Bias in MLE estimates of AR(1)

I have been running some Monte Carlo simulations to see the bias produced by estimating an AR(1) model (x(t)=a+p.x(t-1)+N(0,1), a and p constants) using MLE. The MLE estimates coincide with the OLS estimates, so there is a small sample bias in the estimate for p, and the bias is downwards near p=1. The proof follows quickly from the OLS formula.

In the simulations, there is clear downwards bias for short time periods near p=1, even with large numbers of simulations. For large time periods (1000 or so for quite small differences), the theoretically predicted convergence was observed. Short time periods are not that short - the simulations at 50 time periods gave a clearly biased p parameter. And everything was very sensitive to the starting values of x(0) and a, with higher values tending to increase the estimate of p. For a hundred simulations, 30 periods, a=4, parameter of 0.7, and seed of 8, the mean MLE estimate from all of the simulations was 0.83, with a range of (0.42, 0.95). With the same parameters except a=3, the mean was 0.62 with range (0.19, 0.93). These are the sort of parameter values which occur in economic growth models, so the biases are relevant for reported estimates.