Were they a shock or an opportunity?: The heterogeneous impacts of the 9/11 attacks on refugees as job seekers—a nonlinear multi-level approach

Abstract

This study investigates whether the September 11 terrorist attacks had any impacts on the labor market outcomes of refugees resettled in the United States, who should be distinguished from economic migrants or usual nonnatives. Furthermore, this paper sheds unprecedented light on whether those impacts were heterogeneous depending on a refugee’s ethnicity or religion. In terms of econometric methods, this research attempts to allow for the violation of the conventional condition of independently and identically distributed (i.i.d.) observations and control for cluster-specific unobservables by using nonlinear multi-level models, considering that refugees form unique networks in their resettlement regions and actively interact with one another within their clusters. Due to the binary dependent variable of this study, the incidental parameters problem is also taken into account. The multi-level estimates of this paper suggest that the September 11 attacks did not uniformly shock all sub-populations of refugees: rather, they presented a unique, substantial opportunity for Asian refugees and a serious threat to African and Arab refugees. One unanticipated finding is that the employment probability of European refugees remained stable, whereas that of Asian refugees markedly increased after the attacks. However, in terms of employment quality, measured by real wages, European refugees were the only ones who benefited from the attacks. Possible explanations for such heterogeneous impacts and different patterns of benefits are discussed, including positive versus negative selection into employment.

A Appendix

1.1 A.1 Details on multi-level econometric models

1.1.1 A.1.1 Conditional maximum likelihood model

In general, the fixed effects model is considered more conservative in that it allows for the possible correlation between cluster-specific fixed effects $\alpha _{c}$ and covariates. However, nonlinear maximum likelihood variants of the cluster-specific fixed effects model entail further complications—especially in the case of data with a small number of cluster members $N_{c}$. This is because data with small $N_{c}$ have the problem of too many incidental parameters (i.e., $\alpha _{1},\ldots ,\alpha _{C}$), while the number of observations for estimating each $\alpha _{c}$ is not enough (Neyman and Scott 1948). Unlike linear cases, it is generally not possible to eliminate such nuisance parameters in nonlinear cases (Hall and Severini 1998). This is the reason why the present study, the dependent variable of which is binary, cannot simply use the dummy variable fixed effects model. Cluster-specific dummies, in nonlinear cases with the finite number of cluster members $N_{c}$, fail to properly pick up $\alpha _{c}$ and render maximum likelihood estimates inconsistent (Neyman and Scott 1948; Chamberlain 1980; Lancaster 2000; Greene 2002).

One alternative method for obtaining consistent estimates that eliminates unwanted $\alpha _{c}$ is using the conditional maximum likelihood model (CML). It is based on a log density for the jth individual (in the cth cluster) that conditions on $\sum _{j=1}^{N_{c}}y_{jc}$, which refers to the total number of outcomes equal to one (i.e., employed) for a given cluster (Chamberlain 1980). When a binary logit model with cluster fixed effects $\alpha _{c}$ specifies

$$\begin{aligned} \Pr (y_{jc}=1\mid \mathbf {v}_{jc},\mathbf {i}_{c},\alpha _{c})=\frac{\exp (\mathbf {v}_{jc}^{\prime }\varvec{\delta }+\mathbf {i}_{c}^{\prime }\varvec{\eta }+\alpha _{c})}{1+\exp (\mathbf {v}_{jc}^{\prime }\varvec{\delta }+\mathbf {i}_{c}^{\prime }\varvec{\eta }+\alpha _{c})}, \end{aligned}$$

(11)

where all notations are the same as defined in Section 4, the joint conditional probability for the cth cluster is calculated as follows:

$$\begin{aligned} \begin{aligned}&\Pr {}_{c}(y_{1c},y_{2c},y_{3c},\ldots ,y_{N_{c}c}\mid N_{c}\overline{y}_{c})\\&\quad =\frac{\exp (\sum _{j=1}^{N_{c}}y_{jc}(\mathbf {v}_{jc}^{\prime }\varvec{\delta }+\mathbf {i}_{c}^{\prime }\varvec{\eta }+\alpha _{c}))}{\sum _{d\in \widetilde{B}_{c}}\exp (\sum _{j=1}^{N_{c}}d_{jc}(\mathbf {v}_{jc}^{\prime }\varvec{\delta }+\mathbf {i}_{c}^{\prime }\varvec{\eta }+\alpha _{c}))}\\&\quad =\frac{\exp (\sum _{j=1}^{N_{c}}y_{jc}\mathbf {v}_{jc}^{\prime }\varvec{\delta })}{\sum _{d\in \widetilde{B}_{c}}\exp (\sum _{j=1}^{N_{c}}d_{jc}\mathbf {v}_{jc}^{\prime }\varvec{\delta })}, \end{aligned} \end{aligned}$$

