The powerful D’Agostino-Pearson omnibus (K2) normality test can be applied to the OLS
residuals to confirm these results. This test is based on a standardized combination of the well-known
skewness and kurtosis coefficients, which, on the one hand, allows for rejection of the null hypothesis of
normality on either count (skewness or kurtosis) and, on the other, it avoids the “double jeopardy” of
separate skewness and kurtosis tests. The K2 statistic is 11.93 for the corn price residuals and 8.26 for the
soybean price residuals. Since under the null hypothesis of normality K2 is distributed as a χ2(2), normality is
rejected at the 1% significance level in both cases. Non-normal error term models are strongly justified.
However, the estimated value for the skewness parameter in the corn model (μ c) is very low (-0.0539) and
statistically insignificant at the 20% level. A likelihood ratio test (χ2(1) = -2×(52.54-52.55) = 0.02) confirms
this result. Therefore, μ c is set equal to zero in the final NNSUR-AR(1) model (Table 3).
As explained earlier, μ c=0 implies that the distribution of the error term in the corn model is
kurtotic but not skewed, while the positive sign of the parameter estimate for μ s indicates positive, i.e. right
skewness in the error term distribution of the soybean model. The specific skewness and kurtosis coefficients,
calculated using the formulas in equation (3), are 0 and 22.20 (corn), and 2.08 and 8.61 (soybeans), respectively.
Right skewness, i.e. upward price spikes that are relatively more pronounced than lower price occurrences, is not
a surprising finding, and can be visually perceived on a scatter plot of the soybean price data (Figure 1). The
simulation results are empirically validated in this case: The estimated standard errors for the slope parameter
estimators, are substantially lower under the non-normal model (0.39, 2.81, and 5.30 vs. 0.56, 3.83 and 7.50
under the normal), which means that the proposed partially adaptive estimator allows for much narrower (i.e.
more precise) confidence intervals for the rates of decreases of corn and soybean prices through time.
Confidence intervals for the price occurrences involve the uncertainty in the estimation of all of the
model parameters as well as the estimated error term distribution. These are obtained by applying the
numerical technique of Krinsky and Rob. This technique is based on the asymptotic properties of the ML
estimators, i.e. on the fact that they are normally distributed, consistent, and with known covariance matrix.
It uses the same principle applied to construct confidence intervals for the true parameter values to
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