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75

Illustrative Example

As an illustration, again consider the data in Table 5.1. We saw earlier that regressing the absolute values of the residuals on x (in Glejsers tests) gave the following estimates:

7 = -0.209 5 = 0.0512

Now we regress yjw, on \/w, and xjwf (with no constant term) where w, = 7 + 8jc,. The resulting equation is

- = 0.4843(14) + 0.9176(jc,4.) R- = 0.9886

Vf, (0.1643) (0 0157)

If we assume that cr? = 7o + 7 + yif, the two-step WLS procedure would be as follows.

First, we regress ? on x, and xj. Earlier, using this regression we obtained the estimates as

7o = 0.493 I = -0.071 7 = 0.0037 Next we compute

wj = 0.493 - 0.071X, + 0.0037X? and regress yjwi on 1/w, and / »,. The results were

- = 0.72%(1/ ;,) + 0.9052(x,./h,) R = 0.9982

W, (0.3302) (0 0199)

The Rs in these equations are not comparable. But our interest is in estimates of the parameters in the consumption function

y,= a + Px;

Comparing the results with the OLS estimates presented in Section 5.2, we notice that the estimates of p are higher than the OLS estimates, the estimates of a are lower, and the standard errors are lower.

"The ML estimation of this model is discussed in H. C. Rutemiller and D. A. Bowers, "Estimation in a Heteroscedastic Regression Model," Journal of the American Statistical Association, June 1968.

log L = const. - 2 log(Y + bx,) - S ("tT)

Again, note that the WLS and ML procedures are not the same." A two-step WLS estimation for this model would proceed as follows. Compute the OLS estimates of a and p. Get the estimated residuals and regress the absolute values of these residuals on x to get estimates of and 8. Then use WLS.



5.5 Heteroskedasticity and the Use of Deflators

There are two remedies often suggested and used for solving the heteroskedasticity problem:

1. Transforming the data to logs.

2. Deflating the variables by some measure of "size."

The first method often does reduce the heteroskedasticity in the error variances, although there are other criteria by which one has to decide between the linear and the logarithmic functional forms. This problem is discussed in greater detail in Section 5.6.

Regarding the use of deflators, one should be careful in estimating the equation with the correct explanatory variables (as explained in the preceding section). For instance, if the original equation involves a constant term, one should not estimate a similar equation in the deflated variables. One should be estimating an equation with the reciprocal of the deflator added as an extra explanatory variable. As an illustration, consider the estimation of railroad cost functions by Griliches" where deflation was used to solve the heteroskedasticity problem. The variables are: = total cost, M = miles of road, and X = output.

If = + bX, dividing by M gives C/M = a + biXIM). But if the true relation h = + bX + c, deflation leads to CIM = a + b{XIM) + c{\IM). For 97 observations using 1957-1969 averages as units, regressions were

% = 13,016 + 6.431 = 0.365

M (6218) (0 871) M

= 827 + 6.439 + 3,065,000 -J- R = 0.614

M (5115) (0 682) M (393.000) M

= -1.884M + 6.613Z + 3676 R = 0.945

(2 906) (0.375) (4730)

The coefficient is significant in the second equation but not in the third, and the coefficient of a is not significant in ehher the second or third equation. From this Griliches concludes that there is no evidence that M belongs in the equation in any form. It appears in a significant form in the second equation only because the other variables were divided by it. Some other equations estimated with the same data are the following:

= 2811 + 6.39 R = 0.944

(4524) (0 18)

= 3805 + 6.06 R = 0.826

VM (3713) VM (0 51) VM

"Z. Griliches, "Railroad Cost Analysis," The Bell Journal of Economics and Management Science. Spring 1972.



See E. Kuh and J. R. Meyer, "Correlation and Regression Estimates When the Data Are Ratios," Econometrica. October 1955, pp. 400-416.

The last equation is the appropriate one to estimate if = a + bX + and V(m) = Mu\

One important thing to note is that the purpose in all these procedures of deflation is to get more efficient estimates of the parameters. But once those estimates have been obtained, one should make all inferences-calculation of the residuals, prediction of future values, calculation of elasticities at the means, etc., from the original equation-not the equation in the deflated variables.

Another point to note is that since the purpose of deflation is to get more efficient estimates, it is tempting to argue about the merits of the different procedures by looking at the standard errors of the coefficients. However, this is not correct, because in the presence of heteroscedasticity the standard errors themselves are biased, as we showed earlier. For instance, in the five equations presented above, the second and third are comparable and so are the fourth and fifth. In both cases if we look at the standard errors of the coefficient of X, the coefficient in the undeflated equation has a smaller standard error than the corresponding coefficient in the deflated equation. However, if the standard errors are biased, we have to be careful in making too much of these differences. An examination of the residuals will give a better picture.

In the preceding example we have considered miles M as a deflator and also as an explanatory variable. In this context we should mention some discussion in the literature on "spurious correlation" between ratios.- The argument simply is that even if we have two variables X and Y thai are uncorrelated, if we deflate both the variables by another variable Z, there could be a strong correlation between X/Z and Y/Z because of the common denominator Z. It is wrong to infer from this correlation that there exists a close relationship between X and Of course, if our interest is in fact the relationship between XIZ and Y/Z, there is no reason why this correlation need be called "spurious." As Kuh and Meyer point out, "The question of spurious correlation quite obviously does not arise when the hypothesis to be tested has initially been formulated in terms of ratios, for instance, in problems involving relative prices. Similarly, when a series such as money value of output is divided by a price index to obtain a constant dollar estimate of output, no question of spurious correlation need arise. Thus, spurious correlation can only exist when a hypothesis pertains to undeflated variables and the data have been divided through by another series for reasons extraneous to but not in conflict with the hypothesis framed as an exact, i.e., nonstochastic relation."

However, even in cases where deflation is done for reasons of estimation, we should note that the problem of "spurious correlation" exists only if we start drawing inferences on the basis of correlation coefficients when we should not be doing so. For example, suppose that the relationship we derive is of the form



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