Hat Matrix x Values

Certain values can have more weight in the determination of a regression curve than do other values. We covered this in linear regression (Chapter 3), and the application to multiple linear regression is straightforward. In Chapter 3, 

H also can be used to provide the error term vector, e, where e = (I H)Y, and I is an X identity matrix, or the variance-covariance matrix, (r (e), where 

The elements of ha of the hat matrix, H, also provide an estimate of the leverage exerted by the fth row and the fth column value. By further manipulation, the actual leverage of any particular value set can be known. 

Our general focus will be on the diagonal elements, ha, of the hat matrix, where 

Because the hat matrix is relevant only for the location of observations in X, space, many researchers will use Studentized residual values, S, , in relation to hu values, looking for observations with both large values and large A values. These will be values likely to be strong leverage points. 

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