [Gal99]’s long-run identification intact, this paper presents evidence about the response of hours worked to a TFP shock that is robust to structural changes in the U.S. economy since 1948. Similar to [Gal09], I account for structural change in the data with TVPs and SVs. Including TVPs and SVs gives the SVAR the flexibility to capture any changes in the conditional mean, volatility, and/or persistence in the data. Allowing for changes in the macroeconomic environment is an important step in deriving reliable VAR estimates ([Sto02]; [Gal03]). I estimate TVP-SV-SVARs on ALP and hours worked from 1948Q1 to 2016Q4. The estimates rely on a Metropolis-within-Gibbs algorithm developed by [Can15]. The Canova & Perez-Forero (CPF) TVP-SV-SVAR estimator is useful for at least three reasons. First, the CPF estimator is flexible enough to consistently estimate the BQ decomposition within a TVP-SV-SVAR. The intercepts and slope parameters in this paper are consistent with the long-run identification. The algorithm is a Bayesian Markov chain Monte Carlo (MCMC) posterior simulator that estimates a non-linear state space model. Since long-run identification is non-linear in the reduced-form TVP parameter space, the CPF algorithm imposes restrictions during the Gibbs sampling steps to estimate long-run identified SVARs, instead of at post-estimation after a block by block routine such as [Pri05]. Second, the structural intercepts and slope parameters are adjusted and checked for station- arity at each draw and each date following [Koo11]. According to Koop & Potter, failure to ensure 1In this paper, I refer to technology (non-technology) shock and TFP (demand) shock interchangeably.

2 stationarity of the intercepts and slope parameters at each date leads to erroneous estimates of the posterior of the Bayesian SVAR. Third, the sampler in this paper draws from the correct posterior distribution of the SVAR parameters following the [DN15] correction to sampling SV. Structural change creates non-linearities in ALP and hours worked. The TVPs allow time variation in intercepts and slope parameters of the SVAR that account for structural change in the data. The TVPs pick up changing first moments and co-movement in the ALP and hours worked relationship generated by, for example, the productivity slowdown of the 1970s, the Great Moderation, the technology boom and bust of the late 1990s, and the recent financial crisis. The addition of SVs captures time-varying heteroskedasticity in the structural shocks that account for episodes of changing volatility. The TVP-SV-SVAR provides a parsimonious platform from which to address underlying structural change in the data generating process, without relying on assumptions about the stationarity of hours worked, ad hoc break dates, or filtering low-frequency movements in the data. To the best of my knowledge, only [Gam05], [Gal09] and [Can17] study the ALP-hours worked relationship with a VAR that allows for time variation. [Gam05] uses a TVP-SVAR with static volatility. [Gal09] adapt a [Pri05] framework to analyze the importance of the Great Moderation. They assume stationary hours worked. [Can17] use a static coefficient VAR with rolling-windows of fixed length to account for low-frequency movements in data.2 They find the responses of hours worked to a TFP shock have increased over time. A large body of literature highlights that the U.S. economy has experienced structural change since the 1950s ([Fis88]; [Wat94]; [Nor04]; [Cog05]). In addition to the evidence on the productivity slowdown of the 1970s and the Great Moderation that is often dated to 1984, the most recent financial crisis of 2009 suggests another source of structural change in the economy ([Fer14]; [Rot17]). Much of the business cycle literature ([Ton83]; [Ham89]; [Sim06]; [Nas15]) emphasizes the need to address non-linearities arising from state-dependency in business cycle dynamics due to structural change. Although the identification and extent of this change in the U.S. economy are not the focus of this paper, allowing for structural change is crucial to study the co-movement of TFP shocks and hours worked in the U.S. during the last 70 years. This paper is also related to the TFP literature that addresses structural change as a potential problem of inference in the ALP-hours worked relationship. For example, after accounting for low- frequency movements in hours worked by modeling explicit structural break dates [Fer07] or filtering the data in various ways [Can10], TFP shocks are shown to lower hours worked. [Fra09] extend Galí’s SVAR with HP-filtered hours worked and demographic controls and conclude that hours worked decline in the short-run in reaction to a TFP shock.

2 [Can17] first estimate a SVAR on a sub-sample from 1948Q1 to 1978Q1. They then fix the 87-quarter sample size but move the start date forward by 4 quarters. This results in 39 rolling-window estimates given their total sample from 1948Q1 to 2009Q1.

3 The extant literature has not reached a consensus about the response of hours worked to a TFP shock. Galí’s view of the response of hours worked to a TFP shock is also supported by [Fra03], [Bas06], [Fer07], [Fra09], and [Can10]. For instance, [Fer07] removes the low-frequency movements in hours worked and finds a negative conditional correlation between hours worked and TFP. [Can10], on the other hand, examine the movements in hours worked with periodicity of 52 quarters and higher. They argue that these changes in the long-cycle mean of hours worked distort the dynamics of the SVARs. They propose various detrending methods to account for structural change. These conclusions are not universally accepted. [Chr04] replicate Galí’s exercise with the as- sumption that hours worked is I (0); therefore, hours worked should enter the SVAR as levels. They find hours worked rise in the short-run, as predicted by RBC theory. There is also support for [Chr04] by [Cha09], [Pee09], and [Hol12], among others. My estimates show that the lack of consensus is resolved by estimating a TVP-SV-SVAR. Once structural changes in hours worked and ALP are addressed by TVPs and SV, hours worked falls in response to a TFP shock on impact and at the business cycle horizons. The negative responses appear stable over the entire sample. This result also holds with hours worked in log levels or growth rates. The response of ALP is positive following a demand shock but there is drift in the height of the IRFs over the business cycle horizons across the sample. These findings are in line with [Gal99] that supports new Keynesian theory. On the other hand, my results provide a new challenge to the literature. I report forecast error variance decompositions (FEVDs) on the structural shocks. The TFP shock dominates the fluctua- tions in ALP.The importance of the TFP shock is stable across the sample and robust to how hours worked enters the SVAR. The contribution of the TFP shock to hours worked exhibits substantial time variation especially when hours worked is in growth rates. Dominance of the TFP shock on ALP supports RBC theory. These findings are explained by movements in the structural slope parameters and SV of the ALP regression. In line with [Gam05], there is large time variation and business cycle dependency in the sum of own lags in the ALP regression. I also find a steep decline in the estimated SV of the TFP regression over the sample, which matches the timing of the productivity slowdown of the 1970s, the Great Recession, and other episodes of structural change. These changes in labor productivity are consistent with evidence of movements in productivity at the firm level as discussed in [Com06]. On the other hand, the hours worked regression does not show statistically or economically important drift during the sample. Therefore, my results help to resolve the conflicting estimates in the dynamic response of hours worked to a TFP shock and present a new challenge to the literature. Given episodes of structural change and instability in the structural parameters from aggregate productivity, these results are challenging to explain in the context of either RBC or new Keynesian theories. The remainder of the paper is organized as follows: Section 2 describes the data and the evidence of structural change; Section 3 sets up the TVP-SV-SVAR and outlines the estimation procedure;

4 Sections 4 and 5 address the TFP - hours worked puzzle, present novel challenging results, and discuss the source of structural change from the TVP-SV-SVARs; and Section 6 concludes.

1.2 Data and Instabilities

This section discusses the data used to estimate the SVARs. I present evidence of structural change in ALP and hours worked. I calculate the Wald statistics of the intercepts, slope parameters, standard deviations, and covariance from a reduced-form VAR using a test of structural change with unknow break points of [And93].

1.2.1 Data

The SVARs are estimated on quarterly data. The data are ALP and hours worked ranging from 1948Q1 to 2016Q4. Non-farm real output growth per hours worked of all persons measures ALP.This definition of ALP follows [Fer07] and [Fra09]. Hours worked is measured as the log weekly hours worked per capita. I construct this series using official sources following [Coc18]. Details about the data and alternative measurements for robustness are found in Appendix A.1. Figure 1.1(a) displays ALP.It exhibits changes from pro-cyclicality in the first half of the sample to counter-cyclical behavior after the recessions of the early 1980s. The TFP literature associates this change in the cyclicality of productivity with structural change in the U.S. economy. For in- stance, [Fer16] document stylized facts about these variations in productivity. They show changes in cyclicality are caused by structural change in the macroeconomy. Log weekly hours worked per capita is in Figure 1.2(b). From [Fer07], I include three sub-samples with explicit break dates of 1961Q1 and 1982Q4.3 These break dates are consistent with the pro- ductivity slowdown and the Great Moderation. The upper left line represents average log hours for sub-sample 1948Q1 to 1961Q1; the middle lower line shows the sub-sample mean of 1961Q2 to 1982Q4; and the top right bar is the sub-sample mean between 1983Q1 and 2016Q4. A large jump in the sub-sample means occurs at the start of the Great Moderation in 1983. [Fer07] refers to these changes in the sub-sample means in hours worked as a “high-low-high" pattern. He argues struc- tural changes in hours worked, such as composition changes in school enrollment and government employment, cause these high-low-high movements.

1.2.2 Structural Breaks in a Reduced-form VAR

Instability in ALP and hours worked can affect estimates and inference of the relationship between TFP and hours worked. I assess the statistical importance of structural breaks in ALP and hours

3 [Fer07] adopts the Bai-Perron test to estimate break dates of hours worked. The Bai-Perron test examines an unknown number of breaks in the mean during a sample interval. The test is robust to heteroskedasticity and autocorrelation. A comparable Bai-Perron test result is also found in this paper.

5 worked within a reduced-form VAR. The VAR serves as a vehicle to test the statistical significance of changes in the means, persistence, volatility, and co-movement of ALP and hours worked. The VAR is p X zt = c + Bi zt i + et , et N (0,Ω), p = 2. − i =1 ∼ where z x n contains ALP (x ) and hours worked (n). For this exercise, I estimate the reduced- t = [ t t ]0 form VARs with hours worked in either log levels or growth rates. [And93] offers a Wald test of structural change with unknown break points. I compute the Andrews statistics on the intercepts from c , sum of slope parameters from Bi ’s, and the covariance matrix Ω. The test statistics are computed on a rolling-window basis over an interval of possible break dates for the null that a parameter is constant through the sample. After a trimming of 15% on the data as recommended by [And93], the test is performed on a sample from 1954Q1 to 2011Q4. Details about the Andrews test are in Appendix A.3. The Wald test statistics yield evidence of several break points. Figures ?? and ?? report the Wald test statistics on the reduced-form intercepts and sum of slope parameters when hours worked enters as log levels and growth rates. The sum of own lags in the ALP regression suggests a break point during the Great Recession of 2008-2009. There is instability in the slope parameters of the hours worked regression, but the significance depends on how hours worked enters the VARs. Moreover, the left panels of Figures ?? and ?? display the Andrew’s Wald statistic to test for structural change in the reduced-form intercepts. In the lower left panel of Figure ?? with hours worked in log levels, breaks in the intercept parameters of hours worked occur around the 1973 - 1975 recession and the 1981 - 1982 recession. These breaks are significant at the 10% level using the critical values in [And03]. Compared to the left side panel of Figure ?? with hours worked in growth rates, the 1953 - 1954 and 2007 - 2009 recessions show a significant influence on the instability of the reduced-form intercepts when hours worked is in log levels. The reduced-form VAR appears to have heteroskedastic errors. Figure ?? plots the Wald statistics on the variance-covariance matrix of the reduced-form errors from the ALP and hours worked regressions. Regardless of how hours worked enters the VAR, there is significant evidence on the instability of the reduced-form standard deviations of ALP and hours worked at the 10% confidence level. Furthermore, movements in the Wald statistics on the covariance of the residuals appear to be state dependent on the NBER recession dates. The Andrews test suggests multiple break points to the reduced-form covariance, such as during the first oil price shock and the recent financial crisis in 2009. The instability of the covariance matrix points to the need to relax the restriction of static volatility. In sum, evidence from the reduced-form VAR shows significant time variation in the intercepts, slope parameters, and heteroskedasticity in the forecast innovations of the regressions. The evidence holds whether hours worked enters the VAR as log levels or growth rates. I interpret this result as

6 showing the relationship between ALP and hours worked is not a good description of a time-invariant VAR model.

1.3 A TVP-SV-SVAR and Bayesian Estimation

This section estimates a TVP-SV-SVAR following [Gal99] and [Gal09]. The time-varying feature cap- tures structural change using drifting reduced-form parameters and episodes of SV in productivity and demand shocks. I extend their work with advances in econometrics to consistently impose a long-run restriction. The SVAR is identified with the long-run restriction that a productivity shock is the only source of long-run fluctuation in ALP.I modify CPF’s sampling algorithm to estimate TVP-SV-SVARs with only this long-run restriction.4 This sampler produces structural intercepts and slope parameters that are consistent with the long-run restriction, checks for stationarity step by step following [Koo11], and uses the [DN15] correction to the drawing steps. The sampler obtains the posterior distribution of the TVP-SV-SVAR. The TVPs, SVs, and hyperparameters are drawn sequentially in a Gibbs algorithm.

1.3.1 TVP-SV-SVAR

Consider a p t h order reduced-form VAR

p X zt = ct + Bi ,t zt i + et , et N (0,Ωt ), (1.1) − i =1 ∼ where ct denotes the 2 1 vector of time-varying intercepts, Bi ,t ’s are 2 2 matrices of reduced-form × × slopes that are time-varying, and Ωt represents the time-varying variance-covariance matrices of the Gaussian reduced-form errors, et . Estimates of the reduced-form TVPs and time-varying variance-covariance are treated as latent variables in the TVP-SV-VAR. It is standard to estimate the reduced-form VAR as a state-space model. Rewrite (1.1) as the observation equation

zt = Xt0Bt + et , et N (0,Ωt ), (1.2) ∼ where X I 1, z ,..., z and c , v e c B ,..., v e c B . t0 = M [ t0 1 t0 p ] Bt = [ t ( 1,t )0 ( p,t )0]0 I impose Galí’s⊗ original− identification− on the TVP-SV-VAR of (2.2). The identification assumes the TFP shock is the only source of long-run variation in ALP.The first step in imposing this long-run restriction is to compute: 1 1 2 Γt , = J (I2 Bt )− J 0Ωt , (1.3) ∞ − 4 An alternative estimation algorithm is discussed in [Bog18].

7 where J = [I2,...,02], and Γt , is the matrix of long-run responses of zt to the reduced-form shocks. ∞ i j The elements of Γt , are denoted γt , , which measures the long-run response of variable i to shock ∞ j at quarter t . Imposing the long-run∞ neutrality condition gives the structural cumulative response matrix

– ™ γ11 0 Γ˜ t , , (1.4) t , = 21∞ 22 ∞ γt , γt , ∞ ∞ 11 21 22 ˜ where γt , , γt , , and γt , are computed in (2.3). Note that the Γt , differs from Γt , . Therefore, ∞ ∞ ∞ ∞ ∞ the reduced-form intercepts and slope parameters are not consistent with Γ˜t , . The structural slope ∞ parameters, Bħt , are computed using

1 2 ˜ 1 Bħt = IM Ωt Γt−, , (1.5) − ∞ which is conditional on Ωt and Γ˜t , . Equation (1.5) shows the long-run neutrality restriction creates ∞ non-linearity in Bħt . As a result, the restrictions embedded in (1.4) and (1.5) are placed on each draw from the Gibbs sampler. Details about the sampler appear in Section 1.3.3. Another restriction embedded in the Gibbs sampler is the mapping from the structural shocks,

εt to the reduced-form shocks, et :

– ™ – ™– ™ e1,t 1 h1,t 0 ε1,t = J (I2 Bħt )− J 0 , (1.6) e2,t − δt h2,t ε2,t where , N 0, I , h is the SV of the j t h structural shock, and is the time- εt [ε1,t ε2,t ]0 εt ( ) j,t δt varying covariance.≡ ∼ CPF specify TVPs and SVs as unobserved state variables.5 The state variables are assumed by CPF to evolve as random walks. The time-varying reduced-form intercepts and slope parameters follow a multivariate random walk,

Bt = Bt 1 + vt ; (1.7) − while the SVs in logs are geometric random walks,

l og (hj,t ) = l og (hj,t 1) + ηj,t , j = 1,2, (1.8) − where vt N (0,Q), ηt [η1,t , η2,t ],ηt N (0,W ). The structural shocks and the innovations to the random walks∼ in (2.4) and≡ (1.8) are given∼ a block diagonal covariance matrix

5 [Tay86; Tay94] is the source of the SV setup, which [Kim98] and [Omo07] adopt.

8      εt  I2 0 0 V a r v   0 Q 0 , (1.9) =  t  =   V ηt 0 0 W where Q and W are full rank. If Q = W = 0, the TVP-SV-SVAR of (1.1) reduces to a fixed-coefficient VAR; see Appendix A.2. The SVAR is also capable of accommodating specifications with only TVP or only SV.This entails settingQ = 0 to produce constant reduced-form intercepts and slope parameters, 6 B. Setting W = 0 turns off SV in the SVAR.

1.3.2 Priors

Empirical Bayes priors are given to [B0, l og (hj,0)] and the covariance matrices of the random walks [Q, W ]. The prior distributions for the initial values of the states are assumed to be normal and independent. The empirical Bayes priors use OLS estimates from a fixed-coefficient VAR on the 1948Q1 to 2016Q4 sample.7 Let x¯ denote the OLS estimate, which is the empirical Bayes prior I place on x . Table B.6 shows the priors that initialize the Gibbs sampler.

