metadata: authors: 0: author: Driscoll, John
1: author: Kraay, Aart
docna: 0: docna: Spatial correlations in panel data
docty: Policy Research Working Paper
subtopic: Consumption,Fiscal & Monetary Policy,Investment and Investment Climate,International Trade and Trade Rules,Food Security,Employment and Unemployment
teratopic: Education
abstracts: cdata!: In many empirical applications involving
            combined time-series and cross-sectional data, the residuals
            from different cross-sectional units are likely to be
            correlated with one another. This is the case in
            applications in macroeconomics and international economics
            where the cross-sectional units may be countries, states, or
            regions observed over time. Spatial correlations among such
            cross-sections may arise for a number of reasons, ranging
            from observed common shocks such as terms of trade oil
            shocks, to unobserved contagion or neighborhood effects
            which propagate across countries in complex ways. The
            authors observe that presence of such spatial correlations
            in residuals complicates standard inference procedures that
            combine time-series and cross-sectional data since these
            techniques typically require the assumption that the
            cross-sectional units are independent. When this assumption
            is violated, estimates of standard errors are inconsistent,
            and hence are not useful for inference. And standard
            correction for spatial correlations will be valid only if
            spatial correlations are of particular restrictive forms.
            The authors propose a correlation for spatial correlations
            that does not require strong assumptions concerning their
            form and how show it is superior to a number of commonly
            used alternatives.
repnme: repnme: Spatial correlations in panel data
display_title: Spatial correlations in panel data
listing_relative_url: /research/1995/12/696862/spatial-correlations-panel-data
topicv3: Agriculture and Food Security m1326205 435,Trade m1326924 698,Macroeconomic and Structural Policies
            m1327380 439,Unemployment m1504051 2747,Investment Entry m1504647 2544
publishtoextweb_dt: 2010-07-01T00:00:00Z
historic_topic: Education
origu: Macroeconomics and Growth Division
no_of_pages: 36
model_extractions: I POLICY RESEARCH WORKING PAPER 1553

# Summary findings

In many empirical applications involving combined timeseries and cross-sectional data, the residuals from different cross-sectional units are likely to be correlated with one another. This is often the case in applications in macroeconomics and international economics where the cross-sectional units may be countries, states, or regions observed over time. "Spatial" correlations among such cross-sections may arise for a number of reasons, ranging from observed common shocks such as terms of trade or oil shocks, to unobserved "contagion" or "neighborhood" effects which propagate across countries in complex ways.

inference procedures that combine time-series and crosssectional data since these techniques typically require the assumption that the cross-sectional units are independent. When this assumption is violated, estimates of standard errors are inconsistent, and hence are not useful for inference. And standard corrections for spatial correlations will be valid only if spatial correlations are of particular restrictive forms.

Driscoll and Kraay propose a correction for spatial correlations that does not require strong assumptions - concerning their form and show that it is superior to a number of commonly used alternatives.

Driscoll and Kraay observe that the presence of such spatial correlations in residuals complicates standard

- - This paper a product of the Macroeconomics and Growth Division, Policy Research Department is part of a larger effort in the departmentto study international macroeconomics. Copies of thepaperare available free from theWorld Bank, 1818 H Street NW, Washington, DC 20433. Please contact Rebecca Martin, room Nl 1-059, telephone 202-473-9065, fax 202-522-3518, Internet address rmartinl@worldbank.org. December 1995. (28 pages)

_The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be used and cited accordingly. The findings, interpretations, and conclusions are the authors' own and should not be attributed to the World Bank, its Executive Board of Directors, or any of its member countries._ Produced by the Policy Research Dissemination Center
# **Spatial Correlations in Panel Data'**

John Driscoll Department of Economics Brown University, Box B Providence RI 02912 jd@econ.pstc.brown.edu and Aart Kraay The World Bank 1818 H Street NW Washington DC 20433 akraay@worldbank.org We would like to thank John Campbell, Greg Mankiw, Matthew Shapiro and especially Gary Chamberlain and Jim Stock for helpful comments and suggestions. Financial support from the Earle A. Chiles Foundation (Driscoll) and the Social Sciences and Humanities Research Council of Canada (Kraay) during work on earlier drafts of this paper is gratefully acknowledged.
# **1 Introduction**

