To work or not? Wages or subsidies?: Copula-based evidence of subsidized refugees’ negative selection into employment

Abstract

Despite increasing interest in topics related to refugees, economic literature has remained mostly silent on how refugees make labor supply decisions in their initial resettlement period, during which their host government provides various care and financial assistance. This paper fills that void by applying the copula-based selection model, which is free from the restrictive joint normality assumption, to a unique, high-dimensional data set of refugees who resettled in the US. Its selection parameter estimates suggest that subsidized refugees negatively select themselves into employment in terms of unobserved wage potential, which, according to the theoretical model, should be attributed primarily to the fact that (i) their reservation wages are rigid due to host-provided, non-labor income and (ii) host country employers discount refugees’ unobserved human capital components substantially. As a result, employed refugees’ wages, all observable factors held constant, are lower than the counterfactual wages of non-employed refugees, which contradicts what is usual in conventional labor markets. This devaluation-based skill paradox is more pronounced in regions unfriendly to refugees, and the negative pattern temporarily reversed immediately after the 9/11 attacks, which represented a huge adverse shock to non-natives in the US labor market, suggesting that subsidized refugees’ labor supply decisions are influenced greatly by their expectations regarding future labor market outcomes. Possible explanations are discussed based on a simple theoretical model in the context of the US refugee resettlement system.

Appendix

1.1 Mathematical derivation

The derivation of (16), discussed in Section 2.1, is shown below:

