Figures and data in Imputing individual effects in dynamic microsimulation models an application to household formation and labour market participation in Italy

Abstract

Dynamic microsimulation modelling involves two stages: estimation and forecasting. Unobserved heterogeneity is often considered in estimation, but not in forecasting, beyond trivial cases. Non-trivial cases involve individuals that enter the simulation with a history of previous outcomes. We show that the simple solutions of attributing to these individuals a null effect or a random draw from the estimated unconditional distributions lead to biased forecasts, which are often worse than those obtained neglecting unobserved heterogeneity altogether. We then present a first implementation of the Rank method, a new algorithm for assigning individual effects to the simulation sample. Out-of-sample validation of our model shows that use of the Rank method significantly improves the quality of the forecasts.

This article contains 8 figures and 9 tables.

Figures

**Figure 1.** Distribution of the true and imputed (via the Rank method) individual effects. Parameterization: x~N(−0.5,2),β = β = 1, α~N(0,1),η~N(0,1).

**Figure 2.** Distribution of the true and imputed (via the Rank method) individual effects, first quintile of x. Parameterization: x~N(−0.5,2),β = β = 1, α~N(0,1),η~N(0,1).

**Figure 3.** Probability of graduating, high school (left panel) and university (right panel).

**Figure 4.** Structure of the microsimulation model.

**Figure 6.** The Household formation module (females only).

**Figure 8.** Projected participation rates, individuals aged 17–54.

Tables

Table 1. Application of the Rank method. It is assumed β = β.

	(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
id	y	$x' \hat{β}$	$\hat{y}$	$y - \hat{y}$	$r (y - \hat{y})$	$\tilde{α}$	$\tilde{η}$	$\tilde{ε} = \tilde{α} + \tilde{η}$	$r (\tilde{ε})$	$\tilde{y}$
1	1	−0.49	0	1	1	−1.43	2.01	0.58	1	1
9	1	−0.05	0	1	2	2.36	−2.12	0.24	2	1
8	1	−0.39	0	1	3	0.22	−0.10	0.12	3	0
5	1	−1.38	0	1	4	2.02	−2.45	−0.43	4	0
6	1	−0.77	0	1	5	−0.76	0.16	−0.6	5	0
10	1	0.21	1	0	6	−1.35	0.27	−1.08	6	0
1	0	−2.79	0	0	7	0.29	−1.45	−1.16	7	0
2	0	−0.5	0	0	8	−1.87	0.06	−1.81	8	0
3	0	−0.37	0	0	9	−0.52	−1.85	−2.37	9	0
4	0	0.36	1	−1	10	0.79	−3.33	−2.54	10	0

Table 2. Participation rates by level of education, 2010.

	Italy		EU15
	Men	Women	Men	Women
			%
Low education	64.5	32.6	67.3	46.4
Medium education	80.7	63.1	82.2	71.2
High education	86.5	77.5	90.4	84.0
TOTAL	73.6	51.2	78.8	65.6

Source: our elaboration on Eurostat data

Table 3. Status at age 17.

	In education	Active Males	Unemployed
North	79.8%	17.2%	8.6%
Centre	86.0%	10.4%	15.0%
South	82.3%	12.2% Females	46.9%
North	86.7%	12.1%	19.0%
Centre	86.8%	10.2%	35.0%
South	83.7%	9.9%	55.8%

Source: our elaboration on Italian LFS data (2001).

Table 4. Enrolment.

	Secondary School (Probit)			University (Probit)
	Coef		SE	Coef.		SE
Female	−0.008		0.069	0.113	*	0.055
Age	−11.285	**	2.665	0.691	**	0.214
Age-squared	0.287	**	0.070	−0.016	**	0.004
Centre	0.090		0.098	0.010		0.070
South	0.087		0.081	0.110	**	0.033
Year of birth	0.073	**	0.021	0.036	**	0.014
Constant	−33.024		50.76	−77.856	**	27.901

Source: our elaboration on ECHP 1994–2001 data.

significant at 90% confidence level.

significant at 95% confidence level.

*** significant at 99% confidence level

Table 5. Unemployment and male participation estimates (Table legend).

	Unemployment				Male participation
	Probit		Dynamic RE model		Probit		Dynamic RE model
	Coef.	Robust SE	Coef.	SE	Coef.	Robust SE	Coef.	SE
Lag (participation)	---	---	---	---	1.766 **	0.052	0.799 **	0.077
Lag (unemployment)	1.785 **	0.026	1.123 **	0.039	---	---	---	---
Female	0.356 **	0.022	0.592 **	0.049	---	---	---	---
Age	−0.148 **	0.009	−0.256 **	0.020	0.130 **	0.011	0.326 **	0.026
Age^2	0.002 **	0.0001	0.002 **	0.000	−0.002 **	0.000	−0.004 **	0.000
High education	−0.246 **	0.038	−0.582 **	0.083	0.287 **	0.066	0.341 **	0.115
Medium education	−0.237 **	0.022	−0.368 **	0.045	0.110 **	0.032	0.170 **	0.058
Centre	0.283 **	0.035	0.463 **	0.078	−0.054	0.046	−0.104	0.088
South	0.783 **	0.028	1.508 **	0.065	−0.155 **	0.035	−0.313 **	0.069
Year of birth	---	---	---	---	−0.006	0.007	0.010	0.014
Lag (student)	1.147 **	0.043	0.504 **	0.116	0.205 **	0.061	−0.026	0.135
Unempl.rate	11.354 **	1.585	10.076 **	2.828	---	---	---	---
_cons	−0.167	0.229	1.403 **	0.476	9.376	13.075	−24.420	27.245
σ_α	---		1.256		---		1.071

Source: our elaboration on ECHP 1994–2001 data.

significant at 90% confidence level.

significant at 95% confidence level.

significant at 99% confidence level.

