Manly transformation in quantile regression: A comparison of two transformation parameter estimators

Abstract

This study implements the Manly transformations for normalization of variables in quantile regression analysis.The transformation parameter was estimated using two different methods namely; the maximum likelihood estimation (MLE) method and the two-step estimation method by Chamberlain and Buchinsky(CBTS).The transformation parameters obtained using the two different methods were used for the Manly transformation of data with outliers and data without outliers. The methods were applied to a quantile regression analysis at different quantiles (0.25, 0.50, 0.75, 0.95). Based on our findings, for data without outliers, the 25th quantile model was seen to be the best fit model compared to the other quantiles for the CBTS method with AIC=-43.46279, BIC=20.75212 and MSE=0.70956, while for the MLE the 50th quantile model was seen to be the best fit model with AIC=-348.3657, BIC=20.13548, and MSE=0.00864. Considering data with outliers the 25th quantile model was still seen to be the best fit model compared to the other quantiles for the CBTS method with AIC=-48.5671, BIC=21.8321 and MSE=0.92341, while for the MLE the 50th quantile model was still seen to be the best fit model with AIC=988.6763, BIC=710.09, and MSE=690.7965. Comparison of both methods for data without outliers the study concludes that the estimation of the transformation parameter using the MLE produced better results with lower AIC, BIC and MSE at all quantiles and for data with outliers the study concludes that the estimation of the transformation parameter using CBTS produced better results with lower AIC, BIC and MSE results as is shown in table (3.5) and table (3.6) respectively.