Finite sample distributions of nonlinear IV panel unit root tests in the presence of cross-section dependence

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1 Finite sample distributions of nonlinear IV panel unit root tests in the presence of cross-section dependence Qi Sun Department of Economics University of Leicester November 9, 2009 Abstract his paper presents a response surface analysis for the nite sample distributions of two popular panel unit root tests developed by Chang (2002) and Chang and Song (2005). Numerical distributions illustrate signi cant di erences between nite sample and large sample properties. Dependence of nite sample bias on sample size is investigated. he paper provides 95% con dence intervals for the critical values using David-Johnson estimate of percentile standard deviation. Computation of p value of empirical test statistic is also presented. Key words: Response surface, Monte Carlo, Panel unit root test, Finite sample bias, Numerical distribution. Introduction Ever since the seminal work by Levin and Lin (992) and Im et al. (997), much attention has been given to panel data unit root and cointegration research. However, early work in the literature assumes that panel individuals are independent of each other. More recently, it has been found that cross section correlation (or dependence) widely exists in empirical panels such as exchange rates, interest rates and in ation rates (c.f. O Connell, 998). Intuitively this means that there exists co-movement among panel units. When one unit in the panel experiences a shock, other units will be a ected as well. If the assumption of cross section independence is violated, it could lead to size distortions and low power of the tests. Banerjee et al

2 (2005) using simulations nd that in the presence of cross section dependence the empirical size of panel unit root tests is substantially higher than the nominal level, and the null hypothesis of a unit root is rejected too often even when it is true. A number of papers thereafter suggest various models or methods to account for dependence (cf. Harvey and Bates (2003), Breitung and Das (2005), Jönsson (2005), Chang (2002) and Bai and Ng (2004), Moon and Perron (2004a), Phillips and Sul (2003), Pesaran (2006)). Among these, the former four studies model correlation by covariance matrix of the error terms, and the latter t in dependence by common factors. he two formulations of dependence are referred to as weak and strong forms of dependence by Breitung and Pesaran (2008). Chang and Song (2005) (CS hereafter) test is a most general test that is able to consider both of the situations, and it can also cope with the serious problem of cross unit cointegration within the panel. his situation arises when panel units share common stochastic trend(s) which leads to cointegration of the panel units and forms long run dependence. Based on its early version Chang (2002) (CH hereafter), CS uses the developed nonlinear IV transformations of lagged levels as instruments and estimates the usual augmented Dickey-Fuller (ADF) type regression for each cross section unit. It shows that under the null hypothesis of unit root, the individual IV t-statistics have standard normal distribution asymptotically and are cross-sectionally independent. he panel unit root statistic, which is the standardized sum of individual IV t-statistics, is thus asymptotically normally distributed. 2 Details of the tests are reviewed in section 3. his approach overcomes the weakness of some other methods that solve dependence by de-factoring the model, which may eliminate unit root if unit root exists in the common factors (c.f. Breitung and Pesaran (2008)). he test also gains improvement over CH which can only deal with weak dependence. As panel unit root test statistics usually exhibit standard normal distribution asymptotically, which is the most signi cant advantage against traditional time series unit root tests (c.f. Baltagi (2008)) 3, issues of - See Banerjee et al. (2004, 2005) for the existence of cross unit cointegration in empirical studies and the problems it can cause. 2 Chang and Song (2005) proposes three hypotheses and thus there are three testing statistics. However, for simplicity and comparison with Chang (2002), we only consider the average statistic, i.e. the standardized sum of individual IV t-statistics. 3 ime series unit root tests statistics usually converge to non-standard distributions and are quite often without analytical solution. Consequently their critical values must be computed by simulations experiments. 2

