# Excellent model fit but high VIF

I want to use a predictive model for a time series variable M that is related to an other variable X. I can generate independent scenarios for X and I need to generate corresponding values for M.

Alternative 1: Linear regression

Sample size is over 500. $$R^2$$ is 94% and SE are small, giving highly significant results. First row below are the coefficient estimates, second are the SE.

0.741393543 2.282116981

0.009401482 0.02483862

Alternative 2: autoregressive model

Fit an autoregressive model with X as an exogenous variable:

$$M_{(n+1)} = a M_n +b X_{(n+1)} + c$$

In-sample fit is excellent ($$R^2$$ over 98%) and SE are good. First row below gives the coefficients, second the SE.

0.759621998 0.185402072 0.529660298

0.018255516 0.014094594 0.043751245

However $$M_n$$ and $$X_{(n+1)}$$ are highly collinear (see Alternative 1!). They have a VIF near 20.

Visually, the in-sample fit for Alternative 2 is striking and it's tempting to use this, however the size of the VIF is giving me pause, even though the SE are actually quite good (the very high $$R^2$$ is helping here).

Using Alternative 1 gives coefficient estimates that are much tighter and that should be more robust as they get reestimated over time. Nevertheless, Alternative 2 seems like a solid model despite the high VIF.

What are the risks of using Alternative 2?

Below are the in-sample fits. Top is Alternative 2.

• Did you try some form of cross validation? I wouldn't look to much to the VIF, the sample is what it is and if predictions and se are good, so what? What is the correlation between $M_n$ and $X_{n+1}$? Can you show us a plot? – kjetil b halvorsen Mar 14 at 20:32
• I haven't cross-validated yet but I fully expect the results to be good. The correlation between them is almost 95%. – PMM Mar 14 at 21:34