How can to prove the variance of residuals in simple linear regression? var ⁡ ( r i ) = σ 2 [ 1 − 1 n − ( x i − x ¯ ) 2 ∑ l = 1 n ( x l − x ¯ ) ] my try:using r i = y i − y i ^ var ⁡ ( r i ) = var ⁡ ( y i − y i ^ ) = var ⁡ ( y i − y ¯ ) + var ⁡ ( β 1 ^ ( x i − x ¯ ) ) − 2 Cov ⁡ ( ( y i − y ¯ ) , β 1 ^ ( x i − x ¯ ) )How can I go further?

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How can to prove the variance of residuals in simple linear regression?  var  ⁡  (      r    i    )  =      σ    2        [    1    −          1      n        −                            (                      x            i                    −                              x            ¯                                )            2                                                ∑                          l              =              1                                  n                    (                      x            l                    −                              x            ¯                    )                      ]  my try:using      r    i    =      y    i    −            y      i        ^    var  ⁡  (      r    i    )  =  var  ⁡  (      y    i    −            y      i        ^    )  =  var  ⁡  (      y    i    −      y    ¯    )  +  var  ⁡  (            β      1        ^    (      x    i    −      x    ¯    )  )  −  2  Cov  ⁡  (  (      y    i    −      y    ¯    )  ,            β      1        ^    (      x    i    −      x    ¯    )  )How can I go further?

amilazamiyn · Accepted Answer

since       y    i   and               y      ^        i   are not uncorrelated. Prove this as follows:                    Cov        (        r        )                            =        Cov        (        y        −        P        y        )        ,                P        =        X        (                  X          T                X                  )                      −            1                                    X          T                                                  =        Cov        (        (                  I          n                −        P        )        y        )                                          =        (                  I          n                −        P        )                 Cov        (        y        )                 (                  I          n                −        P                  )          T                                                  =        (                  I          n                −        P        )                           σ          2                          I          n                         (                  I          n                −        P                  )          T                    from which we can conclude that   var  (      r    i    )  =      σ    2    (  1  −      P          i      i        ). It should be quite simple to confirm that your equation is recovered when you let   X be the matrix with a column of   1&#039;s (to represent       x    ¯  ) and a second column of the       x    i  &#039;s.

Jack Ingram · Accepted Answer

Note that                    Var        ⁡        (                  r          i                )                            =        Var        ⁡        (                  y          i                −                                    y            i                    ^                )                                          =        Var        ⁡        (                  y          i                )        +        Var        ⁡        (                                    y            i                    ^                )                                          =                  σ          2                +        Var        ⁡        (                        y          ¯                +                                      β            ^                    1                (                  x          i                −                        x          ¯                )        )                                          =                  σ          2                +        Var        ⁡        (                        y          ¯                )        +        (                  x          i                −                        x          ¯                          )          2                ⋅        Var        ⁡        (                                      β            ^                    1                )                                          =                  σ          2                +                                            σ              2                        n                          +                                                            σ                2                            ⋅              (                              x                i                            −                                          x                ¯                                            )                2                                                                    ∑                                  i                  =                  1                                n                            (                              x                i                            −                                          x                ¯                                            )                2                                                                                      =                  σ          2                          [          1          +                                    1              n                                +                                                    (                                  x                  i                                −                                                x                  ¯                                                  )                  2                                                                              ∑                                      i                    =                    1                                    n                                (                                  x                  i                                −                                                x                  ¯                                                  )                  2                                                              ]

How can I prove the variance of residuals in simple linear regression? var(r_i)=sigma^2[1-1/n-(x_i-x)^2)/(sum_(l=1)^n (x_l-x))

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