## 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.

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## Notes

Examples of specification criteria include changes in the equation

*F*statistic, \( {\overline{R}}^2 \), Mallows’*C*_{p}statistic, Amemiya criteria (PC), Akaike Information Criteria (AIC), Sawa Bayesian Information Criterion and/or the Schwarz Bayesian Information Criterion (BIC) as well as the Jeffreys-Bayes posterior odds ratio, among others (see Burnham and Anderson 2004; Judge et al. 1985; Kuha 2004; Posada and Buckley 2004 for formal discussions)Previous literature on income inequality can be grouped into three broad categories: (1) exploring alternative measures of income and inequality (e.g., Katz 1999; Frank 2014); (2) exploring explanations for increasing inequality (e.g., Partridge et al. 1996; Moller et al. 2009; Florida and Mellander 2016); and (3) seeking to understand the potential outcomes of rising inequality (e.g., Partridge 1997, Partridge 2005; Aghion et al. 1999; Fielding and Torres 2006; Oishi et al. 2018).

In addition, previous studies have found that results vary across different measures of income inequality (e.g., Cancian and Reed 1998). For example, should income be defined to include only earnings or all sources of income; or pre- or post-tax income? Should inequality be measured by a traditional Gini coefficient, some entropy-based measure such as a Theil or Shannon Index, a ratio of high to low income, or as Piketty (2014) suggests, the share of total income going to the top 1% of households? Despite the volume of research on the topic of income inequality measures (e.g., Allison 1978), there are no definitive answers to these questions and we are left with the alternative of testing the sensitivity of our results across different measures.

We explored several variations of income inequality measures ranging from several entropy measures across earnings, family and household income with income measured as both earnings and total income. The three measures selected for analysis tended to be the most consistently correlated with the block of potential measures.

It is important to keep in mind that the SBMA approach provides no insight in the direction of the relationship between the independent variables and income inequality but only the consistency of the underlying data generating processes.

We also tested for multicollinearity in the final specifications of the three models (three measures of income inequality) and while the condition index tends to be modestly high (around 160), the individual variance inflation factors were all below four.

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Earlier versions of this study benefited from the comments of participants at the Annual Meetings of the North American Regional Science Council, Minneapolis, MN November 2016 as well as the participants of the Annual Meetings of the Southern Regional Science Association, Memphis, TN March 2017.

## Methods Appendix: Spatial Bayesian model averaging

### 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 *M*_{1},*…,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:

where *ι*_{n} is an *n* by 1 vector of ones, *ε* = *ρWε* + *u*, *u*~*N*(0, *σ*^{2}*I*_{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,

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

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

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

with *g* = 1/ max {*N*, *K*^{2}}. 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 (MC^{3}) method modeling composition approach introduced by Madigan et al. (1995). Using a random-walk step in every replication of the MC^{3} 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

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.

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Conroy, T., Deller, S. & Watson, P. Regional income inequality: a link to women-owned businesses.
*Small Bus Econ* **56**, 189–207 (2021). https://doi.org/10.1007/s11187-019-00224-y

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DOI: https://doi.org/10.1007/s11187-019-00224-y

### Keywords

- Income inequality
- Women entrepreneurs
- Spatial spillovers
- Model uncertainty

### JEL

- D31
- L26
- R12
- J16