The effects of rurality on mental and physical health

Abstract

The effects of rurality on physical and mental health are examined in analyses of a national dataset, the Community Tracking Survey, 2000–2001, that includes individual level observations from household interviews. We merge it with county level data reflecting community resources and use econometric methods to analyze this multi-level data. The statistical analysis of the impact of the choice of definition on outcomes and on the estimates and significance of explanatory variables in the model is presented using modern econometric methods, and differences in results for mental health and physical health are evaluated.

Appendix A

1.1 A.1. Rural–urban continuum codes

Code Description⁵³

Metro counties (based on the 2003 version of codes):

• 1. Counties in metro areas of 1 million population or more

• 2. Counties in metro areas of 250,000 to 1 million population

• 3. Counties in metro areas of fewer than 250,000 population

Nonmetro counties:

• 4. Urban population of 20,000 or more, adjacent to a metro area

• 5. Urban population of 20,000 or more, not adjacent to a metro area

• 6. Urban population of 2,500 to 19,999, adjacent to a metro area

• 7. Urban population of 2,500 to 19,999, not adjacent to a metro area

• 8. Completely rural or less than 2,500 urban population, adjacent to a metro area

• 9. Completely rural or less than 2,500 urban population, not adjacent to a metro area

A nonmetro county is defined as adjacent if it physically adjoins one or more metro areas, and at least 2% of its employed labor force commute to central metro counties. Nonmetro counties that do not meet these criteria are classed as nonadjacent.

1.2 A.2. Correlated probit

Let $y_{ij}^{*}$ be the latent variable associated with family member j in family i, j = 1, 2,..,J _i , and i = 1, 2,..,n, and assume that

$$ \begin{aligned} y_{ij}^{\ast} &= X_{ij} \beta +u_{i}+\varepsilon_{ij}, \\ u_{i} &\sim iidN\left( 0,\sigma_{u}^{2}\right), \\ \varepsilon_{ij} &\sim iidN\left( 0,\sigma_{\varepsilon}^{2}\right) \end{aligned} $$

where X _ij is a vector of personal characteristics specific to person j , u _i is a family-specific random effect with $u_{i}\sim iidN\left( 0,\sigma_{u}^{2}\right)$, and ɛ _ij is a person-specific effect with $\varepsilon_{ij}\sim iidN\left( 0,\sigma_{\varepsilon }^{2}\right)$. Without loss of generality and in the interest of identification, we set $\sigma_{\varepsilon }^{2}=1$. We define the dependent variable as

$$ y_{ij}=1\hbox{ iff }y_{ij}^{\ast}>0. $$

Then the log likelihood contribution for family i is

$$ L_{i}=\log \int \left( \prod_{j=1}^{J_{i}}\Upphi \left( X_{ij}\beta +u\right)^{y_{ij}}\left[ 1-\Upphi \left( X_{ij}\beta +u\right) \right]^{1-y_{ij}}\right) \frac{1}{\sigma_{u}} \phi \left(\frac{u}{\sigma_{u}}\right) du $$

where $\Upphi(\cdot)$ is the standard normal distribution function and ϕ(·) is the standard normal density function. It can be approximated well with K-point Gaussian quadrature (Butler and Moffitt 1982) as

$$ \log \sum_{k=1}^{K} \omega_{k}\left( \prod_{j=1}^{J_{i}}\Upphi \left( X_{ij}\beta +\sigma_{u}\eta_{k}\right)^{y_{ij}}\left[ 1-\Upphi \left( X_{ij}\beta +\sigma_{u}\eta_{k}\right) \right]^{1-y_{ij}}\right) $$

where $\left\{\omega_{k},\eta_{k}\right\}_{k=1}^{K}$ are the K-point Gaussian quadrature weights and locations available in Stroud and Secrest (1966). The log likelihood function is

$$ L=\sum_{i=1}^{n}L_{i}, $$

and the vector of parameters to maximize with respect to is θ = (β, σ _u ). The value of θ that maximizes $L, \widehat{\theta}$, is the maximum likelihood estimator (MLE) of θ, and it is consistent, efficient, and has an asymptotic distribution of

$$ \begin{aligned} \sqrt{n}\left( \widehat{\theta }-\theta \right) & \sim N\left(0,\Upomega \right) , \\ \Upomega & = plim\left( \frac{1}{n} \sum_{i=1}^{n}\frac{\partial \log L_{i}} {\partial \theta} \frac{\partial \log L_{i}}{\partial \theta^{\prime}}\right)^{-1}. \end{aligned} $$

1.3 A.3. Wald tests

We are interested in two tests:

$$ \begin{aligned} H_{0} &: \beta_{r}=0\hbox{ vs. }H_{A}:\beta_{r}\neq 0; \\ H_{0} &: \beta_{r1}=\beta_{r2}= \cdots =\beta_{rR}\hbox{ vs. } H_{A}:\beta_{r1}\neq \beta_{r2}\neq \cdots \neq \beta_{rR}. \\ \end{aligned} $$

