Silviase/jgov_v1.4_full_ann huggingface dataset:
indicate the natural logarithmic value of potential wage rate if the $ i^{th} $ worker chooses to be self-selected into the for-profit sector or the nonprofit sector. $ Y_{i} $ represents measurable factors and $ v_{i}^{1} $ and $ v_{i}^{2} $ include unobserved factors, determining the $ i^{th} $ worker's utility. And $ \omega $ , $ \gamma^{1} $ , and $ \gamma^{2} $ represent parameters. The $ i^{th} $ worker is expected to be self-allocated in the sector where he or she achieves greater utility, along with the selection rule as follows (Goddeeris (1988)):

$$ \begin{aligned}&\mathrm{Z}_{\mathrm{i}}^{*}=\mathrm{V}_{\mathrm{i}}^{1}-\mathrm{V}_{\mathrm{i}}^{2}=(\mathrm{W}_{\mathrm{i}}^{1}-\mathrm{W}_{\mathrm{i}}^{2})\boldsymbol{\omega}+\mathrm{Y}_{\mathrm{i}}(\gamma^{1}-\gamma^{2})+\mathfrak{v}_{\mathrm{i}}^{1}-\mathfrak{v}_{\mathrm{i}}^{2}\\&\mathrm{Z}_{\mathrm{i}}=1\ \mathrm{if}\ \mathrm{Z}_{\mathrm{i}}^{*}\geq0\ \mathrm{or}\ \mathrm{V}_{\mathrm{i}}^{1}\geq\mathrm{V}_{\mathrm{i}}^{2}\\&\mathrm{Z}_{\mathrm{i}}=0\ \mathrm{if}\ \mathrm{Z}_{\mathrm{i}}^{*}<0\ \mathrm{or}\ \mathrm{V}_{\mathrm{i}}^{1}<\mathrm{V}_{\mathrm{i}}^{2}\\ \end{aligned} $$

$ Z_{i}^{*} $ represents for the $ i^{th} $ worker's unobserved propensity to be allocated to the for-profit sector, such that the observable dichotomous counterpart variable $ Z_{i}=1 $ , meaning the $ i^{th} $ worker actually chooses a for-profit home, if $ Z_{i}^{*}\geq0 $ , and $ Z_{i}=0 $ , implying that the $ i^{th} $ worker chooses a not-for-profit home, otherwise. In equation (2), the expected natural logarithmic wage of the $ i^{th} $ worker can be shown as the following function:

$$ \begin{aligned}&\mathbf{W}_{\mathrm{i}}^{1}=\mathbf{X}_{\mathrm{i}}\boldsymbol{\beta}+\boldsymbol{\varphi}^{1}+\boldsymbol{\tau}_{\mathrm{i}}^{1}\\&\mathbf{W}_{\mathrm{i}}^{2}=\mathbf{X}_{\mathrm{i}}\boldsymbol{\beta}+\boldsymbol{\varphi}^{2}+\boldsymbol{\tau}_{\mathrm{i}}^{2}\\ \end{aligned} $$

$ X_{i} $ includes various factors determining the $ i^{th} $ worker's wage rate, such as human capital characteristics. $ \beta $ , $ \varphi^{1} $ , and $ \varphi^{2} $ are parameters to be estimated, where $ \varphi $ s are selection biases showing the potential wage differential between for-profit and nonprofit firms to be focused on in this study. $ \tau_{i}^{1} $ and $ \tau_{i}^{2} $ show unobservables that would influence his or her wage rates. Substituting (3.1) and (3.2) into (2), we obtain:
$$ \begin{aligned}Z_{i}^{*}&=V_{i}^{1}-V_{i}^{2}=(X_{i}\beta+\varphi^{1}+\tau_{i}^{1}-X_{i}\beta-\varphi^{2}-\tau_{i}^{2})\boldsymbol{\omega}+Y_{i}(\gamma^{1}-\gamma^{2})+\boldsymbol{v}_{i}^{1}-\boldsymbol{v}_{i}^{2}\\&=(\varphi^{1}-\varphi^{2})\boldsymbol{\omega}+Y_{i}(\gamma^{1}-\gamma^{2})+(\tau_{i}^{1}-\tau_{i}^{2})\boldsymbol{\omega}+(\boldsymbol{v}_{i}^{1}-\boldsymbol{v}_{i}^{2})\\=\varphi\boldsymbol{\omega}&+Y_{i}\gamma+\tau_{i}\boldsymbol{\omega}+\boldsymbol{v}_{i}\end{aligned} $$

