Testing for Serial Correlation in Fixed-Effects Panel Data Models

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1 Econometric Reviews ISS: Print Online Journal homepage: Testing for Serial Correlation in Fixed-Effects Panel Data Models Benjamin Born & Jörg Breitung To cite this article: Benjamin Born & Jörg Breitung 206 Testing for Serial Correlation in Fixed-Effects Panel Data Models, Econometric Reviews, 35:7, , DOI: 0.080/ To link to this article: Accepted author version posted online: 20 Oct 204. Published online: 20 Oct 204. Submit your article to this journal Article views: 226 View related articles View Crossmark data Full Terms & Conditions of access and use can be found at Download by: [KU Leuven University Library] Date: 8 July 206, At: 04:57

2 Econometric Reviews, 357:290 36, 206 Copyright Taylor & Francis Group, LLC ISS: print/ online DOI: 0.080/ Testing for Serial Correlation in Fixed-Effects Panel Data Models Benjamin Born and Jörg Breitung 2 University of Mannheim, Mannheim, Germany 2 University of Cologne, Cologne, Germany In this article, we propose various tests for serial correlation in fixed-effects panel data regression models with a small number of time periods. First, a simplified version of the test suggested by Wooldridge 2002 and Drukker 2003 is considered. The second test is based on the Lagrange Multiplier LM statistic suggested by Baltagi and Li 995, and the third test is a modification of the classical Durbin Watson statistic. Under the null hypothesis of no serial correlation, all tests possess a standard normal limiting distribution as tends to infinity and T is fixed. Analyzing the local power of the tests, we find that the LM statistic has superior power properties. Furthermore, a generalization to test for autocorrelation up to some given lag order and a test statistic that is robust against time dependent heteroskedasticity are proposed. Keywords Fixed-effects panel data; Hypothesis testing; Serial correlation. JEL Classification C2; C23.. ITRODUCTIO Panel data models are increasingly popular in applied work as they have many advantages over cross-sectional approaches see, e.g., Hsiao, 2003, p. 3ff. The classical linear panel data model assumes serially uncorrelated disturbances. As argued by Baltagi 2008, p. 92, this assumption is likely to be violated as the dynamic effect of shocks to the dependent variable is often distributed over several time periods. In such cases, serial correlation leads to inefficient estimates and biased standard errors. owadays, cluster robust standard errors are readily available in econometric software like Stata, SAS, or EViews that allow for valid inference under heteroskedasticity and autocorrelation. Therefore, many practitioners consider a possible autocorrelation as being irrelevant, in particular, in large data sets where efficiency gains are less important. On the other hand, a significant Address correspondence to Jörg Breitung, Center of Econometrics and Statistics, University of Cologne, Meister-Ekkehart-Str. 9, Cologne, Germany; breitung@statistik.uni-koeln.de

3 TESTIG FOR SERIAL CORRELATIO 29 test statistic provides evidence that a static model may not be appropriate and should be replaced by a dynamic panel data model. A number of tests for serial error correlation in panel data models have been proposed in the literature. Bhargava et al. 982 generalized the Durbin Watson statistic to the fixed-effects panel model. Baltagi and Li 99, 995 and Baltagi and Wu 999 derived Lagrange Multiplier LM statistics for first order serial correlation. Drukker 2003, elaborating on an idea originally proposed by Wooldridge 2002, developed an easily implementable test for serial correlation based on the ordinary least-squares OLS residuals of the first-differenced model. This test is referred to as the Wooldridge Drucker henceforth: WD test. Inoue and Solon 2006 suggested a portmanteau statistic to test for autocorrelations at any lag order. However, all these tests have their limitations. A serious problem of the Bhargava et al. 982 statistic is that the distribution depends on the number of cross-section units and T the number of time periods and, therefore, the critical values have to be provided in large tables depending on both dimensions. Baltagi and Li 995 noted that, for fixed T, their test statistic does not possess the usual 2 limiting distribution due to the ickell bias in the estimation of the autocorrelation coefficient. The WD test is based on first differences, which may imply a loss of power against some particular alternatives. Furthermore, these tests are not robust against temporal heteroskedasticity and are not applicable to unbalanced panels with the exception of the WD test and the Baltagi and Wu, 999, statistic. Inoue and Solon 2006 proposed an LM statistic for the more general null hypothesis: Eu it u is = 0 for all t < s, where u it denotes the error of the panel regression. This portmanteau test is attractive for panels with small T as it is not designed to test against a particular alternative. If T is moderate or large, however, the size and power properties suffer from the fact that the dimension of the null hypothesis increases with T 2. In this article, we propose new test statistics and modifications of existing test statistics that sidestep some of these limitations. In Section 2, we first present the model framework and briefly review the existing tests. Our new test procedures are considered in Sections 3 5 and the small sample properties of the tests are studied in Section 6. Section 7 concludes. 2. PRELIMIARIES Consider the usual fixed effects panel data model with serially correlated disturbances y it = x it + i + u it u it = u i,t + it, 2 where i =,, denotes the cross-section dimension and t =,, T is the time dimension, y it is the dependent variable, x it is a k vector of regressors, is the