3 nite sample distribution or critical values are thus not given much attention in the literature. Empirical research generally directly applies the nominal critical values of panel unit root tests, however, it is a well known fact that nite sample performance can substantially di er from the corresponding asymptotic properties. In particular, the nite sample distribution of panel unit root statistic can vary across two dimensions (cross section and time dimensions). An e ective method of observing the variation in small sample distributions is running response surface regressions with points simulated by a number of Monte Carlo experiments under a variety of sample sizes. Jönsson (2005a) has tabulated the critical values for a panel corrected standard error (PCSE) based Levin and Lin (992) (LL) homogeneous test by response surface method. he PCSE correction is introduced to deal with cross section correlation. Jönsson (2005b) continues to augment the test with serial correlation and estimates response surfaces for critical values. However, LL test assumes homogeneity across panel, which hardly holds in practice. Due to its robustness to all forms of cross section dependence CS test is a preferable test for applications, while unfortunately its nite sample properties do not provide satisfactory results. able cites the 5% nite sample sizes of CH and CS tests from Chang and Song (2005). CS has reasonable sizes in the presence of strong and long run dependence, but in the case of strong dependence the sizes seriously deteriorate as sample size increases. It also con rms the argument that CH is not able to cope with strong dependence, whereas when only weak dependence exists CH signi - cantly outperforms CS. herefore, CH is also analyzed and is recommended for empirical work where correlation between panel individuals is weak. able : 5% nite sample test sizes of CH and CS tests Weak dependence Strong dependence Long run dependence N CH CS CH CS CH CS :07 0:024 0:26 0:038 0:330 0: :07 0:032 0:20 0:046 0:326 0: :059 0:02 0:288 0:024 0:375 0: :056 0:020 0:292 0:033 0:377 0: :055 0:005 0:374 0:020 0:422 0: :052 0:007 0:378 0:09 0:428 0:048 Note: Citation from Chang and Song (2005); N and denote cross section and time dimensions, respectively. 3

4 Considering the poor nite sample performance this paper applies response surface regressions to analyze the nite sample bias of the two tests %, 5% and 0% signi cant levels. In addition, this study also provides the indecisive range around point critical values caused by the randomness resulting from Monte Carlo simulations and regression errors. Formulas to compute the upper and lower limits of a critical value on certain signi cant level is provided using David-Johnson percentile standard deviation and response surface regression residual standard errors; the numerical distribution functions of the test statistics are calculated to obtain p-value of any given percentile. Finally, graphs of numerical distributions under various sample sizes are given to highlight nite sample properties of the statistics. he rest of this chapter is organized as follows. Section 2 speci es di erent forms of cross section dependence. Section 3 reviews CH and CS tests. Section 4 presents the design of Monte Carlo experiments. Response surface estimation is provided in section 5 and section 6 discusses the results. Section 7 concludes. 2 Speci cation of cross section dependence Without considering residual serial correlation a general form of cross section dependence can be given as where y it = i y i;t + u it ; i = ; :::; N; t = ; :::; ; () u it = 0 if t + it f t = (f t ; f 2t ; :::; f mt ) 0 is an m vector of serially uncorrelated unobserved common factors; i = ( i ; i2 ; :::; im ) 0 is an m vector of factor loadings; it is serially uncorrelated process with mean zero and positive de nite covariance matrix V. For generality set the covariance matrix of f t as identity matrix I m. It is assumed that f t and it are independently distributed. Initially two cases of cross section dependence can be distinguished. (i) Weak dependence. his assumption rules out the presence of unobserved common factors f t ( i = 0). Dependence only arises from spatial correlation among cross section units which is represented by the covariance matrix of it, V. (ii) Strong dependence. In this case unobserved common factors exist and dependence is generated from two sources f t and it. he covariance matrix of error u it is thus given by V = 0 i i + V. 4

5 Given the unit root testing model (), under the null hypothesis of unit root, i.e. i = for all i = ; : : : ; N, in the case of strong dependence y it contains nonstationary common factors f t and nonstationary errors it. If the process built upon it is stationary, e.g. substitute it by it and write u it in (e) as u it = i f t + it (2) then the nonstationarity of is only driven by the nonstationary common factors and therefore serve as common stochastic trends. his leads to cointegration relationship between any pair of and, with linearly independent cointegrating relations among N cross section units. he situation is referred to as cross unit cointegration and it forms the long run dependence. According the form of dependence dealt with, the second generation panel unit root tests can be categorized into two groups. ests solving weak dependence include Harvey and Bates (2003), Breitung and Das (2005) and Jönsson (2005), who adopt OLS or GLS based methods. Bai and Ng (2004), Moon and Perron (2004), Phillips and Sul (2003) and Pesaran (2006) consider strong dependence modeled by common factors. However, most of the tests choose to de-factor the model and are exposed to problem if nonstationarity is casued by common factors. In dealing with all forms of dependence CH and CS apply an alternative novel approach against all other tests. 3 CH and CS tests 3. Review of CH and CS tests o solve weak cross section dependence, instead of using the OLS and GLS based methods, Chang (2002) proposes to use nonlinear IV estimation of the ADF regression for each individual in the panel using nonlinear transformations of the lagged levels as instruments. he basic panel unit root test model is the following rst-order autoregressive regression y it = i y i;t + u it ; i = ; :::; N; t = ; :::; ; (3) where i and t denote panel individuals and time period, respectively; the time length for each individual i may di er across cross-sectional units. he error term u it is assumed to follow an AR(P i ) process, so the unit root 5