φω can be considered as a constant term and $ \tau_{i}\omega + v_{i} $ as $ u_{i} $ , various unmeasured factors that might affect workers' choice on the sectoral allocation. Then, simplifying equation (4), in order to correct the influence of a nonrandom allocation of workers to either the for-profit sector or the not-for-profit sector, the model has to include a reduced equation with a self selection rule, as follows:

$$ \begin{aligned}&Z_{\mathrm{i}}^{*}=\mathrm{Y}_{\mathrm{i}}\gamma+\mathrm{u}_{\mathrm{i}}\\&\mathrm{Z}_{\mathrm{i}}=1\ \mathrm{if}\ \mathrm{Z}_{\mathrm{i}}^{*}\geq0\ \mathrm{or}\ -\mathrm{u}_{\mathrm{i}}\leq\mathrm{Y}_{\mathrm{i}}\gamma\\&\mathrm{Z}_{\mathrm{i}}=0\ \mathrm{if}\ \mathrm{Z}_{\mathrm{i}}^{*}<0\ \mathrm{or}\ -\mathrm{u}_{\mathrm{i}}>\mathrm{Y}_{\mathrm{i}}\gamma\\&\mathrm{u}_{\mathrm{i}}\sim N(0,\sigma_{u}^{2})\\ \end{aligned} $$

where

Again, we specify a separate log-linear wage equation for for-profit and not-for-profit sectors, as follows:

$$ \begin{aligned}&W_{i}^{1}=X_{i}^{1}\pi^{1}+\varepsilon_{i}^{1}\quad&is observable if\quad Z_{i}=1\\&W_{i}^{2}=X_{i}^{2}\pi^{2}+\varepsilon_{i}^{2}\quad&is observable if\quad Z_{i}=0\end{aligned} $$

where

$$ \boldsymbol{\varepsilon}_{\mathrm{i}}^{1}\sim\mathrm{N}(0,\boldsymbol{\sigma}_{1}^{2})\mathrm{~a n d~}\boldsymbol{\varepsilon}_{\mathrm{i}}^{2}\sim\mathrm{N}(0,\boldsymbol{\sigma}_{2}^{2}) $$

In these equations, all the factors that determine workers' allocation do not necessarily influence wage rates. In other words, $ Y_{i} $ and $ X_{i} $ are not essentially the same. For example, variables describing the family backgrounds of workers, such as marital status and number of children, would have significant effects on the sector selection, yet they would not be pertinent to wage rates. In contrast, variables that represent the
workers' status and occupation in homes after the choice was made would influence wage rates, but they might not be relevant to the selection. Since these variables relevant to one equation and irrelevant to another are not available on the data used for the empirical analysis in this study. We will, however, control for the selection and wage equations using the same explanatory variables.

So as to obtain unbiased estimates of parameters, we will apply Heckman's two-stage approach (Lee (1978), Heckman (1979), Willis and Rosen (1979), Greene (1981)). First, maximum likelihood method is used for the probit regression equation to estimate the parameter vector $ \gamma $ of the probability that $ Z_{i}^{*} \geq 0 $ , implying the likelihood for the $ i^{th} $ worker of being allocated to a for-profit home in Equation (5). Then, for each worker, using probit coefficient vector $ \gamma $ , we will be able to obtain $ Z_{i}^{*} $ and hence the inverse of Mill's ratio, $ \lambda_{i} $ or the truncated mean of the normal density with truncation point $ Y_{i} \gamma / \sigma_{u} $ due to selection. Including $ \lambda_{i} $ in the regression along with potential determinants of wage rates will correct truncation and selectivity bias. Therefore, second, rewriting equations (3.1) and (3.2), a linear regression will be performed for $ W_{i} $ on $ X_{i} $ and $ \lambda_{i} $ in the for-profit and nonprofit sector, separately, as follows:

$$ \mathbf{W}_{\mathrm{i}}^{1}=\mathbf{X}_{\mathrm{i}}^{1}\boldsymbol{\delta}^{1}+\lambda_{\mathrm{i}}^{1}\boldsymbol{\theta}^{1}+\boldsymbol{\omega}^{1} $$

$$ \mathbf{W}_{\mathrm{i}}^{2}=\mathbf{X}_{\mathrm{i}}^{2}\boldsymbol{\delta}^{2}+\lambda_{\mathrm{i}}^{2}\boldsymbol{\theta}^{2}+\boldsymbol{\omega}^{2} $$

