Women. Sheng-Kai Chang. Abstract. In this paper a computationally practical simulation estimator is proposed for the twotiered

Transcription

1 Simulation Estimation of Two-Tiered Dynamic Panel Tobit Models with an Application to the Labor Supply of Married Women Sheng-Kai Chang Abstract In this paper a computationally practical simulation estimator is proposed for the twotiered dynamic panel Tobit model originally developed by Cragg (1971). The log-likelihood function simulated through procedures based on a recursive algorithm formulated by the Geweke-Hajivassiliou-Keane simulator is maximized. The simulation estimators are then applied to study the labor supply of married women. The rich dynamic structure of the labor force participation decision as well as hours worked decisions that are conditional on the participation of married women are identified by using the proposed simulation estimators. The average partial effects of the participation and hours worked decisions for married women in response to fertility decisions and increases in the husband s income are also investigated. It is found that the hypothesis that the fertility decision is exogenous and the hypothesis that the husband s income is exogenous to the married women s labor supply function are both rejected in the dynamic and static two-tiered models. Moreover, children aged between 6 and 13 years old may have a negative impact on the hours worked decision for married women that is conditional on their participation. However, these children may provide some positive incentives for married women to participate in the labor force. JEL classification: C15; C23; C24; J13; J22. Keywords: Two-tiered dynamic panel Tobit models; GHK simulator; Correlated random effects; Initial conditions problem. Corresponding author. Department of Economics, National Taiwan University, 21 Hsu-Chow Road, Taipei 100, Taiwan. Tel.: ext 376; fax: address: schang@ntu.edu.tw.

2 1 Introduction In the literature on panel data models, one critical issue is the estimation of limited dependent variable (LDV) models characterized by the presence of lagged dependent variables and serially correlated errors. A dynamic panel Tobit model is a leading example. The conventional techniques used in the estimation of linear panel data models are not applicable to the estimation of dynamic panel Tobit models due to the nature of the Tobit structure. Furthermore, the introduction of lagged dependent variables makes conventional estimation techniques even more difficult to apply. One possible method for estimating the dynamic panel Tobit model is the fixed effects approach. The fixed effects model is valid under weak restrictions on the unobserved individual heterogeneity. For example, Honoré (1993) estimates the panel Tobit model with lagged observed dependent variables through the fixed effects approach by creating orthogonality conditions for method of moments estimators. A set of identification conditions for Honoré s model is provided by Honoré and Hu (2004). See also Hu (2002) for the estimations of the censored panel data model with lagged latent dependent variables by applying the fixed effects method. The other method proposed to handle the dynamic panel Tobit model is the random effects approach. By specifying the distribution of the error conditional on the regressors, the random effects estimators can be obtained through maximizing the corresponding likelihood function. However, the likelihood function of the dynamic panel Tobit model is usually intractable since the dimension of an integral involved in its calculation is as large as the number of censoring periods in the model. Under such circumstances, simulation-based inference methods can be extremely useful. The impact of simulation methods on the analysis of LDV models is profound, especially under recent advances in computing technologies. Various simulation estimation methods and procedures for drawing random variables have been proposed in the econometrics literature. For instance, Lerman and Manski (1981) and Gourieroux and Monfort (1993) suggest adopting the maximum simulated likelihood method (MSL), McFadden (1989) and Keane (1994) propose the method of simulated moments (MSM), and Hajivassiliou and McFadden (1998) present the 2

3 method of simulated scores (MSS). See also Hajivassiliou (1993, 1994) for their applications. Different simulators have also been proposed for simulating multinomial probabilities in LDV models. Among multivariate normal probability simulators, Hajivassiliou et al. (1996) suggest that the Geweke-Hajivassiliou-Keane (GHK) simulator is, in terms of root mean squared errors, the most reliable simulator among the thirteen simulators they examined for approximating the multivariate normal distribution and its derivative. This paper studies a practical, operational and versatile maximum simulated likelihood procedure through the correlated random effects approach for two-tiered dynamic panel Tobit models using GHK simulation estimators. It is known that one potential restriction of the traditional Tobit models is that the choice between the dependent variable y = 0 versus y > 0 and the decision regarding the amount of y given that y > 0 is determined by a single mechanism. This is not always reasonable. Cragg (1971) proposed a two-tiered model to allow the parameters which characterize the decision regarding y = 0 versus y > 0 to be separate from the parameters which determine the decision regarding how much y is given that y > 0. The traditional Tobit models can be viewed as a special case of Cragg s two-tiered model. Moreover, the correlated random effects approach is attractive in several respects. First, time-invariant, time-varying, and time-dummy variables can be incorporated into the model and they can be consistently estimated using the proposed simulation estimators. Most importantly, the approach allows some degree of dependence between the individual unobserved heterogeneity and exogenous explanatory variables. Although stronger distributional assumptions are made for the random effects approach compared with the fixed effects approach, the proposed simulation estimation method with correlated random effects approach allows for complicated dynamics. The introduction of lagged latent dependent variables and lagged observed dependent variables, possibly with more than one lag, is straightforward. It is also easy to accommodate serial correlations in errors. Modifying the estimators to accommodate such specifications is done 3

