Heteroskedasticity, Autocorrelation, and Spatial Correlation Robust Inference in Linear Panel Models with Fixed-E ects

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1 Heteroskedasticity, Autocorrelation, and Satial Correlation Robust Inference in Linear Panel Models with Fixed-E ects Timothy J. Vogelsang Deartments of Economics, Michigan State University December 28, Revised June 2 Abstract This aer develos an asymtotic theory for test statistics in linear anel models that are robust to heteroskedasticity, autocorrelation and/or satial correlation. Two classes of standard errors are analyzed. Both are based on nonarametric heteroskedasticity autocorrelation (HAC) covariance matrix estimators. The rst class is based on averages of HAC estimates across individuals in the cross-section, i.e. "averages of HACs". This class includes the well known cluster standard errors analyzed by Arellano (987) as a secial case. The second class is based on the HAC of cross-section averages and was roosed by Driscoll and Kraay (998). The "HAC of averages" standard errors are robust to heteroskedasticity, serial correlation and satial correlation but weak deendence in the time dimension is required. The "averages of HACs" standard errors are robust to heteroskedasticity and serial correlation including the nonstationary case but they are not valid in the resence of satial correlation. The main contribution of the aer is to develo a xed-b asymtotic theory for statistics based on both classes of standard errors in models with individual and ossibly time xed-e ects dummy variables. The asymtotics is carried out for large time samle sizes for both xed and large cross-section samle sizes. Extensive simulations show that the xed-b aroximation is usually much better than the traditional normal or chi-square aroximation esecially for the Driscoll- Kraay standard errors. The use of xed-b critical values will lead to more reliable inference in ractice esecially for tests of joint hyotheses. Keywords: anel data, HAC estimator, kernel, bandwidth, xed-b asymtotics I am grateful to Todd Elder, Emma Iglesias, Gary Solon and Je Wooldridge for helful conversations, and I thank Silvia Gonçalves and Chris Hansen for suggestions and comments on reliminary drafts. I thank seminar articiants at Cornell, U. Michigan, Michigan State, U. Montreal, Purdue for helful comments. Young Gui Kim rovided excellent research assistance with the simulations and roofread earlier versions of the aer. Portions of this research were suorted by the National Science Foundation through grant SES Corresondence: Deartment of Economics, Michigan State University, Marshall-Adams Hall, East Lansing, MI Phone: , Fax: , tjv@msu.edu

2 Introduction Since the in uential work of White (98) on heteroskedasticity robust standard errors 3 years ago, it has become standard ractice in emirical work in economics to use standard errors that are robust to otentially unknown variance and covariance roerties of the errors and data. In ure cross-section settings it is now so standard to use heteroskedasticity robust standard errors that authors often do not indicate they have used robust standard errors. In time series regression the use of heteroskedasticity and serial correlation robust standard errors is routine with authors usually indicating that they used Newey and West (987) standard errors. In anel models where cross-section individuals are followed over time, the so-called anel cluster standard errors (see Arellano (987)) are aealing because they are robust to heteroskedasticity in the cross-section and quite general forms of serial correlation over time including some nonstationary cases. Even though anel clustered standard errors are covered by graduate level textbooks (see Wooldridge (22)), Bertrand, Du o and Mullainathan (24) found that surrisingly few emirical studies based on anels with relatively many time series observations used clustered standard errors. The situation in the emirical nance literature is similar as reorted by Petersen (29). The validity of anel clustered standard errors requires that individuals in the cross-section be uncorrelated with each other, i.e. no satial correlation in the cross-section. Tyically satial correlation is ignored although sometimes the cross-section can be divided into grous or clusters where it is assumed that individuals within a cluster are correlated but individuals between clusters are uncorrelated. Then, standard errors can be con gured that are robust to the cross-sectional clustering. See Wooldridge (23) for a useful discussion of cluster methods and additional references. A recent aer by Bester, Conley and Hansen (2) rovides an interesting analysis of cluster standard errors in a general setting where the number of clusters is held xed in the asymtotics. In some cases an emirical researcher may have a distance measure for airs of individuals in the cross-section such that the satial correlation is decreasing in distance. In a ure time series setting with stationarity, distance in time is the natural distance measure. When a distance measure is available (or can be estimated) in a satial setting, robust standard errors can be obtained using the aroaches of Conley (999), Kelejian and Prucha (27), Bester, Conley, Hansen and Vogelsang (28) or Kim and Sun (2) which are extensions of nonarametric kernel heteroskedasticity autocorrelation consistent (HAC) robust standard errors to the satial context. For the case of linear anel models with individual and time dummy variables, a recent aer by Kim (2) rovides results on kernel HAC standard errors. A distance measure in the cross-section is needed to imlement the aroach of Kim (2). Suose that a distance measure is either not readily available or is unknown for the crosssection of the anel. If the time series dimension of the anel is stationary, then it is ossible to