$$\begin{aligned} \begin{aligned} \rho _{v_{\text {m}},(v_{\text {m}}-v_{\text {r}})/\kappa }&=\frac{\text {Cov}(v_{\text {m}},(v_{\text {m}}-v_{\text {r}})/\kappa )}{\sigma _{v_{\text {m}}}\sigma _{(v_{\text {m}}-v_{\text {r}})/\kappa }}\\&=\frac{1}{\kappa }\frac{\text {Cov}(v_{\text {m}},(v_{\text {m}}-v_{\text {r}}))}{\sigma _{v_{\text {m}}}\sigma _{(v_{\text {m}}-v_{\text {r}})/\kappa }}\\&=\frac{\sigma _{v_{\text {m}}}^{2}-\text {Cov}(v_{\text {r}},v_{\text {m}})}{\sigma _{v_{\text {m}}-v_{\text {r}}}\sigma _{v_{\text {m}}}\sigma _{(v_{\text {m}}-v_{\text {r}})/\kappa }}\;\;\because \,\;\kappa =\sigma _{v_{\text {m}}-v_{\text {r}}}\\&=\frac{\sigma _{v_{\text {m}}}^{2}-\text {Cov}(\gamma _{\text {r}}\cdot \varepsilon _{s}+\varepsilon _{\text {r}},\gamma _{\text {m}}\cdot \varepsilon _{s}+\varepsilon _{\text {m}})}{\sigma _{v_{\text {m}}-v_{\text {r}}}\sigma _{v_{\text {m}}}\sigma _{(v_{\text {m}}-v_{\text {r}})/\kappa }}\\&=\frac{\sigma _{v_{\text {m}}}^{2}-\{\gamma _{\text {r}}\gamma _{\text {m}}\sigma _{\varepsilon _{s}}^{2}+\text {Cov}(\varepsilon _{\text {r}},\varepsilon _{\text {m}})\}}{\sigma _{v_{\text {m}}-v_{\text {r}}}\sigma _{v_{\text {m}}}\sigma _{(v_{\text {m}}-v_{\text {r}})/\kappa }}\;\;\because \,\;\text {Cov}(\varepsilon _{s},\varepsilon _{\text {r}})=\text {Cov}(\varepsilon _{s},\varepsilon _{\text {m}})=0\\&=\frac{\sigma _{v_{\text {m}}}^{2}-\{\gamma _{\text {r}}\gamma _{\text {m}}\sigma _{\varepsilon _{s}}^{2}+\rho _{\varepsilon _{\text {r}},\varepsilon _{\text {m}}}\sigma _{\varepsilon _{\text {r}}}\sigma _{{\varepsilon _{\text {m}}}}\}}{\sigma _{v_{\text {m}}-v_{\text {r}}}\sigma _{v_{\text {m}}}\sigma _{(v_{\text {m}}-v_{\text {r}})/\kappa }}\\&=\frac{1}{\sigma _{(v_{\text {m}}-v_{\text {r}})/\kappa }}\frac{\sigma _{v_{\text {m}}}^{2}-\{\gamma _{\text {r}}\gamma _{\text {m}}\sigma _{\varepsilon _{s}}^{2}+\rho _{\varepsilon _{\text {r}},\varepsilon _{\text {m}}}\sigma _{\varepsilon _{\text {r}}}\sigma _{{\varepsilon _{\text {m}}}}\}}{\sigma _{v_{\text {m}}-v_{\text {r}}}\sigma _{v_{\text {m}}}}\\&=\frac{\sigma _{v_{\text {m}}}}{\sigma _{(v_{\text {m}}-v_{\text {r}})/\kappa }\sigma _{v_{\text {m}}-v_{\text {r}}}}\frac{\sigma _{v_{\text {m}}}^{2}-\{\gamma _{\text {r}}\gamma _{\text {m}}\sigma _{\varepsilon _{s}}^{2}+\rho _{\varepsilon _{\text {r}},\varepsilon _{\text {m}}}\sigma _{\varepsilon _{\text {r}}}\sigma _{{\varepsilon _{\text {m}}}}\}}{\sigma _{v_{\text {m}}}^{2}}\\&=\frac{\sigma _{v_{\text {m}}}}{\sigma _{(v_{\text {m}}-v_{\text {r}})/\kappa }\sigma _{v_{\text {m}}-v_{\text {r}}}}\left( 1-\frac{\gamma _{\text {r}}\gamma _{\text {m}}\sigma _{\varepsilon _{s}}^{2}}{\sigma _{v_{\text {m}}}^{2}}-\frac{\rho _{\varepsilon _{\text {r}},\varepsilon _{\text {m}}}\sigma _{\varepsilon _{\text {r}}}\sigma _{{\varepsilon _{\text {m}}}}}{\sigma _{v_{\text {m}}}^{2}}\right) \\&=\frac{\sigma _{v_{\text {m}}}}{\sigma _{(v_{\text {m}}-v_{\text {r}})/\kappa }\sigma _{v_{\text {m}}-v_{\text {r}}}}\left[ \frac{\gamma _{\text {m}}^{2}\sigma _{\varepsilon _{s}}^{2}}{\sigma _{v_{\text {m}}}^{2}}\underbrace{(1-\frac{\gamma _{\text {r}}}{\gamma _{\text {m}}})}_{\text {A}}+\frac{\sigma _{{\varepsilon _{\text {m}}}}^{2}}{\sigma _{v_{\text {m}}}^{2}}\underbrace{(1-\frac{\rho _{\varepsilon _{\text {r}},\varepsilon _{\text {m}}}\sigma _{\varepsilon _{\text {r}}}}{\sigma _{{\varepsilon _{\text {m}}}}})}_{\text {B}}\right] \\&=\frac{\sigma _{v_{\text {m}}}}{\sigma _{(v_{\text {m}}-v_{\text {r}})/\kappa }\sigma _{v_{\text {m}}-v_{\text {r}}}}\Psi . \end{aligned} \end{aligned}$$

1.2 Selection framework

This section provides an overview of the selection-into-employment framework and the most common estimation methods. Consensus in labor economics literature suggests that the average wage of working people (i.e., observed wages of  $\{i\mid D_{i}=1\}$) might not measure accurately the wages of all people (i.e., potential wages of $\{i\}$) because working people might not represent a random sample of the entire population. This aspect also applies to refugees. The objective of the current study is investigating refugees’ non-random, systematic self-selection into employment, which requires use of a sample selection framework. Economists have long considered selection-related issues, with selection methods dating to Tobin (1958), Gronau (1974), and Heckman (1974). Diverse sample selection models exist, since many ways exist to generate a ‘selected or truncated’ sample, which refers to a sample based in part, intentionally or unintentionally, on values taken by a dependent variable (i.e., the response from a selection equation) (Cameron and Trivedi 2005). Wage observations in this study’s data represent an archetypal case of such a selected sample in the sense that only employed refugees’ wage levels are observed. Counterfactual wages of non-employed refugees cannot be observed due to their decision to not work.⁶⁰