Table 1.1 Priors

N ,κ2 V Q IW κ2 V ,T B0 (B B B) i ( Q B ) ∼ · 2 ∼ 2 l og (hj,0) N (l og (h j ),κh IM ) Wi IG (κW ,2) ∼ · ∼ Note: N denotes normal; IW denotes Inverted-Wishart; IG denotes In- verse Gamma. κ2 4 and κ2 10 are tightness constants. κ2 0.24, B = h = Q = and 2 1 10 3 are tuning parameters. T 281 is the sample size. κW = − = ×

The time-varying intercepts and slope parameters and SVs are initialized by drawing from a normal prior. The covariance matrices of the innovations to the random walks of the TVPs and SVs are endowed with an inverse Wishart (IW) and an inverse Gamma (IG) distribution. The constants κ2 4 and κ2 10 control the tightness of the priors on the covariances of and l og h . These B = h = B ( j ) constants are calibrated to be weakly informative, as the effective prior variance on B is enlarged by four times the empirical Bayes estimate. Similarly, I magnify the variance of the priors on the SV by 10. According to [Cog05], 10 is large on the logarithm scale to ensure a proper range of values to initialize the Gibbs sampler. Next, I set the tuning parameters 2 0.24, and 2 1 10 3. The κQ = κW = − × 6This is useful in evaluating the data preference on the addition of TVPs and SV to the SVARs. I estimate a TVP-SVAR with static SV, an SV-SVAR with fixed regression coefficients, and a fixed-parameter SVAR to compute and compare their marginal log likelihoods. Details and results are provided in Appendix A.6. 7I also consider traditional prior setting method using a MLE training sample in Appendix A.4.1. The cost of a training sample is the loss of sample size.

9 2 8 calibration of κQ adjusts for the acceptance rate of stationary draws on Bt . With these calibrations, the average acceptance rate of stationary Bt is 37.71% with hours worked in log levels, and 34.10% with hours worked in growth rates.

1.3.3 Sampling Algorithm

Estimation of the TVP-SV-SVAR relies on a Gibbs sampler. I modify the CPF sampler to impose the long-run identification but leave the impact matrix unrestricted. The non-linearity in the SV is approximated with a 10-component mixture normal distribution following [Omo07]. This approach introduces s T as the state of volatility to track the approximation. The Gibbs algorithm sequentially T T T T T samples (Bħ , s ,h1 ,h2 , ) conditional on all data, z . Details about the sampling steps and im- plementing the long-runV restriction in the sampler are deferred to Appendix A.4. However, a brief sketch of the Gibbs sampler follows.

T T T T Step 1 Set the initial values (B0 ,h1,0,h2,0, s0 , 0) and i = 1. V

T Step 2.a Draw the reduced-form intercepts and slope parameters Bi from T T T T T 9 p(Bi si 1,h1,i 1,h2,i 1, i 1) IB(Bi ) using the Carter-Kohn algorithm. The indicator function T| − − − V − · 10 IB(Bi ) truncates the posterior distribution to ensure the stability of the companion form.

T T T Step 2.b Impose the long-run restriction given Bi and h1,i 1,h2,i 1 using equations (2.3) - (1.5). − − T This yields draws of the structural intercepts and slope parameters Bħi that are consistent with T the long-run restriction. Following [Koo11], evaluate the structural Bħ1 at each draw and each date t for roots inside unit circle.

T T T Step 3 Draw h1,i ,h2,i through auxiliary variables si following the 10-component mixture normal 11 approximation in [Omo07].

T T T T Step 4 Draw hyperparameters i given (Bħi ,h1,i ,h2,i , z ). V 8Generally, these tuning parameters do not have significant impact on the VAR within small magnitudes of changes. 2 An excessively large value (e.g., κQ 1) could lead to erroneous time variation attribution in the VAR. [Lub17] and [AA18] provide a discussion on this issue.≥ 9This algorithm relies on the Kalman filter to obtain optimal estimates of the coefficients, and applies a backward algorithm to compute realizations of the coefficients. Hence, this algorithm provides a series of draws of the latent state. Appendix B.6.1 describes the implementation of the Carter-Kohn algorithm in detail. 10 The truncation is important as highlighted by [Koo11]. In absence of such a restriction, even a small amount of posterior weight in explosive regions of the parameter space can lead to impulse responses, which have counterintuitively huge posterior means or standard deviations. 11 2 The parameters of the underlying normal distributions, which are mixed to approximate the l og χ (1) distribution of T T h1 ,h2 , are described in Appendix A.4.3.

10 Step 5 Repeat steps 2 through 4 M times. After discarding the burn-in of N repetitions, the last (M N ) draws, after thinning, result in an approximation to the marginal posterior distributions of− the SVAR coefficients.

In summary, the MCMC algorithm simulates the posterior distribution of the states (B,h1,h2, s ) and the hyperparameters ( ), conditional on the data (z ) and the stationary structural state (Bħ), by V iterating on Steps 2 to 4. I extract M = 400,000 draws in total. Discard the first N = 200,000 draws as the burn-in. The remaining 200,000 simulations are thinned once every 40 draws to correct for serial correlation in the posterior.12 This leaves the posterior with 5,000 draws. Appendix A.5 verifies the successful convergence of the MCMC sampler.

1.4 The Effect of TFP Shocks with Time Variation

This section reports estimates of the TVP-SV-SVAR under the long-run restriction. I present the IRFs of hours worked with respect to a TFP shock, the IRFs of ALP with respect to a demand shock, and output’s IRF to a TFP shock. The IRF of output is constructed by summing the IRFs of ALP and hours worked. Next, FEVDs are reported. The IRFs and FEVDs address the conflicting evidence in the TFP-hours worked literature.

1.4.1 IRFs of Hours Worked to a TFP Shock

Figure ?? plots the IRFs of hours worked in log levels and growth rates to a TFP shock at six selected NBER recession (trough) dates with 95% error bands.13 These dates are 1961Q1, 1970Q4, 1974Q2, 1980Q3, 2001Q2, and 2009Q1. Each IRF shows the response corresponding to a one standard de- viation TFP shock. There are no qualitative differences in the height and persistence of the IRFs whether hours worked is in log levels or growth rates. The IRFs of hours worked with respect to a TFP shock are negative on impact and at the business cycle horizons. There is mean reversion in these IRFs after a peak about five quarters. These IRFs hardly change across the recession dates. In the short-run, the heights of the re- sponses are similar, between -0.1 to -0.17 percentage points in the upper two panels or between -0.08 to -0.13 log points in the lower two panels, across the six sample dates. Therefore, the responses do not appear to have business cycle dependency. The shapes of the IRFs are also comparable across the recession dates, although the hump-shape is more evident when hours worked is in growth

12 It is conventional in the TVP-VAR literature to use a thinning factor to break the autocorrelation across draws. [Cog05] argue the drawback of using a thinning factor is that it increases the variance of ensemble averages from the simulation. In this paper I follow [Owe17] on approximating an optimal thinning factor. However, my results are robust to alternative thinning factors. 13IRFs of NBER recession peak dates are available but do not provide additional information on the results.

11 rates. The long-run effect of a TFP shock on hours worked is zero, despite the cumulative impact is left unrestricted. Figure ?? displays the negative contemporaneous responses of hours worked with respect to a TFP shock with 95% error bands. In the lower panel when hours worked enters the SVAR as log levels, there are several impact IRFs that are statistically indistinguishable from zero. Regardless of how hours worked enters the SVAR, the negative impact responses seem not to be dependent on the state of the economy. The IRFs of hours worked to a TFP shock appear in Figure ??. The IRFs are presented as 3- dimensional plots. The plots show the dynamic responses of ALP and hours worked for a 20-quarter horizon quarter by quarter from 1948Q2 to 2016Q4. The x axis of the IRFs is quarter by quarter across the entire sample. The y axis is the forecast horizon of the IRFs. The z axis gives the magnitude of the IRFs. Figure ?? plots the 3-dimensional IRFs estimated on hours worked in log levels on the left side and in growth rates on the right. The two panels do not show qualitative differences. Both panels exhibit negative impact responses and hump-shaped medium-run responses, which are consistent with new Keynesian theory. These negative IRFs are consistent over the sample regardless of how hours worked enters the SVAR. Whether hours worked should enter the SVAR in log levels or growth rates is resolved by estimating the TVP-SV-SVAR. The conflicting signs of the IRFs of hours worked with respect to a TFP shock disappear once structural change is accounted for by the TVPs and SVs. My results are qualitatively similar to [Gal99], [Gam05], and [Gal09]. Although hours worked falls on impact and over the business cycle fluctuations, the height and the persistence of the responses change over the sample. Two additional points are worth noting about the quantitative differences. First, there is a break in the height of the IRFs over the business cycle horizons around 1975 in Figure ??. After the first oil price shock, the hump shape of the IRFs diminishes when hours worked enters as growth rates. This is not true when hours worked enters in log levels. Second, the top panel of Figure ?? shows that in the 1950s the impact response of hours worked is often smaller than -0.1 while later in the sample it fluctuates around. In the bottom panel of Figure ??, the negative response of hours worked never wanders far from -0.1.

1.4.2 IRFs of ALP to a Demand Shock

Next, I report the IRFs of ALP with respect to a demand shock at selected NBER recession (trough) dates in Figure ??. Whether hours worked enters the SVAR in log levels or growth rates, a demand shock increases labor productivity in the short-run at the NBER recession dates. The upper two panels of Figure ?? show IRFs when hours worked is in growth rates. The IRFs are hump-shaped and peak around the third quarter at the NBER recession dates. However, the heights of theses IRFs are lower in more recent recessions.

12 Figure ?? provides more evidence on the instability in the IRFs of ALP with respect to a demand shock. Similar to Figure ??, these IRFs are plotted in 3-dimensions. In the left panel of Figure ??, the IRFs are positive on impact and remain positive over the business cycle horizons, given the SVAR is estimated with hours worked in log levels. These IRFs display a significant decrease in the height and persistence beginning the productivity slowdown of the 1970s. The IRFs occasionally display negative short-run responses. There is co-movement with NBER’s business cycle dates in the short-run. For example, the height of the IRFs starts declining around the technology boom and bust of the 1990s. In comparison, the right hand side panel of Figure ?? displays the IRF of ALP with respect to a demand shock when hours worked is in growth rates. Although the IRFs exhibit peaks around the third quarter in the right panel, these IRFs show substantial variation over the sample. In most cases, the IRFs on right hand side of Figure ?? follow a positive hump-shaped response with comparable shape and persistence to that of the log-level specification during the early sample. Nonetheless, the IRFs of Figure ?? indicate ALP has a positive short-run response to a demand shock with the peak response drifting down around the productivity slowdown of the 1970s. These IRF results suggest the TFP process and aggregate technology have changed since 1948. This is consistent with the literature that documents changes in technology at the firm-level. For instance, [Com06] document the medium-frequency fluctuations in the persistence and volatility of firm-level productivity data as evidence of structural change.

1.4.3 IRFs of Output to a TFP Shock

Following the TFP - hours worked literature, I report the IRFs of output with respect to a TFP shock on selected NBER recession dates in Figure ?? and on the entire sample in Figure ??. The IRF of output to a TFP shock is computed as the sum of the ALP and hours worked IRFs with respect to a TFP shock. Figure ?? shows output increases with respect to a TFP shock on selected NBER recession dates. These IRFs do not decay over time and settle at a permanently higher steady state after 5 to 8 quarters. The positive long-run response is robust to estimating the SVAR with hours worked in log levels or growth rates. Since the estimated TVP-SV-SVARs are consistent on the effect of hours worked to a TFP shock, SVARs with log-level and growth-rate specifications produce similar responses of output to a TFP shock. Figure ?? plots the 3-dimensional IRFs of output to a TFP shock. The left side panel of Figure ?? displays substantial changes in the height of the hump-shaped impulse responses of output to a TFP shock when hours worked enters the SVAR as log levels. The IRFs of output also exhibit large and frequent long-run movements in the persistence or the slope. The right hand side of Figure ?? plots the IRFs of output with hours worked in growth rates. These IRFs show comparable shape over

13 the sample but a break in the persistence at the peak of the short-run responses around the end of the productivity slowdown of the 1970s. The difference in the IRFs points towards time variation in the intercepts or slope parameters of the ALP regression, particularly when hours worked is in log levels. Since the IRFs of hours worked with respect to a TFP shock appear stable over the sample, it is worth studying the source of the instability in the IRFs of ALP and output. This source of fluctuation is responsible for a larger portion of the time variation in the data especially when hours worked is assumed to be stationary. In sum, the IRFs of hours worked to a TFP shock are consistently negative on impact and in the short-run. These responses are stable over the sample. Whether hours worked is first-differenced or not is not the source of the time variation in the IRFs. My results provide evidence on the importance of addressing structural change in the data generating processes of hours worked and ALP.After structural change is taken into account, the controversy about the sign of the hour worked response to a TFP shock vanishes. The results of the IRFs support new Keynesian theory.

1.4.4 Evidence from FEVDs

The forecast error variance decompositions (FEVDs) of ALP and hours worked appear in Figure ?? and Tables 1.2 and 1.3. Time variation exhibits pronounced effects on the FEVD of hours worked. The FEVDs address the importance of TFP and demand shocks for explaining variation in ALP and hours worked. Figure ?? presents 3-dimensional plots on FEVDs of ALP and hours worked in growth rates (upper panel) and in log levels (lower panel) with respect to a TFP shock. The contribution of a TFP shock to ALP is on the left hand side. The TFP shock dominates and explains more than 90% of the variation in ALP of all horizons across the sample. These FEVD results are also robust to whether hours worked enters the SVARs as log levels or growth rates. I interpret this finding as evidence in support of RBC theory, because the TFP shock dominates fluctuations in ALP (or output) since 1948. The FEVDs of hours worked with respect to a TFP shock appear in the right panel of Figure ??. Unlike that of ALP,the FEVDs of hours worked appear unstable over the sample. With hours worked in log levels, the TFP shock explains less than 40% of the movements in hours worked from 1- to 5-year horizons. The FEVDs of hours worked indicate TFP explains a larger share of the variation in hours worked from the 1970s through the recent financial crisis and the Great Recession. When hours worked is in growth rates, however, the FEVDs of hours worked with respect to a TFP shock exhibit substantial variation across the sample. The fraction of hours worked explained by a TFP shock varies from almost 0 to 90% over the sample.

14 Table 1.2 FEVD at NBER Recession Dates, Hours Worked in Growth Rates

X XXX Dates Variable XXX 1958Q1 1970Q3 1980Q2 1991Q1 2001Q3 2009Q1 Quarters XXX 1 0.9426 0.9741 0.9129 0.9817 0.9489 0.9479 2 0.9621 0.9830 0.9396 0.9854 0.9648 0.9525 ALP to TFP Shock 4 0.9694 0.9855 0.9483 0.9834 0.9664 0.9459 8 0.9717 0.9865 0.9513 0.9824 0.9654 0.9449 20 0.9720 0.9866 0.9516 0.9824 0.9654 0.9448 1 0.0724 0.3945 0.9674 0.6129 0.2505 0.8894 2 0.1275 0.5064 0.8109 0.3753 0.5700 0.3327 HW to TFP Shock 4 0.2426 0.5909 0.8039 0.4041 0.7528 0.3887 8 0.2776 0.6178 0.8075 0.4019 0.7766 0.3821 20 0.2786 0.6209 0.8079 0.4019 0.7771 0.3821 Note: Selected dates are NBER recession trough dates.

Tables 1.2 and 1.3 report the FEVDs at selected NBER recession dates for the contribution of a TFP shock to each variable, estimated with hours worked in growth rates and in log levels. Among the six selected NBER recession dates, the contributions of a TFP shock to ALP appear stable and similar between the two specifications of hours worked. The long-run contribution of a TFP shock to hours worked shows time variation. With hours worked in growth rates, Table 1.2 shows the TFP shock is an important source of variation in hours worked at NBER recession dates. Although the contribution of a TFP shock to hours worked exhibits large time variation in Figure ??, these FEVDs are not related to business cycle fluctuations. In Table 1.3, the FEVDs of hours worked to a TFP shock decline over the six recession dates.

Table 1.3 FEVD at NBER Recession Dates, Hours Worked in Log Levels

XX XXX Dates Variable XX 1958Q1 1970Q3 1980Q2 1991Q1 2001Q3 2009Q1 Quarters XXX 1 0.9810 0.9924 0.9801 0.9859 0.9919 0.9709 2 0.9741 0.9880 0.9728 0.9799 0.9921 0.9639 ALP to TFP Shock 4 0.9713 0.9889 0.9701 0.9784 0.9926 0.9607 8 0.9708 0.9882 0.9695 0.9778 0.9934 0.9605 20 0.9706 0.9882 0.9693 0.9777 0.9947 0.9598 1 0.0102 0.0202 0.0074 0.0091 0.0071 0.0046 2 0.0578 0.1666 0.0406 0.0564 0.0359 0.0128 HW to TFP Shock 4 0.1186 0.3199 0.0834 0.1160 0.0732 0.0220 8 0.1680 0.4026 0.1186 0.1611 0.1057 0.0302 20 0.2323 0.4116 0.1643 0.2030 0.1611 0.0483 Note: Selected dates are NBER recession trough dates.