3 obtain standard errors that are robust to satial correlation. These standard errors remain robust to heteroskedasticity and serial correlation. When information in the time dimension is substantial, relative to the information in the cross-section, the form of the unknown satial correlation can be quite general. Several aroaches aroriate for this situation can be found in the literature. One aroach is to grou the data, estimate the model within each grou and average the estimators across grous. As shown by Ibragimov and Müller (2), t-statistics constructed using these average of estimators will be robust to general forms of correlation within each grou. A secial case of this aroach is the Fama and MacBeth (973) estimator of a anel model. The model is estimated for each time eriod which generates a time series of cross-section estimates for each arameter. The cross-section estimates are then averaged across time and robustness to serial correlation can be obtained using HAC standard errors. The reason this aroach is robust to satial correlation in the cross-section is because the cross-section estimation collases the satial correlation into a single cross-section variance (the variance of a cross-section average) at each time eriod. A second aroach is to estimate the anel regression by ooled OLS and use the robust standard errors roosed by Driscoll and Kraay (998). These standard errors are comuted by taking cross-section averages of roducts of the regressors and residuals and then comuting a HAC estimator with these cross-section averages. Driscoll and Kraay (998) establish consistency of these standard errors as the cross-section and time dimension samle sizes increase under mixing conditions that limit the deendence in the data both in the time and cross-section dimensions. While the Fama and MacBeth (973) and Driscoll and Kraay (998) aroaches deliver robustness to satial correlation and serial correlation in the anel, each aroach has imortant limitations in ractice. If there are individual xed-e ects that are correlated with the regressors, the Fama-Macbeth estimator is inconsistent. For that reason, one rarely sees the Fama-Macbeth estimator used in economics alications. There is a simle aroach very similar in sirit to Fama-Macbeth that easily handles the xed-e ect. If one averages the model across the crosssection, a ure time series regression is obtained in the cross-section averages. As long as an intercet is included in the averaged model, the regression estimators will be exactly invariant to the xed-e ects and OLS will be consistent under usual exogeneity assumtions. The satial correlation and heteroskedasticity is collased by the cross-section averaging and ure time series HAC robust standard errors can be used to handle serial correlation in the time dimension. The drawback to this aroach is that information within the cross-section is lost uon averaging and this leads to less e cient estimates (erhas substantially so) comared to Pooled OLS. Pooled OLS can be made exactly invariant to individual xed-e ects by the inclusion of individ- The Fama-Macbeth aroach was motivated and designed for nance alications. See Petersen (29) for a more detailed discussion. 2

4 ual xed-e ects dummy variables leading to the well known xed-e ects estimator. Unfortunately, the mixing conditions used by Driscoll and Kraay (998) do not hold for the xed-e ects estimator although a recent aer by Gonçalves (2) establishes consistency of the Driscoll and Kraay (998) standard errors for the xed-e ects estimator for general forms of weakly deendent cross-section correlation. No results on the consistency of Driscoll and Kraay (998) standard errors aear available in the literature when time eriod dummies are included in the model. This aer rovides an in-deth analysis of the Driscoll and Kraay (998) heteroskedasticity autocorrelation satial correlation () robust standard errors in linear anel regressions estimated by xed-e ects. Results are also rovided in the case where time eriod dummies are also included in the model. The analysis is carried out using the xed-b asymtotic framework develoed by Kiefer and Vogelsang (25) for HAC based tests. The advantage of the xed-b aroach over the standard asymtotic aroach used by Driscoll and Kraay (998) is that xed-b delivers aroximations for test statistics that deend on the choice of kernel and bandwidth required to imlement the robust standard errors. This is imortant because both choices can have a big imact on the samling behavior of the standard errors. Merely establishing consistency of the standard errors, which can be di cult theoretically, will not cature any of the in uence of choice of kernel or bandwidth on the samling behavior of the standard errors. The main theoretical result of the aer is to show that test statistics constructed using xede ects estimators and the robust standard errors are valid and have the same xed-b asymtotic distributions as found by Kiefer and Vogelsang (25) in ure time series settings. Therefore xed-b critical values can be used for t and Wald tests. The xed-b result requires weak deendence in the time dimension and holds as the time dimension samle size, T, goes to in nity. The cross-section samle size, n, is can be xed or can go to in nity. In the xed-n case, the form of satial correlation in the cross-section can take on very general forms and can be unknown. In the large-n case the satial correlation in the cross-section is required to be weakly deendent. Knowledge of the distance measure itself is not required in either case. This aer does not consider the case where T is xed and n is large, i.e. traditional short anels. With T xed, there is not su cient information in the time dimension relative to the cross-section dimension for the Driscoll and Kraay (998) aroach to work. In this case one would need knowledge of the distance measure in the cross-section to construct valid standard errors. Oosite to the large-t, xed-n case, the satial correlation would need to have weak deendence, but the form of serial correlation could be quite general. The results deend on exogeneity assumtions in imortant ways. In the xed-n, large-t case, only weak exogeneity (in both the cross-section and time dimensions) is required for the xed-e ects estimator. However, if time eriod dummies are also included, strict exogeneity in the cross-section dimension is required but weak exogeneity in the time dimension is allowed. In the large-n, large-t 3

5 case, inclusion of individual xed-e ect dummies requires strict exogeneity in the time dimension although weak exogeneity is still ermitted in the cross-section. If time eriod dummies are also included, strict exogeneity in the cross-section is also needed. For comleteness the aer also rovides a xed-b analysis for a general class of standard errors that include the cluster standard errors of Arellano (987) as a secial case. Results are rovided for large-t and xed-n and generalize/extend Theorem 4 of Hansen (27) to include general kernels and bandwidths while relaxing the strict exogeneity assumtion imlicitly used by Hansen (27). This class of standard errors will only be valid when there is no satial correlation in the cross-section. Because the ultimate goal of this aer is to convince emirical researchers to use robust standard errors along with xed-b critical values in aroriate anel settings, an imortant ractical contribution of the aer is to rovide a simle numerical method for comuting xed-b critical values and -values for t-tests and Wald tests (for testing u to 5 restrictions) for the case of the Bartlett kernel (Newey-West). This numerical method is accurate and removes the need to simulate the xed-b critical values on a case-by-case basis. The remainder of the aer is organized as follows. The next section describes the models and standard errors. Using a time series ersective, the di erences and similarities between traditional cluster standard errors and robust standard errors are highlighted. Section 3 develos xed-b asymtotic results for tests based on the robust standard errors. The nite samle erformance of the various tests is examined in Section 4 using simulation methods. The use of xed-b critical values imroves the erformance of the tests over the standard normal and chi-square aroximations. The resence of some forms of satial correlation causes tests based on the traditional cluster standard errors to severely over-reject under the null hyothesis. The robust standard errors exhibit robustness to satial correlation unless the serial correlation is strong relative to the time dimension samle size. Power simulations show that when there is no satial correlation in the model, the traditional cluster standard errors lead to tests with more ower than those based on robust standard errors. The ower simulations also show that the bandwidth choice has a noticeable imact on ower. Section 5 rovides an emirical alication based on divorce rate data analyzed by Wolfers (26). Several of Wolfers main emirical ndings are shown to be robust when robust standard errors are used although it is shown that inference is very sensitive to the choice of bandwidth when comuting the robust standard errors. Failure to use xed-b critical values would lead to a misleading sense of recision in the estimates of the arameters. Conclusions are given in Section 6 and formal roofs are rovided in Aendix A. Aendix B describes the simle numerical method for comuting xed-b critical values and -values. 4

