The Hausman-Taylor Panel Data Model with Serial Correlation

This paper modifies the Hausman and Taylor (1981) panel data estimator to allow for serial correlation in the remainder disturbances. It demonstrates the gains in efficiency of this estimator versus the standard panel data estimators that ignore serial correlation using Monte


Introduction
The random e¤ects (RE) panel data model assumes that all the explanatory variables are uncorrelated with the random individual e¤ects, while the …xed e¤ects (FE) panel data model assumes that all the explanatory variables are correlated with the random individual e¤ects. Instead of this "all" or "nothing" assumption, the Hausman and Taylor (1981) panel data estimator allows some of the explanatory variables to be correlated with the individual e¤ects. One of the main disadvatages of the …xed e¤ects estimator is that it wipes out the e¤ects of time-invariant variables. In contrast the Hausman and Taylor (HT) estimator recaptures the estimates of these time-invariant variables which are important in empirical applications, see Cornwell and Rupert (1988), Egger and Pfa¤ermayr (2004) and Serelenga and Shin (2007) to mention three applications of this estimator. This paper extends the Hausman and Taylor (HT) estimator to allow for serial correlation in the remainder disturbances of the AR(1) type. The standard …xed e¤ects (FE) and random e¤ects (RE) panel data models with serial correlation in the remainder disturbances have been considered by Bhargava, Franzini and Narendranathan (1982) and Baltagi and Li (1991) to mention a few. While the …xed e¤ects AR(1) estimator (FE-AR(1)) considered by Bhargava, Franzini and Narendranathan (1982) is consistent for the HT model, it does not provide an estimator of the time-invariant variables coe¢ cients which are usually of interest in most economic applications. The Baltagi and Li (1991) random e¤ects AR(1) (RE-AR(1)) estimator provides estimates of the time-invariant variables coe¢ cients, but these will be consistent only if the individual e¤ects are uncorrelated with all the regressors. The modi…ed HT estimator allowing for AR(1) disturbances (denoted by HT-AR(1)) is more e¢ cient than the HT estimator that ignores this serial correlation. Unlike the FE-AR(1) estimator, it captures the e¤ects of time-invariant variables, and unlike the Baltagi and Li (1991) RE-AR(1) estimator it allows for possible correlation between the regressors and the individual e¤ects. The paper performs Monte Carlo experiments that demonstrate the gains in e¢ ciency of this HT-AR(1) estimator over the standard HT estimator in the presence of serial correlation.

The Model and Assumptions
Consider the following Hausman and Taylor (1981) panel data model: where u it = i + it ; and and are unknown vectors. The X it 's are time varying regressors, while the Z i 's are time-invariant. This Hausman-Taylor (HT) model is allowed to have …rst order serial correlation in f it g of the AR(1) type: where " it is a white noise process with variance 2 " . The i 's are independent of the it 's for all i and t. In vector form, Equation (1) can be written as with ; u iT ) 0 and T is a vector of ones of dimension T . Equation (3) can be also written as

The GLS Estimator
Assuming that the individual e¤ects i are random with i iid(0; 2 ) and E ( i jX 0 it ; Z 0 i ) = 0 for all i and t, the resulting random e¤ects (RE ) GLS type estimator correcting for AR(1) remainder disturbances will be the best linear unbiased estimator (BLUE), see Baltagi and Li (1991). In fact, this estimator applies the Prais-Winsten (PW) transformation in the …rst step to transform the remainder AR(1) disturbances into serially uncorrelated classical errors. More speci…cally, one premultiplies equation (4)  to get y = W + u ; where y = (I N C) y, W = (I N C) W and u = (I N C) u. Using the fact that The transformed regression disturbances are given by where v = (I N C) v. As shown in Baltagi and Li (1991), the variance-covariance matrix of the transformed disturbances is given by and " Premultiplying the PW transformed observations by " 1=2 , one gets The least squares estimator of the resulting equation gives uŝ The best quadratic unbiased BQU estimators of the variance components are given by: As suggested by Baltagi and Li (1991), one can estimate by where b it is the within residual. Then^ 2 " and^ 2 can be estimated from equation (10) and (11) by substituting the residualsû from the PW transformed equation using b .