15 In summary, the FEVD results indicate that a TFP shock is the dominant source of variation in ALP over all forecast horizons and across the entire sample. This evidence supports RBC theory. The contribution of the TFP shock to hours worked displays large time variation but is not dependent on the state of the economy. The FEVDs of hours worked differ when hours worked enters the SVAR as log levels compared with when hours worked is growth rates.

1.5 Source of Structural Change

Given the results of substantial time variation in the IRFs and FEVDs, this section investigates the source of these changes from the SVARs. I provide additional evidence from estimates of the SVARs to show that drifts in the structural slope parameters and SV of the ALP regression are important in understanding the dynamic relationship between ALP and hours worked.

1.5.1 Estimates of Stochastic Volatility

This section studies one source of the variations in the IRFs. Figure ?? shows the median posterior estimates of the SVs with 95% posterior tunnels. Overall, estimates of the SVs show no differences whether hours worked enters the SVAR as growth rates (upper panels) or log levels (lower panels).

Substantial time variation is found in the estimated SV of TFP, h1. The top left and bottom left plots of Figure ?? show the SV of TFP declines throughout the sample for hours worked in growth rates or log levels. There is a steep decline during the productivity slowdown of the 1970s through the recent financial crisis and the Great Recession. The falling SV of TFP is evident across the entire sample, despite a small peak of volatility around the 2001 recession of the dot-com bubble. The shrinking variation of the TFP shock suggests instability in the ALP regression of the SVARs.

In contrast, the estimates of SV of the demand shock (h2) stay relatively stable over most of the sample. In the top and bottom right hand panels of Figure ??, I plot the SV of the demand shock with hours worked entered as log levels and as growth rates. Unlike the SV of TFP,movements in SV of the demand shock are mostly due to the labor demand co-movement with the business cycle. For instance, the SV of the demand shock culminates during the Volcker disinflation of the early 1980s, and decreases over the Great Moderation. This pattern is consistent with the impact of the Great Moderation on labor demand (e.g. [Fab17]). Moreover, there is an evident increase in the SV of the demand shock around the 2007 - 2009 Great Recession, when the SV approaches a level similar to that observed in the late 1970s and early 1980s. It is worth noting that since SV of TFP declines faster than that of the demand shock over the sample, the relative volatility in the shocks of labor input rises. These findings reinforce the studies of [Gal17]. They document a decrease in the volatility of the demand shock and an increase in the relative volatility of the demand shock compared with output in postwar data.

16 According to these estimates, the SV of TFP declines over the entire sample while the SV of the demand shock is stable but exhibits business cycle dependence with respect to several NBER recessions. These results are robust to whether hours worked enters the SVAR as log levels or growth rates.

1.5.2 Estimates of Structural Intercepts and Slope Parameters

To provide more insights on the source of time variation in the IRFs and FEVDs, I plot the estimated structural intercepts and slope parameters of the SVARs, Bħt , with 95% error bands in Figures ?? and ??. Regardless of whether hours worked enters the SVAR as log levels or growth rates, the sum of own lags on ALP displays statistically significant drifts over the sample. The time variation is counter- cyclical to most NBER recessions. For instance, there are significant troughs in the estimated sum of own lags on ALP during the 1981-1982 recession and the recent financial crisis in 2009. However, this is not the case for the parameters of the hours worked regression. In Figure ?? when hours worked is in growth rates, the sum of own lags on hours worked displays evident movement over the sample and appears to be pro-cyclical. This time variation disappears when hours worked is in log levels, as shown in Figure ??. Similarly, the sum of other lags on hours worked appears stable across the sample with hours worked in growth rates. When hours worked enters as log levels in Figure ??, the hours worked sum of other lags becomes volatile especially around the Great Recession of 2009. There is no statistically significant time variation of the intercepts in either regression.

I also plot the structural long-run cumulative matrix, Γ˜t , , in Figure ??. The structural long-run ∞ multipliers are computed using equations (2.3) - (1.5). The left (right) panel plots the estimates with hours worked entered as growth rates (log levels). With 95% bands, the estimates of γ11 and γ21 , which are the long-run impact of a TFP shock to ALP and hours worked, show statistically∞ significant∞ time variation over the sample. The time variation of these parameters in early sample displays business cycle dependency with, for example, the 1970 recession and the first oil price shock. The cumulative effect of hours worked from its own shock (γ22 ) exhibits large variation during the early sample that is another source of instability of the IRFs.∞ The value of γ12 , which measures the long-run effect of a demand shock to ALP,is zero in both panels due to the∞ long-run neutrality identification assumption of ALP.

1.5.3 Discussion

This section provides evidence on time variation of the structural slope parameters and SV in the ALP regression. Similar to [Gam05], I argue that instability in the ALP regression is responsible for the time-varying results in the IRFs and FEVDs. Allowing for drifts in the SVAR parameters captures these changes, such as fluctuations in persistence and volatility mentioned in [Can10]. Once one allows drifts in the TVP and SV, whether hours worked is in log levels or growth rates no longer

17 matters for the TFP-hours worked relationship. Estimates of the TVPs and SVs reveal the impact of the productivity slowdown of the 1970s, the first oil price shock, the technology boom and bust of the late 1990s, the recent financial crisis and Great Recession of 2009 on the IRFs and FEVDs. My results indicate the Great Moderation is not an important source of structural change that affects the dynamic relationship between ALP and hours worked. Instead for the dynamic relationship between hours worked and ALP, I find that the productivity slowdown of the 1970s, the first oil price shock, the technology boom and bust of the late 1990s, the recent financial crisis and Great Recession of 2009 are more economically important.

1.6 Conclusion

In this paper, I contribute a solution to the puzzle on the dynamic response of hours worked to a TFP shock. The TFP shock is identified by imposing long-run neutrality on average labor productivity with respect to the demand shock. Following [Gal09], I include time-varying parameters and stochastic volatility in the structural VARs to account for structural change in the data. My approach ensures the structural intercepts and slope parameters are consistent with the long-run restriction, checks for stationarity of these parameters step by step, and adapts the [DN15] correction to the sampler. The estimation is on a quarterly sample of 1948 to 2016 that includes the recent financial crisis and Great Recession. Once I account for structural change, hours worked falls on impact and over the business cycle in response to a TFP shock. These impulse response functions are stable over the entire sample. This result is robust to whether hours worked is stationary in log levels or growth rates. Average labor productivity increases after a demand shock. These findings support new Keynesian theory. Different measures of hours worked or average labor productivity do not alter my results qualitatively. My results are not a complete vindication of [Gal99], [Gam05], and [Gal09]. The forecast error variance decomposition results show the TFP shock dominates variation in average labor productiv- ity across the entire sample. I argue this supports RBC theory. Nonetheless, the error decomposition of hours worked are not stable over the sample. At recession dates in 1975 and 2009, the TFP shock explains as much as 90% of the variation in hours worked, when the structural VAR is estimated with hours worked in growth rates. These forecast error variance decompositions are often near zero at non-recession quarters, or when hours worked enters the structural VAR in log levels. I investigate the sources behind these results. Aggregate labor supply and demand, as represented by hours worked, have been a stable function of the underlying state of the business cycle since 1948. The average labor productivity regression is subject to drifting slope parameters on its own lags. Stochastic volatility of the TFP shock drifts over the entire sample, but stochastic volatility of the demand shock appears stable. Rather than the stationarity assumptions on hours worked, the TVPs and SVs are responsible

18 for obscuring the conflicting evidence in the literature. Episodes of structural change such as the productivity slowdown of the 1970s and the Great Recession are important for understanding the dynamic response of hours worked to a TFP shock. My results suggest future research should focus on the changing character of the aggregate TFP process and aggregate technology in the U.S.

19 Figure 1.1 Average Labor Productivity in Growth Rates. Notes: Sample size ranges from 1948Q1 to 2016Q4. Sub-samples in hours worked are 1948Q1 to 1961Q1, 1961Q2 to 1982Q4, and 1983Q1 to 2016Q4. The horizontal lines represent sub-sample means. Source: FRED, BLS, and NBER.

20 Figure 1.2 Log Weekly Hours per capita. Notes: Sample size ranges from 1948Q1 to 2016Q4. Sub-samples in hours worked are 1948Q1 to 1961Q1, 1961Q2 to 1982Q4, and 1983Q1 to 2016Q4. The horizontal lines represent sub-sample means. Source: FRED, BLS, and NBER.

Figure 1.3 Wald Statistics on Sum of Slope Coefficients, with Hours Worked in Log Levels. Notes: 15% trimmed sample size ranges from 1954Q1 to 2011Q4. Dashed line shows the 95% critical value of 7.12 following [And03]. The Andrews test computes the Wald statistic on structural change of c and bi over the sample against a constant estimate.

21 Figure 1.4 Wald Statistics on Sum of Slope Coefficients, with Hours Worked in Growth Rates. Notes: 15% trimmed sample size ranges from 1954Q1 to 2011Q4. Dashed line shows the 95% critical value of 7.12 following [And03]. The Andrews test computes the Wald statistic on structural change of c and bi over the sample against a constant estimate.

Figure 1.5 Wald Statistics on Standard Deviations and Covariance. Notes: 15% trimmed sample size ranges from 1954Q1 to 2011Q4. Dashed line shows the 95% critical value of 7.12 following [And03]. Shaded are NBER recession dates.

22 Figure 1.6 IRFs of Hours Worked to a TFP Shock: Selected NBER Recession (Trough) Dates. Notes: Median responses (blue line) and 95% confidence bands (red line) of selected dates are reported. The selected dates are NBER recession (trough) dates.

23 Figure 1.7 Impact Response of Hours Worked with respect to a TFP Shock. Notes: Median impact response of hours worked to a TFP shock (blue line) and 95% confidence bands (red line). Grey shaded areas are NBER recession dates. Sample is 1948Q1 to 2016Q4.

Figure 1.8 IRFs of Hours Worked to a TFP Shock: 1948Q1 to 2016Q4. Notes: Median impulse responses of hours worked to a TFP shock on all sample. Sample period is 1948Q1 to 2016Q4. X-axis: quarters after shock; y-axis: magnitude of response; z-axis: sample date.

24 Figure 1.9 IRFs of ALP to a Demand Shock: Selected NBER Recession (Trough) Dates. Notes: Median responses (blue line) and 95% confidence bands (red line) of selected dates are reported. The selected dates are NBER recession (trough) dates.

Figure 1.10 IRFs of ALP to a Demand Shock: 1948Q1 to 2016Q4. Notes: Median impulse responses of average labor productivity to a demand shock on all sample. Sample period is 1948Q1 to 2016Q4. X-axis: quarters after shock; y-axis: magnitude of response; z-axis: sample date.

25 Figure 1.11 IRFs of Output to a TFP Shock: Selected NBER Recession (Trough) Dates. Notes: Median responses (blue line) and 95% confidence bands (red line) of selected dates are reported. The selected dates are NBER recession (trough) dates. IRFs of output to a TFP shock is computed as the sum of responses of ALP and hours worked with respect to a TFP shock.

26 Figure 1.12 IRFs of Output to a TFP Shock: 1948Q1 to 2016Q4. Notes: Median impulse responses of output to a TFP shock on all sample. Sample period is 1948Q1 to 2016Q4. X-axis: quarters after shock; y-axis: magnitude of response; z-axis: sample date. IRFs of output to a TFP shock is computed as the sum of responses of ALP and hours worked with respect to a TFP shock.

27 Figure 1.13 FEVDs with respect to a TFP shock: 1948Q1 to 2016Q4. Notes: 3-dimensional median forecast error variance decomposition of ALP and Hours worked with respect to a TFP shock are reported for 20 quarters post shock. Sample period is 1948Q1 to 2016Q4. X-axis: quarters after shock; y-axis: percentage variation explained; z-axis: sample date.

28 Figure 1.14 Posterior Estimates of Stochastic Volatility. Notes: Median value (blue line) and 95 percent bands (red line) are reported. Shaded areas are NBER recession dates.

29 Figure 1.15 Posterior Estimates of Bħ, Hours Worked in Growth Rates. Notes: Median value (blue line) of the posterior estimates of the structural intercepts and slope parameters and 95 percent bands (red line) are reported. Shaded areas are NBER recession dates.

Figure 1.16 Posterior Estimates of Bħ, Hours Worked in Log Levels. Notes: Median value (blue line) of the posterior estimates of the structural intercepts and slope parameters and 95 percent bands (red line) are reported. Shaded areas are NBER recession dates.

30 Figure 1.17 Estimated Values of Elements of the long-run Cumulative Impact Matrix Γ (1). Notes: Median value (blue line) and 95 percent bands (red line) are reported. Grey shaded areas are NBER recession dates.

Posterior estimates of γ12 are restricted to zero as the long-run neutrality of ALP to the transitory demand shock.

31 CHAPTER

2

THE TIME-VARYING AND VOLATILE MACROECONOMIC EFFECTS OF IMMIGRATION

2.1 Introduction

This paper is about the impact of immigration on the U.S. macroeconomy. Immigration has become an important policy issue in the U.S. because of the increase in immigration to the U.S. since the 1970s. Research on the macroeconomic effects of immigration is scarce. I fill this gap by using structural vector autoregressions (SVARs) to address (i) the macroeconomic effects of immigration on the U.S. labor market, (ii) whether or not the short- and long-run effects of immigration differ, (iii) the impact of aggregate productivity, labor demand, and transitory consumption shocks on immigration, and (iv) whether or not these effects change over time. There is a small literature using SVARs to study the impact of immigration on the aggregate economy. I contribute to this literature by estimating SVARs using a novel approach to identification. Identification of the SVARs begins with [Gal99]. Galí’s approach is useful because he identifies a total factor productivity (TFP) shock by assuming it is the only shock that has a permanent effect on average labor productivity (ALP) in the long-run. The second variable in the SVARs is hours worked. [Gal99] identifies its forecast innovation as a demand shock.

32 Starting from Galí’s SVAR, I introduce immigration to study its impact on the macroeconomy. The identification of the immigration supply shock combines short- and long-run restrictions. One restriction is immigration does not respond to other macroeconomic shocks on impact. This reflects the fact that the decision to immigrate responds to changes in economic conditions only with a lag ([Bor06]). However, this does not completely identify the immigration supply shock. The reason is the decision to migrate responds to labor market shocks and to expectations of the change in lifetime earnings or expenditures from a move. Adding consumption to the SVARs identifies a transitory con- sumption shock. I explore this part of the identification by estimating SVARs in which consumption does and does not respond to an immigration supply shock on impact. A neoclassical model of the labor market predicts immigration causes labor productivity to fall; see [Car01] and [Bor06]. Nonetheless, the microeconomics literature on immigration often reports conflicting empirical evidence about the impact of immigration on labor market outcomes, according to [Ker11]. Given no clear advice, I assume hours worked and ALP respond to immigration supply shocks on impact. Besides the short-run restrictions, two alternative restrictions are considered for the long-run impact of the immigration supply shock on ALP.In the first identification, ALP is assumed to be long- run neutral with respect to an immigration supply shock. The alternative assumes ALP responds to an immigration supply shock in the long-run. The motivation for these identifying restrictions is the lack of consensus on the long-run relationship between immigration and productivity, as documented in [Fey07] and [Ort13], among others. There have been several changes to U.S. immigration policy since 1969. These policy changes could induce structural change to immigration through drifting means, persistence, and/or volatility. Therefore, I estimate a baseline SVAR with time-varying parameters (TVPs) and stochastic volatility (SV). Addressing structural change in data is important in reaching reliable estimates of the SVARs, as stressed by [Sim06] and [Nas15]. I explore preferences the data has with respect to TVP,SV, and the four identification schemes in a Bayesian model selection exercise. To my knowledge, this paper is the first to identify an immigration supply shock using short- and long-run restrictions in a TVP-SV-SVAR. Noteworthy studies on immigration with SVARs include [Kig17]. They use a sign-restricted SVAR to show output per capita falls in the short-run following an immigration shock. [Fur17] assume immigration is a substitute for domestic labor supply in a SVAR with sign restrictions. They conclude that an immigration shock has a negative effect on labor productivity. [Bou13] and [Dal16] identify immigration in a Choleski recursive ordering with GDP per capita and labor input for OECD countries. They find that GDP per capita and other aggregate variables rise in response to an immigration shock. Compared to these studies using fixed- coefficient SVARs, this paper contributes to the immigration literature in documenting substantial time-variation in the impact immigration has on the U.S. macroeconomy in the short-run, the

33 business cycle, and the long-run. This time-variation appears to stem from several episodes of changes in U.S. immigration policy and structural change of the economy since the 1970s. Estimates of the SVARs yield several conclusions about the impact of immigration on the U.S. economy. First, the data strongly favors an identification scheme that immigration does not respond to TFP, labor demand, and transitory consumption shocks on impact. The data also prefers the model to allow ALP to respond to an immigration supply shock in the long-run. This indicates the effect of an immigration supply shock is varying in the long-run. Innovations to immigration are affected by SVs over the sample that match various episodes of U.S. immigration policy changes. For instance, a substantial increase in the SVs occurs at the same time as the Immigration Reform and Control Act of 1986 and the Immigration Act of 1990. These changes to U.S. immigration policy are important in evaluating the dynamics of immigration on the aggregate economy. The impulse response functions (IRFs) of the data-favored SVAR indicate ALP and hours worked decline with respect to an immigration supply shock at the business cycle horizons. An immigration supply shock increases consumption in the short to medium-run. The signs of the IRFs of ALP, hours worked and consumption to an immigration supply shock show changes over the sample that coincide with several changes to immigration policy. The long-run responses of ALP to an immigration supply shock also exhibit co-movement with NBER dated recessions. The long-run responses of hours worked and consumption to an immigration supply shock decline throughout the sample. These findings indicate that the macroeconomic effects of immigration are dependent on the changes to U.S. immigration policy and the current state of the economy. The IRFs of immigration with respect to the TFP and labor demand shocks also display time- variation over the sample. The IRFs of immigration to a TFP shock decline over the business cycle horizons but exhibit sign changes in the long-run response. A transitory consumption shock in- creases immigration over the business cycle horizons and these IRFs are stable. The responses of immigration to a transitory consumption shock are not dependent on immigration policy or the state of the business cycle. The remainder of this paper is organized as follows: Section 2 discusses historical changes in immigration policy and the data; Section 3 sets up the TVP-SV-SVAR and estimation methods; Section 4 provides results; Section 5 concludes.

