The Box-Jenkins model selection procedure in time series analysis begins by looking at the autocorrelation and partial autocorrelation functions of the series. These plots can suggest the appropriate $p$ and $q$ in an ARMA$(p,q)$ model. The procedure continues by asking the user to apply the AIC/BIC criteria to select the most parsimonious model among those that produce a model with a white noise error term.
I was wondering how these steps of visual inspection and criterion-based model selection impact the estimated standard errors of the final model. I know that many search procedures in a cross-sectional domain can bias standard errors downward, for example.
On the first step, how does selecting the appropriate number of lags by looking at the data (ACF/PACF) impact the standard errors for time series models?
I would guess that selecting the model based upon AIC/BIC scores would have an impact analogous to that for cross-sectional methods. I actually don't know much about this area either, so any comments would be appreciated on this point as well.
Lastly, if you wrote down the precise criterion used for each step, could you bootstrap the entire process to estimate the standard errors and eliminate these concerns?
Best Answer
Any model selection procedure will affect the standard errors and this is hardly ever accounted for. For example, prediction intervals are computed conditionally on the estimated model and the parameter estimation and model selection are usually ignored.
It should be possible to bootstrap the whole procedure in order to estimate the effect of the model selection process. But remember that time series bootstrapping is trickier than normal bootstrapping because you have to preserve the serial correlation. The block bootstrap is one possible approach although it loses some serial correlation due to the block structure.