Three Essays: Cointegration Tests and Empirical Studies Using the Dynamic Factor Models

Home , 1970s energy crisis

THREE ESSAYS: COINTEGRATION TESTS AND EMPIRICAL STUDIES USING THE DYNAMIC FACTOR MODELS

by YAN (OLIVIA) LU JUNSOO LEE, COMMITTEE CHAIR VLADIMIR ARCABIC ALLAN TIDWELL XIAOCHUN LIU ROBERT REED

A DISSERTATION

Submitted in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy in the Department of Economics, Finance, and Legal Studies in the Graduate School of The University of Alabama

TUSCALOOSA, ALABAMA

The first essay extends the pioneering cointegration test of Johansen (1991) to allow for structural breaks in a cointegration system. Instead of using usual dummy variables, we utilize a Fourier function to control for an unknown number of multiple breaks in the cointegration system. We provide the limiting distribution of the Johansen-Fourier tests and the corresponding critical values. Monte Carlo simulations show that the new tests display good size and power properties. An empirical application to the Kilian (2009) dataset shows the result of cointegration, while the conventional Johansen cointegration tests indicate no cointegration. The second essay follows the extensive studies on the similarity and synchronization of member states economic fundamentals and conditions triggered by the formation of the Economic and monetary union in Europe. This paper analyzes synchronization in five major macroeconomic variables in the European Union using the dynamic factor model. We do not find significant evidence of synchronization in the Eurozone or EU countries. The degree of synchronization in the Eurozone countries is not greater than that in other countries. Also, we find no significant evidence to show that the EU or Eurozone membership has increased synchronization or similarity within the group over time. Instead, we find that synchronization effects are time-dependent; they are more significant during the financial crisis period. The third and final essay analyzes the co-movements of US housing prices using the state level and metropolitan statistical areas (MSA) data. The objective of the study is to examine the significance and time-varying nature of the co-movements from macroeconomic aspects and determine major factors that drive the movements of the housing prices. Dynamic factor models with time-varying loadings and stochastic volatility (DFM-

ii TV-SV) are employed to estimate the national, regional, and state factors. The results show that the national factor is dominant in explaining the movement of housing prices. On average, the national factor accounts for 79 percent of the variation of housing prices, while its signiﬁcance is the highest during the housing boom and bust periods in many regions and states.

iii DEDICATION

This thesis is dedicated to my advisor Junsoo Lee and my husband Christopher Simp- son.

iv ACKNOWLEDGMENTS

I would like to thank Junsoo Lee for his guidance and advice throughout my graduate school career, and for contributing to all three of the essays in this dissertation. I would like to thank Razvan Pascalau for his contributions to the ﬁrst essay, as well as Walter En- ders, Bruce Hansen, and Joon Park for their comments and feedback on the earlier draft. Thanks are also due to Vladimir Arcabic for his contributions to the second essay, and to Alan Tidwell for his contributions to the third essay. I was fortunate to work with and learn from faculty in the Department of Finance, Economics and Legal Studies in a supportive environment. I am thankful for the time and eﬀort provided by my co-authors on my disseration essays and the opportunities to present my works at the Midwest Econometrics Group Conference in 2018 and the Western Eco- nomic Association 94th Annual Conference (Virtual) in 2020.

v CONTENTS

ABSTRACT ...... ii

DEDICATION ...... iv

ACKNOWLEDGMENTS ...... v

LIST OF TABLES ...... viii

LIST OF FIGURES ...... x

CHAPTER 1: INTRODUCTION ...... 1

CHAPTER 2: JOHANSEN-TYPE COINTEGRATION TESTS WITH A FOURIER FUNCTION ...... 3

2.1 Introduction ...... 3

2.2 Cointegration Model with a Fourier Function ...... 8

2.3 Performance of the Johansen-Fourier Tests ...... 13

2.3.1 Size Properties ...... 14

2.3.2 Power Simulations ...... 15

2.4 Empirical Example ...... 16

2.5 Summary and Concluding Remarks ...... 19

2.6 Appendix ...... 38

CHAPTER 3: HOW INTEGRATED IS EUROPE? ANALYSIS OF CONDITIONS FOR AN OPTIMAL CURRENCY AREA ...... 40

3.1 Introduction ...... 40

3.2 Literature Review and Motivation ...... 42

vi 3.3 Data and Estimation Procedures ...... 44

3.3.1 Dynamic Factor Model ...... 46

3.4 Estimation Results ...... 48

3.4.1 Estimated Factors ...... 48

3.4.2 Relative Contributions of the Estimated Factors ...... 49

3.4.3 Correlation Between Country-speciﬁc Variables with Global or Group Factors ...... 52

3.4.4 Synchronization Eﬀects based on Cross-correlations in Each Group . 53

3.5 Concluding Remarks ...... 55

3.6 Appendix ...... 71

CHAPTER 4: COMOVEMENTS IN U.S. HOUSING PRICES ...... 75

4.1 Introduction ...... 75

4.2 Literature Review ...... 77

4.3 Data and the DFM ...... 81

4.4 Results and Discussion ...... 83

4.4.1 National, Regional, and State Factor Contributions ...... 83

4.4.2 The Synchronization Eﬀect ...... 84

4.4.3 MSA-Level Factor Analysis ...... 85

4.5 Conclusion ...... 86

REFERENCES ...... 99

vii LIST OF TABLES

2.1 Critical Values - Constant Model ...... 20

2.2 Critical Values - Trend Model ...... 21

2.3 Critical Values - Restricted Constant Model ...... 22

2.4 Critical Values - Restricted Trend ...... 23

2.5 Critical Values - Cumulative Frequencies in the Model with a Constant . . . 24

2.6 Critical Values - Cumulative Frequencies in the Model with a Trend . . . . . 25

2.7 Critical Values - Cumulative Frequencies in the Model with Restricted Constant 26

2.8 Critical Values - Cumulative Frequencies in the Model with Restricted Trend 27

2.9 Johansen-cointegration - restricted constant approach ...... 27

2.10 Johansen-Fourier Cointegration test - restricted constant approach ...... 28

3.1 List of Countries in Each Group by Variables and Descriptive Statistics . . . 64

3.2 Average of Factor Contributions in Each group for Each Variable ...... 65

3.2 Average of Factor Contributions in Each group for Each Variable (cont.) . . 66

3.3 Average Correlation Coeﬃcients Between Country-speciﬁc Variables with Global or Group Factors ...... 67

3.4 Pesarans CD Test Results ...... 67

3.5 Average Cross-Country Correlations and Diﬀerences in Two Regions . . . . . 68

3.5 Average Cross-Country Correlations and Diﬀerences in Two Regions (cont.) 69

3.5 Average Cross-Country Correlations and Diﬀerences in Two Regions (cont.) 70

4.1 List of States in Each Region and Descriptive Statistics ...... 90

viii 4.2 Average Factor Contributions to Variation in State-level Housing Prices . . . 94

4.3 R-squared values from the Regression of States on the National Factor . . . 95

4.4 Average Factor Contributions to Variation in MSA-level Housing Prices . . . 96

4.5 Average Factor contributions to Variation of Housing Prices in Selected MSAs 97

4.6 R-squared values from the Regression of Regions on the National Factor . . . 98

ix LIST OF FIGURES

2.1 Break Types used in Simulations ...... 29

2.2 Examples of Non-Cointegrated versus Cointegrated Series...... 30

2.3 Size Performance ...... 31

2.4 Power performance: system of 2 variables...... 32

2.5 Power performance: system of 3 variables...... 33

2.6 Power performance: system of 4 variables...... 34

2.7 Power performance: system of 5 variables ...... 35

2.8 Real Oil Price, Oil Production, and Real Economic Activity ...... 36

2.9 Two Cointegrating Vectors ...... 36

2.10 Impulse Response Analysis ...... 37

3.1 Estimated National and Group Factors ...... 57

3.2 Cross-country correlation between groups (EU and OECD) ...... 59

3.3 Statistical Diﬀerences in Average Cross-country Correlation Between Eurozone and EU non-Eurozone Regions ...... 62

3.3 Statistical Diﬀerences in Average Cross-country Correlation between Eurozone and EU non-Eurozone Regions (cont.) ...... 63

3.4 Variance Decomposition of Each Country in Groups (GDP Growth Rates) . 71

3.5 Cross-country correlation between groups (EU only) ...... 74

4.1 Plot of Housing Price Indices for all states and Washington D.C...... 88

4.2 Plots of the estimates of National and Regional factors ...... 89

x 4.3 Relative Contributions of Each Factor to the Variation of Housing Prices (Blue: national factor, Red: regional factor, and yellow: state-speciﬁc factor.) . . . 91

4.4 Cross-correlations in all states and regions ...... 92

4.5 Plots of the Estimated National and Regional Factors Using MSA Data . . . 93

xi CHAPTER 1 INTRODUCTION

The subject of the dissertation revolves around the topic of “co-movement”. In relating the co-movement of macroeconomic variables of interest over time, three essays are presented. The first essay, “Johansen-Type Cointegration Tests with a Fourier Function,” provides an extended version of the Johansen test, where we control for structural breaks in a cointegration system using a Fourier function. The Johansen test is used for detecting cointegration or co-movements between time series variables of interest. By using a Fourier function, the modified Johansen test has better performance than the standard Johansen test. Through further research on co-movement of economic phenomena over time, the second essay, “How Integrated is Europe? Analysis of Conditions for an Optimal Currency Area,” was developed. This essay analyzes synchronization in five major macroeconomic variables in the European Union using the dynamic factor model. We do not find significant evidence of synchronization in the Eurozone or EU countries. The degree of synchronization in the Eurozone countries is not greater than that in other countries. Also, we find no significant evidence to show that the EU or Eurozone membership has increased synchronization or similarity within the group over time. Instead, we find that synchronization effects are time-dependent; they are more significant during the financial crisis period. The third essay, “Co-movements in U.S. Housing Prices,” analyzes the co-movements of US housing prices using the state level and metropolitan statistical areas (MSA) data. The objective of the study is to examine the significance and time-varying nature of the

1 co-movements from macroeconomic aspects and determine major factors that drive the movements of the housing prices. The results show that the national factor is dominant in explaining the movement of housing prices. On average, the national factor accounts for 79 percent of the variation of housing prices, while its signiﬁcance is the highest during the housing boom and bust periods in many regions and states.

2 CHAPTER 2 JOHANSEN-TYPE COINTEGRATION TESTS WITH A FOURIER FUNCTION

2.1 Introduction

This paper extends the pioneering cointegration test of Johansen (1991) to allow for structural breaks in a cointegration system. Instead of the usual dummy variables approach, we utilize a Fourier function to control for an unknown number of multiple breaks in the cointegration system. Our suggested tests can be useful when the information on the number of breaks and their locations is not known a priori. The tests work better especially when the unknown breaks are smooth and they work reasonably well under various types of breaks. The Johansen (1991) cointegration tests have been widely applied to economic and financial data to analyze the long-run relationship and co-movement of various variables of interest. The reason for the prolific popularity of the Johansen cointegration tests lies with their desirable features. Johansen tests are based on the full information maximum likelihood estimation, and they give the most efficient estimates of the cointegrating vectors and the adjustment parameters. However, one important requirement for these desirable features is that the system of equations needs to be correctly specified in all equations. Omitting structural changes is one common example that can lead to mis-specified models. The seminal work of Perron (1989) suggests that usual unit root tests that fail to account for structural changes will lose power. The same issue could arise in the cointegration tests. Perhaps, the issue would be more severe since usual cointegration tests ignoring existing breaks can exhibit size distortions as well as loss of power.

3 Since the original framework in Johansen (1991) does not allow for structural breaks, a stream of the relevant literature has proposed to consider structural breaks in the cointegrating models. Such efforts mostly rely on dummy variables to allow for structural breaks. Johansen et al. (2000) have modified the original framework to allow for break- points in the deterministic components.1 One practical issue arises in the system based test, however, since these tests require a priori information on the types of deterministic trends (level, trend, nonlinear breaks), the number of breaks, and break locations in each of the equations in the system. In many cases, such information is not available. Even if such information is available, the asymptotic distribution will be complicated by the presence of heterogeneous and multiple breaks which can differ over different variables in the system. As such, some simplified models have been considered. One may assume that the number of breaks and their locations are the same in all equations. Then, if this information is known a priori, applied researchers may use customized computer codes,2 or the corresponding response surface analysis.3 As the break location is not known in many cases, Lutkepohl et al. (2004) attempts to test for the cointegration rank in the system that allows for one common unknown shift in the mean. Then, Kejriwal and Perron (2010) propose the sup-Wald test to test sequentially for multiple structural changes in cointegrated systems, while Qu (2007) proposes a procedure to detect the presence of cointegration when deterministic changes may be present at some unknown dates in the co-integration vector.4 In a similar vein, Hansen (2003) proposes a more generalized reduced rank re-

1Further, Johansen et al. (2000) deploy the toolbox of Johansen and Juselius (1990) to test for the cointegrating rank whereby the break points are assumed to be known in advance. In particular, this study proposes a reduced rank regression, which is a combination of least squares regression and canonical correlation analysis to allow for known breaks in the linear trend of a Gaussian autoregressive model. The study discusses several cointegration rank tests for models with breaks in the level, trend and/or with outlier dummies. 2Examples include the simulation program DisCo by Johansen and Nielson (1993) that the practition- ers can use to obtain the relevant critical values. 3See Johansen et al. (2000) 4This study follows Breitung (2002) to employ a multivariate version of the variance-ratio statistic whose ordered eigenvalues carry information about the cointegrating rank. Qu (2007) orders the eigenvalues resulting from the use of several sub-samples and then employs a linear combination of them. Since

4 gression technique to test for breaks, which can be either full or partial. Further, Lutke- pohl et al. (2004) and Trenkler et al. (2008) propose cointegration tests in the presence of break points, and unlike the previous studies, they employ a generalized least squares procedure.1 The current study provides an alternative procedure for the issue of controlling for breaks in the cointegration system. We follow the framework of Enders and Lee (2012) on unit root tests with a Fourier function and extend the methodology to the Johansen tests for cointegration with breaks.2 Specifically, we use a Fourier function to control for an unknown number of breaks in each equation in the system. We denote the new tests as the Johansen-Fourier tests. Initially, the Fourier function was suggested by Gallant (1981), and it was employed in Enders and Lee (2012) for unit root tests. The Fourier function can work better when structural breaks are smooth and gradual over time. Such breaks may not be well captured with abrupt dummy variables. While abrupt structural breaks can be captured better with binary dummy variables, slowly evolving breaks fit better with a smooth nonlinear function. The approach of capturing non-abrupt regime shifts in a cointegrating vector with the help of nonlinear smooth functions could be beneficial in such cases. Becker et al. (2006) and Enders and Lee (2012) are examples wherein the Fourier series has been successfully used to approximate smooth breaks in stationarity and unit root tests, respectively. Additionally, Perron et al. (2017) employs the Flexible the break dates are assumed to be unknown, Qu (2007) searches over all possible partitions of the sample, which then gives different combinations of the ordered eigenvalues. While the test allows for multiple unknown structural changes and it does not involve an arbitrary normalization of the cointegrating vector, the limiting distribution depends on the process of ordered eigenvalues. 1Lutkepohl et al. (2004) and a series of related papers, including Lutkepohl et al. (2003), Trenkler et al. (2008), and Trenkler (2009), test for the cointegration rank assuming that the break date is unknown. The main feature is to detrend the original series to adjust for deterministic terms in such a way that the asymptotic distribution of the cointegration test under the null is free of the nuisance parameter. This adjustment procedure requires first the estimation of the break date using a full vector autoregressive (VAR) process in levels or trends. Then, the parameters of the deterministic part of the data generation process (DGP) are estimated, and the original series is adjusted for the structural shift such that a cointegrating rank test of the Johansen likelihood ratio (LR) type without breaks can be applied. It remains to be seen if such an approach can be adopted in the presence of a nonlinear trend associated with the Fourier function, since the first difference of the cosine term induces a sine term and vice versa, as we discuss below in Remark (iv) of Theorem 1. We do not consider this approach yet in this paper. 2See also Becker et al. (2006).

5 Fourier transform to test for the presence of a nonlinear deterministic trend in series that could have a stationary or an integrated noise component. It seems that the benefit of using the Fourier function is more significant in the cointegration framework. The underlying presumption of the newly suggested Johansen-Fourier tests is that one or more structural breaks can often be captured by using a small number of low-frequency components from a Fourier approximation. Indeed, structural breaks can be a phenomenon of lower frequencies, and the Fourier function performs well in the aspect of approximating unknown forms of breaks. Unlike the approach based on dummy variables, there is no need to assume that the number of the breaks and their locations are known a priori. Hence, the specification problem of determining the number of breaks, specific break dates, and the form of the breaks is transformed by incorporating the ap- propriate frequency components into the estimating equation. Along with yielding a more simpler and more flexible test, the uncertainty in estimating the parameters for structural changes is reduced drastically. The suggested tests also depend on the nuisance parameter reflecting nonlinear breaks in the deterministic terms, but the frequency parameter is restricted to a few values (one, two, or three in most cases). Also, while we assume for simplicity that the frequency in a Fourier function is the same in each equation, the form of nonlinearity captured by a Fourier function can differ across the equations. Moreover, there is an important advantage of using the Fourier function to approximate unknown forms of nonlinearity. The Flexible Fourier transform provides a parsimonious way to capture the nonlinear trend in a cointegrated system. Even in the presence of multiple breaks, the number of parameters does not increase monotonically. It is a mat- ter of determining the proper value of the number of cumulative frequencies, which can be equal to or less than three. This feature seems helpful, since adding more parameters to capture multiple breaks based on dummy variables will lead to a significant loss of power. As such, the Johansen-Fourier tests will provide a convenient procedure to resolve the issue of dealing with multiple breaks of unknown forms. Thus, the uncertainty in estimat-

6 ing the parameters for structural breaks can be minimized, which can yield a good performance of the tests. The Fourier expansion has also been adopted in a few papers to approximate a slowly evolving nonlinear trend in a cointegrating vector. Banerjee et al. (2017) incorporates the Fourier sums of sines and cosines to control for breaks when testing for cointegration using an autoregressive distributive lag (ADL) test. However, their framework applies to variables that may display a single cointegrating equation rather than a system of cointegrating equations.1 In addition, Tsong et al. (2016) employ the Fourier transform to approximate breaks of an unknown deterministic form when testing for the null of cointegration.2 Their framework allows the Fourier function to approximate possible structural breaks in the deterministic component both under the null and alternative hypotheses. However, their testing approach does not use a cointegration rank testing framework and, therefore, does not consider the possibility of more than one cointegrating equation. The current study adopts the Johansen cointegration rank approach to test for the null of no cointegration against the alternative of one or more cointegrating equations where structural breaks of an unknown form may be present. We use a Fourier function to control for the unknown number of breaks in each of the equations in the cointegration system. We first show that the standard Johansen tests can suffer from size distortions when the breaks are present but ignored. The result is in contrast to the pioneering finding of Perron (1989), who showed the power loss problem of usual unit root tests. Then, we consider the new Johansen cointegration tests with a Fourier function. The tests are free of nuisance parameters related to breaks but depend on the frequency. As such, we provide a set of critical values corresponding to different values of the frequency. Moreover, we consider the procedure where the unknown frequency is determined from the data. Then, we examine the performance of the newly suggested Johansen-Fourier tests under various

1Such examples may include testing for the validity of the purchasing power hypothesis, the expecta- tions hypothesis of the term structure of interest rates, and the Taylor rule, among others. 2Tsong et al. (2016)’s test can be viewed as a multivariate extension of the stationarity test in Becker et al. (2006).

7 scenarios of break types, such as multiple smooth breaks, sharp breaks, trend breaks, and popular forms of nonlinearity. Simulation results show that the new tests perform well in terms of size and power. The rest of the paper is organized as follows: In the next section, we discuss the cointegration model with a Fourier function and suggest the Johansen-Fourier cointegration tests. In Section 3, we examine the performance of the Johansen-Fourier tests, in terms of size and power properties. Section 4 presents an empirical example. Concluding remarks are given in Section 5.

2.2 Cointegration Model with a Fourier Function

This section discusses the cointegration model with a Fourier function. We start with the cointegrated vector autoregressive model as illustrated in Johansen (1991).

∆Xt = ΠXt−k + Γ1∆Xt−1 + .. + Γk−1∆Xt−k+1 + µ + ft + et, t = 1, .., T (2.2.1)

where Xt is a p dimensional process, and et ∼ Np(0, Λ). We have cointegration if Π has reduced rank (r < p) such that we can write Π = αβ0, where α and β are p × r vectors.

0 When µ 6= 0, the non-stationary process Xt has linear trends with coeﬃcients α⊥µ =

0, where α⊥ is a p × (p − r) matrix. We denote the model above as the constant model. However, if the constant µ is restricted to Π only, we denote this model as the restricted constant (RC) model, as suggested in Johansen and Juselius (1990).

