Retirement and Intra-Household Labour Division of Italian Couples: A New Simultaneous Equation Approach

Abstract

The effect of retirement on couple’s labour division has been investigated using cross-sectional and longitudinal data, providing, however, contradictory results. This may be due to the difficulty of specifying the influence of latent factors, such as gender ideology or the bargaining process between partners. The risk of misspecification may be particularly high when referring to Italy, where the life of couples is strongly influenced by familistic norms and gender ideology. We examine the impact of retirement of the man on the market and domestic work of both partners, adopting a Difference-in-Differences estimation procedure and using a correction mechanism to control for misspecifications of latent factors. Our results show that Italian men increase their commitment in domestic activities after retirement. However, the influence of gender ideology and familistic belief significantly hampers this propensity for traditionalist men. Finally, the significativity of estimation results confirms that the correction here introduced leads to robust estimates and allows to use a gender-role score index as a valid predictor.

Appendix: Restrictions Imposed on the Covariances of the Error Terms: Specification and Implementation in the Estimation Procedure

The error terms of each equation are specified taking into account the correlation between equations in correspondence to each observation, and the correlation of each repeated observation over time. We report here the specification of the error terms, respectively, of the equations of paid and unpaid work of women, (1) and (2); and the equations of paid and unpaid work of men, (3) and (4). Indexes and suffixes are the same.

$$\begin{aligned} u_{Lwti} &= \varphi_{Lw1} u_{Dwti} + \varphi_{Lw2} u_{Lmti} + \varphi_{Lw3} u_{Dmti} \\ &\quad+ \rho_{Lw1} u_{Lwt - 1i} + \rho_{Lw2} u_{Dwti - 1} + \rho_{Lw3} u_{Lmt - 1i} + \rho_{Lw4} u_{Dmt - 1i} + \varepsilon_{Lwti} \hfill \\ \end{aligned}$$

(5)

$$\begin{aligned} u_{Dwti} &= \varphi_{Dw1} u_{Lwti} + \varphi_{Lw2} u_{Dmti} + \varphi_{Dw3} u_{Dmti} \\ &\quad+ \rho_{Dw1} u_{Dwt - 1i} + \rho_{Dw2} u_{Lwt - 1i} + \rho_{Lw3} u_{Lmt - 1i} + \rho_{Dw4} u_{Dmt - 1i} + \varepsilon_{Dwti} \hfill \\ \end{aligned}$$

(6)

$$\begin{aligned} u_{Lmti} &= \varphi_{Lm1} u_{Dwti} + \varphi_{Lm2} u_{Lwti} + \varphi_{Lm3} u_{Dmti} \\&\quad+ \rho_{Lm1} u_{Lwti} + \rho_{Lm2} u_{Dwt - 1i} + \rho_{Lm3} u_{Lwt - 1i} + \rho_{Lm4} u_{Dmt - 1i} + \varepsilon_{Lmti} \hfill \\ \end{aligned}$$

(7)

$$\begin{aligned} u_{Dmti} &= \varphi_{Dm1} u_{Lmti} + \varphi_{Lm2} u_{Lwti} + \varphi_{Dm3} u_{Dwti} \\&\quad+ \rho_{Dm1} u_{Dmti} + \rho_{Dm2} u_{Lmt - 1i} + \rho_{Lm3} u_{Lwt - 1i} + \rho_{Dm4} u_{Dwt - 1i} + \varepsilon_{Dmti} \hfill \\ \end{aligned}$$

(8)

The coefficients indicated with the Greek letter φ measure the relationships between the errors of different equations (for instance, between paid work and domestic work of the same subject, or between the domestic work of the subject and that of his/her partner), while coefficients indicated with the Greek letter ρ measure the relationship across time (or across the two Waves) of the errors referred to the same observation (or the same subject). The covariances between the equations and between the repeated observations over time (imposed as non-null) can be easily obtained from the coefficients φ and ρ using the well-known second-order moments relationship.

We assume that the errors u _Lwti , u _Dwti , u _Lmti , u _Dmti are omoskedastic and distributed with zero mean, as well as the disturbances ɛ _Lwti , ɛ _Dwti , ɛ _Lmti , ɛ _Dmti . However, the latter, unlike the former are non-correlated with each other. Considering the restrictions imposed on the covariances, the covariance matrix, Ω, of the error terms is similar to those used in SUR models.

To incorporate the constraints on the errors’ covariances in the estimation procedure, we use the residuals, û _i, of the OLS (uncensored) regressions of Eqs. (2) and (4) and of the Tobit (censored) regressions of Eqs. (1) and (3) that we run separately at the first stage. Then, at the second stage, four OLS regressions using the first-stage residuals are estimated as follows (for the sake of brevity, we report the regression referring to the residuals of Eq. 1 only):

$$\begin{aligned} \tilde{u}_{Lwti} &= \tilde{\varphi }_{Lw1} \hat{u}_{Dwti} + \tilde{\varphi }_{Lw2} \hat{u}_{Lmti} + \tilde{\varphi }_{Lw3} \hat{u}_{Dmti}\\ & \quad+ \tilde{\rho }_{Lw1} \hat{u}_{Lwt - 1i} + \tilde{\rho }_{Lw2} \hat{u}_{Dwti - 1} + \tilde{\rho }_{Lw3} \hat{u}_{Lmt - 1i} + \tilde{\rho }_{Lw4} \hat{u}_{Dmt - 1i} \hfill \\ \end{aligned}$$

(9)

The predicted values, ũ _Lwti, of Eq. (9) are obtained by the OLS regression of the residuals û _Lwti of Eq. (1) on the residuals of Eqs. (2), (3) and (4) and on the lagged residuals, û _t−1,i , of Eqs. (1), (2), (3) and (4).

In this way, the covariances between equations and over-time are incorporated in the estimated OLS coefficients φ and ρ considering the well-known second-order moments properties.

Analogously, we use the second-stage estimated residuals of Eqs. (1), (2), (3) and (4) to compute the following dependent variables, corrected for the common unobserved latent factors such as bargaining and endogeneity of the retirement decision:

$$\begin{aligned} \ln \tilde{L}_{wti} & = \ln L_{wti} - \tilde{u}_{Lwti} ;\quad \ln \tilde{D}_{wti} = \ln D_{wti} - \tilde{u}_{Dwti} \\ \ln \tilde{L}_{mti} & = \ln L_{mti} - \tilde{u}_{Lmti} ;\quad \ln \tilde{D}_{mti} = \ln D_{mti} - \tilde{u}_{Dmti} \\ \end{aligned}$$

(10)

Finally, at the third stage, we substitute the corrected dependent variables (10) into Eqs. (1)–(4), and run OLS and Tobit regressions. To improve the efficiency of estimates, this procedure can be iterated until the estimated coefficients converge.⁸

The result of this correction is satisfactory, as evidenced by the gain of significance of the estimates derived from the iterative process (further details are not given here for the sake of brevity, but are available upon request).