Family-oriented versus career seekers: mixture regression separation

Abstract

Using a finite mixture regression approach, we identify two groups of individuals, family- and career-oriented. Depending on whether the employment status is treated as a binomial or trinomial (with part-time work) variable, 55–70% of women and 4–10% of men belong to the family-oriented group. Children variables are the main source of differences between the two identified groups and between the family-oriented group and the average woman. Treating all women as one group leads to averaged-out estimates of children effects and underestimates the importance of part-time work for family-oriented women. Our results are important for studying women’s labour supply and for designing policies aimed at encouraging women’s labour market involvement.

1 Introduction

In the modern society, women and men face a rich assortment of family, career and social engagements, more so than ever before. Both men and women depart from their traditional family and gender roles, and for either gender the labour market behaviour becomes more heterogeneous. The traditional estimation of labour supply treats women and men as two separate homogeneous cohorts. If the sample is very heterogeneous, say, some women are more career-oriented and some are more family-oriented, such estimates can be of limited use. In this paper, we revisit labour supply estimation without a priori aggregating individuals by gender. Instead, we allow them to “choose” a bundle of family and career objectives. Such kind of estimation produces more reliable estimates, which are crucial for designing a better labour market policy.¹

Typically, heterogeneity in discrete choice models is tackled by either introducing unobservable effects into the panel data, or allowing flexible distribution of the error terms, or relying on the nonparametric methods. Neither of these approaches is geared towards identification of homogeneous groups in a heterogeneous sample, which is the approach taken in this paper. Instead of pooling individuals by gender, we allow them to sort themselves into homogeneous groups with respect to their labour market participation. For this purpose, we employ a mixture regression approach. It provides separate estimates for the identified groups and is, therefore, more relevant for designing labour market policies.

We apply finite mixture regressions model to the data from the Household Income and Labour Dynamics in Australia (HILDA) data set. Australian labour market is representative of a developed economy. In particular, Australian female labour force participation rates are similar to those in the UK; they are somewhat lower than those in Canada and slightly exceed those in the USA. We identify two groups of respondents with very different relationships between having children and labour supply. The employment of the family-oriented group is strongly negatively correlated with children variables. When they have children, the family-oriented are much less likely to be employed than an average woman. In contrast, the employment of the career-oriented is virtually unrelated to whether they have children or not. Their employment pattern resembles that of an average man.

In the binomial employment status model, 55% of women and 4% of men are estimated to be family-oriented. In the trinomial model that allows for part-time work, these shares increase to 70% of women and 10% of men. The trinomial model also suggests that the family-oriented group prefers part-time to full-time work: roughly 45% of the family-oriented is estimated to work part-time, and 35%, full-time. For all women, these numbers are nearly in reverse: 44% is estimated to work full-time, and 33%, part-time. The career-oriented group is estimated to work full-time with probability 80%, which is similar to men. Additional concomitant variables provide further insights into the composition of the family- and career-oriented groups. In particular, both religious men and women are more likely to be family-oriented.

Our results highlight that the most important factor that affects self-selection of women into the family and the career-oriented groups is children. Labour market participation of women is heterogeneous, and pooling all women together, which is currently the common practice, may lead to overgeneralized results and inadequate policy design. Consider a policy that aims to encourage mothers’ employment. To draw mothers to the labour force, the policy has to “compensate" them for the loss of utility they derive from spending time with their children. Naturally, this utility is larger for the family-oriented mothers and smaller for the career-oriented ones. If the policy is based on the estimates for an average woman, it is likely to be insufficient for the family-oriented mothers. It may also be excessive for the career-oriented group who require little or no encouragement to return to labour force after the childbirth. Section 5 further explores how this may affect the design of child care subsidies and rebates.

Our work also suggests that the family-oriented respondents are inclined to work part-time considerably more than the average woman. Hence, a traditional estimation by gender undervalues the efficacy of labour market policies that encourage part-time work and flexible working arrangements. Such policies can be the answer to attempts to encourage the GDP growth without compromising the natural population growth rate, which is already pretty modest in the developed world. This paper contributes to development of such labour market policies.

The rest of the paper is organized as follows: Section 2 describes the finite mixtures approach in general. Section 3 provides the data description. Section 4 presents the results for the binomial model. Section 5 explains the implications of our results for the design of the child care subsidy. Section 6 applies finite mixtures to the trinomial model that includes part-time work, and Sect. 7 concludes.