4 fairly easily, and in an intuitive manner. 1 In addition, the two-tiered Tobit model proposed by Cragg (1971) can be easily extended and adopted by the simulation estimators through the correlated random effects approach. The predictions of the model can also be generated by the estimation results through the correlated random effects approach for two-tiered dynamic panel Tobit models. The proposed simulation estimators are applied to study married women s labor supply. The dynamic panel Tobit model as well as Cragg s two-tiered model are used to study married women s working behavior using the data from the Panel Study of Income Dynamics (PSID). The average partial effects of the participation and hours worked decision for married women in response to the fertility decision and increases in the husband s income are also investigated. It is found that the hypothesis that the fertility decision is exogenous and the hypothesis that the husband s income is exogenous to the married women s labor supply function are both rejected in dynamic and static two-tiered models. Moreover, children aged between 6 and 13 years old may have a negative impact on the hours worked decision for married women that is conditional on the participation. However, these children may provide some positive incentives for married women to participate in the labor force. The Monte Carlo experiments at the end of the empirical section also provide supporting evidence for Cragg s two-tiered model over the traditional Tobit model. The remainder of this paper proceeds as follows. Section 2 proposes a practical simulation estimator for two-tiered dynamic panel Tobit models. The likelihood function simulation through the GHK simulator is presented and then the consistency and asymptotic normality of the simulation estimator is demonstrated. Section 3 applies the simulation estimators to study the married women s labor supply function. Section 4 concludes the paper. 1 It has become quite common to allow rich dynamics in economic models since Heckman (1981) pointed out the importance of distinguishing true state dependence from spurious state dependence. For instance, Hyslop (1999) incorporates state dependence, serial correlation, and individual heterogeneity into the labor force participation model of married women. 4

5 2 Practical Simulation Estimators for Two-Tiered Dynamic Panel Tobit Models 2.1 Correlated Random Effects In a dynamic panel data framework, the Tobit model is described as: y it = x it β + y i,t 1 λ + c i + u it (1) y it = max{y it, 0}, t = 1,..., T i = 1,..., N Note that model (1) is characterized by lagged observed dependent variables. 2 The component c i is an unobserved individual specific random disturbance which is constant over time, and u it is an idiosyncratic error which varies across time and individuals. Throughout this section, I assume that c i and u it are Gaussian conditional on x i1,, x it. Following Chamberlain (1984), the unobserved individual heterogeneity c i can be assumed to be correlated with the exogenous variable x it for all t in a linear way. Thus, c i in (1) can be presented as c i = ω 0 + ω 1 x i1 + + ω T x it + d i (2) where d i in (2) and u it in (1) satisfy the following properties: E[d i x i ] = 0, E[u it d i, x i ] = 0, E[d 2 i x i ] = σ 2 d, E[u 2 it d i, x i ] = σ 2 u, E[d i d j x i, x j ] = 0, E[u it u js d i, d j, x i, x j ] = 0, E[u it u is d i, x i ] = 0 (3) where x i = (x i1,, x it ) and x j = (x j1,, x jt ), and for all i j and t s. Thus, the composite error ϵ it can be represented as ϵ it = d i + u it 2 The dynamic panel Tobit model with lagged latent dependent variables can be obtained from (1) by replacing the first line of (1) with yit = x itβ + yi,t 1λ + c i + u it. The proposed simulation estimator is also applicable to the model with lagged latent dependent variables. Although I present the likelihood simulation based on model (1), the likelihood simulation based on the model with the lagged latent dependent variables can be easily modified from the simulated likelihood function of (1). 5

6 Under such a specification, ω 0 cannot be identified if a constant is included in x it. One way to include a constant term in x it is to assume the independence of time constant x it and the unobserved individual heterogeneity c i. Under such a circumstance, the coefficient of the constant term x it can be consistently estimated using the simulation estimator. Moreover, time dummies are also excluded from x it in c i because they do not vary across i. For a correlated random effects approach of the Chamberlain type, however, the dimension of the parameters to be estimated will increase at the rate T. When the time period T or the dimension of x is small, computation is not an issue for estimating ω 0, ω 1,..., ω T, β and λ. Otherwise, following Mundlak (1978), Hajivassiliou (1985, 1987) and Wooldridge (2002), c i can be assumed to be as follows: c i = ω 0 + ωx i + d i, x i = 1 T T x it (4) t=1 and ω 0 and ω can also be consistently estimated using the proposed simulation estimators. 3 One advantage of using the correlated random effects approach of (4) is that, no matter how large the time period T is, the number of parameters to be estimated will only be affected by the dimension of x i. For a panel data set with a large T, this specification method for unobserved individual heterogeneity will not only allow for correlation between c i and x i, but will also make computation easier for parameter estimation. Equation (4) will be employed to model unobserved individual heterogeneity in the empirical section. 2.2 Simulation Estimation For the random effects plus AR(1) specification, the error term ϵ it can be specified as ϵ it = d i + v it v it = ζv i,t 1 + u it. (5) where d i is defined either by (2) or (4), and u it and d i continue to satisfy all of the conditions in (3). The covariance structure of the random effects plus AR(1) specification is denoted as Σ RE+AR(1). Moreover, the stationarity assumption ζ < 1 is also assumed to be satisfied for the random effects plus AR(1) errors model. 3 The time dummies and time constant variables are also not included in x i. 6