Economists are frequently faced with the problem of drawing inferences from data sets which combine cross-sectional and time-series data. In such situations, it has become standard practice to base inferences on techniques which pool the cross-sectional and time-series dimensions in some way. For such techniques to be valid, it must be the case that the error terms are not correlated across different cross-sectional units, either contemporaneously or at leads and lags. This condition is directly analogous to the usual requirement that the residuals from different observations in a single cross-sectional regression be independent of each other. If this condition is not met, estimates of standard errors will be inconsistent, and will not be useful for inference.

This paper begins with the observation that in many applications, especially in macroeconomics and international economics, the assumption of independent cross-sectional units is inappropriate. While it may be reasonable to assume that cross-sectional units are independent when they are households or individuals chosen according to a well-designed sampling scheme from a large population, this assumption becomes less tenable when the cross-sectional units are countries or regions. Countries or regions are likely to be subject to observable and unobservable common disturbances which will cause the residuals from one cross-section to be correlated with those of another. We will refer to such cross-sectional correlations as "spatial correlations" Spatial correlations may arise for a number of reasons. For example, in applications in which real GDP growth rates are the dependent variable, various channels of interdependence such as trade, capital flows or policy coordination mechanisms will induce cross-country correlations in GDP growth rates.' Unless the regressions of interest include right-hand side variables which correctly specify these channels of interdependence, the residuals from these regressions will be correlated across countries. Similarly, in studies of capital flows to developing countries, common external shocks such as US interest rates, or else unobserved contagion effects

> ' See Kraay and Ventura (1995) for a discussion of the roles of trade and capital mobility in the synchronization of GDP growth rates across countries. Ades and Chua (1993) and Easterly and Levine (1995) provide empirical evidence that policies tend to be correlated among neighbours, leading to correlations of growth rates over long horizons.

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(sometimes dubbed "tequila" effects in aftermath of the Mexican peso crisis) can cause residuals from capital flows regressions to be correlated across countries.

A number of standard corrections for spatial correlations exist, all of which require strong assumptions regarding the form of the spatial correlations. For example, it is common to include time dummy variables in pooled time-series, cross-sectional regressions to capture the effect of common disturbances. This technique is the appropriate correction for spatial correlation only if one assumes that the contemporaneous correlations between any pair of cross-sectional units are equal, and the lagged cross-sectional correlations are zero. Unfortunately, such strong restrictions on the form of the spatial correlations are unlikely to be correct in most applications. For example, different countries may react differently to common disturbances, or contagion effects may spread across countries only after a lag. When the structure of the spatial correlations is misspecified in this way, the properties of the resulting estimator are in general unknown.

Since it is not desirable to impose restrictions on the form of the spatial correlations, it is less clear how to proceed. One alternative is to attempt to parametrically estimate the full unrestricted matrix of spatial correlations for use in a feasible generalized least squares (FGLS) procedure. This procedure, which is a variant of the Seemingly Unrelated Regressions (SUR) technique, will only be effective in a limited set of applications. To see why this is so, suppose that there are N cross-sectional units and T time-series observations. The NxN matrix of contemporaneous cross-sectional correlations has N(N+1)/2 free parameters to be estimated using the NT available observations. Thus, in order to obtain reliable estimates of the matrix of spatial correlations, it must be the case that T>>(N+1)/2. However, in many cross-country applications using annual data, there are many more countries in the sample of interest than there are timeseries observations, so this approach will be infeasible.

In this paper we propose an alternative correction for spatial correlation. Building on the non-parametric heteroskedasticity and autocorrelation consistent (HAC) covariance matrix estimation technique of Newey and West (1987) and Andrews (1991), we show how this approach can be extended to a panel setting with cross-sectional dependence, in addition to serial correlation and heteroskedasticity. We present very weak conditions on the form of the crosssectional and time-series dependence under which a simple variant on the Newey and West 2
estimator yields consistent estimates of standard errors. In particular, we can obtain consistent estimates of standard errors in the presence of arbitrary contemporaneous cross-sectional correlations, as well as lagged cross-sectional correlations which are restricted to become small only as the time interval separating the two observations becomes large. This very general structure is likely to encompass most forms of spatial correlations encountered in practice.

