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Thread: Simple Linear Regression Model

  1. #1
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    Simple Linear Regression Model

    Let a simple linear regression model be:

    $\displaystyle Y_i=\alpha +\beta x_i + e_i \ \ \ (i=1, \cdots ,n)$

    $\displaystyle e_i\sim N(0,\sigma ^{2})$

    Least squares estimator of $\displaystyle \alpha , \beta$ are:

    $\displaystyle \hat{\alpha }=\bar{Y}-\hat{\beta}\bar{x}$, $\displaystyle \hat{\beta}=\frac{S_{(xY)}}{S_{(xx)}}$

    ($\displaystyle S_{(XY)}$ is defined as $\displaystyle \sum_{i=1}^{n}(X_i-\bar{X})(Y_i-\bar{Y})$)

    $\displaystyle \hat{e_i}$ is defined as $\displaystyle Y_i-(\hat{\alpha}+\hat{\beta}x_i)$

    For error sum of squares $\displaystyle SSE=\sum_{i=1}^{n}\hat{e_i}^{2}$ prove the following:

    $\displaystyle E(SSE)=(n-2)\sigma^{2}$

    Can anyone please help me out on this? Thank you.
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  2. #2
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    Re: Simple Linear Regression Model

    Hey piccolo95.

    Hint - What is the distribution of e_i^2? You are finding the expectation of the distribution of the SSE.

    We know e_i ~ N(0,sigma^2) and that Var[e_i] = E[e_i^2] - (E[e_i])^2 but since E[e_i] = 0 it means Var[e_i] = E[e_i^2].

    I'll let you take it from here.
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