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Lifecycle-consistent female labor supply with nonlinear taxes: evidence from unobserved effects panel data models with censoring, selection and endogeneity

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Abstract

This paper uses the Panel Study of Income Dynamics (PSID) from 1979 to 2007 to estimate within-period lifecycle-consistent labor supply elasticities of US females in a two-stage budgeting framework. The paper combines a variety of econometric approaches to estimate unobserved effects panel data models with censoring, selection and endogeneity. The paper finds evidence of substantial upward bias in estimated wage elasticities from pooled panel models which do not account for unobserved effects, as fixed effects and correlated random effects (CRE) specifications yield smaller elasticities. Estimates are also somewhat sensitive to using a lifecycle-consistent specification versus a standard static model. The lifecycle-consistent wage elasticity from a CRE model with instrumental variables is 0.56 on the extensive margin and 0.31 on the intensive margin for an overall wage elasticity of 0.87. The standard static model, on the other hand, yields a wage elasticity of 0.46 on the extensive margin and 0.13 on the intensive margin for an overall elasticity of 0.59.

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Notes

  1. Unless otherwise indicated, all references to female labor supply in this paper refers only to married women, which remains a key group for targeted tax reform policies to increase labor force participation (Guner et al. 2012). For recent research on single women, see Meyer and Rosenbaum (2001), among others.

  2. For example, Hausman (1980, 1981), Moffitt (1984), Eissa (1995), Eissa and Hoynes (2004), Triest (1990), Blomquist and Hansson-Brusewitz (1990), van Soest et al. (1990), Heim (2007, 2009), and Kumar (2010), among others.

  3. Heckman and Macurdy (1980) and Kimmel and Kniesner (1998) estimated intertemporal elasticities for US females using a lifecycle model but ignored taxation. Other papers that estimated lifecycle models for US females ignoring taxes include Eckstein and Wolpin (1989), Jakubson (1988), Lilja (1986), Lundberg (1988), and Zabel (1997). Johnson and Pencavel (1984) is an exception that did account for taxes.

  4. In the first-stage of two-stage budgeting framework proposed by Gorman (1959), the consumer allocates total expenditure across periods to equate the marginal utility of wealth. In the second-stage, she takes the allocation of wealth between periods as given, and chooses between consumption and hours, like a standard static intratemporal problem.

  5. Blundell and Walker (1986), Blundell et al. (1993, 1998) used two-stage budgeting specifications to estimate lifecycle-consistent elasticities for British females. Aronsson and Wikström (1994) estimated lifecycle-consistent family labor supply model with nonlinear taxes using Swedish cross section data. Ziliak and Kniesner (1999) used the PSID to estimate lifecycle-consistent labor supply elasticities in the presence of nonlinear taxes for a sample of US males.

  6. In line with most other studies on female labor supply with nonlinear taxes, this paper estimates a secondary earners female labor supply model, in a unitary rather than collective framework. As secondary earners, wives make labor supply decisions conditional on husbands having made their labor supply choices. See Chiappori (1988), Cherchye and Vermeulen (2008), among others, for collective models of labor supply.

  7. Among recent papers on female labor supply elasticity, Devereux (2004) used repeated cross-section data from the Census IPUMS and used a grouping strategy to estimate static specifications of female labor supply with group fixed effects. Blau and Kahn (2007) and Heim (2007), who find compelling evidence that female labor supply elasticities are in a long term decline and converging towards men, estimated static models using a time series of cross-section data from the CPS. Gelber and Mitchell (2011) used fixed effects panel data model to estimate the hours elasticity with respect to net-of-tax-rate, for single women, of 0.53 using PSID 1975–2004.

  8. Although panel data facilitates life-cycle consistent estimation, it is by no means absolutely necessary. Such specifications can be estimated even using cross-section data (see MaCurdy 1983; Blundell and Walker 1986).

  9. This finding is different from the results for single women in Gelber and Mitchell (2011) who found that elasticities from fixed effects models were 50 % larger than those without fixed effects.

  10. Ziliak and Kniesner (1999) found negative wage elasticity when conditioning on net saving.

  11. To account for nonlinear taxes virtual net saving was calculated as the sum of net saving and a lump sum transfer akin to the one used for virtual income R it i.e. A t  – (1 + r a)A t–1 + {(τ it  × W it  × H it ) − T it }.

  12. More specifically R it  = Y it  + {(τ it  × W it  × H it ) − T it } where T it is the actual tax liability and τ it  × W it  × H it is what it would have been if the entire earnings were taxed at the marginal tax rate.

