Rethinking the Cross Lagged Regression Model

The Cross Lagged Panel Regression Model (CLPM)

• The CLPM is common in many social science applications, particularly in longitudinal, observational research designs

• \(X \rightarrow Y\), \(Y \rightarrow X\), or both?

• The CLPM is frequently used because of its intuitiveness, flexibility, and ease of implementation (Campbell, 1963; Pelz and Andrews, 1966; Rozelle and Thiel, 1969; Abramowitz, 1978; Campbell and Wink, 1979; Iyengar and Kinder, 1987; Jennings and Niemi, 1977; Markus, 1979; Markus and Converse, 1979; Meier, 1979; More recent: Craemer 2008; Claasen 2008; Hatemi, Crabtree and Smith 2019; Hetherington and Globeti 2002; Luttig 2021, Klandermans 2004; Lupu 2015; Goren and Chapp 2017; Layman and Carsey 2002; Putz 2002; Tilley, Neundor and Hobolt 2018; Sibley and Duckitt 2010, among many others)

The CLPM

State and Trait Effects

• The CLPM is a variant of the Autodistributed Lag Model (ADL) (Sims 1980)

• Granger Causality: \(X_t\) Granger causes \(Y\) if \(X_{t-1}\) predicts \(Y_{t}\) better than \(Y_{t-1}\) predicts \(X_t\)

• The CLPM does not differentiate between state and trait effects

• “Unit effects” and stable, individual difference factors, like personality, party, and ideology, over short periods of time (Bakker, Lelkes and Malka 2021; Luttig 2021; Hatemi, Crabtree and Smith 2019)

• Hamaker et al (2015) and A Critique of the Cross-Lagged Regression Model (Hamaker, Kuiper, and Grasman 2015)

CLPMS in Political Science

By Subfield

State and Trait Effects: The RI-CLPM

• In the decade since this publication the number of published political science articles using the method doubled from the prior decade (from 74 to 169)

• Hamaker et al (2015) and the Random Intercept Cross Lagged Panel Model(RI-CLPM) (Hamaker, Kuiper, and Grasman 2015)

• The RI-CLPM is a variant of the CLPM that allows for the separation of state and trait effects, by controlling for stable, unit level variation

The RI-CLPM

The Trait Effect Confounds the State Effect

\[\begin{eqnarray} \begin{bmatrix} x_{it}\\ y_{it} \end{bmatrix} = \begin{bmatrix} \theta_{11} & \theta_{12} \\ \theta_{21} & \theta_{22} \end{bmatrix} \begin{bmatrix} y_{it-1} \\ x_{t-1} \end{bmatrix} + \begin{bmatrix} e_{y,it}\\ e_{x,it} \end{bmatrix} + \begin{bmatrix} \mu_{y,i}\\ \mu_{x,i} \end{bmatrix} \label{eq:ri-clpm_eq} \end{eqnarray}\]

Misspecification

• The CLPM is that its lag structure places a ceiling on how stable a variable can be before the model is mis-specified(Lucas 2023)

• The CLPM assumes that all variables follow a first-order autoregressive process – the correlation between \(x_{t1}\) and \(x_{t3}\) cannot exceed the product of the effects of \(x_{t1}\) on \(x_{t2}\) and \(x_{t2}\) on \(x_{t3}\)

• The correlations between the first two time points, \(r_{x_{t=1},x_{t=1}}\) , then between the first and third – \(r_{x_{t=1},x_{t=3}}\), and so on

• Compare the decay implied by the CLPM with the observed decay, from the test-retest correlations, over time

Misspecification

No Trait Effect

Note: Current realizations \(x\) and \(y\) are a function of autoregressive and cross lagged effects. Cross Lagged Effects: \(\theta_{12} = 0.\) and \(\theta_{2,1} = 0\) Autoregressive Effects: \(\theta_{11}\) from 0 to 1, and fix \(\theta_{22} = 0.5\). \(var(x_{t}\), \(y_t) = 1\), \(N = 5,000\), \(1,000\) replications.

Trait Effects

Note: We vary the degree of “between-unit” variation in both \(x\) and \(y\), from 0.5 to 1.5. We fix the cross lagged effects at 0, \(\theta_{12} = 0\) and \(\theta_{2,1} = 0\), and the autoregressive effects are fixed at 0.5, \(\theta_{11}\), \(\theta_{22} = 0.5\). The “within-subject” wave variance is fixed at 0.5, sample size is \(N = 5,000\).

Ideology and Racial Resentment

• Data: The 2011, 2016, and 2020 Voter Study Group

• 5-point ideology and three item Racial Resentment scale

• The CLPM and RI-CLPM reach different conclusions

The CLPM

The RI-CLPM

Practical Considerations

• The RI-CLPM is difficult to estimate when the vast majority of the variance is “trait” variation.

State-Trait-Occasion (STO)

• The estimates of the CLPM are biased in the presence of time-invariant confounding.
• The State-Trait-Occasion Model (STO) (Cole, Martin, and Steiger 2005).

Variance Estimates and The STO Model

Concluding Remarks

• The CLPM has been widely used in published research in political science

• Confounding due to stable trait variance and unmodeled measurement error lead to spurious estimates of the autoregressive and cross-lagged parameters

• The RI-CLPM is a better alternative to the CLPM, but it is often difficult to estimate in many of the substantive examples that are common in published research, due to excessive “trait” variation

• Alternatives: The Continuous Time Model, the STO model, and dynamic panel models

Supplementary Material: Monte Carlo Results

Note: Each model is tested under the same data generating process in which we vary the “trait level effect,” the random intercept variance. \(N = 5,000\).

Supplementary Material: Monte Carlo Results

Note: Each model is tested under the same data generating process in which we vary the “trait level effect,” the random intercept variance. \(N = 5,000\).