No answers in contemporary textbooks. Answers : When?: Estimating a dynamic model with positively autocorrelated disturbances. Why?: Let’s start with a simple model y(t)= A + Dy(t-1) + e(t) where A and -1.0< D<1.0 are parameters and the error term e(t) is NIID. Then the approximate bias of OLS(=GLS) is given by -(1+3D)/T derived by Kendall(1954) where T is the sample size. Note that the bias decreases with T, making OLS(=GLS) consistent. Let’s call this approximate bias the dynamic effect. The answer to the question Why? is right in this formula, which shows that the sign of the dynamic effect is negative when D is positive, the normal case in practice, which is assumed in the following. How?: Now let’s introduce positive autocorrelation in the disturbance term. Then, e(t) is positively correlated with y(t-1) and makes OLS inconsistent since the effect of the correlation does not disappear as T goes to infinity. Let’s call this effect the correlation effect. For a finite-sample size, the bias of OLS = the dynamic effect+ the correlation effect. Since the dynamic effect is the bias of GLS, it follows that the bias of OLS is smaller than the absolute value of the bias of GLS so long as the correlation effect is smaller than the two times the absolute value of the dynamic effect. My two published papers (1980, 1990) show by Monte Carlo study that this simple heuristic theory is applicable to a model including an exogenous variable, and also explains the relative performance in terms of mean-square error as well.
