Two types of assessment bias

Transcription

1 Methods for Determining Vertical Inequity in Mass Appraisal Justin M. Carter, M.A. The statements made or opinions expressed by authors in Fair & Equitable do not necessarily represent a policy position of the International Association of Assessing Officers. Two types of assessment bias exist in appraisal. The first, horizontal inequity, concerns price differentiation between homes with the same attributes. If two homes with the same attributes (including geographical location) are assessed at different values, horizontal inequity exists. Generally more concerning, however, is vertical inequity. Vertical inequity exists when the ratio of assessed value to sale value changes over the value of properties generally, and it can be either regressive (where low-value homes are regularly assessed at greater ratios to the sale price than the high-value homes in the sample) or progressive (the opposite phenomenon). In general, regressivity tends to occur more often than progressivity, but both are troubling (De Cesare and Ruddock 1998). There exists a plethora of academic articles concerned with the most acceptable way to assess vertical equity for uniformity s sake (e.g., see Bell 1984; Cannaday, Stunard, and Sunderman 1987; Sirmans, Diskin, and Friday 1995; Kochin and Parks 1982a, 1982b). Furthermore, in the early 1990s, Sunderman et al. (1990) and others claimed that there existed, as yet, no measure of vertical inequity [that] is widely accepted. A History of Regression Techniques for Determining Vertical Inequity Before delving into the difference between price-related differential (PRD) and price-related bias (PRB), note that there are numerous hypothesized multiple regression-based techniques for determining vertical equity. Scholars still argue about the choice of dependent and independent variables. Most suggest the use of appraised price as the dependent variable and sale price as the independent variable; some claim the opposite. Table 1 offers a brief history of the formulas in question (De Casare and Ruddock 1998). Once the models have been constructed, there are multiple ways of testing for Table 1. History of techniques for determining vertical inequity Simplified Model Test Citation AV= β 0 (SP) + µ β 0 = 0 Paglin and Fogarty (1972) Log(AV) = β 0 [Log(SP)] + µ β 1 = 1 Cheng (1974) AV/SP = β 0 (SP) + µ β 1 = 0 Almy et al. (1978) AV = β 0 (SP) + β 2 (SP 2 ) + µ Β 0 = β 2 = 0 Bell (1984) AV = β 00 0 (SP) + β 01 (Low) + β 02 (High) 1 (LowSP) Β 00 = β 01 = β 02 = 0 Sunderman et al. (1990) 2 (HighSP) + µ Log(SP) = β 0 [Log(AV)] + µ Β 1 = 1 Kochin and Parks (1982a, b) Notation: AV = appraised value SP = sale price Log = loglinear function µ = error term Fair & Equitable June

2 vertical inequity. The first is to simply analyze the coefficient of the main independent variable: (1) appraised value under the auspices of the suggested models by Kochin and Parks (1982) and De Cesare and Ruddock (1998) and (2) sale price under the auspices of Paglin and Fogarty (1972), Cheng (1974), Almy, Gloudemans, and Denne (1978), Bell (1984), and Sunderman et al. (1990). A coefficient of 1.0 indicates no vertical inequity; a coefficient less than 1.0 is indicative of regressivity; and a coefficient greater than 1.0 is indicative of progressivity. Interestingly, researchers have found that using these modeling structures almost always tends to lead to some variant of vertical bias. When assessed values are implemented as the dependent variable, models generally tend to exhibit regressive tendencies; when sale values are implemented as the dependent variable, models tend to exhibit progressive tendencies. This occurs even when the same data set is used and the independent and dependent variables are flipped (De Cesare and Ruddock 1998). Furthermore, although I have found no mention of this in the literature thus far, these models are inherently problematic. This is because it is a Herculean assumption that sale price causes appraised price or that appraised price causes sale price, which is, perhaps, the main assumption of bivariate or multivariate regression modeling. Consider, for example, the difference in trying to determine sale price through (1) square footage of the home and (2) appraised price. It is very easy to imagine how square footage of the home truly causes variation in sale price; in general, the larger a home (assuming other qualities of homes in question are equivalent), the greater the sale price. But how does a greater appraised price of a home cause a greater sale price? It cannot except through circuitous causation wherein, perhaps, appraised 4 Fair & Equitable June 2016 values become known to the public and begin to influence the sales market, but this is highly unlikely. Instead, what is happening is that attributes of the home (such as square footage, existence of a garage, age of the home, existence of a pool, geographical location of the home, school district in which the home is located, and more) are causing variations in both the sale price and the appraised price. Regression modeling repeatedly stresses that correlation is not equivalent with causation, and