2.2 U.S. Immigration Policy and Data

This section gives a brief review of the changes to U.S. immigration policy that are relevant for this paper. This is followed by a discussion of the data on which the TVP-SV-SVARs are estimated.

34 2.2.1 Background on Recent Changes to Immigration Policy

Since 1969, there has been eight major immigration policy changes. Among these are changes in quota of various categories of immigration, level of border patrol enforcement, and amnesty towards irregular immigrants.1 Frequent and large-scale changes in immigration policy are important to evaluating the macroeconomic dynamics of immigration because these changes have a non-linear effect to the migrating decision. This could be in the form of outliers, structural breaks, or drifting volatility in the immigration data.2 The enactment of the Refugee Act of 1980 and the Immigration Reform and Control Act of 1986 (IRCA) began an era of non-restrictive immigration policies by creating new categories of permanent immigration and a nationwide amnesty of irregular immigration. Coupled with the American Homecoming and Immigration Act of 1990 (IMMACT), which increased immigration quota, these changes in immigration policy raised quarterly immigration influx by almost threefold, peaking at 1990Q1. During 1986 to 1992, the IRCA and IMMACT coincide with the peak of volatility in the immigration series. Accounting for these episodes of structural change is important in conducting consistent inference. Between the mid-1990s and the 2001 dot-com bubble, quota for employment-based visa and permanent residency application was raised three times in 1990, 1998, and 2000 up to 195,000, before it decreased to its pre-1990 level of 65,000 in 2004 through the H-1B Reform Act. This quota is still currently in effect. The re-adjustment of quota generates kinks in at least the employment-based category of immigration. These changes in quota also non-linearly affect the decision to immigrate. It is worth noting that changes in immigration policies affect not only the level of immigration, but the composition of immigration such as demographics or education level, according to [Orr09]. These underlying changes could also induce structural break to the immigration data that would be difficult to detect with standard econometric methods. Therefore, given the episodes of changes to U.S. immigration policy, it is important to examine the macroeconomic dynamics of immigration with an econometrically flexible framework.

2.2.2 Data

I collect quarterly data from 1969 to 2014 on U.S. immigration, ALP,hours worked, and consumption. Data on immigration is collected from the Yearbook of Immigration Statistics published by the DHS.3 This series contains the annual flow of foreign-born civilian admittance to the U.S. by category. Following standard practice in the immigration literature (See [Bla17]), I define immigration as naturalization and permanent residency of foreign-born citizens in the U.S. This definition excludes

1Irregular immigration refers to immigration that is uninspected or undocumented by the Department of Homeland Security (DHS). 2Appendix B.1 provides a detailed description of eight major changes to immigration policy in the U.S. since 1969. 3Early volumes are published as the Statistical Yearbook of the Immigration and Naturalization Service.

35 temporary immigration admittance (non-immigrant visa holders) such as foreign students or visi- tors. The DHS data does not specifically account for irregular immigration. This data may include adjustment of status from irregular immigrants. A problem is that the DHS immigration data is only available annually from 1969 to 2014. I interpolate this immigration data to a quarterly series of 1969Q1 to 2014Q1 using a regression-based 4 approach following [Sil01]. The interpolation is useful in mitigating the over-fitting problems of a large parameter space from estimating structural SVARs with time-variation. Figure ?? plots the interpolated quarterly immigration from 1969Q1 to 2014Q1. Time aggregation and first-differencing may induce artificial serial correlation to the data; see [Wor60]. Eight major immigration policy dates are labeled with arrows. Prior to 1990, immigration displays a positive trend and the fluctuations around trend are small. There is a large peak in immigration between 1990 and 1992, which coincides with the enactment of the 1990 IMMACT. This peak subsides by 1993. Immigration reverts to the pre-1990 trend after the peak but exhibits large volatility around the trend. I test for unit roots in the growth rate and log levels of immigration. The augmented Dickey-Fuller test rejects the existence of a unit root at the 1% level in the growth rate of immigration, but not in log levels. As a result, the interpolated quarterly immigration series is treated as observationally equivalent to an I(1) series.5 However, I note that structural breaks can be responsible for creating random walk like behavior in the quarterly immigration series. This paper includes three other macroeconomic indicators on which the SVARs are estimated. Average labor productivity (ALP) is non-farm real output growth per hours worked of all persons (OPHNFB), collected from the FRED database. Hours worked is log weekly hours per capita following [Coc18]. I follow [Whe02] to construct a constant dollar services and non-durable goods consumption expenditures series using a Fisher ideal index. The index employs current dollar expenses, its chain- weighted price deflator, current dollar non-durable goods expenses, and its chain-weighted price deflator as inputs, obtained from the Bureau of Economic Analysis.6

2.3 TVP-SV-SVARs

This section presents the TVP-SV-SVARs and the identification schemes. I discuss and focus on the identifications of an immigration supply shock. Estimation of the SVARs relies on a Bayesian Markov Chain Monte Carlo (MCMC) algorithm. I set up a Bayesian model selection exercise to evaluate the fitness of the models to the data.

4Details about interpolation and a robust check on different interpolation methods are in Appendix B.2. 5The test results are in Appendix B.2.3. 6See Appendix B.2.4 for details about the Fisher ideal index.

36 2.3.1 TVP-SV-SVAR Methodology

Define zt = [∆l og (xt ), l og (nt ), ∆l og (It ), ∆l og (Et )] as the vector that collects immigration growth, ALP growth, logarithm of hours worked, and consumption growth, respectively. I estimate the TVP-SV-SVAR: p X 1 zt = ct + Bl ,t zt l + A−0,t Σt εt , εt N (0, I ). (2.1) − l =1 ∼ where ct denotes the vector of constants, Bl ,t are matrices of the reduced-form parameters for lags l = 1,...,p and t = 1,...,T , the impact matrix A0,t has ones on the diagonal, and Σt = d i ag σm,t { } collects the square roots of the SVs of the structural shocks εt at t along the diagonal with zeroes everywhere else. Let X I z ... z 1 , v e c B ... v e c B c , the concentrated form of the SVAR t0 = n [ t0 1 t0 p ] Bt = [ ( 1,t )0 ( p,t )0 t ]0 is ⊗ − −

A0,t (zt Xt0Bt ) = Σt εt . (2.2) − The non-diagonal elements in A0,t represent the contemporaneous restrictions. I denote a j k,t as the free elements in A0,t . In the following section, I discuss the restrictions needed to achieve identification of the structural shocks. I also consider long-run identifications. These long-run restrictions are imposed by computing the cumulative impact matrix,

1 1 Dt = J (I2 Bt )− J 0A−0,t Σt , (2.3) − where J = [I2 ...02] is a selection matrix. Denote d j k the elements in Dt , setting d j k = 0 imposes long-run neutrality of variable j to shock k.

Let the restricted long-run cumulative matrix be D˜t . Solve for the restricted structural intercepts ˜ and slope parameters matrix Bt by inverting equation (2.3). This step is important because the reduced-form parameters in Bt are no longer consistent with the restricted long-run matrix. Im- posing the long-run restrictions induces non-linearities to the reduced-form VAR parameter space ˜ because of the inversion of the equation. Therefore, I check the eigenvalues of Bt to ensure station- arity of the SVAR. Also, the implementation of long-run restrictions in (2.3) requires modifying the sampling steps.7

As is customary in the TVP-VAR literature, parameter blocks (Bt , A0,t , σt ) are treated as latent variables that evolve as driftless random walks and geometric random walks:

p(Bt Bt 1,Q) = I (Bt )f (Bt Bt 1,Q), (2.4) | − | − 7Appendix B.6.1.2 discusses details about implementing the long-run restrictions in the Gibbs sampler.

37 a j k,t = a j k,t 1 + ζt , (2.5) − l og (σk,t ) = l og (σk,t 1) + τk,t , (2.6) − where I (Bt ) is an indicator function that rejects unstable draws of Bt , the law of motion f (Bt Bt 1,Q) | − is given by Bt = Bt 1+ηt , and [ηt ,ζt ,τt ] are mean zero, i .i .d . disturbances with variance-covariance − matrices [Q,S,W ]. The TVP-SV-SVAR is estimated in state-space form with (2.2) as the observation equation and (2.4)-(2.6) as the state equations. Putting the blocks of latent variables together yields

    εt I3 0 0 0 η   0 Q 0 0   t    = V a r   =   (2.7) V ζt   0 0 S 0  τt 0 0 0 W where Q,S, and W are full rank. Note that this setup can accommodate different estimations without TVP and/or SV. The estimation of a static-coefficient SVAR requires setting Q = S = W = 0. A TVP- SVAR with no SV sets S = W = 0. A static-coefficient SVAR with SV sets Q = 0.

2.3.2 Identification

In this section, I describe the identification of two I(1) shocks, the TFP and immigration supply shocks, and two transitory shocks, the labor demand and transitory consumption shocks. The identification of the SVARs begins with [Gal99], which identifies a TFP shock and a labor demand shock. I add immigration to Galí’s SVAR. The complication in identifying the immigration innovation stems from the endogeneity of immigration. Forward-looking immigrants self-select into countries and labor markets in which they anticipate their job market prospects are best ([Fri01]; [Bor03]). To identify an immigration supply shock in the macroeconomy, I rely on economic theory and auxiliary assumptions to disentangle the sources and causes of an immigration supply shock. I also discuss the identification of a transitory consumption shock to the SVAR to complete the identification schemes. This section concludes with a comparison of four different identification schemes and specifica- tions on TVP or SV. A Bayesian model selection exercise explores the preference the data has across the competing models.

Short-Run Restrictions

Following [Gal99], I leave the impact responses of hours worked to the TFP shock and the labor demand shock unrestricted for the identification of a TFP shock. The diagonal entries are also

38 unrestricted, which means each variable responds to its own shock. Immigration increases the population of the receiving country. Given the definition of immigra- tion as naturalization and permanent residency, I assume the measure of immigration in this paper proxies for the number of new immigrants entering the U.S. labor market.8 Using this definition, I follow convention in the immigration literature and assume that migration decisions are made prior to the entry date (e.g. [Pis97]; [Car07]; [Dal16]). [Par09] also show that since immigration responds to changes in the economic conditions with a delay, the long-run labor supply curve is more elastic than that of the short-run. Therefore, the response of immigration to macroeconomic incentives lags the date an immigrant enters the labor market.

Imposing this restriction on matrix A0,t that governs the impact relations of the SVAR in equation (2.2) implies

SR1 : a31,t = a32,t = a34,t = 0, where I label this short-run restriction SR1. In other words, immigration responds to changes in macroeconomic conditions only with a lag. Therefore, immigration responds only to its own shock at impact. ALP only responds to the TFP shock on impact. This is the second short-run restriction,

SR2 : a12,t = a13,t = a14,t = 0.

New immigrants respond to a TFP shock only with a lag. The third short-run restriction entails consumption does not respond to the labor demand shock on impact,

SR3 : a42,t = 0.

Restriction SR3 assumes that the effect a labor demand shock has on consumption happens only with a lag. Finally, an inflow of immigration affects consumption through expenditure. There are studies documenting the effect immigration has on the receiving economy through a change in expenditure. For instance, [Cop01] find immigration increases consumption, because immigration enlarges an economy through an immediate increase in the number of households. [Nat14] shows immigrants increase aggregate demand by bringing investment or creating a demand for investment as en- trepreneurs. However, there is little evidence on whether this effect occurs at impact or with a lag.9 I

8Unfortunately, no granular data is available to estimate the exact employed immigration inflow. The immigration series do not necessarily map one-to-one with the number of new labor force entry due to immigration. Most likely, this induces measurement errors in the SVARs. 9 [Hon16] show that immigration affects aggregate demand through shifting long-run labor demand, but movements in aggregate demand can also be attributed to changes in consumption due to immigration.

39 consider two alternatives. Imposing

SR4 : a43,t = 0 assumes that consumption responds to an immigration supply shock with a lag. On the other hand, relaxing SR4 implies an immigration supply shock affects consumption on impact. To sum up, I consider two short-run identifications that differ in the restriction of SR4. The two alternative A0,t matrices are

 1 0 0 0   1 0 0 0  a 1 a a  a 1 a a  AI  21,t 23,t 24,t , and AII  21,t 23,t 24,t , (2.8) 0,t =   0,t =    0 0 1 0   0 0 1 0  a41,t 0 0 1 a41,t 0 a43,t 1

I II where A0,t imposes SR1, SR2, SR3, and SR4 while A0,t imposes SR1, SR2, and SR3. Consumption is allowed to respond to the TFP shock on impact because of the permanent income hypothesis. A TFP shock permanently raises the household income and therefore consumption expenditure. Because of the willingness to smooth consumption over time, the agents take advantage of a higher income by increasing consumption on impact.

Long-Run Restrictions

I incorporate several assumptions into the identification of the long-run dynamics of the structural shocks. The long-run identifications are imposed by restricting the elements of the long-run cumu- lative response matrix, Dt , of equation (2.3) to achieve long-run neutrality of variable j to shock k, d j k,t = 0. An incentive to immigrate is an increase in life time earnings. For example, [Cop01] and [Dam13] argue that the decisions to migrate depend on the economic prosperity of the host country. I assume that this relationship pertains in the long-run. Hence, the long-run restrictions rest on producing variation in life time earnings that potential immigrants use when making the decisions to move to the U.S. Since immigration is observationally equivalent to an I(1) series, I assume the level of immigration is independent of the transitory labor demand and transitory consumption shocks in the long-run. This is restriction

LR1 : d32,t = d34,t = 0.

The cumulative impact of a TFP shock on immigration, which is d31,t , is unrestricted. As a result, estimates of this long-run response is a measure of a macroeconomic incentive to immigrate. There is conflicting evidence about the long-run effect of immigration on labor productivity using micro-level data. Firm-level data show immigration could raise labor productivity through

40 complementing domestic labor force, technological transfers, or innovation engagement; see [Fey07] and [Hun10]. On the other hand, [Ort13] and [Pas13] find no effect on productivity due to immigration. I consider both alternatives in restricting the long-run response of ALP to an immigration supply shock. Following [Gal99] where the long-run fluctuations in ALP is determined only by a TFP shock, the identification entails

LR2 : d12,t = d14,t = 0, and

LR3 : d13,t = 0.

The first set of restrictions imposes LR3, which means an immigration supply shock has no long-run impact on ALP.The alternative is to allow the immigration supply shock to have long-run effects on

ALP.This alternative leaves d13,t unrestricted. Consumption does not respond to the labor demand and transitory consumption shocks in the long-run. These responses are embodied in the restrictions

LR4 : d42,t = d44,t = 0.

Hence, consumption responds to the TFP and immigration supply shocks in the long-run. There is a lack of consensus about the data generating process of hours worked. Common statistical tests on unit roots fail to reject the existence of a unit root in U.S. hours worked data. However, hours worked is bounded above by construction and should not possess a permanent trend. This suggests expressing hours worked in log levels.10 I In summary, I consider two alternative long-run identification schemes, which are labeled Dt II and Dt ,

    d11,t 0 0 0 d11,t 0 d13,t 0 d d d d  d d d d  D I  21,t 22,t 23,t 24,t  and D II  21,t 22,t 23,t 24,t . (2.9) t =   t =   d31,t 0 d33,t 0  d31,t 0 d33,t 0  d41,t 0 d43,t 0 d41,t 0 d43,t 0

I The two long-run identifications differ in imposing LR3 on ALP.In Dt , ALP is long-run neutral to an II immigration supply shock. In Dt , ALP responds to an immigration supply shock in the long-run. The unrestricted elements are associated with the two structural I(1) shocks in the SVAR, which are the TFP and immigration supply shocks.

10As a robustness check, Appendix B.3 examines alternative identifications when hours worked is assumed to be first-difference stationary.

