Suppose we have model Y = X β and first collumn of X consists of 1 1. Why the sum of residuals in the this regression model equals 0? And why in generall it doesn't, when there is no constant term?

Question

Suppose we have model   Y  =  X  β and first collumn of   X consists of 1  1. Why the sum of residuals in the this regression model equals   0? And why in generall it doesn&#039;t, when there is no constant term?

Zariah Fletcher · Accepted Answer

Consider the linear model of a form   Y  =      β    0    +      ∑          j      =      1        p        β    j        x    j    +  ϵ, so to find the OLS estimator you construct      min          β      ∈          B            ∑          i      =      1        n    (      Y    i    −      β    0    +      ∑          j      =      1        p        β    j        x    j        )    2    ,then when you take the derivative w.r.t the intercept term       β    0  , you get the following expression  −  2      ∑          i      =      1        n    (      Y    i    −            β      0        ^    +      ∑          j      =      1        p                β      ^        j        x    j    )  =  2      ∑          i      =      1        n        e    i    =  0.Or, using matrix notations, you get the normal equations that your   β solves, i.e.,      X    ′    X      β    ^    −      X    ′    y  =  0  ,namely,      X    ′    (  X      β    ^    −  y  )  =      X    ′    e  =  0  ,where the first row is      1    T    e  =      ∑          i      =      1        n        e    i

Suppose we have model Y=X beta and first collumn of X consists of 1. Why the sum of residuals in the this regression model equals 0? And why in generall it doesn't, when there is no constant term?

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