7 Let I it be a censored indicator function, such that 1 for yit I it = > 0 0 for yit 0 Therefore, for individual i, if I it = 1 then yit is observed and y it = yit. On the other hand, y it is censored and its value is not observed (i.e., y it = 0) if I it = 0. Thus, the simulated likelihood function with R simulation draws based on the GHK simulator for individual i can be described as: L i = 1 R T [f (r) (y it y i,t 1 )] I it [P (r) (I it = 0 y i,t 1 )] 1 I it (6) R r=1 t=1 By letting l i = ln( L i ), the simulated log-likelihood function can be represented as: l N R = ln( L i ) = i=1 N l i (7) i=1 The error terms ϵ i are assumed to have a normal distribution, that is, ϵ i N(0, Σ RE+AR(1) ), where ϵ i = (ϵ i1,..., ϵ it ) is a T 1 column vector, and so E[ϵ i ϵ i x i1,, x it ] = Σ RE+AR(1). Under this assumption, the simulation estimator based on the GHK simulator is expected to be useful for the dynamic panel Tobit models since it is well recognized that the GHK simulator is very accurate for the simulation of a multivariate normal distribution (Hajivassiliou et al., 1996). Let A be the Cholesky decomposition of Σ RE+AR(1), that is, Σ RE+AR(1) = AA, where A is a lower-triangular matrix. Given this structure, we can write ϵ i = Aη i, where ϵ i = d i + u it, η i N(0, I) and η i = (η i1,..., η it ). Let θ be the vector of parameters of interest in the random effects plus AR(1) model, i.e., θ = (β, λ, σu, 2 σd 2, ζ). Given the sample, suppose that for individual i the total number of instances of censoring is m i and that censoring occurs at time t 1,..., t mi. The random variables ξ (r) it are drawn from the uniform random number generator on [0,1], where t = t 1,..., t mi, i = 1,..., N and r = 1,... R. In order to guarantee the validity of the stochastic equicontinuity condition 4 for the simulation estimator, the random numbers are drawn once and kept fixed when θ varies. 5 4 See Hajivassiliou and McFadden (1998). 5 See McFadden (1989) for further details. 7

8 To simplify the exposition, yi0 = 0 is assumed for all i. Let Φ be the cumulative standard normal function. Then, the truncated normal random variables η (r) it calculated by η (r) it = Φ 1 (ξ (r) it Φ( x t 1 itβ y i,t 1 λ x i ω k=1 A tkη (r) ik A tt )) for t {t 1,..., t mi } t 1 y it x it β y i,t 1 λ x i ω k=1 A tkη (r) ik A tt for t {t 1,..., t mi } Then, the simulated likelihood function (6) can be obtained through can be simulated or f (r) (y it y i,t 1 ) = 1 ϕ( y it x it β y i,t 1 λ x i ω t 1 k=1 A tkη (r) ik ) (9) A tt A tt for t {t 1,..., t mi }, where ϕ is the standard normal density function, and P (r) (I it = 0 y i,t 1 ) = Φ( x itβ y i,t 1 λ x i ω t 1 k=1 A tkη (r) ik ) (10) A tt for t {t 1,..., t mi }. Thus, by combining (9) and (10) into (6), a practical simulation estimator of θ denoted as θ can be obtained for maximizing the simulated log-likelihood function (7). Based on the GHK simulation estimators presented above, the asymptotic variance associated with them can be consistently estimated as follows. Let (8) Ω = 1 N ΣN i=1[ŝ i ( θ)ŝ i ( θ) ] and Ĵ = 1 N ΣN i=1ŝiθ( θ) where ŝ i is the simulated score function associated with (7), and ŝ iθ is the first derivative of ŝ i with respect to θ, that is, Ĵ is simply the Hessian matrix of the simulated log-likelihood function (7). Therefore, the consistent estimator of the asymptotic variance of the simulation estimator θ can be written as Avar( θ) = Ĵ 1 ΩĴ 1. (11) See Hajivassiliou and McFadden (1998) for more details. The average partial effects (APEs) can also be estimated by using the GHK simulation estimators θ under the correlated random effects approach. For the APEs of the decision probability 8

9 related to y = 0 versus y > 0, supposing that there are K regressors in x and x K is a discrete random variable, the APEs with respect to x K when the x K is changed from x K(0) to x K(1) can be written as [N 1 Σ N i=1φ(x 1 i θ/ σ ϵ )] [N 1 Σ N i=1φ(x 0 i θ/ σ ϵ )] (12) where x 1 θ i = x β x K 1 βk 1 + x K(1) βk + y 1 λ + x i ω and x 0 θ i = x β x K 1 βk 1 + x K(0) βk + y 1 λ + x i ω x i = (x i,1,..., x i,t ), and σ ϵ = σ2 d + σ 2 u/(1 ζ 2 ). Moreover, the (x 1,..., x K 1, y 1 ) are any given values of the first K 1 regressors and lagged dependent variables. Normally, the sample averages of these random variables are used to evaluate the APEs. The focus of attention in the Tobit models is also on the expected values E(y x, y > 0) and E(y x). For a given x t, the E(y t x t, y t > 0) and E(y t x t ) can be calculated as follows and E(y t x t, y t > 0) = x t θ + σ ϵ [ ϕ(x tθ/σ ϵ ) Φ(x t θ/σ ϵ ) ] E(y t x t ) = Φ(x t θ/σ ϵ )x t θ + σ ϵ ϕ(x t θ/σ ϵ ) Thus, the APEs of E(y x, y > 0) with respect to x K when x K is a discrete random variable and is changed from x K(0) to x K(1) can be represented as [N 1 Σ N i=1(x 1 θ i + σ ϵ [ ϕ(x1 θ/ σ i ϵ ) Φ(x 1 θ/ σ i ϵ ) ])] [N 1 Σ N i=1(x 0 θ i + σ ϵ [ ϕ(x0 θ/ σ i ϵ ) Φ(x 0 θ/ σ ])] (13) i ϵ ) Moreover, the APEs of E(y x) with respect to x K when x K is a discrete random variable and is changed from x K(0) to x K(1) can be represented as [N 1 Σ N i=1(φ(x 1 i θ/ σ ϵ )x 1 i θ + σ ϵ ϕ(x 1 i θ/ σ ϵ ))] [N 1 Σ N i=1(φ(x 0 i θ/ σ ϵ )x 0 i θ + σ ϵ ϕ(x 0 i θ/ σ ϵ ))] (14) 9

