Asymptotic F Test in a GMM Framework with Cross Sectional Dependence

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1 Asymtotic F Test in a GMM Framework with Cross Sectional Deendence Yixiao Sun Deartment of Economics University of California, San Diego Min Seong Kim y Deartment of Economics Ryerson University First version: October, This version: June, 3 Abstract The aer develos an asymtotically valid F test that is robust to satial autocorrelation in a GMM framework. The test is based on the class of series covariance matrix estimators and xed-smoothing asymtotics. The xed-smoothing asymtotics and F aroximation are established under mild su cient conditions for a central limit theorem. These conditions can accommodate a wide range of satial rocesses. This is in contrast with the standard arguments, which often imose very restrictive assumtions so that a functional central limit theorem holds. The roosed F test is very easy to imlement, as critical values are from a standard F distribution. To a great extent, the asymtotic F test achieves trile robustness: it is asymtotically valid regardless of the satial autocorrelation, the samling region, and the limiting behavior of the smoothing arameter. Simulation shows that the F test is more accurate in size than the conventional chi-square tests, and it has the same size accuracy and ower roerty as nonstandard tests that require comutationally intensive simulation or bootstra. JEL Classi cation: C; C4; C8; C3. Keywords: F distribution, Fixed-smoothing asymtotics, Heteroskedasticity and Autocorrelation Robust, Robust Standard Error, Series Method, Satial Analysis, Satial Autocorrelation. Corresondence to: yisun@ucsd.edu. Deartment of Economics, University of California, San Diego, 95 Gilman Drive, La Jolla, CA , USA. This aer relaces a revious version entitled Asymtotic F Test in the Presence of onarametric Satial Deendence. y minseong.kim@ryerson.ca. Deartment of Economics, Ryerson Univesrity, 35 Victoria Street, Toronto, Ontario M5B K3, Canada.

2 Introduction In this aer, we consider satial data models in a GMM framework. Like time series data, a salient feature of satial data is that the observations are statistically deendent. To cature general and unseci ed deendence structure, we often use heteroskedasticity and autocorrelation robust (HAR) variance estimators. For recent contributions, see, for examle, Kelejian and Prucha (7) and Kim and Sun (). Most commonly used HAR variance estimators are formulated using conventional kernel smoothing techniques. Under some rate conditions, the HAR variance estimator is consistent, and we obtain asymtotic normal and chi-square tests. While convenient in emirical imlementations, consistent HAR rocedures do not cature the randomness of the HAR variance estimator, and the associated test often has large size distortion, esecially when the satial deendence is high. To address the size distortion roblem, Bester, Conley, Hansen and Vogelsang (, BCHV hereafter) extend the xed-b asymtotics in the time series setting to the satial setting. Under this tye of asymtotics, the truncated lag in the kernel HAR variance estimator is set equal to a xed fraction b of the samle size. See for examle Kiefer and Vogelsang (5) and Sun, Phillis and Jin (8). The xed-b asymtotics is in contrast with the conventional asymtotics where b! as the samle size increases. Like the conventional asymtotic distribution, the xed-b asymtotic distributions of the Wald statistic and t statistic are asymtotically nuisance arameter free, albeit nonstandard. BCHV show by simulation that the nonstandard test has better size roerties than the conventional normal or chi-square test. While BCHV make an imortant contribution in extending the xed-b asymtotics to satial settings, several challenging issues remain to be addressed. From a theoretical oint of view, the xed-b asymtotics in BCHV is obtained under a set of assumtions that are very restrictive in the satial setting. As BCHV use a functional central limit theorem (FCLT) for a scaled moment rocess, they imose strong assumtions that may not hold if satial rocesses are heteroskedastic or have satially heterogeneous deendence. BCHV also require a quadrant-wise monotone boundaries condition on the shae of the samling region. This condition may be hard to verify in emirical situations. From a ractical oint of view, imlementation of their xed-b asymtotic test can be comutationally intensive. As the xed-b asymtotic distribution is nonstandard and deends on the samling region, critical values have to be obtained via simulation or bootstra. We confront these challenges by considering the class of series HAR variance estimators, a class of HAR variance estimators that is di erent from but closely related to the class of kernel HAR variance estimators. Both classes of estimators belong to the larger class of quadratic estimators. From a broad ersective, these two classes are analogous to the resective kernel estimators and series/sieve estimators in nonarametric regressions. In the resent setting, the series HAR variance estimator we recommend for ractical use is a kernel HAR variance estimator with a secial kernel function. In general, there is no clear nite samle advantage of

3 one estimator over the other one. However, it is advantageous to use the series HAR variance estimator in develoing a new and more accurate aroximation. The smoothing arameter underlying the series HAR variance estimator is the number of terms K in the series exansion. Deending on whether K is xed or grows with the samle size, we obtain the xed-k asymtotics or the more conventional large-k asymtotics. We establish the xed-k asymtotic theory without using an FCLT or maintaining restrictive assumtions on the samling region. Our basic insight is that an FCLT is not necessary for establishing the xed-k asymtotics. It is su cient to invoke a CLT which can hold under much weaker conditions and therefore can accommodate a wide range of satial rocesses. For examle, we can use the CLT develoed by Jenish and Prucha (9, JP hereafter) which allows the satial rocess to be nonstationary and even to have asymtotically unbounded moments. In addition, a CLT is much less demanding than an FCLT on the shae of the samling region. We imose only a mild boundary condition. In the time series setting and the satial setting of BCHV, the xed-b asymtotic distribution is often reresented by a functional of Brownian motion or Brownian sheet. We will not use this tye of reresentation. Instead, we introduce the notion of asymtotically equivalent distributions. We show that, under the xed-k asymtotics, the Wald statistic is asymtotically equivalent to a quadratic form in a standard normal vector with an indeendent and random weighting matrix. The random weighting matrix catures the estimation uncertainty of the series HAR variance estimator. The asymtotically equivalent distribution is nuisance arameter free. Comared to the reresentation that involves a Brownian sheet, the asymtotically equivalent distribution has a simler reresentation. It is also easier to simulate, as it is a function of only iid standard normal vectors. A further innovation of the aer is that we design a sequence of basis functions so that the asymtotically equivalent distribution becomes a standard distribution. For any given basis functions, we rst center them and then orthonormalize the centered basis functions via the Gram-Schmidt rocedure. We use the transformed basis functions in our series HAR variance estimation. By construction, the transformed basis functions are orthonormal and integrate to zero. These two roerties ensure that the random weighting matrix follows a Wishart distribution and is indeendent of the standard normal vector. So the quadratic form in this standard normal vector follows exactly an F distribution. The transformed basis functions, couled with the xed-k asymtotics, give rise to our asymtotic F test. The F test is very convenient to use in ractice, as critical values from the F distribution can be obtained from statistical tables or software ackages. o comutationally intensive simulation or bootstra is needed. The next ste in using the series HAR variance estimator is to select the number of terms K. We consider the asymtotic mean squared error (AMSE) criterion. The roosed smoothing arameter deends on unknown arameters, which can be estimated by a arametric lug-

