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CHAPTER ELEVEN: HETEROSCEDASTICITY 411

Breusch-Pagan-Godfrey Test.21 The success of the Goldfeld-Quandt test depends not only on the value of c (the number of central observations to be omitted) but also on identifying the correct X variable with which to order the observations. This limitation of this test can be avoided if we consider the Breusch-Pagan-Godfrey (BPG) test.

To illustrate this test, consider the k-variable linear regression model

Yi = fa1 + fa2 X2i + ••• + fakXki + ui (11.5.12)

Assume that the error variance ai2 is described as ai = f («1 + «2 Z2i \-----\~amZmi) (11.5.13)

that is, af is some function of the nonstochastic variables Z's; some or all of the Xs can serve as Z's. Specifically, assume that a2 = a1 + a2Z2i \-----\ amZmi (11.5.14)

that is, af is a linear function of the Z's. If a2 = a3 = ■ ■■ = am = 0, a2 = a1, which is a constant. Therefore, to test whether ai2 is homoscedastic, one can test the hypothesis that a2 = a3 = ■■■ = am = 0. This is the basic idea behind the Breusch-Pagan test. The actual test procedure is as follows.

Step 1. Estimate (11.5.12) by OLS and obtain the residuals ii1, u2,..., un. Step 2. Obtain a2 = J2 ¿<2/n. Recall from Chapter 4 that this is the maximum likelihood (ML) estimator of a2. [Note: The OLS estimator is £u2/(n - k).]

Step 3. Construct variables pi defined as

which is simply each residual squared divided by a2. Step 4. Regress pi thus constructed on the Z's as

where vi is the residual term of this regression.

Step 5. Obtain the ESS (explained sum of squares) from (11.5.15) and define

Assuming ui are normally distributed, one can show that if there is ho-moscedasticity and if the sample size n increases indefinitely, then

21T. Breusch and A. Pagan, "A Simple Test for Heteroscedasticity and Random Coefficient Variation,'' Econometrica, vol. 47, 1979, pp. 1287-1294. See also L. Godfrey, "Testing for Multiplicative Heteroscedasticity," Journal of Econometrics, vol. 8, 1978, pp. 227-236. Because of similarity, these tests are known as Breusch-Pagan-Godfrey tests of heteroscedasticity.

412 PART TWO: RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL

that is, © follows the chi-square distribution with (m - 1) degrees of freedom. (Note: asy means asymptotically.)

Therefore, if in an application the computed © (= x*) exceeds the critical x* value at the chosen level of significance, one can reject the hypothesis of homoscedasticity; otherwise one does not reject it.

The reader may wonder why BPG chose jESS as the test statistic. The reasoning is slightly involved and is left for the references.**

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