2 Finite mixtures and related literature

Mixture regressions are used for identifying homogeneous groups in a heterogeneous sample. In a finite mixture regression framework, the group, to which a respondent belongs, is not observed and is treated as a missing variable. Suppose that observation i belongs to group j. Let $f_{j}\left( y_{i}|X_{i};\beta _{j}\right)$ denote the corresponding conditional probability distribution of the dependent variable $y_{i}$ given the explanatory variables $X_{i}$ and coefficients $\beta _{j}.$ Next, suppose there are J distinct groups, and observation i belongs to group j with probability $\pi _{j},$ a mixture proportion. Whether i actually belongs to group j is unknown a priori and is treated as a missing variable. For that purpose, $\pi _{j}$ is written as $\pi _{j}\left( Z_{i};\gamma _{j}\right)$ with $\sum _{j}\pi _{j}\left( Z_{i};\gamma _{j}\right) =1$ for every i where certain characteristics of i — the concomitant variables $Z_{i}$ — influence the mixture proportions. Given this structure, the probability distribution of $y_{i}$ can be written as

$$\begin{aligned} f\left( y_{i}|X_{i},Z_{i},\mathbf {\beta },\mathbf {\gamma }\right) =\sum _{j=1}^{J}\pi _{j}\left( Z_{i};\gamma _{j}\right) \cdot f_{j}\left( y_{i}|X_{i};\beta _{j}\right) . \end{aligned}$$

(1)

We will further provide parametric functional forms for $f_{j}$ and $\pi _{j}$. Once these are specified, (1) straightforwardly leads to the likelihood function. The unknown $\mathbf {\beta }=\left( \beta _{1},...,\beta _{J}\right)$, the matrix of coefficients on the explanatory variables, and $\mathbf {\gamma =}\left( \gamma _{1},...,\gamma _{J}\right)$ , the matrix of coefficients on the concomitant variables, are to be estimated jointly. For identification purposes, some of the explanatory variables $X_{i}$ do not enter mixture proportions $\pi _{j}$, while some of the concomitant variables $Z_{i}$ do not enter the explanatory relationships $f_{j},$ see Henry et al. (2014). In this respect, finite mixture regression model is similar to Heckman’s sample selection model.

With parametric and nonparametric specifications of f and $\pi ,$ the flexibility of the mixture models has proven to be very useful. Their applications range from labour economics and industrial organization to microeconometric analysis of games. Ahamada and Flachaire (2010), Compiani and Kitamura (2016) and Kitamura and Laage (2018) provide recent surveys.² Wedel and Kamakura (1999) survey the use of mixture regression models in marketing literature. A related class of mixture density models, where the set of the concomitant variables is empty, is used in income inequality studies, e.g., Flachaire and Nunez (2007).

The closest to us is the study of Ahamada and Flachaire (2010). They estimate linear Mincer earnings equations and identify two homogeneous groups with very different relationships between the earnings profile and labour market experience.³ We introduce finite mixtures into nonlinear discrete choice models of labour supply and identify homogenous groups of family- and career-oriented respondents.

A popular alternative to finite mixtures is a quantile regressions method, see Koenker (2005) for a survey. Kordas (2006) estimates the determinants of employment status in such a framework. The relative effects of regressors change as one spans through the quantiles that correspond to different propensities of being employed. In principle, one can associate higher quantiles with the career-oriented and lower quantiles with the family-oriented groups, however, such assignment is necessarily ad hoc. This highlights the advantage of the finite mixture regression approach used in this paper, since the composition of the career- and family-oriented groups is suggested by the data.

3 The data

The date comes from the Household Income and Labour Dynamics in Australia (HILDA) data set, which is nationally representative and contains valuable labour market and family characteristics at the individual level. We use wave 18 collected in 2018. This is the last wave that was collected prior to the Covid-19 pandemic.

Australian labour market is representative of a developed economy. In particular, Australian female labour force participation rates are similar to those in the UK, they are somewhat lower than those in Canada, and in the recent years slightly exceed those in the USA.⁴ For an overview of the variables used for estimation refer to Appendix 1. Here, we discuss the summary statistics for these variables presented in Table 1.

Table 1 Means of variables, by gender and work status

There are 8339 respondents in our sample with 54.4% of women and 45.6% of men. The mean characteristics for them are presented in Table 1 that further sub-divides the sample into working and non-working respondents. We first highlight the differences between the working and non-working groups for the two genders, and then the differences across the genders.

For either gender, the non-working group is older on average, but the working group has more average work experience. The gap in experience between working and non-working men is small, even though the non-working men are 8.7 years older on average. Working women have 5.7 years of extra experience, even though they are 5.2 years younger on average than the non-working women.

