Measuring Vertical Inequity in Property Assessment: A New Approach Using Data from Massachusetts

Transcription

1 Measuring Vertical Inequity in Property Assessment: A New Approach Using Data from Massachusetts Thomas PlaHovinsak Northeastern University Gustavo Vicentini Northeastern University Abstract Several previous studies claim to identify bias in the assessment of housing value for the purposes of property taxes. This chapter highlights some problems with the methodologies used in many of those studies and provides an alternative framework for finding minimum values for assessor error and bias. Using data from the Massachusetts Office of Geographic Information, we build off previous studies in three ways. First, we briefly explain why the errors-in-variables problem can cause biased estimates of vertical inequity in property assessment and how previous solutions to this problem are based on misleading assumptions. Second, we show that a method based on hedonic price estimates using property-level observables can provide a lower bound for the extent of assessor error and bias for Massachusetts towns. Third, we explore if there are differences in vertical inequity across towns in Massachusetts. Our results show that more than 23.3 percent of the variance in the difference between assessment and sale price across Massachusetts is due to assessor error, and that several property-level and town-level features can explain assessor bias. Furthermore, high-value properties across the state are the ones most likely to be under-assessed regardless of whether or not they are located in a high-income town. Keywords: JEL Classifications: Correspondence: Property tax, property assessment, vertical inequity, regressivity H22, H71, H73 Department of Economics Northeastern University 360 Huntington Avenue 301 Lake Hall Boston, MA PlaHovinsak: Vicentini: plahovinsak.t@husky.neu.edu gvicentini@gmail.com 1

2 1. Introduction The distribution of the burden of property taxes has been researched extensively over the past several decades, with many studies exploring issues related to vertical inequity in property assessment. Particular attention has been given to the presence of what the literature calls regressivity, which is a particular type of vertical inequity. Regressivity is defined by both the literature and the International Association of Assessing Officers (IAAO) as a pattern of overassessment of low-value properties in contrast to high-value properties, typically measured by comparing the assessed value of a property to its most recent sale price. 1 Because the assessed value is the basis for calculating the tax levy of a property, regressivity shifts the burden of the tax across property owners. The focus of previous studies has concentrated on two main areas: methodology for measuring regressivity; and identifying the determinants of regressivity. The studies in the first area have proposed new methodological frameworks for measuring regressivity within an assessing jurisdiction as well as testing the validity of the competing methodologies (e.g., Paglin and Fogarty, 1972, Kochin and Parks, 1984, Clapp, 1990, Goolsby, 1997, Fairbanks et al., 2013, Birch and Sunderman, 2014). The second area of research has focused on investigating the causes of regressivity by comparing inter-jurisdictional differences (e.g., Smith et al., 2003, Ross, 2012). Particularly in the first area, a key concern has been how to solve the errors-in-variables problem when estimating the amount of regressivity via regression techniques. Since both assessed value and sale price are likely noisy measures of the market value of a property, authors have had to decide how to best address this issue so as to avoid biased estimates of regressivity. 1 Progressivity, on the other hand, is a pattern of over-assessment of high-value properties in contrast to low-value properties. 2

3 The goal of this paper is to briefly illustrate why previous methods used to measure vertical inequity are flawed, as well as to use a new approach to find minimum values for both assessor error and assessor bias. Specifically, we examine the potential indicators of vertical inequity by regressing the log difference of assessment and sale price on property and neighborhood characteristics. The features of a property (e.g., size, age, proximity to public goods) and of its neighborhood (e.g., proportion of vacant housing) should be determinants of both the assessed value and sale price of a property, but in principle they should be uncorrelated with the difference between these two if assessor error and market error are random. By using this framework, we are able to estimate a lower-bound of both assessor error from the zero-centered R 2 of this regression, and of assessor bias using a hedonic pricing model. We apply this approach to property-level data for single-family residences from 2008 to 2014 for the state of Massachusetts. We find that a number of property features explain assessor bias when controlling for town differences, and that assessor error explains approximately 23.3 percent of the variance in the difference between assessment and sale price. When examining individual towns, 2 we find that properties in highincome towns are much more likely to be under-assessed than those in low-income towns, with the exception of lowest-value properties which are very close to accurately assessed. We also see that the highest-value properties are the most likely to be under-assessed across all towns regardless of the owner s income level. The rest of this paper proceeds as follows. Section 2 describes our proposed methodology and compares it to previous studies. Section 3 describes the data and provides descriptive statistics. Section 4 contains the results, and Section 5 concludes. 2 We note that all references to a town or towns in this paper refer to all types of municipalities within the state of Massachusetts, including both cities and townships. 3