Here, the main point in the present paper is that the Fourier function ft is included in the above model with

n X ft = [Ajcos(2πjt/T ) + Bjsin(2πjt/T )] (2.2.2) j=1 where n is the number of cumulative frequencies. Alternatively, we can consider one par-

8 ticular frequency j with

ft = A cos(2πjt/T ) + B sin(2πjt/T ). (2.2.3)

The key feature of the Fourier function is that it can mimic various types of nonlinearity embodied in each of the equations. We maintain the argument that it can control for multiple smooth breaks of unknown forms in the deterministic term. As such, we have substituted dummy variables with the Fourier function, which can be more ﬂexible and parsimonious in terms of the parameters to estimate. We denote the resulting tests as the Johansen-Fourier tests. Adding a trend function (t) in the above equation can capture a quadratic trend in the level.

∆Xt = ΠXt−k + Γ1∆Xt−1 + .. + Γk−1∆Xt−k+1 + µ + Π1t + ft + et, t = 1, .., T. (2.2.4)

0 In such case, we have Π1 = αγ to eliminate the quadratic trend, and the reduced rank

0 0 0 includes the restriction, (Π, Π1) = α(β , γ ) . We refer to them as the restricted trend (RT) model. The estimation procedure of the newly suggested Johansen-Fourier tests is similar to that of Johansen and Juselius (1990), except that the terms with dummy variables are re-

placed with Ft in this paper. We may rewrite the main model as

∆Xt = ΠXt−k + ΓZt + et (2.2.5)

where Zt = [∆Xt−1, .., ∆Xt−k+1, 1, ft] for the model in (2.1), for example, and Γ denotes

the corresponding coeﬃcient matrices. Here, ft includes the terms for the Fourier function.

Then, we denote R0t as the residuals from the regression of ∆Xt on Zt. Similarly, we de-

note R1t as the residuals from the regression of Xt−k on Zt to give a simpliﬁed regression

9 model, R0t = ΠR1t + et. Then, we compute T −1 X 0 Sij = T RitRjt, (i, j = 0, 1). (2.2.6) t=k+1 It is trivial to obtain the least squares estimators,

ˆ 0 Π = S01S11 (2.2.7)

ˆ −1 Λ = S00 − S01S11 S10. (2.2.8)

It is known that the maximum likelihood estimator of the cointegrating vector β is

obtained in the procedure of ﬁnding the canonical correlation coeﬃcients of R0t and R1t. That is, the maximum likelihood estimator of β is given from the normalized eigen vectors ˆ ˆ ˆ V = (ˆv1, .., vˆp) corresponding to the eigen values λ1 > .. > λp of the following equations

−1 | λS11 − S10S 00S01 | = 0. (2.2.9)

The likelihood ratio test statistic for cointegration with Π = αβ0, p × r , which we denote as the trace test, is given by p X ˆ LR = −T ln(1 − λi). (2.2.10)

i=r1 The likelihood ratio test statistic for testing r versus r +1 cointegation relationships, which we denote as the lambda test, is obtained by

ˆ λmax = −T ln(1 − λr+1). (2.2.11)

The test statistics for the restricted constant (RC) model are obtained in a similar

∗ manner, except that Zt is replaced with Zt = [∆Xt−1, .., ∆Xt−k+1, ft] without the constant, and Xt−k is replaced with (Xt−k, 1) to compute R0t and R1t, and thus Sij. The corresponding test statistics for the restricted trend (RT) model are similarly obtained.

10 Theorem 1. The asymptotic distribution of the likelihood ratio test statistic for the hypothesis of at most r cointegrating relationships is given by

 1  1 −1 1  Z Z Z   0 0 0 LR → tr dB F  FF du F dB (2.2.12)    0 0 0  as T → ∞. Here, B(s) = (B1(s), .., Bp−r(s)) is a (p − r) dimensional standard Brownian motion, and the (p − r) dimensional process F (s) = (F1(s), .., Fp−r(s)) is the projection of the process B(s) on the orthogonal complement of the space spanned by the Fourier funtion, z(s) = [1, cos(2π1s), .., cos(2πks), sin(2π1s), .., sin(2πks)], deﬁned over the interval

ˆ ˆ R 1 ˆ 2 s ∈ [0, 1], such that F (s) = B(s) − z(s)δ with δ = argmin 0 [B(s) − z(s)δ] ds. Also, the lambda statistic and the corresponding LR statistic converge in probability to

 1  1 −1 1  Z Z Z   0 0 0 λmax → Max dB F  FF du F dB (2.2.13) λ    0 0 0 

 1  1 −1 1  Z Z Z   0 0 0 LR → tr dB F  FF du F dB (2.2.14)    0 0 0 

A proof is given in the Appendix. Remark (i): When p = 1, as a special case, the Johansen-Fourier tests become the usual DF type tests with a Fourier function, as suggested by Enders and Lee (2012). The asymptotic distribution will be the same. Remark (ii): For the restricted constant (RC) model, we use F ∗(s)0 = (F (s)0, 1) instead of F (s). Remark (iii): For the model with a trend function, z(s) is replaced with z∗(s) = (s, z(s)). For the restricted trend (RT) model, we use F ∗(s)0and z∗(s) instead of F (s) and z(s) , respectively. Remark (iv): We do not consider a restricted Fourier model where a cointegrating vector can possibly eliminate the Fourier function as in the RC or RT models. Note that we al-

11 low for both sine and cosine functions in the Fourier function. Also note that ∆sin(2πk/T ) =

(2πk/T )cos(2πk/T ) + op(1) and ∆cos(2πk/T ) = −(2πk/T )sin(2πk/T ) + op(1). Thus, allowing for the nonlinear function in the short-run dynamics implies the existence of the nonlinear function in the long-run as well by allowing for both cosine and sine terms. The Fourier function can then exist in the long-run relationship between integrated processes as well as the short-run relationship between the differenced processes. Thus, the asymptotic distributions of the Johansen-Fourier tests depend primarily on the frequency parameter. In this paper, we consider two leading cases. The first case is based on the models using the cumulative frequencies, n. For example, if n = 1, we use the frequency, k = 1. If n = 2, two cumulative frequencies with k1 = 1, k2 = 2 are employed, and so on. When n is large, the power of the tests can decrease as the number of parameters to estimate increases. Overfitting can also be an issue in such cases. Enders and Lee (2013) suggest using n ≤ 3 for most practical cases. The advantage of using the cumulative frequencies is that we do not need to estimate the frequency parameters. In many cases, using one or two cumulative frequencies gives a good approximation to various types of nonlinear breaks. If needed, the number of cumulative frequencies can be determined by using the BIC criteria. The second case is to use a single frequency, k, that minimizes the sum of the squared residuals of the system, and we denote the resulting test

as the MaxkLR test. The asymptotic distribution of the MaxkLR test can be given as the supremum of the limiting distribution of the corresponding LR test.

 1  1 −1 1  Z Z Z   0 0 0 MaxkLR → Max tr dB F  FF du F dB . (2.2.15) k    0 0 0 

The critical values have been obtained by simulations for each of the models; they are shown in Tables 2.1 through 2.8. One question that arises is how the conventional Jo- hansen tests are aﬀected if nonlinear structural breaks are present but ignored. Trenkler (2005) examines the issue of ignoring level shifts on the system of cointegration tests by

12 simulation. We also find that the Johansen tests will exhibit size distortions when the null model holds and loss of power when the alternative model is true. Thus, the nature of the issue of ignoring existing breaks is different from the cases with unit root tests, which would be affected mostly under the alternative, as suggested by Perron (1989). The LR test statistic tends to cause non-negligible over-rejections, even with large samples. At the same time, the Johansen tests ignoring breaks will lose power under the alternative.

2.3 Performance of the Johansen-Fourier Tests

To assess the empirical performance of the new test, this section performs several simulation exercises. In particular, this section compares and contrasts the power and size properties of the trace version of the Johansen-Fourier test with those of the conventional Johansen cointegration test. The simulation exercise uses four diﬀerent sample sizes of 50, 100, 150, and 200 observations while varying the number of variables in the system from 2 to 5 and the number of possible breaks from one to seven. Those seven types of breaks are listed below:

Break(1) = 2 + 0.5 ∗ sin(2π × 1 × t/T ) + 0.5 × cos(2π ∗ 1 ∗ t/T )

Break(2) = 2 + 4 × sin(2π × 1 × t/T ) + 2 × cos(2π × 1 × t/T )

Break(3) = 2 + 10 × sin(2π × 1 × t/T ) + 6 × cos(2π × 1 × t/T )

Break(4) = 2 + 100 × sin(2π × 1 × t/T ) + 100 × cos(2π × 1 × t/T )

Break(5) = 20 × (1 − exp(−0.001 ∗ (1/w2) × (t − 2 × T/3.)2))

Break(6) = 20 + 15/(1 + exp(0.05/w × (t − 0.2 × T )))

−.5 × 15/(1 + exp(0.05/w × (t − .75 ∗ T )))

Break(7) = 30 × I(t ≤ 0.45 ∗ T ∨ t > .75 ∗ T, 3, 1)

13 The first four breaks are various forms of trigonometric breaks, the fifth break is an ESTAR- type break at the 2T/3 point in the sample, the sixth break is an offsetting LSTAR break at the T/5 and 3T/4 points in the sample, and the seventh break is a temporary break. We set w = 1/2. Figure 2.1 shows seven types of breaks used in the simulation to provide a visual understanding of their shapes. In addition, Figure 2.2 shows some examples of cointegrated and non-cointegrated series subject to the seven listed types of breaks. The left side of Figure 2.2 shows that in some cases the non-cointegrated series appear to trend together for a part of the sample period before divergence occurs (e.g., Breaks (1), (2), (3), (4), and (7)), whereas in the other cases the series diverge much faster (e.g., Breaks (5) and (6)). In contrast, the right side of Figure 2.2 shows that the cointegrated series comove together irrespective of the type of break we employ. In all cases, the paper employs 6,000 repetitions, where the error terms are drawn ran- domly from a standard normal distribution. To economize on space, the current version of the paper shows only the results for the model with a restricted constant at the 5% significance level. We emphasize the results for the restricted constant case since the empirical application we employ is suitable to this approach. The rest of the results are available in the online Appendix that accompanies the paper.

2.3.1 Size Properties

Figure 2.3 reports the size performance of the tests using p = 2, 3, 4, 5, and sample sizes of 50, 100, 150, and 200 observations. The left-hand side of Figure 2.3 shows that the conventional Johansen cointegration test suﬀers from severe size distortions that do not attenuate even with large sample sizes. In particular, when using Break (4), the size remains between 0.8 and 1 irrespective of the sample size or the number of variables used. Signiﬁcant size distortions also occur in the cases of Breaks (3) and (5). In contrast, the trace version of our new test shows limited size distortions that attenuate with larger sample sizes. For instance, the trace version test shows size distortions in the case of Break (4) and when the sample size is only 50 observations. However, as soon

14 as the sample size is at least 100 observations, the Johansen-Fourier test shows a size level very close to the theoretical 5%. In addition, when the simulations employ Break (5), the size of the distortion increases with the number of variables in the system. However, provided that the sample size is at least 200 observations, the empirical size converges relatively quickly to the theoretical 5%.

2.3.2 Power Simulations

Figures 2.4, 2.5, 2.6, and 2.7 show the power simulation results in the case of p = 2, 3, 4 and 5, respectively. For instance, for p = 2 the cointegrated series are built in the following way:

x1t = x1t−1 − δ(x1t−1 − x2t−1) + et, t = 1, .., T (2.3.1)

where x1t and x2t are random walks without a drift but that include a Break(j) term from above where j = 1,... 7. In all cases, the delta parameter (i.e., δ = 0.1, 0.2, 0.3, 0.4, 0.5) represents a sequence of progressively higher degrees of cointegration. Thus, a higher delta in absolute value denotes a higher degree of cointegration. The series are built in a similar manner when p = 3, 4 and 5. Due to space limitations and its practical importance, this subsection emphasizes the restricted constant case, wherein a constant appears only in the cointegrating vector. The rest of the power simulations are available in the online appendix that accompanies the study.1 First, Figure 2.4 shows a relatively rapid increase in the power of each test with the magnitude of delta and with the sample size. For instance, power is approximately 1 as soon as delta is 0.2 in absolute value, and the sample size is 200. Overall, it appears that the Johansen test is slightly more powerful than the new Johansen-Fourier test when the sample size is small (i.e., less than 100 observations). These results do not reflect size distortions under the null. But, those differences vanish provided that the delta parameter is sufficiently large (at least 0.2) and the sample size is at least 150 observations.

1We provide detailed critical values and size and power results in an online appendix at https://sites.google.com/view/rpascalau/research/online-appendix/

15 Figure 2.5 yields similar ﬁndings; power increases monotonically with the degree of cointegration and with the sample size. Further, in contrast to the cases where p ≤ 3, Figures 2.6 and 2.7 show that when p ≥ 4, both tests display a low power level in the case of the temporary break (i.e., Break (7)). This poor performance does not improve with the degree of cointegration or with larger sample sizes. To summarize, in the presence of smooth breaks the conventional Johansen test suﬀers from severe size distortions. The unreported results of the size-adjusted power show that the Johansen tests lose power when breaks are present. Therefore, the size and power simulation exercises recommend the use of the new Johansen-Fourier test when the applied researcher suspects the existence of slowly changing and smooth breaks.

2.4 Empirical Example

The empirical example uses the dataset of Kilian (2009) to assess whether the Real Oil Price, Oil Production, and the Real Economic Activity are cointegrated, respectively. In order to highlight the benefits of our new Johansen-Fourier test, we contrast its empirical findings with those from the conventional Johansen test. Figure 2.8 plots the three variables under analysis. A quick visual inspection shows that both the Real Oil Price and the Real Economic Activity lack a long term trend, while the Real Oil Production appears to be trending upwards after mid 1980’s. A battery of unit root tests shows that all three series have a unit root. We have applied the Bai and Perron structural break tests using both levels and first differences. The results are similar. The results using level data indicate the possible existence of three breaks for the real oil price in December 1973, January 1986, and June 2004, three breaks for oil production in July 1988, September 1996, and September 2003, and finally three breaks for the real economic activity in November 1974, May 1982, and March 2003. In- deed, economic history justifies the existence of those breaks. For instance, the early 1970s are associated with the jump in oil prices due to OPEC’s embargo. Similarly, the 1980s

16 are associated with a decrease in oil prices due to the oil glut. In particular, in 1986, oil prices plunged by more than half. Finally, the mid-2000s are associated with an increase in oil prices due to the increased demand from the expanding Chinese economy. Like- wise, as Figure 2.8 illustrates, the break points detected for the real economic activity series coincide with emergence from an NBER dated recession. However, it is likely that those breaks occurred smoothly over time and are therefore suitable to be modeled using a Fourier series. A general-to-specific approach for the lag selection suggests that two lags are sufficient to extract the serial correlation in the system. In addition, a single frequency in the sine and cosine terms appears to minimize the BIC criteria. Therefore, the Johansen-Fourier test employs the corresponding critical values from Table 3 and the online appendix ac- companying the paper.1 As already mentioned, Figure 2.8 does not appear to indicate the need for a linear trend at least in the case of the Real Oil Price and of the Real Economic Activity, respectively. Therefore, we employ the version of the cointegration tests that re- stricts the constant to the cointegrating vector. Table 2.9 shows the results of the conventional Johansen cointegration test. The Jo- hansen cointegration test fails to detect any evidence of cointegration among the three variables under analysis. In contrast, Table 2.10 shows that the Fourier-Jonansen finds up to two cointegrating vectors. The two cointegrating vectors corresponding to the largest two eigen values are the following:

ECT1 = 0.041095y − 0.007007x1 − 0.010444x2 + 0.955787

ECT2 = −0.00284y − 0.00993x1 + 0.04156x2 + 1.54696

where we denote y = Real Oil Price , x1 = Oil Production, and x2 = Real Economic

1We provide detailed critical values in an online appendix at https://sites.google.com/view/rpascalau/research/online- appendix/critical-values-restricted-constant.

17 Activity. According to the theory, both cointegrating vectors should be stationary. How- ever, an application of the Augmented Dickey Fuller and Elliott et al. (1996)’s and Elliott (1999)’s DF-GLS and DF-GLSu root tests reveals that only the second vector is stationary at the conventional signiﬁcance levels. The evidence for stationarity for the ﬁrst vector is weaker.1 Therefore, we use only the second cointegrating vector to estimate the error correction model. We estimate the latter by including the constant only in the cointegrating vector. Therefore, the deterministic component includes only the sine and cosine terms with one frequency only. The system of equations is shown below:

∗∗∗ ∗ ∆yt = 0.398 ∆yt−1 + 0.009∆x1t−1 + 0.023∆x2t−1 − 0.714sin(t) + 0.761 cos(t) + 0.432ECT

∗ ∆x1t = −0.005∆yt−1 − 0.088 ∆x1t−1 + 0.256∆x2t−1 − 2.165sin(t) − 0.106cos(t) + 1.121ECT

∗∗ ∗∗∗ ∗∗ ∗∗∗ ∆x2t = 0.074 ∆yt−1 − 0.004∆x1t−1 + 0.279 ∆x2t−1 + 0.846 sin(t) − 0.001cos(t) − 0.886 ECT

The results above show that only the Real Economic Activity variable is responsible for adjusting the system back to the long-run equilibrium. In contrast, the oil price and oil production appear to be weakly exogenous. It is no surprise then that the deviations from the long-run relationship are long lived. Indeed, Figure 2.10 shows that Oil Prices and Oil Production, respectively only respond to their own shocks, whereas the Real Economic Ac- tivity responds to each one of the three shocks. For example, Real Economic Activity increases following positive one standard deviation shocks to Oil Prices and Oil Production, respectively. While the impulse response following an increase in oil prices is marginally signiﬁcant, it appears that economic activity increases signiﬁcantly over the long run following a positive shock to Oil Production. Finally, one may note the usefulness of incorpo-

1The critical value for the DF-GLS test in the case of the ﬁrst cointegrating vector is -1.60, which is very close to the critical value of the test at the 10% signiﬁcance level at -1.62. The DF-GLSu test yields a test statistic of -2.0, whereas the critical value at the 10% level is -2.46. The corresponding DF-GLSu statistic for the second cointegrating vector is -2.740.

18 rating the sine and cosine terms. Both the Real Oil Price and the Real Economic Activity equations have sine or cosine terms that are significant. This confirms once more that at least those two variables may suffer from slowly changing smooth breaks over time.

2.5 Summary and Concluding Remarks

This paper deals with the issue of controlling for structural breaks in testing for cointegration based on the model framework of the system of equations. We have extended the pioneering Johansen tests to allow for unknown forms of structural breaks with a nonlinear Fourier function. Our suggested procedure can complement the other extended tests, which control for breaks with dummy variables. When the number of breaks, their locations, and the types of breaks in each of the equation in the system is known in advance, the tests utilizing the information can provide a solution. However, such information may not be readily available in many empirical applications. Also, if the break locations are mis-specified and the number of breaks is under-specified, the tests will have the same issue of ignoring breaks. As one alternative remedy, we approximate unknown forms of multiple breaks with a few parameters in the Fourier function. We argue that the Fourier function works well to approximate various forms of multiple smooth breaks and neglected nonlinearity. We recommend using a parsimonious number of parameters in applying the Fourier function to avoid the possibility of over-fitting. Still, they will capture well the effects of multiple breaks of unknown forms. Our simulation exercises show that the new testing procedure can work reasonably well in many different scenarios. In particular, our new procedure can work better in the presence of smooth breaks rather than sharp breaks, for which using dummy variables can lead to better performance. Several Monte Carlo simulation exercises show that the new test displays reasonable size and power properties under different scenarios.