4 in rocedure. We emloy the Matérn model as the aroximating arametric model. As a widely-used model in satial analysis, the Matérn model is very exible in caturing various decaying atterns of satial deendence. Simulation studies show that the xed-k asymtotic tests, which include the F test, are more accurate in size than the conventional chi-square test. They are also as accurate in size as the BCHV test with similar ower roerties. The xed-k asymtotics and xed-b asymtotics may be collectively referred to as the xed-smoothing asymtotics as they e ectively involve smoothing over a xed number of quantities of interest. On the other hand, the conventional large-k asymtotics where K! or the small-b asymtotics where b! may be referred to as the increasing-smoothing asymtotics, as they involves smoothing over an increasing number of quantities. The two seci cations can be viewed as di erent asymtotic devices to obtain aroximations to the nite samle distribution. While we recommend using the xed-smoothing asymtotic aroximation, this does not mean that we advocate the use of an inconsistent HAR variance estimator. Instead, we follow conventional ractices in constructing the HAR variance estimator and the associated test statistic. Only in the last stage of inference that requires a reference distribution do we use the xed-smoothing asymtotic aroximation. When K! or b! ; it can be shown that the xed-smoothing asymtotic aroximation reduces to the conventional chi-square aroximation. So critical values from the xed-smoothing asymtotics are asymtotically valid regardless of whether the amount of smoothing is held xed or grows with the samle size. We can regard the xed-smoothing asymtotic aroximation as a robust aroximation. The series HAR variance estimator has a long history in time series analysis. It can be regarded as a multile-window estimator of Thomson (98). The interest in this tye of variance or sectrum estimators has been recently renewed in time series econometrics by Phillis (5), Sun (6), and Müller (7) with di erent motivations and objectives. Sun (, 3) rovide systematic studies on hyothesis testing using the series HAR variance estimator. Among these aers, the most closely related is Sun (3), which develos an asymtotic F test in the time series GMM framework. The most imortant contribution of this aer, comared to Sun (3), is to accommodate a general samling region, a crucial feature of satial data that is not resent in evenly-saced time series data. As a by-roduct, the aer allows for unevenly-saced time series data or time series data with missing observations. For a recent contribution on this, see Datta and Du (). The rest of the aer is organized as follows. Section describes the roblem at hand and introduces the series HAR variance estimator. Section 3 establishes the xed-k asymtotic theory and F aroximation under transformed basis functions. Section 4 develos a bandwidth selection rocedure. Section 5 discusses ractical issues in imlementing our roosed tests. Section 6 resents simulation evidence. The last section rovides some concluding discussion. Proofs are given in the Aendix. 3

5 GMM Estimation and HAR Inference We are interested in a d vector of arameters R d. Let be the true value and assume that is an interior oint of the comact arameter sace : The moment conditions Ef j () = hold if and only if =, where f j () = f (Y j ; ) is a d f vector of continuously di erentiable functions with d f d; rank E [@f j () =@ j = ] = d; and Y j R d Y at location j. The number of observed locations or the samle size is : is a vector of observations We allow the moments to exhibit general forms of satial correlation where the strength of the correlation deends on some observable distance measure between two locations. For simlicity, we follow Conley (999) and assume that it is ossible to ma the data onto a nite-dimensional integer lattice so that the distance can be exressed in terms of lattice indices. To simlify the resentation further, we consider the two dimensional case so that j = (j ; j ) Z : Extending our results to higher dimensions is straightforward. We assume that the samling region can be reresented by B n = nb \ Z where n = (n ; n ), nb = f(n b ; n b ) : (b ; b ) Bg ; and B is a comact set with ositive Lebesgue measure. As n and n increase, the samling region B n gets in ated in each direction and more observations become available. We allow the ossibility of nonnested samling regions, i.e. B n may not be a subset of B n for n < n and the data generating rocess is understood to be a triangular array data generating rocess. We consider the increasing-domain asymtotics where n! and n! and there is a minimum ositive distance between the locations where the observations are taken. Based on the moment conditions Ef j ( ) = ; the GMM estimator (Hansen, 98) of is given by ^ = arg min 3 4 f j () 5 W 3 4 f j () 5 ; where W is a d f d f ositive de nite and symmetric weighting matrix. Let g () = f j () ; G () j : If lim! G (^ ) = G and lim! W = W for some G and W with full column rank, then under some regularity conditions, ^ satis es ^ = G WG G Wg ( ) + o () = H S ( ) + o () 4

6 where H = G WG; S () = s j () and s j () = G Wf j (), a linear transformation of the original moment functions. To simlify the notation, we write S = S ( ) and s j = s j ( ) : If S ) (; ); where = lim n! s j s j A ; then ^ ) (; H H ): () In the time series literature, is called the long-run variance. We may refer to as the global variance in the satial setting, as it is not a variance associated with a single location but rather a variance contributed by all locations. Since H can be consistently estimated by its samle analog ^H = G (^ )W G (^ ), it su ces to estimate in order to conduct inference about : In this aer, we emloy the series tye estimator for : Let ^s j = s j ^ = G (^ )W f j (^ ): For each k = (k ; k ) (Z + [ fg) (Z + [ fg), de ne A k = A k (^ ) = k ;k ( j ; j )^s j ;j n n (j ;j )B n for some basis function k ;k (; ) = k (; ) that may be comlex and (;) (; ) : The basis functions are used to directly cature the global variation of the satial rocess. notational simlicity, we denote j=n = (j =n ; j =n ) from now on so that we can write k ;k (j =n ; j =n ) = k (j=n): Construct the direct estimator: ^ k = Re (A k A k ) ; For where A k is the conjugate transose of A k: Taking a simle average of the direct estimators yields a new estimator: ^ = ^ k K where K = K n f(; )g, K = (; ; :::; K ) (; ; :::; K ); K and K are smoothing arameters, and K = K K + K + K. The larger K is, the larger kk 5