  13. Analogous to virtual income R it , following Ziliak and Kniesner (1999), the virtual lagged assets are defined as \(A_{t - 1}^{v} = A_{t - 1} + \left\{ {\left( {\tau_{it} \times W_{it} \times H_{it} } \right) - T_{it} } \right\}/r_{it}\). The reason why this adjustment is made to lagged assets and not to current period assets is, as explained in Ziliak and Kniesner (1999), that income on previous period assets figure into tax calculations.

  14. Hereinafter ω denotes after-tax wage of workers as well as nonworkers with the nonworker’s missing wages replaced by after tax predicted wage \(\hat{\omega }\).

  15. A variety of approaches can be used to estimate the parameters of the labor supply Eq. (3). Some methods account comprehensively for piecewise-linear budget set underlying the derivation of the labor supply function (Burtless and Hausman 1978; Hausman 1981; Blomquist and Newey 2002; Heim 2009; Kumar 2008, 2010; Liang 2011). An alternative approach of imputing the effective marginal tax rate from a differentiable smooth budget constraint methodology proposed in MaCurdy et al. (1990) and Ziliak and Kniesner (1999) was also followed but the results were similar.

  16. A Tobit-type model can also be used to estimate (3) if wages for all females were available. Tobit-type labor supply equations, however, are based on the premise of a continuous labor supply schedule that constrains the parameters of the participation decision and the hours of work decision to be identical and will be biased if participation and hours decisions are separate.

  17. Using imputed wages for non-workers is standard in the literature on female labor supply with taxes. For example, see Hausman (1980), Triest (1990), and van Soest et al. (1990), among others. Wage predictions are obtained from a selectivity-corrected wage equation estimated on a sample of workers. Results are robust to other methods used to impute wages e.g. multiple imputations. To account for the fact that wages for part-time workers could be different, interaction terms between a dummy for part-time worker with other variables, was also included in the selection-corrected wage equation. The results were statistically indistinguishable from current ones.

  18. In the literature on nonlinear budget set estimation with taxes using maximum likelihood, gross wage and full income are treated as exogenous. However, both of these could be endogenous in a lifecycle model due to factor such as human capital accumulation.

  19. Other plausible instruments for the tax rate e.g. marginal tax rate on husband’s earnings based on 2,000 h a years were also used as instruments. The results were similar.

  20. Under the assumption of rational expectations, everything in the information set at time t − 1 and before is exogenous. So the twice-lagged value of the gross wage and full income are considered exogenous are valid for making instruments. First lags are not valid instruments if there is first order serial correlation in the error term.

  21. Analogously, \(\hat{\omega }_{it - 2}^{z}\), the second lag of virtual net saving and time dummies are used as instruments in models applicable with linear capital income taxes and \(\hat{\omega }_{it - 2}^{z} , R_{it - 2}^{z}\) and time dummies are used as instruments in the static specifications, where \(R_{it - 2}^{z}\) is constructed by replacing the actual marginal tax rate in R it with the first dollar tax rate.

  22. PSID collects most labor market information for the year before the survey year, so data from 1979 to 2007 waves refer to years 1978–2006. The main sample of PSID, excluding an oversample of low-income families, has 60,368 observations on wives from 1979 and 2007. Restricting the age to 22–60 years olds resulted in 50,675 observations. 14,158 observations were dropped as wife or head was self-employed, head was a farmer, or household had own business, leaving 36,517 observations. Further, 234 person years were excluded as they were deemed outliers using multivariate outlier detection criteria. 5,899 observations from 1979 to 1982 were dropped as the instrument consisted of 4th lag of real asset. Finally 16089 person years were dropped due to missing data on the dependent variable, or the explanatory variables, or the instruments. Note that the instrument set consists of 2nd, 3rd, and 4th lags of assets.

  23. First, liquid assets were calculated by capitalizing the first $200 of annual household asset income using the 1 month CD rate while the amount above $200 was capitalized using the 3-month treasury bill rate. Liquid assets were then added to home equity to calculate total household asset. Home equity was calculated as the difference between self-reported value of the house and the remaining mortgage and principal amount. The remaining mortgage and principal amount was not available for 1982; PSID-CNEF method was followed to impute the amount by adding half the difference between 1983 and 1981 value to the 1981 value.

