Figures and data in Regression-style models for parameter estimation in dynamic microsimulation: An empirical performance assessment

Abstract

Microsimulation models seek to represent real-world processes and can generate extensive amounts of synthetic data. The parameters that drive the data generation process are often estimated by statistical models, such as linear regression models. There are many models that could be considered for this purpose. We compare six potential models, discuss the assumptions of these models, and perform an empirical assessment that compares synthetic data simulated from these models with observed data. We chose six regression-style models that can be easily implemented in standard statistical software: an ordinary least squares regression model with a lagged dependent variable, two random effects models (with and without an autoregressive order 1within-unit error structure), a fixed effects model, a hybrid model combining features from both fixed and random effects models, and a dynamic panel model estimated with system generalised method of moments. The criterion for good performance was the proximity of fit of simulated data to the observed data on various characteristics. We found evidence of violated assumptions in our data for all the models but found that, for the majority of data characteristics assessed, all the models produced synthetic data that were a reasonable approximation to the observed data, with some models performing better or worse for particular characteristics. We hope more modellers will consider and test the assumptions of models used for parameter estimation and experiment with different model specifications resulting in higher quality microsimulation models and other research applications.

This article contains 4 figures and 33 tables.

Figures

**Figure 1.** Flowchart showing the cross validation method used to calculate the simulated means, observed means, and the mean absolute differences reported in the results.

**Figure 2.** Comparison of the observed reading scores and the reading scores simulated under the six models.

**Figure 3.** Cumulative distribution curves of the observed reading scores and the reading scores simulated under the six models.

**Figure 4.** Comparison of observed and simulated mean reading scores by predictors.

Tables

Table 1. Statistical regression techniques used by other dynamic microsimulation models (Table legend).

OLS regression with an LVD	Random effects	Fixed effects
MINT3: Earnings of retirees who choose to work, earnings of social security beneficiaries (Toder et al., 2002)	PenSim2: Earnings (Emmerson, Reed, & Shephard, 2004)	MINT1: Earnings (Toder et al., 2002)
SVERIGE: Earnings (Rephann & Holm, 2004)	MIDAS: Hours worked, wages (Dekkers et al., 2008)	Income Distribution of the Dutch Elderly: Income (Knoef, Alessie, & Kalwij, 2013)
DYNAMOD-2: Earnings (Bækgaard, 2002)	DYNASIM-3: Hours worked, earnings (Favreault & Smith, 2004)	MINT3: Pre-retirement earnings after age 50 (Toder et al., 2002)
LifePaths: Hours worked (Wolfson, 1995)	SAGE: Earnings (Zaidi et al., 2009)
	MINT3: Non-pension wealth, home equity, non-pension assets (Toder et al., 2002)
	SESIM: Earnings, interest paid, size of loan (Klevmarken & Lindgren, 2008)

Description of approach:	Standard model with an LDV.
Estimation:	Ordinary least squares (OLS)
Assumptions on individual effects and errors:	μ_i = 0, ε_i,t = υ_i,t~iid N(0, σ²), Cov(ε_i,t, x_i,t) = 0
Estimates of main effects of time-invariant predictors:	Estimated and included directly in the simulation
Implementation in the simulation:	Coefficients (γ, α, and β) applied to the corresponding individual values to get predicted mean for each individual. Random number drawn from normal distribution with mean equal to the predicted mean and standard deviation equal to σ.
Other advantages and disadvantages:	Simple to estimate the model and implement in the simulation.

Description of approach:	Random child intercepts. No LDV included (α = 0).
Estimation:	Various options but restricted maximum likelihood (REML) used
Assumptions on individual effects and errors:	$μ_{i} ~ i i d N (0, σ_{b}^{2}), υ_{i, t} ~ i i d N (0, σ^{2}), C o v (μ_{i}, x_{i, t}) = 0, C o v (μ_{i}, υ_{i . t}_= 0$
Estimates of main effects of time-invariant predictors:	Estimated and included directly in the simulation
Implementation in the simulation:	As for OLS-LDV except each child assigned an individual effect which is added to the predicted mean
Other advantages and disadvantages:	Must decide how to include the individual effects and treat them during scenario testing

Description of approach:	Random child intercepts. No LDV included (a = 0). Within-child errors assumed to have an AR(1) structure.
Estimation:	Various options but penalized quasi-likelihood used via the R function glmmPQL.
Assumptions on individual effects and errors:	$υ_{i, t} = ρ υ_{i, t - 1} + ω_{i, t}, μ_{i} ~ i i d N (0, σ_{b}^{2}), ω_{i, t} ~ i i d N (0, σ^{2}), C o v (μ_{i}, x_{i, t}) = 0, C o v (μ_{i}, ω_{i, t}) = 0$
Estimates of main effects of time-invariant predictors:	Estimated and included directly in the simulation
Implementation in the simulation:	As for variance components RE, except that the standard deviation of the simulated values is now a random draw from a normal distribution when mean equal to ρυ_i,t-1 and standard deviation σ. In practice, the error (υ_i,t) was simulated first and then added to the individual fitted value (which includes the individual effect, μ_i). This error was saved so that it could be used in simulating the error for the next year.
Other advantages and disadvantages:	As for variance components RE, plus more programming required to implement AR errors

Description of approach:	Uses only within child variation to estimate fixed/within effects for time-variant predictors. Run as standard regression but include the child IDs as a set of dummy variables or use deviation scores for time-variant variables and exclude time-invariant variables. No LDV included (α = 0).
Estimation:	OLS
Assumptions on individual effects and errors:	μ_i estimated, υ_i,t~iid N(0, σ²), Cov(μ_i, x_i,t) ≠ 0, Cov(μ_i, υ_i,t) = 0
Estimates of main effects of time-invariant predictors:	No estimates of main effects. Can include interactions with time-variant predictors.
Implementation in the simulation:	As for random effects.
Other advantages and disadvantages:	Does not provide estimates of time-invariant variables which are crucial for a fully functioning DMSM.

Description of approach:	Estimates fixed effects for time-variant predictors and crosssectional effects for time-invariant predictors. Each predictor decomposed into between-unit variation, Z_i, (repeated individual-specific means for each time point) and within-unit variation, $x_{i, t}^{}$ (the difference between the raw value and Z_i). Model estimated with the outcome in the original form and including the time-invariant predictors, Z_i, $x_{i, t}^{}$ , and random individual effects. No LDV included.
Estimation:	Restricted Maximum Likelihood (REML)
Assumptions on individual effects and errors:	μ_i~iid N(0, $σ_{b}^{2}$ ), υ_i,t~iid N(0, σ²), by construction Cov(μ_i, $x_{i, t}^{}$ ) = 0 (so is not strictly an assumption), Cov(μ_i, υ_i,t) = 0, Cov(z_i, υ_i,t*) = 0
Estimates of main effects of time-invariant predictors:	Estimated and included directly in the simulation
Implementation in the simulation:	As for random effects. For time-invariant variables, the fixed (‘within’) effects are used; the ‘between’ effects of these variables are not used.
Other advantages and disadvantages:	Provides fixed effects estimates and main effects for time-invariant variables. Standard errors not correct.