regression modeling is based upon the assumption that all independent variables included in any model are truly causative of variation in the dependent variable. It is this broken assumption that is so damning to tests for vertical inequity through the use of regression modeling. The Main Alternative Price-Related Differential Therefore, given the prior findings concerning the difficulty of determining vertical inequity through bivariate or multivariate regression modeling techniques, what other techniques exist? The first, and that advocated by the IAAO Standard on Ratio Studies (IAAO 1990, 1999, 2007) is the PRD. This measure was introduced in 1957 by the U.S. Census Bureau and has been both praised and denigrated ever since (Gloudemans 2011). The PRD is calculated thusly: take the mean sales ratio of all homes in question and divide it by the weighted mean of that sample. This means, first, that the analyst must find the sales ratio for each property, sum these ratios, and derive the mean to generate the numerator: (Σ nxi i x). Second, the analyst must take the sum of all appraised values and divide it by the sum of all sale prices (the weighted mean) to generate the denominator: Σ n i AV. Σ n i SP This means that the final equation follows: xi Σ n i x Σ n i AV Σ n i SP where AV = appraised value SP = sale price. According to the IAAO standard (IAAO 1990, 1999, 2007), a PRD score between 0.98 and 1.03 is acceptable. A PRD score less than 0.98 indicates vertical assessment progressivity, while a PRD score greater than 1.03 indicates vertical assessment regressivity. Unfortunately, even IAAO is less than wholly supportive of this measure, stating that the PRD provides only an indication, not a proof of appraisal bias (IAAO 1990), yet again leaving a gap for those in the profession trying to determine whether vertical equity exists. Note that there are some problems with the PRD. First, according to Gloudemans (2011) and others (e.g., Almy, Gloudemans, and Denne 1978; Eckert, Gloudemans, and Almy 1990; Gloudemans 1999), in the vast majority of jurisdictions, the PRD tends to contain an inherent downward bias because of the fact that the mean sales ratio tends to be high because high-value properties skew the mean ratio. The suggested palliative for this bias is to place the comparison point at some level greater than 1.0. In fact, antithetical to the IAAO standard (0.98 to 1.03) (see Eckert, Gloudemans, and Almy 1990; IAAO 1990, 1999, 2007), Almy, Gloudemans, and Denne (1978) suggest standards of 0.95 and 1.10 as signs of regressivity or progressivity, respectively. Even in setting these standards, Gloudemans warns that,

3 When samples are small, have high dispersion, or include properties with extreme value, the PRD may not provide an accurate indication of assessment regressivity or progressivity. (Gloudemans 2011, 4) Second, there is no real sense in the literature of an understanding of what the numbers provided by the PRD actually mean. As Gloudemans states, what does a PRD of a given value, say 1.03, really mean? Or how bad is a PRD of 0.95, 1.05, or 1.15?... The problem is not that the PRD is meaningless. It is simply what its originators meant it to be an easy-to-calculate measure that provides 'some notion of any association' between assessment ratios and property value. While helpful in testing the null hypothesis of no price-related bias, tests indicate only whether it can be concluded that some regressivity or progressivity is measured. The bias may be egregious or it may be benign, as it is well known that even the smallest differences or relationships can be found statistically significant if sample size is large enough. (Gloudemans 2011, 4) Some Data Difficulties in Determining Vertical Inequity The biggest problem, claim many scholars (Cannaday, Stunad, and Sunderman 1987; Clapp 1990; Evans 1995; De Cesare and Ruddock 1998), is that while researchers do indeed have access to both appraised values and sale values (sometimes nondisclosure states may not have access), they do not often have access to market value. Evans (1995), in fact, goes so far as to suggest that neither skilled assessors nor sophisticated statistical methods can predict sales process of properties with any degree of precision, and assumes that the error in predicting market price is likely to be, 'at best,' around 10 percent (De Cesare and Ruddock 1998, 5). This is because customers in the housing market do not act in the economically efficient way that econometric models assume. Econometric models almost always tend to assume that consumers are fully informed and make rational decisions before making a purchase. This assumes that consumers have access to all information about the market, all information about the condition of homes, and all information about mortgage options; weigh all this information against each other; and choose the Nash Equilibrium option when determining which house to purchase and at what price. (In game theory, the Nash Equilibrium is a solution concept of a noncooperative game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing his own strategy [Osborne and Rubinstein 1994].) Even at first