Find the constrained least-squares estimator for a multiple regression modelConsider the multiple regression model Y = X β + ϵwith the restriction that ∑ l n b i = 1I want to find the least squares estimator of β, so I need to solve the following optimization problem m i n ( Y − X β ) t ( Y − X β ) s . t . ∑ l n b i = 1Let's set L = ( Y − X β ) t ( Y − X β ) − λ ( U t β − 1 ) = Y t Y + β t X t X β + − 2 β t X t Y − λ ( U t β − 1 )where U is a dummy vector of ones (and therefore U T β = ∑ l n b i ).Take derivatives d d β = 2 X t X β − 2 X t Y − λ U t = 0 d d λ = U t β − 1 = 0So from the first equation we can get an expression for β, but what should I do with the λ? The second equation doesn't seem to be useful to get rid of it.

Question

Find the constrained least-squares estimator for a multiple regression modelConsider the multiple regression model  Y  =  X  β  +  ϵwith the restriction that       ∑    l    n        b    i    =  1I want to find the least squares estimator of   β, so I need to solve the following optimization problem  m  i  n  (  Y  −  X  β      )    t    (  Y  −  X  β  )  s  .  t  .      ∑    l    n        b    i    =  1Let&#039;s set  L  =  (  Y  −  X  β      )    t    (  Y  −  X  β  )  −  λ  (      U    t    β  −  1  )  =      Y    t    Y  +      β    t        X    t    X  β  +  −  2      β    t        X    t    Y  −  λ  (      U    t    β  −  1  )where U is a dummy vector of ones (and therefore       U    T    β  =      ∑    l    n        b    i  ).Take derivatives      d          d      β        =  2      X    t    X  β  −  2      X    t    Y  −  λ      U    t    =  0      d          d      λ        =      U    t    β  −  1  =  0So from the first equation we can get an expression for   β, but what should I do with the   λ? The second equation doesn&#039;t seem to be useful to get rid of it.

Cindy Walls · Accepted Answer

First, i think in the first derivative the       U    t   should be corrected by   UThe two derivatives can be written as a linear systems of equations as:      [                            2                      X            t                    X                          −          1                                      1                          0                      ]        [                            β                                      λ                      ]    =      [                            2                      X            t                    Y                                      1                      ]    .Solving these equations could lead to the solutions.

Find the constrained least-squares estimator for a multiple regression model

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