A Monte Carlo Study for Pure and Pretest Estimators of a Panel Data Model with Spatially Auto correlated Disturbances

This paper examines the consequences of model misspecification using a panel data model with spatially autocorrelated disturbances. The performance of several maximum likelihood estimators assuming different specifications for this model are compared using Monte Carlo experiments. These include (i) MLE of a random effects model that ignore the spatial correlation; (ii) MLE described in Anselin (1988) which assumes that the individual effects are not spatially autocorrelated; (iii) MLE described in Kapoor, et al. (2006) which assumes that both the individual effects and the remainder error are governed by the same spatial autocorrelation; (iv) MLE described in Baltagi, et al. (2006) which allows the spatial correlation parameter for the individual effects to be different from that of the remainder error term. The latter model encompasses the other models and allows the researcher to test these specifications as restrictions on the general model using LM and LR tests. In fact, based on these tests, we suggest a pretest estimator which is shown to perform well in Monte Carlo experiments, ranking a close second to the true MLE in mean squared error performance.


Introduction
The recent literature on spatial panel data models with error components adopts two alternative spatial autoregressive error processes. One speci…cation assumes that only the remainder error term is spatially correlated but the individual e¤ects are not (Anselin, 1988, Baltagi, Song, and Koh, 2003and Anselin, Le Gallo and Jayet, 2005; we refer to this as the Anselin model). The other speci…cation assumes that both the individual and remainder error components follow the same spatial error process (see Kapoor, Kelejian, and Prucha, 2006; we refer to this as the KKP model). In a companion paper, we introduced a generalized spatial panel data model which nests these two alternative processes in a more general model (see Baltagi, Egger, and Pfa¤ermayr, 2006). 1 The latter paper derived LM tests of the generalized model against its restricted alternatives and studied their size and power performance against LR-tests. This paper compares the performance of ML-estimates of these models under misspeci…cation and suggests a pretest estimator based on the LM-tests derived by Baltagi, Egger, and Pfa¤ermayr (2006). We show that misspeci…ed MLE can cause substantial loss in MSE where as the pretest estimator performs well, ranking a close second to the true MLE.
Section 2 derives the MLE for the various panel data models considered with …rst order spatial autocorrelation in the disturbances. 2 It also describes the algorithm to select the pretest estimator based on a sequence of LM tests derived in Baltagi et al. (2006). Section 3 gives the design of the Monte Carlo experiments and describes the results. These Monte Carlo experiments shed some light on the performance of say the Anselin MLE when the true speci…cation is that of KKP, and vice versa. Also, it shows how robust is the MLE of the general spatial panel model to overspeci…cation, i.e., if the true model is KKP or Anselin. Conversely, how the Anselin and KKP maximum likelihood estimates are a¤ected by underspeci…cation of the general model.
Since the researcher does not know the true model, the Monte Carlo experiments show that the pretest estimator is a viable second best alternative to the true MLE in practice.
2 Maximum likelihood estimators of the alternative models Baltagi, Egger, and Pfa¤ermayr (2006) considered the following generalized spatial error components model: This is a balanced panel, which consists of n = N T observations, where N is the number of unique cross-sectional units, while T is the number of time especially as N gets large. 4 periods. The (n 1) vector y denotes the dependent variable, X is an (n K) matrix of non-stochastic exogenous variables. is the corresponding K 1 parameter vector. Z = T I N denotes the design matrix for the (N 1) vector of random individual e¤ects u 1 . T is a (T 1) vector of ones and I N is an identity matrix of dimension N . The vector of individual e¤ects is assumed to be i:i:d:N (0; 2 I N ), while the (n 1) vector of remainder disturbances is assumed to be i:i:d:N (0; 2 I n ). Furthermore, the elements of and are assumed to be independent of each other. Both u 1 and u 2 are spatially correlated involving the same spatial weight matrix W N for each time period, but with di¤erent spatial autocorrelation parameters 1 and 2 , respectively. W N exhibits zero diagonal elements, the remaining entries are usually assumed to decline with distance. The eigenvalues of W N are bounded and smaller than 1 in absolute value (see Kelejian and Prucha, 2005). The latter assumption holds for the row normalized W N . It also holds for the maximum-row normalized spatial weights matrices. This assumption also implies that all row and column sums of W N are uniformly bounded in absolute value. In addition, we assume that j r j < 1 for r = 1; 2. The data are ordered such that i = 1; :::; N is the fast index and t = 1; :::; T is the slow one. The spatial weights matrix for the panel is then given by W = I T W N ; which is block diagonal and of dimension (n n).