41 Competing Models

Table 2.1 List of Models

Model Short-Run Identification Long-Run Identification Time-variation Model 1 SR1, SR2, SR3, SR4 LR1, LR2, LR4 TVP-SV-SVAR Model 2 SR1, SR2, SR3 LR1, LR2, LR4 TVP-SV-SVAR Model 3 SR1, SR2, SR3, SR4 LR1, LR2, LR3, LR4 TVP-SV-SVAR Model 4 SR1, SR2, SR3 LR1, LR2, LR3, LR4 TVP-SV-SVAR Model 5 SR1, SR2, SR3, SR4 LR1, LR2, LR4 TVP-SVAR Model 6 SR1, SR2, SR3 LR1, LR2, LR4 TVP-SVAR Model 7 SR1, SR2, SR3, SR4 LR1, LR2, LR3, LR4 TVP-SVAR Model 8 SR1, SR2, SR3 LR1, LR2, LR3, LR4 TVP-SVAR Model 9 SR1, SR2, SR3, SR4 LR1, LR2, LR4 SV-SVAR Model 10 SR1, SR2, SR3 LR1, LR2, LR4 SV-SVAR Model 11 SR1, SR2, SR3, SR4 LR1, LR2, LR3, LR4 SV-SVAR Model 12 SR1, SR2, SR3 LR1, LR2, LR3, LR4 SV-SVAR Model 13 SR1, SR2, SR3, SR4 LR1, LR2, LR4 FP-SVAR Model 14 SR1, SR2, SR3 LR1, LR2, LR4 FP-SVAR Model 15 SR1, SR2, SR3, SR4 LR1, LR2, LR3, LR4 FP-SVAR Model 16 SR1, SR2, SR3 LR1, LR2, LR3, LR4 FP-SVAR Note: FP-SVAR denotes Fixed-parameter SVAR.

Given several short- and long-run restrictions, I propose the following SVAR specifications using different combinations of these restrictions. Table F.1 provides a comprehensive list of all I II competing models. The two short-run restrictions (A0,t and A0,t ) and the two long-run restrictions I II (Dt and Dt ) give four possible identification schemes. Models 1 to 4 in Table F.1 estimate the TVP-SV-SVARs with each identification. Appendix B.4 reviews the necessary and sufficient rank conditions for identifications of the 16 SVARs following [RR10]. All the identifications satisfy the necessary conditions, but are only locally identified. I also examine the importance of time-variation in the SVAR parameters and in the SVs of the structural errors. I estimate SVARs with four different identifications by turning the TVPs and SVs on and off. Models 5 to 8 estimate TVP-SVARs with constant SV. Models 9 to 12 have constant intercepts and slope parameters with SV. Models 13 to 16 estimate fixed-parameter SVARs. This yields a total of 16 models to be compared in the Bayesian model selection exercise.

42 2.3.3 Bayesian Estimation

The SVARs are estimated using Bayesian methods on a sample from 1979Q1 to 2014Q1. The goal is to obtain the posterior distribution of the states B, Σ, and A0 using a Gibbs sampler. The regression parameters in (2.4) and covariance matrices in (2.5) are sampled with CPF’s Metropolis-within- Gibbs sampler. SVs are sampled via [Omo07]’s mixture normal routine. Appendix B.5 discusses the selection of priors. A sketch of the sampling algorithm follows.11

T T T T Step 1 Set initial values (B0 , A0,0,Σ0 , s0 , 0) and set i = 1, V

T Step 2 Draw the reduced-form intercept and slope parameters Bi from T T T T T T p(Bi z , si 1,Σi 1, i 1) IB (Bi ) using the Carter-Kohn algorithm, where IB (Bi ) truncates − the posterior| − distribution− V · to ensure the stability of the companion form. Impose long-run T T identifications for each draw of Bi to obtain the structural slope parameters Bħi ,

T T T T T T Step 3 Draw A0,i from p(A0,i z ,Bħi , si 1,Σi 1, i 1), | − − V − T T Step 4 Draw Σi through auxiliary variables si ,

T T T T Step 5 Draw hyperparameters i given (Bħi , A0,i ,Σi , z ), V and

Step 6 Repeat steps 2 through 4 M times. The last N (< M ) draws are engaged to construct the posterior of the SVARs.

I set M = 400,000. The burn-in uses the first 200,000 draws. I apply a thinning factor of 40 on the remaining N = 200,000 iterations to reduce the autocorrelation across draws. The baseline TVP-SV-SVAR model acceptance rates of the Metropolis step and the rate of stationary draws in the Gibbs step are 24.437% and 32.143%. Convergence diagnosis is in Appendix B.7.

2.4 Results

This section presents the Bayesian model selection results. Next, I compute the impulse response functions (IRFs) and forecast error variance decompositions (FEVDs) for the data-preferred SVAR. The time paths of the TVPs and SVs are also reported.

43 Table 2.2 Log Marginal Data Densities and Bayes Factors

Identification TVP-SV-SVAR TVP-SVAR SV-SVAR Fixed-Parameter SVAR I I A0,t & Dt [Model 1] [Model 5] [Model 9] [Model 13] -221.72 -239.42 -272.36 -337.10 (1) (4.86e 7) (9.83e 21) (1.29e 50)

II I A0,t & Dt [Model 2] [Model 6] [Model 10] [Model 14] -248.81 -262.62 -271.27 -337.95 (5.82e 11) (5.79e 17) (3.31e 21) (3.01e 50)

I II A0,t & Dt [Model 3] [Model 7] [Model 11] [Model 15] -223.95 -267.69 -272.91 -337.71 (9.29) (9.22e 19) (1.70e 22) (2.36e 50)

II II A0,t & Dt [Model 4] [Model 8] [Model 12] [Model 16] -259.87 -263.03 -270.35 -338.26 (3.70e 16) (8.72e 17) (1.32e 21) (4.10e 50) I II I II Note: A0,t imposes SR1, SR, SR3, SR4. A0,t imposes SR1, SR2, SR3. Dt imposes LR1, LR2, LR4. Dt imposes LR1, LR2, LR3, LR4. Log likelihoods are reported and calculated following [Gew99]. Bayes factors are calculated with respect to Model 1 and are reported in parentheses.

2.4.1 Bayesian Model Selection Results

I compute the log marginal data densities and Bayes factors of 16 models using the harmonic mean 12 method by [Gel93] and [Gew99]. Table F.2 reports the log marginal data densities and Bayes factors (in parentheses) of all the estimated models. The Bayes factors are in parentheses of Table F.2. All Bayes factors are calculated with respect to Model 1. According to Jeffrey’s criterion, a Bayes factor of >150 represents decisive model selection preference. Data strongly favors TVP-SV-SVARs (e.g. Models 1-4) compared to static-coefficient SVARs (Models 13-16) with an average Bayes factor of 3.37e 50. Adding TVP or SV improves Bayes factors, but the data prefers the addition of TVP (Models 5 - 8) over SV (Models 9 - 12). The data strongly favors Model 1, according to the criterion of [Jef61]. Model 1 imposes short-run neutrality of immigration to the other shocks which are SR1, SR2, and SR3. This leaves the impact response of consumption to the immigration supply shock unrestricted. Model 1 also imposes LR1, LR2, and LR4, but does not restrict the long-run response of immigration and ALP to the TFP and immigration supply shocks. Model 1 also includes TVPs and SVs. Since there are several changes to immigration policy and other structural breaks during the sample period, the TVPs and SVs give Model 1 a better fit to

11 The sampling algorithm closely follows [Can15]. I describe the algorithm in details in Appendix B.6. 12Appendix B.8 describes the technical details in computing log marginal likelihoods.

44 the data. For example, the next section presents impulse response functions with respect to the immigration supply shock that show instabilities across the sample. Given the Bayesian model selection results, I adopt Model 1 to study the immigration dynamics at the macroeconomic level in the following sections.

2.4.2 Structural Dynamic Responses

I use IRFs to explore the time-varying transmission of the immigration supply shocks to the macroe- conomy and the effect of the TFP,labor demand, and transitory consumption shocks on immigration. The IRFs are computed with 95% error bands on selected immigration policy dates and four quar- ters post immigration policy dates. The dates are 1980Q1, 1990Q1, 2004Q1, and 2010Q1, which correspond to the enactment of the Refugee Act, the IMMACT, the H1-B Reform Act, and Operation Streamline. I also plot 3-dimensional IRFs. The x axis represents the sample period from 1979 to 2014. The y axis is the 20-quarter forecast horizon. The z axis denotes the size of the response of variable j to shock k. These IRFs are computed as responses of the specified variable to a one standard deviation shock at quarter t in the sample.

2.4.2.1 Average Labor Productivity

Figure ?? plots the 3-dimensional IRFs of ALP with respect to an immigration supply shock from 1979Q1 to 2014Q1. The immigration supply shock has a minimal effect on ALP at impact. However, the IRFs of ALP with respect to an immigration supply shock have inverse hump-shape paths with troughs around four quarters after the shock. At these horizons, the effect of an immigration supply shock on ALP appears stable. The long-run responses of ALP with respect to an immigration supply shock exhibit substantial drift across the sample. The IRFs of ALP plateau after eight quarters with respect to an immigration supply shock. The long-run response of ALP to an immigration supply shock prior to 1990 is often zero. This changes with the enactment of the 1990 IMMACT. In 1990 and 1991, ALP rises in response to an immigration supply shock in the long-run. Between 1990 and 2008, I find negative long-run responses of ALP to an immigration supply shock. The largest negative response occurs during the enactment of the 2004 H-1B Act. This long-run response is highest towards the end of the sample around the recent financial crisis and the Great Recession. To further illustrate this finding, Panel (a) of Figure 2.3 shows the 2-dimensional IRFs of ALP to an immigration supply shock on selected immigration policy dates with 95% error bands. The negative responses over the business cycle horizons are far from zero in the selected dates. The four IRFs suggest time-variation in the long-run responses from zero, positive, to negative over the policy dates of the Refugee Act (1980Q1), the IMMACT (1990Q1), the H1-B Act (2004Q1), and Operation Streamline (2010Q1).

45 Further, Panel (a) in Figure 2.4 reports IRFs of ALP to an immigration supply shock four quarters after the four changes in immigration policy dates. These IRFs display little movement at the business cycle horizons. In the long-run, the error bands of these IRFs suggest substantial uncertainty about the permanent impact of an immigration supply on ALP.

2.4.2.2 Hours Worked

Figure ?? exhibits the IRFs of hours worked to an immigration supply shock. These IRFs vary across the sample. Hours worked displays minimal impact responses to an immigration supply shock. The impact response, which is left unrestricted in Model 1, is near zero throughout the sample. This suggests hours worked responds to immigration supply with only a lag. However, hours worked shows a substantial drift in the height of the IRFs in the long-run. The negative hump-shaped IRFs have a trough at the 5th quarter. Prior to the change of immigration policy in 1990, the long-run response of hours worked to an immigration supply shock is about zero. This long-run response turns negative after 1990. The largest permanent decrease in hours worked given the immigration supply shock occurs during the Great Recession and the financial crisis. Panel (b) of Figure 2.3 contains the 2-dimensional IRFs of hours worked to an immigration supply shock at the four policy dates. During the change of immigration policy in 1980, hours worked responds negatively in the short-run but the response turns positive over the business cycle horizons. However, the changes in immigration policy of 1990 and 2003 do not have economically important responses at these dates. Further, I plot the 4-quarter post-policy change IRFs of hours worked to an immigration supply shock in Panel (b) of Figure 2.4. These IRFs suggest that changes to immigration policy have economically meaningful effects on the dynamic responses of hours worked to an immigration supply shock. Note the IRFs four-quarter post 1990Q1 and 2003Q1 are negative over the business cycle horizons into the long-run.

2.4.2.3 Consumption

The 3-dimensional IRFs of consumption to an immigration supply shock are plotted in Figure ??. The IRFs display large time-variation in the persistence of the responses of consumption to an immigration supply shock. From the start of the sample to about the 1990 IMMACT, the IRF of consumption driven by an immigration supply shock is positive over the business cycle horizons. During this period, the largest long-run steady state of the IRFs is an increase of up to 4 basis points in 1990Q3. The reason is immigration increases aggregate demand and the U.S. experienced a large influx of immigration during the late 1980s to the early 1990s. However, the sign of the long-run IRFs of consumption to an immigration supply shock turns negative post-1990. Between 1995Q1 and 1999Q1, the long-run IRFs are about zero. By 2000Q1, the long-run impact of an immigration supply shock on consumption turns negative and stays negative

46 for the remainder of sample. Panel (c) of Figures 2.3 and 2.4 reinforce these results.

2.4.2.4 Immigration

Figure ?? plots the 3-dimensional IRFs of immigration with respect to a TFP shock. On impact and over the business cycle horizons, a TFP shock decreases immigration. Over the sample, the hump-shaped responses reach a trough at the fifth quarter and plateau after the eighth quarter. There is time-variation in the long-run IRFs. This instability displays co-movement with changes in immigration policy and the state of the economy. For instance, the drifts in the long-run IRFs of immigration to a TFP shock coincide with the 1990 IMMACT, the 2004 Reform Act, and the Great Recession and the financial crisis. The 2-dimensional IRFs of Figure 2.3 are consistent with these results. As shown in panel (d) of Figure 2.3, the 1990Q1 and 2010Q1 IRFs suggest economically and statistically significant permanent increases in immigration to a TFP shock. An improvement in aggregate productivity increases wages and incentivizes immigration. This evidence complements [Gro12] and [Ort13] that immigrants respond to economic incentives. However, the long-run response of immigration to a TFP shock is about zero except at the time of the 1990 IMMACT, the 2010 Operation Streamline, and the Great Recession. Figure ?? displays the 3-dimensional IRFs of immigration with respect to a labor demand shock. The IRFs of immigration to a labor demand shock has an inverse hump shape with a trough at four to five quarters after the shock. The trough of the IRFs drift from the start of the sample to 1990. This drift diminishes in the post-1990 sample. The plot of the IRFs of immigration to a transitory consumption shock appears in Figure ??. Immigration has a hump-shaped response to a transitory consumption shock. These IRFs peak at five quarters after the shock and plateau after two and half years. The evidence of a positive but transitory IRF of immigration to a transitory consumption shock over the business cycle horizons is in line with the economic theory about the direct and indirect incentives of immigration. First, an increase in aggregate demand through expenditure attracts immigrants because immigrants react to economic incentives ([Luc75]). Second, a larger aggregate demand raises output growth followed by a temporary increase in aggregate supply. Therefore, an increase in labor demand, due to the firm’s willingness to temporarily increase production, also incentivizes immigration. The response of immigration to a transitory consumption shock is the same across the sample. The effects of a transitory consumption shock on immigration do not depend on immigration policy changes or the business cycles of the economy. Therefore, the effect of changes in economic incentive to migrate on immigration is time invariant, while an improvement in productivity or the intensive margin affects immigration differently over the sample depending on the state of the immigration policy and the economy.

47 2.4.3 FEVD of the Structural Shocks

This section reports FEVD to study the importance of the structural shocks. FEVD calculates the percentage of variation in the forecast errors of a variable due to a structural shock. Similar to the IRFs, the time-varying structure of the SVARs allows the composition of the forecast errors to vary over time. I report 3-dimensional FEVDs up to 20-quarter forecast horizons on the sample from 1979Q1 to 2014Q1. Figure ?? plots the 3-dimensional FEVDs of ALP,hours worked, immigration, and consumption with respect to an immigration supply shock. At impact, the immigration supply shock dominates the fluctuations of immigration. This is consistent with the identification that immigration only responds to its own shock on impact. The contribution of an immigration supply shock to immigration decreases over the business cycle horizons into the long-run to about 60% to 75% over the sample. These results indicate that the decision to immigrate is the most dominant factor in explaining variations in immigration. Changes in the macroeconomy also play an important role in driving fluctuations in immigration. The TFP,labor demand, and transitory consumption shocks together account for the remainder of the long-run variations in immigration. However, the immigration supply shock is not economically important in driving the fluctuations in ALP,hours worked, and consumption. The contribution of an immigration supply shock to these variables is less than 15% and does not fluctuate across the sample and over the forecast horizons. In Figure ?? I illustrate the 3-dimensional FEVDs of all variables with respect to a TFP shock. The TFP shock dominates the variations in ALP and contributes up to 30% to the fluctuations in hours worked. This finding is consistent with real business cycle theory about the important role of the TFP shock in driving business cycle fluctuations. The TFP shock contributes the second largest error variance share to the variations of immigration. In the long-run, the TFP shock accounts around 30% of the variability in immigration. Therefore, fluctuations in immigration are mostly explained by the immigration supply and TFP shocks. The dominance of the TFP and immigration supply shocks to variations of immigration is stable over the sample.

2.4.4 Evidence from Time-Varying Parameters and Stochastic Volatility

The lower left window of Figure ?? plots the posterior estimates of the SV of the immigration supply shock with 95% error bands. The SV of the immigration supply shock is acyclical and lags the business cycle. There are three peaks in the SV of the immigration supply shock in early 1980s, early 1990s, and around 2004, which coincide with the IRCA of 1986, the IMMACT of 1990, as well as the H1-B Act of 2004. For instance, the instability caused by the IMMACT of 1990 is well captured by the peak in the SV of the immigration supply shock. The SV of the immigration supply shock drops to an all-sample low after 2004.

48 I show the SV of the TFP shock in the upper left window of Figure ??. It decreases at the beginning of the productivity slowdown in the 1970s with a small peak during the Dot-com boom and bust of 2001. The SV of the labor demand shock in the upper right window follows a similar pattern but increases towards the end of the sample around the recent financial crisis. The SV of the TFP shock declines over the sample while the SV of the transitory consumption shock exhibits co-movement with the business cycles. For example, the lower right window of Figure ?? shows that there are peaks in the SV of the transitory consumption shock around the 1991 recession and the recent financial crisis and the Great Recession.

