Prediction intervals in ARIMAX accounting for forecast uncertainty in future $X$? I have a problem with my SPSS software and ARIMAX forecasts. Consider a series $Y$ that depends on a different series $X$, which is not known in advance with certainty, but must be forecasted itself.
I can easily produce point forecasts for $Y$ using forecasts for $X$ from the model for $X$. However, prediction intervals for $Y$ will not be wide enough, since the ARIMAX process implemented in SPSS presupposes that $X$ is known with certainty.
What would be an easily implemented solution for producing prediction intervals for $Y$? Can I use some form of bootstrapping where known forecast errors are resampled? But it would not produce widening PI's when using more steps into future as it happens when using standard PI's calculated by the software.
 A: First off, I took the liberty of editing your question. Mainly, I changed all "confidence intervals" to "prediction intervals". CIs refer to unobservable parameters in our model. PIs refer to future observable realizations. There is a difference here, although this is often confused.
Second, yes, you could resample your $X$ forecast using its predictive density, then plug it into the ARIMAX model, which would give you empirical forecast densities for $Y$. Of course, you should also account for the parameter uncertainty in the models, both for $X$ and $Y$, so you should not only take the model for $X$ as given and only resample from its error distribution, but you should also sample from the estimated distribution for each parameter in the model for $X$ and from its error distribution, then plug the simulated realization into the model for $Y$ and sample from the estimated distributions for the parameters in the model for $Y$. I think that unless you have a very simple model (like ARIMA(0,0,0)), you will indeed find widening prediction intervals this way. Your PIs will likely get a lot bigger than you want ;-)
(And actually, you should also account not only for parameter uncertainty, but also for model uncertainty, since you also "estimated" the particular ARIMA(p,d,q) model you use for $X$ and $Y$.)
Third, ARIMAX is really aimed at situations where you do know future $X$ with certainty, e.g., future interventions. If you need to forecast $X$, and $Y$ depends on $X$, and $X$ possibly even depends on $Y$, then you may want to look at Vector Autoregressive (VAR) models.
