Kenneth T. Rosen, "The Impact of Proposition 13 on Housing Prices in Northern California: \ Test of the Interjurisdictional Capitalization Hypothesis," Journal of Political Economy, February 1982, pp. 191-200.

property taxes, which led to an increase in housing prices. Rosen studied the impact of reduction in property taxes on housing prices in the San Francisco Bay area. Besides property taxes, there are other factors that determine housing prices and these have to be taken into account in the study. Rosen therefore included the characteristics of the house, such as square footage, age of the house, and a quality index. He also included economic factors such as median income and transportation time to San Francisco. The estimated equation was (figures in parentheses are t-values, not standard errors).

= 0.171 + 7.275 :, + 0.5472 + 0.000733 + 0.06384

(2 97) (2 32) (1 34) (3 26)

- 0.0043x5 + 0.857 = 0.897 n = 64

where

= change in post-Proposition 13 mean house prices X, = post-Proposition 13 decrease in property tax bill on mean house Xj = mean square footage of house = median income of families in the area X4 = mean age of house X5 = transportation time to San Francisco

Xfc = housing-quality index as computed by real estate appraisers

All the coefficients have the expected signs.

The coefficient of x, indicates that each $1 decrease in property taxes increases property values of $7. The question is whether this is about the right magnitude. Assuming that the property tax reduction is expected to be at the same level in the future years, the present value of a $1 return per year is l/r, where r is the rate of interest (also expected to remain the same). This is equal to $7 if /• = 14.29%. The interest rates at that time were around this level and thus Rosen concludes: "The capitalization rate implied by this equation is about 7 which is precisely the magnitude that one would expect with an interest rate of 12-15%."

Example 3

In Table 4.8 data are presented on gasoline consumption in the United States for the years 1947-1969. Let us define

G = 0 - per capita consumption of gasoline in gallons

We will regress G on and y. (The variables are defined in Table 4.8.) The results we get are

G = -117.70 - 0.373P + 0.156F R = 0.953

(-1 14) (0 44) (12 02)

(Figures in parentiieses are /-ratios.) Tiie price F, iias the wrong sign ahhough it is not significant. Only per capita income appears to be the major determinant of demand for gasoline. In log form the equation is

log G = -8.72 + 0.535 log F, + 1.541 log Y R- = 0.940

(-3.00) (1 23) (11 21)

Again Pg has the wrong sign. Instead of population, suppose that we use labor force as divisor and define G as G = KC/ML. This can be justified on the argument that miles driven would be more related to L than to population. The results we get are

G = -248.44 + 0.258F, + 0.406F R = 0.925

(-0 70) (0.09) (9.15)

The variable P still has the wrong sign. In log form the results are

log G = -8.53 + 0.541 log P + 1.636 log F R = 0.907

(-2 18) (0 92) (8.82)

Gilstein and Leamer estimate the equation using the 1947-1960 data and obtain 6 = 799.1 - 2.563P, + 0.0616F

(77.5) (0.706) (0 0015)

(Figures in parentheses are standard errors.) Also, we get, using the 1947-1960 data,

G = -439.4 - 2.241Fg + 0.662F R = 0.979

(-2.92) (-1.64) (22.25)

(Figures in parentheses are /-ratios.) In log form we get

log G = -10.477 - 0.387 log P + 2.472 log Y R = 0.974

(-5 67) (-1.16) (20.04)

In summary, the period 1947-1960 shows a price elasticity of demand slightly negative and an income elasticity of demand significantly greater than unity. Estimation of the same equation over 1947-1969 shows no responsiveness to price but an income elasticity of demand again significantly greater than 1.

Further analysis using these data is left as an exercise. Students can experiment with this data set after going through this and the next two chapters.

Example 4

In Table 4.9 data are presented on the demand for food in the United States. Let us try to estimate the price elasticity and income elasticity of demand for food. A regression of on P and Fgave the following results:

Qc = 92.05 - 0.1421Fo + 0.236F R = 0.7813

(15.76) (-2 13) (7 56)

(Figures in parentheses /-ratios.) The coefficients have the right signs. However, someone comes and says that the variables all have trends and we ought

C. Z. Gilstein and E. E. Leamer, "Robust Sets of Regression Estimates," Econometrica, Vol 51, No. 2, March 1983, p. 330.

to account for this by including T as an extra explanatory variable. The results now are (all variables have the right signs).

= 105.1 - 0.3250 + 0.315 - 0.2247 7? = o.8812

(18 47) (-4 58) (9 85) (3 67)

Now the price variable is more significant than before. Also, a simple regression of Qo on gives a wrong sign for P:

& = 89.97 + 0.107,5 = 0.044

(7 89) (0 91)

We have given four examples, the first two on cross-section data, the other two on time-series data. The results were more straightforward with the cross-section data sets. With the time-series data sets, we have had some trouble obtaining meaningful results. This is a common experience. The problem is that most of the series move together with time.

4.4 Interpretation of the Regression Coefficients

In simple regression we are concerned with measuring the effect of the explanatory variable on the explained variable. Since the regression equation can be written as

- = {x - x) + , this effect is measured by p, where

5„ V{x)

In the multiple regression equation with two explanatory variables x, and we can talk of the joint effect of x, and Xj and the partial effect of x, or Xj upon y. Since the regression equation can be written as

- = p,(x, - X,) + - x,) + ,

the partial effect of x, is measured by p, and the partial effect of Xj is measured by P2. By partial effect we mean holding the other variable constant or after eliminating the effect of the other variable. Thus P, is to be interpreted as measuring the effect of x, on after eliminating the effect of Xj on x,. Similarly, P2 is to be interpreted as measuring the effect of X2 on after eUminating the effect

of X, on Xj.

This interpretation suggests that we can derive the estimator P, of P, by estimating two separate simple regressions:

Step 1. Estimate a regression of x, on X2. Let the regression coefficient be denoted by b- Denoting the residuals from this equation by W„ we have

W, = x„ - X, - bniXb - X2) (4.11)