19 Table 2.1: Critical Values - Constant Model

Freq. = 1 p = 1,..,5 90% 95% 99% 90% 95% 99% Trace Test Lambda Test 1 Trace(1) 11.96591 14.3084 19.15244 11.96591 14.3084 19.15244 2 Trace(1) 25.33234 28.18929 34.17934 19.42429 21.81952 26.89734 3 Trace(1) 42.40214 46.02544 52.73163 25.99134 28.35264 34.30575 4 Trace(1) 63.57798 67.68133 75.99406 32.29822 34.91794 41.03214 5 Trace(1) 89.19402 93.91098 103.47587 38.51377 41.17167 46.46918

Freq = 2 p = 1,..,5 Trace Test Lambda Test 1 Trace(1) 8.47652 10.51359 15.78948 8.47652 10.51359 15.78948 2 Trace(1) 19.73502 22.6004 28.52882 16.55501 19.05594 24.48584 3 Trace(1) 35.74732 38.91366 45.51562 23.88019 26.30747 31.44069 4 Trace(1) 56.6758 60.67533 69.05906 30.8011 33.44158 38.89055 5 Trace(1) 82.08287 86.50896 96.38501 37.55359 40.27457 45.93851

Freq = 3 p = 1,..,5 Trace Test Lambda Test 1 Trace(1) 7.39836 9.4069 14.35656 7.39836 9.4069 14.35656 2 Trace(1) 17.99192 20.71316 26.12183 15.0723 17.35662 22.5481 3 Trace(1) 32.90922 35.77095 42.92628 22.08337 24.69282 29.99618 4 Trace(1) 52.48284 56.38972 64.27689 29.33437 32.11669 37.83719 5 Trace(1) 76.70433 81.37919 90.2707 36.22397 38.74516 45.1397

Freq = 4 p = 1,..,5 Trace Test Lambda Test 1 Trace(1) 7.05204 8.85902 13.35081 7.05204 8.85902 13.35081 2 Trace(1) 17.1943 19.7394 25.06085 14.19221 16.54828 21.3657 3 Trace(1) 31.45236 34.62302 41.07153 21.32647 23.96644 28.6611 4 Trace(1) 50.0063 53.82681 61.7989 28.27393 30.86204 36.18487 5 Trace(1) 73.81947 78.36771 87.3374 35.21214 37.83603 43.96069

Freq = 5 p = 1,..,5 Trace Test Lambda Test 1 Trace(1) 6.84752 8.62743 12.70534 6.84752 8.62743 12.70534 2 Trace(1) 16.85144 19.13382 24.05343 13.84803 16.0725 20.56514 3 Trace(1) 30.56817 33.68254 39.81796 20.5407 22.72801 27.82667 4 Trace(1) 48.91592 52.29388 60.05373 27.21844 30.08076 35.71887 5 Trace(1) 72.02403 76.0114 84.72916 34.04224 36.78717 42.63075 Freq = Maxq p Trace Test Lambda Test 1 Trace(1) 12.77571 15.07919 19.87438 12.77571 15.07919 19.87438 2 Trace(1) 26.10865 28.85249 34.72286 20.63956 22.98041 28.09212 3 Trace(1) 43.32021 46.78274 53.19109 27.42833 29.74012 35.53437 4 Trace(1) 64.83038 68.89566 76.91632 34.05011 36.71878 42.16141 5 Trace(1) 90.67428 95.51971 104.39743 40.20088 42.85702 49.16653 Note: This table shows the critical values from using the Trace and the Lambda Tests with a constant in the model but not in the cointegrating vector. The maxq case denotes the critical values whereby the optimal frequency (over 1 through 5) is chosen such as to maximize the Likelihood Ratio of the Trace statistic.

20 Table 2.2: Critical Values - Trend Model

Freq = 1, p = 1,..,5 90% 95% 99% 90% 95% 99% Trace Test Lambda Test 1 Trace(1) 15.92629 18.09039 23.1309 15.92629 18.09039 23.1309 2 Trace(1) 32.6042 35.55617 42.05774 22.97027 25.46954 30.92691 3 Trace(1) 52.73863 56.49866 64.29932 29.50385 32.22024 37.47534 4 Trace(1) 76.92394 81.12916 89.62183 35.86276 38.25462 43.75503 5 Trace(1) 104.98884 109.17405 118.9881 41.85272 44.55845 50.41063

Freq = 2 p = 1,..,5 Trace Test Lambda Test 1 Trace(1) 13.42129 15.86571 20.85068 13.42129 15.86571 20.85068 2 Trace(1) 28.08645 31.17837 38.31604 21.38229 24.05796 28.93712 3 Trace(1) 46.94701 50.74269 58.20886 27.94128 30.45904 36.13643 4 Trace(1) 69.96733 74.78432 83.11733 34.77303 37.58944 43.42374 5 Trace(1) 97.46749 102.38793 112.22756 41.26979 43.98831 49.73163

Freq = 3 p = 1,..,5 Trace Test Lambda Test 1 Trace(1) 11.82088 14.21382 18.67103 11.82088 14.21382 18.67103 2 Trace(1) 25.22305 28.55989 34.51127 19.64398 22.17946 27.77477 3 Trace(1) 43.02937 46.79052 54.4939 26.69494 29.614 34.98442 4 Trace(1) 65.3013 69.25963 78.00512 33.62306 36.66324 42.41659 5 Trace(1) 91.89456 96.24529 106.04841 39.97296 42.69462 48.66725

Freq = 4 p = 1,..,5 Trace Test Lambda Test 1 Trace(1) 11.16867 13.22626 18.10014 11.16867 13.22626 18.10014 2 Trace(1) 23.76402 26.89691 32.96706 18.56379 20.99576 26.02601 3 Trace(1) 40.46416 44.13739 51.56758 25.33768 28.10436 33.75171 4 Trace(1) 61.94397 66.35955 74.40877 32.38881 35.01388 40.7533 5 Trace(1) 87.85069 92.48682 102.5278 38.97883 41.95523 47.97076

Freq = 5 p = 1,..,5 Trace Test Lambda Test 1 Trace(1) 10.5505 12.58547 17.18919 10.5505 12.58547 17.18919 2 Trace(1) 23.05486 25.83584 31.94371 17.91457 20.27764 25.70681 3 Trace(1) 39.59401 43.00055 49.9499 24.59048 27.17991 33.33271 4 Trace(1) 60.34547 64.33425 72.3245 31.48946 34.32537 40.40508 5 Trace(1) 85.37926 89.88038 99.27682 38.20402 41.09012 46.399

Freq = Maxq p = 1,..,5 Trace Test Lambda Test 1 Trace(1) 17.30189 19.5201 24.27098 17.30189 19.5201 24.27098 2 Trace(1) 33.86703 37.08639 43.06904 24.68907 27.1013 32.45098 3 Trace(1) 54.09262 57.70728 65.05912 31.28083 33.97494 39.39224 4 Trace(1) 78.62984 82.40116 91.30152 37.67745 40.3967 45.9196 5 Trace(1) 106.50309 110.99023 120.87198 43.91339 46.65765 52.83049 Note: This table shows the critical values from using the Trace and the Lambda Tests with a trend in the model but not in the cointegrating vector. The maxq case denotes the critical values whereby the optimal frequency (over 1 through 5) is chosen such as to maximize the Likelihood Ratio of the Trace statistic.

21 Table 2.3: Critical Values - Restricted Constant Model

Freq. = 1 p = 1,..,5 90% 95% 99% 90% 95% 99% Trace Test Lambda Test 1 Trace(1) 12.44508 14.74275 19.32823 12.44508 14.74275 19.32823 2 Trace(1) 26.83763 29.58587 35.48181 19.89873 22.19508 27.22541 3 Trace(1) 45.284 48.92995 55.02927 26.4112 28.89295 34.77758 4 Trace(1) 67.75972 71.81054 80.69072 32.81662 35.39524 41.34965 5 Trace(1) 94.57595 99.11298 108.06549 38.99138 41.54859 47.41295

Freq = 2 p = 1,..,5 Trace Test Lambda Test 1 Trace(1) 9.20649 11.22645 16.01961 9.20649 11.22645 16.01961 2 Trace(1) 21.67897 24.46542 30.38142 17.11164 19.6962 25.25389 3 Trace(1) 38.79007 42.119 48.61202 24.41557 26.90804 31.49482 4 Trace(1) 60.86761 65.00828 73.40683 31.28388 34.01138 39.34604 5 Trace(1) 87.40056 91.92876 101.95271 38.09145 40.76375 46.43124

Freq = 3 p = 1,..,5 Trace Test Lambda Test 1 Trace(1) 8.2839 10.24255 14.84384 8.2839 10.24255 14.84384 2 Trace(1) 20.15245 22.83348 28.0121 15.78246 18.08373 23.02954 3 Trace(1) 36.17527 39.16091 45.83424 22.81981 25.11247 30.69502 4 Trace(1) 56.99358 60.81107 68.827 29.8599 32.64265 38.2623 5 Trace(1) 82.22446 86.83963 95.58291 36.73675 39.36979 45.39216

Freq = 4 p = 1,..,5 Trace Test Lambda Test 1 Trace(1) 7.99887 9.85987 14.22182 7.99887 9.85987 14.22182 2 Trace(1) 19.15091 21.88533 26.82315 15.16224 17.23072 22.30953 3 Trace(1) 34.66447 37.84303 44.1743 22.03466 24.46652 29.42894 4 Trace(1) 54.43354 58.23366 66.30128 28.76678 31.35129 36.61389 5 Trace(1) 79.20521 83.59189 92.67475 35.75293 38.52174 44.24879

Freq = 5 p = 1,..,5 Trace Test Lambda Test 1 Trace(1) 7.77662 9.53509 13.72059 7.77662 9.53509 13.72059 2 Trace(1) 18.92157 21.32864 26.22043 14.70315 16.79633 21.72545 3 Trace(1) 33.77711 36.9486 43.09007 21.17693 23.3704 28.59657 4 Trace(1) 53.0267 56.82161 64.25029 27.88232 30.58855 36.241 5 Trace(1) 77.34222 81.6248 91.50333 34.60705 37.49176 43.31818 Freq = Maxq p Trace Test Lambda Test 1 Trace(1) 13.30296 15.55627 20.3747 13.30296 15.55627 20.3747 2 Trace(1) 27.65232 30.34652 36.0858 21.09088 23.54473 28.55744 3 Trace(1) 46.23075 49.77056 55.65684 27.94093 30.19468 36.01982 4 Trace(1) 68.88544 72.99614 81.20329 34.55318 37.15107 42.73735 5 Trace(1) 96.06927 100.62123 109.79667 40.77776 43.45975 49.86136 Note: This table shows the critical values from using the Trace and the Lambda Tests with a constant restricted to the cointegrating vector only (and not in the model). The maxq case denotes the critical values whereby the optimal frequency (over 1 through 5) is chosen such as to maximize the Likelihood Ratio of the Trace statistic.

22 Table 2.4: Critical Values - Restricted Trend

Freq. = 1 p = 1,..,5 90% 95% 99% 90% 95% 99% Trace Test Lambda Test 1 Trace(1) 16.7816 18.90748 24.1364 16.7816 18.90748 24.1364 2 Trace(1) 34.3846 37.47358 43.64184 23.89548 26.44461 32.01254 3 Trace(1) 55.53018 59.39504 67.08178 30.29101 33.03344 38.17394 4 Trace(1) 80.72425 85.16574 93.36696 36.66105 39.23833 44.89654 5 Trace(1) 109.49544 114.20359 124.53456 42.61192 45.38248 51.21126

Freq = 2 p = 1,..,5 Trace Test Lambda Test 1 Trace(1) 14.07468 16.37934 21.24727 14.07468 16.37934 21.24727 2 Trace(1) 29.69992 33.03774 39.66198 21.98196 24.68322 29.4036 3 Trace(1) 49.85115 53.42384 61.18987 28.52754 30.95725 37.14852 4 Trace(1) 74.19752 78.86507 87.33451 35.34741 38.1791 43.86716 5 Trace(1) 102.74867 107.71686 117.76993 41.89972 44.72682 50.38107

Freq = 3 p = 1,..,5 Trace Test Lambda Test 1 Trace(1) 12.61227 14.82827 19.42158 12.61227 14.82827 19.42158 2 Trace(1) 27.10051 30.20683 36.41345 20.12347 22.90663 28.3612 3 Trace(1) 45.93937 49.80522 57.09804 27.35573 30.23652 35.53448 4 Trace(1) 69.27565 73.87547 82.30167 34.23454 37.2978 43.02077 5 Trace(1) 96.85059 101.74087 111.69513 40.53592 43.44843 49.28561

Freq = 4 p = 1,..,5 Trace Test Lambda Test 1 Trace(1) 11.90545 13.83531 18.74437 11.90545 13.83531 18.74437 2 Trace(1) 25.59554 28.82023 34.84075 19.25288 21.73118 26.64931 3 Trace(1) 43.56009 47.18157 54.76021 26.13052 28.57067 34.22111 4 Trace(1) 66.11489 70.47044 78.46312 32.98635 35.63663 41.35651 5 Trace(1) 92.88905 97.87634 107.81804 39.54687 42.66692 48.34608

Freq = 5 p = 1,..,5 Trace Test Lambda Test 1 Trace(1) 11.39929 13.2854 18.05348 11.39929 13.2854 18.05348 2 Trace(1) 24.94488 27.72151 34.05113 18.73292 20.8269 26.19501 3 Trace(1) 42.54556 45.80354 53.71544 25.21915 27.77769 33.80212 4 Trace(1) 64.48174 68.30843 77.41853 32.14975 35.0607 40.94494 5 Trace(1) 90.91935 95.16862 104.27734 38.78672 41.73858 47.24342 Note: This table shows the critical values from using the Trace and the Lambda Tests with a trend restricted to the cointegrating vector only (and not in the model).

23 Table 2.5: Critical Values - Cumulative Frequencies in the Model with a Constant

Cum. Freq. = 1 p = 1,..,5 90% 95% 99% 90% 95% 99% Trace Test Lambda Test 1 Trace(1) 11.96591 14.3084 19.15244 11.96591 14.3084 19.15244 2 Trace(1) 25.33234 28.18929 34.17934 19.42429 21.81952 26.89734 3 Trace(1) 42.40214 46.02544 52.73163 25.99134 28.35264 34.30575 4 Trace(1) 63.57798 67.68133 75.99406 32.29822 34.91794 41.03214 5 Trace(1) 89.19402 93.91098 103.47587 38.51377 41.17167 46.46918

Cum. Freq. = 2 p = 1,..,5 Trace Test Lambda Test 1 Trace(1) 17.12811 19.73892 25.14218 17.12811 19.73892 25.14218 2 Trace(1) 34.36713 37.92253 44.95216 25.28195 27.76404 33.4209 3 Trace(1) 55.85528 59.84259 66.5754 32.36538 35.04145 40.4307 4 Trace(1) 82.06625 86.58413 95.98911 38.92353 41.93187 48.12253 5 Trace(1) 111.93068 117.05128 127.34426 45.3667 48.45197 54.53528

Cum. Freq. = 3 p = 1,..,5 Trace Test Lambda Test 1 Trace(1) 22.42492 25.34511 30.84669 22.42492 25.34511 30.84669 2 Trace(1) 43.47643 47.78119 55.33764 31.12968 33.8306 39.68737 3 Trace(1) 69.01807 73.55901 82.09909 38.33113 41.42773 47.96842 4 Trace(1) 99.4346 104.83568 115.13299 45.27037 48.46413 55.38 5 Trace(1) 134.49878 140.70307 151.77835 52.09768 55.34988 62.31783

Cum. Freq. = 4 p = 1,..,5 Trace Test Lambda Test 1 Trace(1) 27.3908 30.39381 37.00868 27.3908 30.39381 37.00868 2 Trace(1) 53.08466 57.38707 65.69967 36.94256 39.85131 46.8303 3 Trace(1) 82.47918 87.96552 98.2618 44.71073 47.92928 55.12456 4 Trace(1) 117.64283 123.44607 134.79804 52.26989 55.3712 61.67696 5 Trace(1) 157.52121 163.66453 175.65587 58.9266 62.26346 68.95451

Cum. Freq. = 5 p = 1,..,5 Trace Test Lambda Test 1 Trace(1) 32.16119 35.78632 42.40788 32.16119 35.78632 42.40788 2 Trace(1) 61.68281 67.11283 75.66266 42.79385 45.91888 52.69526 3 Trace(1) 95.77715 102.21303 112.73999 50.98125 54.4122 61.03765 4 Trace(1) 136.06876 142.35412 154.52721 58.50244 61.89791 69.6586 5 Trace(1) 180.00363 187.47393 200.7032 65.21567 69.0094 77.17664 Note: This table shows the critical values from using the Trace and the Lambda Tests with a constant in the model but not in the cointegrating vector.

24 Table 2.6: Critical Values - Cumulative Frequencies in the Model with a Trend

Cum. Freq. = 1 p = 1,..,5 90% 95% 99% 90% 95% 99% Trace Test Lambda Test 1 Trace(1) 15.92629 18.09039 23.1309 15.92629 18.09039 23.1309 2 Trace(1) 32.6042 35.55617 42.05774 22.97027 25.46954 30.92691 3 Trace(1) 52.73863 56.49866 64.29932 29.50385 32.22024 37.47534 4 Trace(1) 76.92394 81.12916 89.62183 35.86276 38.25462 43.75503 5 Trace(1) 104.98884 109.17405 118.9881 41.85272 44.55845 50.41063

Cum. Freq. = 2 p = 1,..,5 Trace Test Lambda Test 1 Trace(1) 22.0948 24.65593 29.4896 22.0948 24.65593 29.4896 2 Trace(1) 43.82382 46.81503 53.76702 29.18822 32.06307 37.91016 3 Trace(1) 68.67275 72.96398 80.86171 36.01288 38.79872 44.19754 4 Trace(1) 98.32984 103.00425 112.93871 42.3176 45.22149 51.82068 5 Trace(1) 131.09695 136.41383 146.91418 48.69856 52.11245 58.27965

Cum. Freq. = 3 p = 1,..,5 Trace Test Lambda Test 1 Trace(1) 27.62038 30.28825 36.58548 27.62038 30.28825 36.58548 2 Trace(1) 54.68141 58.41858 66.32639 35.65251 38.466 44.59761 3 Trace(1) 85.27217 89.82574 99.13845 42.37446 45.62627 51.97475 4 Trace(1) 119.13979 125.17419 135.59561 48.99546 52.60794 59.20915 5 Trace(1) 157.27454 163.80433 175.09341 55.99844 59.17533 65.90978

Cum. Freq. = 4 p = 1,..,5 Trace Test Lambda Test 1 Trace(1) 33.52095 36.91493 43.02022 33.52095 36.91493 43.02022 2 Trace(1) 65.55861 69.48224 78.45906 41.72944 44.757 50.83962 3 Trace(1) 101.34075 106.815 116.32645 49.07747 52.37514 58.71659 4 Trace(1) 140.52655 145.99311 156.98042 56.00045 59.21325 65.66144 5 Trace(1) 183.55658 190.71159 202.59054 62.68861 65.76993 73.5071

Cum. Freq. = 5 p = 1,..,5 Trace Test Lambda Test 1 Trace(1) 39.31262 42.74488 49.29503 39.31262 42.74488 49.29503 2 Trace(1) 76.9944 80.94095 89.49088 47.89613 51.04538 57.69548 3 Trace(1) 117.82788 122.9247 133.94353 55.71479 58.6176 65.84428 4 Trace(1) 161.86362 168.06434 180.33552 62.50166 65.85758 73.45941 5 Trace(1) 210.26792 217.45626 231.53649 69.06417 72.81402 80.94427 Note: This table shows the critical values from using the Trace and the Lambda Tests with a trend in the model but not in the cointegrating vector.

25 Table 2.7: Critical Values - Cumulative Frequencies in the Model with Restricted Constant

Cum. Freq. = 1 p = 1,..,5 90% 95% 99% 90% 95% 99% Trace Test Lambda Test 1 Trace(1) 15.11654 17.36134 22.53505 15.11654 17.36134 22.53505 2 Trace(1) 31.54193 34.86245 41.10625 22.67917 25.09009 30.99055 3 Trace(1) 52.19268 56.15881 63.4624 29.42621 31.81531 38.17176 4 Trace(1) 77.20757 81.32806 90.19446 35.70258 38.66757 44.9486 5 Trace(1) 105.79917 111.08734 121.39147 42.08703 45.12572 51.3927

Cum. Freq. = 2 p = 1,..,5 Trace Test Lambda Test 1 Trace(1) 22.43084 25.51318 30.7509 22.43084 25.51318 30.7509 2 Trace(1) 45.34429 48.98605 56.79718 30.98534 34.10673 39.82854 3 Trace(1) 72.10403 76.33016 84.93196 38.33021 41.3475 47.5348 4 Trace(1) 103.58625 109.20236 118.69695 45.16083 48.25683 55.00137 5 Trace(1) 139.06963 144.88705 156.97414 51.70319 55.09165 61.44896

Cum. Freq. = 3 p = 1,..,5 Trace Test Lambda Test 1 Trace(1) 29.83201 32.82653 39.57299 29.83201 32.82653 39.57299 2 Trace(1) 58.53525 62.79299 71.09629 39.25572 42.475 49.15379 3 Trace(1) 91.86421 96.91238 106.96564 47.08902 50.36196 57.1333 4 Trace(1) 130.05049 135.64276 147.73781 54.31962 57.50662 65.15398 5 Trace(1) 172.31304 179.15233 192.52569 61.1928 64.95132 71.59282

Cum. Freq. = 4 p = 1,..,5 Trace Test Lambda Test 1 Trace(1) 36.81235 40.45318 47.41139 36.81235 40.45318 47.41139 2 Trace(1) 71.71836 77.00024 86.30859 47.25965 50.95535 58.60921 3 Trace(1) 111.50841 116.93661 127.66947 55.7777 59.10846 67.31306 4 Trace(1) 156.52413 162.08337 174.75224 63.20543 67.01418 74.27722 5 Trace(1) 206.02018 212.44297 225.45524 70.40999 73.76121 80.9324

Cum. Freq. = 5 p = 1,..,5 Trace Test Lambda Test 1 Trace(1) 43.79379 47.73399 54.85713 43.79379 47.73399 54.85713 2 Trace(1) 85.18358 90.84489 100.94943 55.5367 59.52646 66.88749 3 Trace(1) 131.27098 137.39421 149.4528 63.99236 67.9576 76.31363 4 Trace(1) 182.90272 189.81457 203.82468 72.04681 75.93939 84.34862 5 Trace(1) 239.19408 246.32093 263.06467 79.17897 83.23137 91.63038 Note: This table shows the critical values from using the Trace and the Lambda Tests with a constant restricted to the cointegrating vector only (and not in the model).