6 2 Linear Panel Models: Estimation and Standard Errors Consider a standard xed-e ects anel model given by y it = x it + a i + u it ; i = ; 2; :::; n t = ; 2; :::; T () where y it, a i and u it are scalars and x it and are k vectors. Often time eriod xed-e ects are included which gives the model y it = x it + a i + f t + u it : (2) A more general model might include individual time trends or, if time eriod xed-e ects are not included, common time trends. The asymtotic results in the aer remain unchanged when additional time regressors are included however the results do not aly to the estimated coe cients on the time trend variables themselves. The focus is on estimation and inference about. Consider the xed-e ects ordinary least squares (OLS) estimator of given by! T T b = ex it ex it ex it ey it (3) where in model () i= t= i= t= ey it = y it y i, ex it = x it x i ; with y i = T P T t= y it and x i = T P T t= x it. In model (2) we have ey it = y it y i n (y jt y j ), ex it = x it x i n j= Plugging in for ey it using () or (2) gives! T T b = ex it ex it ex it u it : i= t= i= t= (x jt x j ): Let ev it = ex it u it and de ne bv it = ex it bu it where bu it are the OLS residuals given by De ne the artial sums of bv it as bu it = ey it ex it b. bv it bs i = where r 2 (; ] and is the integer art of rt. The samle variance-covariance matrix used to obtain the traditional cluster standard errors takes the sandwich form and is given by!! T T T ex it ex it T SiT b S b it ex it ex it! ; i= t= i= 5 t= i= t= j=

7 and it is easy to show that the middle term is a secial case of a more general class of variancecovariance matrix estimators. Let and de ne b i = b(i) b (i) j = T + T j= T t=j+ bv it bv it j; j b (i) k j M + b(i) j which is the nonarametric kernel HAC estimator for cross-section individual i using the kernel, k(x), and bandwidth M. An equivalent exression for b i is given by b i = T T t= s= T K ts bv it bv is; (4) where Consider the samle variance-covariance matrix bv = T i= t= jt sj K ts = k M T ex it ex it!! b i i= i= t= T ex it ex it! (5) where the subscrit,, indicates that the "average" of the n individual-by-individual HAC estimators is used to construct the middle matrix of the sandwich. For the case where k(x) = for jxj and k(x) = otherwise, i.e. k(x) is the truncated kernel, it is well known that b i = T SiT b S b it when M = T giving exactly the variance estimator for the traditional clustered standard errors. Obviously, the traditional cluster standard errors are a secial case of "crosssection averages of HACs" standard errors. Petersen (29) discusses V b in a simulation study of robust standard errors used in nance alications. The statistical roerties of test statistics constructed using the traditional clustered standard errors are well develoed in the econometrics literature beginning with Arellano (987). A recent aer by Hansen (27) rovides a thorough analysis that extends the traditional xed-t, largen results to include large-t, large-n results and large-t, xed-n results. Hansen (27) did not rovide results for the case where time eriod dummies are included as in (2). A key assumtion in showing that the traditional clustered standard errors are valid is an assumtion that the regressors and the error term are indeendent across i. This assumtion rules out the ossibility of correlation across individuals, i.e. satial correlation in the cross-section is not allowed. When there is satial correlation in the cross-section, traditional clustered standard errors are no longer valid. 6

8 As shown by Driscoll and Kraay (998), it is ossible to obtain standard errors in a anel model that are robust to general forms of satial correlation in the cross-section. These standard errors retain robustness to heteroskedasticity and serial correlation although the serial correlation needs to be weakly deendent. Stack the k vectors bv t ; bv 2t ; :::; bv nt into an nk vector v t with transose bv t = [bv t ; bv 2t ; :::; bv nt] :De ne bv t = bv it : i= Note that bv t is n times the cross-section average of bv it. Suose we comute a HAC estimator using bv t as follows: b = b T + j= j k M An equivalent exression for b is given by bj + b j ; b T = T t= s= T K ts bv t bv s; b j = T T t=j+ bv t bv t j: (6) where K ts is de ned by (5). When b is used for the middle term of the estimate variance-covariance matrix, we obtain bv = T! T ex it ex b T it ex it ex it! : i= t= i= t= Note that b V and b V are very similar excet that b V uses the "HAC of the crosssection averages" whereas b V uses "cross-section averages of HACs". Note that utting full weight on all the samle autocovariances is not an otion in ractice for b because in this case using b S T = T bv t = t= T t= i= b = b + bv it = T t= i= T j= bj + b j = T b S b T S T = ; gives an estimator that is identically zero for any data set. ex it bu it = : Using full weights on the samle autocovariances An interesting twist on kernel HAC estimators that are secial cases of b V and b V has been analyzed recently by Bester et al. (2) (BCH). BCH consider covariance matrix estimators in general situations where the data can be divided into clusters that are asymtotically indeendent. In the time dimension the data can be clustered in the following general manner. Divide the time dimension into G contiguous (non-overlaing) exhaustive grous (time clusters) corresonding to the time eriods f; :::; [ T ]g; f[ T ] + ; :::; [ 2 T ]g; :::; f[ G T ] + ; :::; T g where < < 2 < 7