6 test is based on the following ADF regression XP i y it = i y i;t + i;k y i;t k + " it (4) k= where " it are iid (0; P ) random variables; is di erence operator. Cross sectional dependence is dealt with by IV estimation of (4) using instrument generated by a single nonlinear function F as F (y i;t ) F is called the instrument generating function (IGF) by CH and also later in CS. It assumes that F is nonlinear, regularly integrable and satis es Z + xf (x) dx 6= 0. In the Monte Carlo simulations in Chang (2002), the choice of instrument used for the lagged level y i;t F (y i;t ) = y i;t exp ( c i jy i;t j) is generated by IGF where the factor c i is inversely proportional to the sample standard error of y it = u it and =2 i, i.e. c i = K =2 i s y it with s 2 (y it ) = i P i i= (y it) 2 ; K is a constant and the values is xed at 3 for all i = ; : : : ; N. he choice of factor c i and the value of K is decided by the performance of Monte Carlo simulations. For each i = ; : : : ; N, under the unit root hypothesis H 0 : i =, the t-statistic of the nonlinear IV estimator b i is constructed as i = b i se (b i ) (5) where se (b i ) is the standard error of b i. In contrast to the asymptotically non-normal Dickey-Fuller distribution, Chang shows that the nonlinear IV t-statistic i has limiting standard normal distribution under the unit root hypothesis. In addition the statistics are cross-sectionally independent; therefore the test is free from the need of modeling dependence. he panel unit root test statistic is an average of the individual IV t- statistics ( i )s. Under the joint unit root null hypothesis H 0 : i = for all i = ; : : : ; N, the average IV t-statistic is de ned as S = p N 6 N X i= i (6)

7 he average statistic S has standard normal distribution as i!. 4 However, this test works well only with the weak form of cross section correlation. Due to the limitation of Chang (2002) test in the presence of strong dependence, Chang and Song (2005) use a set of orthogonal IGFs instead of a single IGF as in CH. More speci cally, the choice of IGF is a set of orthogonal Hermite functions G k of odd orders k = 2i, i = ; : : : ; N. he Hermite function G k of order k, k = 0; ; 2; :::; is de ned as G k (x) = 2 k k! p =2 Hk (x) exp where H k is the Hermite polynomial of order k given by x 2 =2 H k (x) = ( he IGFs is then de ned by ) k exp x 2 d k F i = G 2i dx k exp x2 for i = ; : : : ; N; i.e. the instrument used for the lagged level y i;t is F i (y i;t ) = G 2i (y i;t ) Based on the panel model in (3), let y t = (y it ; :::; y Nt ) 0 and assume that in the unit root process y t there are N M cointegrating relationships among cross section units represented by cointegrating vectors c j, j = ; :::; N M. he vector autoregression and error correction representation can specify the short-run dynamics of y t as y it = NX XP i NX M a ij y i;t k + b ij c j y t + " it (7) i= k= j= for each cross section unit, where " it are white noise; P i is the order of serial correlation of unit i. Moreover, refer to model (3), under the unit root process y it = u it. herefore, (3) and (7) can be rewritten as XP i X y it = i y i;t + i;k y i;t k + 0 i;k! i;t k + " it (8) k= 4 Due to the independent standard normal distribution of ( i)s, only -asymptotics is using in deriving the limit theory of panel statistic and N may take any value, whereas usual panel unit root tests apply sequential limit theory (see Phillips and Moon (2000) for an overview of asymptotic approaches applied to panel limit theory). Q i k= 7