Our results on consistency are based on asymptotic theory which requires the time dimension, T, to tend to infinity. Thus, our results will only be relevant for panel data sets in which the time dimension is reasonably large (our Monte Carlo simulations suggest that a value of T=20 or T=25 is the minimum). However, our results do not place any restrictions on the size of the cross-sectional dimension, N, and we can even allow the extreme case in which N tends to infinity at any rate relative to T. This implies that our techniques, in contrast to SUR, will be applicable in situations such as cross-country panel data sets where the number of countries is very large.

The rest of this paper proceeds as follows. In Section 2, we first develop the intuitions for our results using a simple ordinary least squares example. We then provide a formal statement of our result, using a mixing random field structure to characterize the permissible extent of crosssectional and time-series dependence. Since this structure is somewhat unfamiliar, we provide some examples of forms of cross-sectional dependence which satisfy the conditions we impose. In Section 3, we consider the finite-sample properties of our estimator using Monte Carlo evidence, and find that our non-parametric estimator performs significantly better than common alternatives such as time dummies or SUR. Section 4 concludes.

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# **2 Consistent Covariance Matrix Estimation with Spatial Dependence**

## 2.1 **Preliminary Discussion**

In order to develop the intuition for the results of this paper, consider the following simple bivariate linear panel regression: To obtain an estimate of ,, it is common practice to pool the cross-sectional and time-series observations and apply OLS to the full set of NT observations. If the errors are independently and identically distributed (i.e. if Q =U2'lr), this will yield consistent estimates of 3 and its standard errors. However, in the presence of spatial correlations, Q is no longer diagonal. In this case, although the OLS estimator of ,B is still consistent, the OLS standard errors will be inconsistent, and hence will not be useful for inference.

We can write the OLS estimator of P in the usual way as follows: To simplify the above expression, denote the term in brackets in the denominator of (2) as QT **_2,_** and define ~~~~~~~~(3)

> **_2_** For the purposes of this illustrative example, we can assume that the x, are constants and that **_QT-_** Q>O as N,T-

4
Substituting into the expression for the OLS estimator, we obtain This change of variables is useful because it reduces the original panel data estimation problem to a simple time-series estimation problem. In other words, by defining a cross-sectional average h4 at every point in time, we have "collapsed" the cross-sectional dimension of the problem to a single time-series observation by averaging over the N cross-sectional units in each period.

Since OLS estimates of P will be consistent even in the presence of spatial correlations, our main concern is with obtaining consistent estimates of the variance of the OLS estimator. Using the above notation, we can write this variance in terms of the h, as The main intuition of the paper is as follows. Given appropriate conditions on the h,, we can apply standard time-series non-parametric covariance matrix estimation techniques such as those employed by Newey and West to obtain a consistent estimate of _ST,_ and hence of VT. These conditions (known as "mixing conditions" in the standard time-series literature) place restrictions on the autocovariances of the h1, requiring the dependence between h, and h, , to become small as the time interval separating them, s, becomes large. Imposing restrictions on the autocovariances of h, will amount to placing restrictions on the contemporaneous and lagged spatial dependence in the residuals, **E[EEj,,e]** ., since the autocovariances of the sequence h, are a weighted average of these covariances, i.e.

In this paper, we show that only very weak restrictions on the form of the spatial correlations are 5
required to ensure that h1 satisfies the regularity conditions necessary for consistent estimation of ST. In particular, we can permit arbitrary contemporaneous correlations, and we require only that lagged cross-sectional dependence declines at a particular rate as the time separation becomes large. As in Newey and West (1987), our asymptotic results rely on a large time dimension. However, we do not need to restrict the size of the cross-sectional dimension, which can tend to infinity at any rate relative to T.