9 ::: < G <. Set = and G =. Let K ts = if time eriods t and s are in the same time eriod cluster and let K ts = otherwise. Plugging into exressions (4) and (6) leads to the BCH-HAC estimators b BCH i = T T t= s= b BCH T = T 2 T G K ts bv it bv is = T 4@ t= s= g= 2 T K ts bv t bv G s = T 4@ g= [ gt ] t=[ g T ]+ [ gt ] t=[ g T ]+ bv it bv t [ gt ] t=[ g T ]+ [ gt ] t=[ g T ]+ bv it bv t 3 A5 ; 3 A5 : These estimators can be viewed as a generalizations of the variance covariance matrix estimator used by Götze and Künsch (996) in the block bootstra. 3 Inference and Asymtotic Aroximations This section de nes the test statistics and derives the asymtotic behavior of the tests under null hyotheses involving linear restrictions on the vector. Results for large-t, xed-n and large- T, large-n are treated searately as they require di erent regularity conditions. Throughout, the symbol ) denotes weak convergence of a sequence of stochastic rocesses to a limiting stochastic rocess and d! denote convergence in distribution. 3. The Test Statistics and De nitions Consider testing linear hyotheses about of the form H : R = r; where R is a q k matrix of known constants with full rank with q k and r is a q vector of known constants. De ne the Wald statistics as W ald = (R b r) h R b V R i (R b r); W ald = (R b r) h R b V R i (R b r): In the case where q = we can de ne the t-statistics t = (Rb r) qr b V R ; t = (R b r) qr b V R : Asymtotic aroximations of the null distributions of the W ald and t statistics are obtained using large-t asymtotics. This asymtotic aroach is convenient and useful in this context because it highlights the crucial role layed by the assumtions of covariance stationarity and weak deendence 8

10 in the time dimension in generating asymtotic invariance to general forms of satial correlation. In addition, the use of large-t asymtotics allows the standard errors to be aroximated within the xed-b asymtotic framework of Kiefer and Vogelsang (25) which catures the choice of kernel and bandwidth in the asymtotic aroximation. With regards to n, results are resented for xed-n and large-n and the assumtions deend on how n is treated in the asymtotic analysis. Because the asymtotic distributions of the statistics deend on the form of the kernel used to comute the HAC estimators, some notation needs to be de ned. Let h > be an integer. The following de nition de nes some random matrices that aear in the asymtotic results. De nition Let B h (r) denote a generic h vector of stochastic rocesses. Let the random matrix, P (b; B h ), be de ned as follows for b 2 (; ]. Case (i): if k(x) is twice continuously di erentiable everywhere, P (b; B h ) = + b Z Z Z b 2 k ( r s )B h (r)b h (s) drds b r k B h ()B(r) + B h (r)b h () dr + B h ()B h () : b Case (ii): if k(x) is continuous, k(x) = for jxj ; and k(x) is twice continuously di erentiable everywhere excet for jxj =, ZZ P (b; B h ) = jr sj<b b 2 k ( r s Z )B h (r)b h (s) drds + k () b B h (r + b)b h (r) + B h (r)b h (r + b) dr b b + Z r k B h ()B h (r) + B h (r)b h () dr + B h ()B h () ; b b b where k () = lim! [(k() k( )) =], i.e. k () is the derivative of k(x) from the left at x =. Case (iii): if k(x) is the Bartlett kernel, k(x) = jxj for jxj and k(x) = for jxj, P (b; B h ) = 2 b b Z Z b B h (r)b h (r) dr b Z b B h (r + b)b h (r) + B h (r)b h (r + b) dr B h ()B h (r) + B h (r)b h () dr + B h ()B h () : De nition 2 For the BCH-HAC estimators de ne the random matrix, P ( ; ::; G ; B h ), as P ( ; ::; G ; B h ) = 3.2 Large-T, Fixed-n Results G (B h ( g ) B h ( g )) (B h ( g ) B h ( g )) : g= This subsection analyzes the asymtotic roerties of the test statistics in the large-t, xed-n case. All limits in this section are taken as T! with n held xed. The following three high level assumtions are su cient for obtaining results for the xed-e ects estimator based on model (). 9

11 Assumtion : T =2 u it = O (): t= T Assumtion 2 : lim T x it = i E(x it ) and lim T ex it ex it = rq i for r 2 (; ] where Q = t= Q i and Q is nonsingular. i= De ne the k vector vt ii = (x it i )u it. Stack the vectors vt ; vt 22 ; :::; vt nn to form the nk vector of time series v t with transose vt = vt ; vt 22 ; :::; vt nn : Assumtion 3 : E(u it jx it ) = and T =2 v t ) W (r), where W (r) is an nk vector of t= standard Wiener rocesses and is the nk nk long run variance matrix (2 times the zero frequency sectral density matrix) of v t. To handle the case where time eriod xed-e ects are also included (model (2)), Assumtion 3 needs to be strengthened as follows. De ne the k vector v ij t = (x it h i )u jt. For a given i j stack the vectors v j t ; v2j t ; :::; vnj t into an nk vector v j t with transose vj t = v j t ; v 2j t ; :::; v nj t and then stack the vectors vt ; vt 2 ; :::; vt n into an n 2 k vector vt ex with transose vt ex = vt ; vt 2 ; :::; vt n where the "ex" suerscrit denotes an extended vector that includes vectors v ij t for i 6= j. Assumtion 4 : E(u it jx jt ) = for all i; j = ; 2; :::; n and T =2 t= vt ex t= ) ex W ex (r), where W ex (r) is an n 2 k vector of standard Wiener rocesses and ex ex is the n 2 k n 2 k long run variance matrix (2 times the zero frequency sectral density matrix) of v ex t. Both sets of assumtions hold under covariance stationarity and weak deendence in the time dimension. Assumtion essentially requires that the regression error satisfy a functional central limit theorem (FCLT). Assumtion 2 requires that the samle mean and samle variance-covariance matrix of the regressors across time have well de ned limits. The form of Q i deends on the form of dummies included in the model. Assumtion 3 allows weak exogeneity in the cross-section and over time and requires that a FCLT holds for v t. Assumtion 4 requires strict exogeneity in the cross-section but allows weak exogeneity 2 over time and a FLCT is needed 3 for the extended vector 2 Note that the exogeneity assumtions rule out cases where lagged deendent variables (time and/or satially lagged) are included in the model and the errors are allowed to have correlation over time and/or in the cross-section. In this situation one would need valid instruments to obtain consistent estimators of and inference could be carried out using results in Kelejian and Prucha (27) rovided a distance measure is available for the cross-section and that instruments and regressors (besides the lagged deendent variables) are nonrandom. 3 A su cient condition for a FCLT to hold for the v t or vt ex vectors is that these vector of time series are covariance stationary and have one-summable autocovariance functions. This includes the class of covariance stationary vector autoregressive moving average models.

