Thus the 95% confidence interval for is

/20(0.07) 20(0.07) I 34.2 9.59

or (0.041,0.146)

(c) The 5% point for the f-distribution with d.f. 2 and 20 is 3.49. Hence using result 6, we can write the 95% confidence region for (3, and P2 as

[5u(P. - P,) + 2S,2(P, - P,)(P2 - P2) + «22(02 - P2« 3.49(2d)

[12(0.7 - (3,) + 16(0.7 - (3,)(0.2 - 2) - 12(0.2 - ( 2)] 3.49(2)(0.07)

Or, dividing by 12 throughout and changing 0, - (3, to (3, - 0, and 02 - 2 to 2 - 02, we get

((3, - 0.7)2 + o.7)(32 - 0.2) + ((3, - 0.2)2 < o.041

This is an ellipse with center at (0.7, 0.2). It is plotted in Figure 4.1. There are two important things to note: first that the ellipse will be slanting to the left if cov(0,, 02) < 0 as in our case, and it will be slanting to the right if cov(0,, 02) > 0. In the case of the two explanatory variables this will depend on the sign of 5,2. The second point to note is that the limits for (3, and ( 2 that we obtain from the ellipse will be different from the 95% confidence limits we obtained earlier for (3, and ( , separately. The limits are (-0.07, 0.47) for ( 2 and (0.43, 0.97) for This is because what we have here are joint confidence

Figure 4.1. Confidence ellipse for 1 and P2 in multiple regression.

limits. If p, and p2 are independent, the confidence coefficient for the joint interval will simply be the product of the confidence coefficients for p, and p, separately. Otherwise, it will be different.

This distinction between separate intervals and joint intervals, and between separate tests and joint tests should be kept in mind when we talk of statistical inference in the multiple regression model. We discuss this in detail in Section 4.8.

(d) To test the hypothesis p, = 1, P2 = 0 at the 5% confidence level we can check whether the point is in the 95% confidence ellipse. But since we do not plot the confidence ellipse all the time, we will apply the F-test discussed in result 6 earlier. We have

F = [5„(P, - Pi) + 25,2(P, - p,)(P2 - P2) + 522(02 - 2

has an F-distribution with d.f. 2 and n - 3. The observed value Fo is

Fo = [12(0.7 - 1.0)2 + 2(8)(0.7 - 1.0)(0.2 - 0) + 12(0.2 - 0)] = 4.3

The 5% point from the F-tables with 2 and 20 d.f. is 3.49. Since Fo > 3.49 we reject the hypothesis at the 5% significance level.

Formulas for the General Case of Explanatory Variables

We have given explicit expressions for the case of two explanatory variables so as to highlight the differences between simple and multiple regressions. The expressions for the general case can be written down neatly in matrix notation, and are given in the appendix to this chapter. If we have the multiple regression equation with regressors, that is,

y, = a - p,x„ - P2X2, + • • • - , +

then we have to make the following changesin the earlier results: In result 2 earlier, in the expressions for V(P,), V(P2), and so on, the denominator now will be the residual sum of squares from a regression of that variable on all the other xs. Thus

var(p,) = for J = 1, 2, . . . ,

where RSS, is a residual sum of squares from a regression of x, on all the other ik - Dxs.

These regressions are called auxiliary regressions. In practice we do not estimate so many regressions. We have given the formula only to show the relationship between simple and multiple regression. This relationship is discussed in the next section.

k&

has an F-distribution with d.f. and {n - - 1). This result is derived in the appendix to this chapter.

Some Illustrative Examples

We will now present a few examples using the multiple regression methods. To facilitate the estimation of alternative models, we are presenting the data sets at the end of the chapter (Tables 4.7 to 4.14).

Example 1

In Table 4.7 data are given on sales prices of rural land near Sarasota, Florida. The variables are listed at the end of the table. Since land values appreciated steadily over time, the variable MO (month in which the parcel was sold) is considered an important explanatory variable. Prices are also expected to be higher for those lots that are wooded and those lots that are closer to transportation facilities (airport, highways, etc.). Finally, the price increases less than proportionately with acreage, and hence price per acre falls with increases in acreage. The estimated equation was (figures in parentheses are standard errors)

logP, = 9.213 + . - 033DA - 0.0057£)75 - 0.203 log A

(0 2W) (0 099) ( ) (0 0142) (0.0345)

+ 0.014 = 0.559

(0.0039)

All the coefficients have the expected signs although some of them are not significant (not significantly different from zero) at the 5% level.

Example 2

In Example 1 we had some notions of what signs to expect for the different coefficients in the regression equation. Sometimes, even if the signs are correct, the magnitudes of the coefficients do not make economic sense. In which case one has to examine the model or the data. An example where the magnitude of the coefficient makes sense is the following.

In June 1978, California voters approved what is known as Proposition 13 limiting property taxes. This led to substantial and differential reduction in

These data have been provided by J. S. Shonkwiler. They are used in the paper of J. S. Shonkwiler and J. E. Reynolds, "A Note on the Use of Hedonic Price Models in the Analysis of Land Prices at the Urban Fringe," Land Economics, February 1986.

In results 3 to 5 we have to change the degrees of freedom from (n - 3) to {n ~ - 1). In simple regression we have = 1 and hence d.f. = - 2. In the case of two explanatory variables = 2 and hence we used d.f. = - 3. In all cases one d.f. is for the estimation of a.

Result 6 is now