In a normal linear model (with intercept), show that if the residuals satisfy e i = a + β x i , for i = 1 … n, where x is a predictor in the model, then each residual is equal to zero.How to do this

Question

In a normal linear model (with intercept), show that if the residuals satisfy       e    i    =  a  +  β      x    i  , for   i  =  1  …  n, where   x is a predictor in the model, then each residual is equal to zero.How to do this

Mckenna Friedman · Accepted Answer

Since your regression model has intercept, we can assume, the   X matrix for the regression has the form  X  =      (                            1                                      x            1            T                                                1                                      x            2            T                                                ⋮                          ⋮                                      1                                      x            n            T                                )  Note that the residual must be orthogonal to every vector in column space of   X. This is because the predicted value,       Y    ^    =      P    X    Y (where       P    X   is the orthogonal projection matrix onto the column space of   X), and hence the residual vector   e  =  Y  −      Y    ^    =  (  I  −      P    X    )  Y. So for any vector   c of appropriate dimension      c    T    e  =  (      c    T    −      c    T        P    X    )  Y  .Now if   c lies in the column space of   X then      P    X    c  =  cor      c    T    =      c    T        P    X    T    =      c    T        P    X         since             P      X         is symmetric  and it follows for any   c in the column space of X  X      c    T    e  =  0.Now, your condition implies   e itself lies in the column space of   X and hence must be orthogonal to itself, i.e.,       e    T    e  =  ‖  e      ‖    2    =  0, i.e.,   e  =  0..

dalllc · Accepted Answer

If our linear model is given by      y    i    =      w    0    +      w    1        x    i    +      e    i  we can substitute       e    i    =      β    0    +      β    1        x    i   to obtain      y    i    =      w    0    +      w    1        x    i    +  (      β    0    +      β    1        x    i    )    ⟹        y    i    =      [          w      0        +          β      0        ]    +      [          w      1        +          β      1        ]        x    i      ⟹        y    i    =              w      ~        o    +              w      ~        1        x    i    +              e      ~        i    ,in which               e      ~        i    =  0. Hence, as the error is not random. The model can capture everything that is present in the error. Residual term               e      ~        i    ≡  0..

In a normal linear model (with intercept), show that if the residuals satisfy e_i=a+beta x_i , then each residual is equal to zero.

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