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74

n - 1 S xf

where RSS is the residual sum of squares from the OLS model. But

£(RSS) = E

S (y, - x.y

Note that if crj = for all /, this reduces to (« - l}cr. Thus we would be estimating the variance of p by an expression whose expected value is

S ? S - S x?a?

in ~ DS xj whereas the true variance is

Thus the estimated variances are also biased. If cr] and x] are positively correlated, as is often the case with economic data so that X xfu] > (l/n) X ? 2 o? then the expected value of the estimated variance is smaller than the true variance. Thus we would be underestimating the true variance of the OLS estimator and getting shorter confidence intervals than the true ones. This also affects tests of hypotheses about p.

Estimation of the Variance of the OLS Estimator Under Heterosledasticity

The solution to the heteroskedasticity problem depends on the assumptions we make about the sources of heteroskedasticity. When we are not sure of this, we can at least try to make corrections for the standard errors, since we have seen that the least squares estimator is unbiased but inefficient, and moreover, the standard errors are also biased.

This expression is of the form a,b,j /X ? 2 where a, = and b, =

. Thus it is less than 1 and is equal to 1 only if a, and b, are proportional, that is, x,z, and xjz, are proportional or is a constant, which is the case if the errors are homoskedastic.

Thus the OLS estimator is unbiased but less efficient (has a higher variance) than the WLS estimator. Turning now to the estimation of the variance of p, it is estimated by

RSS 1



White suggests that we use the formula (5.2) with uf substituted for aj. Using this formula we find that in the case of the illustrative example with data in Table 5.1, the standard error of p, the slope coefficient is 0.027. EarUer. we estimated it from the OLS regression as 0.0253. Thus the difference is really not very large in this example.

5.4 Solutions to tiie

Heterosledasticity Problem

There are two types of solutions that have been suggested in the literature for the problem of heteroskedasticity:

1. Solutions dependent on particular assumptions about of.

2. General solutions.

We first discuss category 1. Here we have two methods of estimation: weighted least squares and maximum likelihood (ML).

If the error variances are known up to a multiplicative constant, there is no problem at all. If V(«,) = ahj, where z, are known, we divide the equation through by z, and use ordinary least squares. The only thing to remember is that if the original equation contained a constant term, that is, y, = a + px, -f-m„ the transformed equation will not have a constant term. It is

= cxl-F P -F V,

Zi Z; Z;

where v, = m,/z,. Now F(v,) = a for all /. Thus we should be running a regression of y,/z, on l/z, and jr,/z, without a constant term.

One interesting case is where V(m,) = ox, and a = 0. In this case the transformed equation is

Hence

that is, the WLS estimator is just the ratio of the means. Another case is where aj = axj. In this case the transformed equation is

= a-+P + v,

X, Xf

White, "A Heteroskedasticity." A similar suggestion was made earlier in C. R. Rao, "Estimation of Heteroskedastic Variances in Linear Models," Journal of the American Statistical Association, March 1970, pp. 161-172.



*S. J. Prais and H. S. Houthakker, Analysis of Family Budgets (New York: Cambridge University Press, 1955), p. 55ff.

Amemiya discusses the ML estimation for this model when the errors follow a normal, log-normal, and gamma distribution. See T. Amemiya, "Regression Analysis When Variance of the Dependent Variables Is Proportional to Square of Its Expectation," Journal of ttie American Statistical Association. December 1973, pp. 928-934.

Thus the constant term in this equation is the slope coefficient in the original equation.

Paris and Houthakker** found in their analysis of family budget data that the errors from the equation had variance increasing with household income. They considered a model cj; = ~[£( ,)]-, that is, cr? = ciia + x,Y. In this case we cannot divide the whole equation by a known constant as before. For this model we can consider a two-step procedure as follows. First estimate a and p by OLS. Let these estimators be a and p. Now use the WLS procedure as outlined earlier, that is, regress y,/(a + px,) on 1/(6: + px,) and x,Ka. + px,) with no constant term. This procedure is called a two-step weighted least squares procedure. The standard errors we get for the estimates of a and p from this procedure are valid only asymptotically. They are asymptotic standard errors because the weights l/(a + Px,) have been estimated.

One can iterate this WLS procedure further, that is, use the new estimates of a and p to construct new weights and then use the WLS procedure, and repeat this procedure until convergence. This procedvire is called the iterated weighted least squares procedure. However, there is no gain in (asymptotic) efficiency by iteration.

If we make some specific assumptions about the errors, say that they are normal, we can use the maximum likelihood method, which is more efficient than the WLS if errors are normal.* Under the assumption of normality we can write the log-Hkelihood function as

log L = -„ log „ - 2 log(„ + p,,) - 2 (ifir")

Note that maximizing this likelihood function is not the same as the weighted least squares that minimizes the expression

/ , - g -

A more general model is to assume that the variance aj is equal to (7 + ,). In this case, too, we can consider a WLS procedure, that is, minimize

/ , - g - px,y

. V 7 + sx, ;

or if the errors can be assumed to follow a known distribution, use the ML method. For instance, if the errors follow a normal distribution, we can write the log-likelihood function as



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