A Generalized Nonlinear IV Unit Root Test for Panel Data with Cross-Sectional Dependence Shaoping Wang School of Economics, Huazhong University of Science and Technology,WuHan, China Peng Wang Department of Economics,New York University, New York, NY, U.S.A. Jisheng Yang School of Economics, Huazhong University of Science and Technology,WuHan, China Zinai Li School of Economics and Management, Tsinghua University, Beijing, China

Introduction Panel unit root test: Survey by Hurlin and Mignon (2004) Assume cross sectional independence Levin and Lin (1992,1993); Levin, Lin and Chu (2002); Harris and Tzavalis (1999); Im, Pesaran and Shin (1997, 2003); Maddala and Wu (1999); Choi (1999,2001)

Panel unit root test : Assuming cross sectional dependence Flôres, Preumont and Szafarz (1995): The test statistics are based on a seemingly unrelated regression (SUR), and follow some nonstandard asymptotic distributions. Tayor and Sarno (1998): They propose a Wald-type test, but the asymptotic distribution of the test statistics is unknown. Breitung and Das (2004): They apply the ADF test to pooled samples with robust standard errors. Introduction

Panel unit root test : Assumming cross sectional dependence Bai and Ng (2001, 2004); Moon and Perron (2004); Phillips and Sul (2003): They all assume that the dependence of the cross- sectional units comes from some common factors, and employ the principal components method to eliminate the common factors (hence the correlation of cross-sectional units), and then apply the ADF type test. Introduction

Panel unit root test : Assumming cross sectional dependence Choi (2002) He models the cross-sectional dependency by time-invariant common factors, and employs the demeaning or detrending method developed by Elliott, Rothenberg and Stock (1996) to eliminate the common factors, and then applies the combining p-value test of Choi (2001). Introduction

Motivation Chang (2002) Bai and Ng (2004) This Paper A Two Step Test Improved the performance This paper, following Bai & Ng (2004) and Chang (2002), (1)apply Bai & Ngs common factor method to eliminate the cross-sectional dependency; (2)employ Changs NIV estimation to the treated data to form the test which convergence to the standard normal distribution.

Bai & Ng (2002, 2004)s Model They model the cross-sectional dependency by common factor: F t is the r×1 common factor among individuals After eliminating the cross-sectional dependency by the method Of principal component, and then employ the combining p-value test of ADF statistics. The statistics has the normal standard limiting Distribution.

Changs Model (1) : coefficient on the lagged dependent variable : error term which follows the AR(p) process: (2) : lag operator : autoregressive coefficient : some integer that is known and fixed

Model Assumptions To ensure the AR(p) process in (2) is invertible Assumption 1: for all and To restrict the distribution of error term Assumption 2: Denote (1) are independent and identically distributed and its distribution is absolutely continuous with respect to Lebesgue measure (2) has mean zero and covariance matrix (3) satisfies for some and has a characteristic function that satisfies for some

NIV Estimation OLS estimation: Under, the asymptotic distribution of obtained from (3) is asymmetric, and not the usual t-distribution NIV estimation: as instrument for, where is some function satisfying Assumption 3: is regularly integrable and satisfy

Under assumption 1-3, Chang draw the key result: as and are asymptotically uncorrelated regardless of the cross sectional dependence And the test statistic has a limiting standard normal distribution Chang Test

Findings about Chang (2002) The bigger the N is, the Smaller the correlation coefficient of cross-sectional units becomes. The test statistic does not fully follow the limiting standard normal distribution when the cross sectional dependence is strong. Chang test perform well in finite samples when when the cross sectional dependence is low.

Our Test: A Two Step Test Step1: eliminate the cross sectional dependence through the method of principal components suggested by Bai & Ng. Step2: Apply Changs NIV estimation to the treated data, and form our test statistics which convergence to the standard normal distribution.

Model Setting Adopting the DGP in Bai and Ng (2004) to model the cross- sectional dependency by common factor: or (3) Where: F t is the r × 1 common factor among individuals. Error term has zero mean with covariance matrix, for. e it is the panel data after eliminating the common factor when the common factor and its loading vector are known.

We are interested to test: for all VS for some Hypothesis

Test procedure: Step one Under the null hypothesis: The differenced common component estimator of is times the eigenvectors corresponding to the largest r eigenvalue of the matrix, the estimated loading matrix is given by..

Step one (contd) the data with weak (or no) cross-sectional dependence where can be set as 0. After eliminating the common factor, model (3) can be rewritten as: (4) Model (4) is of the same form as Changs model (1), but with a different error term.

Step two Denote,,,, where. We have the model with no (weak) cross-sectional dependence (5)

Step Two (contd) Denote, The NIV estimator for (5) is: t-ratio of :, is the variance estimator of. Test statistics:.

The Distribution Suppose that Assumption and Bai & Ng (2004)s assumptions A-E hold. We have: Then, under those assumptions, the t-ratio of auto-regression coefficient in (5) is similar to its asymptotic distribution in Changs model (1).

The Distribution Theorem 1. Suppose that Assumption and Bai & Ng (2004)s assumptions A-E hold. Under the null hypothesis of panel unit root, we obtain, as, for all and, where denote the correlation coefficient. Theorem 2. Suppose that Assumption Bai & Ng (2004)s assumptions A-E hold. Under the null hypothesis of panel unit root and as, we obtain Extend Theorem 1 and 2 to panel data models with individual intercept and/or time trend by de-meaning and/or de-trending schemes

Simulation DGP with General Cross-sectional dependence (6) The covariance matrix of,

DGP with General cross sectional dependence for size evaluation, for power evaluation The number of common factor is set as 1 for eliminating the cross-sectional dependency by method of principal components. Simulation (contd)

DGP with General cross sectional dependence Size no intercept and no linear trend The empirical sizes of BN test and our test in all cases are fairly close to the nominal sizes (we pick up 5%). The distortions of Chang test are more pronounced when the cross sectional dependence is high (e.g., 0.8). Simulation ( contd )

The three tests have reasonably good power in all designs and the power increases as N and T increase. The power of BN test is a little lower than the power of our test in some cases. no intercept and no linear trend ~ Uniform[0.85, 0.99]. General cross sectional dependence Power Simulation (contd)

Simulation DGP of Chang (2002) DGP of Chang (2002) is also based on (6) with the covariance matrix generated randomly. Size Both tests are very close to each other and approximate the nominal test sizes very well. no intercept and no linear trend

Simulation DGP of Chang (2002) Power Changs test is more powerful than our test when sample is small. The empirical power of two tests are high enough. no intercept and no linear trend

The covariance matrix of, for size evaluation, for power evaluation Simulation DGP with one common factor

no intercept and no linear trend The empirical sizes of two tests are almost identical and close to the nominal sizes. Simulation (contd) : Common factor dependence Size

no intercept and no linear trend The empirical power of two tests are high enough. Simulation (contd) : Common factor dependence Power

Idea: remove the cross sectional dependence before applying Chang test Method: first removing the cross-sectional dependence by method of principal component and then applying Chang NIV to the treated data Test Distribution: a limiting standard normal distribution under the null hypothesis of panel unit root Test Performance: (1) much better than Chang test when the cross sectional dependence is moderate to high; (2) as good as Chang test when the cross sectional dependence is low; (3) The finite sample performance is similar to that of BN test. Conclusion

Thank you!