Degrees of Freedom Adjustment for Disturbance Variance Estimators in Dynamic Regression Models

In the classical regression model with fixed regressors the statistic S2, i.e. the sum of squared residuals (SSR) divided by the number of degrees of freedom is an unbiased estimator of the variance of the disturbances. If the model is dynamic and contains lagged-dependent explanatory variables, then the least-squares coefficient estimators are biased in finite samples, and so is S2. By deriving the expectation of the initial terms in an expansion of the expression for SSR in the case of an autoregressive regression model, we prove that the bias in the degrees of freedom adjusted estimator is of smaller order in T, the sample size, than the bias of the unadjusted maximum-likelihood estimator. We also indicate how a further decrease in the bias can be achieved, and what the consequences are for estimating sigma. By simulation insight is provided into the relative numerical magnitude of the bias for various estimators of sigma-squared in some relevant particular cases of this class of model