When testing a hypothesis, should I keep an insignificant lag in ARMA-GARCH model? I am trying to estimate ARMA-GARCH model for my stock returns time series.
I have estimated ARMA model for my series, and found that there exists ARCH, so added GARCH(1,1) term. However I now find previously significant ARMA coefficients being insignificant.
In this case, should I remove the insignificant ARMA terms? I am reluctant in doing so as I read from a book that it is not wise to remove ARMA terms judging from their significance.
Also, as I am not performing any predictions with this model (only using it as a normal return model for event study), I thought it would be unnecessary to do so?
 A: This boils down to the general question of model selection with the goal of explanatory modelling and statistical inference. Indeed, statistical significance may not be a sound approach to feature selection as per Rob J. Hyndman's blog post "Statistical tests for variable selection" and Frank Harrell's "Regression Modelling Strategies" referenced therein (although note that Hyndman is more interested in forecasting than explanatory modelling).
In practice, I would select the model from a reasonable pool of candidates (here probably comprising the model with the insignificant coefficient, the model without it, and perhaps some other models) using information criteria such as AIC or BIC. But I would keep in mind that inference from this model will be conditional on the selected model being the "true" model, which is an uncomfortable qualification (see answers -- if there are any -- to my question "Statistical inference under model misspecification" and check out threads referenced therein; this is a basic and ubiquitous problem in applied modelling that I do not know a good solution for).
