10.1.2.1 CD p Tests for Local Cross‐sectional Dependence

A very flexible way of assessing whether dependence in the cross‐section of a panel dataset is spatially related goes through a particularization of the CD test for general cross‐sectional dependence described in Chapter . The latter is in principle completely a‐spatial, being based on a scaled average of the pairwise correlation coefficients between observations (or residuals). Still, the CD can be restricted to those pairs of observations satisfying one given criterion: most frequently, a contiguity‐based neighborhood one but also that distance be under a given cutoff level.

The local variant of the CD test, called test (Pesaran, 2004), takes into account an appropriate subset of neighboring cross‐sectional units to check the null of no cross‐sectional dependence against the alternative of local cross‐sectional dependence, i.e., dependence between neighbors only. To do so, the pairs of neighboring units are selected by means of a binary proximity matrix, in which zeros correspond to pairs of observations that are not neighbors. The latter is used for discarding the correlation coefficients relative to pairs of observations that are not neighbors in computing the CD statistic. The test is then defined as:

where is the ‐th element of the ‐th order proximity matrix, so that if any pair are not neighbors, and is eliminated from the summation; and is the number of time series observations in common between individuals and ( if the panel is balanced).²

The same procedure can be applied to the LM and SCLM tests described in section 4.3.1. The local version of either test can be computed supplying an matrix (of any type coercible to logical), providing information on whether any pair of observations are neighbors or not, to the w argument of pcdtest. If w is supplied, only neighboring pairs will be used in computing the test; else, w will default to NULL, and all observations will be used. The matrix needs not really be binary, so commonly used “row‐standardized” matrices can be employed as well: it is enough that neighboring pairs correspond to nonzero elements in w³.

10.1.2.2 The Randomized W Test

The test is flexible and well behaved in small samples; moreover it does not suffer the biggest drawback of its global sibling, which does not have any power under zero‐mean dependence and therefore cannot be employed, for example, on cross‐sectionally demeaned data – or equivalently on the residuals of a model containing time fixed effects. Nevertheless, it does not tolerate serial correlation and can be sensitive to non‐spatial types of dependence. In fact, if cross‐sectional dependence of the non‐spatial type is present and a test is performed, it will be based on a subset of spatially related pairs from a population of correlated ones; it is therefore likely to yield a false positive result (a type I error) favoring spatial dependence.

The idea underlying the test, that not all pairs of neighbors are correlated but only those in a specific spatial relationship are and that the latter are identified through the matrix, gives rise to another testing procedure that is remarkably robust to all the above confounding features. The RW test of Millo (2017a) employs a permutation procedure to produce a large number of randomized neighborhood matrices and then compares the statistic under the true spatial ordering with the population of those under the randomized ones. If spatial dependence is absent, the observations must be exchangeable in the cross‐section: then, the true will not take an extreme value with respect to the randomization‐based ones, and the null hypothesis of no spatial dependence will hold. As usual, the share of randomized statistics more extreme than the true one will be the pseudo‐ of the test. In the majority of situations, the alternative hypothesis is of positive spatial dependence. In this case a one‐tailed test will be appropriate. Given a panel‐indexed vector , call the randomized statistic from the ‐th draw, with ; and the one under the true W. If the alternative is positive spatial dependence, the pseudo‐ of the one‐tailed test is then

(10.1)

where is the indicator function. The null of no spatial dependence in would be rejected at, say, 5% significance if , meaning that the actual value is more extreme than the 95th quantile of the distribution of randomized values.

Negative spatial autocorrelation is less common in empirical practice but can be relevant, e.g., in the description of competitive processes (see Griffith and Arbia, 2010; Elhorst and Zigova, 2014). In this case it may happen that the distribution of randomized statistics be shifted in the opposite direction by positive global dependence so that the value of the true test statistic be less extreme, and the one‐tailed procedure would not work. A two‐tailed test is then needed, which is easily accomplished by taking absolute values and cross‐sectionally demeaning the data so that the average of the factors, and hence the average global correlation, is re‐centered on zero:

(10.2)

To take heed of possible asymmetries in the (re‐centered) distribution of randomized statistics, one can go the safest way employing the asymmetric version of the test:

(10.3)

10.2.2.1 Spatial OLS

Ordinary least squares estimation is consistent, under the usual exogeneity conditions on , for models with spatially lagged regressors, in which case it is also efficient provided that the standard hypotheses of homoscedasticity and incorrelation hold; in fact, adding may eliminate the spatial correlation in error terms and effectively make OLS the efficient estimator. Even in the case of the spatial error model, OLS remain consistent, although inefficient, for .

