The behavior of forecast errors from a nearly integrated Ar(1) model as both sample size and forecast horizon become large

We develop asymptotic approximations to the distribution of forecast errors from an estimated AR(1) model with no drift when the true process is nearl y 1(1) and both the forecast horizon and the sample size are allowed to inc rease at the same rate, We find that the forecast errors are the sums of tw o components that are asymptotically independent. The first is asymptotical ly normal whereas the second is asymptotically nonnormal. This throws doubt on the suitability of a normal approximation to the forecast error distrib ution. We then perform a Monte Carlo study to quantify further the effects on the forecast errors of sampling variability in the parameter estimates a s we allow both forecast horizon and sample size to increase.