where $ E(\omega_{i}^{1}) = E(\omega_{i}^{2}) = 0 $ ^{6}

$$ \lambda_{_{i}}^{1}=-\phi(\mathrm{Y}_{_{i}}\gamma/\sigma_{_{u}})/\Phi(\mathrm{Y}_{_{i}}\gamma/\sigma_{_{u}})\quad and\quad\lambda_{_{i}}^{2}=\phi(\mathrm{Y}_{_{i}}\gamma/\sigma_{_{u}})/[1-\Phi(\mathrm{Y}_{_{i}}\gamma/\sigma_{_{u}})] $$

$$ \theta^{1}=\sigma_{_{1u}}/\sigma_{_{u}}=cov(\varepsilon_{_{i}}^{1},u_{_{i}})/\sigma_{_{u}}\quad and\quad\theta^{2}=\sigma_{_{2u}}/\sigma_{_{u}}=cov(\varepsilon_{_{i}}^{2},u_{_{i}})/\sigma_{_{u}} $$
For the above equations for truncated mean $ \lambda_{i} $ , $ \phi $ and $ \Phi $ indicate the standard normal probability density and the standard normal cumulative density functions, respectively. Due to the nature of the model setting and the truncated observations in our data, it is unlikely to estimate the residual in the selectivity equation, $ \sigma_{u} $ . If $ \sigma_{u} $ is standardized and assumed to be 1, then the coefficient of truncated mean adjustment factors for selection bias ( $ \theta^{1} $ and $ \theta^{2} $ ) can be directly interpreted as the covariance between the residuals in the selectivity equation and the residuals in the wage equations, measuring the relation between wage rates and the unobservable characteristics that influence the sector allocation. Consequently, $ \theta^{1} $ and $ \theta^{2} $ are key variables to examine our hypothesis based on asymmetric information caused by the coexistence of the for-profit and nonprofit sectors in the nursing facility industry.

If the selection behavior among workers in the long-term care labor market depends merely on the well-known comparative-advantage hypothesis, then they will choose to work in the sector that rewards their characteristics the most, such as ability, talent, education, and experience. The comparative-advantage argument suggests the condition such that $ \theta^{1}>0 $ if unobserved factors have impacts on both the likelihood of being in the for-profit sector and wage rates in the same direction, while it is $ \theta^{2}<0 $ if unobservables have a positive influence on the probability of being in the for-profit sector and if they have a negative effect on wage rates, or vice versa. If $ \theta^{1}>0 $ and $ \theta^{2}<0 $ are met, then the employee with the highest earnings who is able to provide the best quality of services in the for-profit sector would also do the best performance with the greatest wages in the not-for-profit sector, along with a strictly hierarchical self-allocation pattern based on ability. In other words, $ \theta^{1}>0 $ implies a positive selection bias among workers in the for-profit sector, such that $ E[W_{i}^{1}|Z_{i}^{*}\geq0]>E(W_{i}^{1}) $ , or those who earn greater than predicted wages derived from observed factors of workers in for-profit homes are more likely to be working in the for-profit sector than the predicted probability. In contrast, $ \theta^{2}<0 $ indicates a negative selection bias among nonprofit workers, such that $ E[W_{i}^{2}|Z_{i}^{*}<0]>E(W_{i}^{2}) $ , or those who earn greater than observed predicted wages in the
nonprofit sector, are less likely to be allocated to the for-profit sector than the expected prospect (Willis and Rosen (1979), Holtmann and Idson (1993)).

Instead of the particular directions for $ \theta $ , Trost (1981) claims that $ \theta^{1} > \theta^{2} $ , or the relation between the unobservable characteristics that influence the sector allocation and wage rates is greater in the for-profit than nonprofit sector, as the necessary condition for the comparative-advantage hypothesis. Explicitly, $ E[W_{i}^{1} \mid Z_{i}^{*} \geq 0] > E[W_{i}^{2} \mid Z_{i}^{*} \geq 0] $ , for-profit workers with greater than conditional mean wages of nonprofit employees if they decide to work in the for-profit sector are more likely to be allocated to for-profit homes than the expected probability; and $ E[W_{i}^{2} \mid Z_{i}^{*} < 0] > E[W_{i}^{1} \mid Z_{i}^{*} < 0] $ , not-for-profit workers with greater than conditional mean wages of for-profit workers if they choose to work in the nonprofit sector are less likely to be allocated to the for-profit sector than the expected prospect. The statistical significance of $ \theta $ , such that the hypothesis $ \theta = 0 $ can be rejected by t-test, shows whether or not workers' observed characteristics in the model captures unobservable effects on sector allocation and wage rates better, compared to no sector allocation adjustment in the nursing care industry.