Conventional selection models consist of two sequential equations—one for employment (e.g., selection or participation) and another for outcomes (e.g., wage levels). An employment equation with binary observable outcomes $D_{i}\in \{0,1\}$ can be expressed as:

$$\begin{aligned} D_{i}={\left\{ \begin{array}{ll} \ 1 &{} \text {if}\quad {y_{d_{i}}^{*}>0}\\ \ 0 &{} \text {if}\quad {y_{d_{i}}^{*}\le 0} \end{array}\right. }, \end{aligned}$$

(38)

where $y_{d_{i}}^{*}$ is a latent variable that determines whether to work. From a labor supply viewpoint, $y_{d_{i}}^{*}$ can be construed as the difference between a refugee’s market wage and his or her reservation wage, discussed in Sect. 2.1. If $y_{d_{i}}^{*}>0$, a refugee decides to work, and it can be inferred that the market wage exceeds the reservation wage for that refugee.⁶¹ On the other hand, a resultant wage equation with continuous outcomes can be expressed as:

$$\begin{aligned} y_{w_{i}}={\left\{ \begin{array}{ll} \ y_{w_{i}}^{*} &{} \text {if}\quad {y_{d_{i}}^{*}>0}\\ \ \text {{Unobserved}} &{} \text {if}\quad {y_{d_{i}}^{*}\le 0} \end{array}\right. }. \end{aligned}$$

(39)

Wage equation (39) suggests that a refugee’s wage outcome is observed if and only if a refugee is employed with $y_{d_{i}}^{*}>0$ in (38). The canonical approach specifies linear models with additive error terms, say $\varepsilon _{d}$ and $\varepsilon _{w}$, in the following manner (Cameron and Trivedi 2005).

$$\begin{aligned} {\left\{ \begin{array}{ll} y_{d_{i}}^{*}=\mathbf {x}_{d_{i}}^{\prime }\varvec{\beta }_{d}+\varepsilon _{d_{i}} &{} \text {for employment (i.e., selection)}\\ y_{w_{i}}=\mathbf {x}_{w_{i}}^{\prime }\varvec{\beta }_{w}+\varepsilon _{w_{i}} &{} \text {for wage levels} \end{array}\right. } \end{aligned}$$

(40)

The correlation between $\varepsilon _{d}$ and $\varepsilon _{w}$ in (40) is the key part of sample selection models, which, if overlooked, can cause bias when estimating $\varvec{\beta }_{w}$. In the case of the bivariate ML selection model, also called the Tobin (1958) Type-II estimator, estimation by maximum likelihood (ML) is straightforward, given distributional assumption:

$$\begin{aligned} \begin{bmatrix}\varepsilon _{d}\\ \varepsilon _{w} \end{bmatrix}\sim N\left[ \begin{bmatrix}0\\ 0 \end{bmatrix},\begin{bmatrix}\text {Var}(\varepsilon _{d})=\sigma _{\varepsilon _{d}}^{2} &{} \text {Cov}(\varepsilon _{d},\varepsilon _{w})\\ \text {Cov}(\varepsilon _{d},\varepsilon _{w}) &{} \text {Var}(\varepsilon _{w})=\sigma _{\varepsilon _{w}}^{2} \end{bmatrix}\right] , \end{aligned}$$

(41)

which means that correlated errors are joint normally distributed with homoskedasticity. The normalization of $\text {Var}(\varepsilon _{d})=1$ is used because only the sign of $y_{d_{i}}^{*}$ can be observed. Based on assumption (41), the bivariate ML selection model maximizes likelihood function:

$$\begin{aligned} \begin{aligned}L=\prod \limits _{i=1}^{N}\{\Pr [y_{d_{i}}^{*}\le 0]\}^{1-D_{i}}\{\Pr [y_{d_{i}}^{*}>0]\times f(y_{w_{i}}\mid y_{d_{i}}^{*}>0)\}^{D_{i}}\end{aligned} , \end{aligned}$$