7 the amount of smoothing is. In the de nition of ^; we have exlicitly excluded the case k = k = : We do so because when (;) (r; s) = ; we have A (;) = and hence ^ (;), using the de nition of the estimator ^ : As an examle, consider using the comlex exonential: k (r; s) = ex [ i (k r + k s)] ; k ; k K; () as the basis functions. In this case, A k becomes A k = (j ;j )B n ex k i j + k j s j;j ^ ; n n which is the nite Fourier transform of the satial rocess s j ;j (^ ). elementary maniulations show that Furthermore, some ^ = (j ;j )B n ( ~ j ;~j )B n s j ;j j ~j ^ W ; j ~j s ^ ~j;~j n n where W (x ; x ) = K K K k = k = cos [ (k x + k x )] = K cos [K x + K x ] sin x (K + ) sin x sin x (K + ) sin x : So the series HAR variance estimator is a kernel estimator with a secial kernel weighting function given above. The above formula also shows that ^ deends on n only through K =n and K =n ; which will be seci ed in Section 4. See Sun (, 3) and references therein for more discussions on the series long-run variance estimation in the time series setting. Suose that the null hyothesis of interest is H : R = r and the alternative is H : R 6= r; where R is a q d matrix. The F test version of the Wald statistic for testing H against H is given by F = h R^ When q =, we can construct the usual t statistic i r R ^H ^ ^H R R^ t = r =q: (3) (R^ r) R ^H ^ ^H R = : (4) 6

8 We can also consider nonlinear restrictions; our results remain valid after simle linearization. 3 Asymtotic Proerties of Test Statistics In this section, we establish the asymtotic distribution of the Wald and t statistics under the seci cation that K and K are xed. We maintain the following assumtions. Assumtion. K and K are xed as (n ; n )! (; ). Assumtion. ^! : Assumtion 3. (i) For any oen set A int(b), let j A L = min j : (j ; j ) na \ Z, j A U = max j : (j ; j ) na \ Z and de ne A n (j ) = j : (j ; j ) na \ Z for j A L j j A U : The uniform law of large numbers su su su ( ) Aint(B) j [jl A;jA U ] ja n (j )j j A n(j j H()! holds, where ( ) is an oen neighborhood of, kk is the Euclidean norm, and ja n (j )j is the number of elements in A n (j ) ; (ii) H() is continuous at = ; and H = H( ) is a nonsingular matrix. Assumtion 4. For e j s iid(; I d ) over j B n, we have P = P " P Re k ( j n )s j P Im k ( j n )s j " P Re k ( j n )e j P Im k ( j n )e j # < x k for all k K! # < x k for all k K! + o () uniformly over x k R d where is the matrix square root of ; i.e. =, and is ositive de nite. Assumtion makes it exlicit that our asymtotics is taken under xed K and K and that the samling region exands in each direction but with ossibly di erent seeds. Assumtion is made for convenience. It can be roved under more rimitive assumtions and using standard arguments. The ULL in Assumtion 3 is similar to Assumtion 6 in BCHV. Similar assumtions are made in the time series setting. For a given set A; j A L ; ja U is the range of the rst index in the set na \ Z : The set (j ; j ) : j j A L ; ja U ; j A n (j ) is the same as na \ Z : Assumtion 4 is satis ed if a CLT holds jointly for = P Re k (j=n)s j and = P Im k (j=n)s j. Some rimitive su cient conditions for Assumtion 4 are 7

9 rovided in the Aendix. When Assumtion 4 holds, we write k ( j n )s j a s k ( j n )e j where a s signi es that the two sides are asymtotically equivalent in distribution. Theorem Let Assumtions -4 hold. If k (; ) is continuously di erentiable, then F a s F a ; t a s t a where F a = q K! Re ( k k ) ; t a = kk K P kk Re k k = ; 3 = " j ; k = 4 k ( j n ) k ( j n ) 5 " j ; and " j s iid(; I q ): We rove Theorem under much weaker assumtions than BCHV. As in JP, we allow for very general samling regions. In contrast, BCHV maintain the very restrictive assumtion that the samling region has a quadrant-wise monotone boundary. In addition, our result is established under the CLT for satial rocesses while BCHV require an FCLT indexed by sets. A CLT can hold under very mild conditions to accommodate a wide range of satial rocesses. The satial rocesses can be nonstationary and can even have asymtotic unbounded moments, see JP. In addition to the conditions given in JP, we need a mild homogeneity condition for Assumtion 4 to hold; see Lemma in the Aendix. This condition is weaker than covariance stationarity. In contrast, an FCLT usually requires much stronger assumtions so that the tightness condition can be veri ed. Furthermore, the asymtotically equivalent distribution in Theorem has a simle reresentation. When s j is iid normal, it is exactly equal to the nite samle distribution. This is in contrast to the limiting distribution in BCHV, which is a sum of comlicated functionals of set-indexed Brownian sheet rocesses. In general, the limiting distribution in BCHV is always an aroximation to the nite samle distribution. Since and k are normal and cov(; k ) = P h k (j=n) P i k (j=n) I q = qq for any k K, and k are indeendent. So F is asymtotically equivalent in distribution to a quadratic form in a standard normal vector with an indeendent and random weighting matrix. The random weighting matrix catures the estimation uncertainty of the series HAR variance estimator. Like the nite samle distribution, the asymtotically equivalent distribution deends on K; the basis functions, and the samling region. 8

10 When k (; ) is real, k is normal with mean zero and variance The covariance between k cov ( k ; k ) = var ( k ) = and k is 3 4 k ( j n ) k ( j n ) k ( j n ) k ( j n ) 5 4 k ( j n ) k ( j n ) 5 I q : So the reresentation of F a enables us to see that the asymtotic distribution of F deends on the samling region and basis functions via P k (j=n); P k (j=n); and P k (j=n) k (j=n) or their limiting forms. The following corollary gives the asymtotically equivalent distribution in a secial case. Corollary Let the Assumtions in Theorem hold. If (i) P k (j=n) = o() and (ii) P k (j=n) k (j=n) = fk = k g ( + o ()), then (K q + ) a a F s Fq;K q+ and t s tk q+ : K Corollary shows that the nite samle distribution of qf can be aroximated by K K q + K q+ = (K q + q: ) As K! ; both K q+ = (K q + ) and K=(K q + ) converge to one. As a result, the above limiting distribution reduces to q, the conventional asymtotic aroximation. A direct imlication is that critical values obtained from the F aroximation are asymtotically valid under the conventional asymtotics when K! with the samle size. However, when K is not very large or the number of the restrictions q is large, the F aroximation can be very di erent from the chi-square aroximation. I q : Since both the random denominator K q+ = (K q + ) and the roortional factor K=(K q + ) shift the robability mass to the right, critical values based on the F aroximation are larger than those based on the chi-square aroximation. An examle of Corollary is when B is a rectangle and k (r; s) = cos (k r + k s) or k (r; s) = sin (k r + k s). It is not hard to show that the assumtions in Corollary hold in this case. So when the samling region has a square lattice structure, we can use critical values from the F or t distribution to erform the Wald test or the t test. 9