As a first approximation, and in cases where ML and GM are problematic (one for all, dynamic panels), the so‐called spatial‐OLS method has been advocated: adding the spatial lag of the dependent variable to the right‐hand side regressors. This solution is in general not advisable because is endogenous by construction, and therefore the estimator is hopelessly biased; yet simulation studies have shown how the magnitude of the bias can be limited in real‐world cases, to the point of making this computationally simple solution relatively viable in some applied settings (see Franzese and Hays, 2007).

10.2.2.2 ML Estimation of the SAR Model

An appropriate way to estimate a SAR model, provided the errors are i.i.d. normal, is by ML. Let us start from the cross‐sectional case where is and is a vector of length . Denoting , the model becomes so that . Expressing the usual likelihood function of the linear model in terms of the transformed requires adding the Jacobian of the transformation, i.e., the determinant of , therefore the log‐likelihood becomes:

and this likelihood is to be optimized with respect to and , efficient optimization strategies having been outlined in the seminal book of Anselin (1988). The pure‐SAR panel case, pooling the data without any individual feature, just substitutes for , for and so that it could be estimated with the lagsarlm function from package spdep. Nevertheless, it is always preferable for computational reasons to resort to specific methods for spatial panels when available.

10.3.2.1 Spatial Panels with Independent Random Effects

In a random effects specification, the unobserved individual effects are assumed uncorrelated with the other explanatory variables in the model and can therefore be safely treated as components of the error term.⁸ In this case, , and the error term can be rewritten as:

where . As a consequence, the composite error term becomes

and its variance‐covariance matrix is:

(10.4)

In deriving several Lagrange multiplier (LM) tests, Baltagi et al. (2003b) consider a panel data regression model that is a special case of the model presented above in that it does not include a spatial lag of the dependent variable. Elhorst (2003), Elhorst and Fréret (2009) define a taxonomy for spatial panel data models both under the fixed and the random effects assumptions. Following the typical distinction made in cross‐sectional models, they define the fixed as well as the random effects panel data versions of the spatial error and spatial lag models. However, unlike Case (1991), they do not consider a model including both the spatial lag of the dependent variable and a spatially autocorrelated error term. Therefore, the models reviewed in Elhorst (2003), Elhorst and Fréret (2009) can also be seen as special cases of this more general specification.

Following the treatment in Millo (2014), on which this part of the chapter is based, we label the combined model containing both a spatial lag and a spatial error process SAREM. (This is also often called , because of the two spatial autoregressive processes, one in the response and one in the errors.) If a random individual effect is also part of the composite error term, then we will add the suffix RE. Although SAR and SEM, combined with either FE or RE, are by far the most popular specifications, the literature has also dealt with different types of spatial diffusion processes in the errors other than the autoregressive one, most notably the spatial moving average.⁹ We do not consider them here.

10.3.2.2 Spatially Correlated Random Effects

A different specification for the disturbances was considered in Kapoor et al. (2007). They assume that spatial correlation applies to both the individual effects and the remainder error components. Although the two data‐generating processes look similar, they do imply different spatial spillover mechanisms governed by a different structure of the implied variance‐covariance matrix. In this case, commonly referred to as KKP, the composite disturbance term

follows a first‐order spatial autoregressive process of the form:

It follows that the variance‐covariance matrix of is:

(10.5)

where is the typical variance‐covariance matrix of a one‐way error component model. The variance matrix in (10.5) is simpler than the one in (10.4), and therefore its inverse is easier to calculate, as will be discussed below. As Baltagi et al. (2013) observe, the economic meaning of the two models is also different: in the first model only the time‐varying components diffuse spatially; in the second, spatial spillovers too have a permanent component. Lee and Yu (2012, 2.4) illustrate the difference between this latter specification and through the likelihood of the between model. We label this latter alternative specification , and its extension to including a spatial lag (see Mutl and Pfaffermayr, 2011) .