Figure ?? displays the posterior median values of the impact parameters (a j k,t ’s) in posterior median values with 5% and 95% quantiles of the posterior distribution. These parameters are the estimated elements of the short-run restriction matrix A0,t that control the contemporaneous effect of the respective variable to a one standard deviation structural shock. The estimates of most impact coefficients are small. According to the 5% and 95% quantiles of the posterior distribution depicted in Figure ??, estimates of the impact parameters fluctuate closely to zero, but show co-movements with the NBER recession dates.

There are a few exceptions in Figure ?? worth noting. Parameter a23,t , which measures the contemporaneous response of hours worked to an immigration supply shock, displays a change in cyclicality before and after 1990 with respect to the NBER recession dates. The impact response of hours worked to TFP shocks, a21,t , is zero or negative during the non-recessionary dates. Parameter a43,t represents the impact of immigration supply shock on consumption. The estimated time path of a43,t displays statistically significant troughs during the 1982, 1990, 2001, and 2009 NBER recession trough dates. This evidence suggests that the impact effect of an immigration supply shock on consumption is dependent on the state of the economy. Similar behavior is also observed in a41,t , which denotes the impact response of consumption to a TFP shock.

2.5 Conclusion

This paper studies the dynamic relationship of immigration and the U.S. macroeconomy from 1969Q1 to 2014Q1. I estimate structural VARs with time-varying parameters and stochastic volatility. The structural shocks are identified with a combination of short- and long-run restrictions on average labor productivity, hours worked, immigration, and consumption. A Bayesian model selection exercise is used to find the identification that is preferred by the data. This identification restricts immigration not to respond to TFP,labor demand, and transitory consumption shocks at impact and allows average labor productivity to respond to an immigration supply shock in the long-run. The data also prefers this structural VAR because there is important time-variation in the structural intercepts, slope parameters, and stochastic volatility. Using the structural VAR preferred by the data, I compute impulse response functions and

49 forecast error variance decomposition. An immigration supply shock decreases average labor pro- ductivity over the business cycle horizons, but the sign of the long-run response varies over the sample. The impulse response functions of hours worked and consumption to an immigration supply shock also decline at the business cycle horizons and into the long-run. However, these impulse response functions are state dependent. Changes in these impulse response functions are tied to NBER dated recessions and changes to U.S. immigration policy, especially the IMMACT of 1990. The immigration supply shock explains 60 to 75% of the variation in observed immigration since 1979. The remaining variation is mostly explained by the TFP shock. Over the sample, these forecast error variance decompositions exhibit little change. I interpret this evidence to mean that economic incentives have played a role in the decision to immigrate to the U.S. since 1979. The immigration supply shock is not important for the variations in average labor productivity, hours worked, and consumption. The forecast error variance decompositions of average labor productivity, hours worked, and consumption are dominated by their own shocks. The TFP shock also drives about 30% of variations in hours worked. There is little variation in these forecast error variance decompositions over the sample. My estimates show that the responses of immigration to the identified shocks are dependent on immigration policy regime and the state of the business cycle. This suggests future research should study the impact of immigration with respect to existing immigration policy and the state of the macroeconomy. Another important research issue is to develop models that identify an immigration policy shock. I leave these questions for future research.

50 Figure 2.1 Quarterly Interpolated Immigration Series Notes: Sample from 1969Q1 to 2014Q1. Dates of the immigration policy changes are labeled with arrows. Source: Department of Homeland Security.

51 Figure 2.2 IRF of ALP to an Immigration Supply Shock, Estimated with Model 1 Notes: Median responses of average labor productivity to an immigration supply shock. Sample is 1979Q1 - 2014Q1.

52 Figure 2.3 IRFs on Selected Dates of Changes to Immigration Policy Notes: Selected dates are 1980Q1, 1990Q1, 2003Q1, 2010Q1. Median responses of average labor productivity to immigration supply shocks. Sample is 1979Q1 - 2014Q1. Median value (in blue) and 95% error bands (in red) plotted.

53 Figure 2.4 IRFs at 4 Quarters Post Selected Dates of Changes to Immigration Policy Notes: Selected dates are 1981Q1, 1991Q1, 2004Q1, 2011Q1. Median responses of average labor productivity to immigration supply shocks. Sample is 1979Q1 - 2014Q1. Median value (in blue) and 95% error bands (in red) plotted.

54 Figure 2.5 IRF of Hours Worked to an Immigration Supply Shock, Estimated with Model 1 Notes: Median responses of hours worked to an immigration supply shock. Sample is 1979Q1 - 2014Q1.

55 Figure 2.6 IRF of Consumption to an Immigration Supply Shock, Estimated with Model 1 Notes: Median responses of consumption to an immigration supply shock. Sample is 1979Q1 - 2014Q1.

56 Figure 2.7 IRF of Immigration to a TFP Shock, Estimated with Model 1 Notes: Median responses of immigration to a TFP shock. Sample is 1979Q1 - 2014Q1.

57 Figure 2.8 IRF of Immigration to a Labor Demand Shock, Estimated with Model 1 Notes: Median responses of immigration to a labor demand shock. Sample is 1979Q1 - 2014Q1.

58 Figure 2.9 IRF of Immigration to a Transitory Consumption Shock, Estimated with Model 1 Notes: Median responses of immigration to a transitory consumption shock. Sample is 1979Q1 - 2014Q1.

59 Figure 2.10 Posterior Estimates of Stochastic Volatility, Estimated with Model 1 Notes: Posterior estimates of stochastic volatilities of the TFP,labor demand, immigration supply, and transitory con- sumption shocks. Median value (in blue) and 95% error bands (in red). Sample is 1979Q1 to 2014Q1. Grey shades are NBER recession trough dates.

60 Figure 2.11 Posterior Estimates of Impact Parameters, Estimated with Model 1 Notes: Posterior estimates of the structural impact parameters. Median value (in blue) and 95% error bands (in red). Sample is 1979Q1 to 2014Q1. Grey shades are NBER recession trough dates.

Figure 2.12 FEVD of All Variables to an Immigration Supply Shock, Estimated with Model 1 Notes: Median contribution of an immigration supply shock to all variables. Sample is 1979Q1 - 2014Q1.

61 Figure 2.13 FEVD of All Variables to a TFP Shock, Estimated with Model 1 Notes: Median contribution of a TFP shock to all variables. Sample is 1979Q1 - 2014Q1.

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69 APPENDICES

70 APPENDIX

A

APPENDIX FOR CHAPTER 1

A.1 Data Source and Construction

The baseline SVARs are estimated with hours worked and ALP on a quarterly sample from 1948Q1 to 2016Q4.

A.1.1 Hours Worked

[Coc18] construct data on hours worked from official sources using the following equation

CPSt nt = 4 52, POPt × ÷ where CPSt is total Current Population Survey (CPS) hours worked and POPt denotes non-institutional population from age 16 to 64. Total CPS hours is defined as persons at work1 multiplied by hours worked per week2. Total non-institutional population is specified as civilian population of 16 years and over (BLS Series LNU000000000) less civilian population of 65 years and over (BLS Series

1The CPS definition of persons at work is all employed persons who are not absent from work during the reference week. The BLS defines employed person as all persons who work at least 1 hour as paid employees, work without pay for more than 15 hours, or are not working due to temporary absence. 2For 1948Q1 to 1959Q2, the data are sourced U.S. Bureau of the Census, Current Population Reports. For 1959Q3 to 2016Q4, the data are obtained from the BLS.

71 LNU000000097). Both CPSt and POPt are available monthly and deseasonalized. The monthly data are converted to quarterly through temporal averaging. For robustness, I consider the index of hours of all persons in non-farm business sector (HOANBS) as a different measure of hours worked. I also consider non-agriculture establishment hours [Gal99], and the index of business hours ([Fra03]; [Fra05]), as alternative definitions of hours worked.

A.1.2 ALP Data

Non-farm real output growth per hours worked of all persons measures ALP in this paper. I include alternative data for robustness. I also consider real output per hour of all persons in non-farm business sector (OPHNFB). This ALP definition is used in, for example, [Fra03] and [Fra05]. I obtain these series from the Federal Reserve Bank of St. Louis database.

72 A.2 A Fixed-Coefficient SVAR and Identification

A.2.1 A Fixed-Coefficient SVAR

[Gal99] identifies a fixed-coefficient VAR with a long-run restriction on ALP to study the response of hours worked to a TFP shock. I outline Galí’s reduced-form VAR, structural VAR, and BQ decomposi- tion used in estimation.

A.2.1.0.1 Reduced-form VAR

Consider a standard reduced-form VAR with p lags:

p X zt = Bi zt i + et , et (02 1,Ω2 2), (A.1) − × × i =1 ∼ where – ™ ∆l n(xt ) zt = , ∆l n(nt )

Bi are 2 2 coefficient matrices, for i = 1...p, and et is a 2 1 reduced-form error term. Vector zt × × contains data on ALP (x ) and hours worked (n). [Gal99] assumes that x and n are I (1). Therefore, they enter the VAR as log first-differences. The reduced-form VMA( ) representation of the VAR(p) in (A.1) is ∞ 1 zt = [I2 B(L)]− et Λ(L)et , (A.2) − ≡ where L is the lag operator. I denote Λ(L) the coefficient matrices of the reduced-form error terms.

A.2.1.0.2 SVAR

[Gal99] constructs a SVAR by imposing long-run neutrality on ALP in the reduced-form VAR of (A.1). The SVAR(p) has the following SVMA( ) representation: ∞ – ™ X∞ j γ11,j γ12,j zt = Γ (L)εt , Γ (L) = Γj L , Γj = . (A.3) γ γ j =0 21,j 22,j

Label the first element in the vector of structural innovations as the TFP shock and the second element as the demand shock. Galí assumes that only a TFP shock has a permanent effect on the

73 level of labor productivity. Economic motivation for this long-run neutrality assumption is discussed in Appendix A.2.2. Therefore, the cumulative effect of the demand shock on ALP is zero, or

X∞ Γ12 = γ12,j = 0. (A.4) j =0

Hence, the long-run cumulative impact matrix is

– ™ Γ11 0 Γ (1) = , (A.5) Γ21 Γ22 where the four elements of Γ (1) represent the cumulative impact of the TFP and demand shocks on the level of ALP and hours worked.

A.2.1.0.3 BQ Decomposition

Equating the reduce-form VMA( ) of (A.2) to the structural VMA( ) of (A.3) gives ∞ ∞ 1 εt = Γ (0)− et , Γ (j ) = Λ(j )Γ (0). (A.6)

The impact matrix, Γ (0), needs to be estimated to allow a mapping between the reduced-form error terms and the structural shocks. The BQ decomposition estimates the four elements in Γ (0) in the following way.

1. Estimate a fourth-order reduced-form VAR on quarterly data via equation by equation OLS and obtain Ωˆ 1 PT eˆ eˆ . = T t =1 t t 0

2. Equation (A.6) yields 0 E 0 0 0 . Given the variance-covariance matrix Γ ( ) [εt ε0t ]Γ ( )0 = Γ ( )Γ ( )0 = Ω Ω (and its estimate Ωˆ ) is positive semi-definite, there are three identifying restrictions on Γ (0):

2 2 ˆ γ11,0 + γ12,0 = Ω11,

2 2 ˆ γ21,0 + γ22,0 = Ω22, ˆ and γ11,0γ21,0 + γ12,0γ22,0 = Ω12.

3. The fourth identifying restriction is the zero restriction from the long-run cumulative impact matrix of (A.5),

λ11γ12,0 + λ12γ22,0 = 0, given λ11,λ12.

74 The system is just-identified given the four identifying restrictions are satisfied by the four impact structural parameters. Solving this system of equations gives unique3 estimates of the four elements of Γ (0). Once Γ (0) is computed, the structural IRFs are calculated using equation (A.6).

A.2.2 Identification Assumptions

The long-run identification in this paper assumes that the TFP shock is the only source of unit root in ALP.As in [Gal99] and [Gal04], this identification relies on the following assumptions.

1. The aggregate production function is homogeneous of degree 1:

Yt = F (Kt , At Nt ),

where Y , K , N , and A denote output, capital, labor input, and TFP;

K 2. The capital labor ratio per efficient worker (k AN ) is stationary: ≡

Fk (kt ,1) = rt + δ,

where r and δ denote return on capital and depreciation.

Resting on constant returns to scale, I define ALP,denoted X , as output per efficient labor hour:

Yt Xt = = At F (kt ,1). Nt

Since k is stationary, any shocks that affect k will only have a temporary effect on ALP.Therefore, only shocks that have a permanent effect on A can be a source of unit root in ALP.

3 See [Bla89] a proof.

75 A.3 Details about the Andrews Test

The [And93] structural change test computes a Wald test statistic with unknown break dates. Denote the parameter of interest β, the null hypothesis is

H0 : βt = β0, (A.7) where β0 is the fixed parameter estimate based on the entire sample. The Andrews test constructs X X ˆ t t0 βt 1 the cumulative sums of recursive residuals, Wt − − at each date t by rolling through the = 1 x x p + t0 t sample. In order the start the rolling computation, Andrews suggests a symmetric 15% trim (π0) of the data when there is no knowledge of the change point. [And93] proposes taking the supremum, average, or exponential value of the Wald statistics as inference. The supremum test takes the largest estimated Wald statistics; the average test computes the average of the Wald statistics; the exponential test takes the natural log of the average sample tests. In this paper, I plot the estimated Wald statistics against the asymptotic critical values updated in [And03] with π0 = .15 and a critical value of .10.

76 A.4 Details in Sampling Algorithms

This section describes the technical details in the MCMC sampler, which closely follow [Can15] and [DN15].

A.4.1 Alternative Prior csminwel For robustness, I estimate B and V B with OLS, and h j via Maximum Likelihood using the package from a time-invariant VAR with a training sample of the first τ = 40 observations. csminwel minimizes the negative log likelihood. The search tolerance is set as 1 10 4. The computer code is − available on Christopher Sims’ website. ×

A.4.2 Draw Bħ

A.4.2.0.1 Carter-Kohn Algorithm

Given the concentrated form of the SVAR,

1 2 (zt Xt0Bt ) = Ωt εt , − the reduced-form intercept and slope parameters B are drawn first with the Carter-Kohn algorithm. T T T [Car94] document how the posterior density p(B z ,H , ) can be factored as | V T 1 T T T T T Y− T T p(B z ,H , ) = p(BT z ,H , ) p(Bt Bt +1, z ,H , ). | V | V t =1 | V

This posterior density is evaluated using the following backward recursion algorithm. Condi- T T tional on the previous draws of the mixture normal indicator (si 1), stochastic volatility (Hi 1), − − and hyperparameters ( i 1), the observation equation (5) is Gaussian. The standard Kalman filter V − T T T recursions produce the first element on the right hand side of p(B z ,H , ) = N (BT ,PT ), where | V PT is the precision matrix of BT given by the Kalman filter. Since BT is conditionally normal, the updating equations are:

P P 1 , (A.8) Bt t +1 = Bt t + t t t−+1 t (Bt +1 Bt ) | | | | − and P P P P 1 P . (A.9) t t +1 = t t + t t t−+1 t t t | | | | |

77 T The reduced-form Bi is drawn with the following backward recursion algorithm:

1 Run the Kalman filter from t = 1...T to obtain the mean BT T and variance PT T . Draw BT | | from N (BT T ,PT T ) conditional on the past draw of volatility states and hyperparameters. | | 2 At time T 1, use the updating equations (A.8) and (A.9) of the Kalman smoother to obtain − and P . Draw from N ,P . BT 1 T 1,BT T 1 T 1,BT BT 1 (BT 1 T 1,BT T 1 T 1,BT ) − | − − | − − − | − − | − 3 Repeat step 2 for t = T 2,T 3,...,1. − − This backward recursion algorithm, known as the Carter-Kohn algorithm, delivers a draw of T Bi = [B1,B2,..,BT ] from its conditional posterior distribution. Then, the admissible values of B are restricted by imposing stationarity check through the truncation function IB(B). This implies T explosive draws of B will be discarded if any of its elements has its eigenvalues outside of the unit root.4

A.4.2.0.2 Structural Intercepts and Slope Parameters

Compute the structural intercepts and slope parameters, Bħi , as discussed in Section 3.1. Evaluate eigenvalues for each draw of Bħat each date. Discard the non-stationary draws.

A.4.3 Draw H

T Conditional on the current structural draws of Bħi , the concentrated form of the SVAR can be written as + zt = Ht εt , (A.10) where z + z X . t ( t t0Bħt ) This is≡ approximated− by taking the log of squares on both sides. Equation by equation, this is

+ 2 l og (z j,t + c c ) 2l og (hj,t ) + l og εj,t , (A.11) ≈ where c c is a small constant in case of extremely small values of z +. 2 2 2 However, l og εj,t is not Gaussian and is distributed log χ . Therefore, the distribution of l og εj,t needs to be approximated. I adopt an approximation method with a 10-component mixture of nor- T T T mals following [Omo07]. Auxiliary variables si are introduced and drawn conditional on (Bħi ,Hi 1) − 4 [Koo10] develop a single-move algorithm which does not involve approximation. Despite the advantage in accuracy, [Can15] highlight the higher computational burden of the algorithm. Due to computational efficiency, this study uses the Carter-Kohn algorithm in drawing B.