8 where! it is covariate containing several lagged di erences of other cross section units and linear combinations of the lagged levels of all cross section units, which is suggested by (7). 5 he panel unit root test is then based on the above covariate augmented ADF regression (8). he construction of the t-statistic of nonlinear IV estimator b i is the same as (5) in CH. Chang and Song obtained the same results as CH. As i!, under the null hypothesis that each individual has a unit root, the IV t-statistic of each individual has standard normal limiting distribution and the statistics are independent across panel despite the fact that panel units are either weakly or strongly correlated or cointegrated. he panel unit root test statistic, which is again the average t-statistics de ned as (6) in CH, is asymptotically normally distributed. 3.2 Problems of CH and CS tests he poor nite sample sizes are illustrated in the introduction by able. In addition, Im and Pesaran (2003) argues that the nonlinear IV panel statistic of CH needs a much more restrictive condition for its asymptotic property to hold, i.e. it requires N ln ( ) p! ; as N;! Here for simplicity it is assumed that the panel is balanced, i.e. all panel individuals have the same time dimension. So for the best performance of the test, N needs to be very small relative to to ensure good performance, whereas in practice this is quite often di cult to achieve. Since the only technical di erence between CS and CH tests is the choice of IGF, i.e. a single IGF vs. a set of orthogonal IGFs, a similar problem may also be conjectured to CS test. Nevertheless, the condition term can be included into response surface regression and contributes to the computation of nite sample critical values. Although the nite sample problems are di cult to solve analytically, one can resorts to numerical methods. By simulating critical values with various sample sizes N and and smoothing them through response surface 5 See Chang and Song (2005) for speci cation. he purposes of introducing covariate include increasing the power of the test and helping capture more cross section dependence. However, according to the Monte Carlo simulation results in Chang and Song (2005), although there is some improvement in terms of power performance using covariate, the sizes appear worse than those obtained without covariate. So covariate is not considered in the simulations in this study. 8

9 regression, nite sample distributions and critical values for any sample size can be easily obtained for empirical use. 4 he simulation experiments 4. he DGPs For simulation experiments we use the DGPs in Chang and Song (2005) to formulate non-stationary panels with cross section weak dependence, strong dependence and cross unit cointegration, respectively. hey are referred to DGP, DGP2 and DGP3 hereafter. he basic model considered is y it = i y i;t + u it ; i = ; :::; N; t = ; :::; ; In the simulation experiments balanced panels are used, so each panel cross section unit has the same time length. Under the null hypothesis of unit root y it = u it. he innovations u it are given as following to generate the corresponding dependence DGP : u it = i u i;t + it DGP2 : u it = i u i;t + i t + it DGP3 : u it = i u i;t + i t + it where i is the AR coe cient of u it ; t is common factor and i is the factor loading ; it is innovations with weak cross-sectional dependence generated by symmetric and nonsingular covariance matrix V which is speci ed soon after. Note that in DGP there is weak cross section dependence caused by the covariance matrix of it. Besides it, DGP2 has stronger level of dependence generated by a common stochastic trend built upon t. Under the unit root hypothesis y it contains both nonstationary sum of t and nonstationary sum of it, so the cross section units in the panel are not cointgrated. However, in DGP3 cross unit cointegration is present. Since the sum of it is stationary, the nonstationarity in y it arises only from the nonstationary common stochastic trend built from t. Hence, there is cointegrating relationship between any pair of the N panel individuals with (N ) linearly independent cointegrating relations. he parameters in the DGPs are generated as follows. he AR coe cient of u it, i, is randomly drawn from uniform distribution [0:2; 0:4]. he factor loadings, i, are randomly drawn from uniform [0:5; 3]. he processes t and 9

10 u it are independent and drawn from iid N(0; ) and iid N(0; V ), respectively. he N N covariance matrix V of the innovations it is symmetric positive de nite. o ensure this property, V is generated according to the steps in Chang (2002): () Generate an N N matrix from uniform distribution [0; ]; (2) Construct from an orthogonal matrix H = ( 0 ) =2 ; 6 (3) Generate a set of N eigenvalues, ; :::; N. Let = r > 0, N = and draw 2 ; :::; N from uniform distribution [r; ]; (4) Form a diagonal matrix with ( ; :::; N ) on the diagonal; (5) Construct the covariance matrix using the spectral representation V = HH he Monte Carlo experiments Instead of providing critical values from running one Monte Carlo experiments with extremely large sample size, simulations are conducted for various combinations of N and to obtain quantiles of the empirical distributions of CH and CS statistics. he quantile points are then used to estimate response surface regressions. he sample sizes are 2 f50; 75; 00; 25; 50; 75; 200; 225; 250; 275:300; 325; 350; 375; 400; 425; 450; 475; 500g and N 2 f0; 20; 30; 40; 50; 60; 70; 80; 90; 00g So there are altogether 9 0 = 90 experiments for each DGP. Each experiment consists of np = 0000 replications. For each experiment the 0:0, 0:05 and 0:0 quantiles are recorded. he Monte Carlo simulations are programmed in Gauss 7.0. MacKinnon (2000) provides a detailed discussion of the bene ts of conducting multiple (m) experiments with di erent random seeds for the same samples rather than doing one set of experiments with the same seeds for np m replications. he reason for this is to save computer memories, to protect work against power failure and more importantly to improve and estimate experimental randomness in order to resolve the problem of heteroskedasticity in response surface regressions. his method is widely adopted in the literature of response surface application (c.f. Ericsson and MacKinnon (2000), Jönsson (2005), Sephton(995)). However, the method 6 (M 0 M) =2 is obtained by Cholesky decomposition of the inverse of (M 0 M). 0

