where n = sample size, S = skewness coefficient, and K = kurtosis coefficient. For a normally distributed variable, S = 0 and K = 3. Therefore, the JB test of normality is a test of the joint hypothesis that S and K are 0 and 3, respectively. In that case the value of the JB statistic is expected to be 0.
Under the null hypothesis that the residuals are normally distributed, Jarque and Bera showed that asymptotically (i.e., in large samples) the JB statistic given in (5.12.1) follows the chi-square distribution with 2 df. If the computed p value of the JB statistic in an application is sufficiently low, which will happen if the value of the statistic is very different from 0, one can reject the hypothesis that the residuals are normally distributed. But if
20See C. M. Jarque and A. K. Bera, "A Test for Normality of Observations and Regression Residuals," International Statistical Review, vol. 55, 1987, pp. 163-172.
Gujarati: Basic I. Single-Equation 5. Two-Variable © The McGraw-Hill
Econometrics, Fourth Regression Models Regression: Interval Companies, 2004
Edition Estimation and Hypothesis
CHAPTER FIVE: TWO VARIABLE REGRESSION: INTERVAL ESTIMATION AND HYPOTHESIS TESTING 149
the p value is reasonably high, which will happen if the value of the statistic is close to zero, we do not reject the normality assumption.
The sample size in our consumption-income example is rather small. Hence, strictly speaking one should not use the JB statistic. If we mechanically apply the JB formula to our example, the JB statistic turns out to be 0.7769. The p value of obtaining such a value from the chi-square distribution with 2 df is about 0.68, which is quite high. In other words, we may not reject the normality assumption for our example. Of course, bear in mind the warning about the sample size.
Remember that the CNLRM makes many more assumptions than the normality of the error term. As we examine econometric theory further, we will consider several tests of model adequacy (see Chapter 13). Until then, keep in mind that our regression modeling is based on several simplifying assumptions that may not hold in each and every case.
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