4 292 B. BOR AD J. BREITUG unknown vector of associated coefficients, i is the fixed individual effect, and u it denotes the regression error. In our benchmark situation, the following assumption is imposed. Assumption. i The error it is independently distributed across i and t with E it = 0, E 2 it = 2 i with c < 2 i < c for all i =,, and some finite positive constant c. Furthermore, E it 4+ < for some >0. ii The vector of coefficients is estimated by a consistent estimator ˆ with ˆ = O p /2. iii For the regressors it holds that for fixed T and where C = trc C<. T t= Ex it x it C, All tests are based on the residual ê it = y it x it ˆ = i + u it x it ˆ. Assumption ii allows us to replace the residuals ê it by e it = i + u it in our asymptotic considerations. If x it is assumed to be strictly exogenous, the usual within-group or least-squares dummyvariable, cf. Hsiao, 2003 estimator or the first difference estimator cf. Wooldridge, 2002 may be used. If x it is predetermined or endogenous, suitable instrumental variable or Generalized Method of Moments GMM estimators of satisfy Assumption ii e.g., Wooldridge, It is important to notice that the efficiency of the estimator ˆ does not affect the asymptotic properties i.e., size or local power of tests based on ê it. To test the null hypothesis = 0, Bhargava et al. 982 proposed the pooled Durbin Watson statistic given by T t=2 pdw = û it û i,t 2, T t= û2 it where û it = y it x it ˆ ˆ i, and ˆ and ˆ i denote the least-squares dummy-variable or within-group estimator of and i, respectively. A serious problem of this test is that its null distribution depends on and T and, therefore, the critical values are provided in large tables depending on both dimensions in Bhargava et al Furthermore, no critical values are available for unbalanced panels. Baltagi and Li 995 derive the LM test statistic for the hypothesis = 0 assuming normally distributed errors. The resulting test statistic is equivalent to the LM version of the t-statistic of in the regression û it = û i,t + it 3

5 TESTIG FOR SERIAL CORRELATIO 293 The LM test statistic results as T LM = 2 T T t=2 ûitû i,t T t= û2 it Baltagi and Li 995 show that if and T, the LM statistic has a standard normal limiting distribution. However, if T is fixed and, the application of the respective critical values leads to severe size distortions see Section TEST STATISTICS FOR FIXED T In this section, we consider test procedures that are valid for fixed T and. All statistics have the general form T = ẑti ẑ2 Ti ẑti 2, 4 where ẑ Ti =ê i A T ê i, ê i =[ê i,, ê it ], ê it = y it x it ˆ, and A T is a deterministic T T matrix. The matrix A T is constructed such that it eliminates the individual effects from ê i. Specifically, we make the following assumption with respect to the matrix A T. Assumption 2. i Let T denote the T vector of ones. The T T matrix A T obeys A T T = 0, A T T = 0, and tra T = 0. ii Let X i =[x i,, x it ] and u i =[u i,, u it ]. For all i =,,, it holds that E[X i A T + A T u i]=0. In Lemma A. of the appendix, it is shown that under Assumptions and 2 the estimation error ˆ is asymptotically negligible. ote that, in general, Assumption 2 ii is violated if x it is predetermined or endogenous. The following lemma presents the limiting distribution of the test statistic under a sequence of local alternatives. Lemma. i Under Assumptions and 2, = 0, fixed T, and, the test statistic T has a standard normal limiting distribution.

6 294 B. BOR AD J. BREITUG ii Under the local alternative = c/, it 0, i 2, and Assumptions and 2 the limiting distribution is given by d T c tra T H T,, tra 2 T + A T A T where = m 2 / m 4 with m k = lim k i and H T isat T matrix with t, s-element H T,ts = for t s = and zeros elsewhere. A straightforward extension to unbalanced panel data with T i consecutive observations in group i and ê i A T i ê i yields Ti d c lim 2 i tra T i H Ti lim 4 i tra 2 T i + A T i A Ti, Hence, to accommodate unbalanced panel data, the tests are computed with individual specific transformation matrices A Ti instead of a joint matrix A T. 3.. The WD Test Statistic To obtain a valid test statistic for fixed T, Wooldridge 2002, p. 282f suggests to run a least squares regression of the differenced residuals ê it =ê it ê i,t on the lagged differences ê i,t. 2 Under the null hypothesis = 0, the first order autocorrelation of ê it converges in probability to 05. Since ê it is serially correlated, Drukker 2003 suggests to employ heteroskedasticity and autocorrelation consistent HAC standard errors, 3 yielding the test statistic WD = ˆ + 05 ŝ Here, ˆ denotes the least-squares estimator of in the regression ê it = ê i,t + it, 5 As pointed out by Baltagi and Wu 999, the case of gaps within the sequence of observations is more complicated. 2 Since this test employs the least-squares estimator in first differences, this test assumes Ex it u it = 0 or Ex it u is = 0 for all t and s t, t, t +. 3 This approach is also known as robust cluster or panel corrected standard errors.