4 2. Measuring Assessor Error and Regressivity in Assessment 2.a. The Errors-in-Variables Problem In many jurisdictions, the assessed value of a property is supposed to reflect 100 percent of its market value. The IAAO defines market value as [t]he most probable price which a property should bring in a competitive and open market under all conditions requisite to a fair sale, the buyer and seller each acting prudently and knowledgeably, and assuming the price is not affected by undue stimulus, 3 and the Massachusetts Bureau of Local Assessment ensures that each municipality s properties are assessed at fair market value at least once every three years. 4 As described in Sirmans et al. (2008), many studies measure under or over-assessment by comparing the difference between the assessed value and sale price of a property. For example, if a property sold for $500,000 and its most recent (prior to the sale) assessment was $475,000, then that property is assumed to have been under-assessed by $25,000. In addition, one area of discussion has been whether or not assessed value or sale price is a better reflection of a property s true market value, and hence whether assessed value or sale price should be on the right hand side of a regression. It is reasonable to believe, however, that there will be some degree of error in assessment. This could result from either willful bias on the part of the assessor, the assessor not correctly taking certain observable features of a property into account, or random error in assessment. In spite of these potential sources of error, the assessor s job is to approximate the true, underlying value of a property, analogous to the idea of the fair market price. However, sale price is also likely a noisy measure of fair market value. Sale price may deviate from fair market value due to transient 3 Source: International Association of Assessing Officers (IAAO), Standard on Mass Appraisal of Real Property, January 2012, p Source: Graziano, Joanne. Bureau of Local Assessment. Mass.gov. (accessed May 16, 2016). 4

5 market conditions at the time of sale, differences in bargaining power and discount rate between parties, how eager the seller is to sell the property, and other factors. Hence, there is a potential measurement error issue in both assessment (A) and sale price (P), as outlined in equations (1) and (2). ln(a i ) = ln(v i ) + ε i (1) ln(p i ) = ln(v i ) + μ i (2) Here, V represents the expected value, or the fair market value of property i that the assessor is theoretically trying to determine. We define V as the average price one would obtain if an average seller were able to repeatedly sell the same house on the open market. ε and µ are both assumed to be mean-zero error terms and independent of V. If there were random measurement error only in assessment or only in price, then the variable measured with error could be regressed on the variable that is measured without error to obtain an unbiased estimate of the direction and extent of inequity. Any values other than zero for the constant and zero for the coefficient of the variable measured without error would indicate inequity. But, since both A and P measure V with error, putting either on the right hand side of the regression will lead to biased estimates of inequity. Such a regression could potentially show regressivity where there is none. 2.b. Previous Vertical Inequity Models Sirmans et al (2008) provides an overview of the several methods that past studies have used to test for the presence of vertical inequity in an area as well as contend with the errors-in-variables problem. Building off this overview, Fairbanks et al. (2013) use a Monte Carlo simulation to test the accuracy of these models and makes several recommendations following their results. First 5

6 Fairbanks et al. (2013) suggest that authors attempting to measure vertical inequity try to select areas that appear to have no predisposition to regressivity or progressivity. They also recommend using a graph of the assessment-to-price ratio and sale price as a first pass in determining the existence of vertical inequity. We believe, however, that this approach could lead to incorrect a priori decisions about the presence of vertical inequity. In other words, even if assessment is not actually regressive, because of measurement error in P the figure would display a pattern of regressivity. With the errors-in-variables problem present it is difficult to determine an area s vertical inequity without some form of proper estimation. More specifically, OLS will likely underestimate the slope coefficient, therefore potentially indicating regressivity when there may be none. This is the classical errors-in-variables problem in econometrics, and this downward bias of the slope coefficient is typically known as attenuation bias in the literature. 5 In terms of econometric models, Fairbanks et al. (2013) recommend two models in particular as a result of their simulations. First, they recommend using the model from Clapp (1990), which uses instrumental variables to estimate vertical inequity. Clapp specifies the log of sale price as a function of the log of assessment. 6 ln(p i ) = α 0 + α 1 ln(a i ) + ε i (3) 5 See, e.g., Greene (2012, chapter 8). We note that measurement error in the dependent variable in a regression does not cause attenuation bias in the OLS estimator, although it does reduce the efficiency of the estimator. It is only measurement error in the independent variable that causes the bias. 6 Clapp (1990) justifies specifying his model with assessed value on the right-hand side by citing that in many typical sales-ratio studies, assessed value is determined before sale price, and hence a part of assessed value is negatively capitalized into sale price. See Edelstein (1979) for a full explanation of this justification. 6

