Fractional principal components regression: a general approach to biased estimators

Abstract

Several biased estimators have been proposed as alternatives to the least squares estimator when multicollinearity is present in the multiple linear regression model. Though the ridge estimator and the principal components estimator have been widely used for such problems, it should be noted that their performances in terms of mean square error are dependent upon the orientation of the unknown parameter vector and the magnitude of σ².

By defining the fractional principal components regression model as

y̲ = Zα̲ + 𝛜̲

= ZF⁻α_F + 𝛜̲

where α_F = Fα̲ and F⁻ is a generalized inverse of a diagonal matrix P, the resulting estimators of α̲_F, based on various forms of F, are shown to define the class of the fractional principal components estimators. In the fractional principal components framework, several new estimation techniques are developed. The performances of the new estimators are evaluated and compared with other commonly used biased estimators both theoretically and by simulation studies.