(12)

where ${\widetilde{B}_{c}=\{(d_{1c},d_{2c},d_{3c},\ldots ,d_{N_{c}c})\mid d_{jc}=0}\,\text {or}\,{1,}\,\text {and}\,{\sum _{j=1}^{N_{c}}d_{jc}=\sum _{j=1}^{N_{c}}}y_{jc}{=N_{c}\overline{y}_{c}\}}$ (Hamerle and Ronning 1995; Cameron and Trivedi 2009; Hosmer et al. 2013).³¹ This approach is uniquely possible only with logit by virtue of its feature that $\exp (\cdot )$ appears both in the numerator and in the denominator, which enables two sets of common factors — $\exp (\sum _{j=1}^{N_{c}}y_{jc}\mathbf {i}_{c}^{\prime }\varvec{\eta })=\exp (\sum _{j=1}^{N_{c}}d_{jc}\mathbf {i}_{c}^{\prime }\varvec{\eta })$ and $\exp (\sum _{j=1}^{N_{c}}y_{jc}\alpha _{c})=\exp (\sum _{j=1}^{N_{c}}d_{jc}\alpha _{c})$ conditioning on $N_{c}\overline{y}_{c}$ — to be canceled out in (12).³² Finally, the conditional likelihood is $\prod _{c=1}^{C}\Pr _{c}(y_{1c},y_{2c},y_{3c},\ldots ,y_{N_{c}c}\mid N_{c}\overline{y}_{c})$, the product of (12) over all C clusters, where C refers to the total number of clusters.

Under the between-cluster independence assumption and the conditional independence assumption (CIA) that within-cluster observations $\{y_{1c},y_{2c},y_{3c},\ldots ,y_{N_{c}c}\}$ are independent conditioning on $\mathbf {v}_{jc}$, $\mathbf {i}_{c}$, and $\alpha _{c}$, the conditional maximum likelihood model eliminates $\alpha _{c}$ and yields consistent coefficient estimates of level-one covariates $\mathbf {v}_{jc}$ (Chamberlain 1980). The primary advantage of the conditional maximum likelihood model resides in the fact that it does not rely on any assumptions concerning the distribution of $\alpha _{c}$. This approach is referred to as the Chamberlain’s fixed effects logit model in econometrics and applicable even when cluster sizes vary.

While eliminating $\alpha _{c}$ in (12) is uniquely feasible without the incidental parameters problem, this approach, however, still entails some substantial problems, the most critical point of which lies in the fact that it leads to the loss of observations if $y_{jc}$ is either 0 for all j or 1 for all j. On top of this, all clusters with $N_{c}=1$ are excluded, further impairing efficiency (Cameron and Trivedi 2005). Moreover, while this method is useful for obtaining consistent coefficient estimates, it is generally not possible to estimate additive marginal effects because they depend on eliminated $\alpha _{c}$, unlike linear fixed effects models (Beck 2020). Some researchers report marginal effects evaluated at a certain value $\alpha _{c}=q$: however, that is not much meaningful as where to evaluate $\alpha _{c}$ is a completely arbitrary decision. As an alternative, a common way of interpreting coefficients is using an odds ratio, which measures the probability of $y=1$ relative to the probability of $y=0$. However, such multiplicative interpretation is usually less intuitive than additive marginal effects. This is the point at which the more flexible random effects model should also be considered.

1.1.2 A.1.2 Chamberlain–Mundlak’s correlated random effects probit model

When the simplest model $y_{jc}=\mathbf {x}_{jc}^{\prime }\varvec{\beta }+u_{jc}$ is supposed with the error term decomposition $u_{jc}=\alpha _{c}+\varepsilon _{jc}$, the conventional random effects model assumes

$$\begin{aligned} \alpha _{c}\mid \mathbf {x}_{jc}\sim N(0,\sigma _{\alpha }^{2}), \end{aligned}$$

(13)

and this is undeniably a very strong assumption. It should be carefully noted that (13) requires two conditions to be satisfied concurrently: first, $\alpha _{c}$ and $\mathbf {x}_{jc}$ should not be correlated, and second, $\alpha _{c}$ should be normally distributed with homoskedasticity (i.e., constant variance $\sigma _{\alpha }^{2}$).

