Simulation vs analytic methods to forecast time series process

Question

Just asking if someone knows why the prediction intervals are quite different when one uses a time series analytic method of estimation versus when one simulates such time series.

For example, I used the forecast package's auto.arima function to get the best fit to my data, say it was an ARIMA(1,1,1), and then, on the one hand, I simulated such process doing around 10 thousand simulations and then calculating the 95% percentile with "quantile" function, and on the other hand, I used R's forecast package to do it. So I realized that these different approaches gave prediction intervals with different width (actually, those related with simulation approach are closer than those obtained with forecast package). The way I simulated such time series process is simulating the parameters as random variables distributed normally with mean equal to its estimated value and standard deviation equal to its related standard error. The "white noise" variables related with the Moving Average (MA) part of the process were simulated as normally distributed with mean zero and variance equal to the variance of the residuals.

Thanks in advance for your help.

Answer 1

Analytic prediction intervals for time series usually ignore the uncertainty in the parameter estimation. If you simulate using only the uncertainty due to the noise process, you should get equivalent results (apart from the randomness due to simulations of course).

If you are simulating the parameters, make sure you take account of the correlation between the parameter estimates as they can be highly correlated and that makes a big difference to the results.