26 Table 2.8: Critical Values - Cumulative Frequencies in the Model with Restricted Trend

Table 2.9: Johansen-cointegration - restricted constant approach

Rank Eigenvalue Lambda-max Trace Trace-95% Log-likelihood 0 -4434.7013 1 0.0502 21.4789 32.3136 35.07 -4423.9618 2 0.016 6.7416 10.8347 20.16 -4420.591 3 0.0098 4.0931 4.0931 9.14 -4418.5445

27 Table 2.10: Johansen-Fourier Cointegration test - restricted constant approach

Rank Eigenvalue Lambda-max Trace Trace-95% Log-likelihood 0 -4432.7491 1 0.0663 28.5914 57.3421 48.93 -4418.4534 2 0.0478 20.4043 28.7507 23.16 -4408.2513 3 0.0198 8.3464 8.3464 8.54 -4404.0781

28 Note: The DGPs used in the simulation exercises employ the seven types of breaks displayed above. For instance, Break(1) corresponds to y(t) = 2 + 0.5sin(2πt/T ) + 0.5cos(2πt/T ); Break(2) corresponds to y(t) = 2 + 4sin(2πt/T ) + 2cos(2πt/T ); Break(3) corresponds to y(t) = 2 + 10sin(2πt/T ) + 6cos(2πt/T ); Break(4) corresponds to y(t) = 2 + 100 ∗ sin(2πt/T ) + 100cos(2πt/T ); Break(5) corresponds to y(t) = 20(1 − exp(−0.001 ∗ (1/w2) ∗ (t − 2T/3)2)); Break(6) corresponds to y(t) = 20 + 15/(1 + exp(0.05/w(t − 0.2T ))) − 0.5 ∗ 15/(1 + exp(0.05/w(t − 0.75T ))), and ﬁnally Break(7) corresponds to y(t) = 3 if t ≤ 0.45T or t > 0.75T and 1 otherwise. We set w=1/2.

Figure 2.1: Break Types used in Simulations

29 Note: This table simulates non-cointegaretd versus cointegrated DGPs employing the seven types of breaks displayed above.For instance, Break(1) corresponds to y(t) = 2 + 0.5sin(2πt/T ) + 0.5cos(2πt/T ); Break(2) corresponds to y(t) = 2 + 4sin(2πt/T ) + 2cos(2πt/T ); Break(3) corresponds to y(t) = 2 + 10sin(2πt/T ) + 6cos(2πt/T ); Break(4) corresponds to y(t) = 2 + 100 ∗ sin(2πt/T ) + 100cos(2πt/T ); Break(5) corresponds to y(t) = 20(1 − exp(−0.001 ∗ (1/w2) ∗ (t − 2T/3)2)); Break(6) corresponds to y(t) = 20 + 15/(1 + exp(0.05/w(t − 0.2T ))) − 0.5 ∗ 15/(1 + exp(0.05/w(t − 0.75T ))), and ﬁnally Break(7) corresponds to y(t) = 3 if t ≤ 0.45T or t > 0.75T and 1 otherwise. We set w=1/2

Figure 2.2: Examples of Non-Cointegrated versus Cointegrated Series.

30 Size performance. This table shows the size performance of the Johansen and Johansen-Fourier’s Trace tests respectively, at the 5% signiﬁcance level in the model with a ”restricted constant (RC)”. Break(1) corresponds to y(t) = 2 + 0.5sin(2πt/T ) + 0.5cos(2πt/T ); Break(2) corresponds to y(t) = 2 + 4sin(2πt/T ) + 2cos(2πt/T ); Break(3) corresponds to y(t) = 2 + 10sin(2πt/T ) + 6cos(2πt/T ); Break(4) corresponds to y(t) = 2 + 100 ∗ sin(2πt/T ) + 100cos(2πt/T ); Break(5) corresponds to y(t) = 20(1 − exp(−0.001 ∗ (1/w2) ∗ (t − 2T/3)2)); Break(6) corresponds to y(t) = 20 + 15/(1 + exp(0.05/w(t − 0.2T ))) − 0.5 ∗ 15/(1 + exp(0.05/w(t − 0.75T ))), and ﬁnally Break(7) corresponds to y(t) = 3 if t ≤ 0.45T or t > 0.75T and 1 otherwise. We set w=1/2

Figure 2.3: Size Performance

31 Note: Power performance: system of 2 variables. This table shows the power performance of the Jo- hansen and Johansen-Fourier’s Trace tests respectively, at the 5% signiﬁcance level in the model with a ”restricted constant (RC)”. Break(1) corresponds to y(t) = 2 + 0.5 ∗ sin(2πt/T ) + 0.5cos(2πt/T ); Break(2) corresponds to y(t) = 2 + 4sin(2πt/T ) + 2cos(2πt/T ); Break(3) corresponds to y(t) = 2+10sin(2πt/T )+6cos(2πt/T ); Break(4) corresponds to y(t) = 2+100∗sin(2πt/T )+100cos(2πt/T ); Break(5) corresponds to y(t) = 20(1 − exp(−0.001 ∗ (1/w2) ∗ (t − 2T/3)2)); Break(6) corresponds to y(t) = 20 + 15/(1 + exp(0.05/w(t − 0.2T ))) − 0.5 ∗ 15/(1 + exp(0.05/w(t − 0.75T ))), and ﬁnally Break(7) corresponds to y(t) = 3 if t ≤ 0.45T or t > 0.75T and 1 otherwise. We set w=1/2

Figure 2.4: Power performance: system of 2 variables.

32 Note: Power performance: system of 3 variables. This table shows the power performance of the Jo- hansen and Johansen-Fourier’s Trace tests respectively, at the 5% signiﬁcance level in the model with a ”restricted constant (RC)”. Break(1) corresponds to y(t) = 2 + 0.5sin(2πt/T ) + 0.5cos(2πt/T ); Break(2) corresponds to y(t) = 2 + 4sin(2πt/T ) + 2cos(2πt/T ); Break(3) corresponds to y(t) = 2+10sin(2πt/T )+6cos(2πt/T ); Break(4) corresponds to y(t) = 2+100∗sin(2πt/T )+100cos(2πt/T ); Break(5) corresponds to y(t) = 20(1 − exp(−0.001 ∗ (1/w2) ∗ (t − 2T/3)2)); Break(6) corresponds to y(t) = 20 + 15/(1 + exp(0.05/w(t − 0.2T ))) − 0.5 ∗ 15/(1 + exp(0.05/w(t − 0.75T ))), and ﬁnally Break(7) corresponds to y(t) = 3 if t ≤ 0.45T or t > 0.75T and 1 otherwise. We set w=1/2

Figure 2.5: Power performance: system of 3 variables.

33 Note: Power performance: system of 4 variables. This table shows the power performance of the Jo- hansen and Johansen-Fourier’s Trace tests respectively, at the 5% signiﬁcance level in the model with a ”restricted constant (RC)”. Break(1) corresponds to y(t) = 2 + 0.5sin(2πt/T ) + 0.5cos(2πt/T ); Break(2) corresponds to y(t) = 2 + 4sin(2πt/T ) + 2cos(2πt/T ); Break(3) corresponds to y(t) = 2+10sin(2πt/T )+6cos(2πt/T ); Break(4) corresponds to y(t) = 2+100∗sin(2πt/T )+100cos(2πt/T ); Break(5) corresponds to y(t) = 20(1 − exp(−0.001 ∗ (1/w2) ∗ (t − 2T/3)2)); Break(6) corresponds to y(t) = 20 + 15/(1 + exp(0.05/w(t − 0.2T ))) − 0.5 ∗ 15/(1 + exp(0.05/w(t − 0.75T ))), and ﬁnally Break(7) corresponds to y(t) = 3 if t ≤ 0.45T or t > 0.75T and 1 otherwise. We set w=1/2

Figure 2.6: Power performance: system of 4 variables.

34 Note: Power performance: system of 5 variables. These figures show the power performance of the Jo- hansen and Johansen-Fourier’s Trace tests respectively, at the 5% significance level in the model with a ”restricted constant (RC)”. Break(1) corresponds to y(t) = 2 + 0.5 ∗ sin(2πt/T ) + 0.5cos(2πt/T ); Break(2) corresponds to y(t) = 2 + 4sin(2πt/T ) + 2cos(2πt/T ); Break(3) corresponds to y(t) = 2+10sin(2πt/T )+6cos(2πt/T ); Break(4) corresponds to y(t) = 2+100∗sin(2πt/T )+100cos(2πt/T ); Break(5) corresponds to y(t) = 20(1 − exp(−0.001 ∗ (1/w2) ∗ (t − 2T/3)2)); Break(6) corresponds to y(t) = 20 + 15/(1 + exp(0.05/w(t − 0.2T ))) − 0.5 ∗ 15/(1 + exp(0.05/w(t − 0.75T ))), and finally Break(7) corresponds to y(t) = 3 if t ≤ 0.45T or t > 0.75T and 1 otherwise. We set w=1/2

Figure 2.7: Power performance: system of 5 variables

35 400

300

200

100

-100

-200 1973 1975 1977 1979 1981 1983 1985 1987 1989 1991 1993 1995 1997 1999 2001 2003 2005 2007

Real Oil Price Oil Production Real Economic Activity

U.S. Recessions

Figure 2.8: Real Oil Price, Oil Production, and Real Economic Activity

6 5

4 4

2 2

0 1

0 -2

-1

-4 -2

-6 -3 1973 1976 1979 1982 1985 1988 1991 1994 1997 2000 2003 2006 1973 1976 1979 1982 1985 1988 1991 1994 1997 2000 2003 2006 Note: Error Correction Model corresponding to the largest Eigenvalue Note: Error Correction Model corresponding to the second largest Eigenvalue

Figure 2.9: Two Cointegrating Vectors

36 to Real Oil Price to Oil Production to Real Economic Activity

15.0 15.0 15.0

12.5 12.5 12.5

10.0 10.0 10.0

7.5 7.5 7.5

5.0 5.0 5.0 Real Oil Price 2.5 2.5 2.5

0.0 0.0 0.0

-2.5 -2.5 -2.5

-5.0 -5.0 -5.0 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20

25 25 25

20 20 20

15 15 15

10 10 10 Oil Production 5 5 5

0 0 0

Responses of -5 -5 -5 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20

7 7 7

6 6 6

5 5 5

4 4 4

3 3 3 Real Economic Activity 2 2 2

1 1 1

0 0 0

-1 -1 -1 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 to Real Oil Price to Oil Production to Real Economic Activity Impulse Responses

Figure 2.10: Impulse Response Analysis

37 2.6 Appendix

Proof of Theorem 1 We follow the notations and the procedure in Johansen (1991). The LR test is based on the solutions to the characteristic equation in equation (2.9)

−1 | λS11 − S10S 00S01 | = 0 (2.6.1)

Then, the distribution of the LR statistic for the standard model without any determinstic

terms, ∆Xt = ΠXt−k + et, with et ∼ Np(0, Ω), is given as

 1  1 −1 1  Z Z Z   0 0 0 LR → tr dB B  BB du B dB (2.6.2)    0 0 0 

where B = Ω−1W is the standard Brownian motion with W being the Wiener process. We ﬁrst consider the model with a constant and the Fourier funtion in (2.1). When

0 µ 6= 0, there is a linear trend in the process, but we consider Xt in the p − 1 directions µ⊥.

0 Then, the assumption α⊥µ = 0 implies that the stationary process ∆Xt has expectation zero. We can examine the Granger representation theorem

t X Xt = C (ei + µ + fi) + C(L)(et + µ + ft) + A (2.6.3) i=1

0 0 −1 0 where β A = 0,C = β⊥(β⊥β⊥) α⊥ and A depends on initial conditions. Then, the normalization gives

[uT ] −1/2 0 −1/2 0 X −1/2 0 −1/2 0 T β⊥XuT = T β⊥C (ei + µ + fi) + T β⊥C(L)(et + µ + ft) + T β⊥A (2.6.4) i=1

38 The ﬁrst term goes to a Brownian motion and the second term is zero.

[uT ] −1/2 0 X 0 T β⊥C ei → β⊥CW (u) (2.6.5) i=1

[uT ] −1/2 0 X T β⊥C µ = 0 (2.6.6) i=1 The third term tends to zero

[uT ] −1/2 0 X T β⊥C (A cos(2πki/T ) + B sin(2πki/T )) → 0 (2.6.7) i=1

−1/2 Following Johansen (1994), we deﬁne the normalization matrix, At = (β, T β⊥). We

0 pre- and post-multiply equation (5.1) with At and At, and rearrange the term as in equation (11.16) of Johansen (1994). Then, we deﬁne the standard Brownian motion

0 0 −1/2 0 B = (β⊥CΩC β⊥) β⊥CW (2.6.8)

Also, we deﬁne B(s) = (B1(s), .., Bp−r(s)) as a (p − r) dimensional standard Brownian motion, and the (p − r) dimensional process F (s) = (F1(s), .., Fp−r(s), ), which is the projection of the process B(s) on the orthogonal complement of the space spanned by the Fourier funtion, z(s) = [1, cos(2π1s), .., cos(2πks), sin(2π1s), .., sin(2πks)], deﬁned over the

ˆ ˆ R 1 ˆ 2 interval, s ∈ [0, 1], such that F (s) = B(s) − z(s)δ with δ = argmin 0 [B(s) − z(s)δ] ds. Using this, we can show that the corresponding LR statistic converge in probability to

 1  1 −1 1  Z Z Z   0 0 0 LR → tr dB F  FF du F dB (2.6.9)    0 0 0 

The distribution of the lambda statistic can be obtained in a similar manner.

39 CHAPTER 3 HOW INTEGRATED IS EUROPE? ANALYSIS OF CONDITIONS FOR AN OPTIMAL CURRENCY AREA

3.1 Introduction

Since its formation in 1993, Europe has been gradually expanding the membership in the European Union (EU), and the common European currency was introduced in 1999. Initially, the Eurozone, which is formally called the Economic and Monetary Union (EMU), was made up of 11 member states. As of 2020, 19 nations share the unified monetary system based on the Euro. Forming economic unions and monetary integration in Europe has triggered extensive studies on qualifications for joining the EMU. The 1992 Maastricht Treaty outlined five different convergence criteria, and introduced thresholds for govern- ment debt, deficit, inflation, interest rates, and exchange rate. However, many economists argue that these criteria have little to do with the desired optimal currency area criteria. Mundell (1961), McKinnon (1963), Kenen (1969), Frankel and Rose (1998), Frankel and Rose (2002) and others note that synchronized business cycles, integrated markets, and risk-sharing mechanisms are essential prerequisites for an optimal currency area. They are necessary for the Eurozone members to enjoy the benefits of economic and monetary integration. Thus, a vast literature examined the extent of similarities and synchronizations of member states economic conditions. Mundell (1961) initially suggested that countries with similar business cycles are natural candidates for membership for an optimal currency area (OCA). The reasoning is clear. Common monetary policy would levy costs to member states at different stages of the business cycle. Darvas et al. (2005) further indicate that integrated markets are also relevant criteria. The

40 implications are particularly pertinent given that the EU and Eurozone consider expanding memberships, following recent inquiries by several Eastern European economies. This paper examines the degree of similarity and synchronization in the context of the comovements captured by the dynamic factor model (DFM) with time-varying parameters and stochastic volatility. The DFM approach upgrades the previous measures of synchronization based on simple correlations of cyclical components and other extensions. We use GDP growth rates for the DFM, which permits us to evaluate the intertemporal comovements at the global and group levels. The analysis using GDP growth rates is in line with the literature on the global business cycles, for which the DFM has been used (see Kose et al. (2003) as an example). We extend this analytical thread by using unemployment rates, employment rates, inflation, and interest rates to evaluate similarities and synchronizations from various perspectives. The motivation for using employment and unemployment rates is to gauge similarities in labor markets. Similarly, we establish the implications for monetary integration through inflation and financial integration through interest rates. The overarching question is whether there are significant differences in comovements and synchronizations among the Eurozone, non-Eurozone EU, and non-EU countries. Pol- icy rules such as the Maastricht treaty criteria that apply to Eurozone and EU member states and common institutions are expected to provide a higher degree of synchronization between EU member states. We try to disentangle the EU integration effect on synchronization from the effect of the global business cycle. It is an indirect way to evaluate the efficiency of the EU integration and convergence criteria. Also, we evaluate the differences in the degree of cross-correlation within each group. The paper confirms the presence of significant global comovements and time-varying degrees of synchronization. However, we find that the regional synchronization effects within each group are not significant. More pressing, we find that the degree of synchronization in the Eurozone countries is not greater than that in the non-Eurozone or non-EU countries. Thus, we do not find signifi-

41 cant evidence of a higher level of similarity and synchronization in Eurozone countries economic conditions. At the same time, there is no clear evidence that the EU membership has enhanced similarity and synchronization effects in the group over time. Instead, we find that synchronization effects are time-dependent rather than group-dependent and are generally more significant during financial crises. Overall, our results cast doubt on using the so-called qualification criteria for the union membership. The rest of the paper is organized as follows. Section 2 provides the literature survey and explains how the paper is motivated by and related to existing studies. In Section 3, we discuss the data and methodology. Section 4 gives the estimation results and interpre- tations. Finally, Section 5 gives concluding remarks.

3.2 Literature Review and Motivation

The Maastricht treaty, signed in 1992, introduced convergence criteria for the Euro- pean Union (EU) member states to be admitted to the Eurozone. They are associated with inflation, public debt and public deficit, exchange rate stability, and long-term interest rates. Frankel and Rose (1998) and Frankel and Rose (2002) note that these criteria may not be essential features to maximize a common currency areas advantages. Instead, they argue that synchronized business cycles are more important criteria for the optimal currency area; see also Darvas et al. (2005). The main issue centers around the relevancy and persistence of business cycle synchronization. One strand of the literature has investigated the determinants of business cycle comovements. Figuring out the determinants of business cycle synchronization is of critical interest, and there is an abundance of important papers in the literature on the subject. Frankel and Rose (1998) point out that a countrys suitability to enter a currency union depends on trade intensity with other currency union members. They find that more in- tensive trade makes business cycles more synchronized, suggesting the endogeneity of the optimal currency area or endogenous business cycle correlations. The literature notes sev-

42 eral other components that can affect business cycle synchronization. They include trade openness, industrial similarity, gravity variables, monetary and fiscal similarity, inflation and financial similarity, among others; see Baxter and Kouparitsas (2005), Imbs (2004), Andrle et al. (2017), Inklaar et al. (2008), and others. Another branch of literature looks into the source of business cycle comovements and pays attention to the empirical link between labor market institutions and business cycle synchronization. Fonseca et al. (2010) suggest that cross-country labor market heterogeneity affects business cycle synchronization. That is, divergence in labor market institutions amplifies the differentials in business cycles. Sachs and Schleer (2009) also note that if countries have differently styled labor market institutions, economic shocks can yield distinct consequences, thereby diverging business cycles. Using a nonparametric estimation, Camarero and Ordóñez(2006) demonstrate consistent nonlinear comovements in European unemployment rates. Some papers point out that the factors related to labor markets and similarities in labor market institutions have recently become starker. Fonseca et al. (2010) emphasize the importance of labor market conditions and regulations as they affect the extent of business cycle synchronization. They highlight that remarkable differences still exist in labor markets among countries. Sachs and Schleer (2009) note that a highly inflexible labor market structure can lead to diverging business cycles, and common structural reforms are essential factors. Camarero and Ordóñez(2006) examine the hysteresis hypothesis among European unemployment rates and find nonlinear comovements. Kriˇstićet al. (2019) analyze the convergence of unemployment rates among eurozone member states. They show that the Eurozone integration provides an initial boost for the unemployment convergence. Still, it is stressed that the Eurozone membership itself is not a guarantee for long-term convergence. Quite the contrary, they find unemployment rates to diverge for Italy and some other member states, which have been Eurozone member states since its formation. Estrada et al. (2013) analyze convergence in five macroeconomic variables in Eurozone

43 countries and examine the beta-convergence hypothesis. They argue that if the Eurozone system has been effective, one can observe more evidence of convergence in major macro variables. They conclude that, notably, membership in the Eurozone does not guarantee convergence of all member states. The Eurozone is not homogenous as there are substantial differences between the core and peripheral countries. Whether monetary integration has led to business cycle synchronization is also a question of interest. The degree to which economic unions and monetary integration in Eu- rope affect synchronization in business cycles can be studied through an empirical lens. The literature is somewhat mixed on this issue. Artis and Zhang (1997) note that effective exchange rate regimes impose policy disciplines that are likely to conform to the partic- ipating countries business cycles. On the other hand, Inklaar and De Haan (2001) argue that there is not much evidence to support the view that increased exchange rate stability is related to more synchronized business cycles in Europe. While Baxter and Kouparit- sas (2005), as well as Clark and Van Wincoop (2001), do not consider a currency union as relevant for the determination of business cycle synchronization, Frankel and Rose (2002) report a significantly positive effect of a common currency on the similarity of business cycles. A single monetary policy of the European Central Bank (ECB) can increase business cycle synchronization on some level since all countries are affected by its measures, such as quantitative easing, which started after the sovereign debt crisis in the Eurozone. A single monetary policy may also decrease business cycle synchronization when a one-size-fits-all monetary policy does not work, which could be the case with asymmetric shocks that affect countries differently. Thus, this issue poses an important motivation for this paper.

