Question

Consider the simple linear regression model \[ Y_{i}=\beta_{0}+\beta_{1} x_{i}+\epsilon_{i} \quad i=1,2, \ldots, n \] where the \( \epsilon_{i} \) 's are independent and identically distributed random variables with \( E\left(\epsilon_{i}\right)=0 \). The Sum Squared Error (SSE) is given by \[ \mathrm{SSE}=\sum_{i=1}^{n}\left(y_{i}-\hat{y}_{i}\right)^{2}=\sum_{i=1}^{n}\left(y_{i}-\hat{\beta}_{0}-\hat{\beta}_{1} x_{i}\right)^{2} \] where \( \hat{y}_{i} \) is the predicted value of \( y \) when \( x=x_{i} \). - Derive the following identity: \[ \mathrm{SSE}=S_{y y}-\hat{\beta}_{1} S_{x y} \] where \[ \begin{aligned} S_{y y} & =\sum_{i=1}^{n}\left(y_{i}-\bar{y}\right)^{2} \\ S_{x y} & =\sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)\left(y_{i}-\bar{y}\right) \end{aligned} \] - Consider SSE and \( S_{y y} \), which of the following is correct? \[ \mathrm{SSE} \leq S_{y y} \] or \[ \mathrm{SSE} \geq S_{y y} \] Write down your proof. [Hint: \( \hat{\beta}_{1}=S_{x y} / S_{x x} \); use the above identity]

          Consider the simple linear regression model
\[
Y_{i}=\beta_{0}+\beta_{1} x_{i}+\epsilon_{i} \quad i=1,2, \ldots, n
\]
where the \( \epsilon_{i} \) 's are independent and identically distributed random variables with \( E\left(\epsilon_{i}\right)=0 \).

The Sum Squared Error (SSE) is given by
\[
\mathrm{SSE}=\sum_{i=1}^{n}\left(y_{i}-\hat{y}_{i}\right)^{2}=\sum_{i=1}^{n}\left(y_{i}-\hat{\beta}_{0}-\hat{\beta}_{1} x_{i}\right)^{2}
\]
where \( \hat{y}_{i} \) is the predicted value of \( y \) when \( x=x_{i} \).
- Derive the following identity:
\[
\mathrm{SSE}=S_{y y}-\hat{\beta}_{1} S_{x y}
\]
where
\[
\begin{aligned}
S_{y y} & =\sum_{i=1}^{n}\left(y_{i}-\bar{y}\right)^{2} \\
S_{x y} & =\sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)\left(y_{i}-\bar{y}\right)
\end{aligned}
\]
- Consider SSE and \( S_{y y} \), which of the following is correct?
\[
\mathrm{SSE} \leq S_{y y}
\]
or
\[
\mathrm{SSE} \geq S_{y y}
\]

Write down your proof. [Hint: \( \hat{\beta}_{1}=S_{x y} / S_{x x} \); use the above identity]

Consider the simple linear regression model

Yi=β0+β1 xi+ϵi i=1,2, …, n

where the ϵi 's are independent and identically distributed random variables with E(ϵi)=0.

The Sum Squared Error (SSE) is given by

SSE=∑i=1^n(yi-ŷi)^2=∑i=1^n(yi-β̂0-β̂1 xi)^2

where ŷi is the predicted value of y when x=xi.
- Derive the following identity:

SSE=Sy y-β̂1 Sx y

where

Sy y =∑i=1^n(yi-y̅)^2

Sx y =∑i=1^n(xi-x̅)(yi-y̅)

- Consider SSE and Sy y, which of the following is correct?

SSE≤ Sy y

or

SSE≥ Sy y

Write down your proof. [Hint: β̂1=Sx y / Sx x; use the above identity]

Added by Pranathi C.

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