# Multicollinearity when modeling regression with ARIMA errors

When we fit regression with ARIMA errors, how do we access multicollinearity problem? If it is an lm model, I can use variance inflation factor (VIF) and if the VIF value is greater than 10 (or 5), I would remove predictors with high VIF values. How do we handle it in regression with ARIMA errors.

library(fpp2)
library(forecast)
fit <- auto.arima(uschange[,"Consumption"], xreg=uschange[,c("Income", "Savings")])
fit


Series: uschange[, "Consumption"]

Regression with ARIMA(0,0,0) errors

Coefficients: intercept Income Savings 0.2227 0.8171 -0.0512 s.e. 0.0361 0.0389 0.0027

sigma^2 estimated as 0.1218: log likelihood=-66.98 AIC=141.96 AICc=142.18 BIC=154.88

Multicollinearity is a property of the predictor variables included in a regression model - it is not a property of the errors associated with this model. A predictor variable is said to be collinear with other predictor variables if it can be approximately expressed as a linear combination of these other predictors.

Whether you fit a linear regression model with independent errors or one with ARMA errors, you would assess multicollinearity among predictor variables included in that model in the same way. For example, if your predictors in either model are $$X_1$$, $$X_2$$ and $$X_3$$ (all assumed to be continuous), you could compute the variance inflation factor for each predictor as follows:

• $$VIF_{1} = 1/(1 - R_{1}^2)$$

• $$VIF_{2} = 1/(1 - R_{2}^2)$$

• $$VIF_{3} = 1/(1 - R_{3}^2)$$

where:

• $$R_{1}^2$$ is the multiple coefficient of determination corresponding to the linear regression model obtained by regressing $$X_1$$ on the predictors $$X_2$$ and $$X_3$$;
• $$R_{2}^2$$ is the multiple coefficient of determination corresponding to the linear regression model obtained by regressing $$X_2$$ on the predictors $$X_1$$ and $$X_3$$;
• $$R_{3}^2$$ is the multiple coefficient of determination corresponding to the linear regression model obtained by regressing $$X_3$$ on the predictors $$X_1$$ and $$X_2$$.

Note that all models used to compute these R-squared values assume independent model errors.

Since R already provides functionality for computing VIFs for predictors in linear regression models with independent errors (e.g., a model relating the outcome variable $$Y$$ to the predictor variables $$X_1$$, $$X_2$$ and $$X_3$$), you can use that same functionality to compute VIFs for predictors in linear regression models with ARMA errors. The functionality in question is the vif() function in the car package.

By the way, your R model output seems to suggest that your model errors could actually be assumed to be independent.