Info

'Obtained from the regression: E, = 5.8194 + 0.4590 .

"•Absolute value of the residuals.

Note: The ranking is in ascending order of values.

'Obtained from the regression: E, = 5.8194 + 0.4590 .

"•Absolute value of the residuals.

Note: The ranking is in ascending order of values.

408 PART TWO: RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL

Goldfeld-Quandt Test.17 This popular method is applicable if one assumes that the heteroscedastic variance, of, is positively related to one of the explanatory variables in the regression model. For simplicity, consider the usual two-variable model:

Yi = fa + fa Xi + u Suppose of is positively related to Xi as

where o2 is a constant.18

Assumption (11.5.10) postulates that of is proportional to the square of the X variable. Such an assumption has been found quite useful by Prais and Houthakker in their study of family budgets. (See Section 11.6.)

If (11.5.10) is appropriate, it would mean of would be larger, the larger the values of X;. If that turns out to be the case, heteroscedasticity is most likely to be present in the model. To test this explicitly, Goldfeld and Quandt suggest the following steps:

Step 1. Order or rank the observations according to the values of Xi, beginning with the lowest X value.

Step 2. Omit c central observations, where c is specified a priori, and divide the remaining (n — c) observations into two groups each of (n — c)/2 observations.

Step 3. Fit separate OLS regressions to the first (n — c)/2 observations and the last (n — c) 2 observations, and obtain the respective residual sums of squares RSS1 and RSS2, RSS1 representing the RSS from the regression corresponding to the smaller Xi values (the small variance group) and RSS2 that from the larger Xi values (the large variance group). These RSS each have

where k is the number of parameters to be estimated, including the intercept. (Why?) For the two-variable case k is of course 2.

Step 4. Compute the ratio

If ui are assumed to be normally distributed (which we usually do), and if the assumption of homoscedasticity is valid, then it can be shown that X of (11.5.10) follows the F distribution with numerator and denominator df each of (n — c — 2k)/2.

17Goldfeld and Quandt, op. cit., Chap. 3.

18This is only one plausible assumption. Actually, what is required is that o2 be monotoni-cally related to Xi.

CHAPTER ELEVEN: HETEROSCEDASTICITY 409

If in an application the computed X (= F) is greater than the critical F at the chosen level of significance, we can reject the hypothesis of ho-moscedasticity, that is, we can say that heteroscedasticity is very likely.

Before illustrating the test, a word about omitting the c central observations is in order. These observations are omitted to sharpen or accentuate the difference between the small variance group (i.e., RSS1) and the large variance group (i.e., RSS2). But the ability of the Goldfeld-Quandt test to do this successfully depends on how c is chosen.19 For the two-variable model the Monte Carlo experiments done by Goldfeld and Quandt suggest that c is about 8 if the sample size is about 30, and it is about 16 if the sample size is about 60. But Judge et al. note that c = 4 if n = 30 and c = 10 if n is about 60 have been found satisfactory in practice.20

Before moving on, it may be noted that in case there is more than one X variable in the model, the ranking of observations, the first step in the test, can be done according to any one of them. Thus in the model: Yi = ß1 + ß2X2i + ß3X3i + ß4X4i + ui, we can rank-order the data according to any one of these Xs. If a priori we are not sure whichX variable is appropriate, we can conduct the test on each of the X variables, or via a Park test, in turn, on each X.

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