We use a mixing random field structure to characterize the permissible extent of spatial and temporal dependence. As mixing random fields are somewhat unfamiliar in the econometrics literature, we briefly present the necessary intuitions here, and relegate the details to the appendix. Random fields are simply random variables with multiple indices. For example, returning to Equation (3), we can define the random field **N,t=x,,Ejt,** indexed by i and t. In the standard univariate time-series literature, a time series is described as "mixing" if the dependence between two random variables x, and x,-, becomes small as the time interval separating them, s, becomes large. In this paper, we will analogously describe a random field as being "mixing" if the dependence between h-, and h becomes small as the time interval s becomes large, for any pair of cross-sectional observations i and j.<sup>3</sup> In this way, the standard time-series definition of mixing corresponds to the special case where i=j. Finally, the "size" of a mixing is defined as the rate at which the dependence between two observations must decline as a function of the distance between them.

This particular definition of a mixing random field has the extremely useful property that the cross-sectional averges of this random field, h1 (as defined in Equation (3)), form a univariate This definition of mixing departs from the standard definitions in the random field literature in that it treats the crosssectional and timne-series dimensions asymmetrically. Typically, mixing restriction would require the dependence between hN, and h,,, to become small as the Euclidean distance d=((i-j)<sup>2</sup> +s<sup>2</sup> )"<sup>2</sup> between these two random variables becomes large. This is an unattractive property of standard definitions of mixing random fields for two reasons. First, in our panel data applications, it precludes canonical forms of cross-sectional dependence such as equal contemporaneous cross-unit correlations. To see why this is so, notice that the distance between h, and h, is simply li-jl according to the above definition. Standard definitions of mixing would then rule out equal cross-sectional correlations between any hN and hj, since this correlation will not decline as ji-l becomes large. The second problem is that in order to impose the restriction that observations "far apart" in the cross-sectional ordering be approximately uncorrelated, it is necessary to know what the cross-sectional ordering is. This is problematic, since unlike in the time dimension, in most cases there is no natural ordering in the cross-sectional dimension.

6
mixing sequence of the same size as the underlying random field. This is true for any value of N (the size of the cross-sectional dimension), including the limiting case where N--. If we impose the restriction that the **hit** form a mixing random field of the appropriate size, then h, will be a mixing sequence of the same size, and we can directly apply standard time-series covariance matrix estimation techniques to obtain an estimate of ST in Equation (5). Thus, our results amount to a simple extension of the Newey and West estimator, which may be viewed in the above context as the case in which N= 1.

# 2.2 Results

In this section, we present our main result, which is simply a generalization and formalization of the discussion of the previous section. The theorem is stated in terms of a broad class of Generalized Method of Moments estimators, of which the OLS case discussed above is an example.

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# **Theorem**

_Consider the class of GMM models identified by a pxl vector of orthogonality conditions E[/4(00, zd)]=O, where 019F6& is an axl vector of parameters with as•p, t9is a compact subset of R, z1, is a kxl vector of data, and denote z,=(zi .'. ,zv,) 'and h,=h(O, z) =NA`Xi=,Mb(0, z,) Supposefurther that (1) z, is an a-mixing random field of size 2(r+ o)/(r+ 6-1), as defined in the Appendix;_

- _(2) (a) #r(d, z) is continuously differentiable in 0 and measurable in z,,; (b) E[IVI(fX,_ **_zJ 14r+°l]<J<-_** _VO Fea_

- _(3) For all sequences {_<sup>_0_</sup> _T) such that OT' converges in probability to 00, plimT'((0, zd)/8d) /090rOT*}=plim(T'(dA(0 zd)/80) /l= 0=D'where D 'is offull column rank_

_Then the GM5M estimator_ _is consistent and asymptotically normal and the panel Newey and West (1987) covariance matrix estimator_ _is a consistent estimator of VT as T- for any N (including N-o), where w j,m)_ = _I-j/(m(T) + 1), m(T) =0(T"_<sup>_3_</sup> _) and DT is a consistent estimator of Dr_

# **Proof:** See Appendix.

Before presenting some examples of the forms of spatial correlations encompassed by the theorem, a few comments are in order. First, the random field structure in Assumption (1) is the 8
only substantive assumption required for the above result, as the remaining assumptions are fairly standard conditions required to establish the properties of the GMM estimator. Note that the assumptions of the theorem require no prior knowledge of the form of the spatial and temporal correlations, and place only very weak restrictions on them. Hence this framework subsumes many common forms of spatial dependence, without requiring an explicit (and probably also incorrect) parameterization of the form of the temporal and spatial dependence.