11 When k (r; s) is comlex, we may write K kk Re ( k k ) = K kk R;k R;k + I;kI;k where R;k = P ~ R k (j=n) " j ; I;k = P ~ I k (j=n) " j and ~ R k ( j n ) = Re ~ k ( j n ); ~ I k ( j n ) = Im ~ k ( j n ) ~ k ( j n ) = k( j n ) k ( j n ): So the comlex case is the same as the real case using Re k (r; s); Im k (r; s); k K as the basis functions. The number of basis functions is K: If these basis functions satisfy the assumtions in Corollary, then we immediately have: (K q + ) a a F s Fq;K q+ and t s tk q+ : (5) K When B is a rectangle, a natural choice for comlex valued k (; ) is the comlex exonential given in (). It is easy to check that all assumtions in Corollary hold for Re k (r; s); Im k (r; s); k K : For a general samling region B n and basis functions, the asymtotically equivalent distribution cannot be simli ed in general. However, it can be simulated easily. To obtain a draw from this distribution, we only need to generate iid standard normal vectors. So it is comutationally less intensive than simulating the asymtotic distribution reresented by functionals of Brownian sheets. A more interesting and convenient method to deal with a general samling region is to construct orthonormal basis functions on the region so that Corollary alies. Given the basis function k (; ), we can center it rst and then orthonormalize the centered basis functions via the Gram-Schmidt rocedure. More seci cally, we can follow the three stes below: (i) Center the basis functions to obtain ~ k ( j n ) := ~ k ( j ; j ) = ex n n k j i + k j n n bb n ex k b i + k b n n for all j B n : (ii) For each k; let ~ h i o R k nre be the column vector collecting ~k (j=n) ; j B n and ~ I k n h i o be the corresonding column vector collecting Im ~k (j=n) ; j B n : Concatenate ~ R k and ~ I k to form the K matrix :

12 (iii) Let = Q R be the QR decomosition of ; where Q = ( Q m`) is the K orthonormal matrix. We use the columns of Q as the basis vectors and estimate by ^ Q = K K `= " # " Q m`^sq m m# Q m`^sq (6) m= where ^s Q m is the m-th transformed moment vector with ordering conformable with the columns of : Since the columns of Q are just linear combinations of the centered basis functions, they satisfy the conditions in Corollary. Therefore, we can use the F distribution or the t distribution as the reference distribution in statistical inference. We call the resulting test the asymtotic F test or the asymtotic t test. m= 4 Smoothing Parameter Choice In this section, we select the smoothing arameter based on the mean square error criterion. We focus on the case that k (r; s) = ex [ i (k r + k s)] as it rovides a comlete orthonormal system on L ([; ] ): In addition, for this choice of k (r; s); the series HAR variance estimator reduces to a kernel HAR estimator, so we can use the results in Kim and Sun () to facilitate the mean square error calculation. We consider the case K =n = K =n = and select only one smoothing arameter. We can allow K =n and K =n to take di erent values, but it is often di cult to estimate more than one theoretically-otimal smoothing arameter in nite samles. We de ne ~ as the seudo-estimator that is identical to ^ but is based on the true arameter instead of ^. That is, ~ = Re [A k ( ) A k K ( )] : kk Given a d d weighting matrix V; we de ne the MSE criterion as MSE( ; ~ nh V ) = E vec( ~ ) V vec( ~ io ) ; where vec() is the column-by-column vectorization function. If necessary, we emloy the asymtotic truncated MSE function in Andrews (99). Under the assumtions in Kim and Sun () and following the same arguments, we can show that MSE( ~ ; V ) = 4 [vec (B)] V vec(b) + tr [V ( ) (I d + K dd)] ( + o()) ;

13 where B = () + () + (), () = () = () = 3 Es j ( ) s ~j ( ) j ~ ~j 3 Es j ( ) s ~j ( ) j ~ ~j Es j ( ) s ~j ( ) j ~ ~j j ~j (7) and K dd is the d d commutation matrix. According to the MSE criterion, the MSE-otimal is tr [V ( ) (Id + K dd )] =6 =6 ot = 4 [vec (B)] : V vec(b) When V assigns equal weights only to the diagonal elements of ~, the above formula simli es to all ot =! =6 kdiag ()k =6 kdiag(b)k : When the satial rocess has ositive deendence such that diag(es j ( ) s ~j ( )) for j; ~j ; we can show that kdiag(b)k = diag h () + () + ()i diag 3 h ()i diag + h ()i diag + h ()i diag 3 h ()i h diag + ()i diag h ()i h diag + ()i : = 4 In this case, ot where h diag ()i kdiag ()k =6 diag () + diag () A =6 : diag h ()i To control the asymtotic bias more e ectively, we suggest emloying the lower bound to choose K and K : That is, we take K = bn c and K = bn c where bxc is the integer art of x: This suggestion is line with Sun, Phillis and Jin (8) who show that the asymtotic bias under some testing-oriented criterion should be of smaller order than the asymtotic bias under the MSE criterion. In addition, using instead of ot avoids

14 estimating () ; which is often a di cult task. To imlement the data-driven K and K ; we use an aroximating arametric model to cature the satial deendence. There are two classes of arametric models that are commonly used in the literature. The rst is to model the rocess itself. This aroach is based on the work of Cli and Ord (98) and requires the use of a weight matrix. Kim and Sun () use this aroach. The second aroach is to model the covariance structure directly. In this aroach, rather than starting with the rocess and deriving the covariance matrix, a functional form for the covariance structure is assumed. The arameters of this function are then estimated. Here we use the second aroach as it does not require secifying a weight matrix. We emloy the exible class of Matérn models as the aroximating covariance model. For each comonent s () j ( ) of s j ( ) ; the covariance function is h C (h) = Es () j ( ) s () ~j i ( ) = ( ) khk khk K where > ; > ; h = j ~j ; j ~j, and K is the modi ed Bessel function of the second kind (Abramowitz and Stegun, 965, ). This function has been imlemented in standard rogramming ackages. For examle, in Matlab R the function besselk(nu,z) comutes K (z): The corresonding variogram is given by (h) = C () C (h) = ( ) khk khk K The variogram exression is the same as equation (5.3) in Webster and Oliver (,. 94). For the Matérn class of models, the sectral density for s () `;m ( ) has the form: : f(! ;! ) = +! +! + ; which can be obtained by setting = =, = ( + ), and d = in equation (3) of Stein (999, ). Under the above seci cation, we have diag kdiag ()k = d = h ()i + diag h ()i = d = 4 ; ( + ) 4 ; 3