7 TESTIG FOR SERIAL CORRELATIO 295 and ŝ 2 is the HAC variance estimator computed as ŝ 2 = T t=3 ê i,t ˆ it 2 T t=3 ê2 i,t with ˆ it as the pooled OLS residual from the autoregression 5. In the following theorem, we propose a simplified and asymptotically equivalent version of the WD test. 2, Theorem. Let ẑ Ti = T t=3 ê it 2êi,t 2êi,t 2ê i,t ê i,t 2, and denote by WD the corresponding test statistic constructed as in 4. i Under Assumptions and 2, u it 0, 2 i,t 3, and = c/, it follows that WD d c T 2,, 2T where is defined in Lemma. ii The test statistic WD is asymptotically equivalent to WD in the sense that WD WD p 0 as. Remark. The form of the test statistic results from T t=3 ˆ + 05 = ê itê i,t + 2 ê2 i,t T t=3 ê = ẑti i,t 2 T t=3 ê i,t 2 The WD statistic deviates from WD by computing the HAC variance based on the residuals obtained under the null hypothesis, i.e., 0 it = ê it + 05ê i,t instead of the residuals ˆ it used to compute the original WD statistic. Remark 2. It is interesting to note that under the alternative, we have as ˆ + 05 p r r 2 2 r + OT, where r j is the jth autocorrelation of u it. This suggests that WD and WD is a test against the difference between the first and second order autocorrelation of the errors. Therefore, the test is expected to have poor power against alternatives with r r 2. Remark 3. The test statistic is robust against cross-sectional heteroskedasticity. However, using = 05 requires that the variances do not change in time, i.e., Euit 2 = i 2 for all t. Thus, this test rules out time dependent heteroskedasticity.

8 296 B. BOR AD J. BREITUG 3.2. The LM Test An important feature of the LM test suggested by Baltagi and Li 995 is that the limit distribution depends on T. This is due to the fact that the least-squares estimator of in regression 3 is biased and the errors it in 3 are autocorrelated. Under fairly restrictive assumptions the following asymptotic null distribution is obtained. iid Lemma 2. Under Assumption and u it 0, 2 for all i and t, it holds for fixed T and that d T + T 22 LM + 0, T T 3 It follows that applying the usual critical values derived from the 2 distribution to the two-sided test statistic LM 2 yields a test with actual size tending to unity as. ote also that the limit distribution of the LM statistic tends to a standard normal distribution if T and /T 0. To obtain an asymptotically valid test for fixed T, the transformed statistic T LM = 3 LM + T + T 2 2 T may be used, which has a standard normal limiting null distribution. An important limitation of this test statistic is, however, that the result requires the errors to be normally distributed with identical variances for all i,,. To obtain a test statistic that is valid in more general and realistic situations, we follow Wooldridge 2002, p. 275 and construct a test of the least squares estimator ˆ of in the regression ê it T T ê it = ê i,t T t= T ê it + it 6 t= Let M and M T denote matrices that result from M = I T T T T with I T indicating the T T identity matrix by dropping the first or the last row, respectively. Using this matrix notation, we obtain ˆ p 0 = trm M T /T = T trm T M T T 2 /T = 7 T Thus the regression t-statistic is employed to test the modified null hypothesis H 0 : = 0 = /T. To account for autocorrelation in it, HAC robust standard errors are

9 TESTIG FOR SERIAL CORRELATIO 297 employed, yielding the test statistic LM = ˆ 0 ṽ, where ṽ 2 = ê i M T i i M T ê i 2 = ê im T M T ê i ê i M T M ˆM T ê i ê i M ˆM T M T ê i 2 ê im T M T ê i As for the WD test, we propose a simplified version of this test that is asymptotically equivalent to the statistic LM. Theorem 2. Let T ẑ Ti = ê it T t=2 T ê it ê i,t T t= T ê it t= + ê i,t T T and denote by LM the corresponding test statistic constructed as in 4. T t= 2 ê it, i Under Assumptions and 2, u it 0, 2 i,t 3, and the local alternative = c/, the test statistic LM is asymptotically distributed as 2 c T 3 + T 2 T, ii Under the the local alternative, the statistic LM is asymptotically equivalent to LM A Modified Durbin Watson Statistic The pdw statistic suggested by Bhargava et al. 982 is the ratio of the sum of squared differences and the sum of squared residuals. Instead of the ratio which complicates the theoretical analysis, our variant of the Durbin Watson test is based on the linear combination of the numerator and denominator of the Durbin Watson statistic. Let ẑ Ti =ê i MD DMê i 2 ê i Mê i, 8