Next, a sketch of the proof is as follows. The regularity conditions placed on *(O, Z,,) in Assumption 2(a) are sufficient to ensure that e itself is a mixing random field of the same size as z,t The cross-sectional averages of this random field, h1, will form a univariate mixing sequence as described above. The remainder of the proof is then simply a matter of verifying the standard results for the consistency and asymptotic normality of the GMM estimator and the consistency of the Newey and West covariance matrix estimator.

Finally, as an example of an application of the theorem, consider the simple OLS example of the previous section, which is a particular case of the GMM framework in which the above theorem is cast. To see this, set 00,=P, z4=(yj, x,j) and lr(<sup>3</sup> ,xj)=x (y'-Px,J. Thus to implement this technique, first obtain the usual OLS coefficient estimate. Next construct the sequence of crosssectional averages hk using the estimated residuals from the OLS specification (i.e.

ht=N-'Ej=1 ,(yj,-bx,), where b is the OLS estimate of P) and insert this into the definition of ST to obtain its consistent estimate. Finally, note that a consistent estimate of DT is given by (NT)-Y,= Combining these two expressions gives the consistent estimate of the covariance matrix, VT.<sup>4</sup> TSP and GAUSS codes to perform these calculations are available from the authors.

Another commonly-encountered model is one with unit-specific fixed effects, yi=f+x4P+E. One method to apply the theorem to this model is to transform the data by taking deviations from unit means and rewriting as above i.e. letting 7,t=(y,,-T Yt_,Ty,t,x, -T ' , **TNy** 9
# **2.3 Examples of Spatial Correlation**

As the mixing random field assumption of the theorem may be somewhat difficult to verify in practice, in this section we present some simple examples of forns of spatial correlation which satisfy this assumption. It is most convenient to present some examples of permissible forms of spatial correlation using the simple linear model with fixed scalar regressors of Equation (1). A broad class of spatial correlations can be represented using the following factor structure<sup>6</sup> for the residuals of this regression, i.e.

_f,_ is an Mxl vector of independent, zero mean, unit variance random variables referred to as "factors", while Xi is an Mxl vector of scalar "factor loadings". The vft are random variables with zero mean and variance aYt2 which are independent over time and across units. In addition, they are orthogonal to the factors, and are referred to as the "residuals from the factor structure". In the appendix, we show that this factor structure satisfies the conditions of the theorem.' This result thus provides a convenient way to verify the somewhat more abstract mixing random field conditions of the theorem, and describes a broad class of spatial correlations which have a factor structure representation to which our covariance matrix estimator is robust.

The simplest case of spatial correlations is one in which the contemporaneous crosssectional correlations are all equal. It is straightforward to verify that if M=1, Xi=)X for all i, and **o(t** =X2_o2 for all i and t, then the OLS residuals ch have mean zero, variance a2 and cross-sectional correlations E[ei,EjJ]=)<sup>2</sup> . In this special case, including time dummies in Equation (1) will remove all the spatial correlations from the residuals.

A more interesting case of spatial correlations with a factor structure representation is one 6 We are grateful to Gary Chamberlain for suggesting this approach. See Chamberlain and Rothschild (1983) for a discussion of factor structures in the context of finance theory, and Al-Najjar (1995) for a discussion of factor structures as a method of modelling aggregate uncertainty in a continuum of cross-sectional units.

' The result in the appendix does not reiy on the assumption that the residuals from the factor structure are contemporaneously uncorrelated across units, but allows them to have arbitrary spatial correlations.

