Variables importance on regression with ARIMA errors model

Question

One way to calculate importance of a variable in a regression, is to get the decrease in $RSS$ when we calculate a model without the variable in question.

Is this valid for a regression with ARIMA errors?

Suppose I want to get three measures:

1. Importance of external regressor
2. Importance of ARIMA coefs
3. Importance of Seasonal ARIMA coefs

Would the decrease in $RSS$ if I compare the original $RSS$ with the $RSS$ the following models represent the measures I list above?

1. $ARIMA(p,d,q)(P,D,Q)$ without external regressor
2. $ARIMA(0,0,0)(P,D,Q)$ with external regressor
3. $ARIMA(p,d,q)(0,0,0)$ with external regressor

Answer 1

What you want to do is to express your model as a pure right-hand side model where the right hand side contains lags of the output series and the contemporaneous and lag effects of each of the input regressors and a possible constant. Simply add up the effects to be able to partition the forecast into these separate factors. I have programmed this task in a commercially available piece of software I am involved with. If you need help in programming this I would be glad to help you.

Additional comments: In my opinion it makes no sense breakout the effect of ARIMA vs SARIMA as the combination of them is synergistic reflecting unspecified stochastic series. The way I would approach the problem is to build a transfer function including any needed lag structure for the known X's and include any necessary deterministic structure identfied via Intervention Detection AND set the error process to be white noise i.e. no ar or ma structure at all. As a second step I would ignore the external regressors and incorprate both any necessary deterministic structure identfied via Intervention Detection AND a suitable ar and ma process reflecting ARIMA and SARIMA. The difference in RSS should be based upon a fixed number of fitted points as the incorporation of ar and ma structure often changes the degrees of freedom.