www.ssoar.info A test of cross section dependence for a linear dynamic panel model with regressors Sarafidis, Vasilis; Yamagata, Takashi; Robertson, Donald Postprint / Postprint Zeitschriftenartikel / journal article Zur Verfügung gestellt in Kooperation mit / provided in cooperation with: www.peerproject.eu Empfohlene Zitierung / Suggested Citation: Sarafidis, Vasilis ; Yamagata, Takashi ; Robertson, Donald: A test of cross section dependence for a linear dynamic panel model with regressors. In: Journal of Econometrics 148 (2009), 2, pp. 149-161. DOI: http://dx.doi.org/10.1016/ j.jeconom.2008.10.006 Nutzungsbedingungen: Dieser Text wird unter dem "PEER Licence Agreement zur Verfügung" gestellt. Nähere Auskünfte zum PEER-Projekt finden Sie hier: http://www.peerproject.eu Gewährt wird ein nicht exklusives, nicht übertragbares, persönliches und beschränktes Recht auf Nutzung dieses Dokuments. Dieses Dokument ist ausschließlich für den persönlichen, nicht-kommerziellen Gebrauch bestimmt. 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By using this particular document, you accept the above-stated conditions of use. Diese Version ist zitierbar unter / This version is citable under: http://nbn-resolving.de/urn:nbn:de:0168-ssoar-215764 Accepted Manuscript A test of cross section dependence for a linear dynamic panel model with regressors Vasilis Sarafidis, Takashi Yamagata, Donald Robertson PII: DOI: Reference: S0304-4076(08)00182-6 10.1016/j.jeconom.2008.10.006 ECONOM 3119 To appear in: Journal of Econometrics Received date: 8 May 2006 Revised date: 23 September 2008 Accepted date: 10 October 2008 Please cite this article as: Sarafidis, V., Yamagata, T., Robertson, D., A test of cross section dependence for a linear dynamic panel model with regressors. Journal of Econometrics (2008), doi:10.1016/j.jeconom.2008.10.006 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. IPT ACCEPTED MANUSCRIPT Vasilis Sara…dis} Takashi Yamagata4y Donald Robertson| Discipline of Econometrics and Business Statistics, University of Sydney 4 Department of Economics and Related Studies, University of York | Faculty of Economics, University of Cambridge US } CR A Test of Cross Section Dependence for a Linear Dynamic Panel Model with Regressors DM AN 23 September 2008 Abstract TE This paper proposes a new testing procedure for detecting error cross section dependence after estimating a linear dynamic panel data model with regressors using the generalised method of moments (GMM). The test is valid when the crosssectional dimension of the panel is large relative to the time series dimension. Importantly, our approach allows one to examine whether any error cross section dependence remains after including time dummies (or after transforming the data in terms of deviations from time-speci…c averages), which will be the case under heterogeneous error cross section dependence. Finite sample simulation-based results suggest that our tests perform well, particularly the version based on the Blundell and Bond (1998) system GMM estimator. In addition, it is shown that the system GMM estimator, based only on partial instruments consisting of the regressors, can be a reliable alternative to the standard GMM estimators under heterogeneous error cross section dependence. The proposed tests are applied to employment equations using UK …rm data and the results show little evidence of heterogeneous error cross section dependence. EP Key Words: cross section dependence, generalised method of moments, dynamic panel data, overidentifying restrictions test. JEL Classi…cation: C12; C13; C15; C33. AC C We would like to thank Tom Flavin, Hashem Pesaran and Neville Weber for helpful discussions. We have also bene…ted from the constructive and most helpful comments of Cheng Hsiao and two anonymous referees. Sara…dis gratefully acknowledges full …nancial support from the ESRC during his PhD studies at Cambridge University (PTA-030-2002-00328). Yamagata gratefully acknowledges the …nancial support from the ESRC (Grant No. RES-000-23-0135). y Corresponding author. E-mail: ty509@york.ac.uk 1 1 IPT ACCEPTED MANUSCRIPT Introduction AC C EP TE DM AN US CR During the past decade a substantial literature has been developed analysing the e¤ects of cross section dependence as well as advancing ways of dealing with it in panel data models. Cross section dependence may arise for several reasons often, due to spatial correlations, economic distance and common unobserved shocks. In the case of spatial attributes, where a natural immutable distance measure is available, the dependence may be captured through spatial lags using techniques that are familiar from the time series literature (Anselin, 1988, 2001). In economic applications, spatial techniques are often adapted using alternative measures of economic distance (see e.g. Conley, 1999, Kapoor, Kelejian and Prucha, 2004, Lee, 2004, Lee, 2007, and others). There are several contributions in the literature that allow for time-varying individual e¤ects (Holtz-Eakin, Newey and Rosen, 1988, Ahn, Lee and Schmidt, 2001 and Han, Orea and Schmidt, 2005). Recently, a number of researchers have modelled cross section dependence by restricting the covariance matrix of the errors using a common factor speci…cation with a …xed number of unobserved factors and individual-speci…c factor loadings that give rise to heterogenous cross section dependence (see Forni and Reichlin, 1998, Robertson and Symons, 2000, Phillips and Sul, 2003, Stock and Watson, 2002, Bai and Ng, 2004, Moon and Perron, 2004, Pesaran, 2006, among others). The factor structure approach is widely used because it can approximate a wide variety of error cross section dependence. For example, in a panel data set of …rms we may think of the factors as capturing ‡uctuations in economic activity or changes in regulatory policy for the industry as a whole, and so on. The impact of these factors will vary across …rms, due to di¤erences in size, liquidity constraints, market share etc. In a macro panel data model, the factors may represent a general demand shock or an oil price shock with the factor loadings re‡ecting the relative openness of the economies, di¤erences in technological constraints, and so on.1 In the literature of estimating linear dynamic panel data models with a large number of cross-sectional units (N ) and a moderately small number of time series observations (T ), generalised method of moments (GMM) estimators are widely used, such as those proposed by Arellano and Bond (1991), Ahn and Schmidt (1995), Arellano and Bover (1995) and Blundell and Bond (1998). These methods typically assume that the disturbances are cross-sectionally independent. On the other hand, in empirical applications it is common practice to include time dummies, or, equivalently, to transform the observations in terms of deviations from time-speci…c averages (i.e. to cross-sectionally demean the data) in order to eliminate any common time-varying shocks; see, for example, Arellano and Bond (1991) and Blundell and Bond (1998). This transformation will marginal out these common e¤ects, unless their impact di¤ers across cross-sectional units (heterogeneous cross section dependence). In this case, the standard GMM estimators used in the literature will not be consistent, as shown by Sara…dis and Robertson (2007) and in the current paper. Several tests for cross section dependence have been proposed in the econometric 1 Other examples are provided by Ahn, Lee and Schmidt (2001). 