Regional income inequality: a link to women-owned businesses

Abstract

We assess how women-owned and operated businesses relate to income inequality at the community level. Using U.S. county-level data within the framework of modeling uncertainty, we employ a spatial Bayesian model averaging approach to identify which specific control variables are most consistent with the underlying data generating process for inequality. We find that higher income inequality is linked to larger shares of women-owned and managed businesses. These results are consistent with women-owned businesses being more prevalent at the extremes of the household income distribution where some women are pulled into business ownership at the lower end of the income distribution spectrum and others are driven by opportunities at the higher end of the distribution. We also found meaningful differences in the underlying control variable across our three measures of income inequality. Only a handful of control variables, such as the unemployment rate, rates of college education, and housing costs, are consistent predictors of income inequality.

Methods Appendix: Spatial Bayesian model averaging

Within the economics literature, Bayesian model averaging (BMA) has been introduced to provide a coherent mechanism to account for model uncertainty in terms of what variables should be included in the final specification of the model (Durlauf and Quah 1999; Durlauf et al. 2005). Suppose that there is a set of models all of which may be “reasonable” based on the theory for estimating θ from a given data set y. Suppose further that a particular parameter θ has a common interpretation across all possible models M₁,…,M_k. Instead of using one single model for making inferences about θ, Bayesian model averaging constructs π(θ| y), the posterior density of θ given the data and is not conditional on any specific model (M_i).

Following the lead of (LeSage and Parent 2007) specify the general model as a spatial error model:

$$ y=\alpha {\iota}_n+{X}_k{\beta}_k+\varepsilon $$

where ι_n is an n by 1 vector of ones, ε = ρWε + u, u~N(0, σ²I_n). The number and identity of variables in X_k is unknown so the columns in X_k are taken to be k variables from a larger set (K) of potential explanatory variables contained in X_K with K ≥ k. Any potential model specification is contained in the set of all model possibilities (i.e., ${M}_k\in \mathcal{M}$). The potential number of possible model combinations is 2^K which can become very large in practice.

Inference on the parameters attached to the variables in X_k can be based on the weighted-average parameter estimates of individual models,

$$ p\left({\beta}_j|Y\right)=\sum \limits_{k=1}^{2^K}p\Big({\beta}_j\left|Y,{M}_k\right)p\left({M}_k|Y\right) $$

with Y denoting the data. The spatial lag vector Wy appears in all models as does the intercept term, leaving only the variable vectors in the matrix X subject to change as we compare alternative models. This approach mirrors the one developed by Fernández et al. (2001), where the intercept term appears in all models.

Posterior model probabilities p(M_k| Y) are given by

$$ p\left({M}_k|Y\right)=\frac{p\left(Y|{M}_j\right)p\left({M}_j\right)}{\sum \limits_{k=1}^{2^K}p\left(Y|{M}_k\right)p\left({M}_k\right)} $$

Model weights can be obtained using the marginal likelihood of each individual model after eliciting a prior over the model space. The marginal likelihood of model M_j is given by

$$ p\left(Y|{M}_j\right)={\int}_0^{\infty }{\int}_{-\infty}^{\infty }{\int}_{-\infty}^{\infty }{\int}_{-\infty}^{\infty }p\left(Y|\alpha, {\beta}_k,\rho, \sigma, {M}_j\right)p\left(\alpha, {\beta}_k,\rho, \sigma |{M}_j\right) d\alpha\ d\beta\ d\rho\ d\sigma . $$

Given a model (M_j of dimension k), we can use a noninformative prior on α and σ and a g-prior on the β coefficients we have

$$ p\left({\beta}_k|\alpha, \rho, \sigma, {M}_j\right)\sim N\left({\beta}_k,{\sigma}^2{\left(g{X}_k^{\prime }{X}_k\right)}^{-1}\right) $$

with g = 1/ max {N, K²}. Fernández et al. (2001) show that a great deal of computational simplicity can be found by using Zellner’s g-prior (Zellner 1986) for the parameters b in the SAR model. In addition to simplifying matters, Fernández et al. (2001) provide a theoretical justification for use of the g-prior as well as Monte Carlo evidence comparing nine alternative approaches to setting the hyperparameter g.

The posterior distributions of the β coefficients for the spatial autoregressive specification are calculated as the β which maximizes the likelihood calculated over a grid of ρ values. Building on the prior work of Raftery et al. (1997) as well as Fernández et al. (2001) LeSage and Parent (2007) adopts a Markov Chain Monte Carlo Model Composite (MC³) method modeling composition approach introduced by Madigan et al. (1995). Using a random-walk step in every replication of the MC³ procedure, one can construct an alternative model to the active one in each step of the chain by adding or removing a regressor from the active model. The chain then moves to the alternative model with probability given the product of Bayes factor and prior odds resulting from the beta-binomial prior distribution. The posterior inference is based on the models visited by the Markov chain instead of on the complete model space which is untraceable given a large K (recall the full model space $\mathcal{M}$ is 2^K, if, for example if K = 10, then the full model space has a dimension of 1024). We can more formally define a neighborhood nbd(M) for each $M\in \mathcal{M}$ (the set of all possible models). From there we can define a transition matrix q by setting q(M → M^′) = 0 ∀ M^′ ∉ nbd(M) and q(M → M^′) ≠ 0 ∀ M^′ ∈ nbd(M). If the chain is currently in state M, we can proceed by drawing M′ from q(M → M^′). M′ is accepted with probability

$$ \min \left\{1,\frac{P\left({M}^{\prime }|y\right)}{P\left(M|y\right)}\right\}. $$

Otherwise, the chain remains in state M. Using a Metropolis-Hastings sampling scheme, LeSage and Parent (2007) were able to implement a Markov Chain Monte Carlo routine to move through the modeling space.

There are three ways to use the spatial Bayesian model averaging approach to identify the final set of control variables to include in the income distribution and poverty models. First, use the single model ${M}^{\ast}\in \mathcal{M}$ with the highest posterior probability to determine which variables are to be included in the final set of control variables. Second, look at the frequency of variables entering the top ten models (${M}^{10}\in \mathcal{M}$) ranked by their posterior probability. If a particular variable appears more than, say seven times, in the top ten models, that variable could be included in the final set of control variables. Finally, examine the posterior probability of individual variables and, if the variable has a posterior probability above some threshold, the variable is included in the final set of control variables. In most cases, the three criteria are generally in agreement and the choice of variables is clear. There are, however, a handful of cases where the three methods do not concur and a judgment call is required. For this study, we use each of the three criteria: (1) the variable must be contained in the single model ${M}^{\ast}\in \mathcal{M}$ with the highest posterior probability; (2) the variable must be contained in at least eight of the top ten models (${M}^{10}\in \mathcal{M}$) ranked by their posterior probability; or (3) the posterior probability of the single variable must be greater than 0.80.

Appendix Table A: Descriptive Statistics