10.3.3.1 Spatial Models with a General Error Covariance

Maximum Likelihood estimation with a general error covariance matrix has been outlined in Magnus (1978) (see also Anselin et al., 2008). If the error is distributed as then the log‐likelihood is

Particularizing this likelihood w.r.t. the case at hand, and adding a spatial filter if needed, provides a general framework for ML estimation of the models of interest. Anselin (1988), the classic reference on spatial econometric model estimation by ML, outlines the general procedure for a model with spatial lag, spatial errors, and possibly nonspherical residuals as follows. Let us restrict the analysis, for the moment, to one cross‐section and let our model be:

(10.6)

with and, in general, . Two special cases of this general model are often found in applied literature: if one has the spatial autoregressive (SAR) model, while if , the spatial (autoregressive) error (SEM) model. Both usually include the hypothesis of spherical remainder errors: . Introducing the now‐standard simplifying notation , the model becomes:

where are potentially different spatial weights matrices.¹¹ If there exists such that and , and is invertible, then and the model (10.6) can be written as

or, equivalently,

with a “well‐behaved” error.

Still following Anselin, making the estimator operational requires the transformation from the unobservable to observables. Expressing the likelihood function in terms of requires calculating the Jacobian of the transformation . These determinants are to be added to the log‐likelihood, which becomes

where the difference w.r.t. the usual likelihood of the classic linear model is given by the terms of the Jacobian.¹² The likelihood is thus a function of , , , and parameters in .

It will be convenient for our purposes, and without loss of generality, to scale the overall errors' covariance writing it as (the latter expression is in fact more general, as it does not constrain the heteroscedastic error term to be spatially lagged, through premultiplication by , in its entirety. In our case, only the error covariance of the specification can be separated into a heteroscedastic error term and a spatial filter and therefore straightforwardly written as , while the more common SEM specification cannot). This likelihood can be concentrated w.r.t. and the error variance , by substituting

(10.7)

and a closed‐form GLS solution for and is available for any given set of spatial and other covariance parameters

(10.8)

so that a two‐step procedure is possible that alternates optimization of the concentrated likelihood and GLS estimation. From here on, we explicitly consider the (balanced) panel structure of the data: individuals observed over time periods.

10.3.3.2 General Maximum Likelihood Framework

Building on the framework from Anselin (1988) outlined above, explicitly particularizing and operationalizing it with respect to a number of possible error covariance structures, all specifications outlined above can be estimated without the need to pre‐transform the data as has been customary in the literature since Elhorst (2003). Random effects will instead be considered as one feature of the errors' covariance, just like spatial (or, later on in the chapter, serial) correlation (see Millo, 2014). Considering the spatial dependence features together with all the other sources of heteroscedasticity and correlation instead of separating it clearly, as done in the original Anselin framework, has the advantage of keeping some components of the error term (most notably, the random effects) out of the spatial dependence, which can remain a feature of the idiosyncratic error only, in accordance with most applications in the literature; but also some clear computational disadvantages, as will be discussed below. We will also consider the alternative specification where the individual effects are lagged together with the idiosyncratic errors, as in Kapoor et al. (2007), which one can straightforwardly express in terms of Anselin's original expression , also extending the structure of to include serial correlation. This latter will turn out to be easier to compute, especially on large examples.

First we will discuss the combination of a spatial lag with any error covariance structure; then we will review the most significant among the latter; lastly we will give an example of operationalization through the use of analytical expressions for the inverse and determinant of the error covariance matrix .

Optimization will generally be subject to box constraints according to the following rules: the spatial lag and spatial errors coefficients and will be bounded between and 1, where is the smallest characteristic root of ;¹³ the serial correlation coefficient will be constrained to the usual stationarity condition and the variance ratio of the random effects to be non‐negative.

Spatial Lag

Although both the SAR and the SEM specifications are popular in the literature, estimation generally focuses on one effect only, and there are few applications allowing for both of them to be present in the estimated model, one notable exception being the pioneering work of Case (1991). It is nevertheless straightforward, at least as far as expressing the likelihood is concerned, to combine a spatial lag with any error structure, including spatial dependence ones.