The working group has more respondents with a university degree for every gender. A working man is twice as likely to have a higher education as the non-working man. A working woman is 2.5 times more likely to have a degree than a non-working woman. On the other hand, for either gender, the share of people with professional and trade education or completed high school is similar for working and non-working groups.

The share of respondents with one child is also similar across the working and non-working groups, but among the non-working women 74% have two or more children vs. 53% for the working women. Poor health is clearly more prevalent in the non-working group, especially for men. 65% of the non-working men report a detrimental health condition that prevents them from working normally. Immigrant status is more prevalent among the non-working women, but for men this goes in reverse, an immigrant man is more likely to work. A married respondent is more likely to work, regardless of the gender. Non-labour income is higher for a non-working respondent, again regardless of the gender. Non-working respondents of either gender are also more likely to be religious.

The comparison of the summary statistics across the genders reveals fairly modest differences in the observables. Women have less work experience, but are more likely to have a university degree. Men are more likely to have a trade education. Women have higher non-labour income and are slightly more religious than men. Women are more likely to have two or more children.

It is remarkable how modest are these gender differences in the regressors in comparison to the differences in the employment status of women and men. Almost twice as many women as men do not work: 22.6% of the women vs. 12% of men. Women are also 3.5 times more likely to work part-time: 32.6% of women vs. 9.2 % of men. And finally, almost twice as fewer women work full-time and overtime: 44.8% vs. 78.9%. It appears that the differences in the observable regressors account only for a small portion of the differences in the labour market behaviour between the genders. Much of this difference is therefore due to the unobservable heterogeneity.

Unobservables are often tackled by differencing them out in a panel data framework. Since we are interested in the very effects of the unobservables, in particular, those related to the gender roles, we do not difference them out. Instead, we explore the heterogeneity of the data driven by the unobservables using the mixture regression method. This method allows the individuals to “self-select” into different labour market roles.⁵

We illustrate our approach on the binomial model first and contrast it with a traditional probit estimated on the sub-samples exogenously separated by gender. We then proceed to the model with a trinomial dependent variable.

4 Binomial model

In this section, we use a mixture regression binomial dependent variable model and compare it with the standard gender-based approach, in this case, probit estimation.

In the binomial model, the dependent variable $y_{i}$ takes on two values: $y_{i}=1$ if the respondent works any positive number of hours, and $y_{i}=0$ otherwise. Conditional on observation i being in group j and assuming the usual parametrization for the latent variable, $y_{i}^{*}=X_{i}\beta _{j}+\varepsilon _{i}$ with standard normal $\varepsilon _{i},$ the probability of observing $y_{i}$ is

$$\begin{aligned} f_{j}\left( y_{i}|X_{i};\beta _{j}\right) =\Phi \left( X_{i}\beta _{j}\right) ^{y_{i}}\cdot \Phi \left( -X_{i}\beta _{j}\right) ^{1-y_{i}}, \end{aligned}$$

(2)

where $\Phi$ is the standard normal c.f.d.⁶ Let J be the number of the groups, and let $\pi _{j}\left( Z_{i};\gamma _{j}\right)$ denote the probability that observation i belongs to group j. The likelihood contribution of i is then

$$\begin{aligned} l_{i}\left( y_{i}|X_{i},Z_{i},\beta _{1},\beta _{2},\gamma \right) =\sum \nolimits _{j}\pi _{j}\left( Z_{i};\gamma _{j}\right) \cdot f_{j}\left( y_{i}|X_{i};\beta _{j}\right) . \end{aligned}$$

Finally, assuming $\pi _{j}\left( Z_{i};\gamma _{j}\right) =\exp \left( Z_{i}\gamma _{j}\right) /\sum \nolimits _{j}\exp \left( Z_{i}\gamma _{j}\right) ,$ and independence of $\varepsilon _{i}$ across the observations, the log-likelihood function is $\mathcal {L} =\sum _{i=1}^{n}\ln l_{i}\big ( y_{i}|X_{i},Z_{i},\beta ,\gamma \big ) .$ The vectors of the coefficients $\beta =\left( \beta _{1},...,\beta _{J}\right)$ and $\gamma =\left( \gamma _{2},...,\gamma _{J}\right)$ are estimated jointly via maximum likelihood.

