Variance Function Regressions for Studying Inequality

Regression-based studies of inequality model only between-group differences, yet often these differences are far exceeded by residual inequality. Residual inequality is usually attributed to measurement error or the influence of unobserved characteristics. We present a regression that includes covariates for both the mean and variance of a dependent variable. In this model, the residual variance is treated as a target for analysis. In analyses of inequality, the residual variance might be interpreted as measuring risk or insecurity. Variance function regressions are illustrated in an analysis of panel data on earnings among released prisoners in the National Longitudinal Survey of Youth. We extend themodel to a decomposition analysis, relating the change in inequality to compositional changes in the population and changes in coefficients for the mean and variance. The decomposition is applied to the trend in US earnings inequality among male workers, 1970 to 2005. In studying inequality, we can distinguish differences between groups from differences within groups. Sociological theory usually motivates hypotheses about between-group inequality. For these hypotheses, interest focuses on differences in group averages. For example, theories of labor market discrimination predict whites earn more than blacks, and men earn more than women. Human capital theory explains why college graduates average higher earnings than high school dropouts. Such theories are often tested with a regression where differences in groups means are quantified by regression coefficients. Although theory usually focuses on between-group differences, withingroup variance also contributes to inequality. Within-group inequality can be measured by the residual variance of a regression. Typically the residual is viewed as unexplained, and its variation is not treated as substantively interesting. Although it is often overlooked, residual heterogeneity may vary in substantively important ways. Some groups may be more insecure than others, or vary more in unobserved characteristics. The structure of within-group inequality may be especially important for sociological analysis where the residual variance often greatly exceeds the the between-group variance. We present a statistical model of inequality that captures the effects of covariates on within-group and between-group inequality. Called a variance-function regression, the model features separate equations for the mean and variance of the dependent variable. Regression coefficients for the mean and variance can be estimated with specialized calculations, though we show that they are well-approximated in large samples with standard software. Though variance function regression have a long history in economet-