Rehabilitating the Lagged Dependent Variable With Structural Equation Modeling

Figures & data

Figure 1. (a) Catholic school example from Coleman et al. (1982), without LDV. (b) Catholic school example from Coleman et al. (1982), with LDV.

Figure 2. Lagged DV as collider from Morgan and Winship (2014).

Figure 3. Simulated DGP.

Figure 4. Path model representing m1.

Table 1. Results of m1, all observed variables.

			Label	Estimate	Std. Err.	z-Value
Regressions
$Y_2$	$\leftarrow$	$D$	beta	0.396	0.027	14.608
		$Y_1$	rho	0.304	0.028	10.929
		$V$		1.007	0.048	20.934
$Y_1$	$\leftarrow$	$U$	theta	0.431	0.034	12.747
		$V$		1.321	0.033	39.458
$D$	$\leftarrow$	$U$	gamma	0.572	0.033	17.453

$N=1000,$ ${\chi }^2=0.742\mathrm{ }(3),$ $p=0.863.$

Figure 5. Path model representing m2.

Table 2. Results of m2, LDV with V unobserved.

			Label	Estimate	Std. Err.	z-Value
Regressions
$Y_2$	$\leftarrow$	$D$	beta	0.317	0.032	9.850
		$Y_1$	rho	0.745	0.022	34.444
$Y_1$	$\leftarrow$	$U$	theta	0.412	0.054	7.611
$D$	$\leftarrow$	$U$	gamma	0.572	0.033	17.453

$N=1000,$ ${\chi }^2=22.891 (2),$ $p=0.000.$

Figure 6. Path model representing m3.

Figure 7. Historgam, sampling distribution.

Table 3. Results of m3, LDV with V as latent variable.

			Label	Estimate	Std. Err.	z-Value
Latent variables
alpha	$\to$	$Y_1$		1.000
		$Y_2$		1.000
Regressions
$Y_2$	$\leftarrow$	$D$	beta	0.378	0.042	9.006
		$Y_1$	rho	0.343	0.183	1.879
$Y_1$	$\leftarrow$	$Y_0$	rho	0.343	0.183	1.879
Covariances
alpha	$\leftrightarrow$	$Y_0$		0.931	0.328	2.839
$Y_1$	$\leftrightarrow$	$D$		0.196	0.048	4.126

$N=1000,$ ${\chi }^2=0.151\mathrm{ }(1),$ $p=0.697.$

Figure 8. Weight gain and sex example from Lord (1967); Pearl (2016).

Figure 9. Dining hall example from Pearl (2016).

Coleman, J. S., Hoffer, T., & Kilgore, S. (1982). High school achievement: Public. Catholic, and Private Schools Compared. Basic Books.

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Morgan, S. L., & Winship, C. (2014). Counterfactuals and causal inference: Methods and principles for social research. Cambridge University Press. https://doi.org/10.1017/cbo9781107587991

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Lord, F. M. (1967). A paradox in the interpretation of group comparisons. Psychological Bulletin, 68, 304–305. https://doi.org/10.1037/h0025105.

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Pearl, J. (2016). Lord’s paradox revisited—(Oh Lord! Kumbaya!). Journal of Causal Inference, 4. https://doi.org/10.1515/jci-2016-0021

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