Econometric models are often formulated in terms of the first difference or the percentage change, both of which may be generalized by the Box-Cox difference transformation. Unfortunately, economic theory typically provides little guidance to the proper functional forms appropriate to the specification of the economic relationships. The choice of suitable functional forms thus relies heavily upon established statistical procedures. Existing procedures adopt primarily the classical likelihood approach and mainly confine to a single transformation parameter only, thus seriously limiting the use of more appropriate models. When multi-parameter transformation is allowed, these procedures would require to search over a multidimensional grid of values, rendering them extremely expensive, if not impossible, to find the optimal solution.We have derived an exact test for the parameter vector of transformation in linear models. By utilizing Taylor series approximations this reduces to a choice between two regression equations. The test statistic which has an exact F-distribution can be easily calculated from these two regression equations by least squares estimation, which algorithm is available from the very handy to the sophisticated statistical packages. It is therefore a simple and ready statistical procedure for assessing the suitable choice of the functional forms of the variables, thereby allowing more flexible and appropriate economic relations be formulated and their validity be tested.
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