Econometrics. Journal. The Hausman test in a Cliff and Ord panel model. First version received: August 2009; final version accepted: June 2010

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1 The Econometrics Journal Econometrics Journal 20), volume 4, pp doi: 0./j X x The Hausman test in a Cliff and Ord panel model JAN MUTL AND MICHAEL PFAFFERMAYR,, Institute for Advanced Studies, Stumpergasse 56, 060 Vienna, Austria. mutl@ihs.ac.at Department of Economics, University of Innsbruck, Universitaetsstrasse 5, 6020 Innsbruck, Austria. Michael.Pfaffermayr@uibk.ac.at Austrian Institute of Economic Research, P.O. Box 9, A-03 Vienna, Austria. CESifo, Poschingerstr. 5, 8679 Munich, Germany. First version received: August 2009; final version accepted: June 200 Summary This paper studies the random effects model and the fixed effects model for spatial panel data. The model includes a Cliff and Ord type spatial lag of the dependent variable as well as a spatially lagged one-way error component structure, accounting for both heterogeneity and spatial correlation across units. We discuss instrumental variable estimation under both the fixed and the random effects specifications and propose a spatial Hausman test which compares these two models accounting for spatial autocorrelation in the disturbances. We derive the large sample properties of our estimation procedures and show that the test statistic is asymptotically chi-square distributed. A small Monte Carlo study demonstrates that this test works well even in small panels. Keywords: Hausman test, Panel data, Random effects estimator, Spatial econometrics, Within estimator.. INTRODUCTION The panel literature offers the random effects and the fixed effects models to account for heterogeneity across units. Although the random effects estimator is more efficient than the fixed effects estimator, in many non-spatial empirical applications the random effects model is rejected in favour of the fixed effects model. Often there are plausible arguments for the explanatory variables to be correlated with unit-specific effects. For example, in earnings equations the unobserved ability of individuals may be reflected in both the unit-specific effects and the explanatory variables such as the years of schooling. The estimation of gravity equations to model bilateral trade flows is another important example where the assumptions of the random effects model are often found to be violated. It is perfectly sensible that this issue also comes up in spatial panel models. See also the papers cited in Baltagi 2008) for more examples. Published by Blackwell Publishing Ltd, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA, 0248, USA.

2 The Hausman test in a Cliff and Ord panel model 49 This paper contributes to the literature by introducing a spatial generalized methods of moments estimator for panel data models with Cliff and Ord type spatial autocorrelation and one-way error components. Our work complements the seminal paper of Kapoor et al. 2007), who provide a spatial generalized least squares spatial GLS) estimator for the spatial random effects model. In addition to their work, our model allows for an endogenous spatial lag of the dependent variable. We discuss the proper instrumentation of the endogenous spatial lag and suggest an instrumental variable IV) procedure for both the spatial within estimator and the spatial GLS estimator. To discriminate between the two spatial panel models, we also propose a Hausman test that accounts for spatially autocorrelated disturbances. Specifically, we derive the joint asymptotic distribution of the spatial GLS and the spatial within estimators, as well as the asymptotic distribution of the spatial Hausman test for random versus fixed effects. This test should enable applied researchers to choose between these two models, when spatial correlation of the endogenous variable and/or the disturbances is present. Our paper is not the first that considers spatial within or fixed effects estimators. Case 99) seems to be among the first in estimating spatial random and fixed effects models. Korniotis 200) introduces a bias-corrected estimator for a spatial dynamic panel model with fixed effects. Lee and Yu 200) establish the asymptotic properties of quasi-maximum likelihood estimators for fixed effects spatial autoregressive SAR) panel data models with SAR disturbances, where the time periods and/or the number of spatial units can be finite or large in all combinations except that both are finite see also Yu et al., 2007, 2008). In the next section, we specify our model and spell out the maintained assumptions. Section 3 defines the two estimators under consideration and derives their joint asymptotic distribution. Section 4 introduces the feasible counterparts of the considered estimators based on an initial IV estimator. We show that the initial estimator is consistent and asymptotically normal and derive its asymptotic distribution. We also demonstrate that the true and feasible estimators have the same asymptotic distribution. Section 5 defines the spatial Hausman test that allows to discriminate between the two spatial panel models. It provides its asymptotic distribution under the null and also shows that the test statistic diverges in probability under the alternative hypothesis. In Section 6, we report the results of Monte Carlo experiments that assess both the size and the power of the proposed spatial Hausman test in finite samples. Finally, the last section concludes. Proofs of results are given in the Appendix. Consider the following spatial panel model: y it,n = λ 2. THE SPATIAL PANEL MODEL N w ij,n y jt,n + x it,n β + d i,n γ + u it,n. 2.) j= Index i =,..., N denotes the cross-sectional dimension of the panel while the index t =,..., T refers to the time series dimension of the panel. Throughout we assume that T is fixed; that is, our asymptotic analysis refers to large N. y it,n is the scalar) dependent variable and N j= w ij,n y jt,n denotes the spatial lag of the dependent variable with w ij,n being observable non-stochastic spatial weights. λ is the associated scalar parameter. x it,n denotes a K ) vector of time-varying exogenous variables and β is the corresponding K ) parameter

3 50 J. Mutl and M. Pfaffermayr vector. d i,n is L vector of time-invariant variables, including the constant, with L parameter vector γ. Finally, u it,n is the overall disturbance term. We allow for cross-sectional correlation of the disturbances and, in particular, we assume that the disturbances follow a Cliff and Ord type spatial autocorrelation SAR) in the terminology of Anselin, 988) as proposed by Kapoor et al. 2007): u it,n = ρ N m ij,n u jt,n + ε it,n, 2.2) j= where ρ is a scalar parameter and m ij,n are observable spatial weights possibly the same as the weights w ij,n ). The innovations ε it,n have the following one-way error component structure: ε it,n = μ i,n + ν it,n, 2.3) where ν it,n are independent innovations and μ i,n are individual effects, which can be either fixed or random. We index all variables by the sample size N, because they form triangular arrays. This is necessary because the model involves inverses of matrices whose size depends on N, and, hence, their elements must change with N. Thus, at the minimum y it,n and u it,n are triangular arrays in the present specification. We sort the data so that the fast index is i and the slow index is t. Stacking the model over the N cross-sections for a single period t yields where y t,n = ε t,n = y t,n. y Nt,N ε t,n. ε Nt,N y t,n = λw N y t,n + X t,n β + D N γ + u t,n, u t,n = ρm N u t,n + ε t,n, ε t,n = μ N + ν t,n,, X t,n =, ν t,n = w,n w N,N W N =..... w N,N w NN,N x t,n. x Nt,N ν t,n. ν Nt,N, D N =, M N =, μ N = d,n. d N,N μ,n. μ N,N u t,n =, m,n m N,N m N,N m NN,N Stacking over time periods, we write our model compactly as u t,n. u Nt,N 2.4) 2.5) y N = λw N y N + X N β + D N γ + u N = Z N δ + u N, u N = ρm N u N + ε N, ε N = ι T I N ) μ N + ν N, 2.6)