10 2.3 Two-tiered Dynamic Panel Tobit Models One potential restriction of Tobit models is that of the choice between y = 0 versus y > 0 and the decision regarding the amount of y given that y > 0 is determined by a single mechanism. In other words, the decision related to y = 0 versus y > 0 is inseparable from the decision regarding how much y is given that y > 0 in the traditional Tobit model. However, the mechanisms which determine these two may not be the same in some economic models. Specifically, any variable which increases the probability of a nonzero value must also increase the mean of the positive values in the traditional Tobit models. This is not always the case. For example, as for the application of the married women s labor supply which is further discussed in the empirical section that appears later, the traditional Tobit model indicates that married women who have one more child aged between 6 and 13 years old should always decrease (or increase) both the probability of labor force participation and the mean hours worked. However, by using the more generalized types of Tobit model, it is found that married women with one more child aged between 6 and 13 years old will increase their labor force participation rate but will decrease the mean hours worked that are conditional on the participation. In order to relax the restrictions imposed by the traditional Tobit models, Cragg (1971) proposed a two-tiered model to allow the parameters which characterize the decision regarding y = 0 versus y > 0 to be separate from the parameters which determine the decision regarding how much y is given that y > 0. The traditional Tobit models can be viewed as a special case of Cragg s two-tiered model. 6 In other words, there are basically two assumptions in Cragg s two-tiered model. First, the probability of a zero observation is given by a probit model with the first tier parameters, and then the density of the dependent variable that is conditional on being a positive observation is truncated at zero and characterized by the second tier parameters. Cragg s model is easily extended from the cross-sectional framework to the dynamic panel data models using the simulation estimators proposed earlier. There are alternatives to Cragg s two-tiered models. One of the examples is the Heckman 6 See for example Lin and Schmidt (1984) for a specification test of the Tobit model against Cragg s model. 10

11 (1979) type of sample selection model which can be simply characterized as follows: y 1 = xβ 1 + u 1 y 2 = xβ 2 + u 2 (15) where only the sign of y 2 is observed and y 1 is observed if and only if y 2 > 0. Under such a specification, Tobit model is a special case of the Heckman s model when β 1 = β 2 and u 1 = u 2. The Heckman model (15) is different from Cragg s two-tiered model in two respects. First, it is indicated in the model (15) that there is a positive probability of observing y 1 < 0. Secondly and more importantly, the unobserved y 1 is literally unobserved, rather than observed as being equal to zero. The second difference is in any case fundamental while the first difference can be solved by measuring y 1 in logarithms, but under such a case, the model no longer includes the Tobit models as a special case. Therefore, although Cragg s two-tiered model is more restrictive than the Heckman full selection model, for the economic data set in which the non-zero observations are all positive and zero is a meaningful and common value for the dependent variable, Cragg s two-tiered model might be a better choice than Heckman s sample selection model. For example, for the married women s labor supply data set used in the empirical section, Cragg s two-tiered model on the one hand provides more flexible specifications than the standard Tobit models, and on the other hand it is also more appropriate for characterizing the data set than Heckman s selection models. Specifically for Cragg s two-tiered Tobit models, let Γ 1 = (β 1, λ 1, ω 1 ) be the first-tiered parameters which determine the initial decision between y = 0 and y > 0. Let Γ 2 = (β 2, λ 2, ω 2 ) be the second-tiered parameters which determine the amount of y given that y > 0. y i0 = 0 is assumed again for all i. Then, for the GHK estimators for the two-tiered Tobit models, the truncated normal random variables η (r) it can be simulated from η (r) Φ 1 (ξ (r) it it = Φ( x t 1 itβ 1 y i,t 1 λ 1 x i ω 1 k=1 A tkη (r) ik A tt )) for t {t 1,..., t mi } t 1 y it x it β 2 y i,t 1 λ 2 x i ω 2 k=1 A tkη (r) ik A tt for t {t 1,..., t mi } Notice that the first-tiered parameters Γ 1 = (β 1, λ 1, ω 1 ) are used in the first line, and the second-tiered parameters Γ 2 = (β 2, λ 2, ω 2 ) are used in the second line to obtain η (r) it. 11

12 Then the simulated likelihood function of the two-tiered version of (6) for individual i can be described as: L i = 1 R T [ P (I it = 1 Γ 1, y i,t 1 ) R P (I r=1 t=1 it = 1 Γ 2, y i,t 1 ) f(y it Γ 2, y i,t 1 )] I it [P (I it = 0 Γ 1, y i,t 1 )] 1 I it (16) where the traditional Tobit models can be viewed as a special case of (16) when Γ 1 = Γ 2. Specifically, f(y it Γ 2, y i,t 1 ), P (I it = 1 Γ 1, y i,t 1 ), P (I it = 0 Γ 1, y i,t 1 ) and P (I it = 1 Γ 2, y i,t 1 ) in (16) can be calculated by and and f(y it Γ 2, y i,t 1 ) = 1 ϕ( y it x it β 2 y i,t 1 λ 2 x i ω 2 t 1 k=1 A tkη (r) ik ) (17) A tt A tt P (I it = 1 Γ 1, y i,t 1 ) = 1 Φ( x itβ 1 y i,t 1 λ 1 x i ω 1 t 1 k=1 A tkη (r) ik ) (18) A tt P (I it = 1 Γ 2, y i,t 1 ) = 1 Φ( x itβ 2 y i,t 1 λ 2 x i ω 2 t 1 k=1 A tkη (r) ik ) (19) A tt and P (I it = 0 Γ 1, y i,t 1 ) = 1 P (I it = 1 Γ 1, y i,t 1 ). Thus, a practical simulation estimator of θ can be obtained for maximizing the simulated log-likelihood function by using (16). Moreover, the consistent estimator of the asymptotic variance of the two-tiered simulation estimators can be obtained through (11). The two-tiered dynamic panel Tobit model will be applied to study the married women s labor supply in the empirical section. The APEs of two-tiered Tobit models can also be estimated by using the GHK simulation estimators. Let θ 1 = ( β (1), λ (1), ω (1) ) be the estimators from the first tier, and θ 2 = ( β (2), λ (2), ω (2) ) be the estimators from the second tier. Then the APEs of the decision probability are related to y = 0 versus y > 0, supposing again that there are K regressors in x and x K is a discrete random variable. The APEs with respect to x K when x K is changed from x K(0) to x K(1) can be written as [N 1 Σ N i=1φ(x 1 θ (1) / σ ϵ )] [N 1 Σ N i=1φ(x 0 θ (1) / σ ϵ )] (20) where it is assumed that x 1 θ (1) = x 1 β1 (1) x K 1 βk 1 (1) + x K(1) βk (1) + y λ 1 (1) + x i ω (1) 12