11 is unpractical to the experiments in this study, since the computation of non-linear IV panel unit root test statistics using a set of orthogonal IGFs (Hermite functions) costs signi cantly large amount of time, particularly with large number of cross section units N. he whole computation for the three DGPs can take months in a single standard computer. Although the experiments are not able to repeat m times with m di erent sets of random seeds, jackknife standard error is used to solve the problem of heteroskedasticity. his is shown in the next section. 5 Response surface estimation he early application of response surface estimation was by MacKinnon (994, 996 and 2000). MacKinnon (996, 2000) points out that response surface regression coe cients can help researchers estimate the quantiles for any given sample size, and furthermore we are also able to derive numerical (empirical or nite sample) distribution functions and hence to calculate the empirical p-value for any given percentile. 5. Representation of response surface regression he dependent variable is the di erence between the quantile obtained from Monte Carlo experiment and its corresponding asymptotic quantile, i.e. the nite sample bias. he response surface regression takes the form where q i = + 2 N + 3 q i = eq i q N 2 + N ln ( ) 4 p + " i (9) eq i denotes the quantile obtained from the i th experiment with sample size N and and q is the quantile of standard normal distribution; so the dependent variable qi represents nite sample bias. he parameters in the regression can then assist in obtaining the corresponding nite sample bias to speci ed sample size. he functional form is determined by the goodness-of- t of the regression and the signi cance of variables after a number of experimentations. Ericsson (986) notes that the functional form of response surface regression can be justi ed on the grounds of the signi cant coe cients obtained and the generally high R 2 values. A general form to start with can be q i = N + 4 N N + N ln ( ) 6 p + e i (0)

12 For generalization a uniform speci cation for the %, 5% and 0% signi cant levels is opted for all the three DGPs rather than optimize the functional form for each one. he variable N p ln( ) is chosen according to Im and Pesaran (2003). he estimated standard error of this variable is particularly small for all regressions with di erent combinations of the independent variables in (0). his is shown in the result section and indicates that the coe cient of N ln( ) p is precisely estimated. herefore, N p ln( ) is chosen to remain. Since it is panel data concerned, N is introduced as an interaction term according to Jönsson (2005), whereas the coe cient of this variable is insigni cant for DGP2, 3 for all the 3 percentiles (%, 5% and 0%), and the coe cient of is insigni cant for DGP for all the 3 percentiles. With the presence 2 of N and/or, the values of R 2 are not essentially improved. So the 2 two terms are excluded from the regression. Later it is shown that the chosen functional form has reasonable R 2 and good performance in terms of signi cance of coe cients for all response surface regressions. 5.2 Estimation of response surface regression he response surface regression (9) is estimated by ordinary least squares (OLS). However, the errors in (9) are heteroskedastic due to the increasing values of N and. herefore the variance of the errors depends systematically on the sample sizes (N and ). o account for the heteroskedasticity, the jackknife covariance estimator developed by MacKinnon and White (985) is applied. Let b denote the vector of estimators and X be the matrix of regressors in (9). he covariance estimator of b is given as bv b = n (n ) X 0 X X 0 X b n X 0 bubu 0 X X 0 X () where n is the number of observations in (9), i.e. 9 0 = 90; b is an (n n) diagonal matrix with diagonal elements bu 2 j ; bu j = ( k jj ) be j with k jj as the j th diagonal element of X (X 0 X) X 0 ; (e j )s are residuals of (9) and bu is the vector of bu j. Since the quantiles computed from simulations contain two dimensions (N and ) as panel data, one may consider using panel regression. he suitable panel estimation for (9) is xed-e ect. Nevertheless, if xed-e ect is used, the variables N and will be dropped during estimation process, N 2 which causes the loss of information and results in ine ciency. his can be indicated by the goodness-of- t of the regression. herefore value of R 2 2