The general likelihood for the spatial lag panel model combined with any error covariance structure is a panel version of (10.7):

(10.9)

The usual iterative procedure a la Oberhofer and Kmenta (1974) can be employed to obtain the maximum likelihood estimates. Starting from an initial value for the spatial lag parameter and the error covariance parameters, we obtain estimates for and from the first‐order conditions:

(10.10)

The likelihood can be concentrated and maximized with respect to the parameters in and . The estimated values thereof are then used to update the expression for These steps are then repeated until convergence. In other words, for a specific the estimation can be operationalized by a two‐step iterative procedure that alternates between GLS (for and ) and concentrated likelihood (for the remaining parameters) until convergence.

This general scheme can be applied to the random effects case, where it provides a simple and effective equivalent to the usual partial time‐demeaning procedure, as well as to all the more complicated error covariance specifications discussed in the following.

For example, the spatial autoregressive model with random effects can be written as a combination of spatial filtering on the regressand and a random effects structure in the errors:

hence it can be estimated by “plugging into” the general likelihood (10.9) the particular scaled error covariance characterized by one parameter: , the ratio of the variance of the individual effect over that of the idiosyncratic error.

Error Structures

As already discussed, the spatial error, random effects model gives rise to two possible specifications, depending on the interaction between the spatial autoregressive effect and the individual error components: the specification first analyzed by Anselin (1988) where only the idiosyncratic error is spatially correlated:

with the scaled errors' covariance (denoting and ):

and that of Kapoor et al. (2007) where the same spatial process applies both to the individual and the idiosyncratic error component:

where the scaled errors' covariance is:

10.3.3.3 Generalized Moments Estimation

The computational intensity of ML estimation, which in the simpler models is related mostly to the need to recompute the determinants at each optimization step, has long been a limiting factor in practical applications. Samples of cross‐sectional size in the hundreds were the practical maximum for the simple SAR or SEM models at the end of the 20th century, both because of the difficulty in obtaining a result at all and of the numerical unreliability of the latter if any because of precision problems (Kelejian and Prucha, 1999, Bell and Bockstael, 2000). Today, much more powerful computers have extended the scope of ML methods, but on the other hand the increasing availability of GIS data has brought forward a new generation of estimation problems of ever increasing size (an early survey and examples in Bell and Bockstael, 2000).

This has prompted researchers to explore alternative estimation strategies. Kelejian and Prucha (1999) proposed the generalized moments (GM) method, which, despite being asymptotically equivalent to ML under normality of the errors, is consistent irrespective of the latter; computationally, moreover, it does not require the numerically cumbersome calculation of the determinants.

The GM estimator for the cross‐sectional SEM model (see also Bell and Bockstael, 2000) is based on the following three moments of the error term:

(10.11)

The estimation strategy is based on the idea of estimating the spatial autoregressive coefficient based on the residuals from a consistent estimator (here, OLS) and then using it in a feasible GLS analysis. With respect to maximum likelihood, the GM estimator has the additional advantage of not relying on a normality assumption for the errors. One drawback is that standard errors are not available for the parameter.

The Kelejian and Prucha (1999) GM estimator has first been extended to the panel case by Druska and Horrace (2004), then by Kapoor et al. (2007) who estimated the above described model with RE, a specification which, after them, is known as KKP. In order to perform feasible GLS, one does now need consistent estimates of the spatial autoregressive parameter and the two variance components of the composite error, and . The estimator a la KKP estimates them based on six moment conditions, using the OLS residuals , which are still consistent in this setting:

(10.12)

where , , , and ; and and are, respectively, a time‐demeaning and a time‐averaging matrix.

The moment conditions are now redundant and can be employed in different ways. The simplest is to consider only the first three moment conditions. The second way is to employ all six moments in estimating the three unknown parameters, weighing them through a covariance matrix calculated under the assumption of normally distributed errors. The third and last proceeds like the second, using all available moments but employs a simplified weighting matrix.

GM methods have been extended to the other relevant specifications in spatial econometrics. Spatial fixed effects models can also be estimated in this framework, through a modification of the KKP procedure suggested by Mutl and Pfaffermayr (2011) and consisting in replacing the OLS residuals, inconsistent under the fixed effects assumption, with spatial 2SLS within residuals; the spatial parameter is estimated by an adaptation of the simplified KKP procedure (first three moment conditions only) and used in a spatial Cochrane‐Orcutt transformation of the within‐transformed variables. The GM method has also been extended to the SAR and SAREM models, so that now any combination of spatial lag and error, with individual effects of either random or fixed type, can be estimated through this numerically very efficient method (see Millo and Piras, 2012).