This model encompasses both the KKP model, which assumes that 1 = we de…ne A = (I N 1 W N ) and B = (I N 2 W N ): This allows us to write and This uses the fact that E[u 1 u 0 2 ] = 0 since and are independent by The inverse of u can then be obtained from the inverse of smaller dimension (N N ) matrices as follows: Assuming normality of the disturbances the log likelihood function of the unrestricted model is given by where u = y X . For the special case of 1 = 0, this implies that A = I N and the restricted log likelihood function reduces to the one considered by 6 Anselin (1988, p.154): For the alternative case with 1 = 2 = 6 = 0, A = B and we obtain the log likelihood representation of the KKP estimator: Finally, with 1 = 2 = 0; the log likelihood reduces to the one representing the familiar RE model without any spatial autocorrelation: The pretest estimator is based on a sequence of LM-tests derived by Baltagi, Egger and Pfa¤ermayr (2006). Speci…cally, the following hypotheses were considered: : at least one of the 1 or 2 6 = 0 (10) is not rejected, the pretest estimator reverts to the KKP MLE. Otherwise, 1 6 = 0 or 2 6 = 0 and 1 6 = 2 . Next, we test H C 0 ; 1 = 0. In case H C 0 is not rejected, the pretest estimator reverts to the Anselin MLE. If H C 0 is rejected, the pretest estimator reverts to the MLE of the general model considered by Baltagi, et al. (2006). In other words, where e 2 Here, e u = y X e mle;re denotes the vector of restricted ML residuals under H A 0 . Baltagi, et al. (2006) show that under H A 0 , the LM A statistic is asymptotically distributed as 2 2 . For H B 0 ; the LM-test statistic is given by . Here, u = y X e mle;KKP denotes the vector of restricted ML residuals under H B 0 . The LM B statistic is asymptotically distributed as . The corresponding LM test for H C 0 , which has no simple closed form representation is given by: where Anselin denotes the vector of restricted ML residuals under H C 0 , i.e., the Anselin model, and J 1 33 is the (3,3) element of the inverse of the information matrix described in Baltagi, et al. (2006). ===== Table 1 ===== Table 1  are smaller than 0:2 in absolute value, the pretest estimator is more likely to pick the RE MLE since H A 0 is less likely to be rejected.
===== Table 2 ===== Table 2 summarizes our …ndings regarding the relative mean square error for in (15). Here, relative MSE is always with respect to the MSE of the MLE of the TRUE model. Looking at the results for N = 50, T = 5 and = 0:5; we see that the loss in MSE is less than 1:7% for the misspeci…ed ML estimators when the true model is random e¤ects. However, when the true model is KKP, the loss in MSE for the Anselin MLE is 9% for 1 = 2 = 0:8 and 11% for 1 = 2 = 0:8: The random e¤ects MLE, which ignores the spatial correlation, performs the worst with big loss in MSE. In contrast, the general model MLE, which encompasses the KKP, does well and so does the pretest estimator. When the true model is Anselin, the loss for the KKP MLE in MSE is 9% for 1 = 0 and 2 = 0:8 and 7% for 1 = 0 and 2 = 0:8: The random e¤ects estimator that ignores the spatial correlation again performs the worst with big loss in MSE. In contrast, the general model which encompasses the KKP does well and so does the pretest estimator.
When the true model is the general model, the loss for the KKP MLE in MSE reaches 32% for 1 = 0:8 and 2 = 0:5 and the loss for the Anselin MLE reaches 26% for 1 = 0:8 and 2 = 0:2: The random e¤ects estimator again performs the worst with big loss in MSE. In contrast, the pretest estimator does well with a maximum loss of MSE of 3%. Our Monte Carlo results suggest that the pretest estimator is a practical second best choice no matter what true model generated the data. In most cases considered, it performs only slightly worse than the MLE of the true model. The maximum loss in MSE is 2:7% for 1 = 0:5 and 2 = 0:2: Given that the applied researcher has no hope of knowing the true model, the pretest estimator seems to be a reasonable alternative.
The second panel of Table 2 shows how the relative MSE for gets a¤ected by doubling N to 100, holding T …xed at 5 and = 0:5; while the last two panels show what happens when we alter the testing sequence or the signi…cance level. Table 3 repeats this exercise now doubling T to 10, holding N …xed at 50 and = 0:5. Also, what happens to this relative MSE for when we increase to 0:75 for N = 50, T = 5: Doubling N for a …xed T = 5 and = 0:5; in general, improves the relative MSE performance of the pretest estimator. This is also generally true, if we double T for a …xed N = 50 and = 0:5: The performance of the pretest estimator seems to be robust to increasing from 0:5 to 0:75 for a …xed N = 50, and T = 5: It is also robust to increasing the signi…cance level from 5% to 10%. However, it is sensitive to altering the testing sequence of H B 0 and H C 0 : The loss in MSE for the pretest estimator is at most 9:2% for 1 = 0:2 and 2 = 0:8 for N = 50, T = 5 and = 0:5: =====  Table 4 shows the bias in the MLE of 1 and 2 , under the various models considered, for N = 50, T = 5 and = 0:5: 5 Obviously, when the wrong restriction is imposed on these 's, bias is introduced. For example, if the true model is the General model with 1 6 = 2 ; the bias for the KKP estimators of these 's is huge because KKP imposes falsely 1 = 2 : Surprisingly, the bias for the Anselin estimator of 2 (imposing falsely that 1 = 0); is small not