### 4. Data

The empirical analysis of this study is based on the Jigyosho Ni Okeru Kaigo Rodo Jittai Chosa (Statistical Survey on Nursing Home Employees, administered in September 2000), conducted by the Care Worker Support Center Foundation (Kaigo Roudou Antei Center). This survey is not based on a random sample of the population, including data on various characteristics of approximately 39,261 employees in 1,347 nursing facilities all over Japan, which are self-enumerated by the nursing home's administrator. In this study, we choose a sample of 24,967 employees, due to the lack of information on ownership and various indicators used in our empirical analysis.

There are seven major types of ownership in the industry: for-profit, and six not-for-profit facilities—social welfare cooperation, medical cooperation, authorized Non-Profit Organization (NPO), Co-op, agricultural cooperative, and other charitable
corporation. In this survey, 2,823 (11.3 percent) and 22,144 (88.7 percent) employees are working at proprietary and nonprofit nursing homes, respectively.

Table 1 describes some basic statistics of major characteristics of both employees and facilities that could affect wages in for-profit and nonprofit nursing homes. This indicates significant differences between these sectors in the nursing industry in Japan. Mean daily wages are 4.7 percent higher in nonprofit facilities (the mean wage is 9,634 yen in for-profit homes and 10,217 yen in not-for-profit homes). Not-for-profit facilities are more likely to hire male employees in relatively young age groups from 20 to 49 years old. Also, nonprofit facilities are seen to prefer either full-time or part-time but regular workers. Regarding employees’ job qualifications $ ^{7} $ , a higher percentage of workers who are qualified to perform advanced skills, e.g., physicians, staff nurses, social workers, social care workers, and physical therapists are employed in nonprofit homes. On the contrary, a higher percentage of home helpers with any degree of skill, care managers, and service managers are hired by for-profit homes. Unfortunately, we do not have such information as experience, education, marital and family status, which are controlled for in previous works. However, employee age can be considered as a proxy of years of experience and the qualification may reflect education, which might indicate the selectivity of both employers and employees influencing the mean wage between two sectors in the nursing industry. Instead, using marital and family status as proxy indicators for employees’ stability in previous studies (Holtmann and Idson (1993), and Mocan and Tekin (2000)), we may use employees’ full-time status as a proxy of the stability as well as a facility-level aggregated indicator in the survey directly measuring the stability of full-time and part-time employees at facilities. For-profit homes currently hiring workers claim greater instability of employees for either full-time or part-time status, as compared to nonprofit homes; –13.2 percent versus 7.5 percent for full-time and 21.1 percent versus 14.6 percent for part-time employees, respectively. Also, a higher percentage of for-profit homes are seen to face difficulty in hiring both full-time and part-time workers than nonprofit homes.
We include various facility characteristics $ ^{8} $ that might affect wage rates. First, for-profit facilities tend to hire a relatively large number of employees compared to not-for-profit, though this is not the case for for-profit homes with fewer than 29 employees. Hence, higher wage rates in the nonprofit sector than in the for-profit sector cannot be explained simply by the positive relation between employer size and wage rates that a large number of studies have shown (Brown and Medoff (1989), Idson and Feaster (1990), Troske (1999)). The second notable attribute of facility characteristic is that non-wage benefits–employment insurance, work-related accident compensation insurance, health insurance, and public pensions–are more often offered by for-profit homes than by nonprofit facilities. Probably the cost-sensitive managers in the for-profit sector are motivated to lower wage rates. On the other hand, nonprofit homes are seen to support employees differently from for-profit homes; additional pay to basic salary, professional training inside and outside the facility, health examinations, subsidies for uniforms and equipment, and preventive care for work-related back pain and accidents. Only a few papers pay attention to these factors for non-wage benefits and work-related supports to employees, but they would be key indicators to control for measurement error in wage rates of workers, in particular, on a part-time basis (Montgomery and Cosgrove (1995), and Mocan and Tekin (2000)). Our data shows that, on average, about 70 percent and 50 percent of full-time employees are offered any type of insurance and public pension by for-profit and nonprofit employers, respectively, while the part-time employees of either sector are much less likely to own these non-wage benefits. Although we do not focus on
a wage differential between full-time and part-time employees in this study, non-wage benefits can be considered as labor costs for any firms; they are likely to be significant determinants on wage rates.

