A1 Derivation Of Leastsquares Estimators For Regression Through The Origin

We want to minimize

with respect to p2.

Differentiating (1) with respect to h2, we obtain dEU2 dp?

Setting (2) equal to zero and simplifying, we get

HXiYi

Now substituting the PRF: Yi = p2Xi + ui into this equation, we obtain

Xi2 e XiUi EX?

CHAPTER SIX: EXTENSIONS OF THE TWO-VARIABLE LINEAR REGRESSION MODEL 199

T Xiui

Expanding the right-hand side of (5) and noting that the Xi are nonstochas-tic and the ui are homoscedastic and uncorrelated, we obtain a 2

Incidentally, note that from (2) we get, after equating it to zero

From Appendix 3A, Section 3A.1 we see that when the intercept term is present in the model, we get in addition to (7) the condition Y u = 0. From the mathematics just given it should be clear why the regression through the origin model may not have the error sum, Y u, equal to zero.

Suppose we want to impose the condition that Yu = 0. In that case we have e y = a e xi+e u

= fa ^Xi, since ^^iii = 0 by construction This expression then gives a e Y

Y mean value of Y X mean value of X

But this estimator is not the same as (3) above or (6.1.6). And since the fa of (3) is unbiased (why?), the fa of (9) cannot be unbiased.

The upshot is that, in regression through the origin, we cannot have both Y, uiXi and Y u equal to zero, as in the conventional model. The only condition that is satisfied is that Y uiXi is zero. Recall that

Summing this equation on both sides and dividing by N, the sample size, we obtain

Since for the zero intercept model J2 u and, therefore u, need not be zero, it then follows that

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