Effects of parameter estimation on prediction densities: A bootstrap approach

We use a bootstrap procedure to study the impact of parameter estimation on prediction densities, focusing on seasonal ARIMA processes with possibly non normal innovations. We compare prediction densities obtained using the Box and Jenkins approach with bootstrap densities which may be constructed either taking into account parameter estimation variability or using parameter estimates as if they were known parameters. By means of Monte Carlo experiments, we show that the average coverage of the intervals is closer to the nominal value when intervals are constructed incorporating parameters uncertainty. The effects of parameter estimation are particularly important for small sample sizes and when the error distribution is not Gaussian. We also analyze the effect of the estimation method on the shape of prediction densities comparing prediction densities constructed when the parameters are estimated by Ordinary Least Squares (OLS) and by Least Absolute Deviations (LAD). We show how, when the error distribution is not Gaussian, the average coverage and length of intervals based on LAD estimates are closer to nominal values than those based on OLS estimates. Finally, the performance of the bootstrap intervals is illustrated with two empirical examples.