Show that e ^ and β ^ , the residuals and coefficient vector for the OLS problem y = X β + ϵ, are uncorrelated.

Question

Show that       e    ^   and       β    ^  , the residuals and coefficient vector for the OLS problem   y  =  X  β  +  ϵ, are uncorrelated.

Kaeden Lara · Accepted Answer

The vector       e    ^   of residuals has expectation zero. Here&#039;s the proof: By definition,      e    ^    :=  y  −      y    ^    =  y  −  X              β      ^        T    .Recall Theorem stated that  E  (              β      ^        T    )  =      β    0  where       β    0   is such that   E  (  y  )  =  X      β    0  . Since   X is non-stochastic, it follows that  E  (  X              β      ^        T    )  =  X  E  (              β      ^        T    )  =  X      β    0    =  E  (  y  )  .Or you can prove this directly using the definition              β      ^        T    :=  (      X    ′    X      )          −      1            X    ′    y  E  (  X              β      ^        T    )  =  E  (  X  (      X    ′    X      )          −      1            X    ′    y  )        =          [      A      1      ]        X  (      X    ′    X      )          −      1            X    ′    E  (  y  )        =          [      A      2      ]        X  (      X    ′    X      )          −      1            X    ′    X      β    0    =  X      β    0    =  E  (  y  )  .

Show that epsilon and beta, the residuals and coefficient vector for the OLS problem y=X beta+epsilon, are uncorrelated.

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