4 The Hausman test in a Cliff and Ord panel model 5 where W N = I T W N ), M N = I T M N ), D N = ι T D N ), Z N = D N, W N y N, X N ), δ = γ,λ,β ) and y N = ε N = y,n.. y T,N ε,n.. ε T,N, X N =, ν N = X,N.. X T,N ν,n.. ν T,N, u N =. u,n.. u T,N Throughout, we maintain the following basic assumptions, which follow closely those postulated in Kapoor et al. 2007). Note that we reserve the symbols σν 2,σ2 μ,λand ρ for the true parameter values. ASSUMPTION 2.. The elements of ν N are independently and identically distributed over i and t with finite absolute 4 + δ ν moments for some δ ν > 0. Furthermore, Eν it,n ) = 0 and Eνit,N 2 ) = σ ν 2 > 0. ASSUMPTION 2.2. The spatial weights collected in M N and W N are non-stochastic and a) m ii,n = 0 and w ii,n = 0. b) The matrices I N ρm N ) and I N λw N ) are non-singular. c) The row and column sums of the matrices M N, W N,I N ρm N ),I N λw N ) are uniformly bounded in absolute value; that is, sup j N i= a ij,n k<, where k does not depend on N but may depend on parameters of the model; that is, on ρ or λ, respectively) and a ij,n denotes elements of the above matrices. ASSUMPTION 2.3. The exogenous variables collected in X N and D N are non-stochastic. Their elements are uniformly bounded. Assumption 2. is a restriction on the higher moments of the disturbances required for asymptotic results. Assumption 2.2a) is a typical normalization but is not necessary for our asymptotic results). Assumptions 2.2b) and c) are regularity conditions. Assumptions 2.2b) and c) hold, for example, if the spatial weight matrices W N and M N are maximum)-row normalized and λ k λ < and ρ k ρ <, respectively. From Assumption 2.2c) follows that ρ k ρ < /λ max M N ), λ k λ < /λ max W N ), where λ max ) denotes the largest absolute eigenvalue of a matrix and k ρ and k λ are constants. 2 Assumptions like 2.2b) 2.2c) are typically maintained in spatial models see Kelejian and Prucha, 999) and restrict the extent of spatial dependence among cross-section units. They will be satisfied if the spatial weighting matrix is sparse so that each unit possesses a limited number of neighbours, or if the spatial weights decline sufficiently fast in distance., 3. THE ESTIMATION OF SPATIAL PANEL MODELS 2.7) In their seminal paper, Kapoor et al. 2007) concentrate on the random effects model, assuming that the explanatory variables and the unit-specific error terms are independent. Yet, in applied 2 This follows from Corollary in Horn and Johnson 985) using their Lemma

5 52 J. Mutl and M. Pfaffermayr work this assumption often does not hold and a fixed effects specification is employed instead. Examples in a non-spatial setting, where the unit-specific effects and the explanatory variables may be correlated, include earnings equations. In this setting, the unobserved individual ability of an individual is typically correlated with the years of schooling, which enters the earnings equation as explanatory variable see e.g. Baltagi, 2008, p. 79). Also in models explaining bilateral trade flows, the random effects model is typically rejected in favour of the fixed effects model to mention another example see Egger, 2000). First, we analyse the spatial random effects estimator for this general spatial panel model. 3.. The spatial random effects estimator Under the random effects specification, the unit-specific effects μ i,n are assumed to be random and the following standard assumption is maintained. ASSUMPTION 3. RE). The elements of μ N are independently and identically distributed with finite absolute 4 + δ μ moments for some δ μ > 0 and i) Eμ 2 i,n ) = σ μ 2 > 0. ii) Furthermore, the elements of μ N are independent of the process for ν it,n and Eμ i,n ) = 0 are for all i. The random effects assumptions maintain that the individual effects exhibit homoscedastic variances and that they are orthogonal to each of the explanatory variables. The latter will always hold if the explanatory variables are non-stochastic and E[μ N ] = 0 see Wooldridge, 2002, pp. 257 and 259). The disturbances are generated as u N = I NT ρm N ) ε N. 3.) Under Assumptions 2. and 3. RE), the variance covariance matrix of the disturbances is given by Eu N u N ) = I NT ρm N ) [ σμ 2 ιt ι T I ] ) N) + σ 2 ν I NT INT ρm N = σν 2 u,n, 3.2) [ where u,n = I NT ρm N ) σ 2 ] μ ι σν 2 T ι T I N) + I NT I NT ρm N ). It proves to be useful to apply the notation σ 2 = Tσ2 μ + σ ν 2 and define the following standard within and between transformation matrices Q i,n i = 0, ): Q 0,N = I T ) T J T I N, Q,N = T J T I N, where J T is a T T matrix of unit elements. The matrices Q i,n are the standard transformation matrices utilized in the error component literature but adjusted for the different stacking of the data compare Kapoor et al., 2007, and Baltagi, 2008). The matrices Q i,n are symmetric and idempotent and mutually orthogonal. The variance covariance matrix of the disturbances can then be written as see Baltagi, 2008, p. 8): 3.3) u,n = I NT ρm N ) ε,n INT ρm N), 3.4)