Table 6. Female participation and household formation estimates.

	Female participation				Living in consensual union
	Probit		Dynamic RE model		Probit		Dynamic RE model
	Coef.	Robust SE	Coef.	SE	Coef.	Robust SE	Coef.	SE
Lag (participation)	2.387 **	0.027	1.417 **	0.038	−0.049	0.033	−0.093	0.052
Lag (union)	−0.417 **	0.027	−0.595 **	0.049	3.795 **	0.043	3.010 **	0.075
Lag (children under 3)	−0.159 **	0.032	−0.197 **	0.050	0.347 **	0.098	0.516 **	0.122
Age	0.038 **	0.007	0.087 **	0.015	0.074 **	0.010	0.263 **	0.036
Age2	−0.001 **	0.000	−0.001 **	0.000	−0.001 **	0.000	−0.003 **	0.000
High education	0.808 **	0.047	1.665 **	0.093	−0.001	0.060	−0.170	0.093
Medium education	0.371 **	0.021	0.775 **	0.044	−0.029	0.032	−0.135 *	0.057
Centre	−0.116 **	0.027	−0.355 **	0.064	0.052	0.042	0.145 *	0.073
South	−0.270 **	0.021	−0.738 **	0.052	0.015	0.031	0.003	0.053
Year of birth	0.000	0.004	0.009	0.008	0.002	0.007	−0.026	0.014
Lag(student)	0.549	0.056	0.272 *	0.121	−0.530 **	0.066	−0.706 **	0.090
_cons	−1.326	7.845	−19.025	16.684	−7.667	13.146	44.404	28.285
σ_α	----		1.218		----		0.890

Source: our elaboration on ECHP 1994–2001 data.

significant at 90% confidence level.

significant at 95% confidence level.

significant at 99% confidence level.

Table 7. Birth probability estimates the sample includes only women aged 17–45.

Motherhood	Probit
	Coef.	Robust SE
Lag (children under 3)	−0.571 **	0.050
Lag (no children)	−0.371 **	0.084
Lag (participation)	−0.037	0.046
Lag (unemployment)	−0.087	0.072
Lag (student)	−0.216	0.153
Age	0.287 **	0.043
Age2	−0.006 **	0.001
High education	0.227 **	0.072
Medium education	0.039	0.043
Centre	0.062	0.060
South	0.092	0.073
Fertility rate	11.716 *	5.106
Constant	−4.628 **	0.722

Source: our elaboration on ECHP 1994–2001 data.

significant at 90% confidence level.

significant at 95% confidence level.

significant at 99% confidence level.

Table 8. Validation results: female participation rates (%), individuals aged 17–54.

True data			Simulated data
Dataset			Probit	Null	Rank	Unconditional
ECHP	1994	52.85
	1995	53.22
	1996	53.66
	1997	53.80
	1998	55.52
	1999	55.49
	2000	54.76
	2001	56.24
EUSIC	2005	59.78	61.02	63.41	58.01	57.48
	2006	60.28	61.77	63.83	57.91	57.89
	2007	59.10	61.77	64.26	57.81	58.01
	2008	60.07	62.34	64.06	57.78	57.93
Probability of participate at time t
if not participate at t-1		17.06	14.22	34.69	20.29	20.68
if participate at t-1		91.05	93.79	85.78	89.09	88.41
Probability of not participate at time t
if not participate at t-1		82.94	85.78	65.31	79.71	79.32
if participate at t-1		8.5	6.21	14.22	10.91	11.59

Table 9. Coefficients of lagged endogenous variables (standard errors in parenthesis), same probit models as for input estimation. Estimation is performed on couple of years (the third column reports averages over results from 2002–2003 to 2007–2008).

	ECHP	ECHP/sim	sim
	2000–2001	2001–2002	2002–2008
Union unconditional	3.810 (0.087)	1.648 (0.019)	2.626 (0.034)
Null	3.810 (0.087)	2.285 (0.022)	2.150 (0.036)
Rank	3.810 (0.087)	3.111 (0.026)	2.826 (0.035)
Participation (males) unconditional	2.550 (0.075)	1.426 (0.024)	1.728 (0.033)
Null	2.550 (0.075)	1.901 (0.026)	1.763 (0.051)
Rank	2.550 (0.075)	2.599 (0.028)	1.858 (0.308)
Participation (females) unconditional	2.542 (0.056)	0.638 (0.015)	1.856 (0.024)
Null	2.542 (0.056)	0.974 (0.015)	0.938 (0.024)
Rank	2.542 (0.056)	2.057 (0.017)	1.990 (0.024)