78 T T to keep track of the normal mixtures. The sampling of s is placed after drawing Bħ and before T drawing H , following [DN15]. Specifically, compute

zi∗,k 2l og (hj,t ) ζk + 1.2704 P r (si ,k = k zt∗,l og (hi ,t )) qk φ( − − ), | ∝ × γk for k = 1,...,7, qk is a set of preset weights, and ζk and γk are the mean and standard deviation of the t h k mixture. The selection of parameters qk , ζk and γk are reported in Table ?? below. Next, draw u U 0,1 . If P r s k z ,l og h is larger than its previous (k 1t h ) probability and u, then ( ) ( i ,k = t∗ ( j,t )) ∼ | − set si ,k = k. T T The state-space system is now conditionally Gaussian given the approximation (Bħi , si ). The T draws of Hi are generated and evaluated with the Carter-Kohn algorithm.

Selection of Mixing Distribution Parameters

T Table ?? gives the selection of parameters in the drawing of s , following [Omo07].

Table D.3 Parameters of Mixture Normal Distributions

k qk ζk γk 1 0.00609 1.92677 0.11265 2 0.04775 1.34744 0.17788 3 0.13057 0.73504 0.26768 4 0.20674 0.02266 0.40611 5 0.22715 -0.85173 0.62699 6 0.18842 -1.97278 0.98583 7 0.12047 -3.46788 1.57469 8 0.05591 -5.55246 2.54498 9 0.01575 -8.68384 4.16591 10 0.00115 -14.65000 7.33342

A.4.4 Draw V T T T Draw the hyperparameters i given (Bħi ,Hi , zi ). As mentioned in the main text, Q and W are assumed to have Inverse-WishartV or Inverse-Gamma priors for the Kalman filter within a Gaussian state-space system. Because of the conjugate priors, it is straightforward to sample the hyperparam- eters by drawing from the closed-form posterior.

79 A.5 Convergence Diagnosis

To test for convergence of the sampler, I consider the relative numerical efficiency (RNE) and its associated inefficiency factor (IF) for the evaluation of sampling accuracy of the algorithm. The idea is to examine the ratio between the variance of i .i .d . draws from the posterior and the variance of the sample draws that could be autocorrelated. As suggested by [Gew05], Chapter 4.7, RNE is defined as ˆ VAR(θi ) RNE = S(0) ˆ where VAR(θi ) is the sample variance of the Gibbs draws and S(0) is the variance of the Gibbs sampler. If one can sample directly from the posterior density, then RNE is 1.0. In other words, RNE is the number of draws wound be required to produce the same numerical accuracy if the draws had been made from an i .i .d . sample drawn directly from the posterior distribution.

Table E.1 Inefficiency Factors for Different Sets of Parameters.

Median Mean Min Max 10t h Percentile 90t h Percentile

Bg r 0.924 0.953 0.508 1.927 0.552 1.389 Hg r 0.972 0.993 0.465 1.728 0.753 1.248 Bl l 0.925 0.978 0.570 2.185 0.611 1.390 Hl l 1.072 1.083 0.517 1.699 0.816 1.370 Note: “g r " denotes SVAR estimation results with hours worked in growth rates. “l l " denotes SVAR estimation results with hours worked in log levels.

The inverse of RNE, known as the Inefficiency Factor (IF), is interpreted as an indicator of serial correlation. Large values of IF indicate strong serial correlation across the posterior draws, meaning the sampler is not converged. Table E.1 shows the estimated IFs of Bħand H . The IF values with hours worked in growth rates (log levels) are in the upper (lower) panel. According to [Pri05], IF values of 20 or below are regarded as satisfactory convergence to the ergodic distribution. I also consider the method of evaluating retained draws ([Gel03]). Figure A.1 plots the recur- T T sive means, reported for 20 draws, for all estimated parameters in Bħ and H . If the draws have converged, one should expect to see stable retained draws fluctuating around a stationary mean. In each of the three-dimensional plots, x axis denotes the vectorized coefficients; y axis shows the number of draws from 1 to 20; z axis gives the magnitude of the recursive means of coefficient estimates. All four panels in Figure A.1 show minimal variation, if any, when estimated with hours

80 worked in either log levels or growth rates.

T T Figure A.1 Recursive means for Bħ and H . Notes: The recursive means are recorded 20 draws per estimator. Minimal variation or autocorrelation over the retained draws indicates convergence.

81 A.6 Bayesian Model Selection

I conduct a Bayesian model selection exercise to statistically verify the importance of adding time variation to SVARs in fitting the data. Specifically, I adopt the harmonic mean method, following [Gel93] and [Gew99], to compute and compare log marginal likelihood of different specifications in time variation. Table F.1 displays the eight proposed SVARs. The SVARs are estimated on hours worked in log levels (Models 1-4) or growth rates (Models 5-8).

Table F.1 List of Model Comparison

Model 1 TVP-SV-SVAR with hours worked in log levels Model 2 fixed-coefficient SVAR with with hours worked in log levels Model 3 TVP-SVAR with hours worked in log levels Model 4 SV-SVAR with hours worked in log levels Model 5 TVP-SV-SVAR with hours worked in growth rates Model 6 fixed-coefficient SVAR with with hours worked in growth rates Model 7 TVP-SVAR with hours worked in growth rates Model 8 SV-SVAR with hours worked in growth rates

Table F.2 reports the estimated log marginal likelihoods of the eight SVARs. Overall, the SVARs with both TVP and SV (Models 1 and 5) yield the highest log marginal likelihood. Whether hours worked is in log levels or growth rates, the data strongly prefers the addition of SV. The worst performers are the fixed-coefficient SVARs of Models 2 and 6.

Table F.2 Log Marginal Likelihood

Metrics/Model 1 2 3 4 Log M-L -181.1825 -489.4757 -246.2779 -206.2855

Metrics/Model 5 6 7 8 Log M-L -147.2332 -646.8552 -180.7509 -147.7371

82 A.7 Additional IRFs

Figure A.2 IRFs of hours worked with respect to a TFP shock on selected NBER recession (trough) dates

Figure A.3 IRFs of ALP with respect to a demand shock on selected NBER recession (trough) dates

83 Figure A.4 IRFs of output with respect to a TFP shock on selected NBER recession (trough) dates

84 APPENDIX

B

APPENDIX FOR CHAPTER 2

B.1 Relevant Recent Changes in U.S. Immigration Policy

This appendix describes the eight important immigration policy changes since the 1970s that are relevant to this paper.

1. The Refugee Act of 1980 provided the first clear definition of refugee immigration to the U.S. and allocated a 50,000 cap on refugee admission with unlimited cap for certain protocols. [Ben16] argues that its implication draws marking of a wave of open door policies. Immigration policy fluctuates between being restrictive and permissive, a perception that affects expecta- tion on migration difficulty. Potential immigrants interpret these policy changes differently, which may offers an unintended incentive to migrate ([Han01]; [CC12]).

2. The Immigration Reform and Control Act (IRCA) of 1986 legalized irregular immigrants since 1982 and created a new class of working visa/visa waiver program. The IRCA was considered the largest nationwide amnesty of irregular immigration in history and a strong signal of an era of nonrestrictive immigration policies ([Ver93]; [Bak97]; [Orr03]). The amnesty was followed by a high level of inflow and employment of immigration in the 1990s.

3. The American Homecoming Act and the Immigration Act of 1990 (IMMACT) raised the yearly total immigration cap to 790,000 (a 290,000 addition), provided an additional 140,000 visa cap,

85 and granted special status for immigrants of specific countries of origin. This legislation cre- ated the largest increase in immigration cap since 1910. Together with the IRCA, the IMMACT generated the largest inflow of immigration around 1990; See [Bor94], [Fri95].

4. The enactment of the Illegal Immigration Reform and Immigrant Responsibility Act of 1996 1 is a policy reform on irregular immigration . [Han01] argue that the act was designed to disincentivize irregular immigration by increasing border patrol enforcement, which had a direct negative effect on the inflow of irregular immigrants.

5. The American Competitiveness and Workforce Improvement Act of 1998 increased the annual H-1B cap from 65,000 to 115,000 and required employers to pay for H-1B administration fees and worker benefits in accordance with the same criteria as U.S. workers.

6. The Legal Immigration Family Equity Act (LIFEA) of 2000 drastically improved border patrol tightness, serving as a negative signal to potential irregular immigrants2.

7. The H-1B Visa Reform Act of 2004 reduced the H-1B cap from 195,000 to 65,000 and altered the filing fee structure.

8. The constructed data series shows a number of consecutive large negative spikes post 2005, matching the development and expansion period of Operation Streamline (2006 - 2010). Op- eration Streamline was the largest operation of aggressive enforcement against unauthorized border-crossing to date. Operation Streamline adopted a zero-tolerance policy in the prosecu- tion and deportation of irregular immigration. According to the DHS, the operation resulted in deportation of 1.54 million immigrants from 2007 to 2011.

B.2 Details in Data Construction

B.2.1 Interpolation of Quarterly Immigration

This appendix describes details in the interpolation of the immigration series. There are various methods to conduct temporal disaggregation in the statistics literature. The dynamic smoothing procedure of Lisman and Sandee (1964) generates a smoothed path between the low frequency data points without additional information set. This mechanism closely resembles a “blind" estimate

1The Act lifted the trigger for immediate deportation, announced the right to detain deportees in American jails no less than two years at deportees’ expense, and removed the pardon waiver right for unlawful immigrant for a given period of stay in the country. 2The LIFEA expedited various deportation procedures, including the removal of an immigration judge hearing for deportation. The Act also reduced the petition waiting time for family-based and employment-based immigration.

86 of a cubic spline. Al-Osh (1989) uses an ARIMA model to smooth and predict the high frequency data. The benchmarking method, developed by [Den71] and extended by Cholette and Dagum (1994), generates high frequency data from a low frequency series by minimizing a loss function with respect to a separate high frequency series from a different source (benchmarking a low frequency series against a high frequency series). The key element is to have two distinct sources for the same variable. [Cho71] develop a best linear unbiased regression interpolation method, which built upon the temporal disaggregation literature. The idea is to estimate high frequency series by running a Generalized Least Squares (GLS) regression using multiple “indicator" series. They assume the regression relationship q Y = X β + µ, where Y q is 4n 1 high frequency data of estimation, X is a 4n p matrix that contains p “indicator" × × series of high frequency, and µ is a random vector with mean zero and variance-covariance matrix V y . The low frequency (observed) data regression relationship is

y Y = C 0X β + C 0µ, (B.1) where C is a n 4n frequency converter matrix that converts annual series Y y to quarterly series Y q . Note that equation× B.1 implies that the high frequency variance-covariance matrix, denoted V q , is equal to the low frequency variance-covariance matrix, V y , pre and post-multiplying matrix C (i.e. V q CV y C ). Chow and Lin also include two different forms of C for dealing with flow or = 0 stock variable. Matrix manipulation yields the estimated coefficients

ˆ y 1 1 y 1 y β = [X 0C (C 0V C )− C 0X ]− X 0C (C 0V C )− Y , (B.2) where solving equation B.2 requires knowledge of V y . The Chow-Lin method, which is widely used among statistical agencies such as the BEA or Eurostat, defines the random variable µ to have an AR(1) process. A few extensions of the Chow-Lin method have been done in aim to enhance the interpolation performance. Fernandez (1981) argues that the trend and the cyclical component in economic time-series data should be properly addressed for interpolation. The author uses a random walk µ for interpolation in order to account for potential non-stationarity and/or serial correlation in the data. Litterman (1983) claims that Fernandez’ treatment on data serial correlation fails to remove all serial correlation in certain data series. Litterman’s improvement, labeled “Markov random walk", treats µ as an ARIMA(1,1,0). According to Litterman, the mean squared error could be reduced by up to 13%. I adopt the dynamic model of [Sil01], SSC henceforth, to interpolate the immigration data from

87 annual to quarterly frequency. SSC builds on the above Chow-Lin method in the following form

q q Yt = κYt 1 + Xt β + µt , (B.3) − where SSC adds an autoregressive term, which gives the model the versatility to account for coin- tegration.3 Parameter κ is numerically estimated to maximize the log-likelihood function of the model. After the optimal κ is set, the following estimation is similar to that of Chow-Lin and its extensions.

Table A1 SSC Interpolation Statistics

ˆ Variable Name Estimated Beta St. Dev. Long Run Effect β/(1 κˆ) − Income -0.788 3.106 -4.378 Civilian Labor Force -1.607 1.481 -8.927 Note: The long Household Survey Pop. 5.163 2.463 28.683 Establishment Survey Pop. -2.765 5.104 -15.361 κ 0.820 - -

βˆ run effect ( 1 κˆ ) is equivalent of Chow-Lin regression coefficient. −

y Using equations B.2 and B.3, Yt is a TAnnual 1 vector of the annual DHS immigration data 4 × ranging from 1969-2014, TAnnual = 44. Xt is a TQ ua r t e r l y 4 indicator matrix containing income, civilian labor force, employment household survey population,× and establishment survey population from 1964Q1 to 2014Q1, TQ ua r t e r l y = 176. Income provides a benchmark on the macroeconomic connection of immigration, while civilian labor force, household survey, and establishment survey population measure population change from three different perspectives. If immigration is correctly accounted for, all four variables should be affected. The estimated betas for the four indicator variables are shown in Table A1. These beta coefficients are interpreted as an estimated short- run (1 period lag) sensitivity measure of the indicator variables to immigration. The estimated autoregressive coefficient κ is 0.82, indicating a stationary AR(1) given the SSC model. The estimated long-run effect of the indicator variables (equivalent to the interpretation of β in the Chow-Lin ˆ method), given by β/(1 κˆ), are presented in column 3. − B.2.2 Results of Alternative Interpolation Methods

Figure B.1 displays a comparison of different interpolation methods considered in the construction of quarterly immigration data. For viewability, the Fernandez, Litterman, and SSC series are vertically

3 Note that the SSC model is equivalent to that of Fernandez when κ = 1. 4 q y Yt = Yt C where C is the same frequency converting matrix from Chow-Lin.

88 shifted upwards. Different regression-based interpolation methods produce highly similar results. Although this paper uses the series estimated by the method of [Sil01], using other interpolation methods does not alter the main results qualitatively.

Figure B.1 Comparison across interpolation methods for immigration data. The results from Fernandez, Litterman, and SSC are shifted vertically of equal distance for better viewability.

B.2.3 Unit Root Test Results

According to Table B.2.3, I fail to reject the existence of a unit root in immigration in log levels but rejects the same null at 1% confidence when immigration is in growth rates.

Table B.1 Augmented Dickey-Fuller Test for Unit Root

Test Statistic p-value Immigration in Log Levels -2.528 0.3141 Immigration in Growth Rates -4.056 0.0073 Note: The Dickey-Fuller tests for the null that the immigration series has a unit root. The p-value corresponds to the MacKinnon approximation of the p-value based on the test statistic.

89 B.2.4 Data Construction of Consumption

To construct consumption in non-durables and services, I obtain the following series reported by the Bureau of Economic Analysis: Personal Consumption Expenditure (PCE) in non-durables (c n ), PCE series in services (c s ) from NIPA Table 1.1.5, and their corresponding price indexes (p n , p s ) from NIPA Table 1.1.4. The growth of real PCE in non-durables and services is calculated as a Fisher ideal index ([Fis22]), v u c n p n c s p s c n p n c s p s t t t + t t t t 1 + t t 1 ct = n n s s n n− s −s 1, (B.4) ct 1pt + ct 1pt × ct 1pt 1 + ct 1pt 1 − − − − − − − to correctly account for the addition of chained aggregated NIPA data as suggested by [Whe02].

90 B.3 Alternative Assumption in Stationarity of Hours Worked

This appendix outlines the model selection exercise when hours worked is assumed to be non- stationary. As discussed in the main text, non-stationary hours worked implies

LR5 : d6,t = d8,t = 0, (B.5) while assuming stationary hours worked leaves d6,t and d8,t unrestricted. Table B.3 displays the additional models estimated when hours worked is assumed to be non-stationary.

Table B.3 List of Model Comparison, Hours Worked in Growth Rates

Model Identification Time-Variation Model 17 SR1, SR2, SR3, LR1, LR2, LR4, LR5 TVP-SV-SVAR Model 18 SR1, SR2, SR3, SR4, LR1, LR2, LR4, LR5 TVP-SV-SVAR Model 19 SR1, SR2, SR3, LR1, LR2, LR3, LR4, LR5 TVP-SV-SVAR Model 20 SR1, SR2, SR3, SR4, LR1, LR2, LR3, LR4, LR5 TVP-SV-SVAR Model 21 SR1, SR2, SR3, LR1, LR2, LR4, LR5 TVP-SVAR Model 22 SR1, SR2, SR3, SR4, LR1, LR2, LR4, LR5 TVP-SVAR Model 23 SR1, SR2, SR3, LR1, LR2, LR3, LR4, LR5 TVP-SVAR Model 24 SR1, SR2, SR3, SR4, LR1, LR2, LR3, LR4, LR5 TVP-SVAR Model 25 SR1, SR2, SR3, LR1, LR2, LR4, LR5 SV-SVAR Model 26 SR1, SR2, SR3, SR4, LR1, LR2, LR4, LR5 SV-SVAR Model 27 SR1, SR2, SR3, LR1, LR2, LR3, LR4, LR5 SV-SVAR Model 28 SR1, SR2, SR3, SR4, LR1, LR2, LR3, LR4, LR5 SV-SVAR Model 29 SR1, SR2, SR3, LR1, LR2, LR4, LR5 Fixed-Parameter SVAR Model 30 SR1, SR2, SR3, SR4, LR1, LR2, LR4, LR5 Fixed-Parameter SVAR Model 31 SR1, SR2, SR3, LR1, LR2, LR3, LR4, LR5 Fixed-Parameter SVAR Model 32 SR1, SR2, SR3, SR4, LR1, LR2, LR3, LR4, LR5 Fixed-Parameter SVAR

Table B.4 shows the log marginal likelihood and Bayes factor for the additional SVARs when LR5 is in effect. The result indicates the importance of TVP and SV, because the TVP-SV-SVAR estimates are favored by marginal likelihood regardless of the four identification schemes. Among the additional models, however, Model 1 from the main text is still strongly favored by data, with the next smallest Bayes factor of 6.48e 8 of Model 18. Therefore, assuming non-stationary hours worked does not alter my results, qualitatively.