7 If regressivity is present, then the slope coefficient in this model is greater than one. Due to assessor error, ln(ai) is an imperfect measure of the log of market value, and therefore the OLS estimator would be biased downwards towards progressivity. Clapp proposes using a grouping instrumental variable for ln(ai) that can take three possible values, depending on property i s ranking within the distribution of assessments and sale prices: +1 if property i s assessment falls in the top one-third of assessments and its sale price falls in the top one-third of sale prices; 1 if property i s assessment falls in the bottom one-third of assessments and its sale price falls in the bottom one-third of sale prices; 0 otherwise. Clapp also recommends including some sort of time control in the model. This instrument is likely relevant for market value, because the higher the market value of a property the higher its ranking and therefore the larger the value of the instrument. However, the exogeneity of this instrument relies on the assumption that the ranking observed in assessment and sale price is the same as the ranking that one would observe in market value. This is plausible if measurement error is small enough such that the rankings across assessment, sale price, and market value are all the same, as conceded by Clapp (1990) in his paper. Clapp justifies the use of this method in his paper by claiming that this would still be an improvement over the other leading models at the time. 7 However, if measurement errors are large enough, a property that is (for example) in the middle one-third of market value might be incorrectly allocated in the top onethird of assessment and sale price if its measurement errors are positive. This grouping misallocation would create a correlation between the instrument and the measurement errors, therefore jeopardizing the exogeneity of the instrument. In essence, measurement error would still 7 Specifically, Clapp writes that he believes that his model will be less biased than the Paglin and Fogarty (1972) and the Kochin and Parks (1984) models, which were the leading models for measuring vertical inequity prior to Clapp s model. See Sirmans at al. (2008) for a detailed explanation of these models. 7

8 be present on the right-hand side of Equation (3), and the attenuation bias stemming from the errors-in-variables problem would still be present. The second model that Fairbanks et al. (2013) recommend is a dummy variables regression model from Sunderman et al. (1990) for non-linear estimation of vertical inequity. 8 Their model reverts to having assessed value as the dependent variable, and searches for break-points in the data and measures slope coefficients for each segment of the data with the following equation: Ai = α00 + α10 Pi + α01 LOWi + α02 HIGHi + α11 LOW*Pi + α12 HIGH*Pi (4) where LOW and HIGH are dummy variables indicating if the sale price of property i is less than or greater than the first or second break-point in the data, respectively, and LOW*P and HIGH*P are interaction terms between the dummy variables and the sale price. Other authors have allowed for more variation by using the percentile of sale price on the right hand side. However, although the spline model does provide for a more localized and flexible measure of vertical inequity, it is silent about the potential error in measurement of P. It is not surprising, therefore, that Fairbanks shows that the spline model has a bias towards regressivity as this model does nothing to address the errors-in-variables problem. It is clear, then, that measurement error is still a worrying problem in vertical inequity analysis, as no method to-date has fully addressed how to purge vertical inequity regression analysis of the errors-in-variables problem. 8 We note that the Sunderman et al. (1990) model as specified in Equation (4) is labeled as a spline model by both Sunderman et al. and Fairbanks at al., though it is not strictly a spline model as defined by the econometric literature. See Greene (2012, chapter 7) for a detailed explanation of spline models. 8

9 2.c. Our Methodology To examine potential indicators of assessor bias, we will utilize a model with assessor error ε and market error μ confined to the left-hand side of our regression. We construct a new dependent variable that is the log of the ratio of assessed value to sale price, as shown in Equation (5). We choose to examine the log difference in order to examine the existence of vertical inequity in percentage terms. We estimate ln(a i /P i ) = α 0 + β 1 (PROP i ) + β 2 (X i) + γ i (5) where PROP is a vector of property observables, X is a vector of controls including the month and year in which property i was sold, as well as the town in which property i was located, β 1 and β 2 are conforming vectors of coefficients, α is the constant term, and γ is a stochastic error term. By confining the measurement error in assessment and price to the left-hand side of our regression we eliminate the attenuation bias in our parameters caused by the errors-in-variables problem. 9 Note that what is on the right hand side of Equation (5) will be zero if the assessor exactly predicts the sale price. One can think of ln(a/p) as the percentage difference between the assessment and the sale price. If we can predict any of this difference then the assessor must be making an error as assessors have access to all the same information we do and could have used it to construct their estimates. Hence, since the variance of ln(a/p) is the variance in the difference between assessment and sale price, the zero-centered R 2 of this regression may also be interpreted as the fraction of the variance in our dependent variable ln(a/p) that is due to assessor error. However, since there are many more characteristics of a property than what we observe, adding 9 For example, see Wooldridge for a detailed discussion on running OLS regressions with measurement error and the difference between measurement error being on the left vs. right-hand side of a regression. 9