We can write both of the null hypotheses in the form

$$ A\beta_{r}=0 $$

where β _r is the vector of rural coefficients, R is the number of rural dummies (R = 2 for ADJAC, and R = 3 for POPSIZE), A = I for the first null hypothesis and

$$ A=\left( \begin{array}{ccccc} 1 & -1 & 0 & \cdots & 0 \\ 1 & 0 & -1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & 0 & 0 & \cdots & -1\\ \end{array} \right) $$

for the second null hypothesis. For each of the three types of estimators used, we can represent the asymptotic distribution of the rural estimates $\widehat{\beta }_{r}$ as

$$ \sqrt{n}\left( \widehat{\beta }_{r}-\beta_{r}\right) \sim N\left( 0,\Upomega \right) . $$

This implies that, under H ₀,

$$ \sqrt{n}\left( A\widehat{\beta }_{r}-A\beta_{r}\right) =\sqrt{n}A\widehat{\beta}_{r}\sim N\left( 0,A\Upomega A^{\prime}\right) $$

and

$$ n\left( A\widehat{\beta }_{r}\right)^{\prime }\left[ A\Upomega A^{\prime } \right]^{-1}\left( A\widehat{\beta }_{r}\right) \sim \chi_{K}^{2} $$

where K is the number of restrictions implied by H ₀ (i.e., the number of rows in A).

1.4 A.4. Pseudo lagrange multiplier tests

Consider the linear model,

$$ \begin{aligned} y_{ij} &= X_{ij}\beta +u_{ij},j=1,\ldots,J_{i};i=1,\ldots,I \\ u_{ij} &= e_{i}+\varepsilon_{ij} \\ \end{aligned} $$

where y _ij is an outcome of interest for observation j in county i, X _ij is a vector of observable variables, $e_{i}\sim iid\left( 0,\sigma_{e}^{2}\right)$ is a county-specific random effect, and $\varepsilon_{ij}\sim iid\left( 0,\sigma_{\varepsilon }^{2}\right)$ is a person-specific random effect. We are interested in testing whether there is a county-specific random effect:

$$ H_{0}:\sigma_{e}^{2}=0\hbox{ vs. }H_{A}:\sigma_{e}^{2}>0. $$

(1)

Using residuals defined as

$$ \widehat{u}_{ij}=y_{ij}-X_{ij}\widehat{\beta } $$

where $\widehat{\beta }$ is a consistent estimate of β, we can define

$$ \widehat{u}_{i}=\sum_{j=1}^{J_{i}}\widehat{u}_{ij} $$

as the average county-specific residual. Under H ₀,

$$ \begin{aligned} \hbox{ plim}\frac{1}{I} \sum_{i} \widehat{u}_{i}^{2} & = \hbox{plim}\frac{1}{I} \sum_{i}\sum_{j=1}^{J_{i}}\widehat{u}_{ij}^{2}=\frac{\sigma_{\varepsilon}^{2}} {I}\sum_{i}J_{i} \\ & \Rightarrow \frac{\frac{1}{I} \sum_{i}\widehat{u}_{i}^{2}}{\frac{1}{I} \sum_{i}\sum_{j=1}^{J_{i}}\widehat{u}_{ij}^{2}}\rightarrow 1\\ \end{aligned} $$

asymptotically, while, under H _A ,

$$ \hbox{plim}\frac{1}{I} \sum_{i}\widehat{u}_{i}^{2}=\frac{1}{I} \sum_{i}J_{i}\left( \sigma_{\varepsilon}^{2}+J_{i}\sigma_{e}^{2}\right) , $$

(2)

and

$$ \hbox{plim}\frac{1}{I} \sum_{i}\sum_{j=1}^{J_{i}}\widehat{u}_{ij}^{2}=\frac{1}{I} \sum_{i}J_{i}\left( \sigma_{\varepsilon }^{2}+\sigma_{e}^{2}\right) $$

(3)

$$ \Rightarrow \frac{\frac{1}{I} \sum_{i}\widehat{u}_{i}^{2}}{\frac{1}{I} \sum_{i}\sum_{j=1}^{J_{i}}\widehat{u}_{ij}^{2}}\rightarrow \frac{\frac{1}{I} \sum_{i}J_{i}\left( \sigma_{\varepsilon }^{2}+J_{i}\sigma_{e}^{2}\right)}{ {{1}\over {I}}\sum_{i}J_{i}\left( \sigma_{\varepsilon }^{2}+\sigma _{e}^{2}\right) }=1+\frac{\frac{1}{I}\sum_{i}J_{i}^{2}}{\frac{1}{I} \sum_{i}J_{i}\left( \frac{\sigma_{\varepsilon }^{2}}{\sigma_{e}^{2}} +1\right)}>1. $$

(4)

Breusch and Pagan (1979) use Eqs. 2, 3, and 4 to construct a Lagrange multiplier test statistic for Eq. 4. For nonlinear models such as the probit models used in this paper, we can replace residuals with generalized residuals as described in Gourieroux et al. (1982). Since the Lagrange Multiplier test is a likelihood-based test and it may not be appropriate to assume that generalized residuals are normally distributed, one should simulate the distribution for the critical value (see, for example, Stern 1997).