13 x 1 θ (2) = x 1 β1 (2) x K 1 βk 1 (2) + x K(1) βk (2) + y λ 1 (2) + x i ω (2) x 0 θ (1) = x 1 β1 (1) x K 1 βk 1 (1) + x K(0) βk (1) + y λ 1 (1) + x i ω (1) and x 0 θ (2) = x 1 β1 (2) x K 1 βk 1 (2) + x K(0) βk (2) + y λ 1 (2) + x i ω (2) x i = (x i,1,..., x i,t ), and σ ϵ = σ2 d + σ 2 u/(1 ζ 2 ). Moreover, (x 1,..., x K 1, y 1 ) are any given values of the first K 1 regressors and lagged dependent variables. Notice that only the estimators from the first tier are used. The APEs of E(y x, y > 0) with respect to x K when x K is a discrete random variable and is changed from x K(0) to x K(1) can be represented as [N 1 Σ N i=1(x 1 θ (2) + σ ϵ [ ϕ(x1 θ (2) / σ ϵ ) Φ(x 1 θ (2) / σ ϵ ) ])] [N 1 Σ N i=1(x 0 θ (2) + σ ϵ [ ϕ(x0 θ (2) / σ ϵ ) Φ(x 0 θ ])] (21) (2) / σ ϵ ) Notice that only the estimators from the second tier are used. Moreover, the APEs of E(y x) with respect to x K when x K is a discrete random variable and is changed from x K(0) to x K(1) can be represented as the difference between [N 1 Σ N i=1(φ(x 1 θ (1) / σ ϵ )x 1 θ (2) + Φ(x1 θ (1) / σ ϵ ) Φ(x 1 θ (2) / σ ϵ ) σ ϵϕ(x 1 θ (2) / σ ϵ ))] (22) [N 1 Σ N i=1(φ(x 0 θ (1) / σ ϵ )x 0 θ (2) + Φ(x0 θ (1) / σ ϵ ) Φ(x 0 θ (2) / σ ϵ ) σ ϵϕ(x 0 θ (2) / σ ϵ ))] (23) 3 An Empirical Application: Married Women s Labor Supply 3.1 Motivation One of the most interesting topics in labor economics is the labor supply of married women. Due to fertility decisions and non-labor income, the labor supply of married women is much more complicated than the labor supply of men. Most of the empirical research conducted in this area is focused within the context of cross-sectional data. Since a large fraction of married women have zero working hours, the Tobit model is usually employed to perform the estimation. 13

14 Jakubson (1988) incorporated the life-cycle model into the married women s labor supply equation and estimated the model using four-year panel data. Hyslop (1999) also applied the panel Probit model with a rich dynamic structure to study the labor force participation of married women. It is crucial to incorporate the dynamic structure into the married women s labor supply function as indicated in Hyslop s results. Moreover, in analyzing the interactions between fertility and the labor supply decision for married women, Browning (1992) pointed out that it is important to control for the dynamic structure of their labor supply decisions. In a dynamic structure, it is often argued that individuals who have experienced an event in the past are more likely to experience the same event in the future than individuals who did not experience the event. This is what Heckman (1981) refers to as state dependence. There are two types of state dependence in a dynamic structure. One is true state dependence, and the other is spurious state dependence. True state dependence, as explained by Heckman (1981), is when as a consequence of experiencing an event, preference, prices or constraints relevant to future choice are altered. On the other hand, Heckman (1981) explains spurious state dependence as the phenomenon that individuals may differ in their propensity to experience the event. If individual differences are correlated over time, and if these differences are not properly controlled, previous experience may appear to be a determinant of future experience. Therefore, it is important for economists to distinguish true state dependence from spurious state dependence in the labor supply function of married women. The empirical specification for modeling the labor supply of married women in this section involves the reduced form method. There is also a huge literature on structural models of female labor supply, for example, Heckman and MaCurdy (1980, 1982) and Eckstein and Wolpin (1989). Although the reduced form methods are useful for initial exploratory data analysis and they are relatively easy to implement compared with the structural form methods, there are several advantages in structural estimation. The first advantage of having a structural model behind the estimations is that the parameters estimated can be interpreted in a way that is consistent with the theoretical framework. In other words, the detailed implications of the theory, and not only of the data, can be learned during the process of estimating the structural models. Moreover, the predictions of the impacts of policy changes can be generated more accurately 14