3.3 Data and Estimation Procedures

The dynamic factor model allows the researcher to decompose the variable of interest into its unobserved global, regional (group), and country-speciﬁc components, which has important application and policy repercussions Otrok et al. (2015). Our analysis con-

44 siders group factors that can capture specific features of the Eurozone and non-Eurozone countries, which has not been examined in the literature. We analyze comovements and synchronization of five major macroeconomic variables to gauge the effects in broad sec- tors of the economy: (1) real GDP growth, (2) employment rate, (3) unemployment rate, (4) interest rates, and (5) inflation. All series are seasonally adjusted and reported at a quarterly frequency. The motivation for using GDP growth rates is clear, as the global business cycle estimation is based on GDP growth rates. Unemployment rates can also give information about the global business cycles and shed light on similarities and synchronizations in labor markets. Employment rates may also help to assess labor market conditions than unemployment rates, as their measurement is simple, straightforward, and less affected by methodological changes. The results using inflation rates can have indirect implications on monetary integration. Lastly, interest rates can convey the implication on financial integration effects. The data sources for all variables are the Eurostat and OECD databases. We combine Eurostat and OECD sources using the longer of the two series to obtain a balanced panel of data and maximize observations. For the GDP growth, we use seasonally adjusted real GDP and compute annual growth rates. Employment and unemployment rates include individuals age 15 to 65. Frances employment rate is approximated by the France metropolitan area, as the regular employment rate series started in 2003. Inflation is based on the consumer price index. Eurostat offers a monthly harmonized index of consumer prices, which was averaged to quarterly data. We use annual percentage long term interest rates, one of the EMU convergence criteria. Countries are grouped as Eurozone member states, non-Eurozone EU member states, and non-EU OECD countries (OECD countries that are not EU). OECD countries are included as a control group and can help estimate the trend of the world economy. We will also compare two groups in the EU with this group. Due to data availability, the groups include slightly different countries. We provide a complete list of countries in Table 3.1, presenting descriptive statistics for each variable.

45 3.3.1 Dynamic Factor Model

To examine common factors of ﬁve variables, we use a dynamic factor model (DFM) of Del Negro and Otrok (2008) and Bhatt et al. (2017), who consider the following dynamic factor model:

yi,t = ci + λi,t · Gt + γi,t · Rt + i,t (3.3.1)

where yi,t is the variable of interest of the country i at time t. Here, ci denotes the deterministic term, for which we include a constant. In this model, we decompose the variable into two different factors. First, Gt denotes the global factor that affects all countries con- temporaneously at time t. λi,t is the time-varying factor loading of the global factor. It denotes the heterogeneous response of each country to the global factor. Second, Rt includes group factors at time t, and it is given as a vector considering three groups of the Euro- zone, non-Eurozone, and non-EU OECD countries. The jth element of the vector denotes the jth groups factor, for j = 1, 2, 3. Here, yi,t denotes the time-varying factor loading parameters of group factors; it is a row vector whose elements are non-zero if they corre- spond to country i, and are zero otherwise. Thus, we can evaluate different group factors that are orthogonal to the global factor and other group factors. i,t is the error term that captures country-specific dynamics as well as measurement errors.

0 We let βi,t = [λi,t, γi,t] and assume that it follows a random walk process:

βi,t = βi,t−1 + σηi · ηi,t (3.3.2)

where ηi,t ∼ N(0, Σβ) and is independent across i. We assume that both factors and country-speciﬁc components follow autoregressive processes of order q and pi, respectively:

g g g g Gt = φ1 · Gt−1 + ... + φq · Gt−q + exp(ht ) · vt (3.3.3)

r r r r Rj,t = φj,1 · Rj,t−1 + ... + φj,q · Rj,t−q + exp(hj,t) · vj,t, for j = 1, ..., m. (3.3.4)

46 ei,t = φi,1 · ei,t−1 + ... + φi,pi · ei,t−pi + exp(hi,t) · wi,t (3.3.5)

g 2 r 2 2 where vt ∼ N(0, σg ), vj,t ∼ N(0, σj,r), and wi,t ∼ N(0, σw). We estimate the model assuming a stationary AR(2) process with q = 2 and pi = 2 for the global and regional factors

2 2 and for country-specific component. For the purpose of identification, we let σg = σj,r = 1, j = 1, ..., m. We also assume that each of the time-varying stochastic volatility of the factors and the country-specific components follow a random walk:

g g g g ht = ht−1 + σh · ωt (3.3.6)

r r r r hj,t = hj,t−1 + σj · ωj,t (3.3.7)

h hi,t = hi,t−1 + σi · ωi,t (3.3.8)

g r where wt ∼ N(0, 1), wj,t ∼ N(0, 1) and wi,t ∼ N(0, 1). We assume that volatility shocks in the above are orthogonal to each other, and the initial values of the stochastic volatility

g r terms are zero, h0 = hj,0 = hi,0. The above dynamic factor model allows for time-varying parameters and stochastic volatilities. The usual classical approaches using the MLE or the Kalman filtering can have difficulty in estimating such models. We employ the Bayesian MCMC procedure for which we apply the usual Gibbs-sampling algorithms. The prior distribution for each of the factor loading parameters is N(0,1). We follow Del Negro and Otrok (2008) for the usual assumptions regarding the parameters distributions in the above model. Technical details of the estimation procedures, including the prior distributions of the parameters and the Gibbs sampler, are provided in the Appendix of Del Negro and Otrok (2008)1. The DFM permits us to measure the relative contribution of global, group, and country- specific factors to the variations in the variable of interest in each country, as described in

1Our estimation follows the same model framework of Del Negro and Otrok (2008) and Bhatt et al. (2017). We are grateful for the authors of Bhatt et al. (2017) who shared the Matlab codes that were initially developed for these procedures.

47 Crucini et al. (2011). The variance of the observable data can be decomposed into global, group, and country-speciﬁc component as follows:

2 2 V ar(yi,t) = λi,t · V ar(Gt) + γi,t · V ar(Rt) + V ar(i,t) (3.3.9)

The variance decomposition can be given as a ratio of each of the right-hand side terms in equation (4.3.3) to var(yi,t). For example, the variance decomposition due to the global

2 factor is λi,t var(Gt)/var(yi,t). We believe that the DFM gives a rich set of different measures of synchronization effects. We use the variance decomposition method to determine each factors relative contribution to each variables variation. As noted above, these results are useful to assess the time-varying integration effects of the global factor and the group factors. We are particularly interested in the relative importance of the group factors and possibly different patterns during the financial crisis periods.

3.4 Estimation Results

In the estimation of the dynamic factor model described in the previous section, we consider three distinct groups: Eurozone countries, non-Eurozone EU countries, and non- EU OECD countries. As noted above, we include non-EU OECD countries as a control group. We will refer to them as non-EU countries in this paper. This group can be compared to other groups. For comparison, we have also estimated the same models using the EU countries only, excluding non-EU OECD countries.

3.4.1 Estimated Factors

Figure 3.1(a) presents the estimated global and group factors for five variables: GDP growth rates, unemployment rates, employment rates, interest rates, and inflation rates. The estimated global factor of GDP growth reveals an intuitive fit. We can observe a deep decrease around the periods of the global financial crisis (GFC) and the sovereign debt

48 crisis in the Eurozone. Group factors are less intuitive and show less variation. Looking at the results in Figure 3.1(b), which are based on the EU data only, the Eurozone group factor shows more variation. It exhibits a big U-shaped drop between 2007 and 2015, reflecting the recessions caused by two crises. The global factor of unemployment rates shows a persistently declining trend until the start of the global financial crisis (Figure 3.1(a) and 3.1(b)), reflecting the long period of economic expansion and the great moderation. It increased sharply during the financial crisis. The trend of the non-Eurozone factor differs from that of the Eurozone, indicating different labor market conditions of the two groups. At first, it may seem that the Euro- zones labor market suffered more, as the increasing trend of the Eurozone factor may indicate. Still, as we will show later, the labor market is primarily under the influence of country-specific determinants rather than global or group factors. The results on employment rates show mirror images of those from using unemployment rates. The global factor of interest rates shows undisturbed decline indicating favorable global financial conditions throughout the sample period in Figure 3.1(a) and 3.1(b). On the other hand, the Eurozone factor shows a huge spike during the sovereign debt crisis between 2010 and 2012, as shown in Figure 3.1(a). It is in sharp contrast with the non-Eurozone and non-EU factors, which are fairly stable over time. It suggests that an increase in interest rates was primarily related to the Eurozones financial conditions, and it did not significantly affect other groups. The estimated global factor of the inflation rate shows a double-dip during the GFC and the sovereign debt crisis. It is also much more volatile compared to other variables.

3.4.2 Relative Contributions of the Estimated Factors

Next, we examine the importance of the estimated factors in explaining each major macro variables variation. We utilize the variance decomposition based on equation 4.3.3. The DFM allows us to evaluate time-varying relative contributions of each of the global, group, and idiosyncratic components to the variation of each variable in each country.

49 It is a unique and distinguishing feature of the DFM with time-varying parameters and stochastic volatilities. We summarize the results as average contributions of each factor in each group of countries to save space. A summary of these results is presented in Table 3.2(a) for each variable over four different sample periods. These results reveal interesting findings.1 First, surprisingly, the relative contributions of both Eurozone and non-Eurozone factors are quite low throughout different periods in almost all cases. Looking at the results using GDP growth rates, we see that the global factor is a dominant component. On average, the global factor accounts for 64 percent of the variation of GDP growth rates in all countries in the sample. The contributions of the Eurozone and non-Eurozone factors are almost negligible at only 4 percent. It is unexpected to observe that the group factors contribution is even higher in non-EU OECD countries (10 percent). The combined EU effects are minimal. When we exclude the non-EU OECD countries (Table 3.2(b), the results are qualitatively unchanged. Instead, the country-specific factor accounts for about 27 percent, as shown in Table 3.2(a) (or 15 percent in Table 3.2(b) for the case of EU countries only). Thus, business cycles are largely explained by global comovements. Country-specific shocks are less important than global ones. In short, we do not observe significant synchronization effects in the group factors of either Eurozone or non-Eurozone EU countries. The common movement effects are observed mainly through the global factor, and synchronizations through EU groups are not evident. Second, the results based on unemployment and employment rates also are similar.

1We present the plots of the relative contribution of each factor in Appendix Figure 3.4 for the case of GDP growth rates. Other plots are omitted there to save space, but they are available upon request. Looking at the relative contributions of each factor for GDP growth rates in Eurozone countries, there is significant heterogeneity across countries. While Austria, Belgium, and Germany have dominant global factor contribution (99% overall), Greece, Spain, and Latvia have relatively low global factor contribution, which implies substantial differences between the core and peripheral countries. This result is in line with the findings of decoupling within the EMU bond markets in Bhatt et al. (2017). Factor contribution for all variables using EU and OECD data shows that only unemployment rate decreased for about 5 years after the Euro system is adopted. There has been no change in contributions of the Eurozone factor for other variables, which is consistent with our idea that Eurozone has not played a role in business synchronization in Eurozone countries.

50 At the same time, the effects of the group factors in the Eurozone (20 percent) and non- Eurozone countries (18 percent) are slightly higher than those in non-EU countries (10 percent). Again, the synchronization effects in terms of the significance of the EU group factors are not high. However, more importantly, the country-specific factors explain a great share of the variation in employment and unemployment rates. These findings imply that labor markets are not highly integrated when compared to financial markets. Labor markets are still mainly driven by national determinants and do not show significant integration at the group or global level. Third, we obtain similar findings for the case of interest rates, which can imply financial integration. The same is true for the inflation rate, reflecting the impact of using a single currency for the Eurozone countries. The relative contributions of both Eurozone and non-Eurozone group factors for these variables are quite small. For example, the contribution of the Eurozone factor is a mere 6 percent. The results in Table 3.2 indicate no difference in the group factors effects in these two groups. The effects of country-specific factors are greater for inflation rates, while the global factor explains greater percentage of variation in interest rates. These results are comparable to those of Bhatt et al. (2017). They find a significant global factor of long-term sovereign bond yields before the GFC and substantial heterogeneity of the global and the group factors in the post-crisis period. Overall, our main question is whether the Eurozone formation plays a role in higher synchronization of variables in the countries that adopted the Euro. The answer from the above results is not really. There is also no obvious difference when we compare the Euro- zone factor to Eurozone countries and the non-Eurozone factor to non-Eurozone countries. This finding contrasts with the existing literature that attributes the formation of the Eu- rozone to business cycle synchronization in Eurozone countries. We now investigate the temporal behaviors of factor contributions by looking at sub- samples, as presented in the last three columns of Table 3.2(a) and (b). First, on average, the global factor contributes more during the GFC period in all variables except inter-

51 est rates. Regarding interest rates, the global factor contributes less during the four sub- sample periods. For Eurozone countries, the result is similar: the global factors contribution increases during the GFC period for all variables except interest rates. Second, even though the Eurozone factor contribution is quite insigniﬁcant for most variables, it takes further dips during the GFC for all variables except inﬂation.

3.4.3 Correlation Between Country-speciﬁc Variables with Global or Group Factors

Another way to look at the importance of global and group factors is to examine the correlations between an individual countrys variable and the global factor. In other words,

k k we calculate the correlation between ft and yi,t, where ft is the estimated factor with k denoting the global or group factor, and yi,t, represents each of five macro variables in the country i. This analysis aims to examine the comovements between the country-specific macroeconomic variables and the estimated factors. We present a summary of the results in Table 3.3. The overall results are qualitatively comparable with those from the variance decomposition analysis. The correlations between an individual countrys variable and each of the corresponding EU group factors are generally weak in almost all variables, except for the GDP growth rate. Correlations are especially lower in the labor market variables (employment and unemployment rates), showing that neither global nor group factors are significantly associated with these variables. That implies that domestic determinants still drive the labor market, and it does not show significant effects of global or group integration. Although financial market integration within the groups is more pronounced from the variance decomposition results on interest rates, this finding is less clear since the average correlation coefficient is just 0.184. For the inflation rates, the average correlation coefficient with the group factors is lower. Instead, each variable is more correlated with the corresponding global factor, while the correlation coefficients are not high. However, the correlation with the corresponding

52 global and group factors is higher during the GFC period. In summary, we do not find evidence that each of the five major economic variables in the Eurozone is significantly associated with the groups comovements. The synchronization effects in the EU countries are not significant, judging from the correlations with the group factors. Global and country- specific factors are more pronounced.

3.4.4 Synchronization Eﬀects based on Cross-correlations in Each Group

The synchronization effects can be more directly analyzed via cross-correlations.1 To begin with, we examine first if there are cross-correlation effects in the data set. We adopt the cross-section dependence (CD) tests of Pesaran et al. (2004), which can be applied to various panel data sets. The CD tests are based on average pairwise correlation coefficients. The results given in Table 3.4 show that the null hypothesis of no cross-correlation is rejected clearly in all variables. In the DFM, we assume that the variables are cross- correlated, and they can be captured by the estimated factors capturing the comovements of the variables. Thus, we can have more detailed information from the Bayesian estimation of the DFM on cross-correlations in each group. The main goal is to compare the cross-country correlation in different groups and find the potential role of Eurozone formation in affecting the time-varying cross-country correlation in Eurozone countries. As such, we have presented time-varying cross-country correlations of each variable in Figures 3.2(a)-(e). We report the results in each group as well as in all countries in the sample. We also have provided the 95 percent confidence interval obtained from the Bayesian itera- tion procedure. We can observe some pronounced patterns. The cross-correlations increased significantly during the GFC in all variables except for interest rates. This result reflects the

1Regarding the measure for the degree of business cycle synchronization, correlation-based measures are frequently adopted. De Haan et al. (2001) note that most papers rely on the cyclical component’s cross-correlations to measure business cycle synchronization. Other extensions of using lead and lag correlations, correlations with Germany and the US, rolling correlations, and recession probabilities also are considered. As a dynamic correlation measure of synchronization, spectral based coherence measures are new variations (Croux et al. (2001)). From the DFM, however, we can evaluate the average cross-country correlations among all countries and regions and cross-sectional dispersion in volatility.

53 events in the financial markets before and during the GFC. The Eurozone tried to maintain similar interest rates before the GFC, which might have induced some countries to build excessive debts, such as Greece, Italy, Spain, and Portugal. This is obvious from the very high cross-correlation among the Eurozone countries, close to unity in Figure 3.2(d). However, after the GFC and during the sovereign debt crisis, financial markets differen- tiated between secure and insecure European debt, which resulted in a dramatic surge in interest rates in indebted countries. Such a surge created many variations and decreased cross-correlations, which is again obvious from Figure 3.2(d). For other variables, we also observe a spike increase in the period of the GFC.1 The confidence interval is even nar- rower in this period. Another phenomenon we examine is the cross-country correlation before and after the formation of the Eurozone. Figure 3.3 presents the differences in cross-country correlation between Eurozone and non-Eurozone groups. If synchronization in the Eurozone is higher, then the difference between Eurozone and other groups should be positive and statistically significant. We compute 95% confidence intervals for the differences in cross-country correlations to measure statistical significance. We find no significant differences between Eu- rozone countries and non-Eurozone EU countries in all cases; see Figure 3.3. This implies that the synchronization effects in Eurozone countries and non-Eurozone EU countries are not different. Also, the cross-country correlations in the Eurozone countries are not different before and after the Euro adoption. Summary measures can be helpful, as shown in Table 3.5(a) and (b), which report the differences in cross-country correlations between the two groups during the full sample period and three sub-sample periods. They show clearly that the cross-correlations are not higher in Eurozone countries or EU countries when we compare them in various cases: between the Eurozone and non-Eurozone groups, between the Eurozone and non-EU OECD groups, between the non-Eurozone and non-EU OECD groups, and between all coun-

1In Appendix Figure 3.5, we present the results using the EU data only but excluding non-EU OECD countries. The main results are similar.

54 tries and each group. The differences between these pairs of groups are insignificant in all cases except for interest rates. The results are similar when we exclude the non-EU OECD group; see Table 3.5(b). In short, there is no difference in the synchronization effects mea- sured by cross-correlations. Cross-country correlations in Eurozone, non-Eurozone, and non-EU OECD countries are rather similar. In summary, we conclude from the average cross-country analysis that there is no significant difference in cross-country correlation in different groups. Therefore, our results from major economic variables do not support the notion that the Eurozones formation might have improved synchronization effects within the Eurozone. The Eurozone has not contributed significantly to developing synchronization effects in major economic variables.

3.5 Concluding Remarks

The formation of the Eurozone has triggered a debate on whether it satisfies the optimal currency area conditions. The debate is reopened after the sovereign debt crisis during 2010-2012 and more recently when two new Eurozone candidates, Bulgaria and Croatia, are considering adopting Euro as the currency. This paper analyzes the similarity and synchronization of economic conditions in a multidimensional way by analyzing global business cycles, labor market, financial markets, and monetary policy synchronization. We adopt the dynamic factor model with time-varying parameters and stochastic volatility to identify global, group, and country-specific factors, using five major variables: GDP growth, (un)employment rates, inflation, and interest rates. We find distinct and intuitive global factors of GDP growth, and to some extent, other variables as well. Global factors summarize the events between 1995 and 2018, highlight- ing the economic downturns during the global financial crisis (GFC) and the Eurozones sovereign debt crisis. While the global factor of GDP growth rates explains the economic events very well, the group factors suggest that interest rate increase during the 2010-2012 period was dominantly a local issue affecting only Eurozone countries. Although group

55 factors indicate different labor market conditions between the groups of Eurozone countries and others, they are not shown to be important. They explain just a fraction of variation in the five major macroeconomic variables we consider. On the other hand, global factors are more important in all variables except labor market indicators. Labor markets are still primarily driven by domestic determinants instead, and country-specific factors prove to be the most important. We have tested whether the differences in cross-correlations are significant between the three groups of countries: Eurozone, non-Eurozone EU member states, and non-EU OECD countries. We find significant results for the interest rates only, reflecting the financial markets reaction before and during the GFC. There is no clear evidence that the EU or Eurozone membership has increased synchronization within the group over time. Instead, we find that synchronization effects are time-dependent rather than group-dependent; they are more significant during the GFC periods in all groups. Our results cast doubt on using the so-called qualification criteria for the union membership. Also, to stabilize business cycles through countercyclical economic policies, poli- cymakers should aim in the right direction. In case that business cycles are mainly driven by global factors, national policy has a limited role unless accompanied by global economic reforms. On the other hand, macroeconomic policy coordination at the regional level may be required as a part of free trade agreements when business cycles are mainly affected by regional factors.