6 The Hausman test in a Cliff and Ord panel model 53 where ε,n = E ε σν 2 N ε ) σ 2 N = Q σν 2,N + Q 0,N. 3.5) Furthermore, the inverse of u,n can then be expressed as where u,n = I NT ρm ) N ε,n I NT ρm N ), 3.6) ) 2 ε,n = σ Q,N + Q 0,N. 3.7) σ ν If the parameter values ρ, σν 2 and σ μ 2 and, therefore, σ 2 ) are known, the efficient GLS estimation procedure transforms the model by the square root of u,n given by /2 u,n = /2 ε,n I NT ρm N ). 3.8) This is equivalent to first applying the spatial counterpart of the Cochrane Orcutt transformation I NT ρm N ) that eliminates the spatial correlation from the disturbances and then the familiar panel GLS transformation /2 ε,n that accounts for the variance covariance structure of the innovations induced by the random effects. To simplify the exposition, we collect the parameters of the variance-covariance matrix in a vector ϑ = ρ,σν 2,σ2 ) and use the notation /2 u,n ϑ) to explicitly note the dependence of the GLS transformation on these parameters. Observe that in a balanced panel the order with which the transformations are applied is irrelevant see also Remark A in Kapoor et al., 2007). Because the spatial lag W N y N is endogenous in the transformed) model, we adapt the IV procedure described in Kelejian and Prucha 998). 3 Specifically, we first eliminate the spatial correlation from the error term using the Cochrane Orcutt transformation. Then we apply the IV procedure for random effects models suggested by Baltagi and Li 992), Cornwell et al. 992) and surveyed by Baltagi 2008). In a non-spatial setting, these authors show that the optimal set of instruments for a random effects model with endogenous variables is composed of [Q 0,N X N, Q,N X N, Q,N D N ]. Observe that E[y N ] = I NT λw N ) X N β+d N γ ) [ ] = λ k W k N X N β + D N γ ), k=0 3.9) where W 0 N = I NT. Hence, under the present assumptions, the ideal set of instruments is based on 3 It is possible to use other sets of instruments that are similar to those proposed in Lee 2003) or Kelejian et al. 2004).

7 54 J. Mutl and M. Pfaffermayr /2 u,n ϑ)w NE[y N ] = λ k /2 u,n ϑ)wk+ N X Nβ + D N γ ) k=0 = ) λ k σν Q,N + Q 0,N W k+ N σ X Nβ + D N γ ) k=0 ρ k=0 λ k σν σ Q,N + Q 0,N ) M N W k+ N X Nβ + D N γ ). 3.0) Therefore, the transformed endogenous variable /2 u,n ϑ) W Ny N is best instrumented by H R,N = [H Q,N, H P,N ] = [Q 0,N G 0,N, Q,N G,N ], where G 0,N contains a subset of the linearly independent columns of [ XN, W N X N, W 2 N X N,...,M N X N, M N W N X N, M N W 2 N X N,... ] and G,N contains a subset of the linearly independent columns of [ G0,N, D N, W N D N, W 2 N D N,...,M N D N, M N W N D N, M N W 2 N D N,... ]. The columns in G 0,N and G,N must be chosen so that the columns of H R,N are linearly independent. We do not include the constant and the other time-invariant variables in G 0,N, because Q 0,N W N D N = [ I T T J ) ] T ιt W N D N = 0. In the special case where W N = M N, the set of instruments is based on G 0,N = [X N, W N X N, W 2 N X N,...] and G,N = [G 0,N, D N, W N D N,...]. If the spatial weighting matrices are row normalized, the set of instruments in G,N excludes spatial lags of the time-invariant variables, since in this case M N ι NT = W N ι NT = ι NT, where ι NT is an NT vector of ones. With regard to the choice of the instruments in practical applications, note that it is usually not advisable to use powers of the spatial weight matrices higher than two; see also the discussion of the choice of the instruments in Kelejian and Prucha 998). In the following we assume that the NT p matrix of instruments denoted by H R,N is of the form described earlier. To derive asymptotic properties of the considered estimators, we maintain the following additional assumption for the matrix of instruments and the explanatory variables of the model that are collected in Z N = D N, W N y N, X N ) 4 : ASSUMPTION 3.2. Let Z N ϑ) = /2 u,n ϑ)z N. The matrix of instruments H R,N has full column rank and consists of a subset of linearly independent columns of [Q 0,N G 0,N, Q,N G,N ]. Furthermore, it satisfies the following conditions: a) M HR H R = lim N NT) H R,N H R,N exists and is finite and non-singular; b) M HR Z = p lim N NT) H R,N Z N exists and is finite with full column rank. The spatial random effects estimator of δ = γ,λ,β ) is then defined as δ GLS,N = [ Z N ϑ) Z N ϑ) ] Z N ϑ)ỹ Nϑ), 3.) with Z N ϑ) = P HR,N Z N ϑ), Z N ϑ) = /2 u,n ϑ)z N and ỹ N ϑ) = /2 u,n ϑ)y N. P HR,N = H R,N H R,N H R,N) H R,N is the projection matrix based on the instruments H R,N. 4 Since Z N defined below depends on the parameter vector ϑ, the next assumption is meant to apply to the true parameter values.