  24. The PSID contains more than one measure of the wage rate. One measure can be formed by dividing annual real earnings by the annual hours worked. This measure has been found in the literature to induce division bias in labor supply estimates, yielding parameter estimates inconsistent with theory (Ziliak and Kniesner 1999; Eklof and Sacklen 2000; Engelhardt and Kumar 2007). For hourly workers the hourly wage directly reported by workers was used. For salaried workers, the PSID asked the dollar amount they received in salary and the pay period i.e. once a month, twice a month, or weekly. Assuming that the salaried individual worked 40 hours a week, the dollar amount was divided by the respective number of hours worked during the pay period. Nominal hourly wages are then converted to real 2000 dollars by adjusting with the CPI (U). The log of real wage was used to estimate a selection-corrected wage equation to impute real wages for married women out of the labor force.

  25. Participation elasticities are estimated using three panel methods: standard pooled probit; CRE-Probit methods from Papke and Wooldridge (2008) and Wooldridge (2009); and FE-Logit models. Analogous methods for intensive margin elasticities are standard Heckman model without unobserved effects; selectivity-corrected CRE model from Semykina and Wooldridge (2010); and semiparametric selectivity-corrected hours model with fixed effects from Kyriazidou (1997). A dummy for children less than 7 years was used as an exclusion restriction for the second step selection-corrected hours equation, to calculate hours elasticity in the lower panel of Table 2. The estimation details of various models are available from authors as an online “Appendix”.

  26. A limitation of the fixed effects models is that average partial effects and therefore, elasticities are, in general, not identified, as estimates of the unobserved effects are not available (Wooldridge (2010), Chernozhukov et al. (2009)). For the fixed effect logit, first the predicted probability of participating in the labor force was calculated conditional on a positive outcome for each individual. The mean of this predicted probability was used to calculate the adjustment factor for calculating marginal effects. Having calculated marginal effects, elasticities were calculated by multiplying with the ratio of mean wage to mean labor force participation.

  27. The instrumental variables have significant explanatory power in the first stage regressions for all the three endogenous variables—after-tax wage, lagged asset and current asset - as the p-values on a joint test of instruments were well below 0.05. The p values on the overidentification test, shown in the bottom panel, in columns (2) and (6) suggest that overidentification restrictions cannot be rejected, and therefore the instruments are valid. The p value on the joint test of correlated random effect terms in column (6) indicates rejection of the hypothesis that the terms controlling for correlation between the unobserved heterogeneity and the other model regressors are zero.

  28. Cumulative one period dynamic response has been calculated as suggested in Stock and Watson (2007).

  29. The Hausman test can fail if the difference between variance–covariance matrices of the efficient and consistent specification may not be positive semidefinite. A suggestion in Wooldridge (2010) page 331 is used to calculate the Hausman test statistic for only a subset of coefficients of primary interest, i.e. after-tax wage, virtual lagged asst and current asset.

  30. Due to space constraints robustness results are presented in an extended working paper version of the paper available from http://dallasfed.org/assets/documents/research/papers/2005/wp0504.pdf.

  31. Two alternative sets of grouping instruments were also used. The first set was similar to Blundell et al. (1998) and instruments were formed by interaction of two education groups, 4 year of birth groups, and 19 years. Following Blau and Kahn (2007), wage deciles by year groups were also used as instruments. The estimated participation elasticities are largely robust to use of alternative instruments. p Values on the test of over identifying restrictions on both sets, however, indicated that these grouping instruments are not valid instruments. This could be more due to smaller group sizes in the PSID vis-a-vis larger datasets such as the CPS, rather than due to their validity. Detailed tables on the results of various robustness tests are presented in a working paper version of the paper available from http://dallasfed.org/assets/documents/research/papers/2005/wp0504.pdf.

  32. Analyzing the labor supply of female heads using PSID is complicated due to small sample sizes. Moreover, they are more likely to participate in welfare programs and measuring their marginal tax rates requires a comprehensive welfare benefits calculator not incorporated in the NBER-TAXSIM model.

  33. Using non-PSID datasets, papers such as Cogan (1981), Heckman and Macurdy (1980), and Kimmel and Kniesner (1998) estimated total hours elasticities larger than 2. Estimated elasticities are closer to those in Eissa (1995) who, using the CPS, estimated a total hours elasticity of 0.8 for high income women with roughly half of it on the extensive margin. The estimated participation elasticity of 0.56 from the CRE-IV model is more than twice that of 0.27 found in Eissa and Hoynes (2004) while the intensive margin elasticity is larger than 0.2 reported in Devereux (2004). Both participation and intensive margin elasticities from the CRE specifications are within the range of those in Blau and Kahn (2007) who found that participation elasticities declined from 0.58 to 0.28 from 1980 to 2000 while the hours elasticities dropped from 0.3 to 0.12. Heim (2009) also found a similar decline with participation elasticities declining from 0.66 to 0.03 and hours elasticities shrinking from 0.36 to 0.14.