glance, these assumptions are obviously dubious when the analyst is considering real-world housing market transactions. Trained scientists and economists are often thought to consider incentives in the world, so let us consider the incentive structure in the housing market and determine whether it conforms to the assumptions of econometric models. First, consumers certainly do not have access to complete information about their housing market; this is especially so in nondisclosure states such as Texas, where realtors have a clear incentive not to disclose to the consumer information concerning the housing market and the potential value of homes in that market. Second, sellers of homes, whether the owners themselves, the government and banks (under the auspices of foreclosure sales), and even realtors have little incentive to disclose all information about the condition of a home, unless those conditions will drive up the sale price of a home. Furthermore, banks have little incentive to disclose all possible options for mortgage generation, preferring instead to sell the most profitable mortgage package (the arguably destructive mortgage packages being sold to home purchasers before the economic collapse of 2008 offer further proof of this). Unfortunately, the only party in the process of purchasing a home who does have the incentive to collect all of this information and analyze it to the fullest is the consumer, and, euphemistically, the cards are stacked against the consumer in the incentive structure of the economy. Scholars tend to suggest that sale prices, however, are more accurate than appraised values (Cheng 1974; Bell 1984; Gaston 1984; Benson and Schwartz 1997; De Cesare and Ruddock 1998). Finally, the vast majority of scholars and Improving Real Property Assessment: A Reference Manual (Almy, Gloudemans, and Denne 1978) declare that it is imperative to ensure that a good sales edit increases the accuracy of sales information [and that] lack of good sales analysis procedures affects adversely the measures of assessment performance (De Cesare and Ruddock 1998, 8). This means that the individual or group in charge of analyzing a jurisdiction s data must ensure that the sales included in the analysis are good sales. Sales must reflect the market, and estate sales, corporate sales, foreclosure sales, and non-arm s-length sales must be carefully scrutinized before being included in the final data set used for vertical equity analysis. The Contemporary Alternative Price-Related Bias The PRB has arisen as the newest alternative to the former methods of determining vertical inequity bivar- Fair & Equitable June

4 iate regression models, multivariate regression models, and the PRD. The PRB is, admittedly, simply a variation of the regression techniques discussed above and is inherently plagued, therefore, by some of the same difficulties. It has, however, been widely defended and may become the new standard for determining vertical inequity in the near future. Mathematically, the PRB is expressed as follows: AV Log Median + Sale Price (Sales Ratio Median Sales Ratio) 2 2 = β0 + β Median Sales Ratio 1 Log (2) A simple and edifying explanation of the mathematical underpinnings of the PRB has been published by the Minnesota Department of Revenue (n.d.). The department s tutorial that provides the mathematical interpretation offers the following explanations for each piece of the equation. First, the dependent variable represents the percentage difference between any given sales ratio and the median [sales ratio]. Sales ratios are calculated using current year or next year [appraised values]. By using a percent, the β 1 can be interpreted as a percent change (from the median) in ratios. The independent variable linked with β 1 is adjust[ed] [to] the [appraised] value by dividing the median ratio to ensure the estimated [appraised] value and the sale price [are] equally weighted. Instead of using the sale price as the indicator of value, this side of the formula uses the average of the sale price and the adjusted [appraised value] as a proxy for value. While not a perfect indicator of value, using the average reduces the upward bias of the PRB [more] than if the sale price was used. Using the natural log allows us to interpret increases in property values as a percentage. Given the dependent 6 Fair & Equitable June variable in the analysis is also a percent; it will be interpreted as percent increase in value increases ratios by a percent. Dividing by the natural log of 2 (0.693) permits each doubling of value to be associated with an increment of 1. So, a 100 percent increase in the value increases the estimated ratio by (β 1 100%). This is done for interpretation. If we don t divide by the natural log of 2, the PRB would tell us how ratios change for a 1 percent increase in the value. This would result in very small coefficients which are hard to conceptualize. (Minnesota Department of Revenue n.d.) In addition, use of the natural logarithm helps to minimize the effect of outliers on the coefficient (Gloudemans 2011, 5). Given this understanding of the mathematical concepts underpinning the PRB, how is the