15 by the structural models than by the reduced form models. See Rust (1994) for more details regarding the structural estimation. In this section, the two-tiered dynamic panel Tobit model is employed to study the labor supply of married women. Unlike the models estimated by Cogan (1981) and Mroz (1987), panel data are used in the estimation. In addition, rather than using the static panel Tobit model estimated by Jakubson (1988), rich dynamic structures are incorporated in the model and estimated by the proposed simulation estimators. Moreover, decisions over both participation and hours are considered through the Tobit structure, while Hyslop (1999) only focuses on the labor force participation decision for married women. Since Cogan (1981) and Mroz (1987) have pointed out a possible mis-specification of the simple Tobit specification for the labor supply of married women due to the significant fixed cost involved in their labor force participation decisions, the Cragg type two-tiered Tobit model is also implemented to study the labor supply of married women. 3.2 Data The data used in the estimation consist of a nine-year panel data set from the PSID. The sample contains observations for 1,627 women continuously married between 1984 and 1992, and aged between 19 and 60 in Table 1 presents the summary statistics for the data used. The husband s income in Table 1 is expressed in constant (1985) thousands of dollars, computed as nominal earnings deflated by the consumer price index. The numbers in the parentheses are the standard errors corresponding to the variables. The distribution of years worked during the period for the full sample in Table 1 also suggests the importance of incorporating the dynamic structure for the married women s labor supply. For example, supposing that the individual s participation decision is distributed according to an independent binomial distribution, then about 9.08% of the sample would be expected to work each year if the participation probability were fixed as (the average participation rate during the period), about 13.42% of the sample would be expected to work each year if the participation probability were fixed as 0.8, and about 4.04% of the sample would be expected to work each year if the participation probability were fixed as 0.7. In these three cases, less 15

16 Table 1. Sample Characteristics Variable Full Employed Employed Employed Name Sample 9 Years 0 Years 1 Year Age (1985) (8.769) Education (2.071) Race (Black=1) (0.409) Husband s Income ($1000) (29.16) Children between 1 and (0.466) Children between 3 and (0.516) Children between 6 and (0.891) Hours of Work (887.01) No. of years worked (8.144) (2.031) (0.423) (24.59) (0.419) (0.462) (0.849) (593.45) (9.520) (1.987) (0.427) (42.81) (0.456) (0.523) (0.844) (8.723) (2.077) (0.408) (28.10) (0.467) (0.516) (0.894) (914.41) zero one two three four five six seven eight nine Sample Size

17 than 0.002% of the sample would be expected not to work at all. However, the sample s relative frequencies are 53.47% of the sample who worked each year for 9 years and 5.9% of the sample who did not work at all, accordingly. Therefore, Table 1 indicates that there is significant persistence in the observed annual participation decisions of married women. The differences in the characteristics of the married women across the sub-samples shown in Table 1 can be interpreted as follows. Women who work each year are better educated, are more likely to be black, have lower than average husband s income, and have fewer dependent children. On the other hand, women who have never been employed during the sample period are older, less educated, are more likely to be black, have higher than average husband s income, and have slightly fewer dependent children. 3.3 Estimation Results The dynamic panel Tobit model (1) with the covariance matrix Σ RE+AR(1) under the assumptions in (3) is employed to study the labor supply of married women. The dependent variable is the annual hours of work. The explanatory variables include constants (Cons), years of schooling (Edu), wife s race (Race, black=1), wife s age (Age), square of age (Age 2 ), husband s income (Hinc), number of children aged between 1 and 2 years old (C12), number of children aged between 3 and 5 years old (C35), and number of children aged between 6 and 13 years old (C613). The husband s income is used as a proxy variable for non-labor income for married women. Furthermore, the lagged dependent variable is not included as a regressor in the static models. The numbers in the parentheses are the estimated standard errors corresponding to the estimators. The optimization subroutine used is the ConstrOptim procedure based on the R software, and the BFGS algorithm is used for the maximization. 7 The estimation results are presented in Table 2 for the static models. The pure random effects plus AR(1) model which assumes that c i = d i is presented as model (1) of Table 2. It is shown that better educated or black or older women tend to work more than less educated or white or younger women. In addition, the women who have children, especially children between the ages of 1 and 2, tend to work less than the women who have no children. These results are 7 The same procedure and algorithm are used later in the Monte Carlo experiments subsection. 17

18 Table 2. Static Panel Tobit Models of Married Women s Labor Supply Model (1) (2) (3) (4) Correlated Correlated Correlated Variable RE+AR(1) RE+AR(1) Cragg-1 Cragg-2 Cragg-1 Cragg-2 Cons (1.229) Edu (9.64) Race (51.92) Age (7.132) Age (0.100) Hinc (0.487) C (21.90) C (18.31) C (13.29) (2.797) (11.24) (54.78) (7.522) (0.105) (0.411) (22.69) (19.42) (14.84) Hinc (1.717) C (40.92) C (40.74) C (33.71) (267.7) (9.482) (44.11) (13.42) (0.169) (0.465) (27.76) (22.67) (16.31) (264.7) (10.66) (43.84) (12.56) (0.163) (0.717) (21.49) (18.84) (13.09) (271.3) (10.19) (44.48) (13.63) (0.169) (0.361) (31.44) (27.06) (22.36) (1.464) (119.4) (114.1) (39.24) (272.0) (12.11) (44.74) (12.89) (0.164) (0.509) (21.90) (19.73) (14.33) (2.227) (126.2) (127.2) (37.39) ρ ζ log-like

