How to identify and deal with serial correlation in Structural Equation Model (ML-SEM) and the command 'xtdpdml'

[Tobias Albrecht]

Hi everyone,

in Williams et al. (2018, p.296) the authors say:

"Moral-Benito uses two equations to specify his model. They are:

y_it = λy_it−1 + x'_itβ + w'_iδ + α_i + ξ_t + υ_it (t = 1, . . . , T )(i = 1, . . . , N) (1)

where

...

υ_it is the time-varying error term,

and

E(υ_it | y_i^t−1, x_i^t , w_i , α_i) = 0 ∀i, t (2)

&

"The ML–SEM uses the moment restrictions implied by the assumption that there is no serial

correlation in the error terms in (1)." (p. 299)


My questions are:

1. What would be the most appropriate stata command to detect serial correlation in the error terms in (1)?
2. If serial correlation is detected, what are the options?
   • Is the ML-SEM just not an option for the specific data?
   • Can I try to eliminate the serial correlation by, for example, using first-differences of the variables?

Reference:

Williams, Richard, Paul D. Allison, and Enrique Moral-Benito. "Linear dynamic panel-data estimation using maximum likelihood and structural equation modeling." The Stata Journal 18, no. 2 (2018): 293-326.

[Erik Ruzek]

You might consider a different modeling approach, one that is better suited to dealing with residual correlation. The ML-SEM and first differencing approaches deal with mean-level autoregressive (so-called dynamic) effects whereas you are interested in (or worried about) correlation in the residuals. This is has been addressed recently in the SEM world with such models as the random intercept cross lagged panel model, both univariate and multivariate versions. Statalist threads for univariate and multivariate exist. In the univariate thread, you can see how these models can be implemented in Stata's mixed. There are other approaches to modeling serial correlation, including generalized least squares. See here, for instance.

[Tobias Albrecht]

Thank you for the fast reply and for the suggestions!

Could you also tell me which would be the most appropriate command to test for serial correlation in the error term in case of the ML-SEM?

[Erik Ruzek]

ML-SEM could likely accommodate serial correlation, but you would need to create the model yourself in Stata's sem (look at the threads I linked for how to add residual autocorrelation in the sem syntax) or in Mplus. If you were able to do that, you could run a likelihood ratio test (lrtest in Stata) comparing the vanilla ML-SEM to the ML-SEM with residual autocorrelation. A significant test would indicate that you have residual autocorrelation. Perhaps also worth looking at is this Stata Journal article on testing for autocorrelation in fixed effects models since I assume that is the path you will take with the ML-SEM.