Due to the implausibility of the assumption (13) in the case of the data under investigation, this study does not use the conventional random effects model; instead, the approach of Chamberlain (1980) is leveraged to clustered observations, which had been originally devised in the context of panel data. Based on the method of Chamberlain (1980), the correlation between $\alpha _{c}$ and $\mathbf {x}_{jc}$ can be tolerated by replacing $\alpha _{c}$ with its linear projection onto the cluster-specific means of covariates (i.e., $\overline{\mathbf {x}}_{c}$) including a projection error (Wooldridge 2010). By allowing $\alpha _{c}$ to be determined by $\overline{\mathbf {x}}_{c}$, $\alpha _{c}$ can be expressed as

$$\begin{aligned} \alpha _{c}=\psi +\overline{\mathbf {x}}_{c}^{\prime }\varvec{\xi +}w_{c}, \end{aligned}$$

(14)

where $w_{c}$ denotes the projection error.³³ Then, the Mundlak (1978) version of Chamberlain’s assumption can be written as

$$\begin{aligned} \alpha _{c}\mid \mathbf {x}_{jc}\sim N(\psi +\overline{\mathbf {x}}_{c}^{\prime }\varvec{\xi },\sigma _{w}^{2}). \end{aligned}$$

(15)

This signifies that $\alpha _{c}$ can be correlated with regressors through $\overline{\mathbf {x}}_{c}$: therefore, the inclusion of $\overline{\mathbf {x}}_{c}$ is expected to control for the correlation between cluster-specific heterogeneous features and covariates. However, this assumption (15), albeit more flexible than (13), is still restrictive in that it specifies the conditional distribution of $\alpha _{c}$. In plain terms, (15) means that $\alpha _{c}$ given $\mathbf {x}_{jc}$ should be normally distributed with mean $\psi +\overline{\mathbf {x}}_{c}^{\prime }\varvec{\xi }$ and variance $\sigma _{w}^{2}$.

While the fundamental logic of Mundlak (1978) is to let $\overline{\mathbf {x}}_{c}$ in (14) include all regressors, within-cluster invariant regressors do not provide any information for this projection. Hence, regressors are categorized into two types: within-cluster variant regressors $\mathbf {v}_{jc}$ and within-cluster invariant regressors $\mathbf {i}_{c}$, as previously mentioned with (3) in Sect.  4. Considering this separation, cluster-specific $\alpha _{c}$ can be rewritten as

$$\begin{aligned} \alpha _{c}=\psi +\overline{\mathbf {v}}_{c}^{\prime }\varvec{\tau }\varvec{+}w_{c}, \end{aligned}$$

(16)

where $\overline{\mathbf {v}}_{c}$ refers to the averages of level-one covariates $\mathbf {v}_{jc}$ within each cluster. Naturally, cluster-invariant level-two covariates $\mathbf {i}_{c}$ are still included as exploratory variables. As a result, the generalized linear model along with probit as a binary link function can be defined as

$$\begin{aligned} \Pr (y_{jc}=1\mid \mathbf {v}_{jc},\mathbf {i}_{c},\alpha _{c})=E(y_{jc}\mid \mathbf {v}_{jc},\mathbf {i}_{c},\alpha _{c})=\Phi (\psi +\mathbf {v}_{jc}^{\prime }\varvec{\zeta +}\overline{\mathbf {v}}{}_{c}^{\prime }\varvec{\tau +}\mathbf {i}_{c}^{\prime }\varvec{\varphi }),\qquad \end{aligned}$$

(17)

where $\Phi$ denotes the standard normal cumulative distribution function. This method is called the Chamberlain–Mundlak’s correlated random effects probit model (CRE). The test of $\varvec{\tau }=0$ in (17), which is known as the Mundlak test, can be easily implemented in a bid to figure out whether the assumption of the conventional random effects model (13) is valid (i.e., if $\varvec{\tau }=0$) or not (i.e., if $\varvec{\tau }\ne 0$) (Mundlak 1978). The results of the test applied to this study are discussed in Online Supplement B.2.

The Chamberlain–Mundlak’s correlated random effects probit model yields unbiased estimates of level-one covariates (Mundlak 1978; Neuhaus and Kalbfleisch 1998; Snijders and Berkhof 2008). Hence, this study also uses the Chamberlain–Mundlak’s correlated random effects probit model along with the conditional maximum likelihood model.