19 standard in the literature. The fraction of the variance due to individual heterogeneity, ρ, is 0.568, and the AR(1) coefficient ζ is The unobserved individual heterogeneity c i in the correlated random effects plus AR(1) model (2) of Table 2 is assumed to be as follows: c i = ω 0 + ω 1 Hinc i + ω 2 C12 i + ω 3 C35 i + ω 4 C613 i + d i (24) where Hinc i, C12 i, C35 i, C613 i are time averages of the corresponding variables for individual i. Based on model (2) of Table 2, it is shown later in Table 4 that both the husband s income and fertility decision variables are endogenous in the static model (1) of Table 2 at the 1% significance level. Thus, the impact of education and age is underestimated and the impact of race and younger children is overestimated in model (1) of Table 2 if the endogeneity problem is ignored. Moreover, ρ increases and the AR(1) coefficient ζ decreases from model (2) to model (1) in Table 2. The two-tiered Cragg type panel Tobit model is estimated in model (3) of Table 2 where the decisions on labor force participation and hours worked that are conditional on the participation are estimated separately. Based on model (3) of Table 2, the women receiving higher education are more likely to participate in the labor force. However, being conditional on the participation, education has no statistically significant impact on the decision regarding hours worked. On the other hand, in a way that is conditional on participation in the labor force, black or older women tend to work more than white or younger women, but there is no statistically significant impact of race or age on the labor force participation decision for married women. The husband s income has almost the same negative impact on both the participation and working hours decisions. Thus, for married women, the higher the husband s income, the less likely it is that she will participate in the labor force, and the less hours she will work if she decides to participate in the labor force. The children aged between 1 and 5 years old have a negative impact on the married women in both the participation as well as hours worked decisions, and not surprisingly, the younger the children, the stronger the negative impact will be on both margins. The children aged between 6 and 13 years old also have a negative impact on the working hours decision that is conditional on the participation. However, they may have a positive impact on the labor force 19

20 participation decision although this positive influence is not statistically significant. The ρ and AR(1) coefficient ζ in model (3) are similar to the ones in model (1) of Table 2. The static correlated Cragg s two-tiered model is estimated in model (4) of Table 2 by using (24). The husband s income is still endogenous in both tiers of the model. However, the coefficients of C12 i and C35 i are each insignificant at the 10% level. Moreover, unlike model (3) of Table 2, education has a statistically significant positive impact on both the participation and working hours decisions. The dynamic panel Tobit model is employed to study the labor supply for married women in Table 3. The initial conditions problem is dealt with in Heckman s approach. The true state dependence parameters λ are close to zero and are not statistically significant in the RE+AR(1) model but are marginally statistically significant in the correlated RE+AR(1) model. Moreover, the AR(1) coefficient ζ increases slightly from Table 2 to Table 3. The correlated random effects plus AR(1) model in (2) of Table 3 is based on (24). The coefficients of Hinc i, C12 i, C35 i, C613 i are marginally statistically significantly different from zero at the 5% significance level which indicates that the husband s income as well as fertility decisions are endogenous in the dynamic model. The two-tiered Cragg s type dynamic panel Tobit model is estimated in model (3) of Table 3. As in Table 2, it can be shown that education only has a statistically significant impact on the labor force participation decision, but not on the hours worked decision that is conditional on the participation. On the other hand, in a way that is conditional on the participation in the labor force, black women tend to work more than white women. However, there is no statistically significant impact of race on the labor force participation decision. The husband s income also has an equally negative impact on both participation and the hours worked decision, as shown in Table 2. Moreover, the children between 1 and 5 years of age have a statistically significant negative impact on both participation as well as the hours worked decision. The children aged between 6 and 13 only have a negative impact on the working hours decision that is conditional on the participation, but there is no statistically significant impact for the children aged between 6 and 13 on the labor force participation decision for married women. The dynamic version of the correlated Cragg s two-tiered model is estimated in model (4) 20

21 Table 3. Dynamic Panel Tobit Models of Married Women s Labor Supply Model (1) (2) (3) (4) Correlated Correlated Correlated Variable RE+AR(1) RE+AR(1) Cragg-1 Cragg-2 Cragg-1 Cragg-2 Cons (102.5) Edu (13.33) Race (94.65) Age (24.63) Age (0.238) Hinc (0.554) C (33.01) C (23.52) C (34.31) λ (0.479) (321.7) (12.98) (57.23) (15.19) (0.188) (0.435) (24.12) (21.07) (16.03) (0.019) Hinc (2.002) C (146.3) C (143.6) C (43.05) (302.2) (10.54) (49.67) (14.88) (0.182) (0.502) (28.91) (23.81) (17.26) (0.049) (303.8) (11.89) (47.51) (14.31) (0.182) (0.692) (22.81) (20.01) (14.08) (0.029) (302.4) (11.27) (49.94) (14.85) (0.181) (0.437) (32.50) (28.89) (23.34) (0.042) (1.532) (134.5) (131.8) (42.16) (310.0) (13.22) (48.30) (14.74) (0.185) (0.509) (24.06) (21.52) (15.60) (0.025) (2.205) (139.1) (143.5) (40.37) ρ ζ log-like

