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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 model, called variance function 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 the model 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 U.S. earnings inequality among male workers, 1970 to 2005.