Cross-Lagged and Dynamic Panel Models

Summary

The cross-lagged panel model (CLPM) and dynamic panel model (DPM) both target lagged reciprocal causal effects in longitudinal data. The CLPM does not control for stable time-invariant confounders (traits), causing its cross-lagged paths to absorb between-person trait variance. The DPM (including the random intercept CLPM) extends CLPM to control for these, combining advantages of FE and CLPM. Both models assume no contemporaneous causal effects — an assumption often violated in psychological data.

Cross-Lagged Panel Model (CLPM)

Definition: CLPM Structure (Box 2)

The CLPM models reciprocal lagged effects between $X$ and $Y$:

• $X_{t-1}\to Y_t$ (cross-lagged: past X predicts current Y)
• $Y_{t-1}\to X_t$ (cross-lagged: past Y predicts current X)
• $X_{t-1}\to X_t$ (autoregressive path)
• $Y_{t-1}\to Y_t$ (autoregressive path)
• Unobserved $U$ may confound $X_t$ and $Y_t$

The CLPM targets Granger causality: $X$ Granger-causes $Y$ if past $X$ predicts current $Y$ controlling for past $Y$. Granger causality implies predictability, and implies causation only when additional assumptions are met.

CLPM Bias from Stable Trait Confounders

CLPM Bias (Box 2, Fig. 2b)

If individuals have stable traits $U_X$ and $U_Y$ that persistently influence $X$ and $Y$ respectively (e.g., extroverts are habitually talkative and habitually happier), these create a confounding path:

$$U_X\to X_{t-1}\to Y_t\leftarrow U_Y$$

The CLPM does not control for $U_X$ or $U_Y$. Their influence inflates or deflates estimated cross-lagged paths. The CLPM conflates trait-level between-person associations with genuine lagged within-person causal effects.

Dynamic Panel Model (DPM)

Definition: DPM Structure (Box 3)

The DPM (Lüdtke & Robitzsch 2022; Usami et al. 2019; also implemented as the random intercept CLPM, RI-CLPM — Hamaker et al. 2015) addresses the CLPM’s failure to control for stable traits by:

• Adding person-specific random intercepts (latent means) for both $X$ and $Y$
• These intercepts absorb $U_X$ and $U_Y$, removing stable trait confounding
• Cross-lagged paths then reflect within-person lagged effects, not between-person trait differences

The DPM thus combines: FE model’s control for time-invariant confounders + CLPM’s lagged reciprocal structure.

Comparison of the Three Models

Feature	FE Model	CLPM	DPM / RI-CLPM
Target	Contemporaneous effects	Lagged reciprocal	Lagged reciprocal
Controls time-invariant confounders	Yes	No	Yes
Controls time-varying confounders	No	No	No
Models reciprocal dynamics	No	Yes	Yes
Allows heterogeneous slopes	No	No	No
Key violated assumption	Lagged dynamics	Stable traits confound	Contemporaneous effects

Shared Critical Assumption: No Contemporaneous Effects

All three models assume that X and Y do not causally influence each other at the same time point. In psychology, this is frequently violated:

• Happiness and talkativeness may affect each other within the same day
• Stress and physical symptoms may co-occur simultaneously
• Measurement occasions often span hours or days, during which contemporaneous effects accumulate

If contemporaneous effects exist, the cross-lagged estimates from CLPM/DPM reflect a mixture of lagged and contemporaneous effects, producing biased estimates.

Additional Concerns

1. Time lag misspecification: If the true causal effect operates over a different time scale than the measurement interval, lagged estimates will be attenuated or distorted. High-frequency sampling can overburden participants and may itself interfere with the causal system

2. No heterogeneous slopes: Like FE, neither CLPM nor DPM allows individuals to differ in their within-person $X\to Y$ effect sizes. Population-average cross-lagged paths may not represent any individual’s true dynamics

3. Reciprocal contemporaneous effects create trade-offs: if both $X_t\to Y_t$ and $Y_t\to X_t$, standard model specifications (targeting either contemporaneous or lagged effects) are misspecified

Model Selection Is Not a Substitute for Estimand Specification

Researchers sometimes choose between FE, CLPM, and DPM based on model fit statistics rather than substantive theory. This is a mistake:

• Different models target different estimands
• Choosing post-hoc based on fit constitutes researcher degrees of freedom
• Model fit cannot adjudicate between models targeting different causal quantities

Connections

• Fixed-Effects Model — the contemporaneous version; DPM combines FE with CLPM
• Directed Acyclic Graphs — the causal graphs (Boxes 2b and 3 in the paper) clarify confounding paths
• Estimands in Longitudinal Research — the right model emerges from the estimand, not from model fit
• Cross-lagged effects connect to Carryover Effects and Distributed Lags (Koyck/ADL models in marketing) — same underlying idea of temporal lags in causal effects
• Researcher Degrees of Freedom — post-hoc model choice from the FE/CLPM/DPM menu inflates false positives

See Also

• Fixed-Effects Model — the contemporaneous within-persons approach
• Estimands in Longitudinal Research — choosing the right model starts with the estimand
• Within-Between Persons Causal Inference — the broader context for both models
• Within-Between Persons Distinction - Overview — paper overview