ON AN UNFAIR APPRAISAL OF VERTICAL EQUITY IN REAL ESTATE ASSESSMENT

PETER KENNEDY'

In a recent article in this journal Kochin and Parks (1982), hereafter KP, circum- vent the errors-in-variables problem plaguing econometric evaluation of equity in real estate assessments by exploiting an assumption that assessors' estimates are effi- cient. If correct, this methodology would provide a solution to the measurement errors problem for all cases in which said errors arise from an efficient prediction process, and would therefore be an extremely useful addition to our econometric tool kit. Unfortunately, the KP reasoning involves a crucial additional assumption, not made explicit, which renders their methodology and empirical results unconvinc- ing. The purpose of this note is to clarify the real nature of the difference between the KP paper and the existing literature on the evaluation of equity in real estate assessment.

The traditional approach to econometric evaluation of equity in real estate assessment proceeds as follows. The selling price S, for the iih property of unobserved true value t: is given by

where u, is a random error with zero mean. The sign and magnitude of u, depends on the peculiarities of the sale - the particular buyer and seller and the details of their negotiation, for example. The assessment A, for the ith property of value v is given by

where E , is a random error with zero mean (The minus sign is to retain conformity with K P s notation; the parameter y reflects the fact that often assessed values are intended to reflect some constant fraction of true value.) This error has two compo- nents. The first component, E , , , results from information shortages, as described by KP. The second component, E ~ , , results from subjective peculiarities of the assess- ment - the abilities and personal biases of the particular assessor assigned to make the assessment, the care taken by the assessor, the assessor's mood at the time of the assessment, and impressions created by the weather when the assessment was made, for example. This component is ignored by KP but must nonetheless be included to make the KP analysis comparable to the literature on assessment which they criti- cize. As noted below, it is KP's implicit assumption that e2, does not exist that gener- ates their results.

Combining equations (1) and (2) to eliminate the unobservable V produces the estimating equations

'Simon Fraser University. My thanks, without implication, to Pa0 Cheng, John Herzog, Levis Kochin, Charles Nelson, and Richard Parks for useful comments.

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(3) S, = yA, + (u, + e,) and

to use to test for equity in real estate assessment (equity implies a zero intercept in both equations). Neither A, nor S, is uncorrelated with the composite error term, however, biasing, even asymptotically, regression estimates from both equation (3) and equation (4).

KP attempt to resolve this dilemma by assuming that A, is an efficient predictor of y (and thus of S,). By KP's definition, an assessment A, is efficient.if given the infor- mation available there is no way to improve on A, as a predictor of S, . In particular, a set of assessments is defined to be efficient if the prediction error has zero mean across properties and is uncorrelated across properties with A,. This concept of effi- ciency was borrowed by K P from the context of commodity markets.

then A, and e, must be uncorrelated across the population of properties. They then reason that because A, and e, are uncorrelated, least squares estimation of equation (3) will not suffer from bias, and suggest use of this equation to test for equity in real estate assessment. Unfortunately this argument is fallacious unless an additional, crucial assumption is made, as will now be explained.

In borrowing their concept of efficiency from the commodity markets literature, K P overlook a major difference between that literature and the assessment literature, rendering their analogy incomplete. Consider the KP example of using forward prices F:+ as predictors of future spot prices S, + . They specify that

KP rewrite (2) as V = yA, + E , and note that if A, is an efficient predictor of

S,+, = F/+' + 0,.

F: + is efficient if 0, and F:+'are uncorrelated over time and 0, has expectation zero over time. In KP's analogy an assessment is interpreted as a prediction of a property's selling price, which for y = 1 can be written as

S, = A, + U, + e, .

If A, is an efficient predictor of S, , the A, and the composite error are uncorrelated across houses.

It is important to note that this does not imply that A, and the composite error are uncorrelated in repeated samples (Le., drawing repeated error terms for a given property). In the commodity market example F:"and 0, are uncorrelated in repeated samples because as repeated 0s are drawn, F:+ Iremains unchanged. In the assessment context, however, this is not the case. As repeated u, and e, are drawn A, changes because e, is a component of A , .

To examine this further, consider equation (3). Suppose the raw assessment Ai is correlated with the error term (u, + e,) across properties. KP note that therefore A, is not efficient and can be improved. As an example, they suggest in footnote 15 that by regressing S, on A, to obtain intercept and slope estimates a and b, an improved, efficient assessment A, = a + bA, can be obtained. This produces a new relationship

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(3‘) s, = A, + T i ,

where 7, is the prediction error associated with the improved assessment A,. Because A, is an efficient assessment it is uncorrelated across properties with Z,; this can be verified by showing that E x , 7, = 0 when the expectation is taken across properties. But when the expectation is taken across repeated samples ( i .e. , drawing new errors for the same property) E X , T, = -b2u2, where o2 is the variance of e2,, so that A, and 7, are not uncorrelated in repeated samples. In the presence of subjective assessment errors, it is not possible to construct an efficient assessment that is also uncorrelated with the assessment error in repeated samples.

Since it is lack of correlation in repeated samples that is required for application of the textbook results of unbiasedness in regression, the KP argument fails unless it is also assumed that there is no subjective assessment error ( i .e . , that the assessment procedure is sufficiently objective as to always produce the same assessment for a given property). Making this additional assumption generates the KP results, but at the cost of assuming away the problem as seen by the traditional literature dealing with the evaluation of equity in real estate assessment.

That this literature views e, as varying in repeated samples is readily seen by examining the papers cited by KI! Denne (1977, p. 13-16), International Association of Assessment Officers (1978, p. 152-153), and Reinmuth (1977, p. 51-52) all charac- terize the regression of A, on S, as being cursed by the errors-in-variables problem. From the context of their discussions, it is clear that they are conceptualizing the assessment error as the traditional textbook error associated with the linear regres- sion model, and not, as KP implicitly assume, an error fixed in repeated samples. Cheng’s specification of the assessment error (1976, p. 1252) clearly implies that it is to be interpreted as varying in repeated samples.

Even if the existing literature were ignored, it is unreasonable to assume, as KP implicitly do, that the assessment process is sufficiently objective as to produce in repeated samples the same assessment for a given property. Those familiar with how assessments are made readily admit, even stress, that there is an inescapable subjec- tive element to assessments. One cannot read the International Association of Assess- ment Officers (1978) reference manual on assessment procedures without recogniz- ing that evaluation of such things as quality of construction, traffic density, noise level and quality of view force assessments to be subjective.

Without their additional, implicit assumption, KP’s test is as unfair as the tests they criticize; it is only able to circumvent the errors-in-variables problem by assum- ing it does not exist. The key ingredient in the KP procedure is their suppression of subjective assessment error and not, as they claim, their postulation of “efficient” assessments.

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REFERENCES

Cheng, Pa0 L., “Bias and Error Detection in Property Tax Administration,” Munugement Science, 1976,

Denne, Robert C., “Introduction,” in Analyzing Assessment Equity, International Association of Assess- 22,1251-1257.

ing Officers, 1977, 1-22. International Association of Assessing Officers, Improving Red Property Assessment: A Reference Mun-

uul, 1978. Kochin, Levis A., and Parks, Richard W., “Vertical Equity in Real Estate Assessment: A Fair Appraisal,’’

Reinmuth, James E., “The Measurement of Vertical Inequities in Assessment Practice,” in Anulyzing Economic Inquiry, 1982.

Assessment Equity, International Association of Assessing Officers, 1977,47-60.