Estimation of a2

We shall now consider the estimation of a. If ut) were observable, the most natural estimator of ct2 would be the sample variance T Since {ut} are not observable, we must first predict them by the least squares residuals {ut} defined in (10.2.7). Then a2 can be estimated by

(10.2.35) a2 = ^Xu2

which we shall call the least squares estimator of a. Although the use of the term least squares here is not as compelling as in the case of a and (3, we use it because it is an estimator based on the least squares residuals. Using ct we can estimate Up and Ua given in (10.2.23) and (10.2.24) by

о 9

substituting (T for a in the respective formulae.

We shall evaluate £ct2. From (10.2.7) we can write

(10.2.36) ut = ut – {a — a) — (Э — P)^.

Multiplying both sides of (10.2.36) by u„ summing over t, and using (10.2.8) and (10.2.9) yields

(10.2.37) Ew2 = Ъщщ, from which we obtain

(10.2.38) Ем2 = Ем? – Е(мг – uf.

Using (10.2.36) and (10.2.38), we have

Подпись: lxtu^Lx* ut 2(xf)2 о ‘Lut’Llfut

(10.2.39) Е(м, – щ) = Eu,(m, – ut) =———————– — +

2(1 rf

Подпись: (10.2.40) EL{ut - uf = о2 image530

Taking the expectation of (10.2.39) yields

But multiplying both sides of (10.2.15) by 1* and summing over t yields

(10.2.41) E(l?)2 = Elf

because of (10.2.18). Similarly, multiplying both sides of (10.2.11) by x* and summing over t yields

(10.2.42) E(xf)2 = Exf xt

because of (10.2.17). Therefore, we obtain from (10.2.40), (10.2.41), and


(10.2.42) £E(m( – uf = 2a2.

Finally, from (10.2.38) and (10.2.43) we conclude that (10.2.44) ELut = (T — 2)a2.

Equation (10.2.44) implies that £d2 = T l(T — 2)a2 and hence d2 is a biased estimator of a2, although the bias diminishes to zero as T goes to infinity. If we prefer an unbiased estimator, we can use the estimator defined by (10.2.45) a2


We shall not obtain the variance of a here; in Section 10.3 we shall

indicate the distribution of Xuf, as well as its variance, assuming the normality of {ut}.

One purpose of regression analysis is to explain the variation of {yt} by the variation of {xt}. If {yt} are explained well by (xj, we say that the fit of the regression is good. The statistic ct2 may be regarded as a measure of the goodness of fit the smaller the <r2, the better the fit. However, since cr2 depends on the unit of measurement of {yt}, we shall use the measure of the goodness of fit known as R square, where


(10.2.46) R – 1—————–


Here 52 is the sample variance of {yt}; namely, s2 = T 1Ъ(у( — у)2. This statistic does not depend on the unit of measurement of either {yt} or xt}. Since

Подпись: (10.2.47) and Ter2 = min X(y, — a — fix,)2

a, (З

(10.2.48) 7s2 = minX(y, — a)2,


2 2 2 we have a ^ sy; therefore, 0 ^ R ^ 1.

We can interpret R2 defined in (10.2.46) as the square of the sample

correlation between {yt} and {xt}. From (10.2.5) and (10.2.7),

(10.2.49) ut = yt – у – |3(x, – x).

Therefore, using (10.2.11), we have

(10.2.50) Xu,2 = X(y, – yf + p2X(xf)2 – 2pXxfyt.

Inserting (10.2.12) into the right-hand side of (10.2.50) yields

Подпись: (Zxfyf X(xf)2 (10.2.51) Xuf = Щ – yf

Подпись: Finally, from (10.2.46) and (10.2.51) we obtain Vxfytf (10.2.52) R =


which is the square of the sample correlation coefficient between {yt) and

In practice we often face a situation in which we must decide whether {yt} are to be regressed on {xt} or on another independent sequence st}. That is, we must choose between two regression equations

(10.2.53) yt = ai + Рл + uit and

(10.2.54) yt = a2 + p2st + Uit ■

This decision should be made on the basis of various considerations such as how accurate and plausible the estimates of regression coefficients are, how accurately the future values of the independent variables can be predicted, and so on. Other things being equal, it makes sense to choose the equation with the higher R. In Section 12.5, we shall discuss this issue further.

The statistic a is merely one statistic derived from the least squares residual {ut}, from which one could derive more information. Since {ut} are the predictors of {ut}, they should behave much like {ut}; it is usually a good idea for a researcher to plot {щ} against time. A systematic pattern in that plot indicates that the assumptions of the model may be incorrect. Then we must respecify the model, perhaps by allowing serial correlation or heteroscedasticity in [ut, or by including other independent variables in the right-hand side of the regression equation.

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