1. Cross-lagged Panel Models
Patrick Sturgis,
Department of Sociology, University of Surrey

2. Overview Autoregressive model Cross-lagged panel model
Multiple indicator model
Correlated disturbances
Example: membership and social trust
Limitations
Summary

3. Autoregressive (Simplex) Model
Repeated measures of a variable expressed as function of preceding value plus random disturbance:
Where the represents the overtime stability of
This is a univariate autoregressive model

4. Autoregressive (Simplex) Model
Can be represented as a path diagram:
y1 is treated as predetermined

5. Cross-lagged panel model
Univariate simplex model can be extended to the bivariate case
Cross-lagged panel model (Campbell 1960; Campbell and Kenny 1999; Finkel 1995; Marsh and Yeung 1997)
Also called residualised regression, endogenous regressor models, transition models
Often used for examining questions of reciprocal causality

6. Cross-lagged panel model
Each variable in the system is regressed onto its lagged measure and the lagged measure of the other variable(s) of interest.
Can the history of X predict Y, net of the history of Y (Granger causality)?
Generally estimated via ML, though estimators for non-MVN available
SEM framework enables some powerful extensions to the basic model

7. Cross-lagged panel model

8. Cross-lagged Panel Model= =
Structural parameters may be constrained to equality over time

9. Dealing with Measurement Error
All measurements of abstract concepts will contain error.
Error can be stochastic ( ) or systematic
( ) .
Systematic error biases descriptive and causal inferences.
Stochastic error in dependents leaves estimates unbiased but less efficient.
Stochastic error in independents attenuates effect sizes.
SEM framework allows for error correction models via multiple indicators

10. Correction for Measurement Error
Specify each concept of interest as a latent variable with multiple indicators:
Specify error covariance structure: e1 e2 e3
x11
x21
x31 y2 e4 e5 e6
x12
x22
x32 y1 d1

11. Factorial Invariance = =
Constrain same loading to be equal over time

12. Correlated Disturbances 1
The disturbance terms for the same endogenous variable over time are likely to be correlated
Similarly, the disturbance term for the 2 endogenous variables may be correlated at the same time point.
Caused by unobserved variable bias; a third variable, Z, may be causing both Y variables simultaneously.
Failing to consider these parameters can bias stability and cross-lagged estimates (Williams & Posakoff 1989; Anderson & Williams 1992).

13. Correlated Disturbances 2Y1 Y0 w1 w0 Y2 w2
d21
d22

14. Example: Membership and Trust
Civic participation breeds interpersonal trust (Toqueville 1840, Putnam 2000)
Much x-sectional evidence supports this idea
Membership of civic associations highly correlated with trust
But what if ‘trusters’ are more likely to become members?
A virtuous circle of increasing social capital?

15. Data and Measures
Data come from the British Household Panel Study, waves
Analytical sample = those interviews at all waves from (n=4650).
Membership measured as latent variable from 10 membership indicators using 2 parameter IRT models (Li, Savage and Pickles 2004).
Social trust measured using standard single indicator:
“Generally speaking, would you say that most people can be trusted, or that you can’t be too careful in dealing with people?”

16. Results

17. Limitations
The lagged endogenous variable provides some protection against unit heterogeneity
Fixed characteristics of individuals, some of which may be unobserved…
…and correlated with change on Y and W
Thus, the cross-lagged associations could be spurious, caused by unobserved Z
Can add Time Varying and Time Invariant covariates
But this makes model very complex and does not eradicate unit heterogeneity problem
This pm we look at more effective solution.

18. Summary
Cross-lagged model is useful way of addressing issues of reciprocal causality with panel data
SEM framework allows error correction, factorial invariance, correlated disturbances, overall model fit assessment, categorical endogenous variables, missing data, sample weights and complex sample designs, nested model testing
Provides some protection against unit heterogeneity
But still open to possibility that cross-lagged effects are spurious