8 The Hausman test in a Cliff and Ord panel model 55 The joint asymptotic distribution of the spatial random effects estimator and the spatial within estimator under known nuisance parameter vector ϑ is given in Theorem 3.. This theorem is based on the random effects Assumption 3. RE) and forms the basis of the Hausman test. The asymptotic properties of this test and its feasible counterparts are given in Theorem 5. in Section The spatial within estimator In situations where the random effects assumption might be violated one can use the spatial within estimator that remains unaffected by the possible correlation of the unit-specific effects and the explanatory variables and, hence, a violation of Assumption 3. RE). One can apply the within transformation Q 0,N to wipe out the individual effects see e.g. Mundlak, 978, and Baltagi, 2008). Using Q 0,N I NT ρm N ) = I NT ρm N )Q 0,N, one obtains Q 0,N u N = E T I N )[ρi T M N )u N + ι T I N )μ N + ν N ] = ρi T M N )E T I N )u N + E T I N )ν N = ρi T M N )Q 0,N u N + Q 0,N ν N or Q 0,N u N = I NT ρm N ) Q 0,N ν N, 3.2) where E T = I T T J ) T. Hence, one can apply the Cochrane Orcutt type transformation to the within transformed model to obtain the fixed effects generalized least squares FEGLS) estimator. Note that the parameters of the time-invariant variables, collected in the vector γ, remain unidentified with this approach. More importantly, one can base the method of moment estimator of ρ, σν 2 ) on the initial within transformed residuals of the initial within estimator as given by Q 0,N u N, which are consistently estimated even if the unit-specific effects depend on X N. Obviously, the set of instruments denoted by H Q,N now comprises the linear independent columns of Q 0,N G 0,N. Because the constant and the time-invariant variables are wiped out in the spatial within estimator, we define the NT K) matrixz Q,N = Q 0,N [W N y N, X N ] with the corresponding K ) parameter vector θ = λ, β ). To derive the asymptotic properties of the spatial within estimator for θ, we maintain the following additional assumptions for the matrix of instruments used in the spatial within model. ASSUMPTION 3.3. Let Z Q,N = Q 0,N [W N y N, X N ]. The matrix of instruments H Q,N has full column rank and consists of a subset of linearly independent columns of Q 0,N G 0,N. Furthermore, it satisfies the following conditions: a) M HQ H Q = lim N NT) H Q,N H Q,N is finite and nonsingular with full column rank. b) M HQ Z = p lim N NT) H Q,N I NT ρm N )Z Q,N exists and is finite with full column rank. Again, treating ρ as known, we apply the Cochrane Orcutt type transformation to the within transformed model yielding

9 56 J. Mutl and M. Pfaffermayr Q 0,N I NT ρm N )y N = I NT ρm N )Q 0,N y N = I NT ρm N )Q 0,N Z Q,N + Q 0,N ν N y N ρ) = Z N ρ)θ + ν N, 3.3) where y N ρ) = I NT ρm N )Q 0,N y N, Z N ρ) = I NT ρm N )Z Q,N and ν N = Q 0,Nν N. The spatial within estimator for θ is then obtained by applying IV to the transformed model to obtain θ W,N = [ Ẑ N ρ) Z N ρ)] Ẑ N ρ) y ρ), 3.4) with Ẑ N ρ) = P H Q,N Z N ρ) = P H Q,N I NT ρm N )Z N. P HQ,N = H Q,N H Q,N H Q,N) H Q,N is the projection matrix based on the instruments H Q,N. The following theorem establishes our main asymptotic result concerning the common asymptotic distribution of the spatial random effects and the spatial within estimators under random effects Assumption 3. RE). The Hausman test for spatial panels derived below will be based on this result. Because the random effects estimator includes time-invariant variables including the constant, we define δ GLS,N = γ GLS,N, θ GLS,N ) = γ GLS,N, λ GLS,N, β GLS,N ). THEOREM 3.. Let Assumptions hold. Then ) θ GLS,N θ d NT N θ W,N θ 0, [ ]) GLS GLS, GLS W where W = σ 2 ν M H Q Z M H Q H Q M HQ Z ) and GLS is the lower-right K K block of the matrix σ 2 ν M H R Z M H R H R M HR Z). This theorem forms the basis for the spatial Hausman test derived later under the null hypothesis that Assumption 3. RE) is true. 4. FEASIBLE ESTIMATION The spatial GLS and spatial within estimators defined earlier are based on the unknown parameters ρ, σν 2 and σ μ 2 which have to be estimated. The feasible estimation procedure starts by estimating the within transformed model using the instruments H Q,N = Q 0,N G 0,N as described earlier to obtain initial within IV estimates. This initial estimator is consistent see Baltagi, 2008) and it can be written as θ I,N = [ Ẑ Q,N Q ] Ẑ 0,NZ Q,N Q,N Q 0,Ny N, 4.) where Ẑ Q,N = P HQ,N Q 0,N Z Q,N. The following proposition gives the asymptotic distribution of the initial estimator. PROPOSITION 4.. Let the limit M HQ Z = p lim N NT) H Q,N Z Q,N exist and be finite with full column rank. Let Assumptions and 3.3 hold. Then NT θ I,N θ ) d N0, I ), where I = σ 2 ν M H Q Z M H Q H Q M HQ Z).

10 The Hausman test in a Cliff and Ord panel model 57 The projected residuals then give consistent initial estimates of Q 0,N u N, which can be used in the spatial generalized moments GM) estimator as suggested by Kapoor et al. 2007). These authors use OLS residuals, which are only consistent under the random effects Assumption 3. RE). The spatial GM estimator for ρ,σν 2 can then be based on the first three moment conditions given in Kapoor et al. 2007). Using Q i,n M N = M N Q i,n and the notation ε N = M N ε N, we can formulate the first three moment conditions in terms of Q 0,N u N as E NT ) ε N Q 0,Nε N NT ) ε N Q 0,Nε N NT ) ε N Q 0,Nε N NT ) u N Q 0,N INT ρm ) N INT ρm N )Q 0,N u N = NT ) u N Q 0,N INT ρm ) N M N M N I NT ρm N )Q 0,N u N NT ) u N Q 0,N INT ρm ) N M N I NT ρm N )Q 0,N u N σν 2 = σ 2 ν N trm N M N). 4.2) 0 Under the random effects model, Assumption 3. RE), we add a fourth moment condition: [ ] [ ] E N ε N Q,Nε N = N u N Q,N INT ρm ) N INT ρm N )Q,N u N = σ ) With the solution of the first three moment conditions at hand, one can solve the fourth moment condition to obtain an estimate of σ 2. Theorem in Kapoor et al. 2007) shows that the estimators for ρ, σν 2 and σ 2 based on these moment conditions and some additional assumptions see their Assumption 5) are consistent as long as the initial estimator θ I,N is consistent. Proposition 4.2 demonstrates that the parameters ρ, σν 2 and σ 2 are nuisance parameters and that the feasible spatial random effects and the feasible spatial within estimates have the same asymptotic distribution as their counterparts based on the true values of ρ, σν 2 and σ 2. PROPOSITION 4.2. Let the feasible estimators θ FGLS,N and θ FW,N be based on consistent estimators of ρ, σν 2 and σ 2. Then under Assumptions we have NT θ FGLS,N θ GLS,N ) p. 0, NT θ FW,N θ W,N ) p HAUSMAN SPECIFICATION TEST The spatial within estimator is consistent because it wipes out the unit-specific effects by applying the within transformation. The critical assumption for the validity of the spatial random effects model is that E[μ N ] = 0, implying that the spatial random effects model is inconsistent if the random effects Assumption 3. RE) does not hold. The Hausman test Hausman, 978) suggests comparing these two estimators and testing whether the random effects assumption