coefficient interpreted? It can be interpreted as the expected change in ratios as property values double. If ratios increase as property values increase, the resulting PRB will be positive. For example, a PRB of indicates that if property value doubles, ratios increase by 2.5 percent. A positive PRB indicates that assessments are progressive, meaning high-value properties are over-appraised relative to low-value properties (and the opposite for an equivalent negative coefficient). A PRB greater or less than 3 percent may indicate a bias. A PRB greater or less than 5 percent is cause for greater inspection. (Minnesota Department of Revenue n.d.) Given this review of the PRB, what are the potential problems with this approach to assessing vertical equity? First, because this is simply a variant of bivariate regression, many of the former complaints about regression techniques are applicable, specifically, the colossal assumption of causality between independent and dependent variables. It is questionable, at best, to claim that the independent variable in the PRB is truly causing variation in the dependent variable. Second, the same problems with the incentive structure of the market exist, causing the analyst to question whether sale value is truly representative of the market value of homes in the jurisdiction in question. The Prescription for Determining Vertical Equity Given the difficulties inherent in each of the techniques for determining whether vertical inequity is present in a jurisdiction, this current research would suggest taking the most comprehensive approach possible: use all the techniques mentioned to try to determine whether vertical inequity is problematic. This allows the analyst to gain a deeper understanding of the data and the likelihood of the existence of vertical inequity. Although running more tests is more time-consuming, the analyst can be much more confident in claiming that vertical inequity does or does not exist. Given the importance of ensuring that property values are well-constructed and of protecting the main recipients of the taxes derived from these values (schools and hospitals, at least in the American system), the trade-off between more time spent in analysis and the greater level of confidence derived by running multiple tests seems well worth the effort, at least until the profession comes to a consensus about which measure is best. Data and Summary Statistics The data used in this research come from a West Texas county and were collected from numerous sources including multiple listing services, real estate websites, and city permits. The data consist of 2,690 validated sales of residential single-family homes sold between January 1, 2014, and December 31, Summary statistics are presented in table 2. The range of sale

5 prices varies greatly, from a minimum of approximately $16,000 to a maximum of more than $1,000,000. Appraised Value as Dependent Variable The first test to determine whether vertical inequity exists in the sample data uses bivariate regression, in which the appraised value of each home represents the dependent variable and the sale price represents the independent variable. Table 3 shows the results of this bivariate model. As shown, the statistically significant coefficient of could, arguably, suggest slight regressivity. However, the distance from 1.0 is negligible. Thus it is difficult to suggest that regressivity is problematic. This result does suggest, however, that the profession should come to some agreement about what coefficient can be confidently declared as proof of regressivity or progressivity. some sort of standard should be reached by the profession for declaring whether progressivity or regressivity is or is not problematic). In this case, the data again seem to suggest normality with regard to vertical equity. Price-Related Differential The PRD derived from the data in question suggests a lack of vertical inequity. The PRD of suggests that vertical equity exists in the jurisdiction in question. No evidence for regressivity or progressivity in the data exists, according to the PRD. Price-Related Bias The PRB is also a form of bivariate regression. The results of the analysis are given in table 5. The interpretation of the results is as follows: for every doubling of price (using the median as the arbitrary starting point), the sales ratio will be reduced by 0.7 percent. According to the standards set by the Minnesota Department of Revenue, this coefficient ( 0.007) is indicative of vertical equity. Conclusion The assessment profession must, in the near future, come to some semblance of agreement about (1) which measure of vertical equity is best, (2) the standards for determining whether vertical inequity exists for each measure, and (3) whether it can develop a new measure that does not have the shortcomings of current measures of vertical inequity, such as the problems of causality in regression techniques, the inherent upward bias of the PRD, and the problem with finding regressivity when sale price serves as the dependent variable in bivariate regression and finding progressivity when appraised value is Sale Price as Dependent Variable