I am running a regression in eviews that uses one main x variable and includes AR and MA terms. When I run the Variance Inflation Factor test, high VIF values are produced the AR, MA, and lag terms as well as the main x variable. Are these high VIFs problematic? Although the AR, MA, and lag terms are listed with the x-variables, I don't think they should be treated like them when we talk about VIF. Wouldn't we expect high correlation between the ARMA/lag terms and the x variable? Isn't that the whole point of them? What am I missing here?
Yes, the Variance Inflation Factor (VIF) between lagged terms matters in linear regression analysis. The VIF is used to detect the severity of multicollinearity in the ordinary least square (OLS) regression analysis. Multicollinearity inflates the variance and type II error. It makes the coefficient of a variable consistent but unreliable. VIF measures the number of inflated variances caused by multicollinearity.
No need to check multicollinearity (applying VIF) in such regression. Differently, multicollinearity must exist when model have two same independent variables like in your case.
I presume that FCPIU_US is a price index (or possibly its log). (I may be wrong). If so the variable is probably non-stationary (close to a random walk) and your VIF may arise from the fact that
E[FCPIU_USt+1] = FCPIU_USt
Is FCPIU_USt integrated of order 1 (I(1))? If so you might be better working with its first difference.
Sure, the Variance Inflation Factor (VIF) does matter in linear regression analysis, even between lagged terms.