11 58 J. Mutl and M. Pfaffermayr holds true. The spatial GLS estimator of the random effects model is more efficient than the spatial within estimator under the random effects Assumption 3. RE). Moreover, under the null hypothesis both considered estimators are consistent, while under the fixed effects assumption, the spatial random effects estimator is inconsistent, whereas the spatial within estimator is consistent. We summarize these properties in the following lemma. 5 LEMMA 5.. Under Assumptions we have: a) NT θ GLS,N θ W,N ) d. N0, W GLS ), where W GLS is positive definite at the true parameter values ϑ. b) Furthermore, where and W,N GLS,N p. W GLS, W,N = σ ν,n 2 NT[ Z Q,N INT ρ N M N) PHQ,N I ] NT ρ N M N )Z Q,N GLS,N = σ ν,n 2 NT[ Z N /2 u,n ϑ N )P HR,N /2 u,n ϑ ], N )Z N with ϑ N = ρ N, σ 2 ν,n, σ 2,N ) some consistent estimator of ϑ. The theorem below now defines the Hausman test statistic for spatial panels and provides its asymptotic distribution under the null. THEOREM 5.. and Assume that Assumptions hold and that Ĥ N = NT θ FGLS,N θ FW,N ) W,N GLS,N ) θ FGLS,N θ FW,N ) H N = NT θ GLS,N θ W,N ) W GLS ) θ GLS,N θ W,N ). Then Ĥ N H N p. 0, where H N is asymptotically χ 2 distributed with K degrees of freedom. Given Theorem 2 in Kapoor et al. 2007) and the results in this paper, a feasible estimation and testing procedure can be summarized as follows: ) Calculate a consistent initial IVs within estimator θ I,N which wipes out the individual effects using the within transformation. This estimator ignores the spatial correlation in the disturbances. 5 To operationalize this lemma, we need to provide a consistent estimator ϑ N of ϑ. Our suggestion see the summary of our estimation procedure below) is to construct both W,N and GLS,N using ϑ N = ρ N, σ ν,n 2, σ,n 2 ),where ρ N and σ ν,n 2 are estimated from three moment conditions using the within estimates see 4.2), while σ,n 2 is derived from a fourth moment condition that exploits the between variation.

12 The Hausman test in a Cliff and Ord panel model 59 2) Use the resulting estimated disturbances of the within transformed model in a spatial GM procedure as described in Kapoor et al. 2007) and obtain a consistent) estimator ϑ N = ρ N, σ 2 ν,n, σ 2,N ). 3) Transform the model by the spatial Cochrane Orcutt transformation and then use either the GLS, or the within transformation to obtain the feasible spatial GLS and spatial within estimators θ FGLS,N and θ FW,N, respectively. 4) Calculate the Hausman test statistics Ĥ N to make a decision whether the random or fixed effects specification is more appropriate. Note, once the variables are spatially Cochrane Orcutt transformed standard econometric software can calculate the spatial Hausman test. We now investigate the power properties of the test statistic; that is, its behaviour when the null hypothesis is violated. To do so, we need to specify a particular alternative. Note that all results obtained up to this point were independent of a particular choice of the alternative hypothesis. We now follow Mundlak 978) and assume an alternative fixed effects) model in which the explanatory variables and the unit-specific effects are related in the following way: 6 ASSUMPTION 5. FE). The vector of individual effects is given by μ N = X N π + ξ N, where π 0 K ), and the N K ) matrix X N explanatory variables and is given by X N = ) T ι T I N X N. contains the time averages of the Finally, the elements of the N random vector ξ N satisfy Assumption 3. RE). Note that it would be trivial to generalize our assumption to an alternative model that has been defined by Chamberlain 982), where μ i,n = T t= x it,nπ t + ξ i,n or μ N = T t= X t,nπ t + ξ N. Clearly, setting π t = π yields Mundlak s formulation. T Under the FE assumption, we have [ ) ] ε N = ι T I N ) T ι T I N X N + ξ N + ν N = Q,N X N π+ ι T I N ) ξ N + ν N, u N = I NT ρm N ) [Q,N X N π+ι T I N )ξ N + ν N ]. The individual effects do not depend on the explanatory variables if and only if π = 0 and the random effects model defined under Assumption 3. RE) arises as a special case of Assumption 5. FE). Observe that under Assumption 5. FE) with π 0 the spatial GLS estimator is biased 5.) 6 There are other specifications of the unit effects possible. In the present case, unit effects are decomposed in a spatially correlated error term similar to that defined in Assumption 3. RE) and a systematic component without spatial effects.

