ESTIMATION OF REGRESSION PARAMETERS IN LINEAR REGRESSION MODEL WITH AUTOCORRELATED ERRORS

摘要

A frequent, if not typical, problem in applied Econometrics is the autocorrelated disturbance terms of a linear regression model. We consider the problem of estimation and inference of the regression parameters when the errors are autocorrelated.;For any stochastic model, the general interest is not only to produce the "best" possible estimator of the regression parameter but also to use it to make inference. The finite sample distribution being usually unknown, it has been a common practice in Econometrics to use an estimated asymptotic distribution to make inference in small sample. Obviously, this will give misleading inference unless the estimated asymptotic distribution can well approximate the exact finite sample distribution.;Given the present practice of making inference in the absence of our knowledge of the finite sample distribution of an estimator, if we have to choose an estimator from a set of estimators, it seems one should choose the one for which the estimated asymptotic distribution is closest to its exact distribution.;We examined the five most used estimators of the regression coefficients of the regression model with the error following the first-order autoregressive process, namely Ordinary Least Squares (OLS), Cochrane-Orcutt (CO), Cochrane-Orcutt modified by Prais and Winston (PW), Durbin and Maximum Likelihood (ML) estimators. Adopting the well known measure of distance between two distributions by Kolmogorov and Smirnov as a measure of closeness of the two distributions, we computed the distance between the asymptotic distribution and its small sample estimate for each of these estimators. Due to analytical complexity, we resorted to Monte Carlo study. General conclusion that can be drawn from our study is that OLS should never be preferred, and even though all other estimators are comparable over the entire range of the autocorrelation parameter, PW seems to be preferable. This can be contrasted to the well known conclusion using MSE as the choice criterion that OLS may be preferred when the order of the autocorrelation coefficient does not exceed .30 and for larger values of this coefficient, all other estimators are comparable, but possibly the ML estimator may be preferred.;We also examined the small sample first and second moment properties of these estimators except the ML estimator. Analytical difficulties were the main reasons for not studying the ML estimator. The first and the second moments of the OLS estimators of the regression parameter were easily derived. For the remaining methods, these moments were approximately computed. We found that the CO, PW and Durbin estimators are all unbiased. The computation of the variances of these estimators require numerical integration. Convergence problems which occurred with the Durbin method restricted further our results to the OLS, CO and PW methods only. For these three methods, the variances of both the regression parameter estimators were found to be monotonically increasing function of the true value of the autoregressive parameter. This result contradicts previous studies and general intuition that the variances of the regression parameters should be minimal when the autoregressive coefficient is close to zero. The second important result is that the OLS method is as good or better than any other method when the autoregressive coefficient is small, possibly when its absolute value does not exceed .3. And this agrees with previous studies. But very surprisingly OLS is found to be better than the CO method except possibly for extreme negative values of (rho). Also in agreement with a result from the distance point of view, the PW method seems to be preferable to all other methods.