10.3.4.1 LM Tests for Random Effects and Spatial Errors

Requiring only the estimation of the restricted specification, Lagrange multiplier (LM) tests in the tradition of Breusch and Pagan (1980) are particularly appealing in a spatial random effects setting because of the computational difficulties related to ML estimation of encompassing models.

Baltagi et al. (2003b) derived joint, marginal and conditional tests for all combinations of random effects and spatial correlation. Starting from the random effects model with SEM errors (), the error term can be written as:

(10.13)

and the (unscaled) variance covariance matrix of the errors as:

(10.14)

The hypotheses under consideration are:

1. under the alternative that at least one component is not zero
2. assuming no spatial correlation, under the one‐sided alternative that the variance component is greater than zero
3. assuming no random effects, under the two‐sided alternative that the spatial autocorrelation coefficients is different from zero
4. assuming the possible existence of random effects, under the two‐sided alternative that the spatial autocorrelation coefficient is different from zero
5. assuming the possible existence of spatial autocorrelation and the one‐sided alternative that the variance component is greater than zero

The joint LM test for the first hypothesis of no random effects and no spatial autocorrelation () is given by:

(10.15)

where , , and denotes OLS residuals. The marginal LM test for random effects assuming no spatial correlation is given by:

(10.16)

An alternative standardized version with better finite sample properties can be obtained by centering and scaling the one‐sided LM statistic:

(10.17)

Analogously, the marginal LM test of no spatial autocorrelation assuming no random effects is given by:

(10.18)

which also admits a standardized form with better properties:

(10.19)

and are asymptotically normally distributed as for fixed , under and respectively. Based on the latter, a one‐sided joint test statistic for can be derived as:

(10.20)

which is asymptotically distributed as a standard normal. In practical applications can turn out negative, especially when the random effects variance is small, and the same applies to when the spatial autocorrelation coefficient is small. A test for the joint null hypothesis can therefore be based on the following decision rule:

Under the null the test statistic has a mixed ‐distribution given by:

(10.21)

When using , one is assuming that random regional effects do not exist. However, especially when the random effect variance is actually large, this may lead to incorrect inference. For this reason Baltagi et al. (2003b) derived a conditional LM test for spatial autocorrelation allowing for the random effects variance to be non‐zero. The expression for the test assumes the following form:

(10.22)

where . Also, , and .

Contrarily to previous tests that use OLS residuals, the residuals come from the ML estimation of a one‐way error component model. This last point, on the converse, makes the implementation slightly more complicated. A one‐sided test is simply obtained by taking the square root of 10.22. The resulting test statistics are asymptotically distributed as a standard normal. Similarly, when using , one is assuming no spatial error correlation. This assumption may lead to incorrect inference particularly when it is not the case that is close to zero. A conditional LM test allowing for spatial error correlation can be derived as:

(10.23)

where and . A one‐sided test can be defined by taking the square root of 10.23 based on ML residuals. The test statistic is again asymptotically normally distributed.

10.3.4.2 Testing for Spatial Lag vs Error

If a researcher has a strong reason to expect a spatial‐data‐ generating process to be of the SAR (or, respectively, SEM) kind, then her only problem is to determine whether said spatial effect is present. Then she can either proceed general to specific, estimating the SAR (SEM) model and assessing the significance of the spatial coefficient, or specific to general, testing from the non‐spatial model toward the spatial alternative. In a ML framework, the optimal LM tests for one effect assuming the other out are called marginal. They are dependent on the above hypothesis and will be inconsistent if it is violated; in case only the “other” effect is actually present, they will usually yield a type I error.

As outlined above, although empirical practice has mostly concentrated on either the SAR or the SEM model, estimation of SAREM models containing both a spatial lag and a spatial error is possible. Therefore, if the researcher does not have a strong prior in favor of either, an empirical strategy can be to start from the most general SAREM specification, together with the appropriate kind of individual heterogeneity, and let the data tell us which of the two spatial processes – if any and if not both – did actually generate the observed sample, by looking at the significance diagnostic for either spatial coefficient.

One drawback of this strategy is its computational demands and lesser stability than estimating the simpler models; another is that it does not allow the inclusion of a full set of spatially lagged regressors, a specification approach that has become increasingly popular in recent years.