Description of LDV included. Uses instrumental variables to deal with the approach:	endogeneity of the LDV and any other predictors. The predictors are transformed (by the first-difference or orthogonal deviations transformation) and then instrumented with lagged levels of the original predictors.
Estimation:	Generalised method of moments (GMM)
Assumptions on individual effects and errors:	Cov(y_i1, υ_i,t) = 0 (initial conditions predetermined), Cov(y_i1, μ_i) ≠ 0, Cov(μ_i, x_i,t) ≠ 0, x_i,t allowed to be correlated with υ_i,t and earlier errors but not subsequent errors.
Estimates of main effects of time-invariant predictors:	No estimates provided
Implementation in the simulation:	Change simulated and added to previous value. Estimates applied to the change in the predictors from last time-period.
Other advantages and disadvantages:	Complex estimator with many options

Description of approach:	As for difference GMM DPM but a second ‘levels’ equation is included to form a system of equations. The variables in the levels equation are instrumented with their own first differences.
Estimation:	Generalised method of moments (GMM)
Assumptions on individual effects and errors:	As for difference GMM, plus Cov(μ_i, Δw_i,t = 0, where w is any instrumenting variable (either x or y)
Estimates of main effects of time-invariant predictors:	Estimated and included directly in the simulation
Implementation in the simulation:	As for OLS regression with an LDV.
Other advantages and disadvantages:	Complex estimator with many options, hard to meet additional assumption

Table 2. Predictors used in the regression models.

Variable	Categories/description
Time-invariant Predictors	Values available only at the child’s birth
Gender	Male; Female
Mother’s education	No formal qualifications; Secondary qualification; Tertiary qualification
Father’s education	No formal qualifications, Secondary qualification; Tertiary qualification
Family’s socio-economic position	Semi-skilled, Unskilled, unemployed; Clerical, technical, skilled; Professional, managerial
	Based on the Elley-Irving scale (Elley & Irving, 1976)
Breast-feeding	Duration in months
Time-variant Predictors	Values available at each year
LDV	The lagged dependent variable; The value of the reading score at the previous year
Child’s age	8 to 13 years
Mother’s hours worked	The average number of hours of paid work performed per week by the mother/mother figure
Home ownership	Owned or mortgaged; Rented
Father’s smoking	The average number of cigarettes smoked per day by the father/father figure

Table 3. Assumptions made for each model. ✓A indicates that the assumption is made by the model. Following this may be either the result of testing for the assumption or a comment (A dash (-) indicates that the test is not relevant).

	Assumption	Test used	OLS-LDV	RE	RE-AR(1)	FE	HYBRID	GMM Dynamic Panel
1	No individual effects: μ_i = 0, ε_i,t = υ_i,t	LR test for non-zero $σ_{b}^{2}$ , p-value based on a mixture of chi-squares (COVTEST in the GLIMMIX procedure, SAS 9.3).	✓ A, LDV in model: x²(1) = 0.00, p = .9999; LDV not in model: x²(1) = 7958, p < .0001	-	-	-	-	-
2	Individual effects independent of within-child errors: Cov(μ_i, υ_i,t = 0	Test for Pearson’s correlation between estimated individual effects^a and residuals	-	✓A, $C o r ({\hat{μ}}_{i}, {\hat{υ}}_{i, t}) = 0.079$ , p < .0001			✓A, $C o r ({\hat{μ}}_{i}, {\hat{υ}}_{i, t}) = 0.079$ , p < .0001
3	Individual effects independent of each other: μ_i~iid	Assumed by virtue of the individuals being randomly selected and from different families	-	✓A	✓A	μ_is eliminated by transformation	✓A	μ_is eliminated by transformation
4	μ_i~N(0.σ_b)	Shapiro-Wilk test on the empirical BLUPs^b of individual effects	-	✓A, W = 0.99, p = .0001, but plotting shows distribution close to normal	✓A, W = 0.99, p < .0001, but plotting shows distribution close to normal		✓A, W = 0.99, p = .0001, but plotting shows distribution close to normal
5	Cross-sectional ‘between’ effects equal to within ‘fixed’ ,, . K = λ effects: *k = λ*	Hausman test	✓A, x²(5) = 2.55, p = .7693	✓A, x²(5) = 2.55, p = .7693	✓A, x²(5) = 2.55, p = .7693	Only λ estimated	λ and κ estimated separately	Only λ estimated
6	Exogenous time-invariant predictors (TIPs): Cov(υ_i,t, x_i,t) = 0, Cov(z_i, υ_i,t) = 0,	C statistic from endogtest in stata’s ivreg2^c	✓A, null of exogeneity not rejected for any TIPs^d. LDV endogenous by definition.	✓A, null of exogeneity rejected for most TIPs^e	✓A, no convenient test – refer to tests for RE model	TIPs not used	✓A, null of exogeneity rejected for half of the TIPs^e	Diff GMM: TIPs not used. System GMM: Assumption does not need to be made
7	Exogenous time-variant predictors (TVPs): Cov(ε_i,t, x_i,t) = 0, Cov(μ_i, $x_{i, t}^{*}$ ) = 0	C statistic from endogtest in stata’s ivreg2 or xtivreg2^c	✓A, null of exogeneity rejected for one TVP^d	✓A, null of exogeneity rejected for two TVPs^e	✓A, no convenient test – refer to tests for RE model	✓A, null of exogeneity not rejected for any TVPs^f	✓A, null of exogeneity rejected for one TVP^e	Assumption does not need to be made^g
8	υ_i,t~iid N(0, σ²): Within-child errors serially uncorrelated	System GMM: Arellano-Bond test for AR in first differences Other models: xtserial in stata (Wooldridge 2002)	✓A, F(1, 1003) = 398.5, p < .0001	✓A, F(1, 1043) = 115.5, p < .0001	Assumes an AR(1) structure^h	✓A, F(1, 1043) = 115.5, p < .0001	✓A, F(1, 1043) = 115.5, p < .0001	Assumption not made but choice of instruments depends on degree of serial correlation. instruments depends on degree of serial correlation. AR(2): z = 4.33, p < .0001. AR(3): z = -1.60, p = .109.
9	υ_i,t~iid N(0, σ²) Within-child errors independent across individuals (no cross-sectional dependence)	Pesaran’s test (Pesaran 2004): xtcsd in stata	-	✓A, statistic = 6.62, p < .0001	✓A, no convenient test – refer to test for RE model	✓A, statistic = 6.45, p < .0001	✓A, no convenient test – refer to tests for RE and FE models	✓A, no convenient test – refer to tests for RE and FE models
10	Validity of moment conditions (validity of instruments) for Difference GMM	Hansen test	-	-	-	-	-	✓A, x²(22) = 24.47, p = .323
11	Validity of additional moment conditions for System GMM	Difference in Hansen test	-	-	-	-	-	✓A, x₂(4)=16.27, p=.003
12	Stationarity of within-child errors	Hadri and Larsson (2005)	Z = -2.80, p = .0051	Z = 19.31, p < .0001	Z = 19.46, p < .0001	Z = 19.34, p < .0001	Z = 19.32, p < .0001	System GMM: Z = -2.80, p = .0051

Estimated individual effects were the empirical BLUPs (best linear unbiased predictors) estimated by SAS

BLUP = best linear unbiased predictor. Empirical BLUPs estimated in SAS by requesting fitted values with and without BLUPS and taking the difference.

