The first step in the empirical estimation involves obtaining the parameter estimates of the log-linear wage model. The estimation results are reported in Appendix B. The estimated coefficients of the wage model are largely in line with
a priori expectations.
The majority of the data used as explanatory variables are recoded as binary indicators. To keep variables to scale, the variables of hours worked and hours of parental care are divided by 100 respectively and the variable of household disposable income is divided by 1000. Following Van Soest et al. (2002) the labour supply model is estimated using the maximum simulated likelihood, with 20 Halton draws per individual56. Arguably, this number of draws is sufficient to
provide robust and stable model estimates.
Table 5-2 and Table 5-3 set out the parameter estimates of the labour supply model. A range of specifications and starting values are used, and the estimated coefficients of the significant parameters are found to be stable. The choice of different but reasonable starting values mainly affects the convergence time. Further, a check of the Slustky regularity conditions is conducted using the ex post estimated parameters and 99 percent of individuals in the sample satisfy these regularity conditions.
TABLE 5-2PARAMETER ESTIMATES FOR SINGLE WOMEN WITH YOUNG CHILDREN
VARIABLES TASTES FOR LEISURE
(1)
TASTES FOR PARENTAL
CARE (2)
FIXED COSTS OF WORK
(3) Constant 65.20*** 52.33*** 0.975*** (7.999) (6.710) (0.232) Personal information Age -5.599** (2.508) Age squared 7.989** (3.331) If aged between 18-34 0.514*** (0.164) If aged between 35-40 0.374*** (0.145) If aged between 40-45 0.223* (0.131) If born in Australia -0.306** (0.131) Educational attainment University Vocational or year 12 Household demographics Number of children -1.233*** (0.284) If youngest child aged 0-1 6.287*** 11.37***
(1.393) (1.620) If youngest child aged 2-5 2.255*** 5.557***
(0.736) (0.949) If youngest child aged 6-9 -0.511 1.243
(0.641) (0.853) If informal care is used 0.483
(0.477) Health Poor Health 2.817** (1.225) Health improved -0.486 (0.520) Health worsened 0.0785 (0.414) Other variables
English as first language 1.669** (0.807) Difficulty finding a place in
childcare
-0.501 (0.577) Juggling multiple childcare
arrangements
-0.371 (0.575) Unsatisfied division of childcare
tasks
-0.689 (0.550)
σa 0.036 0.093
(0.250) (0.295) Simulated log likelihood -2,778.27
Observations 738
Notes: Standard errors in parentheses; sample of single mothers with children 0-12 years old; model estimated by Maximum Simulated Likelihood (MSL) using Halton sequences (20 Draws); hours worked and hours of care divided by 100; household disposable income on weekly basis and divided by 1000.
*** significant at 1 percent level, ** significant at 5 percent level, * significant at 10 percent level
a To ensure positive-definiteness of the covariance matrix, the Cholesky elements are instead estimated. The
The estimated coefficients presented in Table 5-2 are not directly interpretable due to the highly non-linear nature of the model. Nonetheless, the signs of the estimated coefficients do indicate the manner in which household characteristics influence the utility function.
The results reported in column (1), ‘Tastes for Leisure’, are discussed first. This equation captures the effects of age, household demographics and health status on household idiosyncratic tastes for leisure. The coefficients of the age variables indicate that the preference for leisure is quadratic in age, peaking at approximately 40 years of age. Further, the estimated coefficients of children age groups indicate that the presence of preschool children significantly reduces household preference for leisure, and such effects are larger if a child less than one year of age is present in the household. This is consistent with a priori expectations since infants typically require significant hours of parenting. The effects of health variables suggest that individuals with poor health have a lower preference for leisure.