12 vt ex. Because Q i is not restricted to be the same for all i and because the forms of and ex ex are not restricted to be block diagonal, the assumtions ermit heterogeneity in the conditional heteroskedasticity and serial correlation while allowing general forms of satial correlation. Stationarity is not required in the cross-section. This is dual to the xed-t, large-n situation where the assumtion of random samling in the cross-section allows general forms of serial correlation in model, including nonstationarity. Assumtions 3 and 4 indicate that the form of exogeneity needed deends on whether time eriod dummies are included in the model. Without time eriod dummies, only weak exogeneity is needed in both the time and cross-section dimensions. When time eriod dummies are included, strict exogeneity is needed in the cross-section while weak exogeneity is still ermitted in the time dimension. Let I h denote a h h identity matrix and let denote an n vector of ones. Let e i denote an n vector with i th element equal to one and zeros otherwise, i.e. e i = (; ; :::; ; ; :::; ). De ne e i = e i ( ) e i = e i n. The following two theorems summarize the theoretical results for the large-t, xed-n case. Theorem Suose the model is estimated with individual xed-e ects but no time eriod dummies are included. Suose Assumtions,2 and 3 hold. For the statistics that use a kernel HAC estimator assume M = bt where b 2 (; ] is xed. For the statistics based on the BCH standard errors assume G is xed. Let B e k i (r) denote a k vector of stochastic rocesses de ned as eb i k (r) = Bi k (r) rq iq B k (); where Bk i (r) = A iw (r), A i is a nonstochastic matrix given by A i = (e i I k) and B k () = () = AW () where A = A i. Let Wq (r) denote a q vector of indeendent standard i= B i k i= Wiener rocesses and de ne f W q (r) = W q (r) bridges. For n xed as T! the following hold: rw q () to be a q vector of standard Brownian T ( b ) d! Q Bk (); d W ald! RQ h B k () RQ C k Q R i RQ Bk (); t d! RQ B k () qrq C k Q R ; W ald d! W q () C q W q (); t d! W () C ;

13 where and for h = q; 8 >< C k = >: C h = eb k i () B e k i () ; i= P (b; B e k i ); i= P ( ; ::; G ; B e k i ); i= ( for traditional clustering for kernel HAC for BCH, P (b; W f h ); for kernel HAC P ( ; ::; G ; W f h ); for BCH. Corollary Suose the BCH standard errors are con gured so that the G grous have the same number of observations, i.e. g = g G for g = ; ; 2; :::; G. Then the the conditions for art (iii) of Proosition of Bester et al. (2) hold in the time dimension and it follows that Wq () Cq Wq () Gq G q F W q;g q; () r G C G t G ; where F q;g q is an F random variable with q degrees of freedom in the numerator and G q degrees of freedom in the denominator and t G is a t random variable with G degrees of freedom. Theorem rovides some interesting insights into using robust standard errors in xed-e ect anel models. The results for the statistics formally show that result of Theorem 4 of Hansen (27) for the traditional cluster standard errors holds under the assumtion of weak exogeneity in the time dimension and generalizes his result to the case where satial correlation is ermitted in the model. The results for the statistics also extend the results of Hansen (27) to general classes of kernels and bandwidths. Theorem shows that the statistics have limiting null distributions that deend on nuisance arameters when there is satial correlation in the model (i.e. the statistics are not asymtotically ivotal). Therefore, the statistics are generally not valid in the resence of satial correlation. This is not surrising given that these statistics are designed for the case where cross-section individuals are uncorrelated with each other. In contrast, the statistics have asymtotic distributions that do not deend on nuisance arameters, i.e. the statistics are asymtotically ivotal, in the resence of satial correlation. Therefore, the statistics have broader robustness roerties with resect to correlation in the model. For the kernel statistics the limiting distributions are identical to the ure time series results obtained by Kiefer and Vogelsang (25). This may not be obvious at rst glance because the form of the random matrix P q (b; f W q ) follows from Hashimzade and Vogelsang (28) and is more comlicated than the form in Kiefer and Vogelsang (25). However, because f W q () =, the formula for P q (b; f W q ) simli es to the exact same form as in Kiefer and 2