15 and so = :55 P d = 4 4 P d = ( + ) 4 8! =6 =6 : (8) In our simulation study below, we set = for all and consider two di erent values of : = =; ; which corresond to the exonential model and the Whittle model. In the former case (h) = f ex ( khk =)g while in the latter case (h) = [ (khk =) K (khk =)] : We estimate the rest of arameters and by tting the theoretical variogram (h) to the emirical estimates ^ (h) based on nonlinear least squares (LS): ^ ; ^ = arg min ; h () khk khk K ^ (h) : In rincile, we can estimate and using more e cient estimators such as the MLE or weighted LS estimator. Here we are content with the above ordinary LS estimator as the aroximating model we seci ed is not necessarily correct. Plugging ^ and ^ into (8) yields the data-driven choices ^, K = bn ^ c, and K = bn ^ c : 5 Emirical Imlementation In ractical situations, observations may not be located on a regular integer lattice. More often than not, the observation locations are characterized by two covariates, say L m = (L m ; L m ) for m = ; :::; : To ma the observation locations to an integer lattice, we can follow the stes below: (i) ormalize L m = (L m ; L m ) to obtain ~ L m = ( ~ L m ; ~ L m ) where ~L m = ~L m = L m min m (L m ) median m (min m;lm 6=L m jl m L m j) ; L m min m (L m ) median m (min m;lm 6=L m jl m L m j) : We emloy the normalization to ensure that ~ L m is invariant to the units of measurement in each coordinate. Other normalizations are ossible. n (ii) On the basis of ~Lm ; m = ; :::; o ; comute n = min max j L ~ m L`j; ~ j ~ o L m L`j ~ ; m6=` which is the minimum of airwise distances. 4

16 (iii) Comute j m = (j m ; j m ) j m = & ~Lm ' ; j m = & ~Lm ' where de is the ceiling function. These stes ma each location (L m ; L m ) in the original sace into a unique oint (j m ; j m ) in the integer lattice B n Z + Z +. The maing is similar to the one that is used in Conley (999). Here we have imlicitly assumed that >, a necessary condition for the increasing domain asymtotics. With the lattice maing, we can follow the stes below to comute the test statistic: (i) Set n = max fj m ; m = ; :::; g and n = max fj m ; m = ; :::; g (ii) For k = ; :::; K and k = ; :::; K ; comute A k ;k ^ = (j ;j )B n ex and ^ = K P K P h i K k = k = Re A k ;k ^ A k ;k ^ : k j i + k j s j;j ^ ; n n (iii) Plugging ^ into (3) or (4), we obtain the test statistic F or t : We can follow a similar rocedure to comute ^ Q and the associated F statistic or t statistic. 6 Simulation Study This section rovides some simulation evidence on the nite samle erformance of our roosed nonstandard test and the F test. The model we consider is a linear regression model: y j = + x j + " j (9) where x j and = ( ; ) are vectors and fj = (j ; j )g indicates the lattice oints where the observations are located. Without loss of generality, we set =. We generate x j and " j according to x j = jaj u x j+a; " j = jaj u " j+a; () jaj where jaj = max (ja j ; ja j) ; u x j s iid(; I d x ), u " j s iid(; ), and fux j g is indeendent of fu " jg: We consider four di erent values of = ; :3; :6; :9. We also consider two di erent 5 jaj

17 samling regions. The rst is a circular lattice given in Figure. The samle size is = 59. The other is a square lattice. For the square lattice, we introduce two di erent designs: one uses a full lattice and the other uses a sarse lattice. For the full lattice case, we take a regular 5 5 lattice and use the data generated at each location for a total samle size of = 65: For the sarse lattice case, we generate the data on the full lattice but then randomly samle (without relacement) 65 of the otential 96 locations. We condition on the same set of 65 locations in each of the simulation relications. Our simulation results are not sensitive to the initial samling of the locations. the circular lattice are resented in Figure. The sarse square lattice together with To exlore the size roerties, we generate the data under the following null hyothesis: H q : R q (; ) = q () where R q = ( q(dx+ q); I q ). Thus, for this testing roblem, r = q : When q = ; the null is = ; when q = ; the null is = = : We set the signi cance level to be 5%, which is also the nominal size. We comute the emirical size based on 5 simulation relications. For the circular lattice, we rst consider the two tests roosed in this aer. The rst uses the comlex exonentials as the basis functions and emloys simulated critical values from the nonstandard asymtotically equivalent distribution. We refer to this test as the series nonstandard test (Series SIM). The other uses the transformed basis matrix via the Gram-Schmidt rocedure and emloys critical values from the standard F distribution. We refer to this test as the orthonormal series asymtotic F test (OS F). For both tests, we consider the corresonding tests that use critical values from the q=q distribution, leading to the series test and the OS test. For comarative uroses, we also consider two alternative tests: the test based on the Gaussian kernel estimator and the nonstandard xed-b asymtotic distribution (BCHV); and the test based on the Gaussian kernel estimator and the q=q distribution (kernel ). The xed-b critical values are simulated using the same DGP as above but with = : The Gaussian kernel we use here is: G(j ; j ) = ex! kj j k ; where d is the smoothing arameter. This Gaussian kernel is also considered by BCHV. We use the data-driven smoothing arameters. data-driven choice of K and K is given in (8). d For the series variance estimator, the For the above Gaussian kernel variance estimator, we can derive the MSE-otimal d: Following Kim and Sun (), we nd that the MSE-otimal d is d? = 6 [vec (BG )] =6 V vec(b G ) =6 () tr [V ( )(I d + K dd )] 6