2 IPT ACCEPTED MANUSCRIPT AC C EP TE DM AN US CR literature. The most widely used test is perhaps the Lagrange Multiplier (LM) test proposed by Breusch and Pagan (1980), which is based on the squared pair-wise Pearson’s correlation coe¢ cients of the residuals. This test requires T being much larger than N . Frees (1995) proposed a cross section dependence test that is based on the squared Spearman rank correlation coe¢ cients and allows N to be larger than T . Recently, Ng (2006) has developed tools for guiding practitioners as to how much residual cross section correlation is in the data and which cross-sectional units are responsible for this in particular, through tests that are based on probability integral transformations of the ordered residual correlations. However, the proposed procedures are valid only in panels p for which T -consistent estimates are available. Pesaran, Ullah and Yamagata (2006) developed bias-adjusted normal approximation versions of the LM test of Breusch and Pagan (1980), which are valid for large-N panel data models but with strictly exogenous regressors only. Pesaran (2004) proposed another test for cross section dependence, called the CD test, which is closely related to Friedman’s (1937) test statistic. Pesaran showed that the CD test can also be applied to a wide variety of models, including heterogeneous dynamic models with multiple breaks and non-stationary dynamic models with small/large N and T . However, as Frees (1995) implied and Pesaran (2004) pointed out, the problem of the CD test is that in a stationary dynamic panel data model it will fail to reject the null of error cross section independence when the factor loadings have zero mean in the cross-sectional dimension. It follows that the CD test will have poor power properties when it is applied to a regression with time dummies or on cross-sectionally demeaned data. This paper proposes a new testing procedure for error cross section dependence after estimating a linear dynamic panel data model with covariates by the generalised method of moments. This is valid when N is large relative to T . Importantly, unlike the CD test, our approach allows one to examine whether any error cross section dependence remains after including time dummies, or after transforming the data in terms of deviations from time-speci…c averages, which will be the case under heterogeneous error cross section dependence. The small sample performance of our proposed tests is investigated by means of Monte Carlo experiments and we show that they have correct size and satisfactory power for a wide variety of simulation designs. Furthermore, the paper suggests a consistent GMM estimator under heterogeneous error cross section dependence. Results on the …nite sample properties of the estimator are reported and discussed. Our proposed tests and estimators are applied to employment equations using UK …rm data, and it is found that there is little evidence of heterogeneous cross section dependence in this data set. The remainder of the paper proceeds as follows. Section 2 reviews some relevant existing tests for error cross section dependence. Section 3 proposes a new test for cross section dependence and a consistent GMM estimator under these circumstances. Section 4 reports the results of our Monte Carlo experiments. Section 5 illustrates an empirical application of our approach. Finally, Section 6 contains concluding remarks. 3 2 IPT ACCEPTED MANUSCRIPT Existing Tests for Cross Section Dependence Consider a panel data model i + 0 xit + uit , i = 1; 2; : : : ; N , t = 1; 2; : : : ; T , CR yit = (1) where the uit may exhibit cross section dependence. The hypothesis of interest is H0 : E(uit ujt ) = 0 8 t for all i 6= j; (2) H1 : E(uit ujt ) 6= 0 for some t and some i 6= j; (3) US vs where the number of possible pairings (uit ; ujt ) rises with N . In the literature several tests for error cross section dependence have been proposed, and some relevant ones are discussed in this section. Breusch-Pagan (1980) Lagrange Multiplier Test DM AN 2.1 Breusch and Pagan (1980) proposed a Lagrange multiplier (LM) statistic for testing the null of zero cross-equation error correlations, which is de…ned as LM = T N X1 N X ^2ij ; (4) i=1 j=i+1 where ^ij is the sample estimate of the pair-wise Pearson correlation coe¢ cient of the residuals PT t=1 eit ejt ^ij = ^ji = ; (5) PT 2 1=2 PT 2 1=2 e e t=1 it t=1 jt 2.2 TE where eit is the Ordinary Least Squares (OLS) estimate of uit in (1). LM is asymptotically distributed as chi-squared with N (N 1)=2 degrees of freedom under the null hypothesis, as T ! 1 with N …xed. Pesaran’s (2004) CD Test AC C EP Recently Pesaran (2004) proposed another test for cross section dependence, called CD test, which allows for a ‡exible model structure, including fairly general heterogeneous dynamic models and nonstationary models. The test statistic is de…ned as 1 0 s N N X1 X 2T @ ^ij A . (6) CD = N (N 1) i=1 j=i+1 4 IPT ACCEPTED MANUSCRIPT uit = i ft + "it , CR For su¢ ciently large N and T , the CD test statistic tends to a standard normal variate under the null of cross section independence.2 The …nite sample evidence in Pesaran (2004) shows that the estimated size of the test is very close to the nominal level for any combinations of N and T considered. As Pesaran (2004) notes, there are two important cases in which the CD test can be unreliable. Firstly, when the distribution of the errors is not symmetric, the CD test becomes invalid and it may not have correct size.3 Secondly, the CD test may lack power towards some directions of alternatives. To see this, consider the following single-factor structure for the error process 3 3.1 DM AN US where i is a factor loading that is …xed and bounded, ft is an unobserved common factor such that ft i:i:d:(0; 1), "it i:i:d:(0; 2 ) and E(ft "it ) = 0 for all i and t. The common factor ft generates error cross section dependence because of the fact that cov(uit ; ujt ) = i j , and the power of the CD test hinges on this non-zero covariance. Now suppose that i i:i:d:(0; 2 ) and i is uncorrelated with ft and "it . In this case, cov(uit ; ujt ) = E( i )E( j ) = 0, even if there does exist (potentially large) error cross section dependence. In the next section, we elaborate on the stochastic properties of the factors and factor loadings, and develop a new cross section dependence test. Sargan’s Di¤erence Tests for Heterogeneous Error Cross Section Dependence in a Linear Dynamic Model with Regressors Model Speci…cation Consider the following model i + yi;t 1 + 0 xit + uit , i = 1; 2; :::; N , t = 1; 2; :::; T , TE yit = (7) where j j < 1, is a (K 1) parameter vector that is bounded and non-zero, xit is a (K 1) vector of regressors with xit = (x1it ; x2it ; :::; xKit )0 , i is a random e¤ect with …nite mean and …nite variance, and uit has a multi-factor structure such that 0 i ft EP uit = + "it ; (8) where i = ( 1i ; 2i ; :::; M i )0 is a (M 1) vector of factor loadings that is assumed to be i:i:d:( ; ) with being a positive semi-de…nite matrix, ft = (f1t ; f2t ; :::; fM t )0 is a AC C 2 As Frees (1995) pointed out, test statistics similar to the CD test of Pesaran were proposed by Friedman (1937), based on the Spearman rank correlation coe¢ cient (which is expected to be robust against non-normality). Although we do not consider the Friedman test in this paper, results that are similar to the CD test would apply for this test. 3 However, the experimental results of Pesaran (2004) illustrate that the CD test is robust to skewed errors. 5 IPT ACCEPTED MANUSCRIPT i + xi;t 1 + xi ft + "i;t 1 + vit , i = 1; 2; :::; N , t = 1; 2; :::; T , US xit = CR (M 1) vector of time-varying common factors that are assumed to be non-stochastic and bounded, and "it is an independently distributed random variable over i with zero-mean and …nite variance 2i . The error (multi-) factor structure has been employed extensively in the economic literature.4 For example, in a macro panel, a common factor could be an unobserved technological shock, and the factor loadings can be thought of as capturing a crosssectionally heterogeneous response to such shock. Note that since our asymptotic is N ! 1 with T …xed, ft is treated as non-stochastic here. In this paper we explicitly employ a random coe¢ cient assumption for the factor loadings. The process for xit is de…ned as (9) DM AN where i = ( i1 ; i2 ; :::; iK )0 is a (K 1) vector of random e¤ects with …nite mean and …nite variance, signi…es the Hadamard product, = ( 1 ; 2 ; :::; K )0 such that j k j < 1 for k = 1; 2; :::; K, xi = ( 1i ; 2i ; :::; Ki )0 with ki = ( ki1 ; ki2 ; :::; kiM )0 , such that xi i:i:d:( x ; x ), = ( 1 ; 2 ; :::; K )0 , and vit is a vector of independently distributed random variables over i with mean vector zero and a …nite variance matrix 2 vi = diag( vki ), k = 1; 2; :::; K. The model de…ned by (7), (8), and (9) is general enough to allow for a large variety of plausible speci…cations that are widely used in the economic literature. Furthermore, this model accommodates more simple processes for xit ; such as those where xit is strictly exogenous, or exogenous with respect to ft : The null hypothesis of interest is then H0 : var( against the alternative i) H1 : as opposed to (2) and (3). = 6= 0, =0 (10) (11) EP TE Remark 1 Observe that error cross section dependence may occur under the null hypothesis when i = for all i, since E(uit ujt ) = 0 ft ft0 , which is not zero unless = 0. However, such error cross section dependence can be eliminated simply by including time dummies, or equivalently by cross-sectionally demeaning the data. This implies that the null hypothesis in (10) can be interpreted as saying that the cross section dependence is homogeneous across pairs of cross-sectional units, against the alternative hypothesis (11) of heterogeneous error cross section dependence. We make the following assumptions: AC C 4 See e.g. Robertson and Symons (2000), Phillips and Sul (2003), Bai and Ng (2004), Moon and Perron (2004), and Pesaran (2006) among others. Notice that these methods are only justi…ed when T is large. For related work that allows for time-varying individual e¤ects in the …xed T , large N case, see Holtz-Eakin, Newey and Rosen (1988), Ahn, Lee, and Schmidt (2001) and Han, Orea and Schmidt (2005). 