13 from OLS is higher than that from xed-e ect estimation Critical values augmented by David-Johnson percentile standard deviation and response surface regression standard errors On smoothing nite sample bias by response surface regressions estimated in the above sections, according to (9) the corresponding critical value is calculated as bq i = q + bq i (2) where bq i = b + b 2 N + b 3 N 2 + b N ln ( ) 4 p bq i is the tted nite sample bias of any given sample size N and. However, there exist two types of uncertainty resulting from Monte Carlo simulation and response surface regressions. Firstly, sample percentiles exhibit certain distribution just as the sample means do. 8 o capture this randomness caused by Monte Carlo experiments the 95% con dence intervals are provided for the quantiles corresponding to the %, 5% and 0% signi cant levels using David-Johnson (954) percentile standard deviations. David and Johnson (954) derived the formula to calculate the standard deviation of percentiles on the basis of its sample size (i.e: the number of replications here). Secondly, since the bias is smoothed by response surface regressions, error terms unavoidably exist. his uncertainty can be easily represented by the standard error of regression residuals. he 95% con dence interval of a sample quantile is given as ulbq i = bq i :96 [se () + se (" )] (3) where bq i is the critical value of percentile obtained from (2) with given sample size N and ; se () is David-Johnson (954) estimate of the standard deviation of a particular percentile; se (" ) is the standard error of the response surface regression residuals in order to take into account the errors incurred by regression. hus, the upper and lower limits of the critical 7 Results are available on request. 8 In Monte Carlo simulation for each replication if we generate Gaussian random numbers with mean 0 and standard deviation, the mean of the random numbers of each replication will also follow Gaussian distribution with mean 0 and standard deviation p N, where n is the number of replications. For the same reason, the percentiles will also following Gaussian distribution with certain mean and standard deviation. 3

14 values are respectively given by ubq i = bq i + :96 [se () + se (" )] lbq i = bq i :96 [se () + se (" )] If the empirical test statistic falls between ubq i null hypothesis is to be rejected or not. and lbq i, it is indecisive if the 5.4 P-values of numerical distributions of CH and CS statistics he response surface coe cients obtained from (9) can also be used to estimate p-values of nite sample distributions. he approximation procedure is according to MacKinnon (996, 2000). MacKinnon (996, 2000) approximate nite sample distribution is by 22 tted values from (2), bq i, for the corresponding 22 s, given the sample size N and. he chosen 22 s are 0:000,0:0002,0:0005,0:00,0:002,: : :,0:0,0:05,: : :,0:99,0:99,: : :,0:999,0:9995,0:9998,0:9999. However, the result of augmented critical values indicates that due to the large response surface regression errors, quantiles smoothed by response surface regressions are not appropriate to use. his is shown is section 6. So the 22 point estimates with certain sample sizes from Monte Carlo experiments are used here. For sample sizes not considered in the experiments, the nearest to those can be applied instead. o interpolate between the 22 points in order to obtain p-value for any given test statistic, the procedure then involves an estimation () = 0 + eq i + 2 (eq i ) (eq i ) 3 + (4) where () is the inverse of cumulative standard normal distribution at, since the asymptotic distributions of CH and CS tests are standard normal; eq i is the point quantile estimate from Monte Carlo experiment for the given sample size N and. he idea of regressing (4) is to use a small number of points in the neighborhood of the test statistic to estimate the relationship between the empirical distribution and standard normal distribution. For example, suppose the distribution of interest is DGP3 with sample size N = 0; = 00, and the empirical test statistic is bq N = 2:72. Among the estimated 22 quantiles from corresponding Monte Carlo experiment, the closest one to this statistic is eq N ( = 0:006) = 2:7326. If 9 points are to be used, (4) is to be estimated with quantiles for = f0:002; 0:003; 0:004; 0:005; 0:006; 0:007; 0:008; 0:009; 0:00g. 9 9 MacKinnon (200) suggests that shown by experiments, 9, and 3 points are reasonable numbers to use and 9 points is a good choice. Hence, 9 points is used here. 4