The tests utilised additional variables not included in any models as excluded instruments (each tested for exogeneity and relevance). Predictors were tested individually with all other variables in the model treated as exogenous. Each model was checked for identification and weak. Significance was set at the .05 level. Full results of the tests and stata code are available in the online Appendix.

LDV included in the model. cluster option not used (panel data structure ignored).

LDV not included in the model. cluster option used to indicate panel data structure.

LDV not included in the model. cluster option used to indicate panel data structure. fe option used.

Although the exogeneity if variables should be investigated so that the analyst has some indication of what variables need to be instrumented and which do not.

A test for whether the AR(1) structure was defensible compared to an unstructured structure was attempted in SAS but the model with the unstructured model would not converge

Table 4. Mean absolute differences (MAD) (smaller is better) and standardised mean absolute differences (SMAD) (more negative is better) between simulated and observed data.

Data characteristic	OLS – LDV	RE)			RE-AR(1)			FE	HYBRID			SYSTEM GMM
	MAD	SMAD	MAD	SMAD	MAD	SMAD	MAD	SMAD	MAD	SMAD	MAD	SMAD
PRESS statistic^a	0.53	−0.11	0.44	−0.66	0.81	1.59	0.39	−0.99	0.44	−0.64	0.68	0.80
Overall distributions	1.47	−0.97	2.66	0.07	4.70	1.86	2.08	−0.44	2.69	0.10	1.87	−0.62
Means across time	0.50	−1.42	0.69	−0.05	0.70	−0.01	0.73	0.23	0.93	1.66	0.64	−0.41
Child-specific correlations (dynamism)	0.09	−1.45	0.23	0.85	0.24	0.92	0.15	−0.46	0.23	0.82	0.14	−0.68
Child-specific standard deviations	0.76	−1.75	1.07	−0.27	1.38	1.19	1.16	0.14	1.13	0.02	1.27	0.66
Time invariant predictors:
Gender	0.72	−1.51	1.16	0.19	1.15	0.15	1.38	1.02	1.37	0.98	0.90	−0.82
Breast-feeding	1.69	−0.77	1.82	−0.42	2.74	1.97	1.77	−0.57	1.97	−0.04	1.92	−0.17
Father’s education	0.85	−1.64	1.19	−0.15	1.51	1.18	1.38	0.64	1.34	0.47	1.11	−0.51
Mother’s education	0.91	−1.46	1.29	−0.04	1.68	1.43	1.35	0.19	1.45	0.56	1.12	−0.67
Family’s socio-economic status	0.94	−1.49	1.06	−0.72	1.38	1.36	1.21	0.28	1.25	0.56	1.17	0.01
Time-variant predictors:
Home-ownership	1.07	−0.64	1.29	−0.45	1.60	−0.18	1.20	−0.53	1.57	−0.21	4.19	2.01
Mother’s hours worked	4.81	−0.61	4.95	−0.49	7.88	1.86	4.63	−0.75	5.01	−0.44	6.10	0.43
Father’s smoking	5.31	−0.80	5.73	−0.51	8.42	1.36	5.34	−0.78	5.80	−0.46	8.17	1.19
Weighted means
Scheme 1b		−1.11		−0.11		1.11		−0.27		0.30		0.08
Scheme 2c		−1.12		−0.07		1.11		−0.28		0.34		0.02

The PRESS statistic is not a mean absolute difference but a standardised measure of the distance between the observed and predicted values. Smaller values are better.

Time-invariant predictors given a weight of 1/5 each, time-variant predictors given a weight of 1/3 each. Other characteristics each given a weight of 1.

Means by predictor variables given a combined weight of 1 with the time-invariant predictors each given a weight of 0.1 and the time-variant predictors each given a weight of 1/6. Other characteristics each given a weight of 1.

Table 5. Overall weighted mean of the standardised mean absolute differences (using weighting scheme 1^a) by age for each model.

			AGE			MEAN OVER AGE^b
MODEL	9	10	11	12	13
OLS – LDV	−1.11	−1.10	−1.10	−1.06	−1.04	−1.11
RE	−0.08	−0.07	−0.06	−0.11	−0.11	−0.11
RE-AR (1)	1.22	1.18	1.09	0.94	0.74	1.11
FE	−0.16	−0.24	−0.27	−0.30	−0.27	−0.27
Hybrid	0.42	0.34	0.28	0.28	0.24	0.30
System GMM	−0.29	−0.09	0.05	0.25	0.44	0.08

Time-invariant predictors given a weight of 1/5 each, time-variant predictors given a weight of 1/3 each. Other characteristics each given a weight of 1.

The weighted mean of the SMADs that were averaged over age. This is the same value as shown in the second-to-last row of Table 4.

Table 6. Coefficients and standard errors estimated on the full dataset for each of the techniques.

Variable		OLS – LDV	RE	RE-AR(1)	FE	Hybrid	System GMM DPM 2 Step ^a	System GMM DPM 1 Step ^a
Reading score previous centred (LDV)		0.90					0.83	0.82
		(0.01)					(0.04)	(0.04)
Reading score previous centred squared (LDV squared)		−0.00					−0.01	−0.01
		(0.00)					(0.00)	(0.00)
Child’s age centred		−0.33	8.02	7.98	8.01	8.02	0.23	0.27
		(0.09)	(0.05)	(0.10)	(0.05)	(0.05)	(0.41)	(0.41)
Child’s age centred squared		0.13	−0.52	−0.06	−0.51	−0.52	−0.01	−0.02
		(0.04)	(0.03)	(0.03)	(0.03)	(0.03)	(0.10)	(0.10)
Gender (reference: Female)		0.07	−3.78	−3.31		−3.79	0.51	0.49
		(0.21)	(0.98)	(0.75)		(0.98)	(0.34)	(0.35)
Father’s education (reference: No formal education)	Tertiary	1.43	6.78	5.10		6.54	0.68	0.51
		(0.39)	(1.79)	(1.37)		(1.81)	(0.58)	(0.57)
	Secondary	0.98	4.60	3.57		4.56	0.67	0.71
		(0.25)	(1.18)	(0.90)		(1.18)	(0.36)	(0.36)
Mother’s education (reference: No formal education)	Tertiary	1.01	7.02	5.16		6.76	1.18	1.05
		(0.32)	(1.53)	(1.17)		(1.55)	(0.55)	(0.54)
	Secondary	0.42	2.47	1.65		2.31	0.47	0.40
		(0.26)	(1.21)	(0.92)		(1.21)	(0.37)	(0.35)
Family’s socio-economic status (reference: Semi-skilled)	Professional	0.22	4.28	3.37		3.76	0.71	0.22
		(0.39)	(1.79)	(1.37)		(1.82)	(0.72)	(0.77)
	Clerical	0.03	2.43	1.97		2.02	0.18	−0.14
		(0.27)	(1.23)	(0.93)		(1.25)	(0.54)	(0.56)
Breast-feeding	0.07	0.26	0.20		0.25		0.05	0.06
	(0.03)	(0.13)	(0.10)		(0.13)		(0.03)	(0.03)
	0.00	0.01	0.00	0.01	(D, M): 0.01, 0.01		−0.02	0.01
Mother’s hours worked	(0.01)	(0.01)	(0.01)	(0.01)	(0.01), (0.04)		(0.04)	(0.04)
Father’s smoking	0.00	−0.03	−0.01	−0.03	(D, M): -0.02, -0.09		−0.16	−0.24
	(0.01)	(0.02)	(0.02)	(0.02)	(0.02), (0.07)		(0.12)	(0.11)
Home-ownership (reference: owned/mortgaged)	−0.72	−1.48	−1.15	−1.30	(D, M): -1.31, -3.75		3.66	2.29
	(0.30)	(0.43)	(0.42)	(0.44)	(0.45), (1.62)		(2.98)	(3.18)

D: For the hybrid technique, estimate of the within / fixed effect calculated from the deviation variable.