14 Vogelsang (25). The limiting distributions are nonstandard, but critical values are easily obtained using simulation methods. A simle and accurate numerical method for comuting critical values and -values for the Bartlett kernel is rovided in Aendix B. Stata code that imlements xed-b critical values and -values for the newey, xtscc and test Stata commands is available on the author s webage: htts:// or by request. As Corollary shows, if the BCH statistics are imlemented using equal sized grous in the time dimension, the limiting distributions are scaled F and t random variables. Therefore, critical values and -values can be comuted using standard tables. Because the statistics are designed for situations where there is no satial correlation in the model, it is instructive to see how the limiting exressions in Theorem simlify when random samling in the cross-section is assumed. In this case would simlify to a block diagonal matrix with n identical k k matrices along the block diagonal and the Q i matrices would be the same for all i leading to the following Corollary. Corollary 2 Suose there is random samling in the cross-section. The results of Theorem simlify as follows: q d W ald! W q () C q W q (); d t! W ()= C ; where W q () = P n i= W q(), i Wq(r) i are q vectors of indeendent Wiener rocesses that are indeendent of each other and for h = ; q 8 cw h i()c Wh i(r) ; for traditional clustering >< C h = >: i= P (b; W c h i); i= P ( ; ::; G ; W c h i); i= for kernel HAC for BCH, where c W i h (r) = W i h (r) rn W h (). For the case of the traditional clustered standard errors, Hansen (27) has shown that W q () C q W q () nq n q F q;n q; W () C r n n t n ; Under random samling in the cross-section, the asymtotic distributions of the statistics are free of nuisance arameters and critical values can be easily tabulated using simulation methods. The results for the traditional clustered case formally extend Corollary 4. of Hansen (27) to allow weak exogeneity in the time dimension. 3

15 When time eriod dummies are also included in the model, the asymtotic distributions of the statistics change whereas the asymtotic distributions of the statistics remain the same. The next theorem and corollary summarize the results for the case where individual xede ects and time eriod dummies are included in the model. Note that Assumtion 3 is relaced with the stronger Assumtion 4. Theorem 2 Suose the model is estimated with individual xed-e ects and time eriod dummies are included. Suose Assumtions, 2 and 4 hold. For the statistics that use a kernel HAC estimator assume M = bt where b 2 (; ] is xed. For the statistics based on the BCH standard errors assume G is xed. For n xed as T! the following hold: (i) the limits of T ( b ) and the statistics take the same form as in Theorem with the de nition of Bk i (r) changed to be where A ex i B i k is the nonstochastic matrix given by A ex i = A ii n (r) = Aex i ex W ex (r); A ij ; where A ij = e i e j I k ; j= and (ii) the limits of the statistics are the same as given by Theorem and Corollary. Corollary 3 Suose there is random samling in the cross-section. The asymtotic distributions of Theorem 2 simlify as follows: d W ald! W f q() C q fw q d q(); t! W f ()= C ; where W f q() = P n f i= Wq ii (), W f q ii (r) = W f q ii (r) W ij n P n l= f W li q (r), f W ij q (r) = W ij q (r) n P n l= W il q (r), q (r) are q vectors of indeendent Wiener rocesses that are indeendent of each other and for h = ; q 8 >< C h = >: ii cfw h () c ii W f h (r) ; i= i= i= P (b; c f W ii h ); P ( ; ::; G ; c f W ii h ); for traditional clustering for kernel HAC for BCH. where c f W ii h (r) = f W ii h (r) rn f W h(). For the case of the traditional clustered standard errors it follows that fw q() C q fw q() (n ) nq (n 2) (n q) F q;n q; fw () C r n n 2 t n : 4

16 The nding that the statistics have the same asymtotic distributions when time eriod dummies are included in the model is useful given that emirical researchers often include both individual and time eriods dummies. The results for the traditional clustered standard errors under random samling are interesting because the only di erence that the inclusion of time eriod dummies makes is the (n )=(n 2) scaling adjustment to the F and t random variables. Obviously, when n is not small, this adjustment is be small. 3.3 Large-T, Large-n Results This subsection analyzes the asymtotic roerties of the test statistics in the large-t, large-n case. All limits in this section are taken as n; T!. The following four high level assumtions are su cient for obtaining results for the xed-e ects estimator based on model (). Assumtion 5 lim ex it ex it = rq, Q exists. i= t= Assumtion 6 E(u it jx is ) = for all t; s. Assumtion 7 (x it i )u it ) W k (; r); where i = E(x it ) and W k (; r) is a k i= t= vector of standard Brownian sheets. Assumtion 8 The rocess (x it random elds indexed by i; t; s. i )u is is a mean zero vector of three dimensional stationary Assumtions 5 and 7 rely on the theory of stationary and ergodic random eld theory. A random eld is simly a random rocess that is indexed by a vector with a ure time series being a secial case of random eld when the index is a scalar. Assumtion 5 is the usual assumtion for the second moments of the regressors and requires the random eld x it to be ergodic, see Adler (98, Section 6.5). Assumtion 7 is a functional central limit theorem for random elds and it requires covariance stationarity of the random eld, see Deo (975) and Basu and Dorea (979). As an examle of a covariance stationary random eld for anel data, suose the cross-section comrises the 5 states of the United States. Suose that over time the data follows a covariance stationary autoregressive model and that cross-section observations i and j have correlation that is an exonentially decreasing function of the hysical distance between the two states. In this examle, the covariance between airs of data is stationary because it is only a function of distance in time and hysical geograhical distance. Assumtion 6 imoses strict exogeneity in the time dimension but allows weak exogeneity in the cross-section dimension. Assumtion 8 relies on Assumtion 6 and stationarity of the random 5