15 Using the estimators in (4), the p-value of an empirical test statistic, bq N, can be computed from p = b 0 + b bq N + b 2 (bq N ) 2 + b 3 (bq N ) 3 (5) MacKinnon (996) con rmed the reliable accuracy of the approximation of p-values. He uses 980 evenly spaced points between 0:0 and 0:99 for two sets of simulated data and compares the percentiles for the original estimated quantiles and the approximated p-values of the quantiles. Since b 0 + b bq N + b 2 (bq N ) 2 + b 3 (bq N ) 3 approximates the cumulative distribution function of the statistic bq N, the approximate density function can also derived by taking the rst derivative of (5) f (bq N ) t b 0 + b bq N + b 2 (bq N ) 2 + b 3 (bq N ) 3 b + b 2 bq N + b 3 (bq N ) 2 6 Results 6. Estimation results of response surface regressions (6) he primary results of response surface estimations are presented in able 2. he table contains the coe cients of variables in the regression and the value of R 2 for each estimation. Since N p ln( ) is an important condition pointed out by Im and Pesaran (2003) for the asymptotic properties of CH test to hold and a similar situation may also be conjectured to CS test as discussed in section 3, the standard errors for b 4 (coe cient of N p ln( ) ) is provided. able 2: Response surface regression estimates b b b 2 b 3 4 R 2 DGP 0:0 5:8576 4: :4385 0:003 (0:0003) 0:640 0:05 6:2600 :8075 0:403 0:0046 (0:0002) 0:7979 0:0 6:8405 0:299 0:9532 0:0057 (0:0002) 0:8363 DGP2 0:0 27:2984 3: :8065 0:08 (0:0085) 0:8455 0:05 20:2953 7:828 24:0752 0:0082 (0:0006) 0:8347 0:0 7:2754 5: :475 0:0066 (0:00047) 0:823 DGP3 0:0 52:5797 2: :0970 0:0302 (0:0022) 0:8309 0:05 72: : :582 0:037 (0:002) 0:849 0:0 44:36 23:07 7:0348 0:008 (0:00084) 0:8040 5

16 he standard errors of b 4 in all the three DGPs (both CH and CS tests) are signi cantly small that it suggests the estimation accuracy of b 4 and implies the important role of N p ln( ) in both the nite and asymptotic properties of the two tests. he goodness-of- t of response surface regressions is reasonably high. Regressions for CS test have better performance than CH test in terms of R 2. he value of R 2 for all the signi cant levels for DGP2 and DGP3 regressions are higher than 0: Finite sample bias Figure and 2 plot the estimates of the %, 5% and 0% quantiles with various sample sizes N and for the three DGPs (for illustration the estimates of 5% quantile in Figure are presented in the text; the rest in Figure 2 are attached in the appendix). he values used to plot Figure and 2, bq i, are computed through response surface regression estimates using N = f5; 5; 25; :::; 05g and = f25; 50; 75; :::; 500g in (2). he combination of N and is due to the consideration of sample sizes used in Monte Carlo simulations. here are a few properties that can be observed from the gure. For all the DGPs as increases it tends to drag down the value of bias, whereas as N increases the value of bias tends to rise. he e ect caused by is similar to all DGPs. When the size of is small, an increase in causes large down movement of the value of bias; as becomes large, the e ect mitigates. When N increases, the response of bias value shows same constant growing pattern in DGP and 2; whereas the growing rate in DGP3 slows down after certain size of N. Moreover, all graphs illustrate signi cant magnitude of bias with small and large N, and suggest that as both N and go large, bias tend to be eliminated. However, Figure and 2 are plotted on di erent scales of N and with growing faster than N. Figure 3 uni es the scale of N and and shows di erent result regarding the magnitude of bias. An addition view for DGP3 from a di erent angle is provided to give better visibility. As N and both increase at the same rate, the size of bias keeps constantly growing. he bias in DGP3 goes in the opposite direction against that of DGP and 2 as shown in Figure 3. he position where bias tends to disappear is where is large and N is small, which again con rms the point of Im and Pesaran (2003). 6

17 6.3 he augmented critical values According to the discussed in section 5.3, considering the uncertainties resulting from Monte Carlo simulations and regression estimation, con dence intervals for percentiles are computed. o see the magnitude of the uncertainties, able 3 presents the gap between the upper limit and lower limit, i.e. (uq lq ). able 3: Gap between upper limit and lower limit (uq lq ) DGP DGP2 DGP3 0:0 0:3437 0:7865 :5630 0:05 0:282 0:596 0:8259 0:0 0:865 0:470 0:630 As expected, the gap increases with decreasing percentiles, since empirical quantiles tend to be more volatile in the extremes. Unfortunately due to the large standard error of response surface regression residuals, the gap is so large that some intervals of neighborhood percentiles (e.g: % and 5%) overlap. his is particularly serious for DGP3. herefore, here response surface estimates are not applicable and point estimates from Monte Carlo simulations are to be applied and construct the 95% percentile con dence interval using only David-Johnson estimate of percentile standard deviation, i.e. eq ul = eq i :96 se () where eq i denotes the quantile obtained from the ith Monte Carlo experiment with sample size N and. 6.4 he numerical distributions Figures 4-6 are the plots the nite sample cumulative distribution functions (CDF) and probability density functions (PDF) for the three DGPs along with the plot of standard normal distribution (for illustration Figure 5 for DGP2 is presented in the text; Figure 4 and 6 for DGP and 3 are attached in the appendix). All gures are constructed using 22 points. he calculation of empirical p-values to form the CDFs is discussed in section 5.4. he values for PDFs are computed using the results of response surface regressions. Plots of N = f0; 50; 00g and = f50; 00; 200; 300; 500g are chosen for illustration as shown in Figures 4-6. o clearly observe the trend as increases, plots of N = f50g and = f50; 00; 200; 300; 500g are extracted and provided in Figures 7-9 (for the same reason, Figure 8 for DGP2 9