M: For the hybrid technique, estimate of the between/cross-sectional effect calculated from the mean variable.

Windmeijer corrected standard errors are displayed

Table 1. Mean absolute differences (MAD) (smaller is better) and standardised mean absolute differences (SMAD) (more negative is better) between simulated and observed data for age 9.

	OLS-LDV		RE		RE-AR (1)		Hybrid		FE		System GMM DPM
Data Characteristic	MAD	SMAD	MAD	SMAD	MAD	SMAD	MAD	SMAD	MAD	SMAD	MAD	SMAD
PRESS statistic^a	0.26	−0.65	0.35	−0.21	0.79	2.01	0.35	−0.19	0.31	−0.43	0.28	−0.53
Overall distributions	1.50	−0.64	1.86	−0.43	5.96	2.01	2.10	−0.28	2.28	−0.18	1.78	−0.48
Means across time	0.50	−1.42	0.69	−0.05	0.70	−0.01	0.93	1.66	0.73	0.23	0.64	−0.41
Within-child correlations (dynamism)	0.09	−1.45	0.23	0.85	0.24	0.92	0.23	0.82	0.15	−0.46	0.14	−0.68
Within-child standard deviations	0.76	−1.75	1.07	−0.27	1.38	1.19	1.13	0.02	1.16	0.14	1.27	0.66
Time-invariant predictors:
Gender	0.82	−1.06	1.13	0.03	1.28	0.57	1.54	1.47	1.17	0.16	0.79	−1.17
Breast-feading	1.39	−0.71	1.57	−0.35	2.69	1.90	1.90	0.31	1.47	−0.55	1.45	−0.6
Father’s education	0.40	−1.24	1.23	−0.03	1.60	1.09	1.51	0.82	1.41	0.52	0.87	−1.16
Mother’s education	0.90	−1.1	1.28	0.02	1.71	1.30	1.56	0.85	1.30	0.09	0.88	−1.17
Family’s socio-economic status	0.92	−1.17	1.26	−0.02	1.56	1.03	1.58	1.11	1.32	0.19	0.93	−1.14
Time-variant predictors:
Home-ownership	1.02	−1.13	1.34	−0.38	1.40	−0.23	1.75	0.57	1.27	−0.54	2.23	1.71
Mother’s hours worked	3.79	−0.56	4.05	−0.35	7.03	2.03	4.17	−0.26	3.85	−0.51	4.05	−0.35
Father’s smoking	3.56	−0.79	4.39	−0.31	8.32	1.98	4.52	−0.24	3.97	−0.55	4.79	−0.08
Weighted means
Scheme 1^b	−1.11		−0.08		1.22		0.42		−0.16		−0.29	−1.11
Scheme 2^c	−1.14		−0.05		1.22		0.42		−0.15		−0.29	−1.14

The PRESS statistic is not a mean absolute difference but a standardised measure of the distance between the observed and predicted values. Smaller values are better.

Time-invariant predictors given a weight of 1/5 each, time-variant predictors given a weight of 1/3 each. Other characteristics were each given a weight of 1.

Table 2. Mean absolute differences (MAD) (smaller is better) and standardised mean absolute differences (SMAD) (more negative is better) between simulated and observed data for age 10.

	OLS-LDV		RE		RE-AR (1)		Hybrid		FE		System GMM DPM
Data Characteristic	MAD	SMAD	MAD	SMAD	MAD	SMAD	MAD	SMAD	MAD	SMAD	MAD	SMAD
PRESS statistic^a	0.41	−0.31	0.38	−0.48	0.81	1.94	0.39	−0.46	0.32	−0.82	0.49	0.12
Overall distributions	1.46	−0.83	2.77	−0.07	6.25	1.95	2.85	−0.02	1.96	−0.54	2.04	−0.49
Means across time	0.50	−1.42	0.69	−0.05	0.70	−0.01	0.93	1.66	0.64	0.23	0.73	−0.41
Within-child correlations (dynamism)	0.09	−1.45	0.23	0.85	0.24	0.92	0.23	0.82	0.14	−0.46	0.15	−0.68
Within-child standard deviations	0.76	−1.75	1.07	−0.27	1.38	1.19	1.13	0.02	1.27	0.14	1.16	0.66
Time-invariant predictors:
Gender	0.66	−1.26	1.05	0.15	1.11	0.4	1.32	1.14	1.21	0.74	0.69	−1.17
Breast-feading	1.49	−0.88	1.77	−0.26	2.75	1.93	1.94	0.12	1.68	−0.45	1.67	−0.46
Father’s education	0.79	−1.49	1.26	0.03	1.57	1.04	1.44	0.6	1.47	0.72	0.97	−0.91
Mother’s education	0.78	−1.4	1.35	0.2	1.70	1.18	1.54	0.71	1.40	0.33	0.92	−1.01
Family’s socio-economic status	0.78	−1.46	1.06	−0.21	1.35	1.08	1.28	0.75	1.26	0.65	0.92	−0.82
Time-variant predictors:
Home-ownership	0.77	−0.67	0.97	−0.47	1.24	−0.19	1.29	−0.15	0.92	−0.52	3.42	2.00
Mother’s hours worked	4.67	−0.58	4.77	−0.49	7.44	1.88	4.88	−0.39	4.45	−0.78	5.72	0.35
Father’s smoking	4.29	−0.81	4.80	−0.49	8.23	1.71	4.83	−0.47	4.52	−0.67	6.68	0.72
Weighted means
Scheme 1^b		−1.10		−0.07		1.18		0.34		−0.24		−0.09
Scheme 2^c		−1.12		−0.04		1.19		0.36		−0.26		−0.12

The PRESS statistic is not a mean absolute difference but a standardised measure of the distance between the observed and predicted values. Smaller values are better.

Time-invariant predictors given a weight of 1/5 each, time-variant predictors given a weight of 1/3 each. Other characteristics were each given a weight of 1.

Table 3. Mean absolute differences (MAD) (smaller is better) and standardised mean absolute differences (SMAD) (more negative is better) between simulated and observed data for age 11.

