35T. Dudley Wallace and J. Lew Silver, Econometrics: An Introduction, Addison-Wesley, Reading, Mass., 1988, p. 265.

36Recall that we have already encountered this assumption in our discussion of the Goldfeld-Quandt test.

Gujarati: Basic Econometrics, Fourth Edition

II. Relaxing the Assumptions of the Classical Model

11. Heteroscedasticity: What Happens if the Error Variance is Nonconstant?

© The McGraw-H Companies, 2004

FIGURE 11.10 Error variance proportional to X2.

CHAPTER ELEVEN: HETEROSCEDASTICITY 419

Hence the variance of vi is now homoscedastic, and one may proceed to apply OLS to the transformed equation (11.6.6), regressing Yi/Xi on 1/Xi.

Notice that in the transformed regression the intercept term p2 is the slope coefficient in the original equation and the slope coefficient P1 is the intercept term in the original model. Therefore, to get back to the original model we shall have to multiply the estimated (11.6.6) by Xi. An application of this transformation is given in exercise 11.20.

If it is believed that the variance of ui, instead of being proportional to the squared Xi, is proportional to Xi itself, then the original model can be transformed as follows (see Figure 11.11):

420 PART TWO: RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL

FIGURE 11.11 Error variance proportional to X.

FIGURE 11.11 Error variance proportional to X.

Given assumption *, one can readily verify that E(v*) = a*, a ho-moscedastic situation. Therefore, one may proceed to apply OLS to (11.6.8), regressing Yi /\[Xi on 1 /*JXi and VX.

Note an important feature of the transformed model: It has no intercept term. Therefore, one will have to use the regression-through-the-origin model to estimate ¡1 and ¡*. Having run (11.6.8), one can get back to the original model simply by multiplying (11.6.8) by

Assumption 3: The error variance is proportional to the square of the mean value of Y.

Equation (11.6.9) postulates that the variance of ui is proportional to the square of the expected value of Y (see Figure 11.8e). Now

Therefore, if we transform the original equation as follows,

Yi 1 Xi ui

CHAPTER ELEVEN: HETEROSCEDASTICITY 421

where Vi = ui/E(Yi), it can be seen that E(v2) = a2; that is, the disturbances Vi are homoscedastic. Hence, it is regression (ii.6.i0) that will satisfy the homoscedasticity assumption of the classical linear regression model.

The transformation (ii.6.i0) is, however, inoperational because E(Yi) depends on Pi and P2, which are unknown. Of course, we know Yi = Pi + P2Xi, which is an estimator of E(Yi). Therefore, we may proceed in two steps: First, we run the usual OLS regression, disregarding the heteroscedasticity problem, and obtain Yi. Then, using the estimated Yi, we transform our model as follows:

where vi = (uiY). In Step 2, we run the regression (ii.6.ii). Although Yi are not exactly E(Yi), they are consistent estimators; that is, as the sample size increases indefinitely, they converge to true E(Yi). Hence, the transformation (ii.6.ii) will perform satisfactorily in practice if the sample size is reasonably large.

Assumption 4: A log transformation such as very often reduces heteroscedasticity when compared with the regression Y = P1 +

This result arises because log transformation compresses the scales in which the variables are measured, thereby reducing a tenfold difference between two values to a twofold difference. Thus, the number 80 is i0 times the number 8, but ln80 ( = 4.3280) is about twice as large as ln 8 (= 2.0794).

An additional advantage of the log transformation is that the slope coefficient P2 measures the elasticity of Y with respect to X, that is, the percentage change in Y for a percentage change in X. For example, if Y is consumption and X is income, P2 in (ii.6.ii) will measure income elasticity, whereas in the original model P2 measures only the rate of change of mean consumption for a unit change in income. It is one reason why the log models are quite popular in empirical econometrics. (For some of the problems associated with log transformation, see exercise ii.4.)

To conclude our discussion of the remedial measures, we reempha-size that all the transformations discussed previously are ad hoc; we are essentially speculating about the nature of ai2. Which of the transformations discussed previously will work will depend on the nature of the problem and the severity of heteroscedasticity. There are some additional problems with the transformations we have considered that should be borne

422 PART TWO: RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL

in mind:

1. When we go beyond the two-variable model, we may not know a priori which of the X variables should be chosen for transforming the data.37

2. Log transformation as discussed in Assumption 4 is not applicable if some of the Y and X values are zero or negative.38

3. Then there is the problem of spurious correlation. This term, due to Karl Pearson, refers to the situation where correlation is found to be present between the ratios of variables even though the original variables are un-correlated or random.39 Thus, in the model Yi = ß1 + ß2Xi + ui, YandXmay not be correlated but in the transformed model Yi/Xi = ßi(l/Xi) + ß2, Yi/Xi and l /Xi are often found to be correlated.

4. When af are not directly known and are estimated from one or more of the transformations that we have discussed earlier, all our testing procedures using the t tests, F tests, etc., are strictly speaking valid only in large samples. Therefore, one has to be careful in interpreting the results based on the various transformations in small or finite samples.40

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