Collinearity Issues in Autoregressive Models with Time-Varying Serially Dependent Covariates

Abstract

First-order autoregressive models are popular to assess the temporal dynamics of a univariate process. Researchers often extend these models to include time-varying covariates, such as contextual factors, to investigate how they moderate processes’ dynamics. We demonstrate that doing so has implications for how well one can estimate the autoregressive and covariate effects, as serial dependence in the variables can imply predictor collinearity. This is a noteworthy contribution, since in current practice serial dependence in a time-varying covariate is rarely considered important. We first recapitulate the role of predictor collinearity for estimation precision in an ordinary least squares context, by discussing how it affects estimator variances, covariances and correlations. We then derive a general formula detailing how predictor collinearity in first-order autoregressive models is impacted by serial dependence in the covariate. We provide a simulation study to illustrate the implications of the formula for different types of covariates. The simulation results highlight when the collinearity issue becomes severe enough to hamper interpretation of the effects. We also show that the effect estimates can be biased in small samples (i.e., 50 time points). Implications for study design, the use of time as a predictor, and related model variants are discussed.

KEYWORDS:

• Psychological dynamics
• autoregressive models
• collinearity
• time series analysis

Notes

1 Note however that the intercept is not the same as the mean, unless one mean-centers the ILD before the analysis

2 implying homoskedasticity (Wooldridge, 2012, p. 243)

3 i.e. as the sample size tends to infinity

4 For the models considered, OLS and maximum likelihood estimation will lead to the same results asymptotically (Lütkepohl, 2005). We verified (but do not report here) that applying maximum likelihood estimation to our simulated data in section 4 leads to highly similar results.

5 In case of perfect collinearity, the model is no longer identifiable since the predictors no longer convey any unique information, and their relative effects can no longer be disentangled (Cohen et al., 2013, p.419)

6 We are thus assuming a specific restricted VAR(1) model in which the lagged effect of the outcome on the covariate is restricted to 0.