18 Circular Lattice Sarse Square Lattice Figure : Circular Lattice and Sarse Square Lattice for B G = () + () ; where () and () are de ned in (7). rocedure as before, we can estimate d? by Using the same lug-in ^d? = 3:968 ( + ) P! d = ^4 ^ 8 =6 P d =6 = :7897 (^ ) ; = ^4 ^ 4 which is a data-driven choice of d when V assigns equal weights only to the diagonal elements of : Table reorts the emirical size of the tests in the circular lattice case. As it is clear from the table, the size distortion tends to increase with satial deendence and the number of restrictions being jointly tested. Comared to the tests based on the xed-smoothing asymtotics, the conventional series and kernel based chi-square tests tend to su er more from over-rejection. For examle, when = :9 and q = ; the emirical tye I errors of the series ; OS and kernel tests with = :5 are.8,.68 and.354 resectively. In contrast, the xed-k asymtotic tests and the BCHV test succeed in reducing the size distortion signi cantly. The emirical tye I errors for the Series SIM, OS F, and BCHV tests are.95,.97 and. resectively. Overall, tests based on the xed-smoothing asymtotics have more or less similar size distortion. Results not reorted here show that the selected K and K values tend to decrease with the satial deendence. So our data-driven smoothing arameter selection rocedure e ectively catures the degree of satial deendence. For the square lattice con guration, we consider two sets of testing rocedures. The rst set consists of the series test and the series F test, which emloys an F distribution as the 7

19 reference distribution based on Corollary. We do not need to conduct any transformation of basis functions because comlex exonentials are orthonormal and integrated to zero on the full square lattice. The second set consists of the BCHV and kernel tests. For the square lattice, we consider a riori xed smoothing arameter choice as well as data-driven smoothing arameter choice. estimator and d = 6 or for the kernel estimator. In the former case, we set K = K = or for the series Table gives the emirical size of the tests when the smoothing arameters are xed a riori. Both full lattice results and sarse lattice results are reorted. We see that the conventional chi-square tests can have large size distortion, esecially when the amount of smoothing is small (small K or large d). As exected, the series F test and the BCHV test are more accurate in size. Comaring the full lattice results with the sarse lattice results, we nd that the size distortion is smaller for the sarse lattice. Having a sarse lattice is analogous to have a weaker satial deendence. Similar results were found by BCHV. Table 3 resents the emirical size as in Table excet that the smoothing arameters are data-driven. The qualitative observations from Tables and remain valid. In Tables and 3, all the xed smoothing asymtotic tests have slightly more accurate size when = = as comared to = : Simulation results not reorted here show that the smaller value of delivers smaller amount of smoothing (smaller K and larger d). Table also shows that the smaller the amount of smoothing is, the more accurate in size the series F test and the BCHV test are. Thus, Tables -3 reveal that the xed-smoothing asymtotic tests are most accurate when the amount of smoothing is small. Finally, as before, the size distortion for the sarse lattice case is smaller comared to the full lattice case. The selected amount of smoothing is larger (larger K and smaller d). The larger the amount of smoothing is, the smaller the di erence between the xed-smoothing asymtotics and the increasing-smoothing asymtotics is. So it is not surrising to see that the xed-smoothing asymtotic tests do not reduce the size distortion of their resective tests by a large margin. Figures -3 resent the nite samle ower of di erent testing rocedures with = :5: We use the circular lattice resented above and emloy the DGP in (9) and (). We consider the following local alternative hyothesis: where c q = H q : R q (; ) = c q = (3) R q E( j j ) P`Bn P mb n E("`" m ` m) E( j j ) R q = ~cq with ` = (; x `) and ~c q is uniformly distributed over a shere with radius ; that is, ~c q = = k k ; ( q ; I q ) : The scaling matrix before ~c q is comuted by simulation. Unrestricted arameters are set at their default value of zero. We comute the ower using the 5% emirical critical values under the null and with data-driven smoothing arameter choice. Thus the nite samle ower is size-adjusted and 8

20 ower comarison among tests with di erent HAR variance estimators is meaningful. Of course, the size adjustment is not feasible in ractice. We consider the testing rocedures with three di erent HAR variance estimators: series estimator (Series), series estimator with the transformed basis functions by the Gram-Schmidt rocedure (OS), and Gaussian kernel estimator (Gauss). The smoothing arameters are data-driven. As illustrated in Figures -3, we do not see any signi cant ower di erence among the testing rocedures. 7 Conclusion The aer studies series HAR variance estimation and inference that are robust to satial autocorrelation. The roosed tests are more accurate in size than the conventional chi-square test because they are based on the xed-smoothing asymtotics that catures the randomness of the variance estimator. We establish the xed-smoothing asymtotic theory under very general conditions that accommodate a wide range of satial data in ractice. Among the tests we roose, the F test is asymtotically valid regardless of the samling region, the satial autocorrelation, and the limiting behavior of the smoothing arameter. The F test is esecially convenient in emirical alications, as the critical values are from standard F distributions. In this aer, we focus on the asymtotic MSE criterion, which may not be most suitable for hyothesis testing or CI construction. It is interesting to extend the methods by Sun, Phillis and Jin (8) on time series HAR variance estimation to the satial setting. The idea of using a CLT rather than a more demanding FCLT in establishing the xed-smoothing asymtotics can be used for both kernel and series HAR variance estimators in the time series setting and for kernel HAR variance estimators in the satial setting. See, for examle, Kim and Sun (). The idea of centering and orthonormalizing may be used to derive simle F aroximations in other scenarios such as when the underlying moment rocesses have local-to-unity ersistence, as considered in the recent aer of Müller (3). 9

21 Table : Emirical size of di erent tests with data-driven smoothing arameters and di erent number of restrictions in the circular lattice case q = Series SIM OS F Gaussian ( xed-b) = = = = = = = = = = :8 :8 :5 :5 :8 :8 = :3 :8 :35 :94 :3 :97 :7 = :6 :7 :4 :85 : :8 :98 = :9 :8 :8 :94 :3 :87 :7 Series OS Gaussian ( ) = = = = = = = = = = :5 :53 :5 :5 :5 :5 = :3 :5 : :5 :5 :4 : = :6 :5 :37 :48 :35 :8 :88 = :9 :57 :47 :53 :45 : :93 q = Series SIM OS F Gaussian ( xed-b) = = = = = = = = = = :5 :5 :53 :55 :5 :5 = :3 :9 :9 :3 :3 :89 :94 = :6 :88 :3 :87 : :89 :4 = :9 :95 :7 :97 :8 : :5 Series OS Gaussian ( ) = = = = = = = = = = :56 :56 :57 :58 :6 :59 = :3 :66 :65 :6 :69 :9 :74 = :6 :75 :35 :58 :8 :346 :34 = :9 :8 :38 :68 :35 :354 :3 ote: Series SIM denotes the series nonstandard test and OS F denotes the orthonormal series asymtotic F test. Gaussian ( xed-b) is the test develoed in BCHV. Series, OS test and Gaussian ( ) are conventional tests.