For identification purposes, the usual exclusion restrictions apply: at least one variable in X should not be contained in Z and vice versa, at least one variable in Z should not be part of X. The first exclusion restriction is easy to satisfy. Having a higher education, in particular, does not necessarily affect the person’s intrinsic desire to work hard or not. For the second exclusion restriction, the concomitant variables Z are chosen in such a way that they affect labour supply decisions and, hence, which style of labour market behaviour people choose but not the market conditions provided by the employers, i.e. they do not affect labour demand. Female dummy and religion dummy in Z serve the purpose due to the anti-discriminatory nature of the Australian labour market.

We have three model specifications for the mixture regression estimation: Model 1 is the base model, with female dummy being the only concomitant variable. Model 2 adds religious status to the concomitant variables. Model 3 further adds female dummy to the explanatory variables while having religious dummy in Z for identification. Changing the set of concomitant variables allows to explore the composition of the identified groups.

Model 3 with the female dummy in X enriches the analysis further as it allows for different employment patterns for men and women within the groups. In other words, instead of two homogeneous groups of career- and family-oriented, there can be a difference between career-oriented men and women (or family-oriented men and women).

All estimation results in this paper are presented as average partial effects (APEs) of the explanatory variables, with the standard errors for all APEs calculated by the $\delta$-method.⁷ It is understood that the decisions on the career path and education, and on how many children to have are interrelated. We use the APEs of regressors to identify the homogeneous groups and as a measure of the difference between the groups, and therefore interpret the results as “correlations” between the explanatory and dependent variables without suggesting any causal inferences.

We identify two homogeneous groups in mixture regressions with binomial dependent variable. Table 2 shows the differences between the standard probit and the mixture regression approaches. Columns (1–2) report the results for women from the probit regression, and the rest contains the results for Group 1 from mixture regressions.

Table 2 Women versus family-oriented group, binomial models, average partial effects, %

Notice that experience, and higher and trade education positively correlate with employment, while age, a detrimental health condition and non-labour income have negative correlation with employment in every estimation.⁸ Having children also negatively correlates with employment in every estimation, but the APEs of children variables in the mixture regression models are much more negative than in the probit regression on women. The APE of having one child is nearly 50–100% larger by the absolute value for Group 1 in mixture regressions than for women in probit, and their APE of having two or more children is 2–3 times as large as that for women.⁹ Group 1 in mixture regressions contains respondents whose labour supply is very sensitive to having children, and we tag this group “family-oriented”.

To understand the difference between the family-oriented and all women, we need to know what percentage of respondents self-select into the family-oriented in the mixture regressions. The groups’ composition is determined by the coefficients on the concomitant variables. In mixture model 1, with female dummy being the only concomitant variable, 55% of women (st. err. 5%) and 4% of men (st. err. 2%) belong to the family-oriented. This corresponds to roughly 2650 of the family-oriented people in the sample of 8339 respondents. Mixture models 2 and 3 add religious status to the concomitant variables. Religiosity implies that a respondent is more likely to be family-oriented. In models 2 and 3, this group contains 5% of religious men (st. err. 2%) and 64–69% of religious women (st. err. 6%). The predominance of religious women may explain the negative correlation between marriage and employment in mixture models 2 and 3. Model 3 adds female dummy to the explanatory variables and suggests that a family-oriented woman is less likely to be employed than a family-oriented man.

Table 3 shows the results of probit estimation for men and for Group 2 in the mixture regressions. Group 2 contains about 95% of men (st. err. 2%). It also contains 45% (st. err. 5%) of all women in Model 1, 36% (st. err. 6%) or religious and 51% (st. err. 5%) of non-religious women in Model 2, and 31% (st. err. 6%) or religious and 42% (st. err. 6%) of non-religious women in Model 3. The relationships between the explanatory and dependent variables for Group 2 resembles that in the probit regression for men. In contrast to Group 1, the employment of Group 2 is unrelated to whether they have children or not, hence we tag this group “career-oriented.” In contract to the family-oriented, the career-oriented are more likely to work if they are married and if they have migrated from overseas. In mixture model 3, the female dummy turns out insignificant. In other words, a career-oriented woman’s employment pattern is at par with that of a career-oriented man, once other observable regressors are controlled for.

Table 3 Men versus career-oriented group, binomial models, average partial effects, %

These comparisons suggest, first, that labour market involvement is more heterogeneous for women than for men. Second, an estimation that pools all women together, as in Column 1 of Table 2, results in averaged out estimates that lie in between the estimates for the family- and career-oriented groups. As a consequence, treating all women as a homogeneous sample can be misleading for policy design. Say a policy attempts to encourage mothers to return to the labour force. To do so it has to compensate the mother for the loss of utility she would have derived by spending time with her children. For a family-oriented mother, such loss is arguably larger than for the average mother.¹⁰ The estimation procedure that pools all women together is likely to underestimate that loss of utility for the family-oriented respondents. If the policy is based on the estimates obtained for an average woman, it would be insufficient to encourage employment of the family-oriented respondents, and it may be unnecessary for the career-oriented. Section 5 explains the implications of this in the canonical child care subsidy model.

