A note on the distribution of the partial correlation coefficient with nonparametrically estimated marginal regressions

AbstractThere has been much interest in the nonparametric testing of conditional indepen-dence in the econometric and statistical literature, but the simplest and potentiallymost useful method, based on the sample partial correlation, seems to have been over-looked, its distribution only having been investigated in some simple parametric in-stances. The present note shows that an easy to apply permutation test based on thesample partial correlation with nonparametrically estimated marginal regressions hasgood large and small sample properties. 1 Introduction Various authors have developed tests of conditional independence without assuming nor-mality of the variables. For example, Kendall (1942), Goodman (1959) and Gripenberg(1992) proposed partial versions of Kendall’s tau. Recently, there has been a focus onincorporating modern nonparametric methods to test for conditional independence (Su W Song, 2009; Huang, 2010; Bouezmarni, Rombouts, & Taamouti, 2010).Conditional independence relations are the building blocks of graphical models, which canbe used to investigate causal relations for economic and other data. Surprisingly however,even though the partial correlation is very well-known, little seems to be known about itssampling distribution unless very strong assumptions are made. The present note fills thisgap in the literature and shows that tests based on the partial correlation are easy to applywhile simulations indicate good small samples properties.Consider the random triple (X,Y,Z), with Y and Z real and X arbitrary, and supposeinterest lies in the question whether Y and Z are conditionally independent given X,denoted Y⊥⊥Z|X. IfY = g(X) +e