10 298 B. BOR AD J. BREITUG where D is a T T matrix producing first differences, i.e., D =, and M as defined above. Using trmd DM = 2T and trm = T, it follows that Ez Ti = 0 for all i, where z Ti = e i MD DMe i 2 e i Me i. The panel test statistic is constructed as in 4, where ẑ Ti is given by 8. The resulting test is denoted by mdw. The following theorem presents the asymptotic distribution of the test statistic. Theorem 3. Under Assumptions and 2, u it 0, i 2, T 3, and the local alternative = c/, the limiting distribution is 3.4. Relative Local Power mdw d c T T 2, T Theorems 3 allow us to compare the asymptotic power of the three different tests. Let WD T = T 2/ 2T denote the Pitman drift of the local power resulting from Theorem i, and denote by LM T and mdw T the respective terms resulting from Theorem 2i and Theorem 3. The test A is asymptotically more powerful than test B for some given value of T if A T/ B T >, where A, B WD, LM, mdw. In what follows, we use the Pitman drift of the WD test as a benchmark and define the relative Pitman drift as LM T = LM T/ WD T and mdw T = mdw T/ WD T. Figure presents the relative Pitman drifts of both tests as a function of T. It turns out that for the minimum value of T = 3 the mdw test is asymptotically more powerful among the set of three alternative tests. For T 4, however, the LM test is asymptotically most powerful, where the difference between the LM and the mdw tests diminish as T gets large. The power gain of both tests relative to the WD test approaches 40% in terms of the Pitman drift of the WD test. ote that the asymptotic power of some test A WD, LM, mdw is a nonlinear function of A T. For small values of c, however, the asymptotic power is approximately linear. In Table we report the asymptotic power of the tests in parentheses for various values of c and T. For example, consider c = 05 and T = 50. The asymptotic power of the LM test relative to the WD test is 0929/0683 = 36 and the relative asymptotic power of the mdw test results as 0923/0683 = 35, which roughly corresponds to the relative Pitman drifts LM 50 and LM 50.

11 TESTIG FOR SERIAL CORRELATIO 299 FIGURE Comparison of local power functions. 4. HIGHER ORDER AUTOCORRELATIO Since the estimation error x it ˆ in the residuals ê it = e it x it ˆ is asymptotically negligible, we simplify our notation and write e it instead of ê it in this section. 4.. The Inoue-Solon Statistic A test for higher order serial correlation in panel data models was proposed by Inoue and Solon Due to the fact that the covariance matrix of the demeaned error vector i = EMe i e i M is singular with rank T, we drop the kth row and column of i yielding the invertible matrix i,k. The null hypothesis of no autocorrelation implies that i,k = 2 i M k, where M k results from M by dropping the kth row and column of M. Denote by S i,k the matrix that results from deleting the kth row and columns of S i = Me i e i M. The LM statistic is given by IS k = s i,k, 0 s i,k s i,k s i,k

12 300 B. BOR AD J. BREITUG TABLE Size and Power of Alternative Tests T WD WD LM LM mdw c = c = c = = 500. Empirical values are computed using 0,000 Monte Carlo simulations. Asymptotic values given in parentheses are computed according to the results in Theorems 3. where s i,k = vechs i,k 2 i M k and vecha denotes the linear operator that stacks all elements of the m m matrix A below the main diagonal into a mm /2 dimensional vector. In empirical practice the unknown variances 2,, 2 are replaced by the unbiased estimate ˆ2 i = T e i Me i.

13 TESTIG FOR SERIAL CORRELATIO 30 An important problem with this test statistic is that it essentially tests a T 2T /2 dimensional null hypothesis. Thus, to yield a reliable asymptotic approximation of the null distribution, should be large relative to T 2 /2. Another problem is that the power of the test is poor if only a few autocorrelation coefficients are substantially different from zero. For example, if the errors are generated by a first order autoregression with u it = 0u i,t + it, then the second and third order autocorrelations are as small as 0.0 and In such cases, a test that includes all autocorrelations up to the order T lacks power. In order to improve the power of the test statistic in such cases, the test statistic should focus on the first order autocorrelations. Inoue and Solon 2006 suggest to construct a test statistic based on the first p autocorrelations by forming the lower-dimensional vector s i p = s ts i t s p with elements s ts i = e it ē i e is ē i + 2 i /T for all t, s,, T with t s p. The LM statistic is obtained as in 0 where the vector s i,k is replaced by s i p. The resulting test statistic is denoted by ISp and has an asymptotic 2 distribution with pt pp + /2 degrees of freedom Bias-Corrected Test Statistics An alternative approach to test against autocorrelation of order k is to consider the kth order autoregression e it ē i = k e i,t k ē i + it, t = k +,, T; i =,, In the following lemma, we present the probability limit of the OLS estimator of k for fixed T and. Lemma 3. Let ˆ k be the pooled OLS estimator of k in. Under Assumption and = 0, we have ˆ k p T, for all k,, T 2. Using this lemma, a test for zero autocorrelation at lag k can be constructed based on the heteroskedasticity and autocorrelation robust t-statistic of the hypothesis k = /T. An asymptotically equivalent test statistic is obtained by letting z k Ti = T t=k+ [ e it ē i e i,t k ē i + ] T e i,t k ē i 2