5 Child care subsidy design

This section shows that a child care subsidy designed for an average woman may be insufficient for the family-oriented. Subsidies that reduce the cost of the paid child care are used in many developed countries. It is the main form of the Australian Government financial support to parents, see Bray et al. (2022).

We first explain the effect of the subsidy in the canonical one-person labour supply model, see Blau (2003). The mother is assumed to be the sole income earner and caretaker of her child. She can purchase child care at the fixed price p per hour. Such paid care is required for every hour the mother works. Her wage rate is fixed at w per hour. Normalizing the price of the consumption good to one, the mother’s budget constraint is $c=y+\left( w-p\right) h,$ where c is consumption of goods, y is non-labour income and h is hours of work. This constraint is labeled “no subsidy” in Fig. 1. The time constraint is $h+\ell =1,$ where $\ell$ is hours of leisure (time spent with the child). The mother’s utility is $u(c,\ell ),$ increasing in both arguments and appropriately concave. Figure 1 shows the indifference curves for the average woman (thin curve) and for the family-oriented (thick curve).

A child care subsidy s reduces the price of child care to $p-s$ and makes the mother’s budget constraint $c=y+\left( w-p+s\right) h$. This constraint is steeper in the $(\ell ,c)$ space. The optimal number of work hours $h^{*}$ is given by the tangency of the mother’s indifference curve and her budget constraint with the subsidy, as in Fig. 1. The key element that our analysis illustrates is that the indifference curves of the family-oriented respondents are steeper than those of the average women. A mother who perceives that she is a high-quality caregiver can be interpreted in this simple model as having a high marginal utility of leisure, see Blau (2003), page 468. A large negative correlation between children variables and employment in Table 2 signals that for a family-oriented mother spending time with her children provides enough joy to compensate for the resulting loss of labour income. With that in mind, a child care subsidy that induces the average woman to work $h^{*}$ hours (see Fig. 1) is insufficient to bring the family-oriented mother to the workplace (they choose a corner solution $h=0$ instead).¹¹

Fig. 1

Child care subsidy designed for the average woman

Clearly any one-size-fits-all policy is bound to be inefficient when applied to a heterogeneous sample. Without assigning a particular utility function, it is hard to quantify the welfare loss associated with sub-optimal policies. Our estimations in Sect. 4 show that over 55% of women, and at least 30% of all respondents in our sample are family-oriented. This implies that an issue at stake is quite large, at least in terms of the number of people affected. Large academic and popular literature on parental burnout suggests this being a major problem even prior to the 2019–21 pandemic.

How much improvement can be achieved by tailoring the policy according to the features of the target groups? This can lead to welfare gains if family or career group “membership” can be accurately predicted by the observable and verifiable respondents’ characteristics. Gender alone reasonably well predicts career-oriented group membership for men. Religious women can be arguably assigned to the family-oriented group. A non-religious woman, however, has only about 50% chance of being family-oriented in the binomial mixture regression models.

This suggests several possible improvements. First, a model that allows for part-time work in Sect. 6, in fact, predicts group membership better even with female dummy being the only concomitant variable. Second, further concomitant variables, like a religious status, add in accuracy of the group membership prediction.

A third possible improvement is an alternative to the first two. Rather than relying on finding observable characteristics to predict group membership, it treats family- vs. career-oriented as an unobservable type and induces the respondent to reveal their type by choosing a child care package from a menu. Suppose a respondent is offered a choice between two packages $\textbf{ c}$ and $\textbf{f}$, shown on Fig. 1. Package $\textbf{f}$ severely limits the number of working hours, but offers a higher per-hour child care subsidy. Package $\textbf{c}$ allows more working hours, but at a lower per-hour child care subsidy.

With the indifference curves as in Fig. 1, a family-oriented respondent will choose package $\textbf{f}$. Since the indifference curves of the career-oriented respondent are flatter, they will choose package $\textbf{c}$ . With this logic in mind, a nonlinear child care subsidy, where the subsidy rate is decreasing with the working hours, is a step in the right direction. A menu of packages such that family- and career-oriented respondents prefer a package intended for their type can always be constructed as long as the MRSs between income and leisure for the two groups are well quantified.