6 Assumption 1: E( i "it ) = 0, E( E (vit "is ) = 0 for all t and s. i "it ) IPT ACCEPTED MANUSCRIPT = 0, and E( i vit ) = 0, for all i, t. Also 0 ) = 0 for all i and t 6= s. Assumption 2: E("it "is ) = 0 and E(vit vis CR Assumption 3: E(yi0 "it ) = 0 and E(xi1 "it ) = 0 for all i and t = 1; 2; :::; T . Assumption 4: Time-varying common factors, fm;t , m = 1; 2; :::; M ; t = 1; 2; :::; T are non-stochastic and bounded. Assumption 5: (b) cov( xi ; i) = 0 and cov( Assumption 6: 0 i ; xi1 ) i) = 0, cov( i ; xi ) = 0, US (a) E("it i ) = 0, E(vit 0i ) = 0, E("it xi ) = 0, cov( i ; cov( i ; 0i ) = 0, cov( 0xi ; i ) = 0, for all i and t. = 0. 6= 0. EP TE DM AN Assumptions 1-3 are standard in the GMM literature; see, for example, Ahn and Schmidt (1995). Assumption 4 ensures that the initial observations of y and x are bounded. Assumption 5(a), a random coe¢ cient type assumption on the factor loadings, allows cross-sectionally heterogeneous inter-dependence among both level and …rst-di¤erenced variables (yit ; x1it ; x2it ; :::; xKit ), as well as cross section dependence of each variable through the common factor ft . This contrasts to a simple time e¤ects assumption, namely i = and xi = x for all i, which is stronger than ours. On the other hand, Assumption 5(a) is stronger than assuming i and xi to be merely bounded, in that the cross-sectionally demeaned variables of (yit ; x1it ; x2it ; :::; xKit ) become cross-sectionally uncorrelated to each other, asymptotically. Assumption 5(b) implies that the cross-sectionally demeaned xi;t 1 and (uit ui;t 1 ) or the cross-sectionally demeaned (xi;t 1 xi;t 2 ) and uit are uncorrelated under the alternative hypothesis (11). This will obviously hold true if the common factor components have no impact on xit namely, when xi = 0 for all i in equation (9). This restriction can be relaxed provided that the direct e¤ect of the factors on yit is uncorrelated with the e¤ect of the factors on xit , which satis…es Assumption 5(b).5 Assumption 6, 6= 0, is also necessary in order to obtain a consistent estimator under the alternative of heterogeneous error cross section dependence, although this is probably not a restrictive assumption in many applications. For further discussion, stacking (7) for each i yields AC C yi = i T + yi; 1 + Xi + ui , i = 1; 2; :::; N , ui = F 5 i + "i , (12) (13) For su¢ ciently large N , the overidentifying restrictions test (de…ned by equation (28) below) is expected to have enough power to detect the violation of Assumption 5(b). 7 IPT ACCEPTED MANUSCRIPT where yi = (yi1 ; yi2 ; :::; yiT )0 , g is a (g 1) vector of unity, yi; 1 = (yi0 ; yi1 ; :::; yiT 1 )0 , Xi = (xi1 ; xi2 ; :::; xiT )0 , ui = (ui1 ; ui2 ; :::; uiT )0 , F = (f 1 ; f2 ; :::; fT )0 , "i = ("i1 ; "i2 ; :::; "iT )0 , and the cross-sectionally demeaned and di¤erenced equation is de…ned by yi; + Xi + ui = F + "i , yi = Wi + ui , 1 i ui , i = 1; 2; :::; N , (14) CR yi = or (15) (16) =( ; US where the underline signi…es that the variables are cross-sectionally demeaned, and “ " denotes the …rst-di¤erenced operator. For example, yi = ( y i2 ; y i3 ; :::; y iT )0 , P y it = (y it y it 1 ), y it = (yit yt ) with yt = N 1 N Wi = ( yi; 1 ; Xi ), i=1 yit , 0 0 ) , with obvious notation. 3.2 DM AN Remark 2 Pesaran’s CD test statistic as de…ned in (6) can fail to reject the null hypothesis of homogeneous error cross section dependence given in (11), when E( i ) = 0. This arises when time dummies are introduced or the data are cross-sectionally demeaned, since E( i ) = 0. On the other hand, the Breusch and Pagan (1980) LM test de…ned as in (4) will have power, although its use is justi…ed only when T is much larger relative to N . Sargan’s Di¤erence Tests based on the First-di¤erenced GMM Estimator De…ne the matrices of instruments 2 y 0 0 0 6 0i0 y y 0 6 i0 i1 6 0 0 0 y i0 ZY i = 6 6 . .. .. .. 6 . . . . 4 . where hy = T (T 0 . ::: y i0 0 ::: y iT 2 1)=2, and 2 6 6 6 =6 6 4 x0i1 0 0 0 0 x0i1 x0i2 0 0 0 0 x0i1 .. .. .. .. . . . . 0 0 0 0 EP ZXi 0 TE 0 .. 3 0 0 0 .. . AC C where hx = KT (T vec(Xi )0 .6 T 1 .. . ::: x0i1 ::: 1)=2. If xit is strictly exogenous 3 0 0 0 .. . x0iT 7 7 7 7 , (T 7 7 5 1 7 7 7 7 , (T 7 5 that is, 1 hy ) (17) 1 hx ) (18) = 0 in (9), ZXi = 6 One can construct ZXi such that both strictly exogenous and predetermined regressors are present. See, for example, Arellano (2003). 8 Proposition 3 Consider two sets of moment conditions 0 Moment Conditions I: E[ZY i ui ] = 0 and 0 (19) (20) CR Moment Conditions II: E[ZXi ui ] = 0, IPT ACCEPTED MANUSCRIPT Proof. See Appendix A. Now de…ne the full set of moment conditions 0 US where ZY i , ZXi , and ui are de…ned as in (17), (18), and (15), respectively. Under Assumptions 1-4, 5(a)-(b), 6 and model (7), both sets of moment conditions given in (19) and (20) hold under the null hypothesis (10). However, under the alternative hypothesis given in (11), (20) holds but (19) does not. E[Zi ui ] = 0, where Zi = ZY i ZXi (T 1 (21) h); (22) TE DM AN and h = hy + hx . Based on Proposition 3, the hypothesis testing for (10) reduces to a test for the validity of the subset of the moment conditions in (21), as given in (19). There are at least three possible approaches. The …rst one is to take a Hausman-type (1978) approach. Newey (1985) showed that the Hausman test may be inconsistent against some local alternatives. In our application, this may happen when hy = T (T 1)=2 > K + 1, which is the most likely empirical situation.7 In addition, Arellano and Bond (1991) examined a Hausman-type test for serial correlation and found that it has poor …nite sample performance, in that it rejects the null hypothesis far too often. The second approach is to adopt Newey’s (1985) optimal test. The third approach is to use Sargan’s (1988) di¤erence test, which we are adopting, since it is relatively easier to compute than the Newey’s test statistic, and furthermore, they are asymptotically equivalent. We consider two versions of Sargan’s di¤erence test. One is based on the Arellano and Bond (1991) two-step …rst-di¤erenced GMM estimator hereafter DIF and the other is based on the Blundell and Bond (1998) two-step system GMM estimator hereafter, SYS. Initially, the test based on DIF is considered. This is de…ned as ! 1 N N N N X X 1X 1X e e 0 0 _ • W Zi Z Wi W0 Z 1 e_ Z0 y ; (23) DIF 2 = EP i i=1 i i i=1 i=1 i i i i=1 AC C P e 0 e e_ 0 Z with u e_ = y where e_ = N 1 N _i u Wi _ DIF 1 , which is the residual i i i i=1 Zi u i vector based on the one-step DIF estimator (denoted by DIF 1), ! 1 N N N N X X X X e_ 1e 1 0 0 Wi Zi Zi Wi W0i Zi e 1 Z0i yi ; (24) DIF 1 = 7 i=1 i=1 i=1 i=1 See Proposition 5 and its proof in Newey (1985), under the assymption of i.i.d. errors. 9 IPT ACCEPTED MANUSCRIPT i=1 CR P 0 where e = N 1 N 1, with 2’s on the i=1 Zi HZi , and H is the square matrix of order T main diagonal, -1’s on the …rst o¤-diagonals, and zeros elsewhere. Now, the two-step DIF estimator based only on the restricted set of moment conditions in (20), which is consistent both under the null and the alternative, is de…ned as ! 1 N N N N X X 1X 1X b b 0 0 _ • W0i ZXi b_ X Wi ZXi X ZXi Wi Z0Xi y ; (25) DIF X2 = i=1 i=1 i i=1 b_ DIF X1 = N X i=1 W0i ZXi b X1 N X Z0Xi Wi i=1 US P b 0 b_ u b0 b_ = y b_ is Wi _ DIF X1 , where u where b_ X = N 1 N u i _ i ZXi with i i i=1 ZXi u i the residual vector based on the one-step DIF estimator that exploits (20), denoted by DIF X1, ! 1 N X i=1 W0i ZXi b X1 N X Z0Xi yi ; (26) i=1 DM AN P 0 where b X = N 1 N i=1 ZXi HZXi . Sargan’s (1958), or Hansen’s (1982) test statistic of overidentifying restrictions for the full set of moment conditions in (21) is based on DIF 2 and is given by ! ! N N X 1 X 0 e 1 0 e e _ SDIF 2 = N u • Zi •i , (27) Z u i i=1 e = where u • i given by i i=1 e Wi •DIF 2 . Sargan’s statistic for (20) that is based on DIF X2 is yi SDIF X2 = N 1 N X i=1 b0 Z u • i Xi ! b_ 1 X N X Z0Xi i=1 b u • i ! ; (28) TE b b = y Wi •DIF X2 . where u • i i We are now ready to state a proposition on Sargan’s di¤erence test of heterogeneous cross section dependence: d Proposition 4 Under Assumptions 1-4, 5(a)-(b), 6 and model (7), SDIF 2 ! d EP under the null hypothesis, SDIF X2 ! native hypothesis, and 2 hx (K+1) DDIF 2 = (SDIF 2 2 h (K+1) under the null hypothesis and the alterd SDIF X2 ) ! 