Question
Show, in the case of a least squares fit to the simple linear regression model$$Y_{i}=\beta_{0}+\beta_{1} x_{i}+\epsilon_{i}, \quad i=1,2, \ldots, n$$that $\sum_{i=1}^{n}\left(y_{i}-\hat{y}_{i}\right)=\sum_{i=1}^{n} e_{i}=0 .$
Step 1
In this model, $Y_{i}$ is the dependent variable, $x_{i}$ is the independent variable, $\beta_{0}$ and $\beta_{1}$ are the parameters of the model, and $\epsilon_{i}$ is the error term. The predicted value of $Y_{i}$ is given by $\hat{y}_{i}=\beta_{0}+\beta_{1} Show more…
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Key Concepts
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Show that in the case of a least squares fit to the simple linear regression model $Y_{i}=a+\beta x_{i}+\epsilon_{i}, \quad i=1,2 \ldots . \mathrm{n}$ that $\sum_{i=1}^{4}\left(y_{i}-\hat{y}_{i}\right)=\sum_{i=1}^{\infty} e_{i}=0$
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Show that, for the simple linear regression model, the following statements are true: (a) $\sum_{i=1}^{n}\left(\mathrm{y}_{i}-\hat{\mathrm{y}}_{i}\right)=0$ (b) $\sum_{i=1}^{n}\left(\mathrm{y}_{i}-\hat{\mathrm{y}}_{i}\right) \mathrm{x}_{i}=0$ (c) $\frac{1}{n} \sum_{i=1}^{n} \hat{\mathrm{y}}_{i}=\bar{y}$
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