17 elds x it and u it and is su cient to make the imact of the xed-e ects demeaning asymtotically negligible. It might be ossible to relax Assumtion 6 to allow weak exogeneity the time dimension and to dro Assumtion 8 but this would likely change and comlicate the asymtotic results. The ushot of Assumtions 5-8 are that they are stronger than the assumtions required when n is held xed and they require stationarity and weak deendence in the cross-section. The next theorem summarizes the results for the xed-e ects estimator. Theorem 3 Suose the model is estimated with individual xed-e ects but no time eriod dummies are included. Suose Assumtions 5-8 hold. As n; T! ( b d )! Q W k (; ) N ; Q Q ; and the asymtotic distributions of W ald and t are the same as given by Theorem. For the case where time eriod dummies are also included in the model, Assumtions 6 and 8 need to be strengthen as follows. Assumtion 9 E(u it jx js ) = for all i; j and t; s. Assumtion The rocess (x it random elds indexed by i; j; t; s. i )u js is a mean zero vector of four dimensional stationary We see that strict exogeneity is added to the cross-section dimension. This is su cient to make the imact of the time eriods dummies asymtotically negligible and leads to the theorem: Theorem 4 Suose the model is estimated with individual xed-e ects and time eriod dummies are included. Suose Assumtions 5, 7, 9 and hold. As n; T! ( b d )! Q W k (; ) N ; Q Q ; and the asymtotic distributions of W ald and t are the same as given by Theorem. Theorems 3 and 4 show that the xed-n, large-t results for the statistics continue to hold in the large-n, large-t case but restrict the satial correlation to be stationary and weakly deendent and require stronger exogeneity assumtions. 4 Finite Samle Proerties In this section the nite samle erformance of the robust standard errors is examined using a simulation study. The asymtotic aroximations given by the theorems are comared and contrasted with the traditional asymtotics. The erformance of the various standard errors are comared 6

18 and contrasted in designs with and without satial correlation. The imact of the strength of the serial correlation is also examined. The data generating rocess used for the simulations is given by y it = x it + 2 x 2it + 3 x 3it + 4 x 4it + u it ; (7) where u it = u i;t + t + " it, u i = ; " it N(; ); t N(; ); For l = ; 2; 3; 4 cov(" it ; " js ) = for t 6= s, cov( t ; s ) = for t 6= s; cov( t ; " js ) = 8t; s; j: x lit = x li;t + lt + e lit ; x li = ; e lit N(; ); lt N(; ); cov(e lit ; e ljs ) = for t 6= s, cov( lt ; ls ) = for t 6= s; cov( lt ; e ljs ) = 8t; s; j: The four regressors are uncorrelated with u it and each other. The regressors and the error are modeled as AR() rocesses with the same autoregressive arameter. The innovations to the regressors and u it have two comonents - a time comonent that is shared by all individuals and an idiosyncratic comonent uncorrelated over time but with otential correlation in the cross-section. The idiosyncratic errors, " it, e lit are con gured to have satial correlation as follows. For a given time, t, n i.i.d. N(; ) random variables are laced on a rectangular grid (usually a square). At each grid oint, " it is constructed as the weighted sum of the normal random variable at that grid oint, the normal random variables that are one ste away in either direction on the grid weighted by and the normal random variables that are two stes away in either direction weighted by 2. Thus, " it is a satial MA(2) rocess with arameters and 2 and the distance measure is maximum coordinate-wise distance on the grid. The e lit shocks are constructed in a similar way. When = and =, there is no satial correlation in the model. When = but 6=, there is equi-satial correlation, sat, between individuals in the cross-section in both the regressors and the idiosyncratic error. The value of that correlation is 2 =( + 2 ). Results are given for samle sizes T = ; 5; 25 and n = ; 5; 25 or n = 9; 49; 256. The latter values of n are used when 6= so that the grid in the cross-section can be con gured as a square. The number of relications is 2, in all cases and the nominal signi cance level is :5. Results are reorted for the Bartlett kernel. Results using other kernels are similar. Unless otherwise stated, xed-e ects OLS is used to estimate the model. The rst set of simulation results are for t-statistics for testing the null hyothesis H : = against the alternative H : 6=. Because the xed-e ects estimate of and its standard errors are exactly invariant to 2 ; 3 ; and 4, these arameters are set to zero without loss of generality. Table reorts emirical null rejection robabilities for the t Bart and tclus statistics. For a small selection of bandwidths are considered. For each statistic rejections are comuted t Bart 7

19 using the traditional N(; ) critical value and using the xed-b asymtotic critical value given by Corollary 2 which are only valid when there is random samling in the cross-section. Table 2 reorts emirical null rejection robabilities for the t Bart statistic. Rejections are comuted using the traditional N(; ) critical value and using the xed-b asymtotic critical value using the method described in Aendix B. Tables and 2 only consider the case of no satial correlation ( =, = ). An obvious attern in both tables is that when n is small, all three statistics have rejection robabilities greater than.5 when the N(; ) critical value is used. This haens even when the regressors and error are i.i.d. and the roblem becomes more ronounced when there is serial correlation in the model. Some of the over-rejection roblem haens because the N(; ) aroximation does not re ect the randomness in the standard error. Because the xed-b aroximation catures some of the randomness in the standard error, the tendency to over-reject is less of a roblem when xed-b critical values are used. The atterns of t clus are similar to the simulations results reorted by Hansen (27). Excet when both n and T are small, t clus has rejections close to.5 esecially when the critical value q is taken from a n n t n random variable ( xed-b columns), and this is true even when the serial correlation becomes strong. In contrast t Bart tends to over-reject when the serial correlation is strong even when xed-b critical values are used. Notice that the tendency to over-reject diminishes as the bandwidth is increased. This is an exected nding. As the bandwidth is increased, more weight is laced on higher order lags and b i becomes closer to T b SiT b S it. From the ersective of size, the traditional cluster standard errors dominate the Bartlett kernel. Unreorted results for other kernels gave similar results. The erformance of t Bart is markedly di erent than t ave as the bandwidth is increased. Notice in Table 2 that when the N(; ) critical values are used, the tendency of t Bart to over-reject substantially increases as the bandwidth is increased. This attern is easy to exlain. Because b is an estimator based on a single time series and because b = when full weight is laced on all the samle autocovariances, it is well known (see Vogelsang (28)) that as the bandwidth increases, bias in b initially falls but then increases as the bandwidth increases further. The variance of b is initially increasing in the bandwidth but eventually becomes decreasing in the bandwidth. Therefore, when a large bandwidth is used, b has substantial downward bias and t Bart tends to over-reject. The xed-b aroximation catures much of the bias and variability in b and thus the over-rejection roblem is less severe when xed-b critical values are used. However, the xed-b aroximation does not cature all of the bias in b. Part of the bias in b deends on the strength of the serial correlation and this bias grows as the serial correlation becomes stronger 4. This is why we see the over-rejection roblem becoming worse as increases. Fortunately this art of the 4 See Vogelsang (28) for a more detailed discussion on the bias and variance of kernel HAC estimators. 8