10 these variables to our regression would only increase the zero centered R-squared and thus our estimate is, asymptotically, a lower bound. Now let us consider what our parameter estimates tell us about the nature of assessor errors. If assessor error and market error are mean zero, then features of a property should not be able to predict differences in ln(a/p), and no coefficients in Equation (5) should be statistically significant. Statistically significant β 1 coefficients would indicate that there are observable features of properties that are not being appropriately accounted for by assessors. For instance, larger properties might be under-assessed if the assessor under-compensates for the size of a house. It could also be, for instance, that assessors intentionally under-assess larger houses to give them a tax break compared to what they would have to pay as a result of accurate assessment. This would imply a systemic error, which is our definition of bias. We would say here that assessors are biased in favor of more expensive homes. 10 Similarly, a non-zero constant term would indicate a constant percentage bias across all properties in a town. Therefore, regressing ln(a/p) on features of a property should not yield any statistically significant coefficients if assessments are unbiased. As such, the zero-centered F-test for our entire regression is a test for the presence of bias in assessment based on our property observables. We note that we do not interpret significant coefficients as indicating systematic market error because market error, by definition, cannot be systematic. Such noise in price may stem from market conditions at the time of sale, differences in bargaining power and discount rates between parties, how eager the seller is to sell the property, and other factors. However, there is little reason to believe that larger properties would receive offers that are consistently above (or consistently 10 We note that another possible reason for positive slope coefficients would be if housing prices are rising in the time between assessment and sale, a possibility we explore later in Section 4. 10

11 below) their market value compared to smaller properties. Consider the second-order hedonic pricing model below: ln(p i ) = α 0 + β 1 (PROP i ) + β 2 (PROP i 2 ) + β 3 (X i) + γ i (6) where our variables are defined the same as in Equation (5). Since price is the expected value of a property given its characteristics, if prices systematically vary with some characteristic in our sample then this cannot be market error. It is exactly this systematic variation that assessors should be taking into account, hence any variation in ln(a/p) must be due to assessor error. 11 In addition, since all of the measurement error in price is confined to the right-hand side of Equation (6), this allows us to generate predicted prices without measurement error introduced by μ. As a result, we may then regress the ln(a/p) that we obtain from Equation (5) on the predicted log price from Equation (6) for each property to determine if there is any bias in assessment: ln(a i /P i ) = α 0 + β 1 (lnp i) + γ i (7) where P is now exogenous. Hence, any explained variance of ln(a/p) in both Equation (5) and Equation (7) must be due to assessor error, and these equations can provide us with an estimate of assessor bias stemming from observables. 11 There is still the possibility of a small part of the explained variance stemming from changes in sales prices within each year, though in a later footnote we demonstrate that this is not a concern. 11

12 The results of Equation (5) are shown Section 4.a. The regression results of equations (6) and (7) may be found in the appendix, while a graph of the results of Equation (7) is shown in Section 4.a. 3. Data and Descriptive Statistics 3.a. Parcel-level Data To estimate equations (5), (6), and (7), we use assessors parcel-level data obtained from the state of Massachusetts Office of Geographic Information ( MassGIS ). 12 For each property in Massachusetts, the data contains information on the address, geographic coordinates, assessed value, most recent sale price and date, and basic property features such as residential area, number of bedrooms, building age, and building style. The data ranges from calendar year 2008 to 2014, with different towns appearing in the data in different years. The data is available for 350 of the 351 towns in Massachusetts, with Boston being excluded. 13 From this original MassGIS dataset, we applied the following cleaning steps to obtain our final working dataset. First, we dropped towns for which data on property features were not available. Next, we kept only the properties that are single-family residences, which is the focus of this paper. Next, we kept only the properties for which a sale took place soon after the assessment. For example, among the properties appearing in the data with a 2012 fiscal year assessment, we restrict attention to only those that were sold during calendar year 2011, because a 2012 fiscal year assessment in Massachusetts means that the assessment was conducted in January of This MassGIS parcel-level dataset was downloaded from the following source and website: MassGIS Data Level 3 Assessors Parcel Mapping, Massachusetts Office of Geographic Information, available at < 13 We again note that towns in this paper refers to municipalties of any type within the state of Massachusetts. 14 In Massachusetts, by state law a property s assessment should reflect the market value of the property as of the January 1 date that immediately precedes the fiscal year of the assessment. Fiscal years in Massachusetts run from 12

13 Therefore, we focus attention on properties that were sold within a short window of time (maximum 12 months) after their date of assessment. Figure 1 illustrates an example of this timeline. Figure 1 Next, in order to isolate only the valid (i.e., arms-length) sales, we matched the MassGIS parcel-level data with a dataset from the Massachusetts Department of Revenue (DOR) that contains a list of all the valid sales that took place in Massachusetts between 2008 and This enabled us to eliminate invalid sales such as within-family transfers, sales resulting from a divorce settlement, and foreclosures. We also dropped any properties with a sale price of less than $10,000 in a further effort to prevent issues of non-valid sales affecting our results. 16 Next, we July 1 to June 30. For example, fiscal year 2012 started in July 1, 2011, and ended in June 30, Therefore, a property s assessment for fiscal year 2012 would have been done in early calendar year 2011 and would have attempted to reflect the market value of the property as of January 1, (Source: Assessment Administration: Law, Procedures and Valuation Chapter 2 Mass Appraisal, Massachusetts Division of Local Services, accessed January 18, 2016, available at: < 15 This DOR valid sales dataset was downloaded from the following source and website: LA3 Parcel Search, Massachusetts Division of Local Services Getaway, available at < 16 We also tried alternative specifications of dropping properties of less than $50,000 and $75,000, but the statistical and substantive results remained unaltered. 13