2 hy (29) AC C under the null hypothesis, where SDIF 2 , SDIF X2 are as de…ned in (27) and (28) respectively, while h = hy + hx with hy and hx de…ned as in (17) and (18) respectively. Proof. See Appendix B. 10 IPT ACCEPTED MANUSCRIPT e ; which Remark 5 The cross section correlation of the sample moment conditions Z0i u • i is induced as a result of the cross-sectional demeaning of the series, is asymptotically negligible,P and standard central limit theorems can be applied to the asymptotic expansion 0 e of N 1=2 N • i ; See proof in Appendix B. i=1 Zi u CR Remark 6 Sargan’s di¤ erence test has non-trivial asymptotic local power; See proof in Appendix B. 3.3 US Remark 7 Proposition 4 holds with unbalanced panel data, so long as min Nt ! 1, where Nt is the number of cross-sectional units for a given time t. In this case, P the crosst 8 sectional demeaning for y t , say, is now de…ned as y t = yt yt , where yt = Nt 1 N i=1 yit . Sargan’s Di¤erence Tests based on the System GMM Estimator DM AN It has been well documented (see, for example, Blundell and Bond, 1998) that the DIF estimator can su¤er from a weak instruments problem when is close to unity and/or the variance of the individual e¤ects is large relative to that of the idiosyncratic errors. Thus, we also consider another version of Sargan’s di¤erence test based on the Blundell and Bond (1998) SYS estimator, which is known to be more robust to the problem of weak instruments under certain conditions. Arellano and Bover (1995) proposed the use of lagged di¤erences as possible instruments for the equations in levels, E (uit wit ) = 0 with wit = (yt 1 ; x0t )0 for t = 3; 4; :::; T , which is valid under the null hypothesis and Assumptions 1-3. In addition, Blundell and Bond (1998, 2000) proposed using an additional condition E (ui2 wi2 ) = 0 under mild conditions upon the initial observations, which would follow from joint stationarity of the y and x processes. Accordingly, we add Assumption 5(c): Assumption 5(c): cov( i ; yi1 ) = 0 and cov( i ; xi2 ) = 0. De…ne ZY i 0 0 ZL Yi TE Z+ Yi = where ZL Y i = diag( y i;t 1 (2(T 1) hys ); ) for t = 2; 3; :::; T , hys = hy + (T ZXi 0 0 ZL Xi EP Z+ Xi = (2(T 1) (30) 1); and hxs ); where ZL 1 K(T 1)) matrix whose sth diagonal raw vector is Xi is a (T s = 2; 3; :::; T , otherwise zeros, and hxs = hx + K(T 1). (31) x0is , AC C Proposition 8 Consider two sets of moment conditions Moment Conditions I : E[Z+0 Yi 8 i] =0 (32) For more details on the computations of the Arellano and Bond (1991) and Blundell and Bond (1998) estimators with unbalanced panels, see Arellano and Bond (1999). 11 and Moment Conditions II : E[Z+0 Xi where Z+ Yi and Z+ Xi i] = 0, are de…ned as in (30) and (31) respectively, + 0 0 = ( u0i ; u+0 i ) ; and ui = (ui2 ; ui3 ; :::; uiT ; ) . (33) (34) CR i IPT ACCEPTED MANUSCRIPT Under Assumptions 1-4, 5(a)-(c), 6 and model (7), both (32) and (33) hold under the null hypothesis in (10). However, under the alternative hypothesis given in (11), (33) holds but (32) does not. US Proof. See Appendix C. In the same manner as DDIF 2 , we de…ne the Sargan’s di¤erence test statistic based on SYS estimator DSY S2 = (SSY S2 SSY SX2 ) (35) DM AN where SSY S2 is Sargan’s test statistic of overidentifying restrictions based on the twostep SYS estimator that makes use of both sets of moment conditions I and II in Proposition 8 (denoted by SY S2), namely E[Z+0 i where Z+ i = i] + Z+ Y i ZXi = 0, (2(T 1) (36) hs ); (37) hs = hsy +hsx , and SSY SX2 is Sargan’s test statistic based on the two-step SYS estimator that exploits only Moment Conditions II , as de…ned in (33), denoted by SY SX2.9 e One-step and two-step SYS estimators based on the moment conditions (36), _ SY S1 and e b b •SY S2 , and those based on (33), _ SY SX1 and •SY SX2 , are de…ned accordingly.10 It is d 3.4 Discussion 2 hys as N ! 1, under the null hypothesis. TE straightforward to see that DSY S2 ! EP The overidentifying restrictions test can be regarded as a misspeci…cation test, in a sense that it is designed to detect violations of moment conditions, which are the heart of GMM methods. Thus, it will have power under the alternative hypothesis of heterogenous error cross section dependence. Nonetheless, the proposed Sargan’s di¤erence test is expected to have higher power than the overidentifying restrictions test, so long as Assumption 6, 6= 0, holds, since the former exploits extra information about the validity of the moment conditions under the alternative hypothesis, which the latter does not use. This AC C 9 With small samples DDIF 2 may not be positive, but it can be patched easily. See, for example, Hayashi (2000, p.220). However, we did not adopt this modi…cation here since one of our aims is to show the properties of a consistent estimator based only on orthogonality conditions E[Z0Xi ui ] = 0 or E[Z+0 Xi i ] = 0. 10 The initial weighting matrix for one-step GMM-SYS estimator is de…ned as a block diagnoal matrix of order 2(T 1), whose diagnoal blocks are H and IT 1 . 12 IPT ACCEPTED MANUSCRIPT TE DM AN US CR also implies that, when = 0, the overidentifying restrictions test should replace our approach.11 Now consider a violation of Assumption 2, E("it "is ) = 0 for t 6= s, no error serial correlation. Under the alternative of error cross section dependence, the composite error uit = 0i ft + "it will be serially correlated, since E(uit uis ) = ft0 E( i 0i )fs 6= 0 for all i. This means that the second-order serial correlation test based on DIF2 or SYS2, the m2 test, proposed by Arellano and Bond (1991), is likely to reject the hypothesis of no error serial correlation, under the alternative of heterogeneous error cross section dependence.12 Then, a question that may arise is how to distinguish between error cross section dependence and serial correlation in the idiosyncratic errors. To answer this question, consider two scenarios. First, suppose that there is …rst-order autoregressive serial correlation but no heterogeneous error cross section dependence, such that "it = i:i:d:(0; 2 ). In this case, the problem can be solved " "it 1 + it with j " j < 1 and it in a straightforward manner by adding a further lag of the dependent variable on the right hand side of (7) and using (up to) yit 3 as instruments for yit 1 and yit 2 . Second, suppose there is both …rst-order autoregressive error serial correlation as above and heterogeneous error cross section dependence. Clearly, the m2 test based on DIFX2 or SYSX2 is likely to reject the null even when E("it "is ) = 0 for t 6= s. Meanwhile, the probability of rejecting the null by the overidentifying restrictions test for the restrictions based only on the subset of Xi (de…ned by (20) or (33)), tends to its signi…cance level when E("it "is ) = 0 for t 6= s, but such a probability goes to one when E("it "is ) 6= 0 for t 6= s. Therefore, the solution given in the …rst case applies, but the test statistic to employ is the overidentifying restrictions test based only on the subset of Xi , not the m2 test based on DIFX2 or SYSX2. Finally, we have shown that the moment conditions (20) and (33) hold under the alternative of error cross section dependence, therefore, the DIFX2 and SYSX2 estimators are consistent. However, in …nite samples there could be a trade-o¤ between e¢ ciency and bias. If the degree of heterogeneity of the error cross section dependence is relatively small, then the bias of the standard GMM estimators exploiting moment conditions including (19) or (20) which are invalid, may be small enough so that these estimators are preferred (in root mean square errors terms) to a consistent estimator based only on the valid moment conditions (20) or (33). We will investigate the …nite sample performance of these estimators in the next section.13 11 AC C EP In Appendix B, it is formally shown that when hx > (K + 1), the DDIF 2 test is asymptotically more powerful than the SDIF 2 test under the local alternatives. 12 Heterogeneity of would also render the error term serially correlated, as discussed in Pesaran and Smith (1995). 13 Other solutions have been proposed in the literature, such as a panel feasible generalized median unbiased estimator, proposed by Phillips and Sul (2003), or the common correlated e¤ects (CCE) estimator proposed by Pesaran (2006). However, both estimators require a larger value for T than that considered in this paper. 