13 60 J. Mutl and M. Pfaffermayr as one obtains δ GLS,N = [ Z N ϑ) Z N ϑ) ] Z N ϑ)ỹ Nϑ) = δ + [ Z N ϑ) Z N ϑ) ] Z N ϑ) /2 u,n [Q,N X N π+ι T I N )ξ N + ν N ]. 5.2) On the other hand, the within transformation wipes out the individual effects and, hence, the within estimator is the same as under H 0 : π = 0. The following proposition shows that the Hausman test statistic is a consistent statistic; that is, the power of the test approaches unity as N for an arbitrary significance level of the test. PROPOSITION 5.. Let Assumptions , 5. FE), 3.2 and 3.3 hold and assume that M HR X = lim N NT) H R,N /2 u,n Q,NX N exists and is finite with full column rank. Let h>0 be some positive constant. Then lim N P H N >h) =. 6. MONTE CARLO EVIDENCE The Monte Carlo analysis investigates the small sample properties of the proposed spatial Hausman test. For this we use a simple spatial panel model that includes one explanatory variable and a constant y it,n = λ N j= w ij,n y jt,n + βx it,n + α + u it,n, i =,...,N and t =,...,T. 6.) We set β = 0.5 and α = 5. The explanatory variable is generated as x it = ζ i + z it with ζ i i.i.d. U[ 7.5, 7.5] and z it i.i.d. U[ 7.5, 7.5] with U[a, b] denoting the uniform distribution on the interval [a, b]. In accordance with Assumption 2.3, x it is treated as a non-stochastic variable and it is held fixed in repeated samples. The individual-specific effects are allowed to be correlated with x i, setting μ i = μ i0 + πx i, where μ i0 is drawn from a normal distribution; that is, μ i0 i.i.d. N0, 0φ) and π is a constant parameter. This mimics the fixed effects Assumption 5. with π 0.Atπ = 0, the random effects Assumption 3. holds and it forms the null for the spatial Hausman test. We normalize μ i so that its mean is 0 and its variance 0φ, where φ = σ μ 2 σμ 2+σ,0< ε 2 φ<, denotes the proportion of the total variance due to the presence of the individual-specific effects. For the remainder error, we assume ε it i.i.d. N0, 0 φ)). This implies that the overall variance of the disturbances is σμ 2 + σ ε 2 = 0. The row normalized spatial weighting matrix uses a regular lattice with 44 and 324 cells, respectively, containing one observation each. The spatial weighting scheme is based on a rook design, where every unit is surrounded by four neighbours. The corresponding spatial weighting matrix is maximum-row normalized following Kelejian and Prucha 2007). We use the same spatial weighting matrix to generate both the endogenous spatial lag and the spatial lag of the error term. The spatial parameters λ and ρ vary over the set { 0.8, 0.4, 0, 0.4, 0.8}. The parameter π takes its values in { 0.2, 0., 0, 0., 0.2}. Based on the discussion earlier

14 The Hausman test in a Cliff and Ord panel model 6 we use the instruments H Q,N = [Q 0,N x N, Q 0,N W N x N,Q 0,N W 2 N x N], while H R,N is composed of [H Q,N, Q,N x N, Q,N W N x N, Q,N W 2 N x N,ι NT, Q,N W N ι NT ]. In each experiment, we calculate the size of the Hausman test, which is given by the share of rejections at π = 0. The power of the spatial Hausman test is given by the share of rejections at π 0. The baseline scenario is reported in Table setting N = 44, T = 5 and φ = 0.5. The results show that the proposed spatial IVGLS estimators work well and that the spatial Hausman test exhibits a good performance for almost all considered parameter configurations. In the experiments reported in Table, the spatial Hausman test comes close to the nominal size of 0.05 in most of the cases. Exceptions are only observed for high values of λ, where the test is slightly oversized. For example, at λ = 0.8 and ρ = 0.8 the size of the spatial Hausman test is At negative values of ρ, this phenomenon is not observed. The power of the test by and large remains unaffected by variations of ρ and λ, although it seems somewhat lower at high absolute values of ρ or high absolute values of λ. A larger cross-section N = 324, NT = 620) improves both the size and the power of the test as expected Table 2), especially the size distortion at high positive values of λ is now reduced. In Table 3, we extend the time series dimension and set T = NT = 584). The size distortion at high values of λ becomes smaller as T increases and this effect seems more pronounced than in an extended cross-section as analysed in Table 2. However, the improvement in power is much smaller than that observed when extending the cross-section dimension. In Table 4, we set N = 44, T = 5 and φ = 0.8, so that σμ 2 = 8 and σ ν 2 is 2. With a larger weight of the variance of the unit-specific effects, we observe a better performance of the spatial Hausman test in terms of its size and the size distortion observed in the baseline scenario now vanishes. Also, the power of the test is significantly higher. We performed a series of robustness checks and the full set of tables with these results is available upon request. In particular, we considered the case where serial correlation in the disturbances is present, but is ignored. Our results show that the size of the Hausman test remains nearly unaffected. However, the spatial Hausman test has lower power especially if the serial correlation is high. In addition, we investigated the performance of the proposed Hausman test when the exogenous variable has weak effects on the dependent variable. In our simulation experiment the variance of the generated explanatory variable is reduced by one-half, which results in weaker instruments as compared to the baseline experiment. The results show that at high positive values of both λ and ρ the spatial Hausman test tends to over-reject, if the explanatory variables are weak. Also, the power of the test is lower compared to the baseline experiment. The tables corresponding to these additional experiments are available upon request from the authors. We also investigated the root mean square error RMSE) and the bias of the estimators of β, λ and ρ for the basic case with N = 44, T = 5 and θ = Following Kapoor et al. 2007) we define the bias as the difference between the respective median of the parameter estimate and its true counterpart, while RMSE = bias 2 + IQ 2,.35) where IQ is defined as the interquantile range, that is the difference between the 0.75 and 0.25 quantile of the simulated parameter distribution. Under a normal distribution the median and the mean coincide and IQ.35 corresponds to the standard deviation up to a rounding error). 7 The tables corresponding to these simulation experiments are available upon request from the authors.

15 62 J. Mutl and M. Pfaffermayr Table. Size and power of the spatial Hausman test. λ ρ π Note: N = 44, T = 5, θ = 0.5, normal disturbances, 2000 replications.

16 The Hausman test in a Cliff and Ord panel model 63 Table 2. Size and power of the spatial Hausman test. λ ρ π Note: N = 324, T = 5, θ = 0.5, normal disturbances, 2000 replications.

17 64 J. Mutl and M. Pfaffermayr Table 3. Size and power of the spatial Hausman test. λ ρ π Note: N = 44, T = 0, θ = 0.5, normal disturbances, 2000 replications.

18 The Hausman test in a Cliff and Ord panel model 65 Table 4. Size and power of the spatial Hausman test. λ ρ π Note: N = 44, T = 5, θ = 0.8, normal disturbances, 2000 replications.