6 Trinomial model

The binomial model reveals the heterogeneity of women’ behaviour in the labour market and, hence, the poor fit of estimation on the exogenously determined sample. Part of the reason is that the binomial model does not take into account part-time work, which is very important for women. As discussed in Sect. 3, as many as 32.7% of women work part-time, compared to 9.2% of men. So, if we are to exogenously determine the sample, a more detailed dependent variable that includes part-time work is called for.

A mixture model with a trinomial dependent variable (full-time work, part-time work and no work) produces a tighter correspondence between gender and group membership. Now more than 70% of women belongs to the family-oriented group, rather than roughly 50% in the binomial model. Around 90% of men belong to the career-oriented group. We use the same mixture regression models in trinomial setting as before: with the female dummy as the only concomitant variable in Model 1, with female and religion dummied as concomitant variables in Model 2 and further adding female dummy to regressors in Model 3. We also allow for more than two homogeneous groups.

In this section $y_{i}$ takes on three values: no work, $y_{i}=0$, part-time work, $y_{i}=1$, and full and over-time work, $y_{i}=2$. Introduce a latent continuous variable $y_{i}^{*}=X_{i}\beta _{j}+\varepsilon _{i}$ with a standard normal $\varepsilon _{i}.$ Employment status suggests an obvious order, in which $y_{i}^{*}$ would be larger for the observations with more employment. Introduce two group specific thresholds $\rho _{j}^{1}\le \rho _{j}^{2}$ for $j=1,2.$ Given the thresholds, respondent $X_{i}$ from group j is predicted to be not working if $y_{i}^{*}\le \rho _{j}^{1},$ working part-time if $\rho _{j}^{1}<y_{i}^{*}\le \rho _{j}^{2},$ and working full or over-time if $\rho _{j}^{2}<y_{i}^{*}.$ Thresholds $\rho$ are estimated together with $\beta _{1}$ and $\beta _{2}$ for the corresponding groups where $\beta$s do not include intercepts.¹²

The probability of observing $y_{i}$ conditional on i being in group j is

$$\begin{aligned} f_{j}\left( y_{i}|X_{i};\beta _{j},\rho _{j}^{1,2}\right)= & {} \Phi \left( \rho _{j}^{1}-X_{i}\beta _{j}\right) ^{I\left\{ y_{i}=0\right\} }\cdot \Phi \left( X_{i}\beta _{j}-\rho _{j}^{2}\right) ^{I\left\{ y_{i}=2\right\} } \\{} & {} \cdot \left( \Phi \left( \rho _{j}^{2}-X_{i}\beta _{j}\right) -\Phi \left( \rho _{j}^{1}-X_{i}\beta _{j}\right) \right) ^{I\left\{ y_{i}=1\right\} }, \nonumber \end{aligned}$$

(3)

where $I\$is an indicator function. This replaces (2) in the likelihood function $\pounds .$

In the trinomial model, for each variable, there are three APEs: for the probability of working full-time, part-time, and not working. By construction, these APEs always add up to 0. The APEs for the ordered probit regressions are presented in Table 4. Table 5 shows the APEs for the trinomial model with female dummy being the only concomitant variables. This mixture regression model with two groups presents the smallest departure from the ordered probit for women and men.

Table 4 Average partial effects, %, trinomial model, exogenous separation by gender

For men and Group 2, the career-oriented, as in the binomial model, there is no correlation between labour supply and children. Having children negatively correlates with the probability of working full-time for Group 1, the family-oriented, and for women. For the family-oriented, the children APEs are larger by the absolute value than for all women. With one or more children, the probability of working full-time is decreased by around 30% for the family-oriented (Table 5) and for all women by around 20% (Table 4).¹³ These reductions are closer to each other than in the binomial model due to the predicted group membership in the trinomial model. Roughly 70% of women are now estimated to be family-oriented, hence, a gender-based estimation that accounts for part-time work approximates the results for the homogeneous groups perhaps “well enough”. We, of course, advocate employing mixture regressions, but if that is not feasible, ordered probit may be a parsimonious alternative.

Table 5 Average partial effects, %, trinomial model, mixture regression model 1*

The trinomial model helps to see where the outflow from the full-time work is directed. For women with children in the ordered probit estimation, the full-time employment probability is reduced by around 20%, for any number of children, and the probability of not working in increased by 12%. That is, the ordered probit model predicts that around 3/5 of women (with children) who are not working full-time, are not working at all, and only 2/5 works part-time, see Table 4. In the mixture regression, having children is accompanied with a 30% reduction in the probability of working full-time. More than half of that outflow (around 55%) from the full-time work feeds into the part-time work, and less than half (around 45%) into the no-work category. Thus, the mixture regression fits the data better than the ordered probit. Indeed, in our sample, 32% of women work part-time and 23% do not work, see Table 1.