13 4 IPT ACCEPTED MANUSCRIPT Small Sample Properties of Cross Section Dependence Tests 4.1 Design DM AN US CR This section investigates by means of Monte Carlo experiments the …nite sample performance of our tests, the Breusch and Pagan (1980) LM test and Pesaran’s (2004) CD test, all based on cross-sectionally demeaned variables. Our main focus is on the e¤ects of (i) the degree and heterogeneity of error cross section dependence, (ii) the relative importance of the variance of the factor loadings and the idiosyncratic errors, and (iii) di¤erent values of and . In order to make the results comparable across experiments, we control the population signal-to-noise ratio and the impact of the ratio between the variance of the individual-speci…c time-invariant e¤ects and the variance of the idiosyncratic errors and the common factor on yit . To this end, we extend the Monte Carlo design of Kiviet (1995) and Bun and Kiviet (2006) to accommodate a factor structure in the error process. Recently Bowsher (2002) reports …nite sample evidence that Sargan’s overidentifying restrictions test exploiting all moment conditions available can reject the null hypothesis too infrequently in linear dynamic panel models. Thus, we only make use of y it 2 and y it 3 as instruments for y it 1 and we use xit 2 and xit 3 as instruments for xit 1 . The data generating process (DGP) we consider is given by yit = i + yi;t uit = i ft 1 + xit + uit ; + "it , i = 1; 2; :::; N ; t = (38) 48; 47; :::; T , TE where i i:i:d:N (1; 2 ) and ft i:i:d:N (0; 2f ).14 "it is drawn from (i) i:i:d:N (0; 2" ) p and (ii) i:i:d:( 21 1)= 2, in order to investigate the e¤ect of non-normal errors. yi; 49 = 0 and the …rst 49 observations are discarded.15 To control the degree and heterogeneity of cross section dependence three speci…cations for the distribution of i are considered: 8 i:i:d:U [ 0:3; 0:7] < Low cross section dependence: i Medium cross section dependence: i:i:d:U [ 1; 2] i : High cross section dependence: i:i:d:U [ 1; 4]: i EP Also, as we change the value of = 0:2; 0:5; 0:8, the long run e¤ect of x on y constant. The DGP of xit considered here is given by xit = xi;t 1 + "i;t 1 + i ft is equated to 1 + vit , i = 1; 2; :::; N ; t = in order to keep 48; 47; :::; T , (39) AC C 14 Note that the unobserved common factor, ft , is randomly drawn to control the signal-to-noise ratio without loss of generality. 15 We do not report the results based on non-normal errors in this paper, since the results were very similar. They are available from the authors upon request. 14 IPT ACCEPTED MANUSCRIPT 4.2 DM AN US CR where = 0:5, i i:i:d:U [ 1; 2], vit i:i:d:N (0; 2v ). is set to 0:5.16 xi; 49 = 0 and the …rst 50 observations are discarded. Since our focus is on the performance of the tests and estimators, we pay careful attention to the main factors that a¤ect it namely, (i) the signal-to-noise ratio, (ii) the relative importance of the variance of the factor loadings and the idiosyncratic errors, and (iii) the impact of the ratio between the variance of the individual-speci…c e¤ects and the variance of the idiosyncractic error and factor loadings on yit . To illustrate, we de…ne the signal as 2s = var (yit uit ), where yit = yit ). Then, denoting the i =(1 (2) 2 (2) 2 2 variance of the composite error by u = var(uit ) = = E( 2i ), we f + " with 2 2 de…ne the signal-to-noise ratio as = s = u . We set = 3. The relative importance (2) 2 2 in terms of the magnitude of the variance of i ft and "it , as measured by f = ", is thought in the literature to be an important factor to control for and we achieve (2) this by changing and applying the normalisation 2f = 2" = 1. As it has been discussed by Kiviet (1995), Blundell and Bond (1998), and Bun and Kiviet (2006), in order to compare the performance of estimators across di¤erent experimental designs it is important to control the relative importance of i and ("it , ft ). We choose 2 such that the ratio of the impact on var(yit ) of the two variance components i and ("it , ft ) is constant across designs.17 We consider all combinations of N = 50; 100; 200; 400, and T = 5; 9. All experiments are based on 2,000 replications. Results EP TE Tables 1 reports the size and power of the tests for T = 5.18 LM denotes Breusch and Pagan’s LM test, as de…ned in (4), and CD denotes Pesaran’s CD test, de…ned in (6), both of which are based on the …xed e¤ects estimator. DDIF 2 is Sargan’s di¤erence test based on the two-step DIF estimator de…ned in (29), and DSY S2 is Sargan’s di¤erence test based on the two-step SYS estimator de…ned in (35). The size of the LM test is always indistinguishable from 100% and therefore it is not recommended. The CD test does not reject the null in all experiments, and has no power across experiments. On the other hand, although the size of DDIF 2 and DSY S2 is below the nominal level for N = 50 (especially for the latter), as N becomes larger the size quickly approaches its nominal size. In addition, our proposed tests have satisfactory power. DSY S2 has more power than DDIF 2 in general, unless DSY S2 rejects the null too infrequently. Di¤erent values of seem to have very little e¤ect on the performance of DDIF 2 and DSY S2 . We now turn our attention to the performance of the estimators. Table 2 reports the bias of the estimators for .19 DIF 1 and DIF 2 are the one-step and two-step DIF AC C 16 xit has zero mean without loss of generality, since we cross section-demean all data before computing the statistics. 17 See Appendix D for the details of the way of controlling these parameters. 18 We do not report the results for T = 9 in this paper, since these were similar to those for T = 5. They are available from the authors upon request. 19 We do not report the performance of the estimators for , since it has a similar pattern to that for ; although it is not as much a¤ected by error cross section dependence. 15 IPT ACCEPTED MANUSCRIPT DM AN US CR estimators respectively, de…ned by (24) and (23), and they are based on the full set of moment conditions I and II in Proposition 3. DIF X1 and DIF X2 denote the one-step and two-step DIF estimators de…ned by (26) and (25), and they are based only on the subset of the moment conditions II. SY S1 and SY S2 are the one-step and two-step SYS estimators respectively, and they are based on the full set of moment conditions I and II in Proposition 8, and SY SX1 and SY SX2 denote the one-step and two-step SYS estimators based only on the subset of moment conditions II . The bias of all GMM estimators under low cross section dependence is not noticeably di¤erent from that under zero cross section dependence. As the degree of error cross section dependence rises, the bias of the GMM estimators based on the full set of moment conditions increases, which is expected as only those estimators based on Moment Conditions II or II are consistent. As a result, the relative bias between those estimators that use the full set of moment conditions and those that use only Moment Conditions II or II increases. Table 3 reports root mean square errors of the estimators for . Under no error cross section dependence and low cross section dependence, DIF 2 and SY S2 outperform DIF X2 and SY SX2 respectively in terms of root mean square error. However, under moderate and high cross section dependence, DIF X2 and SY SX2 have a smaller root mean square error compared to DIF 2 and SY S2 respectively, in most cases. 5 An Empirical Example: Employment Equations of U.K. Firms yit = TE In this section we examine the homogeneity of error cross section dependence of the employment equations using (unbalanced) panel data for a sample of UK companies, which is an updated version of that used by Arellano and Bond (1991), and it is contained in the DPD-Ox package.20 Brie‡y, these authors select a sample of 140 companies that operate mainly in the UK with at least 7 continuous observations during the period 1976-1984. We apply our test to the model speci…cations of Blundell and Bond (1998). The model we estimated is given by i + 1 yit 1 + 0 wit + 1 wit 1 + '0 it + '1 it 1 + uit , 20 EP where yit is log of the number of employees of company i, wit is log of real product wage, it is the log of gross capital stock. Table 4 presents estimation and test results.21 Observe that year dummies are included to remove possible time e¤ects, therefore no cross-sectional demeaning of the series is implemented. Our estimation results based on the full sets of instruments, Zi and Z+ i , as de…ned in (22) and (37) but without cross-sectional demeaning, resemble The data set used is available at http://www.doornik.com/download/dpdox121.zip The GMM estimates of the parameters have been obtained using the xtabond2 command in Stata; see Roodman, D., (2005). xtabond2: