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{\displaystyle {\begin{aligned}&Y_{i}=\beta _{1}+\beta _{2}X_{i}+U_{i}\\&E(U|X)=0\\&Var(U|X)=Var(Y|X)=\sigma ^{2}\\&Y_{i}=\beta _{1}+\beta _{2}X_{i}+U_{i}\\&{\widehat {Y_{i}}}={\widehat {\beta _{1}}}+{\widehat {\beta _{2}}}X_{i}\\&{\widehat {U_{i}}}=Y_{i}-{\widehat {Y_{i}}}\\\end{aligned}}}
Use of
β
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{\displaystyle \beta _{1}}
β
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{\displaystyle \beta _{0}}
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{\displaystyle {\begin{aligned}&{\widehat {\beta _{2}}}={\frac {\sum {(Y_{i}-{\bar {Y}}})(X_{i}-{\bar {X}})}{\sum {(X_{i}-{\bar {X}})^{2}}}}\\&{\widehat {\beta _{1}}}={\bar {Y}}-{\widehat {\beta _{2}}}{\bar {X}}\\&Var({\widehat {\beta _{2}}})={\frac {\sigma ^{2}}{\sum {(X_{i}-{\bar {X}})^{2}}}}\\&Var({\widehat {\beta _{1}}})={\frac {\sum {X_{i}^{2}}*\sigma ^{2}}{n*\sum {(X_{i}-{\bar {X}})^{2}}}}\\&{\widehat {\sigma ^{2}}}={\frac {\sum {\widehat {U_{i}^{2}}}}{n-2}}\\\end{aligned}}}
U and
ϵ
{\displaystyle \epsilon }
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{\displaystyle {\begin{aligned}&S_{2}^{2}={\widehat {Var({\widehat {\beta _{2}}})}}={\frac {\widehat {\sigma ^{2}}}{\sum {(X_{i}-{\bar {X}})^{2}}}}\\&S_{1}^{2}={\widehat {Var({\widehat {\beta _{1}}})}}={\frac {\sum {X_{i}^{2}}*{\widehat {\sigma ^{2}}}}{n*\sum {(X_{i}-{\bar {X}})^{2}}}}\\&S.E.({\widehat {\beta _{2}}})={\sqrt {\widehat {Var({\widehat {\beta _{2}}})}}}\\&S.E.({\widehat {\beta _{1}}})={\sqrt {\widehat {Var({\widehat {\beta _{1}}})}}}\\\end{aligned}}}
S
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{\displaystyle S^{2}}
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{\displaystyle {\begin{aligned}&TSS=\sum {(Y_{i}-{\bar {Y}})^{2}}\\&ESS=\sum {({\widehat {Y_{i}}}-{\bar {Y}})^{2}}\\&RSS=\sum {\widehat {U_{i}^{2}}}\\\end{aligned}}}
TSS may also be presented as SST for the Total Sum of Squares, ESS as SSE (error) and RSS as SSR (residuals).
Depending on the text, ESS and RSS may become very confusing, as there is great variety in the terminology used.
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{\displaystyle R^{2}={\frac {ESS}{TSS}}=1-{\frac {RSS}{TSS}}}
log
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log
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{\displaystyle {\widehat {\log(Y)}}={\widehat {\beta _{1}}}+{\widehat {\beta _{2}}}\log(X)}