Trade and, especially, higher education are associated with a higher probability of working full-time and lower probability of working part-time or not working. Higher education APE for full-time work is larger for women than for men, and also larger for the family-oriented than for career-oriented. For the family-oriented and average women, higher education is associated with a 15% increase in the probability of full-time work. Around 1/3 of this increase is due to the decrease in part-time work, and 2/3 is due to the reduction of not working. For the career-oriented and all men, higher education is associated with 12% and 11% increase in the probability of working full-time. Around 40% of this increase is due to the decrease in part-time work and 60% of it is due to the decrease in not working.

Detrimental health condition is correlated with a drastic decrease in the probability of full-time work for all categories both in the ordered probit and mixture regressions. This decrease is 19% for the family-oriented, 20% for women, 22% for the career-oriented and 27% for men. More than 3/4 of this decrease feeds into the non-working category. Being married is associated with the 6% increase in the probability of full-time work for the career-oriented and with 7% increase for men. Being married is insignificant ( the estimated coefficient is insignificant) for both the family-oriented and women.

Non-labour income is negatively correlated with the probability of working full-time for every category in both the mixture and ordered probit regressions. An extra dollar of non-labour income is associated with 0.21% decrease in the probability of full-time work for the family-oriented and 0.28% decrease for women. Such extra dollar is associated with 0.16% decrease in full-time work for the career-oriented and 0.13% decrease for men. Notice that non-labour income is constructed as the total household income minus the individual’s wage, and men have higher wages than women. So, women in the sample have higher non-labour income than men.

Immigrant dummy is important for women and men in ordered probit and for the career-oriented in the mixture regression. Being an immigrant is associated with 4% to 6% increase in the probability of full-time work. This indicates that sometimes men and sometimes women are the bread winners in immigrant families.

Experience enters each regression in both linear and quadratic terms. Experience APE is positive for all groups in both ordered probit and mixture regressions. An extra year of experience is associated with 2.5–4.5% increase in the probability of full-time work. This may suggest that conditional on experience, Australian men and women have equal employment opportunities.

Age also enters each regression in both linear and quadratic terms. Age APE is negative for the full-time work in mixture regressions. For the career-oriented, an extra year of age is associated with 3% decrease in the probability of working full-time, and 2% increase in the probability of not working. For the family-oriented being one year older decreases the probability of working full-time by 1.4%, and 2/3 of that decreases translates into not working. Our age bracket is from 24 to 64 years of age, and the negative effect of Age means that people work full-time at the beginning of their working career and work less or retire towards the end of it.

As in the binomial model, we can say that the values of the APEs for women (Table 4) are the weighted averages of those APEs for Groups 1 and 2 (Table 5). This is because Group 1, the family-oriented, has 70% (SE is 4%) of women and 10% (SE is 2%) of men.¹⁴ Consequently, 30% of women and 90% of men are in Group 2, the career-oriented. Thus, gender is a more accurate predictor of the group membership in the trinomial mixture regression model. Both the APEs and work probabilities for Group 2, the career-oriented, in the mixture regression and for men in the ordered probit regression are very similar. This is not surprising as men predominantly self-select into Group 2. Women are more heterogeneous and are more dispersed between the groups.

The probabilities with which the family and the career-oriented work full-time, part-time and not work are quite indicative. The family-oriented, works part-time the most, with probability 45%. They work full-time with probability 36% and do not work with probability 19%. For women, the part-time and full-time work probabilities are nearly in reverse, 44% of women are predicted to work full-time and 33% part-time. The career-oriented has a working pattern similar to men in the order probit regression. Around 80% of the career-oriented works full-time, 5% works part-time and 15% does not work. This is very similar to men from the ordered probit regression where 78% work full-time, 10% work part-time and 12% do not work.

Other trinomial mixture models that add religion dummy to the concomitant variables Z, and female dummy to the explanatory variables X while allowing identification via the religion dummy, are in Appendix 2. The APEs for these models are quite similar to the ones in Table 5. Here, we focus on important insight from these models.