Stata module to extend xtabond dynamic panel data estimator. Center for Global Development, Washington. http://econpapers.repec.org/software/bocbocode/s435901.htm AC C 21 16 IPT ACCEPTED MANUSCRIPT 6 DM AN US CR those reported in the last two columns of Table 4 in Blundell and Bond (1998), although the values do not match exactly due to di¤erences in computations and the data set used. First, all m2 tests suggest that there is no evidence of error serial correlation and this implies possibly no heterogeneous error cross section dependence. This is con…rmed by the fact that both Sargan’s di¤erence tests based on DIF and SYS for heterogeneous error cross section dependence safely fail to reject the null hypothesis of homogeneous error cross section dependence. The estimation results based only on partial instruments consisting of the covariates, ZXi and Z+ Xi , as de…ned by (18) and (31) but without crosssectional demeaning, are largely downward biased for the DIF estimator and less so for the SYS estimator. This indicates that the e¢ ciency loss of SYS that does not contain Z+ Y i in the instrument set is much smaller compared to the e¢ ciency loss of DIF. This feature seems to have some e¤ect on the testing results. For example, the p-value of the …rst-order serial correlation test, m1 , for DIF with the full set of instruments, Zi , is zero up to three decimal points, but it goes up to 0.028 with the subset of instruments ZXi . On the other hand, the p-value of m1 for SYS is zero up to three decimal points in both cases. Concluding Remarks AC C EP TE This paper has proposed a new testing procedure for error cross section dependence after estimating a linear dynamic panel data model with regressors by the generalised method of moments (GMM). The procedure is valid when the cross-sectional dimension is large and the time series dimension of the panel is small. Importantly, our approach allows one to examine whether any error cross section dependence remains after including time dummies, or after transforming the data in terms of deviations from time-speci…c averages, which will be the case under heterogeneous error cross section dependence. The …nite sample simulation-based results suggest that our tests perform well, particularly the version based on the Blundell and Bond (1988) system GMM estimator. On the other hand, the LM test of Breusch and Pagan (1980) overrejects the null hypothesis substantially and Pesaran’s (2004) CD test lacks power. Also it is shown that the system GMM estimator, based only on partial instruments consisting of the regressors, can be a reliable alternative to the standard GMM estimators under heterogeneous error cross section dependence. The proposed tests are applied to employment equations using UK …rm data, and the results show little evidence of heterogeneous error cross section dependence. 17 IPT ACCEPTED MANUSCRIPT Appendices A Proof of Proposition 3 t j j = i X2 t j ` + t j 1 xi1 + `=0 and for t 0 i X2 t j ` ft j ` X2 + `=0 "i;t `=0 i t 1 X j t + y i0 + t 1 X j xi;t j 0 i + j=0 j=0 t 1 X + 1 ` ` v i;t j `; (40) `=0 1 y it = j X2 2 + j, t j ` j ft + t 1 X j US xit CR To simplify the analysis without loss of generality we consider the case where K = 1. For t given j such that 0 j t 1, it can be shown that j j=0 "i;t j: (41) j=0 Firstly we consider E(y i;t s uit ) for s t T , given 2 s T , under the alternative hypothesis of 6= . But this is equivalent to considering E(y it ui;t+s ) for 0 t T s, given 2 s T . Initially we focus on the case of s = 2. When t = 0, E(y i0 ui2 ) = E(y i0 0i ) f2 by Assumption 3, which is not necessarily zero under the alternative. When t 1, using (40) and (41), together with uit = 0i ft + "it , E(y it ui;t+2 ) = DM AN i t E y i0 0 i ft+2 + 0 ft+2 E i 0 i t 1 X j j=0 ft j (42) 6= 0; under Assumptions 1-4, 5(a)-(b), 6 and model (7). A similar approach for the case of s > 2 leads to the conclusion that E(y it ui;t+s ) 6= 0 for t 0, s 2 under the alternative, as required. Under the null of , it follows immediately that E(y it ui;t+s ) = 0 for t 0, s 2. i = Now we consider E(xi;t s uit ) for s < t T , given 1 s T 1, under the alternative, which is equivalent to considering E(xit uit+s ) for 1 t < T s, given 1 s T 1. Initially we focus on 2 and using the case of s = 1. When t = 1, E(xi1 ui2 ) = 0 due to Assumption 3 and 5(b). For t (40), together with uit = 0i ft + "it , we have E(xit uit+1 ) = 0, under Assumptions 1-4, 5(a)-(b), 6 and model (7), under the alternative. A similar approach for the case where s > 1 leads to the conclusion that E(xit ui;t+s ) 6= 0 for t 0, s 1 under the alternative, as required. Under the null of i = , it follows immediately that E(y it ui;t+s ) = 0 for t 0, s 1, which completes the proof. In addition, it is straightforward to show that E(xi;t 1 y i;t 1 ) 6= 0. Proof of Proposition 4 TE B d Firstly we establish that SDIF X2 ! 2hx (K+1) under the alternative of H1 : P 0 b gives de…ned in (28). Rewriting p1N N ui u • i i=1 ZXi EP N N X 1 X 0 b = p1 p ZXi u • Z0Xi ui i N i=1 N i=1 N 1 N X i=1 b 1=2 PN 0 b gives Next, orthogonally decomposing _ X • i i=1 ZXi u 1=2 X 1 b p b_ X Z0Xi u • i N i=1 AC C N p e Z0Xi Wi N •DIF 2 . (43) 1=2 X 1 ^ b_ 1=2 X 0 b + p1 M b ^ B b_ X p P ZXi u • Z0Xi u • B X i i N N i=1 i=1 N = 6= 0, where SDIF X2 is 1 ^ b_ 1=2 X 0 b, p M ZXi u • B X i N i=1 N N = 18 (44) ^ 0, M ^B =I B b_ 1=2 N X ^B, B ^ = P 1 PN i=1 Z0Xi Wi , and the last line follows b = 0 by the de…nition of the GMM estimator. Substituting (43) into (44) u • i 1=2 X 1=2 X 1 b = p1 M ^ B b_ X p b_ X • Z0Xi u Z0Xi ui , i N N i=1 i=1 N N (45) CR 1 ^B = B ^ B ^ 0B ^ where P 1=2 P N 0 ^ 0 b_ X from B i=1 ZXi yields IPT ACCEPTED MANUSCRIPT ^ BB ^ = 0. since M We can express the instruments as deviations from their cross-sectional averages: ZXi = ZXi ZX ; x0it s (46) ZoXi = ZXi US where ZXi is de…ned similarly to ZXi but all are replaced with xit , and ZX is de…ned similarly but all non-zero elements are replaced with their cross-sectional averages, xt . Also de…ne the instruments in terms of deviations from their mean as and uoi = where mZX = E(ZXi ), F( i "i )+ i ). Using (46)-(48) DM AN = E( N 1 X 0 p ZXi ui N i=1 N 1 X o0 p ZXi ( ui N i=1 = ZX (47) mZX ; mZX 0 (48) (49) u) N 1 X p ( ui N i=1 u) . It is easily seen that the second term of (49) is asymptotically negligible. Consider the …rst term of (49). Reminding ourselves that ui = F i + "i , we have N 1 X o0 p ZXi ( ui N i=1 = N 1 X o0 p ZXi F ( N i=1 i N 1 X o0 p ZXi ( "i N i=1 ) i:i:d:( ; ), "it i:i:d:(0; 2i ) above, PN o = O (1). Then it follows that Z p i=1 Xi ") . = Op (N TE By the assumptions i Op (N 1=2 ) as well as N u) 1=2 ) and " = 1=2 N N 1 X 0 1 X o0 p ZXi ui = p ZXi uoi + op (1). N i=1 N i=1 (50) EP P 1=2 b o0 o ^ B = op (1) where B = Since _ X = op (1), B N 1 N and X X i=1 ZXi Wi P N 1 o0 o o0 o = p lim N E(Z u u Z ) with obvious notations, together with (50), (45) can X N !1 Xi i i Xi i=1 be written as N N 1=2 X X 1 1=2 1 o b = p1 MB p b_ X p Z0Xi u • Zo0 (51) i Xi ui + op (1). X N N N i=1 i=1 AC C o As Zo0 Xi ui are independent across i, a suitable Central Limit Theorem ensures that 1=2 1 p X N N X i=1 d o Zo0 Xi ui ! N (0; Ihx ). 19 SDIF X2 = (K + 1) we have ! N 1 X o Z uo0 i Xi N i=1 d 2 hx ! 1=2 MB X N X 1=2 X o Zo0 Xi ui i=1 ! (K+1) , under the alternative hypothesis of H1 : + op (1) CR Noting that rank(MB ) = hx IPT ACCEPTED MANUSCRIPT 6= 0, as required. Under the null hypothesis of H0 : (52) = 0, d (52) follows immediately. Also it is straightforward to establish that SDIF 2 ! 2h (K+1) , where SDIF 2 is de…ned as in (27), in line with the proof provided for (52). Now we provide the asymptotic distribution of SDIF 2 SDIF X2 . Consider the local alternative i = i + N 1=4 , US HN : where 0 < jj i jj < 1 for all i, which are assumed to be non-stochastic for expositional convenience. Here the analysis is based on the instruments in terms of deviations from their true mean, rather than from the cross-sectional average, since we have already shown that the e¤ect of such replacement is asymptotically negligible. Without loss of generality, consider ZoY i = diag(y oi;t 2 ), t = 2; 3; :::; T , hy = T 1. Also de…ne Zoi = ZoXi , 1 = p lim N N !1 N X uoi uo0 Zoi , E Zo0 i where = (53) i=1 DM AN ZoY i Y YX XY X , (54) with block elements that are conformable with ZoY i and ZoXi . By using (42) we have ! N X 1=2 o0 o ZY i u i = N , E N i=1 P 0 = O(1) is a (T 1 1) vector whose …rst element is N 1 N 1)th i=1 E(y i0 i ) f2 and the (t P P t 1 j N 0 0 0 1 t ft+2 + ft+2 i i ft j , for t = 2; 3; :::; T 1. De…ne elements are N E y i0 i j=0 i=1 where N Ihy 0 Zoi = Zoi L0 , with L = TE where L is non-singular, so that N 1 X o p Zi N i=1 where 0 N uoi 0 with Y = EP = L L0 = Y YX SDIF 2 = XY . It follows that ! N X 1 N B (B 0 B ) AC C with MB = I 1 X o uo0 i Zi 1=2 YX d ), ! N (0; 0 Y 0 ; X 1=2 MB N X Zoi 0 i=1 i=1 1 1 X Ihx B 0, B = BY B = 1=2 N 1 N X i=1 20 Zoi 0 Woi , uoi ! + op (1) so that SDIF X2 = i=1 N X 1 N o uo0 i ZXi i=1 N X 1 N = ! o uo0 i Zi ! o uo0 i Zi i=1 1=2 MB 0 0 Zoi 0 i=1 ! 