Power Transformations and Unit Mean and Constant Variance Assumptions of the Multiplicative Error Model: The Generalized Gamma Distribution

Aims: To study the implications of power transformations namely; inverse-square-root, inverse, inverse-square and square transformations on the error component of the multiplicative error and determine whether the unit-mean and constant variance assumptions of the model are either retained or violated after the transformation.
Methodology: We studied the distributions of the error component under the various distributional forms of the generalized gamma distribution namely; Gamma (a, b, 1), Chi-square, Exponential, Weibull, Rayleigh and Maxwell distributions. We first established the functions describing the distributional characteristics of interest for the generalized power transformed error component and secondly applied the unit-mean conditions of the untransformed distributions to the established functions.
Results: We established the following important results in modeling using a multiplicative error model, where data transformation is absolutely necessary;(i) For the inverse-square-root transformation, the unit-mean and constant variance assumptions are approximately maintained for all the distributions under study except the Chi-square distribution where it was violated. (ii) For the inverse transformation, the unit-mean assumptions are violated after the transformation except for the Rayleigh and Maxwell distributions. (iii) For the inverse-square transformation, the unit-mean assumption is violated for all the distributions under study. (iv) For the square transformation, it is only the Maxwell distribution that maintained the unit-mean assumption. (v) For all the studied transformations the variances of the transformed distributions were found to be constant but greater than those of the untransformed distribution.
Conclusion: The results of this study though restricted to the distributional forms of the generalized gamma distribution, however they provide a useful framework in modeling for determining where a particular power transformation is successful for a model whose error component has a particular distribution.