David L. Jensen
Manatron, Inc.
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ABSTRACT
Within the field of appraisal the Price Related Differential (PRD) is a statistic frequently computed as a quantified measure of “regressivity”, where lower-value properties are over-assessed relative to higher-value properties, or “progressivity”, where the lower-value properties are under-assessed relative to higher-value properties.
This statistic is computed as the ratio of the mean assessment-to-sale ratio to the weighted mean assessment-to-sale ratio, where the ratios are each weighted by their corresponding sale prices. It is intended to measure the relative “fairness” of appraised values relative to actual sale prices (used as estimates of fair market values) of the properties across the range of sale prices as opposed to the accuracy of the appraisals themselves.
Jensen (2009) proved mathematically and demonstrated graphically that heterogeneous variance (“heteroscedasticity”) in the sale prices can drastically reduce the reliability of the computed PRD. Heterogeneous variance is the natural consequence when the variances in the sale prices reflect percentage differences rather than uniformly random differences, which is oftentimes the case for a wide range of sale prices such as those for commercial properties or those in a severely-escalating market. This paper presents several alternatives to the PRD for testing for regressivity/progressivity.
It begins by summarizing the computation of the PRD and the effect of heterogeneous variance on this statistic. It also summarizes the geometrical comparison of the slope of the best-fit line to that of the equality line as an indicator of regressivity/progressivity and the destabilizing effect of heterogeneous variance on the slope of the best-fit line. It then discusses ways of reducing the heterogeneity of the variance including restricting the range of the data analyzed and the use of weighted regression. Subsequently, it presents Fisher’s t test and variants of it as a means of quantitatively testing the equality of the intercept and the slope of the best-fit line to those of the equality line. Finally, it proposes the resistant line as a means of representing the slope of the best-fit line without the perturbing effects of extreme outliers, and then testing the equality of the intercept and slope of this resistant line to those of the equality line using bootstrapped estimates of its parameters as a better (and perhaps the best) test for regressivity/progressivity.