Religiosity makes a respondent more likely to be family-oriented. Our sample includes 915 religious respondents (more than 10% of the sample). We estimate that 75% (SE 4%) of religious and 68% (SE 4%) of non-religious women are in the family-oriented group. This group is also estimated to include 13% (SE 3%) of religious and 9% (SE 2%) of non-religious men. Gender added to the explanatory variables ends up insignificant for the career-oriented group, that is the career-oriented women work on par with the career-oriented men. Women from the family-oriented group, however, are more likely to work full-time than the family-oriented men.

We have also explored the possibility of identifying more than two homogeneous groups with a trinomial dependent variable. With three groups, 94% (SE 1%) of men is career-oriented. In addition, 26% (SE 3%) of women belongs to the career-oriented group with the labour supply relationship similar to men. Roughly 50% (SE 6%) of women is family-oriented, they are very unlikely to work full-time when they have children. The remaining 24% (std. err. 5%) of women would never work full-time, regardless of whether they have children or not. This result only reinforces our main message: labour supply of women is very heterogeneous, and estimation on the sample that pools all women together, which is currently the common practice, may lead to misleading results.

7 Conclusion

Traditionally, labour supplies of women and men are studied separately since family and productivity characteristics influence their labour market involvement differently. Our results suggest that such gender stratification is misleading for labour supply analysis since the modern society organization largely departs from the traditional gender roles. The mixture regression approach explored here is a useful way of improving upon the exogenous stratification by gender since the composition of the groups is suggested by the data and is, therefore, endogenous.

We identify two homogenous groups: family- and career-oriented. For the career-oriented group, the labour supply relationship resembles that of an average man. In contrast, children (negatively) correlate with the employment prospects of the family-oriented respondents, much more so than for an average woman. In the binomial mixture regression, 55% of women and 4% of men are family-oriented, while 45% of women and 96% of men are career-oriented. In the trinomial model with part-time work, roughly 70% of women and 10% of men are family-oriented. The family-oriented group is less likely to work full time and is more likely to take part-time jobs than the average woman.

Our results also shed a new light on the policies intended to encourage women’s labour market participation. A significant share of women, the ones we identify as the family-oriented, is much harder to encourage to work full-time than the average woman. They are, however, much more inclined to take part-time jobs than the average woman. Creation of such jobs and providing flexible working arrangements is likely to have more positive effect on women’s labour supply than a traditional gender-based estimation could suggest.

Appendices

Appendix 1: HILDA data

Here, we describe the construction of variables used in all the regressions in this paper.

Employment status is either working or not working for the binomial model and has an additional option of part-time work in the trinomial model. A person is considered working if they report positive number of hours of work, is considered working part-time if that number is less than 35 h per week and works full-time otherwise.

Experience is accumulated as years in paid work. It does not accumulate over the periods of unemployment and accumulates the corresponding fraction if a person reports not working for a portion of a year. Age and Experience enter regressions in both linear and quadratic terms.

The three education dummy variables (University Degree, Trade Certificate, and Year 12) are mutually exclusive and indicate the highest educational level achieved. The base for all three variables is incomplete high school. “University Degree” dummy is 1 for a bachelor or higher degree holder. “Trade Certificate” dummy is 1 for an advanced diploma, trade school certificate, diploma or Certificate III or IV holder. “Year 12” dummy is 1 for a Certificate I or II, or for completed Year 12 holder.

The two children variables, “Ever had one child” and “Ever had two or more children” are also mutually exclusive. “Detrimental Health Condition” dummy is equal to 1 if “Does your condition limit the type or amount of work you can do?” is reported as “Yes”. “Immigrated to Australia” dummy is 1 for those born overseas. “Married” dummy variable is 1 for those in a legal or de facto marriage.

“Non-Labour Income” is constructed by subtracting the individual’s “Financial year gross wages & salary” from “Household financial year gross total income”, it is then equivalized by dividing it by the square root of the number of people in the household. For estimation purposes, we also scale down the Non-Labour Income using inverse Hyperbolic Sine function. The APEs are calculated for the actual (unscaled) Non-Labour Income.

For the concomitant variables, female dummy is equal to 1 if the individual is a woman, and 0 otherwise. A person is identified as religious if “Importance of religion” for them is equal to 9 or 10 on the scale from 0 (one of the least important things in their life) to 10 (one of the most important things in their life).

Appendix 2: APEs for other trinomial mixture models

See Tables 6, 7 and 8.

Table 6 Average partial effects, %, trinomial model, mixture regression model 2*

Table 7 Average partial effects, %, trinomial model, mixture regression model 3*

Table 8 Average partial effects, %, trinomial mixture regression model, three groups*