Assume a linear regression model: y i = β 0 β 1 x i 1 +...+ β p x i p + ϵ i C o v ( e , Y ^ )where: e = the residuals vector Y ^ = the predicted vector of YUse the fact that X T e

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Assume a linear regression model:       y    i   =       β    0        β    1        x          i      1       +...+       β    p        x          i      p       +       ϵ    i    C  o  v  (  e  ,      Y    ^    )where:   e = the residuals vector      Y    ^   =  the predicted vector of   YUse the fact that       X    T    e

fairymischiefv9 · Accepted Answer

You can write your estimator:      Y    ^    =  X      β    ^  Therefore, and by rules of   C  o  v, you can take the constant matrix out (right side with transpose):  C  o  v  (  e  ,      Y    ^    )  =  C  o  v  (  e  ,  X      β    ^    )  =  C  o  v  (  e  ,      β    ^    )      X    T  Now you can proove that:  C  o  v  (  e  ,      β    ^    )  =  0Since:  C  o  v  (  e  ,      β    ^    )  =  C  o  v  (  Y  −      Y    ^    ,      β    ^    )  =  C  o  v  (  Y  ,      β    ^    )  −  C  o  v  (      Y    ^    ,      β    ^    )1.   C  o  v  (  Y  ,      β    ^    )  =  C  o  v  (  Y  ,  (      X    T    X      )          −      1            X    T    Y  )  =  C  o  v  (  Y  ,  Y  )  ×  (  (      X    T    X      )          −      1            X    T        )    T    =      σ    2    I  X  (      X    T    X      )          −      1        =      σ    2    X  (      X    T    X      )          −      1      2.  C  o  v  (      Y    ^    ,      β    ^    )  =  C  o  v  (  X      β    ^    ,      β    ^    )  =  X  [  C  o  v  (      β    ^    ,      β    ^    )  ]  =  X  [      σ    2    (      X    T    X      )          −      1        ]  =      σ    2    X  (      X    T    X      )          −      1      Therefore:  C  o  v  (  Y  ,      β    ^    )  −  C  o  v  (      Y    ^    ,      β    ^    )  =      σ    2    X  (      X    T    X      )          −      1        −      σ    2    X  (      X    T    X      )          −      1        =  0Both equations rely on knowledge about the distribution of       β    ^    ∼  N        (    β  ,      σ    2    (      X    T    X      )          −      1              )   and that       Y    ^    =  X      β    ^    =  X  (      X    T    X      )          −      1            X    T    Y

Assume a linear regression model: y_i = beta_0 + beta_1 x_(i1)..... + beta_p x_(ip)+ epsilon_i. Prove that: Cov(e,Y) = 0 where: e = the residuals vector, Y = the predicted vector of Y

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