  34. The estimated elasticities can also be used to recover the λ-constant (Frisch) elasticity- by using the formula \(e_{\lambda } = e_{W}^{comp} - e^{IS} e_{{A_{t} }}^{2} \left( {WH/A_{t} } \right)\), where e IS is the intertemporal substitution elasticity (Browning 2005). An estimate of the intertemporal substitution elasticity is needed to recover e λ . With \(e_{{A_{t} }}^{2} \left( {WH/A_{t} } \right) = 0.008\), e λ is not much different from the compensated elasticity \(e_{W}^{comp}\) for most plausible estimates of e IS. Using an estimated e IS of −0.69 from Blundell et al. (1993), the implied e λ for the lifecycle-consistent CRE specificationis 0.92, about the same as the compensated elasticity.

  35. The compensated elasticity on labor supply can be recovered using the formula \(e_{W}^{comp} = e_{W}^{uncomp} - (WH/A_{t} )e_{{A_{t} }}\) where \(e_{W}^{comp} , e_{W}^{uncomp} , e_{{A_{t} }}\) are the uncompensated wage, compensated wage, and wealth elasticity, respectively. As the married womens’ earnings relative to assets (WH/A t ) = 0.19, the compensated elasticity from the CRE model with IV is 0.91.

  36. Deadweight loss equals \((1/2)\tau^{2} e_{W}^{comp} WH/(1 - \tau ),\) where τ, is the marginal tax rate. This expression equals 0.028WH if τ = 0.25 and 0.034WH if τ = 0.27.

  37. The calculation attempted here is at best a crude measure of deadweight loss from tax changes. Eissa et al. (2008) showed that the relevant tax rates for welfare cost calculations on the participation margins may be different. While the effective marginal tax is valid for the intensive margin, average rate is appropriate for the participation margin. Further, the calculation using labor supply elsticity ignores other margins of behavioral response i.e. effort, avoidance etc. A more comprehensive measure can be calculated using response on taxable income (Saez et al. 2012).

  38. A vector of residuals, \(V_{it} = (v_{it}^{{\omega_{it} }} , v_{it}^{{A_{it} }} , v_{it}^{{A_{it - 1}^{v} }} )\), from the first stage linear fixed effects regressions of each of the endogenous variables ω it , \(A_{it - 1}^{v}\), and A it on instruments \(\hat{\omega }_{it - 2}^{z} , A_{t - 3}^{vz} , and \, A_{it - 4}\) and other exogenous variables X it is included in the model.

  39. Typically such CRE specifications are applicable only for balanced panels. However, following Wooldridge (2009), CRE can be adapted for unbalanced panels by assuming that in addition to strict exogeneity of covariates selection into the panel is also strictly exogenous i.e. dropping out of the panel is not systematically correlated with ϵ it .

  40. Note that time means of endogenous variables are not included in the second stage CRE specification. Instead the time means of exogenous variables X it and instruments Z it are included.

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Acknowledgments

I am grateful to Gary Engelhardt for his constant encouragement and generous support during this research. I thank Sonia Oreffice, Soren Blomquist, Monica Singhal, Tom Kniesner, Dan Black, Duke Kao, Jeffrey Kubik, Jan Ondrich, Michael Weiss, two anonymous referees and seminar participants at Syracuse University, Miami University, Federal Reserve Bank of Dallas, RAND Corporation, Institute of Social Research at the University of Michigan, conference participants at the 17th Annual Meetings of the Society of Labor Economists (SOLE) in Chicago, 99th Annual Conference on Taxation in Boston, and the 13th International Panel Data Conference in Cambridge, England for helpful discussions and insightful comments on previous versions.

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    Anil Kumar

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Correspondence to Anil Kumar.

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Previous versions of this paper were also circulated under two different titles: (1) “Lifecycle Consistent Estimation of Female Labor Supply” and (2) “Taxes, Deadweight Loss and Intertemporal Labor Supply: Evidence from Panel Data”. The views expressed here are those of the author and do not necessarily reflect those of the Federal Reserve Bank of Dallas or the Federal Reserve System.

