is estimated for each of the six models with the two sets of annual and quarterly data, using the full-information maximum likelihood estimator in SAS, which allowed us to impose nonlinear restrictions presented in . The results for price and quantity models are presented in and , respectively.

Parameter estimates for the price models (comparing annual with quarterly data).

Parameter estimates for the quantity models (comparing annual with quarterly data).

The own-price coefficients, and , for all the models in have the expected negative sign with both annual and quarterly data sets. is significantly different from zero for the Bertrand and the Stackelberg models with US price leadership, while is statistically significant for the Bertrand and the Stackelberg models with Iran price leadership, in both data sets. Although the magnitude of estimated coefficients is smaller using the quarterly data set, comparing the two sets of results does not show much qualitative differences. presents estimation results for the three quantity models using both annual and quarterly data sets.

It is important to note that in and several variables are not statistically significant, raising some doubts about the overall validly of the results. However, these results are consistent with the results in Carter and MacLaren (1997) and Gasmi, Laffont, and Vuong (1992), and the key variables including own prices in the demand equations are statistically significant and have signs consistent with the theory. The competing models have simple specifications because the focus is not merely to estimate the demand equations, but to get to the LRs for each equation. Gasmi, Laffont, and Vuong (1992) emphasize that the competing models do not need to be correctly specified in order for the LR tests to be valid. The possible misspecification problem has also been addressed by both Carter and MacLaren (1997) and Gasmi, Laffont, and Vuong (1992). Carter and MacLaren (1997) note that there is a possibility that the best of the models specified may not be the correct model since the actual behavior of the market may differ from the choice of the game-theoretic equilibrium.

The own-price elasticities of demand, at the average point of the annual data set for Bertrand model for Iran and the United States are –0.94 and –1.24 for the annual data and –0.97 and –1.22 using the quarterly data, respectively. Same elasticities for the Stackelberg model with Iran’s leadership are –0.72 and –1.26, and for the US leadership they are –0.95 and –4.18 with the annual data. Considering estimated coefficients using quarterly data, own-price elasticities for the Stackelberg model with Iran’s leadership are –1.86 and –1.22 for Iran and the United States, respectively. These elasticities for the model with the US leadership are –0.97 and –1.48. Log-likelihood statistics are very close to each other for the models using similar data sets. Likelihood function for the third model has a smaller absolute value. Income elasticities for Iran’s pistachio for these three models are 2.28, 2.07, and 2.34 with the annual data, respectively, which shows that Iranian pistachio is a luxury commodity, and for US pistachio are –1.73, –3.3, and –3.48. The negative sign for US pistachio income elasticity implies that an increase in Japanese income makes them switch to a more luxurious good, i.e., Iranian pistachio.

Although it seems that the results of quantity models, in term of statistical significance and parameter signs, are not as good as price models, the Cournot model has the highest likelihood statistics in absolute value among all those six models, using both data sets. In the quantity models, similar to the price models, we expected to get a negative sign on income parameters for US pistachio demand equations, indicating the substitution of lower quality (taste) good with more luxurious one.

Results for the model comparison based on the normalized LR statistics are presented in . Each column shows two sets of statistics based on the type of data used for the estimation. Apart from a slight difference in magnitude of the statistics, overall the results are the same using annual and quarterly data sets. To calculate the statistics, the model in the row is subtracted from the model in the column. Hence, a negative sign implies that the model in the column is preferred to the model in the row. Models included in rows are M1 through M5, while the models in columns are M2 through M6. For each comparison of models, we have two sets of statistics, considering annual and quarterly data. Overall, the table consists of five rows and ten columns of statistics.

Table 5Normalized LR statistics.

Considering 0.05 level of significance, the critical values of the standard normal variable would be –1.96 and 1.96 for the annual data set. All price models (M1, M2, and M3) are significantly better than the Cournot model (M4) because the LR statistics are positive for both annual and quarterly data sets (see values in column 5 and 6). The quantity Stackelberg models (M5 and M6) are also preferred to the Cournot model (M4) because the LR statistics are negative and statistically significant (–11.91 and –4.90 for the annual data set, and –17.36 and –19.52 for the quarterly data set, respectively). Using the annual data sets, the difference between models M6 and M4 is statistically significant at the 5% significance level, but the difference between M6 and models M1 and M3 would be statistically significant at the 20% significance level. M6 is better than M5 because of the negative LR statistics for both annual and quarterly data set, although not statistically significant. All in all it seems that M6 is preferred to all the other models within this data set, because of the negative LR statistics in the last column. Also, none of the price models has statistical significance superiority to the other price models.

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