14 kept only properties that are at least two years old, in order to avoid cases of property tear-downs and reconstruction, and cases of new construction taking place on parcels previously used for a different purpose. Finally, we kept only towns with at least 20 valid sales within a year. After the application of all these filters, the final dataset contains valid sales for 25,971 single-family residences across 177 towns, ranging from calendar years 2008 through Figure 2 shows the geographic distribution of the Massachusetts towns that are included in our final cleaned sample, illustrating that we are able to capture data across much of the state, though towns are more concentrated in the eastern half of the state. This is mainly due to dropping a number of towns in western Massachusetts from our sample for having less than twenty valid sales. Figure 2 14

15 Table 1 (found on page 27) lists descriptive statistics for our cleaned sample of valid sales. 17 Given that our original (un-cleaned) dataset provides the assessment value for the stock of all single-family properties in the state, whether they experienced a valid sale or not, we are able to check how our cleaned sample of valid sales compares to the overall stock of single-family properties. The average price in our sample is greater than the average assessed value. A t-test for difference in means shows a clear disparity with a test p-value < While properties in our cleaned sample have a mean assessed value of $411,674, the mean in the overall stock is $314,781. This finding is not surprising if the owners of properties that have an assessed value greater than its perceived value are the most likely to sell. However, this just strengthens our claim to be providing a lower-bound estimate of the importance of assessor error. Since our sample is truncated compared to the population of houses in Massachusetts, and we are seeing an over-representation of properties that are accurately or over-assessed (since they have higher-than-average assessed values), this would likely bias β 1 in Equation (7) towards zero, leading us to underestimate the systemic assessor bias. This is not a strict arithmetic lower bound in the same manner as is the zero-centered R 2 from Equation (5); however, this indicates that we may safely interpret our findings in Equation (7) as conservative estimates of overall assessor bias. 3.b. Other Data We also use census block and block-group level data from the 2010 Census in order to obtain neighborhood characteristics for each single-family residence in our sample. Specifically, each 17 Since it appeared odd that some single-family residences would be comprised of a single room, we also ran alternative specifications omitting properties with one reported room. However, this again did not change the statistical or substantive results of our analysis. 15

16 single-family residence from our cleaned sample was matched to its census block and/or blockgroup to obtain socio-economic data for its surrounding area. We first attempted to match each residence to demographic data at the block level. When data were not available at the block level, we matched the residence to socio-economic data from its respective block-group. At the block level we have data on population count, percentage of the population that is black, median age of population, and the percent of housing units that are vacant. At the block-group level we have data on median income. In addition to the parcel-level data described above, MassGIS has additional datasets containing the current location (as of 2014) of various public goods across the state of Massachusetts. The data contains the location of police stations, fire stations, hospitals, libraries, schools, town halls, and train stations. We then calculated the distance of each single-family residence in our cleaned dataset to its closest police station, its closest train station, and so on. However, most of these variables were not statistically significant in Equation (5), with only two being significant at the α =.1 level and our R2 remaining unchanged when they were added to the model. Hence, they were omitted from the final analysis. 16

17 Table 1 Descriptive Statistics Single-Family Residences Mean Std. Dev. Skewness Min 25th Percentile 50th Percentile 75th Percentile Max Assessed Value (A) $411,674 $309, $43,900 $242,400 $330,200 $474,500 $7,429,500 Sale Price(P) $435,352 $335, $45,000 $249,000 $347,900 $509,500 $6,800,000 Log (A/P) House Age (in years) Lot Size (in sq. ft.) 29, , ,712 15, , ,090,880 Residential Area (in sq. ft.) 1, ,867 Number of Rooms Observations Source: MassGIS Data - Level 3 Assessors' Parcel Mapping 1

18 4. Econometric Results 4.a. Regressivity in Massachusettes towns Table 2 presents the results a first and second-order version of Equation (5), regressing ln(a/p) on property observables. As seen in the table, several factors explain the variance in the difference between assessment and sale price, indicating that there is bias in assessment. In Model (1), both lot size and the number of rooms can predict changes in ln(a/p), as well as the black share of the population of a census block. However, a higher-order model might be more appropriate for capturing differences in mis-assessment. In Model (2), we can see that several variables have nonlinear impacts on the log difference in assessment. Interestingly, both the overall size of the property (lot size) and the size of the livable area (residential area) are both statistically significant, indicating that these are appropriately evaluated separately by assessors, though still in a biased manner. Also, house age and residential area both have a U-shaped influence on ln(a/p). An F-test confirms that the second-order terms introduced in Model (2) are jointly statistically significant. 18 Nonetheless, we are cautious in our interpretation of these variables without consulting our hedonic price estimates (see Appendix, Section A for a table of those results). 18 The cluster-robust F-stat generated using the centered R 2 is 7.23 with a p-value of <

