ascertain the direction and magnitude of the impact of the proposed explanatory variables on the basis, with
reasonable statistical certainty.
The NHAR(4) model is then extended to account for error term non-normality [NNHAR(4)]. In this
case, Ramirez and Shonkwiler ’s proposed procedures are expanded by letting the first non-normality parameter
(Θ) shift by Θsd, Θpdi, and Θpd2, according to the seasonal and policy dummies. Since Θ affects both Skew[U]
and Kurt[U] [equation (3)], this model specification assumes an error term distribution that can have a different
variance, skewness and kurtosis depending on the season and policy period. The slope parameter estimators under
this NNHAR(4) model exhibit substantially lower standard errors than their normal-error counterparts (Table 5),
empirically validating the previously discussed Monte Carlo simulation results again. Final [FNHAR(4) and
FNNHAR(4)] models are obtained by excluding the transportation cost (RRI) and 1996-2001 policy period
(PD2) variables from the regression equation, since they show the lowest levels of statistical significance in both
of the initial models and their corresponding parameter estimates bear incorrect signs.
All parameter estimates in the final models have the signs that would be expected from theory (Nelson).
The estimates for the standard errors of the slope parameter estimators are again all higher under the normal
model. Eight of the eleven standard error estimates are over 70% higher; and six are more than twice as high as in
the non-normal model. On average, the standard error estimates under the NHAR(4) model are 99.6% higher. As
a result, in the NNHAR(4) model, six regression parameters are statistically significant at the 1% level, and two
more at the 5% level, while only one regression parameter is statistically significant under the NHAR(4) model.
These remarkable estimation efficiency gains are related to the substantial 70.47 increase in the
maximum log-likelihood function value obtained by the modeling of non-normality. To put this number into
perspective, recall that an increase of just χ2(4,0.01)÷2 = 6.64 is required for the likelihood ratio test to reject the null
hypothesis that all non-normality parameters are equal to zero and conclude that the NNHAR(4) model represents
a statistically significant improvement over the NHAR(4) model. Such a large log-likelihood value increase is an
unmistakable indication that the observed basis data (Figure 3) is much more likely to have been generated by the
non-normal than by the normal-error model. The D’Agostino-Pearson normality test applied to the OLS
19