a Derive the following identity:

Chapter 11, Problem 15E

(choose chapter or problem)

a Derive the following identity:

\(S S E=\sum_{i=1}^{n}\left(y_{i}-\tilde{y}_{i}\right)^{2}=\sum_{i=1}^{n}\left(y_{i}-\tilde{\beta_{0}}-\tilde{\beta}_{1} x_{i}\right)^{2}\)

\(=\sum_{i=1}^{n}\left(y_{i}-\bar{y}\right)^{2}-\widetilde{\beta}_{1} \sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)\left(y_{i}-\bar{y}\right)=S_{y y}-\tilde{\beta_{1}} S_{x y}\)

Notice that this provides an easier computational method of finding SSE.

b Use the computational formula for SSE derived in part (a) to prove that \(S S E \leq S_{w}\). [Hint: \(\tilde{\beta_{1}}=S_{p y} / S_{x x}\) .]

Equation transcription:

Text transcription:

S S E=\sum{i=1}^{n}\left(y{i}-\tilde{y}{i}\right)^{2}=\sum{i=1}^{n}\left(y{i}-\tilde{\beta{0}}-\tilde{\beta}{1} x{i}\right)^{2}

=\sum{i=1}^{n}\left(y{i}-\bar{y}\right)^{2}-\widetilde{\beta}{1} \sum{i=1}^{n}\left(x{i}-\bar{x}\right)\left(y{i}-\bar{y})=S{y y}-\tilde{\beta{1}} S{x y}

S S E \leq S{w}

\tilde{\beta{1}}=S{p y} / S{x x}