19 Depedent Variable: Log(A/P) (1) (2) Coefficient Std. Err. Coefficient St. Err. House Age ( ) *** ( ) Lot Size (in 1000s, sq. ft.) * ( ) *** ( ) Residential Area (in 1000s, sq. feet) ( ) *** ( ) Number of Rooms *** ( ) ( ) Med hhd inc in Block Group (in 1000s) ( ) ** ( ) Population of Block (in 1000s) ( ) ( ) Median age of Block Population ( ) ( ) Black Share of Block Population *** ( ) *** ( ) Share of Houses Vacant in Block ( ) ** ( ) [House Age] *** ( ) [Lot Size (in 1000s, sq. ft.)] ** (4.60e-08) [Residential Area (in 1000s, sq. feet)] *** ( ) [Number of Rooms] ( ) [Med hhd inc in Block Group (in 1000s)] *** ( ) [Population of Block (in 1000s)] ( ) [Median age of Block Population] ( ) [Black Share of Block Population] ( ) [Share of Houses Vacant in Block] *** ( ) Constant (0.0154) (0.0236) Observations Zero-centered R-squared Zero-centered F-stat ⱡ Source: MassGIS Data Level 3 Assessors Parcel Mapping Census Data from 2010 Census Standard errors clustered by town Month, year, and town dummies included * p<0.10 **p<0.05 ***p<0.01, two-tailed test Table 2 ⱡ Calculated using robust standard errors due to cluster restrictions 19

20 The results of our hedonic pricing model show that house age has a non-linear, U-shaped influence on the price. This means that Model (2) could be capturing the diminishing influence of increased house age on the price of the property, and so focusing on second-order turning points in this case could be potentially mis-leading. However, residential area has a positive first derivative and positive second derivative based on our hedonic pricing model, so in this case we are more certain that a second-order term is appropriate to capture the impact of residential area on mis-assessment. Specifically, a property with a residential area of greater than 3,355 square feet (much larger than sample average of 1,983 square feet) is predicted to have a positive relationship with ln(a/p). So, while it is clear from an F-test that mis-assessment and property observables have a non-linear relationship, it is difficult to definitively say that a second-order model best captures the variance of mis-assessment (as opposed to a higher-order model). As for neighborhood characteristics, the median income of the block, as well as the share of the black population and share of houses that are vacant are all statistically significant. The signs of the first and second derivative of median income seem to indicate that higher-income residents could be enjoying more of a discount on their property tax bill compared to lower-income residents, although those living in the highest income areas might be getting less of a discount. Again, however, we are cautious about interpreting a specific turn around point based on these findings below considering that median income also has a negative first derivate and a positive second derivative in the results of our hedonic pricing model. We further explore the effect of area income levels later in the paper. We also see from our results that there could be some level of racial discrimination if areas with more black residents or houses owned by black residents are being over-assessed more often than those of white residents, controlling for median income. Harris (2004) had similar findings when he examined home sales in twenty-one neighborhoods in 20

21 New Haven, Connecticut, from 2000 to 2001 and found that minority-majority neighborhoods were over-assessed on average compared to other neighborhoods. Lastly, as mentioned previously, the zero-centered R 2 of our regression is an asymptotic lower bound on the fraction of the variance in the difference between assessment and sale price, or ln(a/p), that is due to assessor error. As seen in Table 2, the zero-centered R 2 for Model (2) is.233, so approximately 23.3 percent of the variance in ln(a/p) can be explained by our model, and hence that is the asymptotic lower-bound on the variance in this difference due to assessor error An overall F-test (also zero-centered) confirms with a p-value of <0.001 that our model is statistically significant, indicating that we can indeed predict error in assessment based on our observables, and that assessors exhibit some degree of bias in assessment. We also find that the town dummies are statistically significant based on an F-test with a p-value of < Hence, at least some of the variation in mis-assessment can be explained by town-specific fixed effects. Interestingly, the constant term in both models is not statistically different from zero, indicating that there is no evidence of systematic, constant assessor error across towns. It seems clear, then, that basic features of a property such as size and age play an important role in mis-assessment. Older and larger houses are much more likely to be under-assessed than 19 For reference, the centered R 2 for Model (1) is.108, and for Model (2) is We are also interested in the possibility that changes in housing prices could affect our measure of regressivity. Since our sample consists of houses that were assessed in January of a given year, and then sold at point during that corresponding calendar year, it is possible that house prices could change during that period. According to data at the Federal Housing Finance Agency ( housing prices in Massachusetts rose on average about one percent within a given calendar year. To examine if this had an impact on our estimation, we ran Model (2) from Table 2 while using the month variable as a linear time trend, and found that it was not statistically significant even at the α = 0.1 level. We also ran Model (2) using quarter dummies instead of town dummies using quarter 1 as the base. If housing prices were rising during the year, then we would expect the coefficient on later quarters to be statistically significant and negative, since houses sold later in the year would appear in the data to have been mis-assessed in a more regressive manner as prices increase. However, the coefficient on quarter 4 was not statistically significant at the α = 0.05 level, which to us is evidence that changing housing prices do not have a significant impact on our regressivity estimates. Together, we believe these robustness checks demonstrate that the results of our slope coefficients are driven by bias in assessment, and not housing price changes occurring within the calendar year. 21