91 Table B.4 Log Marginal Likelihood and Bayes Factor of All Estimated Models

TVP-SV-SVAR TVP-SVAR SV-SVAR Fixed-Parameter SVAR [Model 17][Model 21][Model 25][Model 29] -252.97 -257.49 -272.37 -336.86 (3.73e 13) (3.43e 15) (9.93e 21) (1.01e 50)

[Model 18][Model 22][Model 26][Model 30] -242.01 -270.47 -270.36 -337.04 (6.48e 8) (1.49e 21) (1.33e 21) (1.21e 50)

[Model 19][Model 23][Model 27][Model 31] -253.10 -278.13 -275.50 -337.79 (4.25e 13) (3.15e 24) (2.27e 23) (2.56e 50)

[Model 20][Model 24][Model 28][Model 32] -242.16 -277.72 -277.31 -339.89 (7.53e 8) (2.09e 24) (1.39e 24) (2.09e 51)

Note: Log likelihoods are reported and calculated following [Gew99]. Bayes factors are calculated with respect to Model 1 (TVP-SV-SVAR with identification 2) from the main text and are reported in parentheses.

B.4 Necessary and Sufficient Conditions of Identification Schemes

The SVARs are identified with short- and long-run restrictions. Table B.5 below revisits the preferred identification scheme of Model 1, suggested by the Bayesian model selection exercise. Following [RR10] (RRWZ), I perform a global identification check on the proposed restrictions. From Table B.5, the number identified shocks is n = 4, the number of column restrictions q1 = 1, q 6, q 1 and q 5. The total number of column restrictions (P4 q 11) is greater than 2 = 3 = 4 = j =1 j = n(n 1)/2 = 6. Following [Rot71] and Theorem 7 of [RR10], this set of restrictions is over-identified. All proposed− identification schemes satisfy the necessary rank condition. As an example, I verify the RRWZ global identification conditions for the selected identification scheme for Model 1. The intuition is to check the rank condition of the restriction matrix, denoted

K0, to confirm that observational equivalence does not exist. It must satisfy necessary and sufficient conditions outlined below.

92 Table B.5 Impact and Long-run Restrictions of Model 1

∆l og (x )  0 0 0 − l og (n)   ∆l og I −0 −0 − −0  IR  ( )   0 ∆l og c  0 −  IR = ( )   ∆l og x − 0 − −0 ∞ ( )   l og n − −  ( )   ∆l og (I ) − −0 − −0 ∆l og (c ) − 0 − 0 − −

Note: “-" denotes unrestricted entries; “0" denotes zero restricted entries. The columns, from left to right, denote TFP shock, labor demand shock, immigration supply shock, and transitory consumption shock.

Proof. First, I import K00 :   1 0 0 0   1 1 1 1   0 0 1 0   1 0 1 1 K 0   (B.6) 0 =   1 0 1 0   1 1 1 1     1 0 1 0 1 0 1 0

Then, I calculate Qi = Ri K00 i j matrix for i = 1...4. The detailed calculation step of Q1 is shown as example. Notice that:

     1 0 0 0 0 0 0 0 1 0 0 0 0 0      0 0 1 0 0 0 0 01 1 1 1 0        0 0 0 1 0 0 0 00 0 1 0 1 0      0 0 0 0 1 0 0 01 0 0 1 0 0             =   (B.7) 0 0 0 0 0 0 1 01 0 1 0 0 0      0 0 0 0 0 0 0 11 1 1 1 0 0           0 0 0 0 0 0 0 01 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0

93 Hence,      1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0      0 0 1 0 0 0 0 01 1 1 1 0 0 1 0      0 0 0 1 0 0 0 00 0 1 0 1 0 0 1      0 0 0 0 1 0 0 01 0 0 1 1 0 1 0      Q1 =    =   (B.8) 0 0 0 0 0 0 1 01 0 1 0 1 0 1 0      0 0 0 0 0 0 0 11 1 1 1 1 0 1 0           0 0 0 0 0 0 0 01 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 ” — Now, stack Q1 on top of 1 0 0 0 to yield M1:

  1 0 0 0   1 0 0 0     0 0 1 0   1 0 0 1   M1 = 1 0 1 0 (B.9)     1 0 1 0   1 0 1 0     0 0 0 0 0 0 0 0

Following similar procedure, I calculate M2...M6:

1 0 0 0 0 1 0 0     1 0 0 0     0 0 1 0   1 0 0 1 M2 =   (B.10)   1 0 1 0   1 0 1 0   1 0 1 0     0 0 0 0 0 0 0 0

94   1 0 0 0   0 1 0 0   1 0 1 0     1 0 0 0     0 0 1 0   M3 = 1 0 0 1 (B.11)   1 0 1 0     1 0 1 0   1 0 1 0     0 0 0 0 0 0 0 0

1 0 0 0 0 1 0 0     0 0 1 0   0 0 0 1     1 0 0 0     0 0 1 0 M4 =   (B.12) 1 0 0 1     1 0 1 0   1 0 1 0   1 0 1 0     0 0 0 0 0 0 0 0

It is straightforward to verify that the ranks of matrices M1 and M2 are 3 and the ranks of M3 and

M4 are 4. Therefore, K0 is locally identified.

95 B.5 Priors and Estimation Setup

This section describes the priors and pre-estimation setup of the TVP-SV-SVARs. Table B.6 shows the conjugate priors for three sets of initializations. Let x¯ denote the OLS/ML estimate of parameter x . I use the first τ = 40 observations (1969Q1 - 1978Q4) as a training sample 5 to estimate prior values B,VB,a, and l og (h) via OLS or MLE. Block B’s posterior distribution is truncated to ensure stationary draws, as seen in (2.4). I also choose priors for the covariance matrices of the innovations that govern the law of motions for the aforementioned TVP-SV-VAR parameters. They control the size of the search steps in the MCMC samplers. Reported in Table B.6, the calibration of the tightness and re-scaling constants is set to allow for proper acceptance rate for convergence, while accommodating necessary time-variation.

Table B.6 Priors

2 B0 N (B,κB VB) Qi IW (κQ VB,τ) ∼ · ∼ 2 a0 N (a,κa I3) Si IW (κs I3, 1 + d i m a) ∼ · ∼ · 2 l og (h0) N (l og (h),κh I3) Wi IG (κW ,2) ∼ · ∼ Note: N denotes normal; IW denotes Inverted-Wishart; IG denotes Inverse Gamma. κB = 4, κa = 4, and κh = 10 are tightness constants. 2 0.24, 2 5 10 3, and 2 1 10 3 are re-scaling factors. κQ = κs = − κW = − × ×

5 4 I adopt Christopher Sims’ csminwel package with a search tolerance of 1 10− in obtaining the ML estimator. ×

96 B.6 Details in the Sampling Algorithm

I provide details of the Metropolis-within-Gibbs sampler, which closely follows [Can15].

B.6.1 Draw Time-Varying Coefficients

B.6.1.1 Draw Reduced-form Coefficients, B

The drawing of the reduced-from coefficients B relies on the Carter-Kohn algorithm. [Car94] show that the combination of the Kalman filter and a backward recursion algorithm gives an efficient way of solving for the states of a state-space model. Particularly, the Carter-Kohn algorithm contains the following steps:

1 The Kalman filter provides an estimate of the mean BT T and variance PT T . | |

2 Given the Kalman estimates, take a random draw of a multivariate normal with mean BT T | and covariance PT T . | 3 At T 1, use Kalman update equations to recursively obtain and P . Draw BT 1 T 1,BT T 1 T 1,BT − | − − | − −from N ,P . BT 1 (BT 1 T 1,BT T 1 T 1,BT ) − − | − − | − 4 Repeat step 2 for t = T 2,T 3,...,1. − − Therefore, the Carter-Kohn algorithm yields a series of draws of B for t = 1, ,T . ···

B.6.1.2 Draw Structural Coefficients, B˜

Given the current draw of the reduced-form coefficient Bi , the last draw of the contemporaneous matrix A0,i 1, and the last draw of the standard deviations of the structural shocks Σi 1, I impose − − long-run restrictions. As discussed, long-run restrictions create non-linearity in the VAR parameter space. Therefore, long-run restrictions need to be imposed in the sampler during each draw. Compute the long-run cumulative impact matrix

T T 1 T T Di = J (I2 Bi )− J 0A0,i 1Σi 1, (B.13) − − − T where J = [I2 ...02] is a selection matrix. Denote d j k the elements in Di . Setting the respective d j k to zero achieves the long-run restriction.

Let the restricted long-run cumulative matrix be Di , solve for the draws restricted structural ˜ coefficient matrix Bi by reversing equation B.13. Lastly, evaluate the eigenvalues of B and discard the draws that have eigenvalues outside the unit circle.

97 B.6.2 Draw A0 ˜ Given Bt , the state-space model can be re-written as

˜ A0,t (zt Xt0Bt ) A0,t zˆt = Σt εt . − ≡

Let v e c (At ) = SA f (at )+ sA, where SA and sA are selection matrices of ones and zeros. Reparametrize the model as

(zˆt 0 I )(SA f (at ) + sA) = Σt εt . ⊗ Therefore the system of regression only contains linear restrictions, with observation equation

(zˆt 0 I )sA = (zˆt 0 I )SA f (at ) + Σt εt . ⊗ − ⊗ and state equation

f (at ) = f (at 1) + ηt . − Given initial f (at )0 0 and P0 0, the Extended Kalman Filter (EKF) gives updates of f (at ) and its | | covariance matrix. The smoothed estimates are denoted f (at )∗T T and PT∗ T . | | The algorithm of drawing f (a) is:

1 Given z T ,V i 1 , compute f a i 1 and P i 1. ( − ) ( t )∗T −T T∗ T− | | 2 For t 1,...,T , draw a candidate f a + from p f a f a i 1 . = ( t ) [ ( t ) ( t − )] ∗ | f a + T p f a f a i 1 T p( ( ) ) [ ( t ) ( t − )] 3 Compute θ = T × ∗ T | i 1 . p(f (a) ) p [f (a ) f (at − )] × ∗ | i T T 4 Draw µ U (0,1). Accept new draw as f (a ) = f (a +) if µ < θ ; otherwise keep old draw. ∼ 5 Given z T , f a i T , draw V i IW v,V¯ 1 , where V¯ 1 is the sum of squared errors in the state ( ( ) ) ( − ) − ∼ equation of f (a).

B.6.3 Draw Σ

˜ T T Conditional on the current draws (Bi ,ai ), the left hand side of the model equation A(at )yt = Σt εt is known. Taking square and log each element of this vector yields a linear state-space system

zt∗ = 2l og (σt ) + 2l og (εt );

l og (σt ) = l og (σt 1) + ηt , −

98 where z l og z 2. However, this system is not Gaussian, because each element of 2log is dis- t∗ ( t ) (εt ) ≡ tributed log χ2. These shocks are approximated with a 7-component mixture of normals. Following T ˜ T T T [Kim98], auxiliary variables si are introduced and drawn conditional on (Bi ,ai ,σi 1) to keep track − ˜ T T T of the normal mixtures. The state-space system is Gaussian given the approximation (Bi ,ai ,σi ). T The draws of σi are generated and evaluated with the Carter-Kohn algorithm.

B.6.4 Draw V ˜ T T T T Draw the hyperparameters i given (Bi , A0,i ,Σi , zi ) according to the specified distributions shown in Table B.6. V

99 B.7 Convergence Diagnosis

I adopt two methods to investigate the convergence of the sampling routine to the ergodic dis- tribution. First, I plot the sequence of retained draws in Figures B.2 for the selected SVAR. If the sampler has converged, there should be minimal variation for each coefficient estimates between the retained draws. All three panels exhibit minimal and trendless fluctuations up to 20 draws. Next, I follow [Gew05] to compute the numerical accuracy of the sampler by computing the Inefficiency Factor (IF) for each block of draws. Large values in the IF suggest a high degree of serial correlation between draws, meaning the MCMC algorithm has not converged. [Pri05] points out that IF values under 20 are usually regarded as successfully converged. Table B.7 displays the descriptive statistics of IF values for each sampling block. The low IF values in Table B.7 suggest statistical support of satisfactory convergence of the samplers.

Table B.7 Inefficiency Factors for Different Sets of Parameters.

Median Mean Min Max 10t h Pct 90t h Pct

Bi 1.161 1.168 0.570 2.044 0.892 1.563 A0,i 15.554 10.379 0.696 19.221 1.006 18.479 Σi 1.133 1.178 0.662 2.701 0.815 1.533

100 Figure B.2 Sequence of Retained Draws.

101 B.8 Computation of Log Marginal Likelihood

This section outlines the details in computing the log marginal likelihood of a SVAR, post estimation, using the harmonic mean estimator following [Gel93] and [Gew99]. Z p(z Mk ) = p(z θk ,Mk )p(θk Mk d θk ) (B.14) | | | is the marginal likelihood of model k. For the class of TVP-SV-SVARs outlined in this paper, an analytical expression of the marginal likelihood is unavailable due to the degree of parameterization and non-linearity. The Gibbs sampler provides a convenient approximation. Following [Gel93] and [Gew99], I compute the marginal likelihoods of the competing models using the harmonic mean method. They show that given any distribution g , the inverse of the marginal likelihood p z 1 (θ ) ( )− can be written as the expected value of the product of the likelihood p(z θ ) and the prior p(θ ) for the model | 1 g (θ ) p(z )− = E [ z ]. (B.15) p(z θ )p(θ )| | In practice, I estimate log marginal likelihood of equation (B.15) as follows. First, I use the Kalman filter to compute the log prior density. This is done by summing all log prior densities of the latent variables (B, A0, Σ) and their respective hyperparameters (Q, S, W ) according to the probability distribution functions (PDFs) of the priors. p(z θ ) is evaluated by updating the | measurement equation of the Kalman filter. [Gew99] suggests using a truncated normal distribution for g (θ ). The following sections detail the computation of each elements in equation (B.15).

B.8.1 Estimating p(z θ ) | The likelihood for a state space model p(z θ ) can be evaluated with the Kalman filter. For a state space model of the following representation,|

1 zt = X 0βt + (At )− Σt εt , εt N (0, I ), (B.16) ∼ and

βt = F βt 1 + ut , ut N (0, ), (B.17) − ∼ V the linear-Gaussian Kalman filter recursion is given by

βt t 1 = F βt 1 t 1 (B.18) | − − | −

Pt t 1 = FPt 1 t 1F 0 + , (B.19) | − − | − V ηt t 1 = zt Xt βt t 1, (B.20) | − − | −

102 ft t 1 = XPt t 1X 0 + ηt η0t , (B.21) | − | − 1 K = Pt t 1X 0 ft−t 1, (B.22) | − | − βt t = βt t 1 + K ηt t 1, (B.23) | | − | − and

Pt t = Pt t 1 KXPt t 1. (B.24) | | − − | − The first two equations (B.18) and (B.19) are the prediction equations that provide the value and the estimated variance of the state variable one period ahead. Equation (B.20) computes the prediction error, whereas equation (B.21) computes the variance of the prediction error. The Kalman gain, usually considered as the weight attached to the prediction error, is calculated in equation (B.22). The last two equations (B.23) and (B.24) are the updating equations that give one period ahead estimates of the states. Given estimates from the Kalman filter, the log likelihood is computed as

T T k 1 X 1 l n p z θ l n 2π P η P 1η . (B.25) [ ( )] = 2 ( ) 2 t 2 t t− 0t | − − t =1 | | −

B.8.2 Estimating p(θ )

The prior densities p(θ ) of states and hyperparameters are computed directly from their probability density functions. Given the Gaussian states and the inverse-Wishart priors of hyperparameters, p(θ ) is equal to the sum of all individual densities. In the case of TVP-SVARs with no SV and SV-SVARs with no TVP,the log density of the respective missing states is 1.

B.8.3 Estimating p(z)

As in equation B.15 in the main text, the marginal likelihood of a model is defined as

1 g (θ ) p(z )− = E [ z ]. (B.26) p(z θ )p(θ )| | In practice, this is estimated as

M ˆ 1 1 X g (θ ) p(z )− = , (B.27) M p z θ p θ j 1 ( ) ( ) = | where M is the number of retained draws. The distribution of g (θ ) is selected as a truncated normal. [Gew99] elaborates the details of the selection of g (θ ).

103