The ‘Tastes for Parental Care’ equation (column (2)) indicates that both the number and age of children have a significant effect on parental care utilization. Households with multiple children are found to have a lower preference for parental care compared to those with a single child. This may be related to parenting specialisation in the presence of multiple children, where mothers can often provide supervision on a one-to-many basis. As such, hours of parental care do not necessarily increase linearly with the number of children. Nonetheless, the presence of a child aged less than one year is found to have large positive effect on preference for parental care, as does the presence of a two to five-year old child, and a six to nine-year old child. Part of the explanation most likely relates to the lower childcare needs as children age and the possibility that older siblings may provide care for younger dependents. Finally, the results indicate that where
the parent speaks English as the first language, her preference for own parenting activities is higher, compared to individuals from non-English speaking backgrounds.
Most of the coefficients in the fixed costs of work equation (column (4) of Table 5-2) are significant, suggesting that the inclusion of the fixed costs of work is appropriate. The coefficient on Australian-born is negative and significant, suggesting that for Australian-born mothers the fixed costs to participate in the labour market are lower compared to those born outside of Australia.
Table 5-3 presents the remainder of the utility parameters and the elements of the correlation matrix. The utility parameters have no clear interpretation but they are indispensable components in the expression for marginal effects. Note that none of the elements in the correlation matrix is significant.
TABLE 5-3ESTIMATED PARAMETERS OF THE UTILITY FUNCTION
(a) UTILITY PARAMETER
Parameters Coef. Std. Err. Parameters Coef. Std. Err.
𝛼1 0.275 (0.229) 𝛼4 2.427** (1.186) 𝛼2 -26.55*** (3.078) 𝛼5 2.370* (1.291) 𝛼3 -27.43*** (3.501) 𝛼6 -51.890*** (6.233) 𝛽1 -0.0812 (1.175) (b) CORRELATION MATRIXa Errors 𝜀1 𝜀2 0.092 (7.640) Notes: Standard errors in parentheses.
*** significant at 1 percent level, ** significant at 5 percent level, * significant at 10 percent level
Table 5-4 reports the actual and the fitted means of hours of working and parental care alongside the probabilities associated with each discrete choice. Overall, the estimated model provides a reasonable fit to the sample data as indicated by the consistent sample and fitted means.
TABLE 5-4ACTUAL AND FITTED MEANS AND PROBABILITIES
HOURS WORKED ACTUAL FITTED PARENTAL CARE ACTUAL FITTED
Mean Mean Std.Err. Mean Mean Std.Err. Hours per week 18.260 17.538 (1.550) Hours per week 24.402 26.233 (2.192) Probability (in percent) Probability (in percent)
0 36.314 36.554 (3.967) 0-10 24.551 26.348 (3.898) 0-6 1.626 2.242 (0.929) 10-20 22.069 21.006 (1.585) 6-12 5.013 4.598 (0.870) 20-30 25.655 16.432 (1.947) 12-18 8.672 7.628 (0.648) 30-40 11.586 12.393 (1.714) 18-24 10.569 10.312 (1.573) 40-50 7.310 9.145 (1.179) 24-30 9.621 11.416 (2.264) 50-60 4.414 6.643 (1.061) 30-36 6.504 10.611 (1.837) 60-70 3.586 4.750 (1.433) 36-42 13.415 8.314 (1.113) 70+ 5.655 3.283 (1.742) 42-48 5.149 5.413 (1.517) 48+ 3.117 2.910 (1.650)
Notes: The standard errors are constructed by averaging over the corresponding demographic group for each of the 100 independent draws from the asymptotic distribution of the estimated parameters.
In the actual hours worked, bunching is observed at two discrete-hour points, zero and 36-42 hours, which correspond to non-working and full-time employment respectively. The model has identified the large proportion of non-working individuals (36.6 percent), where the actual rate is 36.3 percent. However, the probability of the observed bunching around full-time employment is underpredicted. The effects of underprediction are partially offset by overpredicted probabilities associated with discrete choices of lower hours, which results in the expected hours worked being close to the sample average. In relation to parental care, the predicted choice probabilities of most alternatives are well in line with actual figures. The largest discrepancy occurred in the smallest two discrete choices where the weekly time-spent in parental activities is less than 20 hours per week. The model has provided a reasonable fit over the sample data for the remaining discrete choices.