22 newer, smaller houses, and we have evidence that these effects are non-linear. Again, if variation in the difference between assessment and sale price were random, then our model would not be able to predict any of the variation in the gap. Since it does, however, it indicates that there is indeed a systematic bias in the gap stemming mainly from basic features of the property. From an assessment perspective, these may be easier biases to correct for when assessing a property compared to neighborhood characters since property features are more easily ascertained by a quick visual scan of the property. While the above estimation would highlight potential indicators of assessor bias, we will also measure vertical inequity more directly by regressing ln(a/p) on the predicted price of each property, as outlined in Equation (7). As explained earlier, graphing ln(a/p) against the sale price of a property could yield a false impression as to the existence of regressivity given that price is a noisy measure of value, since it is essentially regressing assessed value on price. A better alternative is to approximate the true value of the property, V, using the property features from Equation (5) in a hedonic pricing model shown in Equation (6). In doing so, we confine market error μ from Equation (2) to the left-hand side of the regression, allowing us to generate predicted prices without measurement error introduced by μ. Figure 4 displays the results from Equation (7) using a local polynomial smoothing function. The actual regression results may be found in Section B of the appendix. As seen in Figure 4, there is clear evidence of increasing regressivity across most of the price spectrum. There is also evidence that regressivity begins to decrease for the highest-value properties, creating the non-linear shape of the regressivity, and corroborating the statistically significant second-order terms from the results in Table 2. In addition, Figure 4 again shows us 22

23 that there is no evidence of constant assessor error, indicating that assessor error is due more to property observables than some systematic amount of bias. Figure 3 Single Family Residences in Massachusetts Calendar Years Predicted Log of Sale Price Predicted Log(Price) Local Poly. Smoothing Epanechnikov kernel function bandwidth = 0.1 To reiterate, this estimate of the bias in assessment can be interpreted as a conservative estimate of regressivity. Because of the greater mean assessed value in our sample versus the population of single-family residences in Massachusetts, our truncated sampled would be biased towards zero, meaning that regressivity in assessment across the state could potentially be larger. We also want to stress that a graph such as this using predicted mis-assessment and predicted price is a much better, unbiased way to visually assess the existence of vertical inequity in an area due the 23

24 exogeneity of P. A graph that simply uses the difference in assessment and sale price or the log of (A/P) will be predisposed to showing a regressive picture because of the errors-in-variables problem. 4.b. Regressivity within-town Each town in Massachusetts has its own board of assessors, which is responsible for assessing all properties within the town. Assessing boards are typically comprised of three assessors plus assisting staff. Each board member is required to take a course on adequate assessing practices and then pass a certification exam within two years of taking office. 21 Both the course and the exam are administered by the state s DOR. Moreover, the DOR also monitors towns assessments on a regular basis in order to ensure assessment equity. 22 Despite all this state supervision and guidance, and given that town dummies were statically significant in Model (2) of Table 2, one might still expect to observe some heterogeneity in the amount of regressivity across towns. For example, it could be the case that some towns experience regressivity in assessment and others experience progressivity, with a net result of regressivity when pooling the data. To investigate this possibility, we regress ln(a/p) on property features for each property i in town j, as outlined in Equation (8) below. ln(a ij /P ij ) = α j + β 1j (PROP ij ) + β 2j (PROP ij ) 2 + ε ij (8) 21 Source: Assessment Administration: Law, Procedures and Valuation Introduction, Massachusetts Division of Local Services, accessed January 18, 2016, available at: < 22 Source: Assessment Administration: Law, Procedures and Valuation Chapter 1 Assessing Administration, Massachusetts Division of Local Services, accessed January 18, 2016, available at: < 24

25 We then generate a predicted ln(a/p) for each property in our sample based on the estimates obtained above. Next, we break up our data into 16 cells based on predicted price from Equation (6), and town median income. We match each predicted ln(a/p) from Equation (8) with one of these cells, and compute the mean value for each cell. To improve readability, we convert these 16 cells into percentages and change the sign, so that a positive number indicates underassessment/regressivity. The results are shown in Figure 5 below. Figure 4 The results above tell an interesting story about vertical inequity when comparing towns across Massachusetts. According to the income-axis, single-family properties in higher-income towns in 25