	OLS-LDV		RE		RE-AR (1)		Hybrid		FE		System GMM DPM
Data Characteristic	MAD	SMAD	MAD	SMAD	MAD	SMAD	MAD	SMAD	MAD	SMAD	MAD	SMAD
PRESS statistic^a	0.54	−0.01	0.43	−0.65	0.82	1.55	0.43	−0.65	0.36	−1.05	0.69	0.81
Overall distributions	1.87	−0.93	3.70	0.2	6.32	1.81	3.67	0.18	2.37	−0.63	2.37	−0.63
Means across time	0.50	−1.42	0.69	−0.05	0.70	−0.01	0.93	1.66	0.73	0.23	0.64	−0.41
Within-child correlations (dynamism)	0.09	−1.45	0.23	0.85	0.24	0.92	0.23	0.82	0.15	−0.46	0.14	−0.68
Within-child standard deviations	0.76	−1.75	1.07	−0.27	1.38	1.19	1.13	0.02	1.16	0.14	1.27	0.66
Time-invariant predictors:
Gender	0.57	−1.5	1.13	0.27	1.14	0.33	1.26	0.71	1.38	1.11	0.76	−0.92
Breast-feading	1.84	−1	2.14	−0.22	2.99	1.93	2.21	−0.04	2.10	−0.32	2.09	−0.35
Father’s education	0.83	−1.58	1.33	0.07	1.65	1.08	1.40	0.29	1.57	0.85	1.09	−0.71
Mother’s education	0.93	−1.4	1.65	0.3	1.97	1.07	1.72	0.48	1.79	0.63	1.06	−1.08
Family’s socio-economic status	0.86	−1.66	1.06	−0.57	1.38	1.22	1.22	0.31	1.26	0.55	1.19	0.16
Time-variant predictors:
Home-ownership	1.04	−0.79	1.57	−0.4	2.16	0.04	1.78	−0.24	1.34	−0.57	4.71	1.96
Mother’s hours worked	5.17	−0.45	5.06	−0.52	8.79	1.9	5.13	−0.48	4.70	−0.76	6.35	0.31
Father’s smoking	5.42	−0.88	5.90	−0.51	8.22	1.26	5.96	−0.47	5.65	−0.70	8.27	1.30
Weighted means
Scheme 1^b		−1.10		−0.06		1.09		0.28		−0.27		0.05
Scheme 2^c		−1.10		−0.03		1.09		0.34		−0.30		0.01

The PRESS statistic is not a mean absolute difference but a standardised measure of the distance between the observed and predicted values. Smaller values are better.

Time-invariant predictors given a weight of 1/5 each, time-variant predictors given a weight of 1/3 each. Other characteristics were each given a weight of 1.

Table 4. Mean absolute differences (MAD) (smaller is better) and standardised mean absolute differences (SMAD) (more negative is better) between simulated and observed data for age 12.

	OLS-LDV		RE		RE-AR (1)		Hybrid		FE		System GMM DPM
Data Characteristic	MAD	SMAD	MAD	SMAD	MAD	SMAD	MAD	SMAD	MAD	SMAD	MAD	SMAD
PRESS statistic^a	0.65	0.19	0.48	−0.74	0.82	1.05	0.48	−0.73	0.42	−1.06	0.86	1.29
Overall distributions	2.00	−1.1	3.72	0.37	5.23	1.66	3.68	0.33	2.61	−0.58	2.48	−0.69
Means across time	0.50	−1.42	0.69	−0.05	0.70	−0.01	0.93	1.66	0.73	0.23	0.64	−0.41
Within-child correlations (dynamism)	0.09	−1.45	0.23	0.85	0.24	0.92	0.23	0.82	0.15	−0.46	0.14	−0.68
Within-child standard deviations	0.76	−1.75	1.07	−0.27	1.38	1.19	1.13	0.02	1.16	0.14	1.27	0.66
Time-invariant predictors:
Gender	0.78	−1.49	1.18	0.18	1.09	−0.18	1.31	0.76	1.45	1.33	0.99	−0.61
Breast-feading	1.92	−0.62	1.95	−0.52	2.76	1.94	2.06	−0.19	1.88	−0.73	2.17	0.13
Father’s education	0.92	−1.71	1.16	−0.37	1.46	1.25	1.25	0.11	1.33	0.56	1.26	0.16
Mother’s education	1.03	−1.57	1.31	−0.22	1.67	1.53	1.42	0.33	1.38	0.11	1.32	−0.18
Family’s socio-economic status	1.04	−1.09	1.01	−1.32	1.26	0.97	1.17	0.16	1.18	0.27	1.26	1.01
Time-variant predictors:
Home-ownership	1.44	−0.49	1.51	−0.44	1.75	−0.29	1.76	−0.29	1.38	−0.52	5.43	2.03
Mother’s hours worked	5.14	−0.57	5.30	−0.44	8.14	1.82	5.29	−0.44	4.78	−0.85	6.45	0.48
Father’s smoking	7.48	−0.71	7.80	−0.48	9.13	0.51	7.86	−0.43	7.49	−0.71	10.88	1.82
Weighted means
Scheme 1^b		−1.06		−0.11		0.94		0.28		−0.30		0.25
Scheme 2^c		−1.08		−0.05		0.95		0.34		−0.32		0.16

The PRESS statistic is not a mean absolute difference but a standardised measure of the distance between the observed and predicted values. Smaller values are better.

Time-invariant predictors given a weight of 1/5 each, time-variant predictors given a weight of 1/3 each. Other characteristics were each given a weight of 1.

Table 5. Mean absolute differences (MAD) (smaller is better) and standardised mean absolute differences (SMAD) (more negative is better) between simulated and observed data for age 13.

	OLS-LDV		RE		RE-AR (1)		Hybrid		FE		System GMM DPM
Data Characteristic	MAD	SMAD	MAD	SMAD	MAD	SMAD	MAD	SMAD	MAD	SMAD	MAD	SMAD
PRESS statistic^a	0.78	0.25	0.56	−0.75	0.83	0.5	0.56	−0.73	0.52	−0.93	1.08	1.66
Overall distributions	1.98	−1.43	3.82	0.6	4.39	1.23	3.78	0.56	3.20	−0.09	2.49	−0.87
Means across time	0.50	−1.42	0.69	−0.05	0.7	−0.01	0.93	1.66	0.73	0.23	0.64	−0.41
Within-child correlations (dynamism)	0.09	−1.45	0.23	0.85	0.24	0.92	0.23	0.82	0.15	−0.46	0.14	−0.68
Within-child standard deviations	0.76	−1.75	1.07	−0.27	1.38	1.19	1.13	0.02	1.16	0.14	1.27	0.66
Time-invariant predictors:
Gender	0.74	−1.65	1.32	0.22	1.12	−0.42	1.40	0.48	1.68	1.36	1.26	0.01
Breast-feading	1.81	−0.38	1.68	−0.75	2.52	1.67	1.73	−0.61	1.69	−0.72	2.21	0.78
Father’s education	0.86	−1.39	0.98	−0.72	1.27	0.81	1.11	−0.06	1.12	−0.01	1.37	1.37
Mother’s education	0.93	−0.62	0.87	−0.85	1.36	1.08	1.01	−0.27	0.89	−0.78	1.45	1.43
Family’s socio-economic status	1.09	−0.26	0.89	−1.08	1.32	0.74	1.01	−0.57	1.03	−0.48	1.54	1.63
Time-variant predictors:
Home-ownership	1.07	−0.48	1.07	−0.48	1.46	−0.24	1.28	−0.36	1.09	−0.47	5.15	2.03
Mother’s hours worked	5.27	−0.79	5.57	−0.56	7.98	1.3	5.60	−0.54	5.40	−0.69	7.95	1.27
Father’s smoking	5.79	−0.51	5.75	−0.53	8.22	0.7	5.83	−0.5	5.07	−0.88	10.23	1.71
Weighted means
Scheme 1^b		−1.06		−0.11		0.94		0.28		−0.30		0.25
Scheme 2^c		−1.08		−0.05		0.95		0.34		−0.32		0.16

The PRESS statistic is not a mean absolute difference but a standardised measure of the distance between the observed and predicted values. Smaller values are better.