1.1.3 A.1.3 Linear probability model with cluster fixed effects

As previously mentioned in Sect. 4.3, if we use the identity link for g in (7), the simple linear probability model with cluster fixed effects can be defined as shown below.

$$\begin{aligned} \Pr (y_{jc}=1\mid \mathbf {v}_{jc},\mathbf {i}_{c},\alpha _{c})=E(y_{jc}\mid \mathbf {v}_{jc},\mathbf {i}_{c},\alpha _{c})=\mathbf {v}_{jc}^{\prime }\varvec{\delta }+\mathbf {i}_{c}^{\prime }\varvec{\eta }+\alpha _{c} \end{aligned}$$

(18)

Despite its simplicity and computational convenience, the simple linear probability model with cluster fixed effects has some shortcomings as follows. First, the linear probability model, which uses neither a cumulative distribution function nor a latent variable model, has no structural room for error terms. Second, some predicted probabilities based on the linear probability model may have nonsensical values that are less than zero or greater than one (i.e., outside the unit interval). Third, the linear probability model can even lead to negative variances (Greene 2002).

However, on a more positive note, the linear probability model requires no distributional assumptions of disturbances, the violation of which can make maximum likelihood estimates inconsistent (Bera et al. 1984). Hence, this study also uses the linear probability model for comparisons. It is also used when a maximum likelihood function fails to converge because of the large number of parameters to be estimated.

1.2 A.2 Bivariate selection model

Conventional selection models are composed of two sequential equations—one for employment (i.e., also often called selection or participation) and the other for wage outcomes (Amemiya 1985). An employment equation with binary outcomes can be expressed as shown below, where $y_{d}^{*}$ is a latent variable that determines whether to work or not.

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

(19)

While Cameron and Trivedi (2005) explains that $y_{d}^{*}$ is construed as the unobserved desire or propensity to work, Heckman (1974), from a labor supply standpoint, notes that it can be regarded as the difference between a refugee’s market wage (i.e., wage offer) and his or her reservation wage. On the other hand, a resultant market wage equation with continuous outcomes can be expressed as follows, where latent $y_{w}^{*}$ determines how much to work.

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

(20)

The market wage equation (20) denotes that the wage outcome of a refugee is observed if and only if a refugee is employed (i.e., $y_{d_{i}}^{\text {Employment}}=1$) with $y_{d_{i}}^{*}>0$ in (19). In other words, whether we can observe a refugee’s market wage level or not entirely depends on his or her labor supply decision. The canonical approach for modeling selection specifies linear models with additive error terms in the following manner (Cameron and Trivedi 2005).

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

(21)

The correlation between $\varepsilon _{d}$ and $\varepsilon _{w}$ in (21) is the core of sample selection models, which, if overlooked, can cause problems in estimating $\varvec{\beta }_{w}$ (Greene 2002). In the case of the bivariate sample selection model, suggested by Tobin (1958), estimation by maximum likelihood is straightforward given the additional assumption of

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

(22)

which, in plain terms, means that the correlated errors are joint normally distributed with homoskedasticity (Cameron and Trivedi 2005).³⁴ Gronau (1974), Heckman (1979), and Keane et al. (1988) are among previous studies using this bivariate normality assumption: for further details on this assumption, see Amemiya (1985) and Moffitt (1999). Based on the assumption (22), the bivariate sample selection model maximizes the likelihood function

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

(23)

and the use of probit as a link function leads to

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

(24)

where $\rho$ refers to the correlation coefficient between $\varepsilon _{d}$ and $\varepsilon _{w}$. As is customary, $\Phi$ denotes the standard normal cumulative distribution function, whereas $\phi$ is the standard normal probability density function. The log of (24) is the objective function of the bivariate sample selection model. However, in this context, the selection parameter of European refugees and that of non-European refugees should be separately estimated, and if they are estimated in a single likelihood function, they can be directly compared. Therefore, (24) is modified as follows.

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

(25)

While all notations are the same as defined hitherto, $S_{i}=1$ means that an individual i is a European refugee (i.e., $i\in S$).³⁵ Thus, $\rho _{1}$ refers to the correlation between $\varepsilon _{d}$ and $\varepsilon _{w}$ of European refugees, and $\sigma _{1}$ denotes the standard deviation of European refugees’ $\varepsilon _{w}$. Likewise, $\rho _{2}$ refers to the correlation between $\varepsilon _{d}$ and $\varepsilon _{w}$ of non-European refugees, and $\sigma _{2}$ denotes the standard deviation of non-European refugees’ $\varepsilon _{w}$. The likelihood function (25) is flexible in the sense that the coefficients of $\mathbf {x}_{d}$ and $\mathbf {x}_{w}$ are allowed to be different in those two groups (i.e., $\varvec{\beta }_{d}$, $\varvec{\beta }_{w}$, $\varvec{\gamma }_{d}$, $\varvec{\gamma }_{w}$).³⁶