22 of Table 3. The hypothesis of the exogeneity of the husband s income can be rejected at the 5% significance level in both the first and second tier of the estimation. The impacts of race and age on the married women s labor supply in model (4) of Table 3 are similar to those in model (3) of Table 3. However, education now has a statistically significant positive impact on both the labor force participation decision and the mean hours worked decision conditional on the participation. Most interestingly, in the dynamic correlated Cragg s two-tiered model it is found that the children aged between 6 and 13 years old will give the married women a positive incentive to participate in the labor force. The estimators of the true state dependence parameter λ are all negative and range from to in Table 3. They are statistically significant only in the correlated random effects models. On the other hand, the estimators of the spurious state dependence parameter ζ are slightly larger in Table 3 than in Table 2. The estimators of the ρ in Table 3 are similar to the ones in Table 2. Thus, based on Cragg s two-tiered models that adopt the correlated random effects approach, there is a significant true state dependence in both the participation and hours worked decision for the married women s labor supply. The ignorance of the true state dependence parameter, λ, will lead to an overestimation of the spurious state dependence parameter, ζ, as shown in Tables 2 and 3. In order to investigate the exogeneity of the fertility decisions and the husband s income to the married women s labor supply function in various models, the Wald statistics and t statistics of the following hypotheses are performed for the correlated random effects models in Tables 2 and 3. The hypothesis that the fertility decisions are exogenous to the married women s labor supply function can be written as H 0 : ω 2 = 0, ω 3 = 0, ω 4 = 0 (25) where ω 2, ω 3 and ω 4 are defined in (24). The fertility decisions are endogenous to the married women s labor supply function if the null hypothesis (25) is rejected. This null hypothesis can be tested based on the Wald statistics. Moreover, the hypothesis that the husband s income is exogenous to the married women s 22

23 labor supply function can be represented as H 0 : ω 1 = 0 where ω 1 is defined in (24). This null hypothesis can be tested using t statistics for the traditional Tobit model and Wald statistics for Cragg s two-tiered model. The husband s income is endogenous to the married women s labor supply function if H 0 : ω 1 = 0 is rejected. The test results are reported in Table 4. At the 1% significance level, the hypothesis that the fertility decisions are exogenous to the married women s labor supply function is rejected in both the dynamic and static models. Moreover, the hypothesis that the husband s income is exogenous to the married women s labor supply function is also rejected in all models at the 1% significance level. Therefore, there is strong evidence that both fertility decisions and the husband s income may not be exogenous variables in both the static and dynamic models of the married women s labor supply function. 3.4 Average Partial Effects In order to investigate the impact of the fertility decisions and the change in the husband s income on the married women s participation decisions and expected hours worked, the APEs of the fertility decisions and husband s income changes for married women are presented using equations (12)-(14) and equations (20)-(23). By employing both standard Tobit models and Cragg s two-tiered models, the following four events are considered: 1. having a child aged less than 2 years old, 2. having a child aged between 3 and 5 years old, 3. having a child aged between 6 and 13 years old, and 4. 10% increases in the husband s income. Two types of APEs of expected hours worked for married women are considered, that is, one unconditional and another conditional on the labor force participation. It is assumed that a married woman has no children in the base model. The age and husband s income of the married woman are set equal to the sample average, that is Age = 38 and Hinc = 31, 000. Moreover, two education levels are considered for estimating the average partial effects, one being Edu = 12 (possibly high school graduates), and the other Edu = 16 (possibly college graduates). Three types of APEs are reported in Tables 5-7. P.P. stands for the labor force participation 23

24 Table 4. Tests of Exogeneity of Fertility Decisions and Husband s Income H 0 : ω 2 = 0, ω 3 = 0, ω 4 = 0 (exogeneity of fertility decisions) M odel T est Statistics p value Static RE+AR(1) Dynamic RE+AR(1) Static Cragg (1st tier) Static Cragg (2nd tier) Static Cragg (both tiers) Dynamic Cragg (1st tier) Dynamic Cragg (2nd tier) Dynamic Cragg (both tiers) H 0 : ω 1 = 0 (exogeneity of husband s income) M odel T est Statistics p value Static RE+AR(1) Dynamic RE+AR(1) Static Cragg (1st tier) Static Cragg (2nd tier) Static Cragg (both tiers) Dynamic Cragg (1st tier) Dynamic Cragg (2nd tier) Dynamic Cragg (both tiers)

25 probability for married women, E(y x) represents the expected hours worked given the personal characteristics x, and E(y x, y > 0) refers to the expected hours worked given the personal characteristics x and their being conditional on the participation. These three types of APEs are reported for the differences from the base model in response to the fertility decision and the change in the husband s income, namely, events 1-4. Both the random effects and correlated effects models are employed in Tables 5 and 6. In Table 7, only correlated effects models are used. The APEs of the fertility decision and the increase in the husband s income for married women with 12 years of education are reported in Table 5 for both the two-tiered and Tobit models. For the labor force participation decision, it is shown in both the Tobit and two-tiered models with either random effects or correlated random effects that blacks have higher labor market participation rates than whites for married women with the same personal characteristics. Moreover, blacks also have higher expected hours worked than whites with the same personal characteristics, that are conditional or unconditional on the participation. Overall, the random effects models which assume the exogeneity of fertility decisions and the husband s income tend to overestimate participation decisions and expected hours worked as well as the impacts of fertility decisions and the increase in the husband s income than the correlated random effects models. In comparing Cragg s two-tiered models with the standard Tobit models, it is found that the Tobit models tend to overestimate the participation probability and to underestimate the expected working hours. One interesting observation in the correlated random effects two-tiered models is that the married women, blacks or whites, will increase their participation probability in the labor market but will decrease their expected hours worked if they have a child aged between 6 and 13 years old. For all other models, the participation probability and expected hours worked will both decrease if the married woman has a child aged between 6 and 13 years old. It is impossible for the Tobit models to detect evidence like that since those models are restricted in that the direction of the changes in response to the fertility decision on the participation probability and expected hours worked should always be the same. The impacts of the increase in the husband s income on the participation probability and expected hours 25