Model: y_i=B_0+sum_(i=0)^p B_k X_(ik)+e_i, looking to show the sum of squared residuals is zero if p=(n−1)

roletatx

roletatx

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2022-08-21

Model:
y i = B 0 + i = 0 p B k X i k + e i
show the sum of squared residuals is zero if p = ( n 1 )

Answer & Explanation

Emmy Snow

Emmy Snow

Beginner2022-08-22Added 4 answers

If we write the model as y = β X + ε, then we have
S S R = β ^ X y 2
If we have the same number of regressors as the number of observations, we have that X is a square non-singular matrix, under the standard assumptions. Since
β ^ = ( X X ) 1 X y
we have that
β ^ X = ( X X ) 1 X X y = y
plugging in yields
S S R = β ^ X y 2 = y y 2 = 0

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