26 Massachusetts are more likely to be under-assessed, and that most residents enjoy a discount on their property taxes just by virtue of living in that town. However, there seems to be an exception for the lowest-value houses in located in the highest-income towns, which are very close to being accurately assessed. On the other hand, the price-axis of Figure 5 indicates that higher-value properties in general are much more likely to be under-assessed no matter where they are located, and that the difference in under-assessment seems to be greater when comparing the highest and lowest-value houses than when comparing the highest and lowest-income towns. Hence, it appears while the income level of a town does affect mis-assessment, the value of the property still plays a significant role, and the owners of the highest-value properties enjoy the largest discount on their property taxes regardless of location. Conversely, those with the lowestvalue houses always get the smallest break, with owners of low-value houses in high-income towns actually receiving no discount on their property taxes. These results, coupled with the evidence from our regressions in the previous section, are a clear indication that: 1) assessor error ε from Equation (1) is not random and there is indeed bias in assessment across towns; and 2) systematic assessor bias stems from town-level observables, as well as property-level observables. 5. Conclusion Past studies concerning property taxes and mis-assessment have so far been unable to adequately deal with the errors-in-variables problem present in housing data stemming from measurement error in both the assessed value and market price of a house. In particular, we argue that the Clapp (1990) method that many studies have relied on is a flawed measure of vertical inequity since his instrument keeps measurement error on the right-hand side of the regression and is most likely not an exogenous measure of assessor error. We propose several new methods that 26

27 effectively deal with this problem as well as demonstrate how they may be used to estimate minimum values for assessor error and bias. By regressing the log difference between assessed value and sale price on property features, the slope coefficients provide estimates of assessor bias based on these observables. We find that there is bias in assessment based on several observables such as house age, residential area, and median income of the census block-group, and that there is a nonlinear relationship between these observables and mis-assessment. We also argue that the zero-centered R 2 is an asymptotic lowerbound on how much of the variance in mis-assessment is due to assessor error (and not withinyear variance in price), which we estimate to be approximately 23.3 percent. We also use a hedonic price estimates to confirm that there is bias in assessment across most of the housing price spectrum in Massachusetts, and show that this bias is indeed regressive. Lastly we show that there is bias in assessment based on town-observables as well, and highlight that properties in higher-income towns being more likely to be under-assessed than those in lower-income towns. However, across the income spectrum, higher-value properties are still more likely to under-assessed. So, no matter where they live, higher-income residents are much more likely to get a property tax break compared to the rest of the population. Given the availability of property data on state websites and commercial websites such as Zillow, it should be reasonable for assessment boards and organizations to obtain the property features data necessary for the estimation we propose. States themselves can also use these methods to not only determine if vertical inequity is present in an area, but also determine which aspects of the house and neighborhood are the most likely causes of assessor bias as well as the degree to which assessor error is responsible for any mis-assessment. We believe that the framework described in this paper will allow future researchers to better measure the impact of 27

28 assessor error and bias on mis-assessment, which will help assessors be more accurate and ensure that residents are not over-paying or under-paying on their annual property tax bill. References Birch, J., and M. Sunderman, Regression Modeling for Vertical and Horizontal Property Tax Inequity, Journal of Housing Research, 23(1). Clapp, J., A New Test for Equitable Real Estate Tax Assessment, Journal of Real Estate Finance and Economics, 3, Edelstein, R., An Appraisal of Residential Property Tax Regressivity, Journal of Financial and Quantitative Analysis, 14(3). Fairbanks, J., Goebel, P., Morris, M., and W. Dare, A Monte Carlo Exploration of the Vertical Property Tax Inequity Models, Journal of Real Estate Literature. Greene, W., Econometric Analysis, Seventh Edition, Pearson. Goolsby, W., Assessment Error in the Valuation of Owner-Occupied Housing, Journal of Real Estate Research. Harris, L., Assessing Discriminaiton: The Influence of Race in Residential Property Tax Assessments, Journal of Land Use, 20(1). International Association of Assessing Officers (IAAO), Standard on Mass Appraisal of Real Property, January 2012, p. 17. Johnson, M., Assessor Behavior in the Presence of Regulatory Constraints, Southern Economic Journal, 55(4): Kochin, L., and R. Parks, Testing for Assessment Uniformity: A Reappraisal, Property Tax Journal, 3: Paglin, M., and M. Fogarty, Equity and the Property Tax: A New Conceptual Performance Focus, National Tax Journal, 25(4): Ross, J. M., Assessor Incentives and Property Assessment, Southern Economic Journal, 77(3):