The Within-GLS Estimator
A critical assumption for the GLS estimator is that E ( i jX 0 it ; Z 0 i ) = 0. Otherwise^ GLS would be biased and inconsistent. In fact, the classic paper by Mundlak (1978) assumes that the i 's are explicitly formulated as a function of the means of all the regressors over time, in this case X i : . The result is that under this Mundlak model, GLS su¤ers from omitted variable bias, i.e., the omission of X i :, while the within transformation wipes out this source of endogeneity and remains consistent. However for the AR(1) remainder disturbances, the within estimator will not be best linear unbiased estimator (BLUE). This can be easily recti…ed using a within-GLS estimator that uses the Prais-Winsten transformation to correct for serial correlation in the remainder error in the …rst step, then applying a quasi-within transformation to wipe out the transformed ; i s in the second step. Premultiplying equation (5) by (I N E T ), one gets using E T T = 0. Applying the least squares estimator to the resulting equation gives us the FE-PW GLS estimator^ One can easily verify that this estimator is equivalent to the one suggested in Bhargava, Franzini and Narendranathan (1982). When there is no serial correlation, i.e., = 0, the FE-GLS estimator in equation (14) reduces to the usual FE estimator^ where E T = I T J T and J T is a T T matrix with all elements equal to 1=T . In fact, when = 0; C = I T , = 1, T = T and E T = E T .
Despite several warnings not to omit the …rst observation when correcting for serial correlation, see Parks and Mitchell (1980), researchers still use the FE estimator using the Cochrane-Orcutt (CO) transformation to correct for serial correlation in the remainder error. The Monte Carlo experiments in Section 6 demonstrate the loss in e¢ ciency in using this FE-Cochrane-Orcutt rather than FE-PW.

The Hausman-Taylor Estimator
Hausman and Taylor (1981) split the time varying X's and the time invariant Z's into two sets of variables: X = (X 1 ; X 2 ) and Z = (Z 1 ; Z 2 ) where X 1 and Z 1 are assumed exogenous in that they are not correlated with i and " it , while X 2 and Z 2 are endogenous because they are correlated with i , but not " it . The Within transformation would sweep the i 's and remove the bias, but in the process it would also remove the Z i 's and hence the Within estimator will not give an estimate of which in most empirical economic applications is important for policy purposes. In this section, we modify the Hausman and Taylor (1981) estimator to allow for AR(1) remainder errors. This can be done as follows: Step 1: From the assumptions of our model, one can estimate consistently using the FE-PW GLS estimator in equation (14) rather than the usual FE estimator used by Hausman and Taylor (1981) in the absence of serial correlation. Averaging the residuals from this regression over time as in HT but now weighting the initial period di¤erently from the rest as in Baltagi and Li (1991), one getŝ d y X ^ F E GLS : where y = I N J T y and X = I N J T X .
Step 2: Run 2SLS ofd on Z with the set of instruments A = (X 1 ; Z 2 ). This yieldŝ where the projection matrix P A = A (A 0 A ) 1 A 0 .
Step 3: Estimate~ 2 " and~ 2 from equation (10) and (11) Step 4: Once the variance components estimates are obtained, the model in Equation (8) is transformed using^ 1=2 as follows: The Hausman-Taylor estimator is a 2SLS estimator of Equation (16) using A HT = X ; X 1 ; Z 1 as instruments, whereX = (I N E T ) X and X 1 = I N J T X 1 . More speci…cally, 4

Case 2-A Random
E¤ects world, where Z 2;i and X 2;it are not correlated with i by construction: where i and it are the same de…ned as in case 1. It is clear that the Z 2;i variable is still correlated with X 11;it (by the term i ) with X 12;it (by the term i ).
The sample sizes (N; T ) are (100; 5) and (500; 10). For each experiment, we perform 1,000 replications. For each replication we estimate the model using FE, RE and HT with and without serial correlation in the disturbances. Table 1 and 2 report the root mean square error (RMSE) of 2 and 2 , the coe¢ cients of the endogenous time-varying and time-invariant variables, respectively. This is done for various values of over the range (0; 0:2; 0:4; 0:6; 0:8; 0:9). Following Baltagi, Bresson and Pirotte (2003), we focus on the coe¢ cients of the endogenous regressors X 2;it and Z 2;i , i.e. 2 and 2 , respectively.
In case 1, the Hausman-Taylor world, as shown in Table 1, the RE estimator su¤ers from endogeneity bias and hence has a much larger RMSE than the FE and HT estimators. This is true for both sample sizes (N; T ) = (100; 5) and (500; 10). The FE estimator performs well as it wipes out the endogeneity caused by correlation of the regressors with the individual e¤ects. However, it does not yield an estimator of 2 . Note that the FE estimator with Cochrane-Orcutt transformation performs worse than the FE estimator ignoring the autocorrelation in the error term for both sample sizes (N; T ) = (100; 5) and (500; 10). The FE-PW estimator yields a lower RMSE than FE-CO and emphasizes the gains from not dropping the N observations of the initial time period. A practice still in use in standard econometrics software today. The HT estimator yields an estimator of 2 , and this has lower RMSE than the RE estimator. Comparing FE, HT with their AR(1) counterparts, we note that the RMSE improves as long as > 0:2.
In case 2, the Random E¤ects world, as shown in Table 2, the RE estimator controlling the autocorrelation in the error term is the e¢ cient estimator by construction. FE and HT have larger RMSE than RE. The FE estimator with Cochrane-Orcutt transformation has larger RMSE than the FE estimator with Prais-Winsten transformation. Comparing RE, FE, HT with their AR(1) counterparts, we note that the RMSE improves as long as > 0:4.

Conclusion
This paper shows how to modify the HT estimator to allow for AR(1) disturbances. Monte Carlo experiments show that this estimator works well in terms of RMSE.