14 302 B. BOR AD J. BREITUG and LM k = zk2 Ti zk Ti zk Ti 2 2 Along the lines of the proof of Theorem 2, it is not difficult to show that, under the null hypothesis, LM k has a standard normal limiting distribution. In empirical practice it is usually more interesting to test the hypothesis that the errors are not autocorrelated up to order p. Unfortunately, the probability limit of the autoregressive coefficients,, p in the autoregression e it ē i = e i,t ē i + + p e i,t p ē i + it is a much more complicated function of T, since the regressors are mutually correlated. Instead of presenting the respective probability limits, we therefore suggest a simple transformation of the regressors such that the autoregressive coefficients can be estimated unbiasedly under the null hypothesis. Specifically, we propose an implicit correction that eliminates the bias. In matrix notation, the transformed autoregression is given by where Me i = L e i + 2 L 2 e i + + p L p e i + i, 3 L k e i = 0 k e i ē i e i,t k ē i + T k e T 2 i, T where the first vector on the right-hand side represents the lagged values of the demeaned residuals, whereas the second vector is introduced to remove the bias. It is not difficult to show that T k Ee i ML ke i = T 2 i + T T T k 2 i = 0 T for all k =,, p and, therefore, under the null hypothesis, the least-squares estimators of,, p converge to zero in probability for all T and. The null hypothesis = = p = 0 can therefore be tested using the associated Wald statistic Qp = ˆ Ŝ ˆ, 4

15 TESTIG FOR SERIAL CORRELATIO 303 where Ŝ = Z i Z i Z iˆ iˆ i Z i Z i Z i is the HAC [ estimator ], for the covariance matrix of the vector of coefficients with ˆ = ˆ,, ˆ p Zi = L e i,, L p e i, and ˆ i = Me i Z i ˆ. The Qp statistic is asymptotically equivalent to the statistic Qp = [ ] e i MZ i Z i Me ie i MZ i Z i Me i e i MZ i Z i Me i Along the lines of Theorem 2, it is straightforward to show that the statistics Qp and Qp are asymptotically equivalent and have an asymptotic 2 distribution with p degrees of freedom. 5. A HETEROSKEDASTICITY ROBUST TEST STATISTIC An important drawback of all test statistics considered so far is that they are not robust against time dependent heteroskedasticity. This is due to the fact that the implicit or explicit bias correction of the autocovariances depends on the error variances. To overcome this drawback of the previous test statistics, we construct an unbiased estimator of the autocorrelation coefficient. The idea is to apply backward and forward transformations such that the products of the transformed series are uncorrelated under the null hypothesis. Specifically, we employ the following transformations for eliminating the individual effects: ẽ f it =ê it êit + +ê it, T t + ẽ b it =ê it êi + +ê it t The hypothesis can be tested based on the regression ẽ f it = ẽ b i,t + it, t = 3,, T, 5 or, in matrix notation, V 0 ê i = V ê i + w i,

16 304 B. BOR AD J. BREITUG where the T 3 T matrices V 0 and V are defined as V 0 = T T 2 T 2 T 2 T 2 T 2 and 0 0 T V = T 2 T 2 T 2 T 2 T 2 T 4 T 3 T 3 T T Under the null hypothesis of no first order autocorrelation, we have = 0. Since the error vector w i is heteroscedastic and autocorrelated, we again employ robust HAC standard errors. The resulting test statistic is equivalent to the test statistic where HR = ẑ Ti =ê i A T T ê i = ẑti ẑ2 Ti ẑti ẽitẽ f b i,t t=3 2, with A T = V 0 V Since V 0 T, V T = 0, and trv 0 V = 0, it follows from Lemma i that the test statistic HR has a standard normal limiting null distribution. 6. MOTE CARLO STUDY This section presents the finite sample performance of the test statistics for serial correlation and assesses the reliability of the asymptotic results derived in Section 3. We also conduct experiments with different forms of serial correlation and heteroskedasticity. The benchmark data generating process for all simulations is a linear panel data model of the form y it = x it + i + u it,