20 bias decreases as the bandwidth is increased and this is why the over-rejection roblem lessens as the bandwidth is increased when xed-b critical values are used. To summarize, as the bandwidth increases, the art of the bias of b that deends on the strength of the serial correlation decreases. But, the increase in the bandwidth increases the other art of the bias of b. This second bias is catured by the xed-b aroximation. This is why the over-rejection roblem of t Bart is lowest when the bandwidth is set equal to the samle size (M = T ) and xed-b critical values are used. The over-rejection roblem of t Bart dissiates as the time dimension samle size increases. When T is small, larger n hels reduce the over-rejection roblem but not substantially. Overall, it is the size of the time dimension samle size relative to the strength of the serial correlation that matters for the robustness of t Bart. The stronger the serial correlation, the larger T needs to be for t Bart to have nite samle null rejection robabilities close to the desired nominal level. Table 3 considers a case where there is satial correlation in the model with = :5. Critical values for t clus and tbart are still based on Corollary 2 to show the extent to which these tests breakdown when satial correlation is in the model 5. Emirical null rejection robabilities are reorted and in all cases xed-b critical values were used. Not surrisingly, the t clus and tbart statistics substantially over-reject across the board. The over-rejection roblem becomes worse as either n or T increase. In contrast, t Bart erforms much better as long as T is not too small and/or the serial correlation is not too strong. When T =, t Bart tends to over-reject regardless of the serial correlation and the rejections do not change much as n gets bigger. This is not surrising because b is being estimated with a univariate time series with observations and T is too small for the xed-b aroximation to be accurate. When T = 5, the rejections are close to :5 when the serial correlation is weak ( = ; :3) and a large bandwidth is used. When T = 25 rejections are close to :5 when is as large as :6 and when = :9 the over-rejection roblem is not severe. If larger values of T were considered, we would see rejections closer to :5 when = :9. Conversely, for a given value of T, the value of could be ushed closer to resulting in over-rejections. Again, it is the size of T relative to the strength of the serial correlation that matters 6 for t Bart. Tables 4 and 5 rovide some results on the ower of the tests. Table 4 considers the case without satial correlation whereas Table 5 has satial correlation. Rejections are comuted for = : using two-tailed tests of H : =. Because of the over-rejection roblem under the null, size-adjusted ower was comuted. This was accomlished by rst simulating nite samle null critical values of the statistics for each DGP. Size-adjusted ower is useful for making theoretical 5 The results of Theorem could be used to simulate asymtotic critical values for t clus(n) and t given satial correlation in the model. But, such an exercise would require knowledge of the form of satial correlation and would not be feasible in ractice (even if it is feasible in a simulation with known DGP). 6 Simulations reorted by Gonçalves (2) suggest that the moving blocks bootstra can further reduce the over-rejection roblem of t, relative to using xed-b critical values, when the serial correlation is strong. 9

21 comarisons of nite samle ower. Unfortunately, this size adjustment is not feasible in emirical alications where the DGP is unknown. Table 4 reveals some interesting atterns. Power of t and t Bart tend to decrease as the bandwidth increases. A similar attern in ower was found by Kiefer and Vogelsang (25) in a ure time series setting. Power of t clus tends to be lower than t Bart. This is not surrising given that tclus e ectively uses a bigger bandwidth than tbart (it uts more weight on higher order samle autocovariances). Therefore, when emirical researchers use t clus rather than use the Bartlett kernel, they are giving u ower in exchange for greater robustness of the test to serial correlation (in terms of over-rejections). Power of t Bart is lower than t clus and tbart. This is to be exected since the variation between the individuals in the cross-section is averaged out to form b. For all three statistics, ower increases as n or T increase and ower decreases as decreases. Table 5 only reorts ower for t Bart given that tbart and tclus are invalid when there is satial correlation in the model. Again, we see that ower tends to fall as the bandwidth increases. Power decreases as increases and ower is increasing in n and/or T. One notable feature of Table 5 is that ower is lower than in Table 4 across the board. Clearly satial correlation in the model negatively imacts ower. This is exected because satial correlation reduces the information in the model available from the cross-section. When 6= and satial correlation is generated by a random time e ect common to all individuals in the samle, an obvious solution to the satial correlation roblem is to include time eriod xed-e ect dummies and then use t clus. In fact, this will work quite well because including time dummies will make all three statistics exactly invariant to the satial correlation in the model. However, the time dummies only do the job when the satial correlation is generated by a common random time e ect and the emirical researcher knows that is the source of the satial correlation. If the satial correlation is driven by another source, time dummies will not make t clus or tbart valid. To illustrate this, the cross-section was divided into two grous of equal size. The DGP within each grou is given by (7) with = 3 and = which generates a correlation of.75 for airs of individuals within each grou. The idiosyncratic errors are uncorrelated between the two grous but the random time comonents are con gured to have correlation of.25 between the two grous. A simle calculation shows that the correlation between individuals across the two grous is : = 3 6 : Table 6 reorts emirical rejection robabilities for t Bart and tclus for the two grou case. Results for t Bart are qualitatively similar to tclus and are not reorted to save sace. Results are given for both the case where no time dummies are included in the model and the case where time dummies are included. As the table clearly shows, t clus systematically over-rejects whether or not time dummies are included and the over-rejections become quite large as n increases. On 2