Appendix 1: Estimation details

Appendix 1: Estimation details

1.1 Pooled panel data models

As a benchmark for comparison with models with unobserved effects, the paper first estimates simple pooled panel data models assuming that the dual error term α i  + ϵ it , in the labor supply Eq. (3), is a normally distributed random error that is uncorrelated with other regressors. The participation equation is estimated using pooled Probit. The selection-corrected version of the labor supply Eq. (3) on pooled data is estimated using a Heckman-type framework with inverse mills ratio from a first step labor force participation equation (Heckman 1979). Endogeneity of ω it , \(A_{it - 1}^{v}\), and A it is addressed using conventional two-stage procedure. Estimates from pooled models will be biased if any of the right hand side variables or the instruments are correlated with α i An option is to use a first-differencing or fixed effects specification.

1.2 Fixed effect models

1.2.1 Labor force participation models with fixed effects

Parameters of labor force participation equation are consistently estimated using fixed effects Logit models (Andersen 1970; Chamberlain 1980). Endogeneity is corrected by extending the control function approach proposed in Blundell and Powell (2004).Footnote 38

1.2.2 Panel selection-corrected hours equations with fixed effects

This paper follows the selection-correction set-up in Heim (2009), except that the model incorporates individual specific fixed effects in both the selection equation and the hours equation. The first-step selection equation is a labor force participation equation with fixed effects:

$$D_{it}^{LFP} = X^{S} \pi_{1} + \eta_{i} + e_{it} ,$$
(4)

where \(D_{it}^{LFP}\) is a dummy variable for whether the individual participates in the labor force, X S is a vector of covariates included in the selection equation, and η i are individual specific unobserved effects. The second step selection-corrected hours equation conditional on labor force participation can be written as:

$$H_{it} = \beta_{0} + \beta_{1} \omega_{it} + \beta_{2} A_{it} + \beta_{2} A_{it - 1}^{v} + X_{it} \gamma + \varphi (X_{it}^{S} \hat{\pi }_{1} + \eta_{i} ) + \alpha_{i} + \epsilon_{it}$$
(5)

where \(\varphi (X^{S} \hat{\pi }_{1} + \eta_{i} )\) is the inverse mills ratio obtained from the first step. Although, individual specific effects α i can be eliminated by differencing, η i enters the equation nonlinearly through the inverse mills ratio and cannot be eliminated by differencing.

The paper applies a semiparametric selection correction method proposed in Kyriazidou (1997). First, parameters \(\hat{\pi }_{1}\) of the selection Eq. (4) are estimated using a consistent estimator e.g. fixed effect Logit. Then for any two time periods, t − 1 and t in which the female participates in the labor force, if \(X_{it}^{S} \hat{\pi }_{1} = X_{it - 1}^{S} \hat{\pi }_{1}\) i.e. \(\Updelta X_{it}^{S} \hat{\pi }_{1} = 0\) ,then first differencing (5) would difference away not only α i but also η i and \(\varphi (X_{it}^{S} \hat{\pi }_{1} )\). Because \(X_{it}^{S}\) has both continuous and discrete elements, \(\Updelta X_{it}^{S} \hat{\pi }_{1}\) is unlikely to equal zero. However, estimation can be restricted to observations for which, \(\Updelta X_{it}^{S} \hat{\pi }_{1}\) is small. Better still, using all observations and estimating the following weighted least squares equation for labor force participants with weights going to zero if \(|\Updelta X_{it}^{S} |\) increases, yields consistent estimates:

$$\hat{\xi }\Updelta H_{it} = \beta_{1} \hat{\xi }\Updelta \omega_{it} + \beta_{2} \hat{\xi }\Updelta A_{it} + \beta_{2} \hat{\xi }\Updelta A_{it - 1}^{v} + \hat{\xi }\Updelta X_{it} \gamma + \hat{\xi }\Updelta \epsilon_{it} ,$$
(6)

where \(\hat{\xi } = \left( {\frac{1}{{{\text{h}}_{\text{n}} }}} \right){\text{K}}\left( {\frac{{\hat{\pi }_{1} \Updelta X_{it}^{S} }}{{{\text{h}}_{{\rm n}} }}} \right)\) is a kernel weight function with hn, the bandwidth.

Endogeneity in (6) is dealt with following a straightforward application of the two-stage procedure adopted in Charlier et al. (2001) with \(\Updelta \hat{\omega }_{it - 2}^{z} , \Updelta A_{t - 3}^{vz} , and \, \Updelta A_{it - 4}\) used as instruments.