56 a. Global factor b. Eurozone factor c. non-Eurozone EU factor d. non-EU OECD factor

5 5 5 5

0 0 0 0

-5 -5 -5 -5

GDP Growth Rate Growth GDP -10 -10 -10 -10 2000 2005 2010 2015 2000 2005 2010 2015 2000 2005 2010 2015 2000 2005 2010 2015 e. Global factor f. Eurozone factor g. non-Eurozone EU factor h. non-EU OECD factor 60 60 60 60 40 40 40 40 20 20 20 20 0 0 0 0 -20 -20 -20 -20

Unemployment Rate Unemployment 1995 2000 2005 2010 2015 1995 2000 2005 2010 2015 1995 2000 2005 2010 2015 1995 2000 2005 2010 2015 i. Global factor j. Eurozone factor k. non-Eurozone EU factor l. non-EU OECD factor

20 20 20 20

0 0 0 0

-20 -20 -20 -20 Employment Rate Employment 2000 2005 2010 2015 2000 2005 2010 2015 2000 2005 2010 2015 2000 2005 2010 2015

m. Global factor n. Eurozone factor o. non-Eurozone EU factor p. non-EU OECD factor 10 10 10 10

5 5 5 5

0 0 0 0

Interest Rate Interest -5 -5 -5 -5 1995 2000 2005 2010 2015 1995 2000 2005 2010 2015 1995 2000 2005 2010 2015 1995 2000 2005 2010 2015

q. Global factor r. Eurozone factor s. non-Eurozone EU factor t. non-EU OECD factor 5 5 5 5

0 0 0 0

-5 -5 -5 -5 Inflation Rate Inflation -10 -10 -10 -10 1995 2000 2005 2010 2015 1995 2000 2005 2010 2015 1995 2000 2005 2010 2015 1995 2000 2005 2010 2015 (a) EU and OECD

Figure 3.1: Estimated National and Group Factors

57 a. Global factor b. Eurozone factor c. non-Eurozone EU factor

5 5 5

0 0 0

-5 -5 -5

GDP Growth Rate Growth GDP -10 -10 -10 2000 2005 2010 2015 2000 2005 2010 2015 2000 2005 2010 2015 d. Global factor e. Eurozone factor f. non-Eurozone EU factor 40 40 40

20 20 20

0 0 0

-20 -20 -20

-40 -40 -40 Unemployment Rate Unemployment 2000 2005 2010 2015 2000 2005 2010 2015 2000 2005 2010 2015 g. Global factor h. Eurozone factor i. non-Eurozone EU factor

20 20 20 10 10 10 0 0 0

-10 -10 -10 Employment Rate Employment 2000 2005 2010 2015 2000 2005 2010 2015 2000 2005 2010 2015

j. Global factor k. Eurozone factor l. non-Eurozone EU factor 10 10 10

0 0 0

Interest Rate Interest -10 -10 -10 1995 2000 2005 2010 2015 1995 2000 2005 2010 2015 1995 2000 2005 2010 2015

m. Global factor n. Eurozone factor o. non-Eurozone EU factor 5 5 5

0 0 0

-5 -5 -5 Inflation Rate Inflation -10 -10 -10 1995 2000 2005 2010 2015 1995 2000 2005 2010 2015 1995 2000 2005 2010 2015 (b) EU only

58 All countries Eurozone 1 1

0.9 0.9

0.8 0.8

0.7 0.7

0.6 0.6

0.5 0.5

0.4 0.4

0.3 0.3

0.2 0.2

0.1 0.1 1995 2000 2005 2010 2015 2020 1995 2000 2005 2010 2015 2020

EU non-Eurozone non-EU 1 1

0.9 0.9

0.8 0.8

0.7 0.7

0.6 0.6

0.5 0.5

0.4 0.4

0.3 0.3

0.2 0.2

0.1 0.1 1995 2000 2005 2010 2015 2020 1995 2000 2005 2010 2015 2020 (a) GDP Growth Rates All countries Eurozone 0.8 0.8

0.7 0.7

0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1

0.1 0

0 -0.1 1990 1995 2000 2005 2010 2015 2020 1990 1995 2000 2005 2010 2015 2020

EU non-Eurozone non-EU 1 1

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0 0

-0.2 -0.2 1990 1995 2000 2005 2010 2015 2020 1990 1995 2000 2005 2010 2015 2020 (b) Unemployment Rates

Figure 3.2: Cross-country correlation between groups (EU and OECD)

59 All countries Eurozone 0.6 0.6

0.5 0.5

0.4 0.4 0.3 0.3 0.2 0.2 0.1

0.1 0

0 -0.1 1995 2000 2005 2010 2015 2020 1995 2000 2005 2010 2015 2020

EU non-Eurozone non-EU 0.6 0.8

0.7 0.5 0.6 0.4 0.5

0.3 0.4

0.2 0.3 0.2 0.1 0.1 0 0

-0.1 -0.1 1995 2000 2005 2010 2015 2020 1995 2000 2005 2010 2015 2020 (c) Employment Rates All countries Eurozone 1 1

0.9 0.9

0.8 0.8

0.7 0.7

0.6 0.6

0.5 0.5

0.4 0.4 1990 1995 2000 2005 2010 2015 2020 1990 1995 2000 2005 2010 2015 2020

EU non-Eurozone non-EU 1 0.95

0.9

0.9 0.85

0.8 0.8 0.75

0.7 0.7 0.65

0.6 0.6 0.55

0.5 0.5 1990 1995 2000 2005 2010 2015 2020 1990 1995 2000 2005 2010 2015 2020 (d) Interest Rates

60 All countries Eurozone 0.5 0.8

0.45 0.7

0.4 0.6 0.35 0.5 0.3 0.4 0.25 0.3 0.2 0.2 0.15

0.1 0.1

0.05 0 1990 1995 2000 2005 2010 2015 2020 1990 1995 2000 2005 2010 2015 2020

EU non-Eurozone non-EU 0.7 0.35

0.6 0.3

0.5 0.25

0.4 0.2

0.3 0.15

0.2 0.1

0.1 0.05

0 0 1990 1995 2000 2005 2010 2015 2020 1990 1995 2000 2005 2010 2015 2020 (e) Inﬂation Rates

61 GDP Growth Rates Unemployment Rates 1 0.4

0.2 0.5 0

-0.2 0 -0.4

-0.6 -0.5 2000 2005 2010 2015 1995 2000 2005 2010 2015

Employment Rates Interest Rates 0.5 0.6

0.4

0 0.2

-0.5 -0.2 2000 2005 2010 2015 1995 2000 2005 2010 2015

Inflation Rates 0.4

0.2

-0.2

-0.4

-0.6 1995 2000 2005 2010 2015 (a) EU and OECD

Figure 3.3: Statistical Diﬀerences in Average Cross-country Correlation Between Eurozone and EU non-Eurozone Regions

62 GDP Growth Rates Unemployment Rates 0.5 0.6

0.4 0 0.2

0 -0.5 -0.2

-1 -0.4 2000 2005 2010 2015 2000 2005 2010 2015

Employment Rates Interest Rates 0.6 0.6

0.4 0.4

0.2 0.2

0 0

-0.2 -0.2

-0.4 -0.4 2000 2005 2010 2015 1995 2000 2005 2010 2015

Inflation Rates 0.2

-0.2

-0.4

-0.6 1995 2000 2005 2010 2015 (b) EU only

Figure 3.3: Statistical Diﬀerences in Average Cross-country Correlation between Eurozone and EU non-Eurozone Regions (cont.)

63 Table 3.1: List of Countries in Each Group by Variables and Descriptive Statistics

GDP Growth Unemployment Rate Employment Rate Interest Rate Inﬂation Rate Dates 1996:1-2018:4 1993:2-2019:4 1997:2-2019:3 1992:4-2019:4 1992:1-2019:4 Mean 2.79 7.62 66.06 4.65 3.80 Max 29.16 27.83 78.3 25.4 78.40 Min -20.78 1.9 48.5 -.78 -6.13 Std 3.72 4.15 6.50 3.26 5.79 Count 43 27 33 20 37 Groups Euro Non-Euro Non-EU Euro Non-Euro Non-EU Euro Non-Euro Non-EU Euro Non-Euro Non-EU Euro Non-Euro Non-EU Coutnries AUT DNK AUS AUT CZE AUS AUT DNK AUS AUT CZE AUS AUT CZE AUS BEL SWE CAN BEL DNK USA BEL GBR CAN BEL DNK CAN BEL DNK CAN DEU GBR SRB DEU HUN CAN DEU CHE CYP HUN USA DEU POL CHL ESP CZE ZAF ESP POL CHE FRA ZAF DEU POL ISR ESP SWE USA FIN POL IDN FIN SWE CHL GRC JPN ESP ROU JPN FIN GBR JPN FRA HUN ISL FRA GBR CHN IRL NOR EST SWE NOR FRA KOR GRC BUL JPN GRC COL ITA NZL FIN GBR NZL GRC MEX IRL HRV KOR IRL CRC LUX USA FRA IRL NOR ITA ROU MEX ITA ZAF NLD GRC ITA NZL LUX NOR SVK IDN PRT IRL LUX NLD NZL LUX IND ITA NLD PRT TUR PRT ISL LTU PRT SVK USA NLD ISR LUX SVK LTU CHE JPN LVA LVA CRC KOR MLT SVN ARG MEX LND CYP NOR PRT EST NZL SVK SVN

64 Table 3.2: Average of Factor Contributions in Each group for Each Variable

(a) EU and OECD

Full Sample Before GFC During GFC After GFC GDP Growth Rates Global Group Country Global Group Country Global Group Country Global Group Country Average 64% 4% 27% 61% 4% 30% 82% 3% 14% 61% 5% 28% Eurozone 71% 2% 23% 71% 2% 24% 86% 0% 12% 67% 3% 25% non-Eurozone 70% 1% 26% 63% 1% 32% 85% 1% 11% 72% 1% 23% non-EU 51% 10% 32% 50% 10% 34% 73% 6% 18% 45% 12% 35% Unemployment Rates Global Group Country Global Group Country Global Group Country Global Group Country Average 32% 16% 41% 31% 17% 41% 51% 13% 30% 29% 15% 45% Eurozone 24% 20% 44% 20% 19% 47% 41% 12% 39% 25% 22% 42% non-Eurozone 40% 18% 34% 38% 20% 34% 56% 21% 20% 37% 14% 38% non-EU 34% 10% 47% 35% 13% 43% 55% 5% 33% 25% 9% 55% Employment Rates Global Group Country Global Group Country Global Group Country Global Group Country Average 28% 10% 54% 25% 10% 57% 37% 7% 49% 27% 11% 54% Eurozone 29% 10% 53% 27% 10% 54% 38% 8% 48% 27% 11% 53% non-Eurozone 28% 10% 54% 22% 9% 60% 34% 7% 52% 31% 13% 49% non-EU 27% 10% 57% 26% 10% 56% 40% 7% 47% 24% 10% 60% Interest Rates Global Group Country Global Group Country Global Group Country Global Group Country Average 83% 5% 10% 87% 3% 9% 82% 7% 10% 77% 7% 12% Eurozone 82% 7% 9% 90% 2% 6% 81% 10% 7% 69% 12% 14% non-Eurozone 92% 2% 5% 95% 1% 4% 92% 2% 5% 89% 4% 6% non-EU 74% 7% 17% 74% 6% 18% 74% 8% 17% 74% 7% 17% Inﬂation Rates Global Group Country Global Group Country Global Group Country Global Group Country Average 32% 11% 52% 28% 10% 57% 42% 10% 44% 35% 12% 48% Eurozone 42% 11% 42% 37% 9% 48% 54% 10% 32% 46% 13% 36% non-Eurozone 33% 11% 52% 28% 12% 56% 45% 8% 43% 37% 11% 48% non-EU 21% 11% 64% 18% 10% 68% 26% 12% 57% 23% 12% 60% Note: Before GFC: 1991 Q1 - 2006 Q4, During GFC: 2007 Q1 - 2009 Q4; After GFC: 2010 Q1 - 2018Q4. Non-Euro includes EU countries that do not belong to the Eurozone.

65 Table 3.2: Average of Factor Contributions in Each group for Each Variable (cont.)

(b) EU only

Full Sample Before GFC During GFC After GFC GDP Growth Rates Global Group Country Global Group Country Global Group Country Global Group Country Average 78% 3% 15% 72% 3% 20% 93% 1% 5% 81% 3% 12% Eurozone 80% 5% 11% 77% 6% 12% 94% 1% 4% 78% 6% 12% non-Eurozone 77% 1% 19% 67% 1% 28% 91% 0% 6% 84% 1% 12% Unemployment Rates Global Group Country Global Group Country Global Group Country Global Group Country Average 38% 18% 34% 34% 20% 34% 33% 17% 38% 43% 17% 31% Eurozone 30% 17% 40% 25% 19% 42% 28% 15% 44% 36% 17% 37% non-Eurozone 46% 19% 27% 43% 21% 27% 39% 20% 33% 51% 17% 25% Employment Rates Global Group Country Global Group Country Global Group Country Global Group Country Average 24% 9% 60% 21% 9% 63% 28% 7% 59% 26% 10% 57% Eurozone 23% 8% 61% 22% 9% 62% 29% 7% 58% 23% 9% 60% non-Eurozone 25% 9% 59% 19% 9% 64% 27% 7% 60% 29% 10% 54% Interest Rates Global Group Country Global Group Country Global Group Country Global Group Country Average 80% 5% 13% 86% 3% 10% 80% 6% 12% 72% 8% 18% Eurozone 79% 6% 12% 87% 3% 8% 80% 9% 9% 68% 10% 17% non-Eurozone 81% 3% 15% 85% 2% 12% 80% 4% 15% 76% 5% 18% Inﬂation Rates Global Group Country Global Group Country Global Group Country Global Group Country Average 44% 10% 42% 37% 11% 48% 53% 10% 35% 50% 9% 37% Eurozone 49% 6% 41% 42% 6% 47% 60% 6% 31% 56% 5% 35% non-Eurozone 38% 15% 43% 32% 16% 48% 46% 13% 38% 44% 13% 38% Note: Before GFC: 1991 Q1 - 2006 Q4, During GFC: 2007 Q1 - 2009 Q4; After GFC: 2010 Q1 - 2018Q4. Non-Euro includes EU countries that do not belong to the Eurozone.

66 Table 3.3: Average Correlation Coeﬃcients Between Country-speciﬁc Variables with Global or Group Factors

Variable Group Full Sample Before GFC During GFC After GFC GDP Global 0.57 0.45 0.643* 0.629 Eurozone 0.381 0.385 0.547* 0.323 Non-Eurozone 0.081 -0.156 0.107 0.238* Non-EU 0.064 -0.002 0.016 0.126* Inﬂation Global 0.1 0.067 0.213* 0.099 Eurozone 0.05 0.0407 0.230* 0.005 Non-Eurozone -0.092 -0.144 0.024* -0.074 Non-EU 0.165 0.078 0.326* 0.203 Interest rate Global 0.184 0.218 0.251* 0.133 Eurozone -0.015 -0.035 0.117* -0.035 Non-Eurozone 0.028 0.02 0.262* -0.034 Non-EU -0.004 -0.129 0.004 0.110* Employment rate Global 0.128 0.083 0.157* 0.141 Eurozone 0.015 0.089 -0.472 0.129* Non-Eurozone 0.082 -0.032 0.345* 0.057 Non-EU -0.01 -0.118 0.126* 0.001 Unemployment rate Global -0.035 -0.004 0.158* -0.12 Eurozone -0.068 -0.071 -0.419 0.040* Non-Eurozone -0.238 0.095* -0.48 -0.456 Non-EU -0.107 -0.011 -0.001* -0.223 Note: * denotes the case where the maximum correlation occurs in each variable. Be- fore GFC: 1991 Q1 - 2006 Q4, During GFC: 2007 Q1 - 2009 Q4; After GFC: 2010 Q1 - 2018Q4. Non-Euro includes EU countries that do not belong to the Eurozone.

Table 3.4: Pesarans CD Test Results

Variable GDP Growth rates Employment Rates Unemployment Rates Inﬂation Rates Interest Rates CD stat 63.89 28.45 25.75 51.56 80.66 p-value 0.000 0.000 0.000 0.000 0.000 Note: The null hypothesis assumes no cross-sectional correlation.

67 Table 3.5: Average Cross-Country Correlations and Diﬀerences in Two Regions

(a) EU and OECD GDP Growth Rates Correlation Full Sample Before GFC During GFC After GFC All countries 0.562 0.529 0.76 0.537 Eurozone 0.654 0.64 0.82 0.615 non-Eurozone 0.617 0.531 0.803 0.66 non-EU 0.477 0.439 0.693 0.452 Difference Eurozone - non-Eurozone 0.037 0.109 0.016 -0.044 All countries - Eurozone -0.092 -0.111 -0.06 -0.078 All countries - non-Eurozone -0.055 -0.002 -0.043 -0.123 All countries - non-EU 0.085 0.091 0.067 0.085 Eurozone - non-EU 0.177 0.202 0.126 0.164 non-Eurozone - non-EU 0.14 0.093 0.11 0.208 Unemployment Rates Correlation Full Sample Before GFC During GFC After GFC All countries 0.169 0.121 0.386 0.169 Eurozone 0.18 0.114 0.317 0.23 non-Eurozone 0.277 0.169 0.549 0.343 non-EU 0.157 0.132 0.456 0.1 Difference Eurozone - non-Eurozone 0.097 0.055 0.232 0.113 All countries - Eurozone -0.011 0.007 0.07 -0.06 All countries - non-Eurozone -0.108 -0.048 -0.162 -0.173 All countries - non-EU 0.012 -0.011 -0.07 0.069 Eurozone - non-EU -0.023 0.0184 0.139 -0.13 non-Eurozone - non-EU -0.121 -0.037 -0.092 -0.242 Employment Rates Correlation Full Sample Before GFC During GFC After GFC All countries 0.164 0.107 0.272 0.186 Eurozone 0.169 0.123 0.261 0.184 non-Eurozone 0.162 0.033 0.235 0.266 non-EU 0.153 0.116 0.328 0.136 Note: Before GFC: 1991 Q1 - 2006 Q4, During GFC: 2007 Q1 - 2009 Q4; After GFC: 2010 Q1 - 2018Q4. Non-Euro includes EU countries that do not belong to the Eurozone, while non-EU includes OECD countries that do not belong to the EU. 68 Table 3.5: Average Cross-Country Correlations and Differences in Two Regions (cont.)

(b) EU and OECD Difference Eurozone - non-Eurozone -0.006 -0.09 -0.026 0.082 All countries - Eurozone -0.005 -0.016 0.011 0.002 All countries - non-Eurozone 0.002 0.074 0.037 -0.08 All countries - non-EU 0.011 -0.009 -0.056 0.05 Eurozone - non-EU -0.015 -0.007 0.067 -0.048 non-Eurozone - non-EU -0.009 0.083 0.093 -0.13 Interest Rates Correlation Full Sample Before GFC During GFC After GFC All countries 0.771 0.822 0.776 0.699 Eurozone 0.813 0.888 0.83 0.703 non-Eurozone 0.929 0.952 0.934 0.895 non-EU 0.753 0.756 0.758 0.747 Difference Eurozone - non-Eurozone 0.116 0.064 0.104 0.192 All countries - Eurozone -0.042 -0.066 -0.054 -0.004 All countries - non-Eurozone -0.157 -0.13 -0.158 -0.196 All countries - non-EU 0.019 0.066 0.017 -0.048 Eurozone - non-EU -0.06 -0.132 -0.071 0.044 non-Eurozone - non-EU -0.176 -0.196 -0.175 -0.148 Inflation Rates Correlation Full Sample Before GFC During GFC After GFC All countries 0.225 0.177 0.295 0.274 Eurozone 0.414 0.32 0.55 0.511 non-Eurozone 0.281 0.224 0.404 0.328 non-EU 0.148 0.112 0.175 0.193 Correlation Full Sample Before GFC During GFC After GFC Eurozone - non-Eurozone -0.132 -0.096 -0.145 -0.183 All countries - Eurozone -0.189 -0.143 -0.255 -0.236 All countries - non-Eurozone -0.056 -0.047 -0.109 -0.054 All countries - non-EU 0.076 0.064 0.12 0.081 Eurozone - non-EU -0.265 -0.207 -0.374 -0.317 non-Eurozone - non-EU -0.133 -0.112 -0.229 -0.135 Note: Before GFC: 1991 Q1 - 2006 Q4, During GFC: 2007 Q1 - 2009 Q4; After GFC: 2010 Q1 - 2018Q4. Non-Euro includes EU countries that do not belong to the Eurozone, while69 non-EU includes OECD countries that do not belong to the EU. Table 3.5: Average Cross-Country Correlations and Differences in Two Regions (cont.)

