| In the context of model building in data analysis and statistical inference, it is often necessary to transform the observed variables to a new scale so as to have a better representation of the real world. In this paper, we study the problem where the independent variables in a linear regression model are parametrically transformed. Asymptotic theory and Monte Carlo simulations are used to study whether we can transform the data using data-based transformations and proceed to do statistical inference on the resulting linear model as if the transformations were known. We find that for some cases the variances of the estimates of the linear model parameters are much larger when the transformation is unknown than when it is known. Correspondingly, variances of prediction estimates increase only moderately. However, by considering an equivalent model where the transformed independent variables are standardized, the inflation in variances of the estimates of the parameters in the new model is not as severe, and in the simple regression case, is nil. It is also found that restricting the number of possible transformations to a finite set in general gives no improvement in the performance of the estimates of the parameters. Least squares estimates, robust Huber-type M-estimates, and a hybrid of the two where the transformation is estimated by least squares and the linear model parameters by M, are considered. M-estimates generally perform; much better for heavy tailed error distributions and slightly worse for normal errors.; *This research was supported in part by National Science Foundation Grants MCS78-01422 and MCS81-02349. |