Lagrange multiplier tests for SAR (SEM) can be either of the conditional type, allowing for the presence of SEM (SAR) tout court, or of the locally robust type, allowing for a limited deviation from zero of the SEM (SAR) coefficient. The former are optimal under the standard assumptions of the ML framework detailed above, and provided the general SAREM model holds; and they require residuals from the restricted SEM (SAR) model. The second kind have suboptimal statistical properties with respect to the optimal conditional tests, and under the above hypotheses on the data‐generating process, they are not guaranteed to hold if misspecification is “too far away,” i.e., if the SAR (SEM) coefficient is of sizable magnitude (and how far is far, i.e., whether 0.1 or 0.4 is tolerable, is an empirical question); moreover the currently available robust LM tests have been developed in a cross‐sectional framework and do not explicitly incorporate panel features. On the other hand, they are computationally simpler being based on the residuals of the non‐spatial model, and they allow including spatially lagged regressors; hence their remarkable success in applied practice.

Marginal Spatial LM Tests

In the context of pooled cross sections, without allowing for any correlation feature across either time or cross section (i.e., setting and in equation ), any cross‐ sectional test can be straightforwardly applied to the pooled dataset. The LM tests of Anselin et al. (1996) (LM) are simply rewritten for the pooled dataset, stacked by cross section, and drawing on an enlarged version of the weights matrix obtained by replicating the cross‐sectional over the main diagonal so that (see Anselin et al., 2008). The pooled test becomes:

(10.24)

where are the OLS residuals and

and

(Elhorst, 2010, Formulae 11 to 13). In turn, the pooled test is:

(10.25)

Locally Robust Spatial LM Tests

The robust LM tests of Anselin et al. (1996) can in turn be straightforwardly adapted to the (pooled) panel case, as per Elhorst (2014, Ch. 2.3):

using, again, the OLS residuals (Elhorst, 2010, Formulae 14‐15).

Moreover, according to Bera et al. (2009), the LM test for the joint null hypothesis versus or is equal to the sum of the marginal test for one effect and the locally robust test for the other:

so that the RLM tests can also be obtained indirectly by subtracting the marginal test for the “other” effect from the joint test.

The slmtest function, specifying test = 'lml' ('lme') will perform either the marginal test for SAR (SEM) assuming no SEM (SAR) component in the data‐generating process or the locally robust version if specifying test = 'rlml' ('rlme').

Wald Tests

Wald‐type tests are ‐tests for significance of the relevant parameter in the encompassing model. Thus, from ML estimates of the general SAREM‐RE model,

and symmetrically for . Importantly, the test can be made conditional to (i.e., valid in the presence of) individual random effects by including them in the specification. As observed, fixed individual effects can be eliminated through data transformation in two ways, both familiar from the spatial panel literature: either through time‐demeaning (within transformation) (Elhorst, 2003) or by forward orthogonal deviations (Lee and Yu, 2010a). The former induces residual serial correlation, which can nevertheless be considered (i.e., estimated out) in the encompassing model; while the latter preserves the features of the original errors covariance matrix (Debarsy and Ertur, 2010, p. 7).

LR Tests

Likelihood ratio tests are based on the likelihoods from the general and the restricted model. The test statistic is a simple transform of the difference in likelihoods:

where is the full vector of ML parameter estimates from the unrestricted model and from the restricted one, and the number of restrictions. Thus,

and symmetrically for . Again, including random effects in the estimated models makes the test conditional to these effects, while fixed effects can be transformed out as detailed in the previous paragraph but always keeping in mind the effects of the transformation on the error properties.

10.4.1.1 Serial and Spatial Correlation in the Random Effects Model

Generalizing the structure of the errors further by introducing serial correlation in the remainder of the error term, together with spatial correlation and random effects, Baltagi et al. (2007) derived a number of conditional and marginal LM tests for the different effects, possibly allowing for the presence of the other ones. Based on their work, Millo (2014) extended the model to include a SAR term. The errors of the SAREM model are specified as in the previous paragraph, so that the full model is:

To derive the likelihood, Baltagi et al. (2007) suggest a Prais‐Winsten transformation of the model with random effects and spatial autocorrelation. Following their simplifying notation, define: with:

(10.27)

then the expression for the scaled error covariance matrix can be written as

While in principle the inverse and determinant of can be calculated by brute force, in practice it is convenient, and often necessary, to rely on simplified analytical expressions to reduce the computational burden and extend the range of feasible sample sizes. Baltagi et al. (2007) derived expressions for the inverse and determinant of the error covariance matrix:

where and . They can be plugged in the general likelihood (10.9) to estimate the model.