22 Table : Emirical size of di erent tests with a riori xed smoothing arameters in the square lattice case Series F Series Gaussian ( xed-b) Gaussian ( ) K i = K i = K i = K i = d = 6 d = d = 6 d = Regular Lattice = :47 :45 :9 :65 :46 :43 :66 : = :3 :6 :74 :8 :98 :7 :6 :3 :49 = :6 :7 :95 :35 : :98 :74 :9 :7 = :9 :79 :9 :43 :36 :9 :79 :4 :74 Sarse Lattice = :58 :54 :4 :7 :47 :6 :68 :89 = :3 :56 :63 : :85 :6 :66 :86 :3 = :6 :6 :76 :7 :99 :78 :75 :3 :5 = :9 :63 :83 : :4 :86 :8 : :9 ote: Series F is the test develoed in this aer. Gaussian ( xed-b) is the test develoed in BCHV. Series and Gaussian ( ) are conventional tests. Table 3: Emirical size of di erent tests with data-driven smoothing arameters in the square lattice case Series F Series Gaussian ( xed-b) Gaussian ( ) = = = = = = = = = = = = Regular Lattice = :45 :45 :47 :47 :4 :39 :45 :45 = :3 :76 :9 :97 :7 :67 :7 : :6 = :6 :7 :79 :36 :7 :78 :78 :7 :6 = :9 :8 :95 :4 :35 :8 :84 :73 :66 Sarse Lattice = :53 :5 :54 :53 :5 :5 :55 :54 = :3 :7 :84 :83 :8 :69 :63 :9 :86 = :6 :7 :78 :3 :98 :7 :75 :7 :8 = :9 :8 :86 :8 :5 :73 :8 :3 : ote: See notes to table.

23 ower ower ower ower ower ower ower ower (a) γ =. (b) γ = Series. OS Gauss δ.8 (c) γ = Series. OS Gauss δ.8 (d) γ = Series. OS Gauss δ.6.4 Series. OS Gauss δ Figure : Size-adjusted ower of di erent testing rocedures with q = and = = in the circular lattice case (a) γ =. (b) γ = Series. OS Gauss δ (c) γ =.6.4 Series. OS Gauss δ (d) γ = Series. OS Gauss δ.4 Series. OS Gauss δ Figure 3: Size-adjusted ower of di erent testing rocedures with q = and = = in the circular lattice case

24 8 Aendix Primitive Conditions for Assumtion 4 In this section, we use the CLT by JP to rovide rimitive su cient conditions for Assumtion 4. We consider the case when s j is a scalar and the distribution of = P k (j=n)s j for a given k: The vector case and the joint asymtotic equivalence over k K can be dealt with using the Cramér-Wold device. De nitions and below are from JP who rovide motivations and detailed discussions of these de nitions. De nition For U B n and V B n, let n (U) = ( j : j U) ; n (U; V ) = ( n (U) ; n (V )) and n (U; V ) = ( n (U) ; n (V )) ; where (; ) and (; ) are -mixing and -mixing coef- cients resectively. Then, the generalized -mixing and -mixing coe cients for the random eld f j : j B n g are de ned as follows: where l ;l ;n (r) = su ( n (U; V ) ; juj l ; jv j l ; (U; V ) r) ; l ;l ;n (r) = su ( n (U; V ) ; juj l ; jv j l ; (U; V ) r) ; (U; V ) = inf f (i; j) : i U and j V g : and for i = (i ; i ) and j = (j ; j ) ; (i; j) = max fji j j ; ji j jg : We further de ne l ;l (r) = su l ;l ;n (r) ; n l ;l (r) = su l ;l ;n (r) ; n with l ; l ; r; n Z + : De nition (i) The uer-tail quantile function Q : (; )! [; ) is de ned as Q (u) = inf ft : P ( > t) ug : (ii) For the non-increasing sequence of f ; (r)g r=, set ; () = and de ne its inverse function inv (u) : (; )! Z + [ fg as De nition 3 Let B int n inv (u) = max fr : ; (r) > ug : = j B n : lim n! P `B n E (s j s `) = and int = B int : n Lemma Assume (i) there exists an array of ositive real constants fc j;n g such that " lim su su C! k( j n )s j=c j;n k ( j n )s # j=c j;n > C = ; n E where () is the indicator function, (ii) lim inf n! Mn n > ; where M n = max jbn c j;n and n = var P jbn k (j=n)s j ; (iii) f k (j=n)s j : j B n g is either -mixing satisfying 3

25 (a) lim C! lim n! su jbn R inv (u) Q jk (j=n)s j j(j k (j=n)s j j>c) du = ; (b) P r= r l ;l (r) < for l + l 4; (c) ; (r) = O r " for some " > ; or -mixing satisfying (a) P r= r = ; (r) < ; (b) P r= r l ;l (r) < for l + l 4; (c) ; (r) = O r " for some " > : If k (; ) is continuously di erentiable with resect to ; P`B n PmB n (`; m) ke (s`s m )k = O(); and int =! ; then k ( j Z n )s j! d n n ; lim n! k (x)dx : jb xb n Proof of Lemma. Without loss of generality, we assume that k () is real. As the asymtotic normality of = P k (j=n)s j holds by JP (Theorem ), it su ces to show Let lim var n! `;m = E (s`s m ) ; then k ( j n )s ja = = `B n `B n mb n : = I + I : k ( j n )s j k ( ` n ) k( m n ) `;m mb n k ( ` n ) k( m n ) A = lim n! k ( ` n ) Z n n k (x)dx: xb `;m + k ( ` n ) `;m mb n Let n min = min (n ; n ). First, I = o () because there exists a ositive constant C such that ji j k( ` n ) k( m n ) k( ` n ) j `;mj `B n mb n C ` n ; m j `;mj n `B n mb n = C j` m j ; j` m j j `;mj n n max `B n mb n C n min `B n `B n mb n (`; m) j `;mj = o () ; 4

26 as n increases. Second, for I ; we have where ow for some constant C; I = I = ji j C I = I + I `B int n `=B int n m=b int n k ( ` n ) mb n `;m k ( ` n ) `;m: mb n `B n j `;mj = o () : because int = = o () : Using Fubini s Theorem, we have lim I = lim n! n! = lim n! = lim n! = lim n! `Bn int k ( ` n ) lim n! mb n k ( ` n ) k ( ` n ) A `B n `=B n k n n ( ` n ) A `B n Z (x) dx: n n n n k B = lim n! Combining the above analyses yields the desired result. 4 lim n! int `B int n 3 k ( ` n ) 5 int `=B int n k ( ` n ) 5

