Identifying Heterogeneity in Dynamic Panel Models with Individual Parameter Contribution Regression

Figures & data

FIGURE 1 Path diagram of a bivariate autoregressive and cross-lagged panel model for three waves of data.

FIGURE 2 Demonstration of iterated IPC regression. 1000 individuals were sampled in equal shares from a two-group exponential distribution with group-specific rate parameters ${\lambda }_1=0.5$ and ${\lambda }_2=1.5$. Iterated IPC regression with a dummy variable indicating grouping was used to estimate the group difference in the rate parameter. On the left side, initial and re-estimated IPC regression estimates are shown. Red dots are estimates of ${\lambda }_1$ and blue dots are estimates of the difference ${\lambda }_2-{\lambda }_1$. Dashed lines mark the corresponding maximum likelihood estimates. Clearly, the initial IPC regression estimates are biased. After just two iterations, however, the iterated IPC regression estimates approach the corresponding maximum likelihood estimates. The log-likelihood function is shown on the right side. The iterative reduction of the bias in the IPC regression estimates leads to an increase of the log-likelihood.

TABLE 1 Group-specific Population Parameter Values for the Dynamic Panel Models in Discrete and Continuous Time

Time	$\theta$	Group 1	Group 2	$\theta$	Group 1	Group 2
Discrete	${\beta }_{xx}$	0.700	0.450	${\varphi }_{yx}$	0.300	1.000
	${\beta }_{yx}$	0.000	0.300	${\varphi }_{yy}$	1.000	2.000
	${\beta }_{xy}$	0.000	0.300	${\psi }_{xx}$	0.510	1.145
	${\beta }_{yy}$	0.700	0.450	${\psi }_{yx}$	0.153	0.168
	${\varphi }_{xx}$	1.000	2.000	${\psi }_{yy}$	0.510	1.145
Continuous	$a_{xx}$	– 0.357	– 1.092	${\varphi }_{yx}$	0.300	1.000
	$a_{yx}$	0.000	0.805	${\varphi }_{yy}$	1.000	2.000
	$a_{xy}$	0.000	0.805	$q_{xx}$	0.713	2.760
	$a_{yy}$	– 0.357	– 1.092	$q_{yx}$	0.214	– 1.034
	${\varphi }_{xx}$	1.000	2.000	$q_{yy}$	0.713	2.760

FIGURE 3 Boxplots of the bias of the IPC regression estimates for the discrete-time dynamic panel model. Red: vanilla IPC regression, blue: iterated IPC regression.

FIGURE 4 Boxplots of the bias of the IPC regression estimates for the continuous-time dynamic panel model. Red: vanilla IPC regression, blue: iterated IPC regression.

TABLE 2 Population Parameter Values for the Dynamic Panel Models Used in Simulation II.

Discrete time				Continuous time
$\theta$	Value	$\theta$	Value	$\theta$	Value	$\theta$	Value
${\beta }_{xx}$	0.500	${\varphi }_{yx}$	0.300	$a_{xx}$	– 0.780	${\varphi }_{yx}$	0.300
${\beta }_{yx}$	0.200/0.300	${\varphi }_{yy}$	1.000	$a_{yx}$	0.424/0.546	${\varphi }_{yy}$	0.300
${\beta }_{xy}$	0.200	${\psi }_{xx}$	0.650	$a_{xy}$	0.424	$q_{xx}$	1.306
${\beta }_{yy}$	0.500	${\psi }_{yx}$	0.013	$a_{yy}$	– 0.780	$q_{yx}$	– 0.379
${\varphi }_{xx}$	1.000	${\psi }_{yy}$	0.650	${\varphi }_{xx}$	1.000	$q_{yy}$	1.306

Note. That the cross-lagged effects ${\beta }_{yx}$ and $a_{yx}$ differ between the two groups.

FIGURE 5 Power to detect that the population group difference in the cross-lagged effect ${\beta }_{yx}$ of the discrete-time model is non-zero. Black crosses: MGSEM, red squares: vanilla IPC regression, blue pluses: iterated IPC regression.

FIGURE 6 Power to detect that the population group difference in the drift parameter $a_{yx}$ of the continuous-time model is non-zero. Black crosses: MGSEM, red squares: vanilla IPC regression, blue pluses: iterated IPC regression.

TABLE 3 Proportions of Type I Errors for the Parameters Estimates of the Discrete-time Dynamic Panel Model

	Vanilla IPC			Iterated IPC
Group size	$\beta$	$\varphi$	$\psi$	$\beta$	$\varphi$	$\psi$
100	5.483	5.047	5.353	6.097	5.047	5.437
125	5.167	5.163	5.133	5.663	5.163	5.277
150	5.037	5.143	5.010	5.500	5.143	5.207
175	5.173	4.940	5.150	5.600	4.940	5.090
200	5.093	4.900	4.850	5.443	4.900	4.983

TABLE 4 Proportions of Type I Errors for the Parameters Estimates of the Continuous-time Dynamic Panel Model

	Vanilla IPC			Iterated IPC
Group size	$a$	$\varphi$	$q$	$a$	$\varphi$	$q$
100	5.333	5.083	5.353	5.250	5.083	4.477
125	5.103	5.007	5.263	5.183	5.007	4.620
150	5.207	5.090	5.143	5.247	5.090	4.657
175	5.077	5.193	4.993	5.157	5.193	4.580
200	4.810	5.207	4.803	4.920	5.207	4.463