Testing Functional Forms of Market Share Models Using the Box-Cox Transformation and the Lagrange Multiplier Approach

The most commonly used functional forms in measuring market response functions are the linear and log-linear (double-log) specifications. Although the two models are mutually non-nested, they are both nested within the class of Box-Cox regression models. This enables one to test the statistical validity of these two models using nested tests, the power characteristics of which are better established relative to non-nested hypotheses tests, at least in large samples. In this paper, an application of the Lagrange multiplier (LM) test to determine the validity of linear, log-linear, and attraction-type formulations of market share models is illustrated using marketing data. The test is easy to compute and involves running only one extra linear regression. A Monte Carlo simulation is performed to study the properties of the test for samples of varying size and different levels of error variance. The simulation results indicate that the LM test should not be used with samples of less than 100 observations. We also compare the performance of the LM test to that of the PE test developed by MacKinnon, White, and Davidson for non-nested models. The results show that the PE test has a lower probability of a type 1 error for all sample sizes and different error levels. The power of the LM test, however, is greater when the error variance of the true model is high, given a fixed sample size.