17 TESTIG FOR SERIAL CORRELATIO 305 where i =,, and various values of T. We set to in all simulations and draw the individual effects i from a 0, 25 2 distribution. To create correlation between the regressor and the individual effect, we follow Drukker 2003 by drawing xit 0 from a 0, 8 2 distribution and computing x it = xit i. The regressor is drawn once and then held constant for all experiments. In our benchmark model, the disturbance term follows an autoregressive process of order : u it = u i,t + it, where it 0, and we discard the first 00 observations to eliminate the influence of the initial value. 5 Table compares the size and power of alternative test statistics where, according to our theoretical analysis, we let = c/. The empirical values are computed using 0,000 Monte Carlo simulations, while the asymptotic values given in parentheses are computed according to the results given in Theorems 3. All tests exhibit good size control i.e., for c = 0 the actual size is close to the nominal size. They reject the null hypothesis of no serial correlation slightly more often than the nominal size of 5% but are never above 6%. In all cases, the LM statistic has superior power compared to the competing statistics. While the modified Durbin Watson statistic performs similar to the LM statistic, the WD test exhibit considerably less power. To evaluate the small-sample properties of the test for higher-order autocorrelation Qp, we simulate a model with disturbances following a second-order autoregressive process: AR2:u it = u i,t + 2 u i,t 2 + it, where again it 0, and we discard the first 00 observations to eliminate the influence of the initial value. The results for three different AR2 processes are presented in Table 2. The Q2 statistic has good power properties in all configurations considered. ot surprisingly, the WD test exhibit high power when = 003 and 2 = 003 or = 0 and 2 = 008. As mentioned in Remark 2, however, the WD and WD statistic is a test of the difference between the first and second order autocorrelation of the errors. As a consequence, it also has power against most AR2 alternatives. However, choosing = 2 = 003, which implies equal first- and second-order autocorrelations of u it, leads to a complete loss of power of the WD test, confirming our considerations in Remark 2. The LM statistic has less power than the Q2 statistic, in particular for the setup with = 0 and 2 = The cross-sectional dimension is held constant at = 500, results for smaller are qualitatively similar. Values of c = 0, c = 05, and c = therefore imply = 0, = 00244, and = 00447, respectively. 5 As suggested by a referee, we also consider non-gaussian error distributions. The results were very similar and can be obtained on request.

18 306 B. BOR AD J. BREITUG TABLE 2 AR2 Processes T WD WD LM LM mdw Q2 IS k IS2 = 0, 2 = na na = 003, 2 = na na = 2 = na na 0.6 = 0, 2 = na na.000 = 500. Rejection frequencies are computed using 0,000 Monte Carlo replications. ote that the IS test statistics cannot be computed if the degrees of freedom are larger than na. In the last two columns of Table 2 we present the original portmanteau statistic IS k with k = and the IS2 statistic that is based on all autocorrelation up to to the lag order p = 2 see Section 4. ote that for a sample with = 500 the inverse of IS k does not exist for T > 20. It turns out that for the AR2 alternatives considered in this Monte Carlo experiment 6 the Q2 statistics has substantially more power than the variants of the IS statistic. ext, we consider four different types of time-dependent heteroskedasticity by using the following error process: u it = h t it, 6 We also considered a wide range of other combinations of and 2. In all cases, the Q2 turns out to be more powerful than the IS statistics.

19 TESTIG FOR SERIAL CORRELATIO 307 TABLE 3 Size under Temporal Heteroskedasticity T WD LM mdw HR WD LM mdw HR break in variance u-shaped variance negative exponential trend 0.2 positive exponential trend = 500. Rejection frequencies are computed using 0,000 Monte Carlo replications. where it 0,. The first model implies a break in the variance function, i.e., h t = 0 for t =,, T/5, and h t = otherwise. The second specification is a U-shaped variance function of the form h t = t T/ The third and fourth types of heteroskedasticity are positive and negative exponential variance functions, i.e., h t = exp 02t and h t = exp02t, respectively. Table 3 shows that the modified Durbin Watson test has massive size distortions under all four types of heteroskedasticity. While the WD test has the proper size in case of a U-shaped variance function, the break in the variance function and the exponential variance functions lead to large size distortions. The modified LM statistic does surprisingly well under heteroskedasticity, but for small T we observe also some size distortions, especially for the negative exponential variance function. On the other hand, our proposed heteroskedasticity-robust test statistic retains its correct size under all types of heteroskedasticity considered here. 7. COCLUSIO This article proposes various tests for serial correlation in fixed-effects panel data regression models with a small number of time periods. First, a simplified version of the test suggested by Wooldridge 2002 and Drukker 2003 is considered. The second test is based on the LM statistic suggested by Baltagi and Li 995, and the third test is a modification of the classical Durbin Watson statistic. Under the null hypothesis of no serial correlation, all tests possess a standard normal limiting distribution as and T is fixed. Analyzing the local power of the tests, we find that the LM statistic has superior power properties. We also propose a generalization to test for autocorrelation up to some given lag order and a test statistic that is robust against time dependent heteroskedasticity.