19 66 J. Mutl and M. Pfaffermayr The simulation exercises reveal a negligible bias for β and a somewhat higher efficiency of the random effects estimator under H 0. The gain in efficiency is especially large at high positive values of λ and at high absolute values of ρ. A similar pattern can be found for the RMSE of λ, although the efficiency loss of the spatial within estimator is much higher compared to that for β. Under H the random effects estimator is inconsistent, leading to large biases in both β and λ. The bias of the slope parameter β is hardly affected by different degrees of spatial dependence as represented by the parameter values of λ. However, the bias is negative at low and negative values of ρ and turns positive if ρ gets high. With respect to the estimates of λ, we find that the bias is negative if λ or ρ take on negative values, but that it declines in λ and/or ρ.atλ = 0.8 or ρ = 0.8 the bias nearly vanishes. The estimates of ρ remain unaffected by deviations from H 0 as expected. These estimates are based on the spatial within estimator which is consistent under both H 0 and H. We also assess the performance of the spatial Hausman test for non-normal disturbances against the baseline case in Table. 8 In particular, we follow Kelejian and Prucha 999) and assume lognormal remainder disturbances assuming ε it = eξ it e 0.5, where ξ e 2 e it i.i.d. N0, ). Alternatively, we maintain that the distribution of the remainder error exhibits fatter tails than the normal and ε it i.i.d. t5). In both cases, the performance of the spatial Hausman test is comparable to that under normal disturbances. However, under the t5) error distribution the power of the test is smaller. To summarize, the small Monte Carlo study shows that the proposed spatial Hausman test works well even in small panels. In this spatial setting, the test is able to detect deviations from the assumption that unobserved unit effects and the explanatory variables are uncorrelated, which is critical for the validity of spatial random effects models. 7. CONCLUSIONS In this paper, we study spatial random effects and spatial fixed effects models. We note that in many non-spatial applications the critical assumption maintained under the random effects specification, namely that unit-specific effects and explanatory variables are uncorrelated, does not hold. This also seems a possibility in a spatial setting and should be tested, since the estimates of spatial random effects are inconsistent if this assumption fails to hold. Using a spatial Cliff and Ord type model as analysed in Kapoor et al. 2007), but augmented by an endogenous spatial lag, we introduce feasible) IVs estimators for both the spatial random effects model and a spatial fixed effects model. We derive the asymptotic distributions of these estimators as well as those of their feasible counterparts. In addition, we propose a spatial Hausman test to compare these two models, accounting for spatial autocorrelation in the disturbances. A small Monte Carlo study shows that this test works well even in small panels. ACKNOWLEDGMENTS We would like to thank Robert Kunst, Ingmar Prucha, the editor and two anonymous referees for helpful comments and suggestions. 8 The corresponding tables are available from the authors upon request.

20 The Hausman test in a Cliff and Ord panel model 67 REFERENCES Anselin, L. 988). Spatial Econometrics: Methods and Models. Boston, MA: Kluwer. Baltagi, B. H. 2008). Econometric Analysis of Panel Data 4th ed.). Chichester: John Wiley. Baltagi, B. H. and Q. Li 992). A note on the estimation of simultaneous equations with error components. Econometric Theory 8, 3 9. Case, A. C. 99). Spatial patterns in household demand. Econometrica 59, Chamberlain, G. 982). Multivariate regression models for panel data. Journal of Econometrics 8, Cornwell, C., P. Schmidt and D. Wyhowski 992). Simultaneous equations and panel data. Journal of Econometrics 5, 5 8. Egger, P. 2000). A note on the proper econometric specification of the gravity equation. Economics Letters 66, Greene, W. 2008). Econometric Analysis 6th ed.). Englewood Cliffs, NJ: Prentice Hall. Hausman, J. A. 978). Specification tests in econometrics. Econometrica 46, Horn, R. A. and C. R. Johnson 985). Matrix Analysis. Cambridge: Cambridge University Press. Kapoor, M., H. H. Kelejian and I. R. Prucha 2007). Panel data models with spatially correlated error components. Journal of Econometrics 40, Kelejian, H. H. and I. R. Prucha 998). A generalized spatial two-stage least squares procedure for estimating a spatial autoregressive model with autoregressive disturbances. Journal of Real Estate Finance and Economics 7, Kelejian, H. H. and I. R. Prucha 999). A generalized moments estimator for the autoregressive parameter in a spatial model. International Economic Review 40, Kelejian, H. H. and I. R. Prucha 2007). Specification and estimation of spatial autoregressive models with autoregressive and heteroskedastic disturbances. Journal of Econometrics 57, Kelejian, H. H., I. R. Prucha and Y. Yuzefovich 2004). Instrumental variable estimation of a spatial autoregressive model with autoregressive disturbances: large and small sample results. In J. LeSage and R. K. Pace Eds.), Advances in Econometrics, 8, New York: Elsevier. Korniotis, G. M. 200). Estimating panel models with internal and external habit formation. Journal of Business and Economic Statistics 28, Lee, L. 2003). Best spatial two-stage least squares estimators for a spatial autoregressive model with autoregressive disturbances. Econometric Reviews 22, Lee, L. and J. R. Yu 200). Estimation of spatial autoregressive panel data models with fixed effects. Journal of Econometrics 54, Mundlak, Y. 978). On pooling time series and cross section data. Econometrica 46, Mutl, J. 2006). Dynamic panel data models with spatially correlated disturbances. Ph.D. dissertation, University of Maryland. Schmidt, P. 976). Econometrics. New York: Marcel Dekker. Wooldridge, J. 2002). Econometric Analysis of Cross Section and Panel Data. Cambridge, MA: MIT Press. Yu, J. R., R. de Jong and L. Lee 2007). Quasi-maximum likelihood estimators for spatial dynamic panel data with fixed effects when both n and T are large: a nonstationary case. Working paper, Ohio State University. Yu, J. R., R. de Jong and L. Lee 2008). Quasi-maximum likelihood estimators for spatial dynamic panel data with fixed effects when both n and T are large. Journal of Econometrics 46, 8 34.