To examine the effects of these factors on wage determination, we first run a simple semi-logarithmic human capital wage function for a pooled full sample, using a dummy variable that indicates whether or not the ownership is proprietary. Then, to examine our suspicion that the labor market in the nursing home industry may be completely segregated between two sectors on account of not the difference in the quality but contents of services they are producing, we perform the same analysis separately by employees' qualification $ ^{9} $ . The results appear in Table 2. After adjusting for all characteristics in Table 1, we found a negative and statistically significant for-profit wage differential by -1.6, -5.3, -3.7, -2.9, and -9.3 percentage points for full-sample, helpers, care managers, staff nurses, and service managers, respectively, that is consistent with previous findings for the nursing home industry (Borjas, Frech III, and Ginsburg (1983), Holtmann and Idson (1993), Ruhm and Borkoski (2000)).

Consequently, a wage differential among employees with the same qualification is seen to be in favor of not-for-profit workers, regardless of whether the qualification requires advanced or less-advanced skills and education. Therefore, we may still pursue an examination of the relation between wage rates and the quality of service in the for-profit and nonprofit sectors. Note that adjusted for -profit wage differentials for helper and staff nurses become even larger than unadjusted ones. This implies that the standardized percentage difference for either home helpers or staff nurses with similar characteristics between two sectors is even larger than the gross for-profit effects.

For evaluating wage differentials across ownerships in the nursing home industry, however, simple least square regressions are not appropriate since the allocation of workers with various characteristics in our data to the for-profit and nonprofit sectors is nonrandom. When examining sample selection in our data, a Chow test also proves that we can reject
the hypothesis that equations for for-profit and nonprofit homes are the same at more than 99.9 percent probability level where the F-statistic is 119.5. Therefore, applying for Heckman's two-stage approach (Lee (1978), Heckman (1979)) is an appropriate procedure to obtain reliable estimates on the effects of a variety of individual and firm factors on the sectoral allocation and wage rates. There are only a few studies that address self-selection of workers into different sectors, in particular, with control for firm effects.

### 5. Results

Table 3 shows the estimated coefficients of the for-profit selection equation and wage equations, including workers with any qualifications. The results of the sector allocation probit regression in the first column basically replicate unadjusted comparisons between the two sectors in Table 1. Workers in the age groups from 40 to 49 years old, who are supposed to have relatively long labor market experience and relatively advanced levels of skill or education, remain less likely to self-select into proprietary homes, which is consistent with the simple comparison between the two sectors. Since full-time workers are more likely to be self-allocated into the nonprofit sector, profit-seeking facilities might face in-short and unstable supply of full-time workers needed so that they are more likely to be hiring full-time workers.

The difference in various facility characteristics such as firm sizes, professional training and supports to employees between two sectors largely reflects our simple comparison analysis. However, the probit sectoral allocation setting underlines the for-profit nursing homes' characteristics of the compensation policy for the employees more clearly than our summary statistics. Basically, employees are more likely to be offered any type of insurance as non-wage compensation in proprietary homes, while this is not the case for public pensions, used mainly for supporting living expenses after retirement. According to the “attenuated property rights” assumption, efficient managers in for-profit facilities who offer solely work-related non-wage insurances to attract stable workers in the liquid labor market would not be tolerated for non-wage benefits that are not directly
relevant to current job performance such as public pensions; they might even have incentives to lower wages instead.

The remainder of columns in Table 3 show the estimated coefficients of wage equations for proprietary (7.1) and not-for-profit homes (7.2), corrected for self-selection bias. Although, in general, male and/or full-time employees are seen to be more appreciated in both sectors, the direction, magnitude, and significance of a number of important indicators' effects on wage rates vary between for-profit and not-for-profit facilities. Wage rates are seen to increase proportionally along with the age until 59 years old in either the for-profit or nonprofit sector. Assuming workers' age as a proxy of years of experience, the effect is larger in the not-for-profit than the for-profit settings, which is consistent with the results of previous studies.

It is interesting that, regardless of stability, the hiring status of full-time and part-time workers has different influences on wage rates between the for-profit and nonprofit sectors. As we discussed, due to the short and unstable supply of full-time workers for-profit facilities might be facing, full-time workers are more appreciated than part-time workers within the profit-seeking sector. Compared to the nonprofit sector, however, the proprietary sector rewards full-time employees less and part-time employees more $ ^{10} $ . As regards qualification dummies, home helpers with the first or third degrees of skill, physicians, staff nurses, social care workers, physical and occupational therapists who are less likely to self-select into the for-profit sector are valued relatively high in nonprofit.