A Modelling Framework for Regression with Collinearity
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This study addresses a fundamental, yet overlooked, gap between standard theory and empirical modelling practices in the OLS regression model y=Xβ+u with collinearity. In fact, while an estimated model in practice is desired to have stability and efficiency in its "individual OLS estimates", y itself has no capacity to identify and control the collinearity in X and hence no theory including model selection process (MSP) would fill this gap unless X is controlled in view of sampling theory. In this paper, first introducing a new concept of "empirically effective modelling" (EEM), we propose our EEM methodology (EEM-M) as an integrated process of two MSPs with data (y^o,X) given. The first MSP uses X only, called the XMSP, and pre-selects a class D of models with individually inefficiency-controlled and collinearity-controlled OLS estimates, where the corresponding two controlling variables are chosen from predictive standard error of each estimate. Next, defining an inefficiency-collinearity risk index for each model, a partial ordering is introduced onto the set of models to compare without using y^o, where the better-ness and admissibility of models are discussed. The second MSP is a commonly used MSP that uses (y^o,X), and evaluates total model performance as a whole by such AIC, BIC, etc. to select an optimal model from D. Third, to materialize the XMSP, two algorithms are proposed.
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