(c) EU only GDP Growth Rates Correlation Full Sample Before GFC During GFC After GFC All countries 0.721 0.664 0.898 0.731 Eurozone 0.74 0.722 0.913 0.704 non-Eurozone 0.696 0.568 0.88 0.79 Difference non-Eurozone - Eurozone 0.044 0.154 0.034 -0.086 All countries - Eurozone -0.019 -0.058 -0.016 0.027 All countries - non-Eurozone 0.025 0.096 0.018 -0.059 Unemployment Rates Correlation Full Sample Before GFC During GFC After GFC All countries 0.182 0.097 0.183 0.265 Eurozone 0.158 0.082 0.171 0.229 non-Eurozone 0.26 0.129 0.176 0.414 Difference Full Sample Before GFC During GFC After GFC Eurozone - non-Eurozone -0.102 -0.048 -0.005 -0.185 All countries - Eurozone 0.024 0.016 0.011 0.036 All countries - non-Eurozone -0.078 -0.032 0.007 -0.149 Employment Rates Correlation Full Sample Before GFC During GFC After GFC All countries 0.12 0.046 0.191 0.169 Eurozone 0.117 0.057 0.194 0.153 non-Eurozone 0.136 0.003 0.186 0.252 Difference Eurozone - non-Eurozone -0.019 0.054 0.008 -0.099 All countries - Eurozone 0.002 -0.01 -0.003 0.017 All countries - non-Eurozone -0.017 0.044 0.005 -0.082 Interest Rates Correlation Full Sample Before GFC During GFC After GFC All countries 0.766 0.838 0.782 0.661 Eurozone 0.768 0.84 0.79 0.66 EU non-Eurozone 0.815 0.855 0.82 0.757 Correlation Full Sample Before GFC During GFC After GFC Eurozone - non-Eurozone -0.047 -0.015 -0.03 -0.097 All countries - Eurozone -0.002 -0.002 -0.007 0.001 EU non-Eurozone - Eurozone 0.047 0.015 0.03 -0.093 Inflation Rates Correlation Full Sample Before GFC During GFC After GFC All countries 0.394 0.315 0.509 0.475 EU non-Eurozone 0.308 0.248 0.393 0.372 Eurozone 0.429 0.338 0.559 0.525 Correlation Full Sample Before GFC During GFC After GFC All countries - EU non-Eurozone 0.085 0.067 0.116 0.103 All countries - Eurozone -0.036 -0.023 -0.051 -0.05 EU non-Eurozone - Eurozone -0.121 -0.09 -0.166 -0.153 Note: Before GFC: 1991 Q1 - 2006 Q4, During GFC: 2007 Q1 - 2009 Q4; After GFC: 2010 Q1 - 2018Q4. Non-Euro includes EU countries that do not belong to the Eurozone, while non-EU includes OECD countries that do not belong to the EU. 70 3.6 Appendix

Austria Belgium Cyprus 1 1 1

0.5 0.5 0.5

0 0 0 2000 2010 2020 2000 2010 2020 2000 2010 2020 Germany Spain Estonia 1 1 1

0.5 0.5 0.5

0 0 0 2000 2010 2020 2000 2010 2020 2000 2010 2020 Finland France Greece 1 1 1

0.5 0.5 0.5

0 0 0 2000 2010 2020 2000 2010 2020 2000 2010 2020 Ireland Italy Lithuania 1 1 1

0.5 0.5 0.5

0 0 0 2000 2010 2020 2000 2010 2020 2000 2010 2020 Luxembourg Latvia Netherlands 1 1 1

0.5 0.5 0.5

0 0 0 2000 2010 2020 2000 2010 2020 2000 2010 2020 Portugal Slovakia Slovenia 1 1 1

0.5 0.5 0.5

0 0 0 2000 2010 2020 2000 2010 2020 2000 2010 2020

(a) Eurozone

Figure 3.4: Variance Decomposition of Each Country in Groups (GDP Growth Rates)

71 Bulgaria Czechia Denmark 1 1 1

0.5 0.5 0.5

0 0 0 2000 2010 2020 2000 2010 2020 2000 2010 2020 United Kingdom Croatia Hungary 1 1 1

0.5 0.5 0.5

0 0 0 2000 2010 2020 2000 2010 2020 2000 2010 2020 Poland Romania Sweden 1 1 1

0.5 0.5 0.5

0 (b)0 non-Eurozone EU 0 2000 2010 2020 2000 2010 2020 2000 2010 2020

72 Argentina Australia Canada 1 1 1

0.5 0.5 0.5

0 0 0 2000 2010 2020 2000 2010 2020 2000 2010 2020 Switzerland Costa Rica Indonesia 1 1 1

0.5 0.5 0.5

0 0 0 2000 2010 2020 2000 2010 2020 2000 2010 2020 Iceland Japan South Korea 1 1 1

0.5 0.5 0.5

0 0 0 2000 2010 2020 2000 2010 2020 2000 2010 2020 Mexico Norway New Zealand 1 1 1

0.5 0.5 0.5

0 0 0 2000 2010 2020 2000 2010 2020 2000 2010 2020 Serbia Turkey United States 1 1 1

0.5 0.5 0.5

0 0 0 2000 2010 2020 2000 2010 2020 2000 2010 2020 South Africa 1

0.5

0 2000 2010 2020

73 All countries Eurozone EU non-Eurozone 1 1 1

0.8 0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.4 0.2

0.2 0.2 0 2000 2005 2010 2015 2020 2000 2005 2010 2015 2020 2000 2005 2010 2015 2020 (a) GDP Growth Rates All countries Eurozone non-Eurozone 0.6 0.6 1

0.8 0.4 0.4 0.6

0.4 0.2 0.2 0.2

0 0 0 2000 2005 2010 2015 2020 2000 2005 2010 2015 2020 2000 2005 2010 2015 2020 (b) Unemployment Rates All countries Eurozone non-Eurozone 0.4 0.4 0.6

0.3 0.3 0.4

0.2 0.2 0.2

0.1 0.1 0

0 0 -0.2 2000 2005 2010 2015 2020 2000 2005 2010 2015 2020 2000 2005 2010 2015 2020 (c) Employment Rates All countries Eurozone EU non-Eurozone 1 1 1

0.8 0.8 0.8

0.6 0.6 0.6

0.4 0.4 0.4 1990 2000 2010 2020 1990 2000 2010 2020 1990 2000 2010 2020 (d) Interest Rates All countries Eurozone EU non-Eurozone 0.8 0.8 0.6

0.6 0.6 0.4 0.4 0.4

0.2 0.2 0.2

0 0 0 1990 2000 2010 2020 1990 2000 2010 2020 1990 2000 2010 2020 (e) Inﬂation Rates

Figure 3.5: Cross-country correlation between groups (EU only)

74 CHAPTER 4 COMOVEMENTS IN U.S. HOUSING PRICES

4.1 Introduction

Attempts to empirically examine the ripple effect and long-run convergence of regional or national U.S. housing prices have yielded mixed results.1 Studies differing in selection of methodology, model specification, data appropriation yield contradictary results. The historical volatility and cyclical nature of the U.S. housing market and cross-state hetero- geneity2 limit the efficacy of traditional empirical mechanisms to analyze the comovements of housing prices. The motivation of this study is to test for and examine comovements of state and MSA housing prices with factors exogenous to the local environment and economic conditions accounting for time-varying factor loadings and stochastic volatilities common in housing price index data. For this, we employ a dynamic factor models with time-varying factor loadings and stochastic volatilities (DFM-TV-SV), a first in real estate studies. Dynamic factor models (DFM) have been previously employed to analyze cycles and volatilities in housing prices. These models have the advantage of decomposing dynamics of observed variables into factors that are unobservable. Stock and Watson (2008) use a dynamic factor model with stochastic volatility (DFM-SV) to analyze US housing construction. In their analysis, they use national and regional factors to explain housing con-

1The transmission of shocks across regional housing markets, known as the ripple eﬀect, will result in housing prices moving together in the long-run (Meen (1999)). 2Since January 2012 (post-housing recession), state housing prices have increased by more than 100% in some states, such as Washington, Arizona, Nevada, and Florida according to the Housing Price In- dex (HPI); however, in some states, such as West Virginia, Alaska, Connecticut, and Mississippi, housing prices have increased by less than 25%.

75 struction activity. Fu et al. (2007) use a similar approach of dynamic factor model with stochastic volatility (DFM-SV) to analyze the US housing market of major U.S. metropolitan areas and confirm the presence of national and regional factors. Del Negro and Otrok (2007) also use a dynamic factor model with stochastic volatility to disentangle the housing prices into common, region-specific, and state components. Their analysis finds the local component drives housing price movements historically, but the increase in the housing prices prior to the financial crisis is a national phenomenon. However, these methodologies have difficulty dealing with time variation, as they do not allow for time-varying factor loadings. The time-varying factor loadings and stochastic volatility features can enrich the dynamic factor model structures.1 For example, Gupta et al. (2018) employ the DFM-TV-SV to show that economic policy uncertainty (EPU) increases common volatility of the U.S. states leading to a significant negative effect on the common economic activity. Del Ne- gro and Otrok (2008) employ DFM-TV-SV models to study the evolution of international business cycles. Scaffolding on earlier real estate studies examining the contribution of national and regional factors on state- and MSA-level housing market prices and volatility, this empirical study extends the literature on the ripple effect for US housing prices. We adopt the DFM-TV-SV to account for time-varying factor loadings and stochastic volatilities.2 This approach can be more effective in terms of model construction and more flexible, allowing stochastic volatility and time-varying factor loadings to better explain our data series. We decompose the observed variation in state- and MSA-level housing prices into latent components: a national factor, a regional factor, and idiosyncratic factors. We find

1For example, McConnell and Perez-Quiros (2000) show that there is a decline in output volatility in the US since the mid 1980s. Crucini et al. (2011) use dynamic factor models to analyze business cycles for G-7 countries. Cogley and Sargent (2005) use a DFM to show that the dynamics of inflation has evolved in the post war period. 2We use the Bayesian Markov Chain Monte Carlo estimation procedure and find that the national factor has been fairly stable over most of the sample period; but it became much more volatile during the financial crisis period. The regional factors have generally been more volatile, particularly in the 1980s and during the Global Financial Crisis (GFC).

76 that the national factor is generally the dominant determinant aﬀecting house price movements at both the state (79%) and MSA (85%) levels. Although subdued relative to the national factor, the regional factor also accounts for some variation, 7% for both the state and MSA housing prices. The idiosyncratic component contribution was only 11% at the state-level and 7% at the MSA-level as the observations become more aggregated. These results imply that the national factor plays an important role in explaining the changes in state- and MSA-level housing prices, particularly during times of increased national real estate volatility. The structure of the paper is as follows. Section 2 presents the literature review. Sec- tion 3 describes the data and methodology. Section 4 includes a discussion of the results from the DFM-TV-SV estimation. Section 5 concludes.

4.2 Literature Review

There has been an abundance of literature analyzing the U.S. housing prices. For example, Nneji et al. (2013) examine the impact of macro-economic changes on the dynamics of the housing market. Del Negro and Otrok (2007) show that expansionary monetary policy may be responsible for a potential housing bubble in the U.S. Balagyozyan et al. (2016) investigate housing price cycles and business cycle movement in the U.S. using a vector Markov switching model. They find mixed evidence of a connection between housing cycles and business cycle movements. Similarly, many studies examine the co-movements or convergence in housing prices in different states and regions in the U.S. For example, Holmes et al. (2011) investigate the convergence in U.S. state housing prices by using pairwise probabilities test statistics for convergence and found results supportive of a long run convergence, while speed of adjustment towards the long-run equilibrium is inversely related to distance distance between states. Payne (2012) tests the ripple effect and the long-run convergence of housing prices in 9 U.S. regions using an Autoregressive-Distributed Lag apporach, and finds evidence in

77 the the cointegration relations across the regional housing markets. However, the authors acknowledge that there is variation in the degree of the ripple effects and speed of adjustment towards the long-run equilibrium in different regions. Zohrabyan et al. (2008) find mixed evidence of regional housing prices cointegration relations using Johansen’s ML procedure and symmetric error-correction model techniques. They find the presence of four cointegration relations in four regions (East South Central, West South Central, South Atlantic, New England) out of nine. In terms of cluster convergence, Apergis and Payne (2012) use a club convergence and clustering procedure of Phillips and Sul (2007) and find 3 convergence clubs concentrated in different BEA regions in the U.S. Canarella et al. (2012) use unit root tests accomodating structural changes to provide implications on the ripple effects of U.S. housing prices and find partial evidence of the ripple effect. Specifically, they find that housing prices in the U.S. are influenced by the east and west coast metropolitan areas. Barros et al. (2014) use fractional integration and autoregressive models accomodating persistence and seasonality to examine the degree of persistence in the ratio of state housing prices to U.S. housing price indices and find mixed evidence on the degree of convergence in housing prices across the U.S. Holly et al. (2010) account for unobserved common factors using common correlated estimators before their second procedure of measuring the driving forces of variation between state level housing prices. Their results show evidence of a comovment component in state-level housing prices while examining the heterogeneity across states. However, Barros et al. (2012) found neither cointegration between U.S. state housing prices and the overall U.S. housing prices nor cointegration relations among 8 state housing prices that display unit root, using a similar method to Barros et al. (2014), but without incorporating persistence and seasonality. Also, Nissan and Payne (2013) use a σ- convergence test on the U.S. housing prices at the regional and state levels and find that the majority of 41 states fail to converge to the overall U.S. housing prices, and that four regions (Great Lakes, Plains, Southeast, Rocky Mountains, and Southwest) out of eight di-

78 verge from the U.S. overall housing prices. They conclude that housing prices are highly persistent and housing prices across states are influenced more by state-specific economic and demographic factors than national shocks. Regarding the heterogeneity effects in U.S. state housing prices, there is literature investigating the driving forces behind the differences in different state and regional housing markets. For example, in Holmes et al. (2011), the authors argue that the speed of adjustment towards the housing prices long-run equilibrium in different states is inversely related to the distance between states. That is, the speed of housing price convergence de- clines with distance. Holly et al. (2010) examine the extent to which real house prices at the state level are driven by real per capita disposable income. They identify a significant negative effect for a net borrowing cost variable and a significant positive effect for the state level popupation growth on changes in real housing prices. Specifically, they conclude that real housing prices rise with state level disposable income, implying no national housing bubbles and that real house prices are higher in states with a higher rate of population growth. Various frameworks of the dynamic factor models have a long history to be applied to analysis the co-movements in various time series variables. For example, McConnell and Perez-Quiros (2000) show that there is a decline in output volatility in the U.S. since the mid 1980s. Crucini et al. (2011) use dynamic factor models to analyze business cycles for G-7 countries. Cogley and Sargent (2005) use a DFM to show that the dynamics of inflation has evolved in the post-war period. Since the assumption of structural stability does not stand for most macroeconomic variables, models with fixed parameters might not be useful to analyze macroeconomic data. For example, Del Negro and Otrok (2008) measure the co-movement of global business cycles using the dynamic factor model that allows time-varying factor loadings and stochastic volatility. Indeed, many macroeconomic datasets have been proven to have stochastic time-series behaviors. A dynamic factor model that allows time varying factor loadings and stochastic volatility is believed

79 to be a good tool to use. Dynamic factor models (DFM) have also been previously employed to analyze cycles and volatilities in housing prices. For example, Stock and Watson (2008) use a dynamic factor model with stochastic volatility (DFM-SV) to analyze U.S. housing construction. In their analysis, they use national and regional factors to explain housing construction activity. Fu et al. (2007) use a similar approach of dynamic factor model with stochastic volatility (DFM-SV) to analyze the US housing market of major U.S. metropolitan areas and confirm the presence of national and regional factors. Del Ne- gro and Otrok (2007) also use a dynamic factor model with stochastic volatility to disentangle the housing prices into common, region-specific, and state components. However, each of these attempts either is not applied on state-level housing prices or did not employ the most flexible dynamic factor models that allow for time-varying factor loadings or stochastic volatility. To summarize the literature review, there is mixed evidence in terms of comovements in state level U.S. housing prices. Each of the streams of literature examining comovements in the housing prices have contradicting results based on methodology selection, model specification, and data appropriation choices. The motivation of this paper is to recognize the presence of comovements of state housing prices caused by external factors that are exogenous to local environment and economic conditions of each state and examine the degree of these comovements on state housing prices. Then, we examine the factors that are related to the local conditions of state housing markets. The use of dynamic factor models allows me to decompose the observed variation in housing prices for U.S. states into latent components: a national factor, a regional factor (divided geographically), and an idiosyncratic state-specific factor. This is an effective way to examine the comovements in the housing prices while accounting for the heterogeneity across states.

80 4.3 Data and the DFM

We use the seasonally adjusted House Price Index (HPI) of Freddie Mac for U.S. state- level data, from January 1975 to March 2020. The HPI is constructed using transactions on single-family detached and townhome properties. We follow the deﬁnitions of regions from the Census Bureau and consider eight diﬀerent regions. The descriptive statistics of our data in each region are provided in Table 4.1. This paper employs the dynamic factor model (DFM) with time-varying factor loadings and stochastic volatility of Del Negro and Otrok (2008). In the DFM-TV-SV framework, change of housing price in all states is decomposed into three components: common (national) factor, regional factor, and the idiosyncratic term:

n r yi,t = λi,t · ft + γi,t · ft + i,t (4.3.1)

n r where yi,t indicates the change prices of each state i at time t, ft and ft represents the national and regional factors, respectively, and λi,t and γi,t are the loadings to the national

r factor and the regional factor. ft is a row vector that contains a non-zero value in the po- sition corresponding to the region that state i belongs to, and zeros elsewhere. These regional factors capture the housing prices in eight geographic regions: New England, South- east, Mideast, Great Lakes, Plains, Rocky Mountains, Southwest, and Far West. The idiosyncratic term i,t includes state-speciﬁc shocks and random errors. Factor loadings measure the responses of the housing price of each state to the corresponding factors. We assume that they follow a random walk process without drift:

λi,t = λi,t−1 + ηi,t (4.3.2)

γi,t = γi,t−1 + ηi,t

where ηi,t ∼ i.i.d.N(0, 1) and is independent across i. The national and regional factors are assumed to be orthogonal to each other. They

81 are also assumed to be uncorrelated with the state-speciﬁc idiosyncratic term. Under the framework of the above model, we can consider the variance decomposition analysis. The variance of the housing price of each state at time t is given as the sum of the variances of three components in equation 4.3.1.

2 n r 0 V ar(yi,t) = λi,t · V ar(ft ) + γi,t · V ar(ft ) · γi,t + V ar(i,t) (4.3.3)

We can then evaluate each factor’s relative contribution to the variation of the housing price over time. Thus, the DFM permits us to evaluate time-varying contributions of each factor, unlike the principal component analysis, where they are assumed to be constant throughout the sample period. The factors ft are assumed to follow a stationary AR(p) process with time-varying stochastic volatility:

f f f f f ft = φ1 ft−1 + φ2 ft−2 + ... + φp ft−p + exp(ht ) · vt (4.3.4)

f f where vt ∼ i.i.d.N(0, σn) The time carrying stochastic volatility is modeled as a random walk process:

f f h f f ht = ht−1 + σf · ωt , ωt ∼ i.i.d.N(0, 1) (4.3.5)

The state-speciﬁc factor is assumed to follow a stationary AR(q) process, and the stochastic volatility is also modeled as random walk:

i,t = φi,1i,t−1 + φi,2i,t−2 + ... + φi,qi,t−q + exp(hi,t) · vi,t (4.3.6)

h hi,t = hi,t−1 + σi · ωi,t (4.3.7)

2 where vi,t ∼ i.i.d.N(0, σi ), and ωi,t ∼ i.i.d.N(0, 1).

82 4.4 Results and Discussion

We use the Bayesian Markov Chain Monte Carlo estimation procedure and generate successive draws from the joint posterior distribution of factors and parameters, based on 12,000 iterations, including the initial 2,000 burnouts. We then present the plots of the estimated national and regional factors in Figures 4.2. Since the model is estimated using data in first-differences, the extracted factors are plotted in the first differences in Figure 4.2. The result shows that the national factor has been fairly stable over the sample period, we observe that it is more volatile during the financial crisis period. The regional factors have been more volatile, particularly in the 1980s and the Global Financial Crisis (GFC).

4.4.1 National, Regional, and State Factor Contributions

To examine the national and regional factors efficacy in explaining state housing prices over time, we use Equation 4.3.3, which employs the DFM-TV-SV to decompose the variances in housing prices into national, regional, and state-level factors. The DFM-TV-SV allows for each factor’s time-varying contributions, unlike the principal component analysis, where the factors contributions are assumed to be constant and remain the same for all cross-sectional units. Figure 4.3 presents the time-varying factor variance decomposi- tions in each state over time, grouped by region. We find that the national factor is generally the dominant determinant affecting price movements. The summary of results presented in Table 4.2 shows that the national factor contributes on average 79.1% of the variation in state-level housing prices. Regional and state-specific factors account for only 7.3% and 10.8%, respectively. Table 4.2 docu- ments additional results on the average factor contributions over four sub-sample periods, which we refer to as a pre-boom period (January 1975 December 2000), a housing boom period (January 2001 March 2007), the bust period (April 2007 February 2012), and the recent period (March 2012 March 2020). We find that the national factor’s contribution

83 increases in times of larger price volatility, such as the housing boom period (85.3%) and the ﬁnancial crisis (bust) period (86.6%). These results imply that the national factor plays a key role in explaining the changes in housing prices in each state, particularly during times of increased national real estate volatility. Our DFM model results diﬀer from those found in previous papers that claim that regional and state-level factors are more dominant. However, the results are consistent with the claim of Del Negro and Otrok (2008) that the housing boom was indeed a national phenomenon.

