Chapter 6 Covariates

Often we want to explain the variability in our data using covariates or exogenous variables. We may want to do this in order to create forecasts using information from the covariates in time step \(t-1\) or \(t\) to help forecast at time \(t\). Or we may want to understand what causes variability in our data in order to help understand the underlying process.

We can include covariates in the time-varying regression model and the ARIMA models. We cannot include covariates in an exponential smoothing model. That doesn’t make sense as a exponential model is a type of filter of the data not a ‘process’ model.

In this chapter, I show a number of approaches for including covariates in a multivariate regression model (MREG) with temporally independent errors. This is not a time series model per se, but rather a multivariate regression applied to time-ordered data. MREG models with auto-regressive errors and auto-regressive models with covariates will be addressed in a separate chapter.

I illustrate a variety of approaches for developing a set of covariate for a MREG model. The first approach is variable selection, which was the approach used by Stergiou and Christou for their MREG models (6.3). The other approaches are penalized regression (6.4), relative importance metrics (6.5), and orthogonalization (6.6). These approaches all deal with the problem of selecting a set of covariates to include in your model.

Before discussing models with covariates, I will show a variety of approaches for evaluating the collinearity in your covariate set. Collinearity will dramatically affect your inferences concerning the effect of your covariates and needs to be assessed before you begin modeling.