1=2 MB X N X 1=2 X 1=2 1=2 M N X Zoi 0 0 MB M11 M21 = M12 M22 2 (hy ; ) ! (55) ! + op (1) ! + op (1) , US d uoi i=1 which is a symmetric and idempotent matrix of rank hy . Finally, we have ! N N X 1 X o 0 1=2 1=2 Z M SDIF 2 SDIF X2 = uo0 Zoi i i N i=1 i=1 ! uoi o Zo0 Xi ui i=1 where M = MB N X 1=2 CR SDIF 2 N X 1 N IPT ACCEPTED MANUSCRIPT 0 uoi ! (56) + op (1) DM AN which is a non-central chi-squared distribution with hy degrees of freedom and non-centrality parameter d 1=2 1=2 M11 Y = p limN !1 0N Y N > 0, so long as N 6= 0. Therefore, the result SDIF 2 SDIF X2 ! 2 hy readily follows under the null hypothesis, as required. d Furthermore, using (55) and (56), it is easily seen that SDIF 2 ! 2 (h (K + 1); ) under the local alternatives, where is the same non-centrality parameter of the asymptotic distribution of (SDIF 2 SDIF X2 ). Therefore, when h (K + 1) > hy , or subtracting hy from both sides hx > (K + 1), the Sargan’s di¤erence test is locally more powerful than the overidentifying restrictions test. C Proof of Proposition 8 In line with the proof of Proposition 3 in Appendix A, consider E( xit ( i + uit )). For t 3 and using (40) together with uit = 0i ft + "it , we have under the alternative hypothesis of i 6= , E( xit ( i + uit )) = 0, under Assumptions 1-4, 5(a)-(c), 6 and model (7). A similar line of argument proves that E( xi2 ( i + ui2 )) = 0. However, for t 2 and using (40) and (41) we have under the alternative i + uit+1 )) = t E TE E( y it ( y i1 0 i 0 ft+1 + ft+1 E i 0 i t 1 X j=0 j ft j 6= 0; D EP under Assumptions 1-4, 5(a)-(c), 6 and model (7). A similar line of argument will prove that E( y i1 ( i + ui2 )) 6= 0 under the alternative. Finally, under the null hypothesis, it is also easily seen that E( y it ( i + uit+1 )) = 0 for t 1 and E( xit ( i + uit )) = 0 for t 2, which completes the proof. Furthermore, it is straightforward to show that E( xi;t 1 xit ) 6= 0, E( xi;t 1 y i;t 1 ) 6= 0, E( y i;t 1 xit ) 6= 0, E( y i;t 1 y i;t 1 ) 6= 0. Derivations of Parameters in Monte Carlo Experiments AC C Using the lag operator, L, we can write yit and xit as yit = i + 1 xit = 1 L L 1 L i xit + "it + 1 i 1 21 L L ft + ft + 1 1 1 1 L L vit "it (57) (58) and thereby substituting (58) into (57) yields i + 1 De…ne yit = yit vit L) (1 (1 L) + i (1 L) (1 L) 1+( ft + L) (1 L) (1 ), such that (57) can be rewritten as i =(1 yit = yit = var (yit + xit + uit 1 2 2 s = u; and let the signal-to-noise ratio be denoted by 2 s + i (1 where 2 s )L "it . L) (59) (60) CR yit = IPT ACCEPTED MANUSCRIPT is the variance of the signal, 2cov (yit ; uit ) . uit ) = var (yit ) + var (uit ) 2 s (61) 2 s where a1 = 2( (1+ (1 2 )(1 ) 2 )(1 2 = ) 2 v 2 f b1 + , b1 = + 2 " b2 2 + a1 2 = )( 2 " = 1, substituting 2 u (1 + ) + 2 2 s = 2 u + ) =a1 )2 + and b2 = 1 + ( , and solving for DM AN 2 v = 2( 2 u, 1+ 1+ 2 f 2 f 2 2 + )( + ) . Applying the normalisation US varies across designs, with the aim being to keep the signal-to-noise ratio constant over changes in and the distribution of i , so that the explanatory power of the model does not change. In particular, we set 2s = 2u = = 3, where 2u = var (uit ). We normalise 2f = 2" = 1 and we keep the total signal-to-noise ratio …xed by modifying 2s accordingly through changes in 2v . It can be shown that 2 v yields (b1 + b2 ) . In line with the simulation design of Bun and Kiviet (2006), we choose 2 such that the ratio of the impacts on var(yit ) of the two variance components i and ("it , ft ) is 2 . By (59) var(yit ) = var 1 +var i + (1 i (1 L) (1 2 = Now de…ne = 2 (1 such that 2 (1 )2 = 2 2 ( +( 2 f b1 ) (b1 + b2 )a1 : We choose 2 + 2 v (1 2 f b1 + 2 " b2 )a1 . = 1. EP AC C L) (1 L) vit L) 1+( )L ft + var "it L) (1 L) (1 L) TE set 2 2 )2 (1 + var i 1 22 + 2 " b2 )a1 . By applying the normalisation 2 " = 2 f = 1, we IPT ACCEPTED MANUSCRIPT References [1] Ahn, S.G., Lee, Y.H. and P. Schmidt, 2001, GMM estimation of linear panel data models with time-varying individual e¤ects. Journal of Econometrics 102, 219-255. CR [2] Ahn, S.C and P. Schmidt, 1995, E¢ cient Estimations of Models for Dynamic Panel Data. Journal of Econometrics 68, 5-28. [3] Anselin, L., 1988, Spatial Econometrics: Method and Models, Kluwer Academic Publishers, Oxford. US [4] Anselin, L., 2001, Spatial Econometrics, in: B. Baltagi, (Eds.), A Companion to Theoretical Econometrics, Blackwell, Oxford, pp. 310–330. [5] Arellano, M., 2003, Panel Data Econometrics, Oxford University Press, Oxford. DM AN [6] Arellano, M. and S. Bond, 1991, Some tests of speci…cation for panel data: Monte Carlo evidence and an application to employment equations. Review of Economic Studies 58, 277-297. [7] Arellano, M. and S. 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Journal of Business & Economic Statistics 20, 147–162. 25 IPT ACCEPTED MANUSCRIPT Table 1: Size and Power of the Cross Section Dependence Tests in the Case with Predetermined Regressors with T = 5 = 0:2 100 200 400 LM 100.00 100.00 100.00 100.00 CD 0.00 0.00 0.00 0.00 DDIF 2 3.55 4.20 5.30 5.35 DSY S2 3.65 5.20 5.30 5.15 LM 100.00 100.00 100.00 100.00 CD 0.00 0.00 0.00 0.00 DDIF 2 4.80 9.10 18.15 29.25 DSY S2 4.00 10.90 21.65 36.40 DM AN LM 100.00 100.00 100.00 100.00 CD 0.00 0.00 0.00 0.00 DDIF 2 38.65 70.10 86.40 94.00 DSY S2 29.80 77.25 92.35 96.95 = 0:5 100 200 400 Size: i = = 0 100.00 100.00 100.00 100.00 0.00 0.00 0.00 0.00 4.40 4.45 4.60 5.40 4.15 5.35 5.55 5.80 Power: i i:i:d:U [ 0:3; 0:7] 100.00 100.00 100.00 100.00 0.00 0.00 0.00 0.00 6.00 9.90 18.70 32.10 5.35 10.75 22.95 41.60 Power: i i:i:d:U [ 1; 2] 100.00 100.00 100.00 100.00 0.00 0.00 0.00 0.00 43.25 74.45 88.45 95.25 34.50 79.75 93.30 97.80 Power: i i:i:d:U [ 1; 4] 100.00 100.00 100.00 100.00 0.00 0.00 0.00 0.00 67.70 90.30 96.40 98.55 56.35 94.10 98.30 99.65 50 LM 100.00 100.00 100.00 100.00 CD 0.00 0.00 0.00 0.00 DDIF 2 63.30 89.70 95.15 98.20 DSY S2 55.25 93.80 98.60 99.30 50 = 0:8 100 200 400 100.00 100.00 100.00 100.00 0.00 0.00 0.00 0.00 6.45 7.05 6.30 5.40 5.10 6.25 6.35 6.30 CR 50 100.00 100.00 100.00 100.00 0.00 0.00 0.00 0.00 7.65 13.95 21.75 35.30 5.50 13.55 28.05 46.55 US Test,N 100.00 100.00 100.00 100.00 0.00 0.00 0.00 0.00 46.25 77.85 91.30 95.85 36.80 83.30 95.90 98.80 100.00 100.00 100.00 100.00 0.00 0.00 0.00 0.00 69.00 92.65 98.00 99.25 62.20 96.35 98.95 99.80 AC C EP TE Notes: LM and CD denote the Breusch-Pagan LM test and Pesaran’s (2004) CD test, respectively. Both are based on the residuals of the Fixed E¤ects estimator. DDIF 2 and DSY S2 denote Sargan’s di¤erence tests based on the two-step Arellano Bond (1991) DIF estimator, and on the two-step Blundell and Bond (1998) SYS estimator respectively. The data generating process (DGP) is yit = i + yi;t 1 + xit + 48; 47; :::; T with yi; 49 = 0. The initial 49 observations are discarded. i ft + "it , i = 1; 2; :::; N , t = i:i:d:N (1; 2 ), and i are as speci…ed in the Table, = 1 , ft i:i:d:N (0; 1), "it i:i:d:N (0; 1); i xit = xi;t 1 + "i;t 1 + i ft + vit , i = 1; 2; :::; N , t = 48; 47; :::; T with xi; 49 = 0 and the initial 50 observations being discarded. = = 0:5, = 0:5; i i:i:d:U [ 1; 2], vit i:i:d:N (0; 2v ), 2v is chosen such that the signal-to-noise ratio equals 3. 2 is chosen such that the impact of the two variance components i and (ft , "it ) on var(yit ) is constant. All variables are cross-sectionally demeaned before computing statistics. All experiments are based on 2,000 replications. 