20 308 B. BOR AD J. BREITUG Proof of Lemma APPEDIX: PROOFS as i Let y i =[y i,, y it ] and X i =[x i,, x it ]. The residual vector can be written ê i = e i X i ˆ The following lemma implies that the estimation error in the residuals ê it is asymptotically negligible. Lemma A.. i ii Proof. where Under Assumptions and 2, the following equations hold: ê i A T ê i = e i A T e i + O p ; ê i A T ê i 2 = e i A T e i 2 + O p /2. i Let ê i A T ê i = e i A T e i + i, i = e i A T + A T X iˆ + ˆ X i A T X i ˆ = u i A T + A T X iˆ + ˆ X i A T X i ˆ We have e i A T + A T X iˆ Using Assumption 2 and Chebyshev s inequality yields u i A T + A T X i ˆ u i A T + A T X i = O p /2, and therefore, from Assumption c, we conclude u i A T X i ˆ =O p

21 TESTIG FOR SERIAL CORRELATIO 309 ii Consider ê i A T ê i e i A T e i i, ê i A T ê i 2 e i A T e i 2 2 i u i A T u i + 2 i, with { [ ] 2 [ ] } 2 2 i 2 u i A T + A T X iˆ + ˆ X i A T X i ˆ We have u i A T + A T X iˆ 2 A T + A T 2 and It follows that Since = O p [ ] 2 ˆ X i A T X i ˆ AT 2 i u i A T u i we finally obtain = O p 2 2 i = O p 2 i /2 u i X i 2 ˆ ˆ X i X i 2 ˆ 4 /2 u i A T u i 2 = O p /2, ê i A 2 T ê i e i A T e i 2 = O p /2 Since Eu i A T u i = 2 i tra T = 0 and there exist some real number M such that /M < varu i A T u i <M, it follows from the Lindeberg Feller central limit theorem that T = ẑti + o p d 0, ẑ2 Ti

22 30 B. BOR AD J. BREITUG ii Under the sequence of local alternatives, we have 2 i = Eu i u i = 2 2 T T 2 T T 2 T 3 = i 2 c/ I c i 2 T + c/ H T + O C T = 2 i I T + c 2 i H T + O C T, where C T and C T are T T matrices with bounded elements and H T is a matrix with elements h ts = if t s = and zeros elsewhere. Since tra T = 0 it follows that Eu i A T u i = tra T = 2 i tra T + 2 i c tra T H T + O = 2 i c tra T H T + O Using standard results for the variance of quadratic forms of normally distributed random variables 7 e.g., Searle, 97, we have varu i A T u i = 4 i tra2 T + A T A T + O /2 It follows from the Lindeberg Feller central limit theorem u i A p T u i cm 2 tra T H T, m 4 tra 2 T + A T A T, where m 2 and m 4 are defined in Lemma. For the second moment, we obtain from the law of large numbers ê i A T ê i 2 = = u i A T u i + O p /2 2 u i A T u i 2 + O p /2 p m 4 tra 2 T + A T A T 7 ote that for non-symmetric A T the expression can be rewritten in a symmetric format as u i A T u i = 05[u i A T + A T u i].

23 TESTIG FOR SERIAL CORRELATIO 3 Furthermore, under the sequence of local alternatives, 2 ê i A T ê i = O p, and therefore, the second term of the denominator in 4 does not affect the limiting distribution. It follows that the limiting distribution of the test statistic is given by T = u i A T u i Proof of Theorem u i A T u i 2 + o p d c tra T H T tra 2 T + A T A T, i Let D k denote a T 2 T matrix that results from dropping the kth row of the first difference matrix D defined in 9. Using Lemma A., we have /2 ẑti = /2 z Ti + O p, where For T 3, A T = z Ti = e i D T D + 05D T D T e i e i A T e i For example, 0 A 3 = [ 0 ] + 05 [ 0 ] 0 05 = 05 [ 0 ] =

24 32 B. BOR AD J. BREITUG Obviously, tra T = 0, A T T = 0, and T A iid T = 0. If u it 0, i 2, the variance of the quadratic form e i A T e i = u i A T u i is i 4trA2 T + A T A T, where tra 2 T = 3 T 3, 2 tra T A 7T 3 T = It follows that vare i A T e i = i 4 [2T 3 + 3]. Furthermore, tra T H T = T 2 and, by using Lemma, we obtain WD d T 2 c, 2T ii The original version of the WD test can be written as WD = T = = = ẑti t=3 ê i,t ê it ˆê i,t T T ẑti t=3 ê i,t ê it + 05ê i,t ˆ + 05ê 2 t ẑti + O p /2 2 t=3 ê i,t ê it + 05ê i,t ẑti + O p /2, ẑ2 Ti 2 2 where ˆ + 05 = O p /2 under the null hypothesis and local alternative. Furthermore, by using the results in the proof of Lemma, WD = ẑti 2 ẑ2 Ti ẑti = ẑti + O p /2 ẑ2 Ti