Time-invariant predictors given a weight of 1/5 each, time-variant predictors given a weight of 1/3 each. Other characteristics were each given a weight of 1.

Table 2. t-values from paired t-tests (df=29) comparing the absolute differences (between the simulated and observed percentiles of the distribution of reading scores). Values greater than 2.46 indicate (unadjusted) significance at the .01 level.

	Percentile
Comparison	10th	25th	50th	75th	90th
OLS-LDV – RE	−31.08	−19.56	−28.23	−24.00	−14.27
OLS-LDV – RE-AR(1)	−9.90	−18.06	−17.57	−8.52	−6.81
OLS-LDV – Hybrid	−2.11	0.11	2.12	−1.35	−0.61
OLS-LDV – System GMM DPM	1.45	−10.74	−22.55	−7.51	−8.31
OLS-LDV – FE	−29.93	−9.64	−6.25	−29.91	−13.54
RE – RE-AR(1)	−2.63	−2.30	1.66	1.75	−3.02
RE – Hybrid	2.41	9.73	25.19	8.85	2.46
RE – System GMM DPM	11.01	−0.14	12.25	9.27	−2.54
RE – FE	20.03	9.72	3.66	31.28	10.41
RE-AR(1) – Hybrid	14.97	13.80	27.01	22.03	10.03
RE-AR(1) – System GMM DPM	30.17	5.88	14.38	26.04	10.18
RE-AR(1) – FE	3.34	10.18	16.87	6.51	3.00
Hybrid – System GMM DPM	11.02	0.92	5.88	5.89	0.31
Hybrid – FE	2.72	−8.57	−18.00	−4.00	−3.29
System GMM DPM – FE	−31.08	−19.56	−28.23	−24.00	−14.27

Table 2. t-values from paired t-tests (df=29) comparing the absolute differences (between the simulated and observed means of reading scores). Values greater than 2.46 indicate (unadjusted) significance at the .01 level.

	Age
Comparison	9	10	11	12	13
OLS-LDV – RE	−1.84	−1.95	−3.48	−0.90	−1.83
OLS-LDV – RE-AR(1)	−1.85	−1.65	−3.75	−1.04	−2.31
OLS-LDV – Hybrid	−3.39	−3.59	−4.38	−3.43	−3.07
OLS-LDV – System GMM DPM	1.80	0.21	−2.02	−1.49	−2.69
OLS-LDV – FE	−1.53	−2.34	−3.70	−1.38	−2.05
RE – RE-AR(1)	−0.05	1.98	−1.10	−0.28	−0.74
RE – Hybrid	−3.47	−2.94	−1.75	−2.64	−1.98
RE – System GMM DPM	2.16	2.12	1.38	−0.77	−1.75
RE – FE	0.96	−0.75	−1.87	−1.57	−1.15
RE-AR(1) – Hybrid	−3.43	−3.13	−1.32	−2.44	−1.72
RE-AR(1) – System GMM DPM	2.18	1.80	1.68	−0.71	−1.56
RE-AR(1) – FE	0.91	−1.55	−1.12	−1.07	−0.38
Hybrid – System GMM DPM	3.65	3.63	2.56	1.05	−0.76
Hybrid – FE	3.54	2.09	0.78	1.79	1.16
System GMM DPM – FE	−1.85	−2.38	−2.02	0.29	1.52

Table 3. t-values from paired t-tests (df = 29) comparing the absolute differences (between the simulated and observed percentiles of the distribution of child-specific correlations between current and lagged reading scores). Values greater than 2.46 indicate (unadjusted) significance at the .01 level.

	Percentile
Comparison	10th	25th	50th	75th	90th
OLS-LDV – RE	−37.88	−82.01	−93.94	−89.52	−83.17
OLS-LDV – RE-AR(1)	−30.27	−44.09	−55.95	−62.67	−61.41
OLS-LDV – Hybrid	−40.68	−88.06	−96.87	−87.17	−78.17
OLS-LDV – System GMM DPM	−14.53	−13.64	−13.83	−13.49	−12.42
OLS-LDV – FE	−14.39	−83.70	−111.84	−104.30	−95.05
RE – RE-AR(1)	−14.68	0.23	22.05	40.07	47.60
RE – Hybrid	2.32	2.40	2.45	2.49	2.21
RE – System GMM DPM	3.64	33.06	65.29	75.15	72.97
RE – FE	52.68	51.85	49.02	46.74	44.81
RE-AR(1) – Hybrid	14.35	0.36	−18.04	−30.84	−34.43
RE-AR(1) – System GMM DPM	10.04	25.17	40.75	48.78	49.55
RE-AR(1) – FE	34.24	24.66	14.86	4.77	−2.32
Hybrid – System GMM DPM	3.50	34.37	69.56	75.74	71.70
Hybrid – FE	51.39	48.19	44.55	41.19	37.90
System GMM DPM – FE	9.43	−14.40	−47.04	−63.95	−67.09

Table 4. t-values from paired t-tests (df = 29) comparing the absolute differences (between the simulated and observed percentiles of the distribution of within-child standard deviations of reading scores). Values greater than 2.46 indicate (unadjusted) significance at the .01 level.

	Percentile
Comparison	10th	25th	50th	75th	90th
OLS-LDV – RE	−9.99	−7.43	−7.45	10.71	7.70
OLS-LDV – RE-AR(1)	6.32	1.89	−12.79	−22.29	−17.18
OLS-LDV – Hybrid	−10.20	−7.87	−7.97	9.05	8.06
OLS-LDV – System GMM DPM	−8.23	−5.83	−1.88	−11.14	−13.73
OLS-LDV – FE	−9.84	−5.31	−2.92	0.78	−0.01
RE – RE-AR(1)	41.95	42.88	−23.07	−24.19	−15.65
RE – Hybrid	−2.70	−2.81	−2.74	−4.01	0.82
RE – System GMM DPM	3.10	2.52	6.10	−16.29	−15.82
RE – FE	−0.02	9.12	9.48	−5.53	−19.46
RE-AR(1) – Hybrid	−31.43	−22.63	8.32	24.63	16.44
RE-AR(1) – System GMM DPM	−10.40	−6.09	11.37	6.68	4.14
RE-AR(1) – FE	−35.78	−26.65	20.58	9.90	8.05
Hybrid – System GMM DPM	3.54	3.13	6.71	−14.82	−16.49
Hybrid – FE	2.17	7.17	8.74	−3.88	−13.89
System GMM DPM – FE	−3.12	−0.91	−1.21	6.90	6.82

Table 5. t-values from paired t-tests (df = 29) comparing the absolute differences (between the simulated and observed mean reading scores by gender). Values greater than 2.46 indicate (unadjusted) significance at the .01 level.