26 Table 2: Bias ( 1000) of Fixed E¤ects and GMM estimators for Predetermined Regressors with T = 5 FE -115.62 -114.70 DIF1 -23.68 -10.76 DIF2 -22.83 -10.11 DIFX1 -20.58 -10.07 DIFX2 -19.93 -10.18 SYS1 13.43 7.31 SYS2 11.58 6.06 SYSX1 9.57 4.71 SYSX2 7.94 3.37 FE -107.82 -106.72 DIF1 -22.29 -10.66 DIF2 -20.85 -9.55 DIFX1 -18.53 -8.89 DIFX2 -17.71 -8.66 SYS1 12.24 6.24 SYS2 10.12 4.93 SYSX1 9.36 4.72 SYSX2 7.18 3.42 -87.17 -30.36 -27.14 -14.25 -12.92 -0.15 -1.04 6.00 3.96 -85.41 -21.66 -16.75 -6.81 -5.71 -2.22 -1.85 3.00 2.14 FE -63.49 -34.91 -30.25 -9.91 -8.17 -0.25 -1.63 9.23 4.93 -61.52 -29.04 -22.18 -4.75 -3.38 -3.56 -3.71 4.76 2.10 EP DIF1 DIF2 DIFX1 DIFX2 SYS1 SYS2 SYSX1 SYSX2 = 0:5 = 0:8 400 50 100 200 400 50 100 200 No Cross Section Dependence: i = = 0 -116.66 -115.46 -187.03 -183.89 -185.06 -184.63 -392.65 -389.05 -388.91 -7.76 -2.46 -46.94 -24.22 -10.82 -5.29 -205.25 -105.79 -59.39 -7.65 -1.98 -48.67 -24.47 -10.48 -5.25 -223.06 -115.40 -61.99 -7.70 -2.30 -65.99 -32.32 -15.45 -7.80 -438.04 -326.91 -209.79 -7.62 -2.11 -66.16 -33.28 -14.50 -7.79 -457.19 -339.03 -214.87 1.92 2.04 6.89 2.96 1.59 1.42 -12.35 -3.91 -5.17 0.34 1.51 7.24 4.18 2.20 1.42 -4.91 2.04 -0.20 0.54 1.47 5.67 5.55 3.01 2.37 -50.06 -42.01 -30.40 -0.59 1.45 10.07 8.00 6.87 4.32 -37.91 -28.31 -15.51 Low Cross Section Dependence: i i:i:d:U [ 0:3; 0:7] -108.15 -107.89 -171.34 -168.69 -169.18 -168.50 -325.46 -321.39 -321.04 -7.80 -3.68 -42.94 -23.58 -11.97 -6.50 -140.41 -75.11 -45.60 -7.44 -2.92 -43.64 -23.37 -11.12 -6.08 -147.90 -77.45 -46.13 -7.33 -2.29 -60.43 -29.10 -13.46 -7.00 -353.16 -251.67 -159.44 -7.26 -2.05 -60.16 -29.79 -12.58 -6.89 -365.60 -254.65 -158.36 1.06 0.53 2.80 -0.67 -1.42 -0.59 -18.63 -11.68 -10.26 -0.41 0.12 3.05 1.22 -0.17 -0.37 -12.05 -5.93 -5.40 0.38 1.27 2.25 3.82 2.44 1.75 -53.49 -43.00 -30.35 -0.89 1.29 6.42 6.01 6.09 3.68 -43.81 -31.09 -18.55 Moderate Cross Section Dependence: i i:i:d:U [ 1; 2] -85.48 -87.94 -143.05 -141.46 -140.35 -138.30 -281.73 -278.38 -277.75 -18.81 -19.59 -60.73 -50.04 -42.90 -35.16 -202.24 -165.42 -151.49 -15.14 -14.83 -57.18 -44.98 -36.37 -29.22 -204.32 -163.72 -150.64 -5.99 -1.91 -51.31 -24.03 -11.00 -5.72 -289.33 -199.89 -124.44 -5.65 -1.29 -46.85 -21.95 -8.43 -4.37 -281.22 -184.10 -106.67 -6.09 -9.19 -15.90 -19.55 -20.10 -15.63 -50.84 -48.19 -46.00 -6.40 -8.35 -13.94 -16.42 -17.16 -13.89 -45.48 -43.31 -41.70 -0.28 0.67 -3.87 1.04 1.15 0.76 -56.04 -40.74 -28.56 -1.32 1.02 0.77 3.29 4.38 2.87 -43.59 -25.70 -16.02 High Cross Section Dependence: i i:i:d:U [ 1; 4] -60.98 -64.34 -119.29 -118.17 -116.32 -113.69 -374.01 -371.82 -372.45 -26.18 -29.00 -83.57 -77.34 -72.34 -64.51 -459.56 -434.21 -438.01 -20.73 -22.36 -76.68 -68.73 -62.60 -54.55 -466.73 -441.78 -447.79 -4.25 -1.22 -41.29 -19.10 -8.88 -4.72 -453.74 -344.77 -244.99 -3.57 -0.52 -33.56 -14.40 -4.84 -2.42 -436.91 -313.41 -198.66 -8.64 -13.43 -17.11 -26.10 -29.33 -25.59 -79.86 -95.60 -100.75 -8.97 -12.56 -15.61 -24.64 -27.43 -24.05 -77.14 -94.71 -100.97 1.29 1.22 11.12 9.72 5.95 3.11 -5.01 2.03 0.03 -0.78 0.87 12.32 8.56 6.71 3.56 4.83 18.18 17.69 TE FE DIF1 DIF2 DIFX1 DIFX2 SYS1 SYS2 SYSX1 SYSX2 ; in the Case with CR = 0:2 100 200 US 50 DM AN Te st,N IPT ACCEPTED MANUSCRIPT 400 -388.59 -26.46 -27.85 -116.42 -118.03 -1.46 0.54 -13.39 -2.08 -319.85 -24.33 -24.42 -87.67 -85.55 -6.72 -3.88 -15.23 -6.48 -274.77 -136.35 -135.37 -68.86 -55.71 -48.00 -42.90 -16.86 -5.77 -366.96 -414.08 -427.26 -148.10 -106.04 -114.51 -113.50 1.38 22.97 AC C Notes: See notes to Table 1. FE is the …xed e¤ects estimator, DIF1 and DIF2 are the Arellano and Bond (1991) one-step and two-step …rst di¤erenced GMM (DIF) estimators, respectively. DIFX1 and DIFX2 are the one-step and two-step DIF estimators, respectively, which are based on the instruments consisting of subsets of Xi only. SYS1 and SYS2 are the Blundell and Bond (1998) one-step and two-step system GMM (SYS) estimators, respectively. SYSX1 and SYSX2 are the one-step and two-step SYS estimators, respectively, which are based on the instruments consisting of subsets of Xi only. 27 IPT ACCEPTED MANUSCRIPT Table 3: Root Mean Square Errors ( 1000) of Fixed E¤ects and GMM estimators for ; in the Case with Predetermined Regressors with T = 5 FE 13.70 DIF1 4.26 DIF2 4.72 DIFX1 6.38 DIFX2 7.27 SYS1 3.78 SYS2 3.95 SYSX1 6.57 SYSX2 7.05 FE 11.92 DIF1 7.20 DIF2 6.34 DIFX1 5.14 DIFX2 5.11 SYS1 7.18 SYS2 6.26 SYSX1 5.23 SYSX2 4.87 9.38 7.95 6.24 3.80 2.86 9.28 7.63 4.18 2.93 EP FE DIF1 DIF2 DIFX1 DIFX2 SYS1 SYS2 SYSX1 SYSX2 See Notes to Table 2. AC C 50 = 0:8 100 200 = =0 159.49 153.85 152.75 78.33 29.52 12.20 98.83 37.00 14.03 315.62 204.34 106.17 362.43 230.90 115.28 8.64 5.38 3.16 9.23 5.41 2.68 41.18 32.94 22.05 47.67 37.96 23.18 i:i:d:U [ 0:3; 0:7] 110.86 105.91 104.80 44.25 19.17 9.75 53.34 21.98 10.06 219.80 133.09 71.42 248.84 143.90 74.07 6.91 5.07 3.37 7.32 4.65 2.64 33.21 29.45 20.42 36.72 29.89 20.00 i:i:d:U [ 1; 2] i 93.31 89.68 88.77 98.86 83.78 81.96 102.60 81.07 77.90 176.18 98.94 52.60 186.10 97.06 47.46 22.59 22.96 24.48 20.06 19.19 19.07 32.33 27.07 18.39 32.23 21.75 14.97 i:i:d:U [ 1; 4] 185.46 183.37 184.96 364.79 357.52 380.38 378.32 369.79 395.92 362.81 233.76 141.53 370.67 223.07 119.02 59.51 67.97 79.84 57.14 64.93 74.88 43.40 34.61 28.51 46.88 34.68 26.44 i 28 400 151.86 5.26 5.66 52.48 54.88 1.66 1.26 13.32 12.41 CR = 0:5 400 50 100 200 400 No Cross Section Dependence: 14.24 14.27 13.75 37.79 35.41 35.12 34.69 1.98 1.09 0.47 9.06 4.18 1.81 0.92 2.34 1.18 0.50 11.00 4.82 1.97 0.97 3.48 1.91 0.89 22.98 11.60 5.92 3.16 3.91 1.99 0.91 27.12 12.61 6.30 3.23 1.84 0.94 0.48 4.82 2.48 1.23 0.65 1.84 0.88 0.39 5.14 2.50 1.18 0.58 3.44 1.83 0.95 15.09 8.64 4.80 2.58 3.66 1.80 0.84 16.82 9.33 4.83 2.49 Low Cross Section Dependence: i 12.54 12.43 12.12 32.16 30.03 29.56 29.09 1.99 1.13 0.57 8.33 4.04 1.94 1.16 2.29 1.17 0.56 9.97 4.47 2.02 1.11 3.17 1.70 0.81 20.57 9.96 5.19 2.73 3.53 1.74 0.83 24.14 10.70 5.50 2.74 1.95 1.04 0.63 4.58 2.55 1.44 0.95 1.88 0.95 0.50 4.88 2.41 1.25 0.71 3.19 1.65 0.86 13.60 7.63 4.27 2.29 3.31 1.59 0.77 14.81 8.02 4.26 2.19 Moderate Cross Section Dependence: 10.98 10.77 10.63 27.41 25.10 24.52 23.50 5.57 4.98 4.40 18.59 14.22 12.47 10.62 4.24 3.44 2.91 16.57 11.09 8.86 7.00 2.54 1.32 0.65 17.80 8.07 4.06 2.12 2.40 1.15 0.56 17.92 7.35 3.74 1.78 6.56 5.99 5.53 12.54 11.26 10.69 10.06 4.70 3.90 3.38 10.61 7.93 6.74 5.93 2.60 1.31 0.67 11.61 6.48 3.54 1.91 2.25 1.06 0.53 10.82 5.52 2.95 1.53 High Cross Section Dependence: i 8.73 8.62 8.33 24.88 22.38 21.96 20.60 6.81 6.48 6.03 28.58 24.28 23.09 19.77 4.69 4.26 3.92 24.40 19.06 17.33 14.08 1.81 0.95 0.47 15.21 6.73 3.29 1.74 1.30 0.62 0.30 11.89 4.63 2.31 1.05 9.28 9.10 8.47 19.20 18.75 18.68 17.70 6.62 6.10 5.55 16.35 14.20 13.37 12.27 2.02 1.02 0.52 10.96 6.02 3.22 1.68 1.27 0.60 0.29 8.95 4.09 2.08 1.00 US FE 15.45 DIF1 4.42 DIF2 5.03 DIFX1 7.01 DIFX2 8.05 SYS1 3.91 SYS2 4.11 SYSX1 7.09 SYSX2 7.79 = 0:2 100 200 DM AN 50 TE Test,N 103.60 5.98 5.60 33.30 33.74 2.46 1.71 12.20 10.27 87.15 72.86 66.73 25.59 21.20 24.15 18.60 12.57 8.52 180.72 358.02 376.19 69.36 48.29 81.53 76.46 22.76 17.26 IPT ACCEPTED MANUSCRIPT Table 4: Homogeneity Error Cross Section Dependence Tests and Estimates of Employment Equation, 140 Firms with 9-Year Observations (79) m1 m2 DDIF 2 -4.46 -0.17 26:84 (28) US CR A: Two-Step DIF Estimator, 1976-84 Estimation Results Based on Zi Based on ZXi Coef. Std.Err. Coef. Std.Err. yi;t 1 0.679 (0.084) 0.401 (0.124) wit -0.720 (0.117) -0.551 (0.130) wit 1 0.463 (0.111) 0.347 (0.112) 0.454 (0.101) 0.447 (0.110) it -0.191 (0.086) -0.079 (0.105) it 1 cons 0.005 (0.017) 0.003 (0.014) Test Results statistics p-values statistics p-values Sargan 88:8 [0.211] 62:0 [0.140] (51) [0.000] [0.866] [0.527] -2.19 -0.47 - [0.028] [0.641] - DM AN B: Two-Step SYS Estimator, 1976-84 Estimation Results Based on Z+ Based on Z+ i Xi Coef. Std.Err. Coef. Std.Err. yi;t 1 0.873 (0.044) 0.825 (0.071) wit -0.780 (0.116) -0.717 (0.105) wit 1 0.527 (0.168) 0.560 (0.149) 0.470 (0.071) 0.395 (0.088) it -0.358 (0.072) -0.253 (0.092) it 1 cons 0.948 (0.390) 0.720 (0.402) Test Results statistics p-values statistics p-values Sargan 111:6 [0.201] 77:3 [0.142] (100) -5.81 -0.15 34:30 (35) (65) [0.000] [0.883] [0.502] TE m1 m2 DSY S2 -5.19 -0.12 - [0.000] [0.906] - AC C EP Notes: The estimated model is yit = i + t + 1 yit 1 + 0 wit + 1 wit 1 + '0 it + '1 it 1 + uit , where yit is the log of the number of employees of company i, wit is the log of real product wage and it is the log of gross capital stock. Year dummies are included in all speci…cations. The standard errors reported are those of the robust one-step GMM estimator. The …rst row of the test results reports Sargan’s statistic for overidentifying restrictions. m1 and m2 are the …rst-order and secondorder serial correlation tests in the …rst-di¤erenced residuals. DDIF 2 denotes Sargan’s Di¤erence test for heterogeneous error cross section dependence based on the two-step Arellano and Bond (1991) DIF GMM estimator. DSY S2 denotes Sargan’s Di¤erence test based on the two-step Blundell and Bond (1998) SYS GMM estimator. Sargan test and Sargan’s di¤erence test are distributed as 2 under the null with degrees of freedom reported in parentheses. Instruments used in each equation are for DIF: yi;t 2 ; yi;t 3 ; :::; yi;0 ;wi;t 2 ; wi;t 3 ; :::; wi;0 ; i;t 2 ; i;t 3 ; :::; i;0 , and for SYS: yi;t 1 , wi;t 1 , i;t 1 . + Zi , ZXi , Z+ i , ZXi are de…ned by (22), (18), (37), (31) but without cross-sectional demeaning, respectively. 29

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