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Notes and Comments: Spurious Correlation Due to Deflating Variables

Albert Madansky

Abstract

This paper shows that when a homogeneous linear regression of a normally distributed variable Y on two normally distributed variables X and Z is deflated by Z, and when X and Y are uncorrelated, the deflated dependent variable Y/Z and independent variable X/Z are either uncorrelated or perfectly correlated. Thus, existing approximations to the covariance of these deflated variables are poor. A new approximation to this covariance is given that has the same defect for normally distributed variables, but that should otherwise be better than existing ones.