By David A. Belsley
Presents practising statisticians and econometricians with new instruments for assessing caliber and reliability of regression estimates. Diagnostic strategies are built that reduction within the systematic situation of information issues which are strange or inordinately influential, and degree the presence and depth of collinear family members one of the regression info and aid to spot variables concerned about each one and pinpoint anticipated coefficients probably so much adversely affected. Emphasizes diagnostics and comprises feedback for remedial motion.
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Presents practising statisticians and econometricians with new instruments for assessing caliber and reliability of regression estimates. Diagnostic suggestions are built that reduction within the systematic situation of knowledge issues which are strange or inordinately influential, and degree the presence and depth of collinear family members one of the regression information and aid to spot variables desirous about each one and pinpoint envisioned coefficients in all likelihood such a lot adversely affected.
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Extra info for Regression Diagnostics: Identifying Influential Data and Sources of Collinearity (Wiley Series in Probability and Statistics)
For smallp, 2p/n tends to call a few too many points to our attention, but it is simple to remember and easy to use. In what follows, then, we call the ith observation a leveragepoint when hi exceeds 2 p / n . The term leverage is reserved for use in this context. Note that when hi= 1, we havepi=yi; that is, ei=O. This is equivalent to saying that, in some coordinate system, one parameter is determined completely byy, or, in effect, dedicated to one data point. A proof of this result is given in Appendix 2A where it is also shown that det[X‘(i)X(i)]=(I -hi)det(XTX).
1 THEORETICAL FOUNDATIONS 11 Finally we consider generalized distance measures (like the Mahalanobis distance) applied to the Z matrix. These distances are computed in a stepwise manner, thus allowing more than one row at a time to be considered. 2. Single-Row Effects We develop techniques here for discovering influential observations? Each observation, of course, is closely associated with a single row of the data matrix X and the corresponding element of y? ) than is the case for most of the other observations.
We again consider the diagnostic technique of row deletion, this time in a comparison of the covariance matrix using all the data, a2(X'X)-', with the covariance matrix that results when the ith row has been deleted, u2[XT(i)X(i)]-'. Of the various alternative means for comparing two such positive-definite symmetric matrices, the ratio of their determinants det[XT(i)X(i)]-'/ det(XTX)-' is one of the simplest and, in the present application, is quite appealing. Since these two matrices differ only by the inclusion of the ith row in the sum of squares and cross products, values of this ratio near unity can be taken to indicate that the two covariance matrices are close, or that the covariance matrix is insensitive to the deletion of row i.