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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.

Alternative 2

Alternative 1

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  • $\begingroup$ 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? $\endgroup$ – kjetil b halvorsen Mar 14 at 20:32
  • $\begingroup$ I haven't cross-validated yet but I fully expect the results to be good. The correlation between them is almost 95%. $\endgroup$ – PMM Mar 14 at 21:34
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In general, multicollinearity isn't really an issue if your goal is a predictive model. The predictions remain unbiased if MC is the only "issue" because none of the assumptions needed for unbiasedness and consistency are violated.

In a time series, however I don't have much predictive modeling experience. R-squared will generally be very high until you partial out time, and you ought to be careful as autocorrelated errors will likely be the case and violate some important assumptions for unbiasedness and consistency in ordinary linear regression-- an autoregressive model would be a possible other option.

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  • $\begingroup$ Thanks, LSC. It's the time series prediction with the multicollinearlity for the autoregressive model that I'm unsure about. $\endgroup$ – PMM Mar 18 at 14:31
  • $\begingroup$ The error in the coefficient estimates will start to accumulate as we iterate forward to get the time series predictions. $\endgroup$ – PMM Mar 18 at 14:42
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    $\begingroup$ I would not put any confidence in R square in time series because of spurious regression. To me the best way to access models for time series is AIC or a hold out data set (in the later case if you predict correctly). $\endgroup$ – user54285 Mar 18 at 22:56

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