4.4.2 The Synchronization Eﬀect

The previous models document heterogeneity across states relating to the national house price factor and state-level housing price changes. In this section, we address the synchronization effect of housing prices among states and the national factor. Following Marcellino et al. (2000), we employ time-series regressions of each state’s housing price on the estimated national factor. The estimated R-squared values provide information on how significantly each state-level price tends to comove with the national factor. The results are shown in Table 4.3 and Table 4.6 for the full sample and four sub-sample periods. The R-squared values are greater than 50% in most states while observing heterogeneous responses in each state over different periods. In some states, these values are close to or greater than 80%, including Arizona (AZ), California (CA), Florida (FL), New York (NY), and Nevada (NV). Notably, these are states where housing prices are relatively higher than in other states. On the contrary, the R-squared values are lower in states that contain large rural areas. States with R-squared values lower than 40% include: Arkansas (AK), North Dakota (ND), Oklahoma (OK) and West Virginia (WV). We again observe that state-level housing prices are more closely associated with the national factor when price movements experience larger fluctuations such as the boom and bust periods. Before the boom and after the bust, state-level housing prices synchronize less with the national trend than during these periods.

84 Next, we examine the comovement of state-level housing prices. Figure 4.4 presents the time-varying cross-state correlations for all states and regions. The cross-correlation among all states hit a historic low in the early 1980s, during which time the 1980s oil glut occurred following the 1970s energy crisis, and there was a six-year decline in the price of oil. This phenomenon is observed in almost all eight regions. However, the cross-correlation among states increased in the1990s, suggesting housing prices in the U.S. became more integrated. The state-level cross-correlations peaked around 2008 at approximately 0.8.

4.4.3 MSA-Level Factor Analysis

In this section, we analyze the average DFM-TV-SV factor contributions to the variation in metropolitan statistical areas (MSA) housing prices. There are 382 MSAs, and the sample period is again from January 1975 to March 2020. Figure 4.5 shows that the estimated national and regional factors have similar trends at the MSA-level relative to the state-level. Still, the regional factors tend to be more volatile than the estimated regional factors from state-level data, with the exception of the Southeast region. Similar to the state-level findings, we find that at the MSA-level the national factor is again the dominant factor in explaining house price variability, results are reported in Ta- ble 4.4. The national factor contributes on average 85.3% of MSA house price variability. This is higher than the 79.1% previously reported using state-level housing prices. Con- sidering the MSA-level data only includes housing prices in the metropolitan areas, this result is reasonable. Due to the higher price aggregation at the MSA level, the regional and MSA factor are negligible, at about 7% on average. There is great variability, ranging from 1.1% to 72.6%, in the MSA factor compared to the state factor; see Table 6 for a list of the MSAs with the largest and smallest national factor. We conducted the sub-sample analysis for factor contributions for MSA-level data, as we did for the state-level data. In contrast with the state-level results, there is a general declinging trend in terms of national factor contribution during different sub-sample periods. It decreased steadily from 89.5% in the pre-boom period to 74% in the recent period.

85 A similar trend is presented in the regional factors. The sub-sample analysis of time-varying cross-state correlations for all states and regions for the MSA-level housing prices delivers the same messages as for the state-level housing prices. On average, the MSAs are correlated with each other at 0.57 for the full sample period, and they are more correlated during the boom and bust periods. The regions follow this general trend as well, with the exception of the New England and Great lakes. This result shows that the synchronization on MSA-level housing prices are more profound during the housing boom and bust periods, similar to the sate-level prices.

4.5 Conclusion

Since the assumption of structural stability does not hold for most macroeconomic variables, we employ the dynamic factor model with time-varying factor loadings and stochastic volatility to enrich dynamic factor model structures with parameter instability on housing prices in the U.S. The main results from the DFM-TV-SV provide evidence that the national factor is the dominant factor for both state and MSA-level housing price movements in the U.S. On average, the national factor accounts for about 79 percent of the variation in housing prices at the state level and 85 percent at the MSA-level. Regional, and state- or MSA-specific factors, although subdued relative to the national factor, also account for a some amount of price variability. The regional factor accounts for another 7 percent of state housing price variability and MSA housing price variability. The MSA-level idiosyncratic variability is also small at 7 percent, compared to the state level variability of 11 percent. The difference in variability is due to similar prices at the MSA-level yielding more price movements captured at the national or regional levels. The heterogeneity in the synchronization effects of state housing prices on the national factor are also documented using a regression on the DFM-TV-SV results. We observe that state-level housing prices are more closely associated with the national factor when national house price movements experience larger fluctuations such as real estate price

86 boom and bust periods. Additionally, cross-correlations among all states have increased over time from about 0.20 in the 1980s to about 0.60 in 2020, suggesting housing prices have become more integrated (the sample peak appeared to be around 2008 at approximately 0.8).

87 Housing prices in U.S. states and Washington, D.C 400

350

300

250

200 HPI

150

100 MI

0 1975 1980 1985 1990 1995 2000 2005 2010 2015 2020 Year Figure 4.1: Plot of Housing Price Indices for all states and Washington D.C.

88 National factor Regional factor New England Regional factor Mideast 15 15 15 10 10 10 5 5 5 0 0 0 -5 -5 -5 -10 -10 -10 -15 -15 -15

1980 1990 2000 2010 2020 1980 1990 2000 2010 2020 1980 1990 2000 2010 2020

Regional factor Great Lakes Regional factor Plains Regional factor Southeast 15 15 15 10 10 10 5 5 5 0 0 0 -5 -5 -5 -10 -10 -10 -15 -15 -15

1980 1990 2000 2010 2020 1980 1990 2000 2010 2020 1980 1990 2000 2010 2020

Regional factor Southwest Regional factor Rocky Mountains Regional factor Far West 15 15 15 10 10 10 5 5 5 0 0 0 -5 -5 -5 -10 -10 -10 -15 -15 -15

1980 1990 2000 2010 2020 1980 1990 2000 2010 2020 1980 1990 2000 2010 2020

Figure 4.2: Plots of the estimates of National and Regional factors

89 Table 4.1: List of States in Each Region and Descriptive Statistics

Far West Southeast Southwest Rocky Mountains New England Mideast Plains Great Lakes Min 98.5144 84.6008 97.8580 86.2110 90.5879 99.8860 87.2204 73.1138 Max 129.0469 106.5516. 100.2449 106.6558 102.6895 136.6514 110.6150 86.9363 Mean 106.3761 94.0295 98.8976 97.6863 97.5650 107.8639 92.9662 82.5986 Std 11.5785 6.7513 1.0797 7.9175 4.9800 14.5309 8.1778 5.6340 Count 6 12 4 5. 6 6 7 5 States AK AL AZ CO CT D.C. IA IL CA AR MN ID MA DE KS IN HI FL OK MT ME MD MN MI NV GA TX UT NH NJ MO OH OR KY WY RI NY ND WI WA LA VT PA NE MS SD NC SC TN VA WV

90 England.pdf England.pdf CT MA ME NH RI VT 1 1 1 1 1 1 0.5 0.5 0.5 0.5 0.5 0.5 0 0 0 0 0 0 2000 2000 2000 2000 2000 2000 (a) New England AL AR FL GA KY LA 1 1 1 1 1 1 0.5 0.5 0.5 0.5 0.5 0.5 0 0 0 0 0 0 2000 2000 2000 2000 2000 2000 MS NC SC TN VA WV 1 1 1 1 1 1 0.5 0.5 0.5 0.5 0.5 0.5 0 0 0 0 0 0 2000 2000 2000 2000 2000 2000

(b) Southeast DC DE MD NJ NY PA 1 1 1 1 1 1 0.5 0.5 0.5 0.5 0.5 0.5 0 0 0 0 0 0 2000 2000 2000 2000 2000 2000 (c) Mideast Lakes.pdf Lakes.pdf IL IN MI OH WI 1 1 1 1 1 0.5 0.5 0.5 0.5 0.5 0 0 0 0 0 2000 2000 2000 2000 2000 (d) Great Lakes IA KS MN MO ND NE 1 1 1 1 1 1 0.5 0.5 0.5 0.5 0.5 0.5 0 0 0 0 0 0 2000 2000 2000 2000 2000 2000 SD 1 0.5 0 2000

(e) Plains Mountains.pdf Mountains.pdf CO ID MT UT WY 1 1 1 1 1 0.5 0.5 0.5 0.5 0.5 0 0 0 0 0 2000 2000 2000 2000 2000 (f) Rocky Mountains AZ NM OK TX 1 1 1 1 0.5 0.5 0.5 0.5 0 0 0 0 2000 2000 2000 2000 (g) Southwest West.pdf West.pdf AK CA HI 91 NV OR WA 1 1 1 1 1 1 0.5 0.5 0.5 0.5 0.5 0.5 0 0 0 0 0 0 2000 2000 2000 2000 2000 2000 (h) Far West

Figure 4.3: Relative Contributions of Each Factor to the Variation of Housing Prices (Blue: national factor, Red: regional factor, and yellow: state-speciﬁc factor.) All states New England Mideast

0.8 0.8 0.8

0.6 0.6 0.6

0.4 0.4 0.4

0.2 0.2 0.2

0 0 0

1980 1990 2000 2010 2020 1980 1990 2000 2010 2020 1980 1990 2000 2010 2020

Great Lakes Plains Southeast

0.8 0.8 0.8

0.6 0.6 0.6

0.4 0.4 0.4

0.2 0.2 0.2

0 0 0

1980 1990 2000 2010 2020 1980 1990 2000 2010 2020 1980 1990 2000 2010 2020

Southwest Rocky Mountains Far West

0.8 0.8 0.8

0.6 0.6 0.6

0.4 0.4 0.4

0.2 0.2 0.2

0 0 0

1980 1990 2000 2010 2020 1980 1990 2000 2010 2020 1980 1990 2000 2010 2020

Figure 4.4: Cross-correlations in all states and regions

92 National factor Regional factor New England Regional factor Mideast

5 5 5

0 0 0

-5 -5 -5

-10 -10 -10 1980 1990 2000 2010 1980 1990 2000 2010 1980 1990 2000 2010

Regional factor Great Lakes Regional factor Plains Regional factor Southeast

5 5 5

0 0 0

-5 -5 -5

-10 -10 -10 1980 1990 2000 2010 1980 1990 2000 2010 1980 1990 2000 2010

Regional factor Southwest Regional factor Rocky Mountains Regional factor Far West

5 5 5

0 0 0

-5 -5 -5

-10 -10 -10 1980 1990 2000 2010 1980 1990 2000 2010 1980 1990 2000 2010

Figure 4.5: Plots of the Estimated National and Regional Factors Using MSA Data

93 Table 4.2: Average Factor Contributions to Variation in State-level Housing Prices

Full Sample Pre-boom Boom Bust Recent National Regional State National Regional State National Regional State National Regional State National Regional State Average 79.09% 7.28% 10.76% 75.44% 9.79% 11.16% 85.28% 4.64% 8.39% 86.55% 3.22% 8.9% 81.44% 3.79% 12.42% New England 75.67% 15.27% 6.21% 71.02% 20.85% 5.04% 91.67% 4.82% 2.64% 84.85% 6.45% 7.34% 72.66% 10.88% 11.94% Mideast 79.04% 5.13% 12.48% 73.64% 7.84% 13.82% 94.8% 0.58% 4.02% 91.83% 1.33% 6.08% 76.42% 2.29% 18.56% Great Lakes 87.61% 5.34% 4.84% 86.51% 6.5% 4.43% 84.73% 6.31% 6.64% 90.78% 4.18% 3.75% 91.43% 1.6% 5.42% Plains 76.18% 0.99% 20.81% 73.55% 1.38% 22.34% 80.16% 0.58% 17.98% 75.53% 0.55% 22.65% 81.85% 0.34% 17% Southeast 81.78% 2.47% 13.54% 80.5% 3.38% 13.32% 83.32% 1.69% 13.33% 87.34% 1.05% 10.72% 81.32% 1.03% 16.1% Southwest 79.11% 3.93% 13.41% 75.12% 5.58% 14.39% 85.04% 2.28% 10.64% 87.02% 2.22% 9.17% 82.47% 1% 14.95% Rocky Mountains 79.28% 13.06% 5.12% 76.85% 15.99% 4.34% 74.97% 16.32% 5.97% 85.03% 7.23% 5.86% 86.84% 4.77% 6.53% Far West 74.04% 12.06% 9.68% 66.34% 16.78% 11.61% 87.58% 4.58% 5.86% 90% 2.73% 5.65% 78.5% 8.44% 8.9% Note: Pre-boom period: January 1975 - December 2000, a housing boom period: January 2001 - March 2007, the bust period: April 2007 - February 2012, and the recent period: March 2012 - March 2020 94 Table 4.3: R-squared values from the Regression of States on the National Factor

State Full Pre-boom Boom Bust Recent Average 0.62 0.21 0.56 0.62 0.41 AK 0.27 -0.04 0.83 0.2 -0.18 AL 0.68 0.31 0.44 0.41 0.42 AR 0.62 0.28 0.71 0.64 0.15 AZ 0.85 0.41 0.87 0.92 0.49 CA 0.82 0.27 0.91 0.76 0.59 CO 0.5 0.14 0.46 0.59 0.46 CT 0.68 0.5 0.75 0.77 0.57 DC 0.54 0.27 0.65 0.69 0.22 DE 0.78 0.38 0.85 0.58 0.33 FL 0.85 0.18 0.96 0.86 0.56 GA 0.78 0.36 0.39 0.83 0.62 HI 0.6 0.04 0.91 0.79 0.31 IA 0.46 -0.08 0.47 0.57 0.39 ID 0.68 0.01 0.51 0.62 0.43 IL 0.81 0.06 0.78 0.83 0.37 IN 0.54 0.14 0.48 0.63 0.45 KS 0.5 0.11 0.32 0.45 0.33 KY 0.59 0.18 0.53 0.59 0.48 LA 0.5 0.09 0.36 0.54 0.23 MA 0.65 0.43 0.41 0.74 0.7 MD 0.84 0.41 0.95 0.9 0.51 ME 0.65 0.45 0.61 0.48 0.17 MI 0.62 0.11 0.43 0.78 0.57 MN 0.76 0.21 0.51 0.83 0.62 MO 0.61 0.09 0.6 0.62 0.46 MS 0.48 0.18 0.22 0.31 0.28 MT 0.59 0.01 0.48 0.31 0.49 NC 0.66 0.28 0.3 0.38 0.49 ND 0.15 0.16 0.49 0.56 -0.23 NE 0.46 -0.02 0.53 0.56 0.49 NH 0.69 0.48 0.51 0.74 0.4 NJ 0.79 0.48 0.85 0.76 0.64 NM 0.72 0.23 0.47 0.67 0.4 NV 0.84 0.16 0.82 0.85 0.6 NY 0.73 0.5 0.66 0.67 0.6 OH 0.64 0.19 0.49 0.68 0.51 OK 0.35 0.22 0.31 0.28 0.17 OR 0.74 -0.19 0.62 0.79 0.58 PA 0.72 0.39 0.8 0.61 0.52 RI 0.76 0.47 0.57 0.82 0.67 SC 0.67 0.25 0.46 0.51 0.4 SD 0.41 0.06 0.4 0.27 0.26 TN 0.63 0.13 0.38 0.68 0.53 TX 0.4 0.17 0.02 0.56 0.3 UT 0.62 0.05 -0.09 0.8 0.81 VA 0.82 0.51 0.95 0.81 0.5 VT 0.63 0.39 0.85 0.43 0.36 WA 0.73 0.01 0.52 0.8 0.69 WI 0.66 0.05 0.64 0.56 0.58 WV 0.3 0.16 0.62 0.26 -0.21 WY 0.44 -0.11 0.05 0.44 0.04 Note: Pre-boom period: January 1975 - December 2000, a housing boom period: January 2001 - March 2007, the bust period: April 2007 - February 2012, and the recent period: March 2012 - March 2020

95 Table 4.4: Average Factor Contributions to Variation in MSA-level Housing Prices

Full Sample Pre-boom Boom Bust Recent National Regional State National Regional State National Regional State National Regional State National Regional State Average 85.34% 6.8% 6.73% 89.52% 5.75% 4.09% 86.12% 4.8% 7.83% 81.14% 6.83% 10.34% 74% 11.67% 12.08% New England 77.4% 14.88% 6.5% 79.69% 15.01% 4.33% 82.94% 8.43% 7.54% 81.65% 7.5% 9.48% 63.36% 23.84% 10.81% Mideast 90.22% 4.22% 4.23% 94.17% 2.7% 2.66% 89.66% 5.08% 3.87% 83.56% 6.92% 6.95% 82.13% 6.78% 7.85% Great Lakes 83.92% 11.05% 3.61% 87.39% 10.34% 1.76% 81.86% 9.49% 6.14% 79.08% 11.56% 6.45% 77.41% 14.18% 5.86% Plains 85.75% 6.32% 7.05% 90.23% 5.41% 3.77% 86.98% 2.31% 9.34% 79.11% 6.65% 13.39% 74.61% 12.11% 11.9% Southeast 90.33% 3.22% 5.55% 93.8% 2.14% 3.62% 91.92% 1.69% 5.4% 87.62% 2.11% 8.88% 79.72% 8.47% 9.79% Southwest 87.74% 4.62% 6.57% 90.84% 4.44% 3.9% 89.89% 2.07% 7.16% 82.28% 6.36% 9.85% 79.52% 6.08% 12.6% Rocky Mountains 81.06% 5.26% 12.81% 89.85% 2.87% 6.84% 78.08% 6.15% 15.12% 69.8% 9.22% 19.4% 62.24% 9.8% 26% Far West 86.27% 4.86% 7.48% 90.18% 3.08% 5.82% 87.62% 3.21% 8.04% 86.02% 4.33% 8.3% 72.98% 12.1% 11.83% Note: Pre-boom period: January 1975 - December 2000, a housing boom period: January 2001 - March 2007, the bust period: April 2007 - February 2012, and the recent period: March 2012 - March 2020 96 Table 4.5: Average Factor contributions to Variation of Housing Prices in Selected MSAs

Full Sample Pre-boom Boom Bust Recent National Regional MSA National Regional MSA National Regional MSA National Regional MSA National Regional MSA Staunton-Waynesboro, VA 98.47 % 0.25 % 1.07 % 99.04 % 0.15 % 0.68 % 98.92 % 0.18 % 0.75 % 97.74 % 0.51 % 1.41 % 96.78 % 0.46 % 2.34 % Flagstaﬀ, AZ 98.44 % 0.03 % 1.48 % 98.3 % 0.03 % 1.62 % 99.28 % 0.01 % 0.69 % 98.95 % 0.01 % 1.03 % 97.94 % 0.04 % 1.93 % Richmond, VA 98.06 % 0.88 % 0.84 % 98.93 % 0.33 % 0.6 % 99.28 % 0.07 % 0.55 % 97.92 % 0.47 % 1.43 % 94.45 % 3.51 % 1.43 % Lynchburg, VA 97.74 % 0.5 % 1.41 % 99.07 % 0.14 % 0.65 % 98 % 0.55 % 1.11 % 96.22 % 0.39 % 2.86 % 94.28 % 1.68 % 3.17 % Beckley, WV 50.36 % 0.4 % 48.05 % 57.58 % 0.24 % 41.49 % 55.81 % 0.23 % 43.31 % 48.75 % 0.62 % 48.41 % 24.24 % 0.88 % 72.29 % Midland, MI 47.51 % 46.94 % 3.12 % 37.88 % 59.09 % 1.65 % 58.99 % 26.28 % 8.59 % 48.12 % 46.29 % 3.09 % 68.91 % 24.59 % 3.62 % Kahului-Wailuku-Lahaina, HI 26.13 % 0.51 % 71.55 % 32.8 % 0.48 % 65.02 % 5 % 0.74 % 92.13 % 10.69 % 0.63 % 86.07 % 30.42 % 0.37 % 67.8 % Urban Honolulu, HI 24.89 % 0.61 % 72.62 % 27.6 % 0.61 % 70.09 % 8.44 % 0.52 % 89.35 % 13.33 % 0.69 % 83.58 % 35.83 % 0.65 % 61.21 % Note: Pre-boom period: January 1975 - December 2000, a housing boom period: January 2001 - March 2007, the bust period: April 2007 - February 2012, and the recent period: March 2012 - March 2020 97 Table 4.6: R-squared values from the Regression of Regions on the National Factor

State Full Pre-boom Boom Bust Recent Average 0.57 0.5 0.73 0.83 0.26 New England 0.47 0.31 0.3 0.8 0.39 Mideast 0.63 0.38 0.88 0.83 0.62 Great Lakes 0.63 0.67 0.46 0.85 0.56 Plains 0.58 0.56 0.62 0.85 0.42 Southeast 0.58 0.67 0.98 0.95 -0.06 Southwest 0.54 0.33 0.92 0.83 0.23 Rocky Mountains 0.58 0.54 0.81 0.82 0.13 Far West 0.55 0.52 0.83 0.73 -0.16 Note: Pre-boom period: January 1975 - December 2000, a housing boom period: January 2001 - March 2007, the bust period: April 2007 - February 2012, and the recent period: March 2012 - March 2020

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