The bias of Ordinary least squares variance estimators when the disturbances follow a stationary first-order autoregressive scheme

The bias of Ordinary Least Squares estimators of the variance of first-order autocorrelated errors and of the variance of the coefficient estimator is investigated for four particular and some typical econometric models. First it is shown that the hypothesis of no autocorrelation is frequently accepted, and hence OLS is applied, even thought in fact the autocorrelation coefficient is high. Then the direction and magnitude of the bias of the OLS variance estimators is analysed. Approximative formulas for the bias of these variance estimators as given by Malinvaud are shown to be exact; analytical and numerical results are presented. These results indicate that in rather typical econometric models OLS underestimates the variance. When the explanatory variables have a smooth evolution, values of the autocorrelation coefficient from 0.4 up to 0.8 cause the real variance of the coefficient estimates to be twice till ten times as large as the estimate obtained by the OLS procedure.