Question

According to (1.17), lei = 0 when regression model (1.1) is fitted to a set of n cases by the method of least squares. is it also true that l e:i = o? comment.

Answer 1

solution:

\sum ei =0

that is  

\sum (yi-yiˆ)=0

means,

\sumyi = \sum yiˆ

yi = \alpha +\beta xi+ei

yiˆ=\alphaˆ+\betaˆxi

so,

\sumei=0

hence proved

Answer 2

Answer:

Properties of Fitted Regression Line

Step-by-step explanation:

We know that,

$\sum{e_{i} } = 0$

In turn we understand that

$e_{i}= Y_{i}-Y'_{i}$

The third property of Fitted Regression Line tells us that: The sum of the observed values $Y_{i}$ equals the sum of the fitted values $Y'_{i}$, so:

$\sum{Y_{i}} = \sum{Y'_{i}}$ (1)

We further understand that the values given for $Y_{i}$, is equivalent to:

$Y_{i}= \beta_{0} + \beta_{1}X_{i}+\epsilon_{i}$ (2)

On the other hand for the definition of the value for the regression function of $Y'_{i}$ is,

$Y'_{i}= \beta_{0}+\beta_{1}X_{i}$ (3)

By replacing (3) and (2) in (1), we get that

$\sum{ (\beta_{0} + \beta_{1}X_{i}+\epsilon_{i})} = \sum{(\beta_{0}+\beta_{1}X_{i})}$

Since the sum is distributive

$\sum \beta_{0} + \sum \beta_{1}X_{i}+\sum \epsilon_{i} = \sum\beta_{0}+ \sum \beta_{1}X_{i}$

Equal values on opposite sides of an equation are canceled, we get that

$\sum \epsilon_{i} = 0$