Showing sum of squared residuals is zero?

Question

I have the model $$y_i = B_0+\sum\limits_{i=0}^pB_kX_{ik} + e_i$$ I'm looking to show the sum of squared residuals is zero if $p = (n-1)$. I have tried expanding it quite in depth and I haven't been able to come anywhere near a solution.

Answer 1

If we write the model as $y= \beta X + \varepsilon$, then we have $$SSR = \|\hat{\beta}X - y\|^2$$ 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 $$\hat{\beta} = (X'X)^{-1}X'y$$ we have that $$\hat{\beta}X = (X'X)^{-1}X'Xy=y$$ plugging in yields $$SSR=\|\hat{\beta}X-y\|^2=\|y-y\|^2=0$$