ecoanuncios7x

## Answered question

2022-09-30

Linear Regression:
$Y=a+bX+ϵ$
For $R$ squared in linear regression, in the form of ratio between $\left({y}_{i}-{y}^{bar}\right)$, or in terms of
$\left({S}_{xy}{\right)}^{2}/\left({S}_{xx}{S}_{yy}\right)$
Not sure if you guys come across this form:
${R}^{2}=\frac{Var\left(bX\right)}{V\left(bX\right)+V\left(ϵ\right)}$?

### Answer & Explanation

Lohre1x

Beginner2022-10-01Added 8 answers

Assume that the process that generates the i.i.d data $\left\{\left({x}_{i},{y}_{i}{\right\}}_{i=1}^{n}$ is ${y}_{i}=a+b{x}_{i}+{ϵ}_{i}$, where $E{ϵ}_{i}=0$ and $E{ϵ}_{i}^{2}={\sigma }^{2}<\mathrm{\infty }$. Now, recall that
${R}^{2}=\frac{\sum \left({\stackrel{^}{y}}_{i}-\overline{y}{\right)}^{2}}{\sum \left({y}_{i}-\overline{y}{\right)}^{2}}=1-\frac{\sum \left({\stackrel{^}{y}}_{i}-{y}_{i}{\right)}^{2}}{\sum \left({y}_{i}-\overline{y}{\right)}^{2}}=1-\frac{\sum \left({\stackrel{^}{y}}_{i}-{y}_{i}{\right)}^{2}/n}{\sum \left({y}_{i}-\overline{y}{\right)}^{2}/n}=1-\frac{{\stackrel{^}{\sigma }}_{ϵ}^{2}}{{\stackrel{^}{\sigma }}_{Y}^{2}}.$
Namely, the sample-based measure ${R}^{2}$ is a biased estimator of a population parameter that is
$\rho =1-\frac{{\sigma }_{ϵ}^{2}}{{\sigma }_{Y}^{2}}.$
So,
$\rho =1-\frac{Var\left({ϵ}_{i}\right)}{Var\left({y}_{i}\right)}=1-\frac{Var\left({ϵ}_{i}\right)}{Var\left(b{x}_{i}\right)+Var\left({ϵ}_{i}\right)}=\frac{Var\left(b{x}_{i}\right)}{Var\left(b{x}_{i}\right)+Var\left({ϵ}_{i}\right)}.$

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