(a) An investigator looks at the drop in R and concludes that the first equation is better. Is this conclusion valid?

(b) What would the equation to be estimated be if var(M,) = crx, instead of crxp How would you determine which of these alternative hypotheses is the better one?

(c) Comment on the computation of R from the transformed equation and the R from the equation in terms of the original variables.

9. In discussion of real estate assessment, it is often argued that the higher-priced houses get assessed at a lower proportion of value than the lower-priced houses. To determine whether such inequity exists, the following equations are estimated:

1. A, = a -I- (35, -I- M,

2. AJS, = 7 + 65,-1- m;

3. log A, = e + X log 5, +

where A, is the assessment for property / and 5, is the observed selling price. In a sample of 416 houses in King County, Washington during the period 1977-1979, the following results were obtained:

1. A, = 7505.40 + 0.33825, R = 0.597

(559.2) (0.0136) standard errors

(13.42) (24.79) -ralios

2. AjSi = 0.7374 - 4.5714 x 10-*5, R = 0.2917

(0 0144) (3 5005 X 10 ) standard errors

51 38 -13 06 -ratos

3. log A, = 2.8312 + 0.6722 log 5, R = 0.6547

(0.2513) (0 0240)

It has been suggested that it is more appropriate to answer this question by estimating a reverse regression:

residuals against \lx showed that there was no such relationship. Hence the assumption made was

var(M,) = ax]

The estimated equation was

- = 0.121 + 3.803(l/x) = 0.03

X (0 009) (4 570)

In terms of the original variables, we have

= 3.803 + 0.121 : The estimation gave the following results:

Appendix to Chapter 5

Generalized Least Squares

In Chapter 4 we assumed that the errors were independent with a common variance or E{uu) = lu. This assumption is relaxed in Chapters 5 and 6. We start with the model

= Zp -I- M E{uu) =

where is an arbitrary positive definite matrix. Premultiplying by 17""2, we get

y* = Z*p + w*

where

u* = n-«M

Then E{u*u*) = n-"2£(Mw) -" = /. Hence by the result in the Appendix to Chapter 4, we get the BLUE of p as

Pols = (X*Z*)-(X*y*) = {Xn~X)"Xa~y

4. 5, = -Yo + + -n,

5. A./S. = Po + P,A, + Tl,

The estimation now gave the results:

6. 5, = 2050.07 + 1.7669 , = 0.597

(1527 93) (0 0713) standard errors

(1 3417) (24 79) -ralos

7. AJS, = 0.5556 + 3.8288 x , = 0.0004

(0 0203) (9 506 X 10 ) standard errors

(27 26) (0 403) -ra"os

The average value of AiS was 5.6439.

(a) Interpret what the coefficients in each of these equations mean to answer the question of whether there is inequity in real estate assessment.

(b) Given some arguments as to why equations (4) and (5) may be more appropriate to look at than equations (1) to (3).

(c) Also, explain why equations (2) and (5) may be more appropriate than equations (1) and (4) respectively.

(The results are from L. A. Kochin and R. W. Parks, "Testing for Assessment Uniformity: A Reappraisal," Property Tax Journal, March 1984, pp. 27-54.)

We use the subscript GLS to denote "generalized least squares." The variance of this estimator is given by

V(0GLs) = iX*X*r = iXn-X)-

By contrast, the oruinary least squares estimator is given by

0OLS = (XX)-Xy = + iXX)-Xu

Since E{u) = 0 this estimator is still unbiased. But its variance now is given by

V(0oLs) = { )-* { ) ( = iXX)XilX(XX)

This is the general expression corresponding to equation (5.2) in Section 5.3. In this chapter we considered the case where is a diagonal matrix with the fth diagonal element cr;. It is important to note thatPoLs is not necessarily inefficient la. Rao* showed that a necessary and sufficient condition that the OLS and GLS methods are equivalent is that fi be of the form

fi = XCX + ZDZ + la

where Z is a matrix such that XZ = 0 and and D are arbitrary nonnegative definite matrices. As an example consider the following model:

y, = a + hx, + u, i = 1, 2, . . . , /J var(m,) = 1 cov(m,m,) = p for / 5 j

Thus the errors u, are not independent. They are equicorrelated. In matrix form we can write this model as

where

= (1 - p)/ + pee

where e is an x 1 vector with all Is (first column of X). Thus

fi = XCX + lo

with CT = (1 - p) and =

P 0 0 0

. Hence, in this model, even if the errors

are correlated and fi 5 Ict, the OLS and GLS estimators are identical.

*C. R. Rao, "Least Squares Theory Using an Estimated Dispersion Matrix," Proceedings of the Fifth Berkeley Symposium (Berkeley. Calif.: University of California Press, 1967), Vol. 1, pp. 355-372.