Normality of the Disturbances

If the disturbance are not normal, OLS is still BLUE provided assumptions 1-4 still hold. Nor­mality made the OLS estimators minimum variance unbiased MVU and these OLS estimators turn out to be identical to the MLE. Normality allowed the derivation of the distribution of these estimators and this in turn allowed testing of hypotheses using the t and F-tests consid­ered in the previous chapter. If the disturbances are not normal, yet the sample size is large, one can still use the normal distribution for the OLS estimates asymptotically by relying on the Central Limit Theorem, see Theil (1978). Theil’s proof is for the case of fixed X’s in repeated samples, zero mean and constant variance on the disturbances. A simple asymptotic test for the normality assumption is given by Jarque and Bera (1987). This is based on the fact that the normal distribution has a skewness measure of zero and a kurtosis of 3. Skewness (or lack of symmetry) is measured by

S [E(X — Ц)3]2 Square of the 3rd moment about the mean

[E(X — a)2]3 Cube of the variance

Kurtosis (a measure of flatness) is measured by

E(X — a)4 4th moment about the mean

[E(X — a)2]2 Square of the variance

Подпись: JB Подпись: n image133

For the normal distribution S = 0 and к = 3. Hence, the Jarque-Bera (JB) statistic is given by

where S represents skewness and к represents kurtosis of the OLS residuals. This statistic is asymptotically distributed as x2 with two degrees of freedom under H0. Rejecting H0, rejects normality of the disturbances but does not offer an alternative distribution. In this sense, the test is non-constructive. In addition, not rejecting H0 does not necessarily mean that the distribution of the disturbances is normal, it only means we do not reject that the distribution of the disturbances is symmetric and has a kurtosis of 3. See the empirical example in section

5.3 for an illustration. The Jarque-Bera test is part of the standard output using EViews.

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