(42)

and using probit as a link function leads to:

$$\begin{aligned} \begin{aligned}L=&\prod \limits _{i=1}^{N}\{\Phi (-\mathbf {x}_{d_{i}}^{\prime }\varvec{\beta }_{d})\}^{1-D_{i}}\times \\&\left[ \sigma _{\varepsilon _{w}}\phi \left( \frac{y_{w_{i}}-\mathbf {x}_{w_{i}}^{\prime }\varvec{\beta }_{w}}{\sigma _{\varepsilon _{w}}}\right) \Phi \left\{ \frac{\mathbf {x}_{d_{i}}^{\prime }\varvec{\beta }_{d}+(y_{w_{i}}-\mathbf {x}_{w_{i}}^{\prime }\varvec{\beta }_{w})\rho /\sigma _{\varepsilon _{w}}}{\sqrt{1-\rho ^{2}}}\right\} \right] ^{D_{i}}, \end{aligned} \end{aligned}$$

(43)

where $\rho$ refers to:

$$\begin{aligned} \rho (\varepsilon _{d},\varepsilon _{w})=\frac{\text {Cov}(\varepsilon _{d},\varepsilon _{w})}{\sigma _{\varepsilon _{d}}\sigma _{\varepsilon _{w}}}=\text {Corr}(\varepsilon _{d},\varepsilon _{w}), \end{aligned}$$

(44)

the correlation coefficient between $\varepsilon _{d}$ and $\varepsilon _{w}$. As is customary, $\Phi$ is the cumulative distribution function of the standard normal distribution, and $\phi$ is the standard normal probability density function. The log of (43) is the objective function of the bivariate ML selection model.

When using the bivariate ML selection model, it is important to note that assumption (41) is restrictive and difficult to test (Puhani 2000; Bushway et al. 2007).⁶² Since the bivariate ML selection model relies heavily on (41), its estimates are inconsistent if normality fails (Vella 1998). Thus, economists commonly prefer to use the two-step estimator from Heckman (1976; 1979), which is based on weaker assumption:

$$\begin{aligned} \varepsilon _{w_{i}}=\delta \varepsilon _{d_{i}}+\eta _{i}, \end{aligned}$$

(45)

where $\eta _{i}$ is independent of $\varepsilon _{d_{i}}$. This less restrictive assumption suggests that error term $\varepsilon _{w_{i}}$ in the wage equation is a multiple of error term $\varepsilon _{d_{i}}$ in the employment equation, with additive noise $\eta _{i}$. In addition to (45), when $\varepsilon _{d}\sim N(0,\sigma _{\varepsilon _{d}}^{2}=1)$ is assumed for using probit as a link function, the Heckman two-step estimator can be defined in the form of an augmented ordinary least squares regression:

$$\begin{aligned} y_{w_{i}}=\mathbf {x}_{w_{i}}^{\prime }\varvec{\beta }_{w}+\rho (\varepsilon _{d},\varepsilon _{w})\cdot \sigma _{\varepsilon _{w}}\cdot \lambda (\mathbf {x}_{d_{i}}^{\prime }\widehat{\varvec{\beta }}_{d})+v_{i}, \end{aligned}$$

(46)

where $\lambda (\mathbf {x}_{d_{i}}^{\prime }\widehat{\varvec{\beta }}_{d})$ is the (estimated) inverse Mills ratio $\phi (\mathbf {x}_{d_{i}}^{\prime }\widehat{\varvec{\beta }}_{d})/\Phi (\mathbf {x}_{d_{i}}^{\prime }\widehat{\varvec{\beta }}_{d})$. In this context, $\lambda (\mathbf {x}_{d_{i}}^{\prime }\widehat{\varvec{\beta }}_{d})$ proxies for a refugee’s participation in employment, the addition of which measures the sample selection effect (Dolton and Makepeace 1986).

