One Does Not Simply Correct for Serial Dependence

Abstract

Serial dependence is present in most time series data sets collected in psychological research. This paper investigates the implications of various approaches for handling such serial dependence, when one is interested in the linear effect of a time-varying covariate on the time-varying criterion. Specifically, the serial dependence is either neglected, corrected for by specifying autocorrelated residuals, or modeled by including a lagged version of the criterion as an additional predictor. Using both empirical and simulated data, we showcase that the obtained results depend considerably on which approach is selected. We discuss how these differences can be explained by understanding the restrictions imposed under the various approaches. Based on the insight that all three approaches are restricted versions of an autoregressive distributed lag model, we demonstrate that accessible statistical tools, such as information criteria and likelihood-ratio tests can be used to justify a chosen approach empirically.

Keywords:

• Intensive longitudinal data
• psychological dynamics
• serial dependence
• time series analysis

Ethical approval

This manuscript uses data collected during an experimental study. This study was approved by the local ethics committee (Social and Societal Ethics Committee at the KU Leuven; case number G-2020-2772-R2(MIN)).

Disclosure statement

The authors declare the absence of any financial, intellectual, or other conflicts of interest which may have biased any aspect of this manuscript. No potential conflict of interest was reported by the author(s).

Data availability statement

The materials used in this manuscript are available online through the open science framework. These supplementary materials include the scripts used to generate and analyze simulation data, the experimental data, and scripts used to analyze the experimental data.

Notes

1 The term $(1-{\beta }_{Ly}L)$ is a lag polynomial, and if $\left|{\beta }_{Ly}\right|<1,$ it is invertible (Hamilton, 1994, p. 28–29). It is well known that inverting an AR(1) process yields an equivalent infinite order moving average (MA) process: $y_t=v_t{(1-{\beta }_{Ly}L)}^{-1}.$ Here, ${(1-{\beta }_{Ly}L)}^{-1}=(1+({\beta }_{Ly}L)+{({\beta }_{Ly}L)}^2+\dots$), and as such inverting the polynomial corresponds to solving by recursive substitution.

2 Ordinary least squares is the classical approach for estimating the parameters of the ADL, LR, and LCV models. Generalized least squares is the classical method employed for the ACLR (Aitken, 1936; Cochrane & Orcutt, 1949).

3 In general however, the inclusion of irrelevant predictors is known to decrease estimation precision (Wooldridge, 2012, p. 88).

4 These residuals were extracted using the lm() function in R.

5 The terminology “innovation process” is usually used for error processes in time series contexts, we use the error terminology here due to its familiarity for researchers working with linear regression models.

6 In terms of error variances, we get: (12) $$\begin{array}{l}{\sigma }_{ϵ}^2=E({ϵ}_t^2)\\ {\sigma }_{ϵ}^2=E({({\beta }_x{\zeta }_t+v_t)}^2)\\ {\sigma }_{ϵ}^2=E({\beta }_x^2{\zeta }_t^2+2{\beta }_x{\zeta }_t{v}_t+v_t^2)\\ {\sigma }_{ϵ}^2={\beta }_x^{2}E({\zeta }_t^2)+2{\beta }_{x}E({\zeta }_t{v}_t)+E(v_t^2)\\ {\sigma }_{ϵ}^2={\beta }_x^2{\sigma }_{\zeta }^2+{\sigma }_v^2.\end{array}$$(12)

Additional information

Funding

This work was supported by a research fellowship from the German Research Foundation (DFG) awarded to Janne Adolf, by research grants from the Fund for Scientific Research-Flanders (FWO, Project No. G.074319N), the Research Council of KU Leuven [C14/19/054 and iBOF/21/090], awarded to Eva Ceulemans.