1.5 A.5. Multiple-argument one-sided test statistics

The hypotheses proposed at the end of Sect. 5.2 involve multiple one-sided restrictions. While the methodology for multiple restrictions is straightforward (and described in Appendix 9.3 above, and the methodology for single, one-sided restrictions is also straightforward and can be done with a t-test statistic, the methodology for multiple, one-sided restrictions is significantly more difficult. Consider a parameter vector ${\theta}{\in}\Uptheta$, and consider the null hypothesis $H_{0}:\theta \in \Uptheta_{0}$ against $H_{A} :\theta \notin \Uptheta_{0}$ where $\Uptheta_{0}\subset \Uptheta$ with positive measure. Now consider a Wald-type test statistic of the form $W=\left\| \widehat{\theta}-\theta_{0}\right\|_{\widehat{\Upomega}}$ for some $\theta_{0}\in \Uptheta_{0}$ where $\widehat{\theta }$ is an unrestricted consistent estimate of $\theta, \widehat{\Upomega}$ is a consistent estimate of the covariance matrix of $\widehat{\theta}$, and $\|x\|_{A}$ is the quadratic form, x′A ⁻¹ x. If we think of H ₀:θ = θ₀, then W is a Wald statistic and has an asymptotic χ² distribution. There are two problems with this approach. First, H ₀:θ = θ₀ is not the correct null hypothesis, and it is not obvious how to choose $\theta_{0}\in \Uptheta_{0}$. Kudo (1962), Gourieroux et al. (1982), and Kodde and Palm (1986) suggest defining a different test statistic,

$$ W^{\ast }=\min_{\theta_{0}\in \Uptheta_{0}}\left\| \widehat{\theta}-\theta_{0}\right\|_{\widehat{\Upomega }}, $$

(5)

and Kodde and Palm (1986) show that

$$ \lim_{N\rightarrow \infty }\Pr \left[ W_{N}^{\ast }>c\right] =\sum_{k=0}^{K}\omega \left( K,k,\Upomega \right) \Pr \left[ \chi_{k+1}^{2}>c \right] $$

where N is the sample size, K is the number of restrictions under the null hypothesis,

$$ \begin{aligned} \omega (K,k,\Upomega) & = \Pr \left[ R\left( v\right) =k\right] , \\ v & \sim N (0, \Upomega), \\ \end{aligned} $$

R(v) is the number of elements of v > 0. While it is not feasible to analyze the distribution of W ^* analytically, Wolak (1987) provides a simple method to simulate the distribution’s critical values, and Stern (1995) uses it in a problem similar to the one in this paper.

1.6 A.6. Nonparametric estimation of spatial correlation

We can construct a nonparametric estimate of the correlation of two county-specific generalized residuals $\widehat{u}$ as

$$ \widehat{\rho }[d] =\frac{\sum_{m}\sum_{i,k;i\neq k}c\left( \widehat{u}_{mi_{m}},\widehat{u}_{mk_{m}}\right) K\left( \frac{d-d\left( i_{m},k_{m}\right) }{b}\right) }{\sum_{m}\sum_{i,k;i\neq k}K\left( \frac{ d-d\left( i_{m},k_{m}\right) }{b}\right)} $$

where $\widehat{u}_{mi_{m}}$ is the generalized residual for county i _m of site m,

$$ \begin{aligned} c\left( \widehat{u}_{mi_{m}},\widehat{u}_{mk_{m}}\right) &=\frac{\left( \widehat{u}_{mi_{m}}-\widehat{u}_{m}\right) \left( \widehat{u}_{mk_{m}}- \widehat{u}_{m}\right)}{\sqrt{\frac{1}{M_{m}-1}\sum_{l}\left( \widehat{u} _{ml_{m}}-\widehat{u}_{m}\right)^{2}}}, \\ \widehat{u}_{m} &= \frac{1}{M_{m}}\sum_{l}\widehat{u}_{ml_{m}}, \\ \end{aligned} $$

M _m is the number of counties in site m, and $K\left(\frac{\cdot}{b}\right)$ is a kernel function with bandwidth b. Note that

$$ Var\left( \widehat{\rho }[d] \right) =\frac{\sum_{m}\sum_{i,k;i \neq k}\left[ c\left( \widehat{u}_{mi_{m}},\widehat{u}_{mk_{m}}\right) - \widehat{\rho }[d] \right]^{2}\left[ K\left( \frac{d-d\left( i_{m},k_{m}\right) }{b}\right) \right]^{2}}{\left[ \sum_{m}\sum_{i,k;i\neq k}K\left( \frac{d-d\left( i_{m},k_{m}\right)}{b}\right) \right]^{2}}, $$

which can be used to construct a confidence interval.