	Gender
Comparison	Girls	Boyx
OLS-LDV – RE	−3.97	−3.21
OLS-LDV – RE-AR(1)	−3.75	−4.12
OLS-LDV – Hybrid	−4.51	−3.65
OLS-LDV – System GMM DPM	−2.48	−1.64
OLS-LDV – FE	−5.08	−4.03
RE – RE-AR(1)	0.97	−0.91
RE – Hybrid	−2.88	−0.98
RE – System GMM DPM	2.22	1.52
RE – FE	−3.70	−3.81
RE-AR(1) – Hybrid	−2.44	0.07
RE-AR(1) – System GMM DPM	1.71	2.30
RE-AR(1) – FE	−2.16	−0.98
Hybrid – System GMM DPM	3.35	1.85
Hybrid – FE	1.12	−1.13
System GMM DPM – FE	−3.41	−2.84

Table 6. t-values from paired t-tests (df = 29) comparing the absolute differences (between the simulated and observed mean reading scores by duration of breast-feeding). Values greater than 2.46 indicate (unadjusted) significance at the .01 level.

	Breast-feeding Duration (months)
Comparison	0	3	6	9	12
OLS-LDV – RE	−0.35	−2.91	−1.63	−5.11	−0.45
OLS-LDV – RE-AR(1)	−2.99	−4.71	−3.66	−4.12	−3.40
OLS-LDV – Hybrid	−2.15	−2.66	−2.87	−4.09	−1.10
OLS-LDV – System GMM DPM	−2.89	−2.52	−2.01	−4.44	−1.10
OLS-LDV – FE	−0.33	−2.46	−2.96	−4.79	−0.96
RE – RE-AR(1)	−3.08	−3.82	−3.46	−0.95	−2.69
RE – Hybrid	−2.76	−1.43	−2.58	3.03	−1.26
RE – System GMM DPM	−2.07	−0.09	−0.91	0.93	−0.09
RE – FE	−0.04	−0.12	−2.42	0.05	−1.61
RE-AR(1) – Hybrid	0.64	2.70	1.78	1.97	2.44
RE-AR(1) – System GMM DPM	0.52	2.42	1.73	1.61	2.82
RE-AR(1) – FE	2.74	3.00	2.53	0.88	1.88
Hybrid – System GMM DPM	−0.10	0.57	0.43	−0.10	0.49
Hybrid – FE	2.38	1.20	0.87	−2.05	−0.10
System GMM DPM – FE	1.95	0.04	0.01	−0.89	−0.47

Table 7. t-values from paired t-tests (df = 29) comparing the absolute differences (between the simulated and observed mean reading scores by father's education at the child's birth). Values greater than 2.46 indicate (unadjusted) significance at the .01 level.

	Father’s education
Comparison	Tertiary	Secondary	None
OLS-LDV – RE	−3.39	−2.57	−2.37
OLS-LDV – RE-AR(1)	−5.48	−4.60	−2.30
OLS-LDV – Hybrid	−3.19	−3.14	−3.59
OLS-LDV – System GMM DPM	−4.01	−2.57	−1.27
OLS-LDV – FE	−4.96	−2.38	−2.83
RE – RE-AR(1)	−3.49	−3.00	−0.10
RE – Hybrid	−0.38	−0.79	−3.00
RE – System GMM DPM	0.38	0.55	0.65
RE – FE	−4.00	0.17	−1.61
RE-AR(1) – Hybrid	3.59	2.01	−2.10
RE-AR(1) – System GMM DPM	3.24	3.47	0.67
RE-AR(1) – FE	0.26	2.65	−0.59
Hybrid – System GMM DPM	0.55	0.96	2.07
Hybrid – FE	−2.62	0.86	1.97
System GMM DPM – FE	−2.54	−0.43	−1.03

Table 8. t-values from paired t-tests (df = 29) comparing the absolute differences (between the simulated and observed mean reading scores by mother's education at the child's birth). Values greater than 2.46 indicate (unadjusted) significance at the .01 level.

	Mother’s education
Comparison	Tertiary	Secondary	None
OLS-LDV – RE	−6.45	−0.29	−2.34
OLS-LDV – RE-AR(1)	−6.12	−3.75	−3.28
OLS-LDV – Hybrid	−4.41	−2.06	−3.56
OLS-LDV – System GMM DPM	−3.33	−2.50	−0.42
OLS-LDV – FE	−7.30	0.12	−2.20
RE – RE-AR(1)	−2.01	−4.99	−1.28
RE – Hybrid	0.79	−2.13	−2.76
RE – System GMM DPM	2.81	−1.62	1.36
RE – FE	−3.19	0.97	0.06
RE-AR(1) – Hybrid	2.57	2.54	−1.27
RE-AR(1) – System GMM DPM	5.07	2.15	1.95
RE-AR(1) – FE	0.66	4.18	1.06
Hybrid – System GMM DPM	2.17	−0.06	2.67
Hybrid – FE	−2.45	2.72	2.64
System GMM DPM – FE	−3.92	2.02	−1.34

Table 9. t-values from paired t-tests (df = 29) comparing the absolute differences (between the simulated and observed mean reading scores by family's socio-economic status at the child's birth). Values greater than 2.46 indicate (unadjusted) significance at the .01 level.

	Socio-economic status
Comparison	Professional	Clerical	Semi-skilled
OLS-LDV – RE	−2.90	−0.90	0.34
OLS-LDV – RE-AR(1)	−4.91	−2.14	−2.62
OLS-LDV – Hybrid	−2.41	−2.75	−1.37
OLS-LDV – System GMM DPM	−3.18	−1.14	−1.70
OLS-LDV – FE	−4.24	−1.22	0.09
RE – RE-AR(1)	−2.34	−2.12	−3.34
RE – Hybrid	0.00	−3.11	−2.46
RE – System GMM DPM	0.13	0.16	−1.90
RE – FE	−3.99	−1.24	−0.46
RE-AR(1) – Hybrid	2.31	−1.35	1.06
RE-AR(1) – System GMM DPM	2.46	1.29	0.55
RE-AR(1) – FE	−0.30	1.18	2.84
Hybrid – System GMM DPM	0.11	2.10	−0.48
Hybrid – FE	−2.50	2.51	1.83
System GMM DPM – FE	−2.19	−0.52	1.71

Table 10. t-values from paired t-tests (df = 29) comparing the absolute differences (between the simulated and observed mean reading scores by home-ownership status). Values greater than 2.46 indicate (unadjusted) significance at the .01 level.

	Home-ownership status
Comparison	Owned	Rented
OLS-LDV – RE	−4.37	−0.56
OLS-LDV – RE-AR(1)	−4.74	−3.13
OLS-LDV – Hybrid	−4.72	−1.70
OLS-LDV – System GMM DPM	−4.57	−7.86
OLS-LDV – FE	−3.93	0.52
RE – RE-AR(1)	−1.13	−4.04
RE – Hybrid	−2.20	−2.74
RE – System GMM DPM	−2.40	−7.80
RE – FE	−0.59	2.66
RE-AR(1) – Hybrid	−1.83	1.12
RE-AR(1) – System GMM DPM	−2.00	−6.59
RE-AR(1) – FE	0.12	4.75
Hybrid – System GMM DPM	−0.81	−7.30
Hybrid – FE	1.63	3.87
System GMM DPM – FE	2.13	8.07

Table 11.