Semiparametric fixed effects methods solve the problem of correlated unobserved heterogeneity, heteroscedasticity, and nonnormality of the error distributions. However, an important limitation is that average partial effects and therefore, elasticities are, in general, not identified, as the unobserved effects are not estimated (Wooldridge (2010)). Papke and Wooldridge (2008) and Wooldridge (2009) showed that estimating average partial effects is feasible in a correlated random effects framework.

1.2.3 Correlated random effects (CRE) probit models

Letting α i be a linear function of time means of \(\omega_{it} ,A_{it} ,A_{it - 1}^{v} , and \, X_{it},\) so that \(\alpha_{i} = \zeta_{0} + \zeta_{1} \bar{\omega }_{i} + \zeta_{2} \bar{A}_{i} + \zeta_{3} \bar{A}^{v}_{t - 1} + \varsigma \bar{X}_{i} + a_{i}\) and substituting in (3) yields the CRE specification,

$$H_{it}^{*} = \kappa_{0} + \beta_{1} \omega_{it} + \beta_{2} A_{it} + \beta_{2} A_{it - 1}^{v} + \gamma X_{it} + \zeta_{1} \bar{\omega }_{i} + \zeta_{2} \bar{A}_{i} + \zeta_{3} \bar{A}^{v}_{t - 1} + \varsigma \bar{X}_{i} + u_{it} ,$$
(7)

Assuming strict exogeneity and normality of the composite error term u it  = a i +ϵ it , (7) is estimated using a Probit.Footnote 39

Control function approach proposed in Papke and Wooldridge (2008) and Wooldridge (2009) is used to account for endogeneity. Letting X it , represent the exogenous variables in the model and Z 1it , the instruments \((\hat{\omega }_{it - 2}^{z} , A_{t - 3}^{vz} , and \, A_{it - 4} )\), so that Z it  = (X it , Z 1it ) represents the vector of all exogenous variables and instruments. Also let \(\bar{Z}_{i} , \bar{X}_{i} ,\bar{Z}_{1it}\) be the time means of all exogenous variables, included exogenous variables, and excluded instruments, respectively. The first stage in (7) consists of regressing each endogenous variable, \(\omega_{it} , A_{it} , A_{it - 1}^{v}\), on Z it and \(\bar{Z}_{i}\) and getting the vector of residuals \((v_{it}^{{\omega_{it} }} , v_{it}^{{A_{it} }} , v_{it}^{{A_{it - 1}^{v} }} )\). In the second stage, hours or labor force participation is regressed on \(\omega_{it} , A_{it} , A_{it - 1}^{v} ,X_{it} ,\bar{Z}_{i} ,v_{it}^{{\omega_{it} }} , v_{it}^{{A_{it} }} , v_{it}^{{A_{it - 1}^{v} }}\).Footnote 40

1.3 Correlated random effect models with selection correction

Semykina and Wooldridge (2010) proposed a parametric method to account for correlated individual effects in the selection Eq. (4) as well as the main Eq. (5), which combines the typical Heckman-type selectivity correction with correlated random effects models similar in spirit to (7). In the specification without endogeneity, in the first step, year-specific inverse mills ratio, \(\hat{\lambda }_{it}\) are obtained by running a Probit of \(D_{it}^{LFP}\) on \(X_{it}^{S}\) and \(\bar{X}_{i}^{S}\) for each t. In the second step, the specification in (7) is augmented with \(\theta \hat{\lambda }_{it} + \mathop \sum \limits_{t = 1}^{T} \theta_{t} dt \times \hat{\lambda }_{it}\).

To account for endogeneity, Semykina and Wooldridge (2010) proposed 2SLS estimation similar in spirit to Papke and Wooldridge (2008) and Wooldridge (2009). In the first step a Heckman-type selection equation is estimated by running a reduced form Probit of labor force participation on Z it and an exclusion restriction dkidsu7 for each year separately and the year-specific inverse mills ratio for each labor force participant, λ it , saved. In the second step hours is regressed on \(\omega_{it} , A_{it} , A_{it - 1}^{v} ,X_{it} ,\bar{Z}_{i} ,\hat{\lambda }_{it} ,\) \(dt \times \hat{\lambda }_{it}\) in a two stage least squares framework treating \(\omega_{it} , A_{it} , A_{it - 1}^{v}\) as endogenous.

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Kumar, A. Lifecycle-consistent female labor supply with nonlinear taxes: evidence from unobserved effects panel data models with censoring, selection and endogeneity. Rev Econ Household 14, 207–229 (2016). https://doi.org/10.1007/s11150-013-9217-6

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