Building and interpreting power-exponential regression models

Abstract

In our research, we developed and studied new structural specifications for regression models. The article presents two generalisations of the Cobb-Douglas production function: power exponential regression with powers presented in the form of linear combinations of explanatory variables and in the form of linear combinations of natural logarithms of explanatory variables. The approximation qualities of these models are always on a par with those of the Cobb-Douglas production functions. Models of the first type are perfect for predicting the dependent variable values, but the interpretation of their estimates is problematic. For the second type of models it has been proven that the coefficient of elasticity of the regression function for a specific explanatory variable does not depend on its value and is a linear combination of the natural logarithms of the remaining variables. Therefore, the estimates of such models are quite interpretable. In addition, models of the second type with the given values of elasticity coefficients can be used to determine predicted values of the explanatory and the response variable. In our study, we built four regression models for the dependence of the railway passenger volume in the Irkutsk Region on the number of personal vehicles and the cost of train and airplane tickets. The article presents a detailed interpretation of the power exponential regression with powers given in the form of linear combinations of natural logarithms of explanatory variables. The model was used to make graphs of changes in the elasticity of railway passenger volume for each explanatory variable over time. The predicted values of the explanatory and the response variable were obtained.