Statistics (Regression Analysis): Show that the residuals from a linear regression model can be expressed as e = ( I − H ) ϵThe bold represents vectors or matrices.I know that e = y − H y So I tried expanding this to, e = X β + ϵ − H X β − H ϵ At this point I can see how to derive the more traditional, e = ( I − H ) y how to solve the original problem?

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Statistics (Regression Analysis): Show that the residuals from a linear regression model can be expressed as   e  =  (  I  −  H  )  ϵThe bold represents vectors or matrices.I know that   e  =  y  −      H    y  So I tried expanding this to,  e  =  X  β  +  ϵ  −      H    X    β    −    H    ϵ  At this point I can see how to derive the more traditional,      e    =    (    I    −    H    )    y  how to solve the original problem?

recepiamsb · Accepted Answer

Recall that   H is an orthogonal projection onto the space spanned by the columns of   X, hence   H  X  β  =  X  β, thus                    e                            =        Y        −                        Y          ^                                                  =        X        β        +        ϵ        −        H        Y                                          =        X        β        +        ϵ        −        H        (        X        β        +        ϵ        )                                          =        X        β        −        X        β        +        ϵ        −        H        ϵ                                          =        (        I        −        H        )        ϵ        .

Statistics (Regression Analysis): Show that the residuals from a linear regression model can be expressed as e=(I−H) epsilon. The bold represents vectors or matrices.

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