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

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

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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

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

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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

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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

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Cross-lagged panel model

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

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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

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Correction for Measurement Error Specify each concept of interest as a latent variable with multiple indicators: y1 e1e2e3 x11x21x31 y2 e4e5e6 x12x22x32 Specify error covariance structure: d1

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

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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).

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Y1Y1 Y0Y0 w1w1 w0w0 Y2Y2 w2w2 d 21 d 22 d 11 d 12 Correlated Disturbances 2

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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?

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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 cant be too careful in dealing with people?

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Results

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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.

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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

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