The second test for determining vertical inequity again implements bivariate regression modeling, but flips the dependent variable (sale price) and the independent variable (appraised value). Table 4 shows the results of this new model. Table 4 suggests the opposite of the results in table 3. When the results are compared, they provide an affirmation of the findings of De Cesare and Ruddock (1998) that simply flipping the dependent variable can lead to completely opposing results. When sale price is implemented as the dependent variable, the model points to slight progressivity (coefficient equals 1.015), and when appraised value is implemented as the dependent variable, the model points to slight regressivity (coefficient equals 0.958). Again, although the coefficient is statistically significant, it is only marginally greater than 1.0, so the analyst should be very careful in attributing progressivity to the results (and again suggests that Table 2. Summary statistics for 2,690 sales of single-family homes in a West Texas county, January 21, 2014, to December 31, 2014 Mean Median Standard Deviation Minimum Maximum Sale price $174,697 $142,000 $111,290 $16,100 $1,075,000 Appraised value $172,046 $138,889 $108,138 $14,811 $910,011 Sales ratio Table 3. Appraised value as dependent variable Coefficient Standard Error t-value Significance Sale price Constant Adjusted R N = 2,688 Table 4. Sale price as dependent variable Coefficient Standard Error t-value Significance Appraised value Constant Adjusted R N = 2,688 Table 5. Price-related bias Coefficient Std. Error t-value Significance Independent variable Constant Adjusted R N = 2,688 Fair & Equitable June

6 implemented as the dependent variable (when using the exact same data). Because of all these inherent problems and the lack of agreement about which technique is best, the analyst must hedge his or her bets and use all these techniques when trying to determine whether vertical inequity exists in the data sample. This will make the analyst far more confident in declaring that vertical inequity exists or does not exist, at least until the profession finally sets some standard for this determination. References Almy, R.R., R.J. Gloudemans, and R.C. Denne Improving Real Property Assessment: A Reference Manual. Chicago: International Association of Assessing Officers (IAAO). Bell, E.J Administrative Inequity and Property Assessment: The Case for the Traditional Approach. Property Tax Journal 3: Benson, E.D., and A.L. Schwartz, Jr Vertical Equity in the Taxation of Single Family Homes. The 13th Annual Meeting of the American Real Estate Society, Sarasota, Florida. Cannaday, R.E., E.W. Stunard, and M.A. Sunderman Property Tax Assessment: Measures and Tests of Uniformity Applied to Chicago Condominiums. Illinois Business Review 44 (2): Cheng, P.L Property Taxation, Assessment Performance, and Its Measurement. Public Finance XXIX: Clapp, J.M A New Test for Equitable Real Estate Tax Assessment. Journal of Real Estate Finance and Economics 3: De Cesare, C.M., and L. Ruddock A New Approach to the Analysis of Assessment Equity. Assessment Journal 5 (2): Eckert, J., R. Gloudemans, and R. Almy, eds Property Appraisal and Assessment Administration. Chicago: IAAO. Evans, A.E The Property Market: Ninety Percent Efficient? Urban Studies 32 (1): Gaston, P Redeeming the Assessment Ratio Equity Test. Property Tax Journal 3: Gloudemans, R.J Mass Appraisal of Real Property. Chicago: IAAO. Gloudemans, R.J The Coefficient of Price-Related Bias: A Measure of Vertical Equity. Fair and Equitable 9 (8): 3 8. IAAO Standard on Ratio Studies. Chicago: IAAO. IAAO Standard on Ratio Studies. Chicago: IAAO. IAAO Standard on Ratio Studies. Kansas City, MO: IAAO. Kochin, L.A., and R.W. Parks. 1982a. Vertical Equity in Real Estate Assessment: Fair Appraisal. Economic Inquiry XX: Kochin, L.A., and R.W. Parks. 1982b. Testing for Assessment Uniformity: A Reappraisal. Property Tax Journal 3: Minnesota Department of Revenue. n.d. Sales Ratio Tutorial The Price Related Bias. Sales%20Ratio%20Tutorial%20-%20 PRB.pdf. Osborne, M.J., and A. Rubenstein A Course in Game Theory. Cambridge, MA: MIT Press. Paglin, M., and M. Fogarty Equity and the Property Tax: A New Conceptual Focus. National Tax Journal 25 (4). Sirmans, G.S., B.A. Diskin, and H.S. Friday Vertical Inequity in the Taxation of Real Property. National Tax Journal 48 (1): Sunderman, M.A., J.W. Birch, R.E. Cannaday, and T.W. Hamilton Testing for Vertical Inequity in Property Tax Systems. Journal of Real Estate Research 5 (3): Justin Carter graduated from Texas Tech University with honors with a bachelor's degree in political science and a minor in philosophy. He continued his education with a master's degree in military conflict, diplomacy, and reconciliation, and another master's degree in political science with a focus on comparative politics and statistics, graduating summa cum laude in both. He taught political science at Texas Tech University from 2012 to He recently won Tyler Technologies' Annual Excellence in Public Service award for his statistical modeling. Mr. Carter can be contacted at justincarterma@yahoo.com. 8 Fair & Equitable June 2016