variance of multiple regression coefficientsIf I consider universal kriging (or multiple spatial regression) in matrix form as: V = X A + R where R is the residual and A are the trend coefficients, then the estimate of A ^ is: A ^ = ( X T C − 1 X ) − 1 X T C − 1 V (as I understand it), where C is the covariance matrix, if it is known. Then, the variance of the coefficients is:VAR( VAR ( A ^ ) = ( X T C − 1 X ) − 1 ???How does one get from the estimate of A ^ , to its variance? i.e. how can I derive that variance?

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variance of multiple regression coefficientsIf I consider universal kriging (or multiple spatial regression) in matrix form as:            V      =      X      A      +      R      where       R   is the residual and       A   are the trend coefficients, then the estimate of                               A          ^                     is:                              A          ^                      =  (            X              T                    C              −        1              X          )              −        1                    X              T                    C              −        1              V  (as I understand it), where       C   is the covariance matrix, if it is known. Then, the variance of the coefficients is:VAR(  VAR  (                              A          ^                      )  =  (            X              T                    C              −        1              X          )              −        1            ???How does one get from the estimate of                               A          ^                    , to its variance? i.e. how can I derive that variance?

datganuhl · Accepted Answer

First, recall that  var  ⁡  (  M  V  )  =  M            (        var  ⁡  (  V  )            )            M    T    .so                          var        ⁡        (        (                  X          T                          C                      −            1                          X                  )                      −            1                                    X          T                          C                      −            1                          V        )                            (1)                                  =        (                  X          T                          C                      −            1                          X                  )                      −            1                                    X          T                          C                      −            1                                                (                          var        ⁡                  V                                      )                                                (                          (                  X          T                          C                      −            1                          X                  )                      −            1                                    X          T                          C                      −            1                                                              )                                T                .            Then, recall that   (  A  B      )    T   (with   A to the left of   B) is equal to       B    T        A    T   (with   A to the right of   B). With       X    T        C          −      1        X, one cannot invert all three matrices and multiply in the opposite order, since   X is not a square matrix. But that matrix is symmetric, i.e. it is its own transpose. And   C is also symmetric, and so is       C          −      1      . So we get:            (        (      X    T        C          −      1        X      )          −      1            X    T        C          −      1                          )              T    =      C          −      1        X  (      X    T        C          −      1        X      )          −      1        .Then   (  1  ) becomes                                  (                  X          T                          C                      −            1                          X                  )                      −            1                                    X          T                          C                      −            1                                                (                          var        ⁡                  V                                      )                                    C                      −            1                          X        (                  X          T                          C                      −            1                          X                  )                      −            1                                                            =        (                  X          T                          C                      −            1                          X                  )                      −            1                                    X          T                          C                      −            1                                                (                          C                              )                                    C                      −            1                          X        (                  X          T                          C                      −            1                          X                  )                      −            1                                                            =        (                  X          T                          C                      −            1                          X                  )                      −            1                                    X          T                          C                      −            1                          X        (                  X          T                          C                      −            1                          X                  )                      −            1                                                            =        (                  X          T                          C                      −            1                          X                  )                      −            1                          .

If I consider universal kriging (or multiple spatial regression) in matrix form as: V=XA+R

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