Despite less-restrictive assumption (45), this study does not use the Heckman two-step estimator as its primary econometric method for several reasons. First, ML-based selection methods (e.g., copula selection model) are more efficient than the Heckman two-step estimator, and the loss of efficiency caused by using the Heckman two-step estimator is often large (Leung and Yu 1996; Stolzenberg and Relles 1997; Moffitt 1999; Puhani 2000; Bushway et al. 2007). Second, the objective of this study is investigating selection patterns of refugees into employment, which makes $\rho$ in (43) the main parameter of interest. Using the Heckman two-step estimator, it is impossible to estimate $\rho$ directly in (46) because it is not $\rho$ but $\rho (\varepsilon _{d},\varepsilon _{w})\cdot \sigma _{\varepsilon _{w}}$ that is estimated as the coefficient of selection correction term $\lambda (\mathbf {x}_{d_{i}}^{\prime }\widehat{\varvec{\beta }}_{d})$. The Heckman version correlation coefficient, $\rho ^{\text {Heckman}}$, can be estimated only indirectly using additional stages.⁶³ Third, the asymptotic properties of the standard error of $\rho ^{\text {Heckman}}$ have not been investigated extensively, making it challenging to test statistical significance. Fourth, using the Heckman two-step estimator, an exclusion restriction is practically necessary, though theoretically unnecessary, which refers to the requirement that at least one regressor, say z, in the employment equation should be excluded from the wage equation so that $\mathbf {x}_{d}=(\mathbf {x}_{w}^{\prime },z)^{\prime }$ holds.⁶⁴ Its importance amplifies when $\mathbf {x}_{d_{i}}^{\prime }\widehat{\varvec{\beta }}_{d}$ in (46) has little variation. In that case, $\varvec{\beta }_{w}$ in (46) is identified only weakly because the Heckman two-step estimator was designed to exploit the non-linearity of $\lambda (\mathbf {x}_{d_{i}}^{\prime }\widehat{\varvec{\beta }}_{d})$. If it is approximately linear over a range of its argument, this intended mechanism does not operate properly.⁶⁵ Since it is difficult to find a defensible exclusion restriction, this aspect should be regarded as a substantial limitation of the Heckman two-step estimator (Puhani 2000).⁶⁶ For more on limitations of the Heckman (1976; 1979) two-step estimator, see Stolzenberg and Relles (1997) and Bushway et al. (2007). Despite such shortcomings, this paper uses the Heckman two-step estimator for a robustness check in Sect. 6 due to its less-restrictive assumption, at the expense of less efficiency. It is also used when a log-likelihood function does not converge due to a limited number of observations and a large number of parameters estimated. Bushway et al. (2007) and Wooldridge (2010) discuss that it is often difficult for ML-based selection methods to converge.

1.3 Econometric details

In this paper, several copulas are used, and Archimedean copulas are especially practical, with mathematical properties that are easy to deal with (Smith 2003). The mathematical properties of Archimedean copulas are captured by an additive generator function, $\varphi :\mathbf {I}=[0,1]\rightarrow [0,+\infty )$, which is continuous, convex, and strictly decreasing (i.e., $\varphi ^{\mathrm {\prime }}(t)<0$ and $\varphi ^{\prime \prime }(t)>0$ for $0<t<1$), with terminal $\varphi (1)=0$ (Smith 2003).⁶⁷ Generator function $\varphi$ maps the interval [0, 1] onto the non-negative real line. According to Smith (2003), in a bivariate case with two continuous random variables $\varepsilon _{w}$ and $\varepsilon _{d}$ and their marginal CDFs $F_{1}(\varepsilon _{w})=u_{w}$ and $F_{2}(\varepsilon _{d})=u_{w}$, the means by which $\varphi$ generates the copula are based on:

$$\begin{aligned} \varphi \left( C\{u_{w},u_{d};\theta \}\right) =\varphi \left( u_{w}\right) +\varphi \left( u_{d}\right) . \end{aligned}$$

(47)

For all Archimedean copulas, $C\{u_{w},u_{d};\theta \}$ is recovered by:

$$\begin{aligned} \begin{aligned}C\{u_{w},u_{d};\theta \}&=\varphi ^{-1}\left\{ \varphi \left( u_{w}\right) +\varphi \left( u_{d}\right) \right\} \\&=\varphi ^{-1}\left\{ \varphi \left( C\{u_{w},u_{d};\theta \}\right) \right\} . \end{aligned} \end{aligned}$$