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in which the cross-sectional units are divided into m=l,...,M groups, and the within-group correlations are equal for all the members of the group. Such groups might be regions, geographic "neighbours" or any other grouping based on observable or unobservable characteristics. To represent this as a factor structure, define Im as a set which consists of the indices of the members of group m. Then, if ;Xmi=Am for ieI1l, **j,'= -a',** .0j2 and the vi, are again independent across units, it is immediate to verify that the OLS residuals will have mean zero, variance a2 and cross-sectional correlations E[eitEjt]=Xm2 for i,jel, and zero otherwise.

The most general case of contemporaneous spatial correlations is one in which the crosssectional correlations are arbitrary. This would be a natural structure in the case where there is a common factor to which cross-sectional units react differently. If we introduce the assumption that the residuals from the factor structure, vi, have arbitrary contemporaneous cross-sectional correlations, then we can somewhat trivially write this as a factor structure in which all the factor loadings are zero. In this case, there will be spatial correlations in the residuals from Equation (1) even after time dummies are included in the specification.

Note that in these examples, we have used a factor structure to characterize the contemporaneous spatial dependence in the residuals. We can easily extend this to introduce dependence over time as well. For example, suppose that there are arbitrary contemporaneous cross-sectional correlations, zero lagged cross-sectional correlations, and within units the disturbances follow an AR(M) process. To give this a factor structure representation, letfm have an autocovariance function which is I at lag m, and zero otherwise. Then if we set ).^=E[Ei,Eit-$] the factor structure will replicate exactly this combination of cross-sectional and temporal dependence. Along the same lines, it is possible to generate much more complicated forms of lagged cross-sectional dependence using this factor structure.

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# **3 Monte Carlo Results**

In this section we use Monte Carlo experiments to examine how well our estimator (which we will refer to as the HAC estimator) performs relative to common alternative corrections for spatial correlations such as SUR and OLS and time dummies.<sup>8</sup> We generate large numbers of samples of artificial data with various forms of spatial correlations, and obtain coefficient estimates and standard errors using OLS, SUR and our HAC estimator. We can then evaluate the relative performance of these estimators by reporting a "coverage rate" for each estimator, which is the fraction of samples in which two standard deviation confidence intervals contain the true parameter values. For the HAC procedure, this fraction equals .95 as T-o, as it does not rely on a correct parameterization of the temporal and spatial correlations. In finite samples (T<o), however, this coverage rate will inevitably be somewhat smaller. OLS with time dummnies and SUR will in general be misspecified, and their asymptotic properties when misspecified are generally unknown. However, by reporting coverage rates, we can get a rough idea of the severity of the impact of the misspecification of these procedures.

We consider linear models such as <!-- Start of picture text --> Y21 2 E2<br>p + ~~~~~~~~~~~(10)<br>YNt XNt ENt<br><!-- End of picture text --> Without loss of generality, we set ,=0 in Equation (10). To introduce a rich structure of 8 Most other techniques are versions of feasible GLS which impose various zero restrictions on the variance-covariance matrix (for example, Case (1991) and Keane and Runkle (1992)). Elliot (1993) discusses the case when there is a single cross-section, with one observation per geographical location. There are two classes of interesting exceptions to this. One class, used in finance, assumes there is no serial correlation and applies a technique similar to the White (1980) correction for heteroskedasticitv (for example, Fama and MacBeth (1973), Lehmann (1990), and Froot (1989)). A second alternative has been offered by Conley (1994), who proposes a nonparametric variance-covariance matrix estimator when the "distance" between cross-sectional units is known.