18 is presented in the text; Figure 7 and 9 for DGP and 3 are attached in the appendix). he nite sample distributions of all the three DGPs suggest signi cant di erence from standard normal distribution and each DGP has its own feature. Keep constant and let N increase, the numerical CDF and PDF of DGP move to the right (Figure 4); CDF of DGP2 tends to move anticlockwise and PDF grows taller and thinner (Figure 5); the left tail of CDF in Figure 6 for DGP3 is particularly unstable and move toward that of standard normal distribution. Let grow given constant N, the numerical CDF and PDF of DGP both shift to the right slowly rst and then back toward the left as shown in the enlarged views in Figure 7. he numerical CDF of DGP2 tends to move to the left and PDF is pressed atter, just in the opposite direction to that as N increases (Figure 8); the CDF of DGP3 also shifts in the opposite direction to that as N increases, left tail goes away from standard normal distribution, and PDF follows the same pattern as that of DGP2 (Figure 9). In general, increase in N or only drives nite sample distributions away from that of standard normal distribution; whereas growth in the other tends to o set the e ect and bring the numerical distributions back to standard normal. However, the speed of convergence is seriously slow. he graphs in section provide additional evidence that simply applying critical values from standard normal distribution for CH and CS tests is highly unreliable. 7 Conclusion his paper has assessed the nite sample performance of newly developed panel unit root tests Chang (2002) and Chang and Song (2005) using response surface analysis. Although comparing with traditional time series unit root tests panel unit root testing statistics exhibit standard normal distribution asymptotically, the nite sample performance can substantially di ert from asymptotic properties. Simply apply critical values given by asymptotic distribution may thus lead to serious mistakes. A number of Monte Carlo experiments are conducted on CH and CS tests. Results are used to estimate response surface regressions to study nite sample bias of the two tests. Formulas to compute 95% con dence intervals of the critical values for each percentile are provided using David- Johnson percentile standard deviation. In addition, following MacKinnon (996, 200), computation of p-values of empirical test statistic is also available. Numerical distributions for some certain sample sizes are presented 22

19 in Figures. he plots of both nite sample bias and distributions suggest that empirical quantiles are systematically dependent on sample size (N and ), i.e. empirical critical values signi cantly di er from asymptotic ones. herefore our results have satisfactorily demonstrated the signi cant di erences between nite sample and large sample properties and highlight their important roles in empirical applications. he results also seem to con rm the statement in Im and Pesaran (2003) that N ln( ) p is an important condition for the asymptotic properties of CH in the response surface regressions is very precisely estimated. he plots of nite sample bias also provide strong support for the existence of this problem. Nevertheless, by applying the nite sample critical values, this problem should not pose a danger to applications. Although the results are in the favour of empirical application, they do show some limitation due to the intermediate goodness-of- t of response surface regressions. he standard errors of residuals are comparatively large and in turn enlarge the gap between a percentile s upper and lower limits (see equation (3) and able 3). his also re ects the limitations of Monte Carlo simulations. A general notable limitation for unit root tests is that the idiosyncratic term in the model is assumed to be normally distributed, whereas this assumption is hardly the case for empirical data. If normality is relaxed in the model, the corresponding asymptotic distributions can be seriously distorted. An extended analysis of relaxation of the assumption is of interest in our further studies. test to hold, given that the coe cient of N ln( ) p References [] Bai, J. and S. Ng (2004), A Panic Attack on Unit Roots and Cointegration. Econometrica, 72, [2] Baltagi, B.H. (2008), Econometric Analysis of Panel Data, Fourth Edition. John Wiley & Sons. [3] Banerjee, A., M. Marcellino and C. Osbat (2005), esting for PPP: Should we use Panel Methods?. Empirical Economics, 30, [4] Breitung, J. and S. Das (2005), Panel Unit Root ests Under Cross Sectional Dependence. Statistica Neerlandica, 59,

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