10.4.1.2 Serial and Spatial Correlation with KKP‐Type Effects

As an alternative to the specification, Millo (2014) presents an extension of the errors a la Kapoor et al. (2007) to serial correlation in the remainder errors. As in the case, the random effects are spatially lagged together with the idiosyncratic ones, while the remainder errors in turn are serially correlated:

This alternative specification assumes that individual effects follow the same spatial diffusion process as the idiosyncratic errors do. By analogy, it is termed . Just as in the case, the error covariance is then again of the form (see Section 10.3.3.1), which simplifies computations considerably. In fact, the (scaled) error covariance for this model is:

and, by the properties of Kronecker products, its inverse is

so that there is no need for the numerically demanding and unstable inversion of .¹⁴

10.4.2.1 Tests for Random Effects, Spatial, and Serial Error Correlation

Baltagi et al. (2007) derive the joint, marginal, and conditional LM tests for the model with serial correlation.

They consider all possible combinations of joint, marginal and conditional tests:

• the joint test for , (J)
• the marginal tests for and assuming in turn that the other two are zero (M.1‐3)
• the joint tests for any combination of two of the parameters assuming the third one is zero (M.4‐6)
• the marginal tests for and assuming in turn that the other two may or may not be zero (C.1‐3)
• the joint tests for any combination of two of the parameters assuming the third one may or may not be zero (C.4‐6)

M.1‐3 are well‐established testing procedures in the literature (as observed in Baltagi et al., 2007). M.1 (test for ) is the LM test for spatial error correlation derived by Anselin (1988) in the context of a pooled model with no serial correlation or individual effects. On the other hand, M.2 (test for ) is analogous, for large , to the well‐known Breusch (1978), Godfrey (1978) serial correlation test. Finally, M.3 is simply the Breusch and Pagan (1980) random effects test.

Baltagi et al. (2007, Appendix A.3) show that the test statistic for the joint hypothesis M.4 () assuming no random effects is simply the sum of the marginal tests M.1 () and M.2 (). Additionally, M.5 is the Baltagi et al. joint test outlined in section 10.3.4.1; and M.6 is the joint test for random individual effects and serial correlation derived in Baltagi and Li (1995) (see section 4.3.2).

As a result of the previous discussion, we only consider the three‐way joint test J and the one‐way conditional tests C.1‐3.

The corresponding null hypotheses are:

1. under the alternative that at least one component is not zero (J)
2. , assuming : test for spatial correlation, allowing for serial correlation and random individual effects (C.1)
3. , assuming : test for serial correlation, allowing for spatial correlation and random individual effects (C.2)
4. , assuming : test for random individual effects, allowing for spatial and serial correlation (C.3)

The joint LM test for is given by:

(10.28)

where, , , , , is a matrix with bidiagonal elements equal to one and denotes OLS residuals. Under , is distributed as .

The conditional C.1 test for gives rise to the following statistic, asymptotically distributed as under :

(10.29)

where

is the score vector (evaluated at the null), a vector of ML residuals obtained from the estimation of the model with individual error components and serial correlation, , and has been defined above.

The conditional C.2 test for is based on the following statistic, asymptotically distributed as under the null:

(10.30)

where is the corresponding element of the information matrix,¹⁵

(10.31)

with the score evaluated at the null and the vector of ML residuals from the estimation of a panel model with individual error components and serial correlation. Both and assume the same expression as before, while as usual .

The conditional C.3 test for is based on the following statistic:

(10.32)

where

is the score evaluated at the null, is the corresponding element of the information matrix¹⁶ and is the vector of estimated residuals from the ML estimation of a panel model with spatially and serially correlated errors but no individual error components. The test statistic is asymptotically distributed as under :

10.4.2.2 Spatial Lag vs Error in the Serially Correlated Model

Testing for spatial lag vs spatial error in a model allowing for random effects and/or serially correlated errors can be done via the Wald approach, from the encompassing specification.¹⁷