[since var(yo) = var(yo - y) = + crVn and cov(yo, p) = 0]. Substituting the estimate 6 of derived earlier into (3.14), we get the estimated variance. Thus

t= 3o - ep

Vct4i + 1/ ) + e46-V5,,)

has a /-distribution with degrees of freedom (n - 2). This can be used to construct a confidence interval for 6, which is what we need. For instance, to get a 95% confidence interval for , we find a value / of t (from the /-tables for n-2 degrees of freedom) such that

Prob(/2 < tl) = 0.95

To get the limits for we solve the quadratic equation

/ 1 „2\

o - )2 -tMt+- + - =0 (3.15)

\ « S,J

The roots e, and 6, of this equation give the confidence limits.

As an illustration, suppose that we have estimated a consumption function on the basis of 10 observations and the equation is"

= 10.0 + 0.9x

and X = 200, = 185, 6 = 5.0, and S,, = 400. Consider the predicted value of x for = 235. This is cleariy x = 250.

"C. E. Fieller, "A Fundamental Formula in the Statistics of Biological Assay and Some Applications " Quarterly Journal of Pharmacy, 1944, pp. 117-123.

"We have changed the illustrative example slightly from the one we discussed in Section 3.7 because with cr- = 0.01 and 5„ = 4000 we get a symmetric confidence interval for 6. We have changed the example to illustrate the point that we need not get a symmetric confidence interval.

The main problem is that of obtaining confidence limits for xq because both yo and p are normally distributed. One can use a method suggested by Fieller for this purpose.This method can be used to obtain confidence intervals for ratios of estimators. These problems also arise in our discussion of simultaneous equation models and distributed lag models in later chapters. Another problem that falls in this same category is to estimate the value ofx at which the value of is zero. This is given by

and a and p have joint normal distribution.

Let us define e = £(yo/p) (6 is just xq and is thus a constant). Then the variable yo - is normally distributed with mean zero and variance

+ 62- (3.14)

3.11 Stochastic Regressors

In Section 3.2 we started explaining the regression model saying that the values x„ Xj, . . . , x„ of X are given constants, that is, x is not a random variable. This view is perhaps valid in experimental work but not in econometrics. In econometrics we view x as a random variable (except in special cases where x is time or some dummy variable). In this case the observations x,, , . . . , x„ are realizations of the random variable x. This is what is meant when we say that the regressor is stochastic.

Which of the results we derived earlier are valid in this case? First, regarding the unbiasedness property of the least squares estimators, that property holds good provided that the random variables x and are independent. (This is proved in the appendix to this chapter.) Second, regarding the variances of the estimators and the tests of significance that we derived, these have to be viewed as conditional on the realized values of x.

To obtain any concrete results, v,ith stochastic regressors, we need to make some assumptions about the joint distribution of and x. It has been proved that when and x are jointly normally distributed, the formulas we have derived for the estimates of the regression parameters, their standard errors, and the test statistics are all valid. Since the proof of this proposition is beyond the scope of this book, we will not pursue it here.

"In the case of multiple regression, proof of this proposition can be found in several papers. A clear exposition of all the propositions is given in: Allan R. Simpson, "A Tale of Two Regressions," Journal of the American Statistical Association, September 1974, pp. 682-689.

To get the confidence interval for Xq we solve equation (3.15). Note that y = 235 - 185 = 50 and = 2.306 from the /-tables for 8 degrees of freedom. Thus we have to solve the equation

(50 - 0.99)2 (2.306)45)1 + ] + ) = «

2500 - 906 + 0.8102 = 29.25 + .

0.74462 - 906 + 2470.75 = 0

The two roots are 6, = 42.12 and 62 = 78.85. Thus the 95% confidence interval for xo is (42.12, 78.85) or for x it is (242.12, 278.85). Note that the confidence interval obtained by the Fieller method need not be symmetric around the predicted point value (which in this case is Xo - 250).

If the samples are large, we can use the asymptotic distributions and this will give symmetric intervals. However, for small samples we have to use the method described here.

3.12 The Regression Fallacy

In the introduction to this chapter we mentioned a study by Gallon, who analyzed the relationship between the height of children and the height of parents. Let

JC = mid-parent height

= mean height (at maturity) of all children whose

mid-parent height is jc .

Galton plotted against x and found that the points lay close to a straight line but the slope was less than 1.0. What this means is that if the mid-parent height is 1 inch above jc, the childrens height (on the average) is less than 1 inch above y. There is thus a "regression of childrens heights toward the average."

A phenomenon like the one observed by Galton could arise in several situations where and x have a bivariate normal distribution and thus is a mere statistical artifact. That is why it is termed a "regression fallacy." To see this, first we have to derive the conditional distributions \ ) and f(y\x) when x and are jointly normal. We will show that both these conditional distributions are normal.

The Bivariate Normal Distribution

Suppose that X and are jointly normally distributed with means, variances, and covariance given by

E(X) = E(Y) = m, { ) = al V{Y) =

co\{X, Y) = pcrcTj, Then the joint density of Zand is given by

/(jc, y) = (2-rra,a,vTIV)- exp(0

where

r X - m

2(1 - p)

2 / V 2n

ci. (i,

ZJ}h\

Now completing the square in x and simplifying, we get

Q I - - m\ 1 / - m\

2(1 - p) V ct, ) 2\ ay j

Thus, with stochastic regressors the formulas we have derived are valid if and X are jointly normally distributed. Otherwise, they have to be considered as valid conditional on the observed xs.