Testing Linear and Log-Linear Regressions for Functional Form

Abstract

It is often a problem to know what functional form to choose when specifying an econometric model since economic theory does not usually provide a very precise guide. The choice of functional form may, however, have important implications for subsequent statistical tests, for forecasts and for policy analysis, e.g. see Hall (1978)and Mizon (1977). Due to their simplicity, the specifications most commonly used are the linear and log-linear models. Sometimes the estimates of these two variants are compared with a view to choosing one of them as the correct representation. Although this comparison may be of interest in certain cases, in others it may be more appropriate to test linearity or log-linearity against a more general functional form rather than against each other. In this paper, we discuss two approaches to testing the adequacy of the linear and log-linear specifications against the more general alternative of the extended Box-Cox (1964) regression model considered by Savin and White (1978). The first of these procedures is based on the Lagrange multiplier approach discussed by Breusch and Pagan (1980) and by Godfrey and Wickens (1980), while the second is derived from work by Andrews (1971) on the selection of data transformations.1 Both approaches lead to tests which are easy to compute and which can reject both models as well as being capable of selecting one form rather than the other. The paper is set out as follows. In Section 2 we compare some existing procedures for testing the functional form of linear and log-linear models. In Section 3 we derive new large sample tests which are based on the Lagrange multiplier approach. The possibility of using small sample tests is discussed in Section 4 and a numerical example to illustrate the use of our new tests is given in Section 5.