	Mother’s Hours Worked (hours per week)
Comparison	0	16	27	40
OLS-LDV – RE	−3.05	−4.92	4.61	−2.85
OLS-LDV – RE-AR(1)	−4.95	−8.76	1.44	−6.15
OLS-LDV – Hybrid	−4.15	−3.22	4.51	−3.18
OLS-LDV – System GMM DPM	−4.11	−4.30	−1.35	−4.52
OLS-LDV – FE	−2.78	−4.31	3.26	−2.57
RE – RE-AR(1)	−4.54	−7.13	−5.52	−3.86
RE – Hybrid	−3.32	1.02	−0.95	−1.29
RE – System GMM DPM	−2.10	0.47	−5.83	−2.89
RE – FE	0.60	−0.90	−2.07	0.23
RE-AR(1) – Hybrid	1.63	7.43	5.58	1.87
RE-AR(1) – System GMM DPM	0.31	6.53	−2.63	−1.68
RE-AR(1) – FE	3.56	6.24	3.37	3.26
Hybrid – System GMM DPM	−0.50	−0.03	−5.73	−2.22
Hybrid – FE	2.70	−1.94	−1.73	1.34
System GMM DPM – FE	2.35	−0.82	4.07	2.86

Table 12.

	Father’s Smoking (cigarettes per day)
Comparison	0	4	20
OLS-LDV – RE	−3.20	−5.84	−0.03
OLS-LDV – RE-AR(1)	−3.62	−6.38	−0.18
OLS-LDV – Hybrid	−4.33	−5.50	−1.36
OLS-LDV – System GMM DPM	−6.99	−1.60	−6.97
OLS-LDV – FE	−3.02	−3.51	−1.91
RE – RE-AR(1)	−1.42	−5.41	−0.24
RE – Hybrid	−2.69	−0.63	−2.72
RE – System GMM DPM	−4.60	1.81	−7.37
RE – FE	−1.06	0.51	−2.47
RE-AR(1) – Hybrid	−2.03	5.17	−1.46
RE-AR(1) – System GMM DPM	−4.47	5.56	−7.67
RE-AR(1) – FE	0.11	4.33	−1.49
Hybrid – System GMM DPM	−2.77	2.16	−6.86
Hybrid – FE	1.74	0.72	0.15
System GMM DPM – FE	3.88	−1.13	6.95

Table 3. Exogeneity tests for predictors for OLS-LDV, RE, RE-AR(1), and FE models.

	OLS-LDV				RE an RE-AR (1)				FE
	C statistic: x²(1)	p	Under-identification Test Rejected^a	Cragg-Donald Wald F	C statistic: x²(1)	p	Under-identification Test Rejected^a	rk statistic^b	C statistic: x²(1)	p	Under-identification Test Rejected^a	rk statistic^b
Age	6.93	.0085	Yes	13.60	2.67	.1021	Yes	10.47
Age squared	4.83	.0280	No	0.71	4.76	.0291	No	1.49
Time-invariant:
Father’s education – tertiary	0.53	.4273	Yes	5.90	0.05	.8202	No	1.75
Father’s education – secondary	0.01	.9324	Yes	9.85	0.44	.5062	Yes	2.24
Breast-feeding	0.96	.3263	Yes	28.94	0.31	.5777	Yes	7.23
Mother’s education – tertiary	0.67	.4125	Yes	18.96	5.94	.0148	Yes	5.27
Mother’s education – secondary	0.78	.3766	Yes	24.78	4.87	.0273	Yes	5.77
Socio-economic status – professional	0.80	.3702	Yes	36.90	4.30	.0382	Yes	8.30
Socio-economic status -clerical	1.43	.2297	Yes	30.21	6.29	.0122	Yes	5.13
Gender – male	1.94	.1637	Yes	3.20	3.00	.0831	No	0.51
Time-variant:
Father’s smoking	0.00	.9615	Yes	41.87	0.12	.7262	Yes	10.14	0.27	.6052	Yes	12.10
Home-ownership	1.24	.2664	Yes	97.35	4.03	.0447	Yes	16.51	0.24	.6278	Yes	6.96
Mother’s working hours	5.64	.0175	Yes	67.08	10.81	.0012	Yes	26.95	0.25	.6190	Yes	20.41

Kleibergen-Paap rk LM test. Rejection based at the .05 level.

Kleibergen-Paap rk Wald F test.

Table 2. Exogeneity tests for predictors for the hybrid model.

	C statistic: x²(1)	p	Under-identification Test Rejected^a	rk statistic^b
Age	0.80	.3726	Yes	8.03
Age squared	3.19	.0741	No	1.15
Time-invariant:
Father’s education – tertiary	0.00	.9863	No	1.60
Father’s education – secondary	0.37	.5457	Yes	2.25
Breast-feeding	0.44	.5063	Yes	7.08
Mother’s education – tertiary	4.76	.0291	Yes	4.62
Mother’s education – secondary	4.12	.0423	Yes	5.29
Socio-economic status – professional	3.45	.0633	Yes	7.31
Socio-economic status -clerical	5.28	.0216	Yes	4.20
Gender – male	2.48	.1153	No	0.48
Father’s smoking means	1.50	.2200	Yes	5.84
Home-ownership means	3.89	.0486	Yes	14.23
Mother’s working hours means	9.16	.0025	Yes	18.47
Time-variant:
Father’s smoking deviation score	1.95	.1627	Yes	6.36
Home-ownership deviation score	0.57	.4524	Yes	3.34
Mother’s working hours deviation score	7.05	.0079	Yes	11.05

Kleibergen-Paap rk LM test. Rejection based at the .05 level.

Kleibergen-Paap rk Wald F test.

Table 3. Glossary of variable names and labels used in stata code.

Variable name	Variable label
burt	Burt reading score
agecent	Age centred
agecentsq	Age centred squared
gender	Gender
meduc1	Indicator variable for mother having tertiary education at the child’s birth
meduc2	Indicator variable for mother having secondary education at the child’s birth
feduc1	Indicator variable for father having tertiary education at the child’s birth
feduc2	Indicator variable for father having secondary education at the child’s birth
ses1	Indicator variable for family’s socio-economic status to be professional at the child’s birth to
ses2	Indicator variable for family’s socio-economic status to be clerical at the child’s birth to
breast	Duration of breast-feeding
mhrswrk	Average number of hours worked per week by the mother
homeown	Indicator variable for owning the home
fsmoke	Average number of cigarettes smoked per day by the father
ga	Gestational age
single	Indicator variable for child living in a single-parent family
welfare	Indicator variable for child’s family receiving a benefit
fhrswrk	Average number of hours worked per week by the father
chpar	Indicator variable for whether the child experienced a change in parents in the last year
kids	Number of children in the household
chres	Number of changes in residence over the last year
pregalc	Average number of alcoholic drinks consumed per week during pregnancy
bw	Birth weight
single0	Indicator variable for the child being born into a single-parent family
accom	Indicator variable for the child living in a detached house
fage	Father’s age at the child’s birth
pregsmk	Average number of cigarettes smoked per day during pregnancy
overcrowd	Indicator variable for the child living in over-crowded conditions
mage	Mother’s age at the child’s birth
msmoke	Average number of cigarettes smoked per day by the mother
id	Individual child id