(48)

Differentiating (48) with respect to $u_{w}$ yields:

$$\begin{aligned} \frac{\partial }{\partial u_{w}}C\{u_{w},u_{d};\theta \}=\frac{\varphi ^{\mathrm {\prime }}(u_{w})}{\varphi ^{\mathrm {\prime }}(C\{u_{w},u_{d};\theta \})}, \end{aligned}$$

(49)

where $\varphi ^{\mathrm {\prime }}(\cdot )$ refers to the derivative of $\varphi (t)$. Substituting (49) into (36) leads to:

$$\begin{aligned} \begin{aligned}L&=\prod \limits _{i=1}^{N}\{\Pr [y_{d_{i}}^{*}\le 0]\}^{1-D_{i}}\left\{ f_{1}(\varepsilon _{w})\times \left( 1-\frac{\varphi ^{\mathrm {\prime }}(u_{w})}{\varphi ^{\mathrm {\prime }}(C\{u_{w},u_{d};\theta \})}\right) \right\} ^{D_{i}},\end{aligned} \end{aligned}$$

(50)

which is the likelihood function with an Archimedean copula. Estimating using ML is straightforward.

Among various Archimedean copulas, this study uses the Frank (1979) copula, due primarily to the fact that only the Frank copula is comprehensive in terms of dependence coverage (i.e., $-\infty<\theta <\infty$).⁶⁸ In addition, it has weaker tail dependence. In the current investigation, it is also preferred because of its lowest information criterion values, as measured by both the Akaike information criterion (AIC) and the Bayesian information criterion (BIC).⁶⁹ In this study, with two marginal distributions, the Frank copula, the generator function of which is:

$$\begin{aligned} \varphi (t)=-\log \left( \frac{e^{-\theta t}-1}{e^{-\theta }-1}\right) , \end{aligned}$$

(51)

can be expressed as:

$$\begin{aligned} \begin{aligned}C_{{-\infty<\theta <+\infty }}^{\text {Frank}}\{u_{w},u_{d};\theta \}&=-\theta ^{-1}\log \left\{ 1+\frac{(e^{-\theta u_{w}}-1)(e^{-\theta u_{d}}-1)}{(e^{-\theta }-1)}\right\} .\end{aligned} \end{aligned}$$

(52)

When using (52), one complication arises; the parameter space of $\theta$ ranges from $-\infty$ to $+\infty$, which makes $\theta$ less informative than $\rho$ in the bivariate ML selection model, bounded in the interval $[-1,1]$. Thus, Kendall’s $\tau$ is often used because it is also bounded in the interval $[-1,1]$ (Smith 2003). For an Archimedean copula with its generator function $\varphi$, Kendall’s $\tau$ can be calculated by:

$$\begin{aligned} \tau =1+4{\displaystyle \int _{0}^{1}}\frac{\varphi (t)}{\varphi ^{\mathrm {\prime }}(t)}\mathrm{d}t. \end{aligned}$$

(53)

Kendall’s $\tau$ for the Frank copula can be calculated by:

$$\begin{aligned} \tau ^{\text {Frank}}=1-\frac{4}{\theta }\left\{ 1-\frac{1}{\theta }{\displaystyle \int _{0}^{\theta }}\frac{t}{e^{t}-1}\mathrm{d}t\right\} . \end{aligned}$$

(54)

By this calculation, $\theta$ is converted to $\tau ^{\text {Frank}}$, bounded in the interval $[-1,1]$ and facilitating its interpretation.⁷⁰ The closer $\tau$ is to $-1$ $(+1)$, the stronger the negative (positive) dependence between $\varepsilon _{w}$ and $\varepsilon _{d}$. $\tau =0$ indicates no dependency. For details on Archimedean copulas, see Nelsen (1999). The unique aspect of the current study is that the selection parameters are of primary interest, unlike other studies, in which selection parameters function only as selection-correction terms.

1.4 Additional table

See Table 4.

Table 4 Summary statistics (key variables only)