On Ranks of Regression Errors and Residuals

Details On Ranks of Regression Errors and Residuals On Ranks of Regression Errors and Residuals

2012-02-09

arxiv.org/abs/1202.1964v2

Mathematics

Statistics Theory

Version 2: Improved section "Heuristics"

Scientific paper

Consider the standard linear regression model $Y = D \theta + \epsilon$ with given design matrix $D$ ($n \times p$), unknown parameter $\theta$ ($p \times 1$) and unobserved error vector $\epsilon$ ($n \times 1$) with i.i.d.\ centered Gaussian components. Motivated by an application in economics, we compare the ranks $R_i$ of the errors $\epsilon_i$ with the ranks $\hat{R}_i$ of the residuals $\hat{\epsilon}_i$, where $\hat{\epsilon} = Y - D \hat{\theta}$ with the least squares estimator $\hat{\theta}$. Exact and approximate formulae are given for the rank distortions $\sqrt{E \, (\hat{R}_i - R_i)^2}$.

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