27 Proofs of Main Results Proof of Theorem. Under Assumtion 3, we have k ( j n )^s j = ( + o ()) k ( j n )G Wf j (^ ) ( = ( + o ()) k ( j n )G W f j ( ) j( ~ ) ) i = ( + o ()) k ( j n )G W f j ( ) j( ) h^ 8 39 < = ( + o ()) : k ( j n )s j k ( j n )G j( H 4 = s ~ 5 j ; ~ 8 39 < = ( + o ()) : k ( j n )s j k ( j n )@s j( H 4 = s ~ 5 j ; : ~ Let j min = min fj : (j ; j ) B n g and j max = max fj : (j ; j ) B n g : For each j [j min ; j max ] ; we artition f(j ; j ) : (j ; j ) B n g into maximal subsets such that each set has consecutive second coordinates j : That is, f(j ; j ) : (j ; j ) B n g = [ jmax j =j min [ L j `= n h io (j ; j ) : j J `;j min ; J `;j max for some integers L j ; J `;j min and J `;j max that deend on j. We ensure that J `+;j min 6= J `;j j and ` = ; :::; L j : If this haens, we combine the two adjacent sets [J `;j min ; J `;j [J `+;j min ; J `+;j max ] into one larger set. ow for each j and `; the number of elements in or roortional to n : Let S (`; j ; j ) = j m=j `;j min max for all max] and n h io j : j J `;j min ; J `;j max is either (j ;m)( for j = J `;j min ; :::; J `;j max and S `; j ; J `;j min = : By Assumtion 3, S (`; j ; j ) = O() or S (`; j ; j ) = j J `;j min + H ( + o ()) uniformly 6

28 over `; j and j : Using this observation, we have k ( j n )@s j( = = = = + = + j max L j j =j min `= j max L j j =j min `= j max L j j =j min `= j max L j j =j min `= j max L j j =j min `= j max j =j min `= j max J `;j max j =J `;j min J `;j max j =J `;j min J `;j max j =J `;j min J `;j max j =J `;j min k ( j ; J `;j max )S n n L j L j j =j min `= J `;j max j :=J `;j min k ( j ; J max `;j )H n n k ( j ; j (j ;j )( ) n k ( j n ; j n ) [S (`; j ; j ) S (`; j ; j )] k ( j n ; j n )S (`; j ; j ) j max L j j =j min `= k ( j ; j ) k ( j ; j + ) S (`; j ; j ) n n n n `; j ; J `;j max J `;j max j =J `;j min k ( j n ; j + n )S (`; j ; j ) k ( j ; j ) k ( j ; j + ) H j J `;j min n n n n + ( + o ()) J `;j max J `;j min + ( + o ()) But and j max L j j =j min `= C j max j =j min `= J `;j max j =J `;j min L j j max j =j min `= J `;j max j =J `;j min k ( j ; j ) k ( j ; j + ) n n n n L j k( j J `;j max J `;j min + n ; J max `;j ) n n H H j J `;j min + jbn j = O = O () J `;j max J `;j min + = O () ; 7

29 so k ( j n )@s j( = + = H j max L j j =j min `= j max L j j =j min `= j max j =j min `= J `;j max j =J `;j min k ( j ; J max `;j )H n n L j J `;j max j =J `;j min = k ( j n )H + o () : k ( j ; j ) k ( j ; j + ) H j J `;j min n n n n + J `;j max J `;j min + + o () k ( j n ; j n ) + o () Therefore 8 k ( j < n )^s j = ( + o ()) : k ( j n )s j = ( + o ()) It now follows from Assumtion 4 that F k ( j n j k ( j n ) ~ s ~ j A : 4 ~ s ~ j 3 8 a s 4R ^H (;) ( j < n )e j5 : R ^H Re 4 k ( j K kk n j k ( j n ) j e j A 5 ^H R 4RH ; (;) ( j n )e j5 =q 3 8 = d 4 (;) ( j < n )" j5 Re 4 : k ( j K kk n j " j A 39 3 k ( j n ) j " j A 5 4 ; (;) ( j n )" j5 =q = K! Re ( k k ) =q: kk The roof for t is similar and is omitted here. 39 = 5 ; e j A 8

30 a Proof of Corollary. Under assumtions (i) and (ii) in the corollary, k s iid(; Iq ) and P kk kk a s W(I q ; K) a Wishart distribution. So F is asymtotically equivalent in distribution to Hotelling s T-squared distribution (Hotelling, 93): qf a s T (q; K): Using the well-known relationshi between the T-squared distribution and the F distribution, we have (K q + ) a F s Fq;K q+ : K The roof for the t statistic is similar. References [] Abramowitz, M. and Stegun, I. (965): Handbook of Mathematical Functions, Dover Publications, ew York. [] Andrews, D. W. K. (99): Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation. Econometrica, 59(3), [3] Bester, A., Conley, T., Hansen, C., and Vogelsang, T. J. (): Fixed-b Asymtotics for Satially Deendent Robust onarametric Covariance Matrix Estimators. Working aer, Michigan State University. [4] Cli, A. D., and Ord, J. K. (98): Satial Processes: Models and Alications. London: Pion. [5] Conley, T. G. (999): GMM Estimation with Cross Sectional Deendence. Journal of Econometrics, 9, 45. [6] Datta, D. D. and Du, W. (): onarametric HAC Estimation for Time Series Data with Missing Observations. International Finance Discussion Paers #6, Board of Governors of the Federal Reserve System. [7] Hansen, L. P. (98): Large Samle Proerties of Generalized Method of Moments Estimators. Econometrica 5, [8] Hotelling, H. (93): The Generalization of Student s Ratio. Annals of Mathematical Statistics,, [9] Jenish,. and Prucha, I. R. (9): Central Limit Theorems and Uniform Laws of Large umbers for Arrays of Random Fields. Journal of Econometrics, 5, [] Kelejian, H. and Prucha, I. R. (7): HAC Estimation in a Satial Framework. Journal of Econometrics, 4():3 54. [] Kiefer,. M. and Vogelsang, T. J. (5): A ew Asymtotic Theory for Heteroskedasticity-Autocorrelation Robust Tests. Econometric Theory,,