25 TESTIG FOR SERIAL CORRELATIO 33 It follows that WD WD = O p /2. Proof of Lemma 2 For the first order autocorrelation, Cox and Solomon 988 derive the asymptotic approximation T t=2 u it ū i u i,t ū i T t= u it ū i 2 d T T + T 22, T 2 T 2 Using this result, the limiting distribution of the LM statistic is easily derived. Proof of Theorem 2 i As shown in the proof of Lemma, we have ẑ Ti = e i A T e i + o p. Using the matrix notation introduced in Section 3.2, A T = M M T + M T T M T. Let S k denote a selection matrix that is obtained from dropping the kth row of I T. Since M T T = S T M T = 0 and M T = S M T = 0, it follows that A T T = 0 and A T T = 0. Since trm M T = trms S T M = trs T MS = T /T and trm T M T = trs T MS T = T T /T it follows that tra T = 0. Thus, z Ti = e i A T e i = u i A T u i The variance of z Ti is obtained as varz Ti = 4 i tra2 + AA by using the following results: trm M T M M T = T 2 2T /T 2, trm M T M T M = trm T M T M T M T = T 3 2T 2 + /T 2, trm M T M T M T = trm T M T M M T = trm T M T M T M = T 2 T /T 2 It follows that varz Ti = 4 i 2 T 3 + TT Furthermore, trm M T H T = trs T MH T MS = trs T H T S 2T trs T T T H T S + T 2 trs T T T H T T T S

26 34 B. BOR AD J. BREITUG Using trs T H T S = T, trs T H T S T = 0, trs T T T H T S = trs T T T H T S T = 2T, trs T T T H T T T S = trs T T T H T T T S T = 2T 2 yields tra T H T = tr M M T H T T M M 2 T H T = T 3 + TT, which is identical to tra 2 T + A T A T. With these results the limiting distribution follows. ii Using ˆ + T = O p /2 and ê i = T T t= êit the LM statistic can be written as LM = [ T = = ẑti ] 2 t=2 ê i,t ê i ê it ˆê i,t ˆê i [ T ẑti + O p /2 ẑ2 Ti ẑti t=2 ê it ê i ê i,t ê i + T T t=2 ê it ê i 2 + O p /2 Furthermore, under the null hypothesis and local alternative, LM = z Ti + O p /2 z2 Ti ] Proof of Theorem 3 By Lemma A., we have /2 ẑ Ti = /2 e i MD D 2I T Me i + O p = u i A T u i + O p,

27 TESTIG FOR SERIAL CORRELATIO 35 where D D 2I T = is a symmetric T T matrix. Since M T = M T = 0, it follows that A T T = A T T = 0. Furthermore, tr[md D 2I T M] =trd D 2I T T D D 2I T T = = 0. The variance is obtained as Furthermore, varz Ti = 2 4 i tra2 T = 4 4 i T 2 tra T H T = T 2T 2 T It follows from Lemma that under the local alternative Proof of Lemma 3 mdw Rewrite Eq. in matrix terms as d c T T 2, T L 0 Mê i = k L k Mê i + i, where L 0 = L k =

28 36 B. BOR AD J. BREITUG are T k T matrices. Under Assumption and using the results [ trm L k L 0 M = tr L k ] T T T L k L 0 it follows, as, for all k,, T 2. = tr L k L 0 T tr T L k L 0 T = T k T, [ trm L k L k M = tr L k ] T T T L k L k ˆ k = tr L k L k T tr T L k L k T = T T k T, p trm L k L 0 M trm L k L k M == T, REFERECES Baltagi, B. H Econometric Analysis of Panel Data. 4th ed. ew York: Wiley & Sons. Baltagi, B. H., Li, Q. 99. A joint test for serial correlation and random individual effects. Statistics and Probability Letters 3: Baltagi, B. H., Li, Q Testing AR against MA disturbances in an error component model. Journal of Econometrics 68:33 5. Baltagi, B. H., Wu, P. X Unequally spaced panel data regressions with AR disturbances. Econometric Theory 506: Bhargava, A., Franzini, L., arendranathan, W Serial correlation and the fixed effects model. Review of Economic Studies 494: Cox, D. R., Solomon, P. J On testing for serial correlation in large numbers of small samples. Biometrika 75: Drukker, D. M Testing for serial correlation in linear panel-data models. Stata Journal 32: Hsiao, C Analysis of Panel Data. 2nd ed. Cambridge: Cambridge University Press. Inoue, A., Solon, G A Portmanteau test for serially correlated errors in fixed effects models. Econometric Theory 2205: Searle, S. 97. Linear Models. ew York: Wiley. Wooldridge, J. M Econometric Analysis of Cross Section and Panel Data. Cambridge, MA: MIT Press.