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contemporaneous and lagged spatial correlations into the residuals, we generate them according to the following autoregressive scheme: <!-- Start of picture text --> E2 =R e 2 + +, R = pIN , {E[iErPEj] = 2(1)<br>ENr ENt- I<br><!-- End of picture text --> This specification ensures that the residuals in Equation (10) exhibit both contemporaneous and lagged spatial correlations. In the simple case in which p=0, there are only contemporaneous spatial correlations in the residuals in Equation (10), and these are given by E. In this case, both the HAC and the SUR are correctly specified, and their coverage rates can be compared directly. Finally, we use the same structure to generate the regressors, xh Before we can perform Monte Carlo experiments, we need to parameterize the spatial correlation matrix E and the serial correlation parameter, p. We allow the parameter p to range over the values _(_ 0, .1, .3, .5 }, which correspond to the moderate degree of serial correlation likely to be present in most applications. Selecting the matrix of contemporaneous spatial correlations is more difficult. One alternative is to choose a simple parameterization for this matrix, and report Monte Carlo results as these parameters vary. While this approach is useful in that it allows us to vary the degree of spatial correlations directly, it has the disadvantage that such simple parameterizations are unlikely to capture the complicated forms of spatial correlations which are likely to be encountered in practice.<sup>9</sup> Instead, we use a data-based method of selecting the matrix 9 This approach was taken in an earlier draft of the paper. We performed a large number of simulations, allowing the magnitude of spatial correlations and the sizes of the cross-sectional and time-series dimensions to vary. The main findings of this exercise were: 1) OLS with time dummies performs poorly when there is heterogenous response to a common factor and when there are lagged cross-sectional correlations. 2) SUR perforrns well only when N is very small relative to T and 3) the HAC estimator performs well in all cases, even for values of T as low as 25, and its performance does not depend on the size of the cross-sectional dimension which may be arbitrarily large. The results of these experiments are available from the authors upon request.

13
of spatial correlations. We estimate an AR(1) process for real output growth for 20 U.S. states'<sup>0</sup> and for 24 O.E.C.D. economies, and use the observed variance-covariance matrix of the residuals as 2 to generate spatial correlations in our artificial data.

Table 1 reports the coverage rates for this specification for the cases T=25 and T=50 for each of the three estimators (OLS with time dummies, SUR and HAC). From Table 1, it is clear that both OLS and SUR do quite poorly even when T=50, having coverage rates which never rise above .694 and .679 respectively. Even when SUR is correctly specified (in the first column where p=O), its performance is quite poor. The reason for this is that even when T=50, the estimates of the contemporaneous spatial correlations matrix which it uses in a FGLS procedure are very imprecise, as it attempts to estimate a very large number of free parameters. In contrast, our non-parametric spatial correlation consistent HAC estimator performs much better than both OLS and SUR, with coverage rates which range from .818 to .903.

'° Of course, a drawback of this approach is that the estimates of elements of E will not be very precise, as the crosssectional dimension in these regressions is large relative to the time dimension. For this reason, we use only 20 states because annual data for gross state product is only available for 23 years, from 1963-1986. To give some idea of the magnitude of the spatial correlations, note that the average cross-sectional correlation for the U.S. state data is .193, with a maximal value of .629, while for the O.E.C.D., the corresponding figures are .312 and .761.

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# 4 Conclusion

Spatial and other forms of cross-sectional correlation are likely to be an important complicating factor in many empirical studies. We have argued that they are especially likely to arise in macroeconomics and international economics applications in which the cross-sectional units are countries or regions. Standard techniques for dealing with this problem such as the introduction of time dummies or SUTR require either restrictive parameterizations of the form of the correlation or pre-estimation of a large number of parameters. In this paper, we have shown that non-parametric covariance matrix estimators of the type proposed by Newey and West (1987) have a simple analog in the panel data case. Asymptotic theory indicates that this technique can accommodate a wide variety of spatial correlations, and moreover, that the size of the cross-sectional dimension is no obstacle to obtaining large-T asymptotic results. This suggests that our technique is applicable to a broad class of empirical studies which look at large cross-sections of countries, states or regions observed over time. Monte Carlo experiments demonstrate that the finite-sample properties of this estimator are good, and are often superior to those of other commonly used techniques.

Finally, we note that this paper has relied exclusively on large-T asymptotics to deliver consistent covariance matrix estimates in the presence of cross-unit correlations. However, when T is small or when there is only a single cross-section, the problem of consistent non-parametric covariance matrix estimation appears much less tractable. The reason for this is that, unlike in the time dimension, there is no natural ordering in the cross-sectional dimension upon which to base mixing restrictions, and hence it is not possible to construct the pure cross-sectional analogs of time-series HAC estimators. Thus, it would appear that consistent covariance matrix estimation in models of a single cross-section with spatial correlations will have to continue to rely on some knowledge of the form of these spatial correlations.

15