21 68 J. Mutl and M. Pfaffermayr APPENDIX ) μ Proof of Theorem 3.: We denote the T + )N vector of i.i.d. 0, ) innovations as ζ N = N σ μ, ν N σ ν and we write the stacked estimators as 9 ) δ GLS,N δ = P N F N ζ N, θ W,N θ where with and ) ) P R,N 0 FR,N F R2,N P N =, F N = 0 P Q,N 0 F Q,N P R,N = [ Z N H R,N H R,N H ) ] R,N H R,N Z N Z N H ) R,N H R,N H R,N, P Q,N = [ Z Q,N H ) Q,N H Q,N H Q,N H Q,N Z Q,N Z Q,N H ) Q,N H Q,N H Q,N, ] F R,N = σ μ H R,N /2 ε,n ι T I N ) ) = σ μ H σν R,N Q,N + Q 0,N ι T I N ), σ F R2,N = σ ν H R,N /2 ε,n ) = σ ν H σν R,N Q,N + Q 0,N, σ F Q,N = σ ν H Q,N Q 0,N. By Assumptions 3.2 and 3.3 it follows that the sequences of the stochastic matrices NT) P R,N and NT) P Q,N converge in probability; that is, p. NT) P R,N [ ] M H R Z M H R H R M HR Z M H R Z M H R H R, p. NT) P Q,N [ ] M H Q Z M H Q H Q M HQ Z M H Q Z M H Q H Q. Next we apply the central limit theorem for vectors of triangular arrays given in Theorem A in Mutl 2006) to NT) 2 F N ζ N. By Assumptions 2. and 3., the vector of random variables ζ N satisfies the assumptions of the central limit theorem. Observe that the matrix F N is non-stochastic and that Assumptions 2.2, 2.3 and 3.2 imply that the row and column sums of F N are bounded in absolute value. Hence, it remains to be demonstrated that the matrix NT) F N F N has eigenvalues uniformly bounded away from zero. 9 We use the properties of the Q 0,N and Q,N transformation matrices see e.g. Kapoor et al., 2007, Remark A, and Baltagi, 2008). In particular, we have /2 ε,n = σ Q,N + σν Q 0,N.

22 The Hausman test in a Cliff and Ord panel model 69 One can show that 0 ) F N F N = σ 2 ν p+2q) p+2q) H R,N H R,N H R,N H ) Q,N H Q,N H R,N H Q,N H Q,N ) ) ) H = σ 2 R,N 0 INT I NT HR,N 0 ν, 0 H Q,N I NT I NT 0 H Q,N and, hence, ) [ NT) λ min FN F N min NT) λ min H R,N H ) R,N, NT) λ min H Q,N H Q,N ) ] [ )] INT I λ NT min. I NT I NT ) Observe that NT) H R,N H R,N and NT) H Q,N H I Q,N and NT I NT are symmetric. By Assumptions 3.2 I NT I NT and 3.3 the first two matrices have full rank p + q and q, respectively. Note that the third matrix has trivially full rank as well. Hence, NT) λ min F N F N ) is uniformly bounded away from zero. Therefore, by the central limit theorem it follows that ) NT) /2 d. F N ζ N N 0, lim N NT F NF N. From Assumptions 3.2 and 3.3 we also have ) δ NT) 2 GLS,N δ θ W,N θ d. N0, ), where = p lim N NT) P N F N F N P N. Fairly straightforward calculation shows that ) 2 = 2, 22 with ) ) = σ 2 ν M γ,gls γθ,gls H R Z M H R H R M HR Z =, I K γθ,gls ) ) 2 = σ 2 ν M 0L K H R Z M H R H R M HR Z 22 = σ 2 ν M H Q Z M H Q H Q M HQ Z ) = W. K+L) K GLS γθ,gls 0 Recall that the NT q) matrix of within transformed instruments H Q,N = Q 0,N G 0,N has full column rank q and that Q 0,N H Q,N = H Q,N. Furthermore, H R,N = [H P,N, H Q,N ], where H P,N = Q,N G,N with dimension NT p) and full column rank p. Because Q 0,N Q,N = 0, it follows that = GLS ) H R,N Q 0,N H Q,N = H R,N H Q,N ) ) G,N Q,N 0p q = G 0,N Q Q 0,N H Q,N = 0,N H Q,N H. Q,N Note that it can be demonstrated that lim N NT) F N F N exists.

23 70 J. Mutl and M. Pfaffermayr We have ordered the elements of the vector of parameters δ such that the first elements correspond to the time-invariant variables so that the asymptotic variance covariance matrix of the stacked estimators becomes γ,gls γθ,gls γθ,gls = γθ,gls GLS GLS, γθ,gls GLS W and, hence, γ GLS,N γ NT θ GLS,N θ d. N 0, ). θ W,N θ Proof of Proposition 4.: Note that we have θ I,N θ = [ Z Q,N H Q,N H Z Q,N H Q,N Q,N H Q,N H Q,N H Q,N = P Q,N H Q,N Q 0,Nν N. ) H Q,N Z ] Q,N ) H Q,N Q 0,Nν N Given Assumption 3.3 and the condition in the proposition, it follows that and NT) P Q,N p. [M H Q Z M H Q H Q M HQ Z] M H Q Z M H Q H Q NT)λ min H Q,N Q 0,NQ 0,N Q,N) H = NT)λmin HQ,N H Q,N), which is uniformly bounded away from zero. Given Assumption 2., the conditions of Theorem A in Mutl 2006) are satisfied and in light of Assumption 3.3a), we have the desired result. Proof of Proposition 4.2: The proof follows closely that of Theorem 4, part 2, in Kapoor et al. 2007) and Theorem 3 in Mutl 2006). In particular, it will be sufficient to show that see e.g. Schmidt, 976): and G,N = NT) [ Z N ϑ) Z N ϑ) Z N ϑ) Z N ϑ) ] p 0, W,N = NT) [ Ẑ N ρ) Z N ρ) Ẑ N ρ) Z N ρ) ] p 0, G2,N = NT) /2 Z N ϑ)ũ N ϑ) Z N ϑ)ũ Nϑ) ) p 0, W 2,N = NT) /2 Ẑ N ρ) u ρ) Ẑ N ρ) u ρ) ) p 0, where ϑ and ρ are any) consistent estimators of ϑ and ρ. Note that Z N ϑ) = P HR,N Z N ϑ) andẑ N ϑ) = P HQ,N Z N ϑ), where Z N ϑ) = = /2 ε,n I NT ρm N )Z N Q,N + σ ν σ Q 0,N ) I NT ρm N )Z N,