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\begin{document}


\begin{center}
\textbf{CLASSICAL MODEL}

\fbox{OLS:} \underline{Ass: $E\left( Y|X\right) =X\beta \qquad V\left(
Y|X\right) =\sigma ^{2}I_{n}\qquad \Pr \left[ rk\left( X^{\prime }X\right) =k%
\right] =1$}

$\widehat{\beta }_{OLS}=\left( X^{\prime }X\right) ^{-1}X^{\prime }Y\qquad 
\hat{Y}=PY\qquad \hat{e}=MY\qquad X^{\prime }\hat{e}=0\qquad $nx $\ \widehat{%
\beta }_{1}=\left( X_{1}^{\prime }M_{2}X_{1}\right) ^{-1}X_{1}^{\prime
}M_{2}Y$

$r=\frac{\sum \left( X_{i}-\overline{X}\right) \left( Y_{i}-\overline{Y}%
\right) }{\sqrt{\sum \left( X_{i}-\overline{X}\right) ^{2}}\sqrt{\sum \left(
Y_{i}-\overline{Y}\right) ^{2}}},$ 1x $\widehat{\beta }_{1}=r_{xy}\frac{s_{y}%
}{s_{x}},$ $R^{2}=r^{2},$ 2x $\ \widehat{\beta }_{2}=\frac{\widehat{\beta }%
_{12}-\widehat{\beta }_{13}\widehat{\beta }_{23}}{1-\widehat{\beta }_{23}%
\widehat{\beta }_{32}},$ $\widehat{\beta }_{ij}=r_{ij}\frac{s_{i}}{s_{j}}%
\qquad $

$R_{UC}^{2}=1-\frac{\widehat{e}^{\prime }\widehat{e}}{Y^{\prime }Y}=\frac{%
\widehat{Y}^{\prime }\widehat{Y}}{Y^{\prime }Y}\in \left[ 0,1\right] \qquad
R^{2}=R_{C}^{2}=1-\frac{\widehat{e}^{\prime }\widehat{e}}{Y^{\prime }M_{0}Y}%
\in \left[ 0,1\right] $ if c $R_{adj}^{2}=1-\left( 1-R^{2}\right) \frac{n-1}{%
n-k}$

$\overline{\frame{Properties}:\Gamma \widehat{\beta }-BLUE:E\left( \widehat{%
\beta }|X\right) =\beta ,Var\left( \widehat{\beta }|X\right) =\sigma
^{2}\left( X^{\prime }X\right) ^{-1},Cov\left( \widehat{\beta },\widehat{e}%
|X\right) =0\qquad }$

\underline{$Var\left( \widehat{\beta }_{1}|X\right) =\sigma ^{2}\left(
X_{1}^{\prime }M_{2}X_{1}\right) ^{-1},\widehat{\sigma }_{OLS}^{2}=\frac{%
\widehat{e}^{\prime }\widehat{e}}{n-k}-$unbiased$,V\left( \widehat{Y_{0}}%
|X\right) =\sigma ^{2}\left( I+X_{0}\left( X^{\prime }X\right)
^{-1}X_{0}^{\prime }\right) $}

Ass: \underline{Joint Normality: $Y|X\thicksim N_{n}\left( X\beta ,\sigma
^{2}I_{n}\right) $}$\qquad \Longrightarrow \qquad $RandSamp: $\left(
X,Y\right) _{i}\thicksim i.i.d.$

$\Gamma \widehat{\beta }|X\thicksim N_{k}\left( \Gamma \beta ,\sigma
^{2}\Gamma \left( X^{\prime }X\right) ^{-1}\Gamma ^{\prime }\right) ,$ $%
W=\left( \Gamma \widehat{\beta }-\Gamma \beta \right) ^{\prime }\left(
\sigma ^{2}\Gamma \left( X^{\prime }X\right) ^{-1}\Gamma ^{\prime }\right)
^{-1}\left( \Gamma \widehat{\beta }-\Gamma \beta \right) \thicksim \chi
^{2}\left( rk\left( \Gamma \right) \right) $

$\frac{\Gamma \widehat{\beta }-\Gamma \beta }{\sqrt{\widehat{\sigma }%
^{2}\Gamma \left( X^{\prime }X\right) ^{-1}\Gamma ^{\prime }}}\thicksim
t\left( rk\left( \Gamma \right) \right) ,$ $\frame{$\frac{1}{p}$}\left(
\Gamma \widehat{\beta }-\Gamma \beta \right) ^{\prime }\left( \widehat{%
\sigma }^{2}\Gamma \left( X^{\prime }X\right) ^{-1}\Gamma ^{\prime }\right)
^{-1}\left( \Gamma \widehat{\beta }-\Gamma \beta \right) \thicksim F\left(
rk\left( \Gamma \right) ,n-k\right) $

\frame{Hyp} SignLevel=$\Pr \left( \text{Rejected}|\text{null}\right) =\Pr
\left( I\text{ error}\right) ,$ \ \ Power=$\Pr \left( \text{Rejected}|\text{%
alter}\right) =1-\Pr \left( II\text{ error}\right) $

\underline{$R:\Gamma \widehat{\beta }=\gamma ,F_{0}=\frac{RSS_{R}-RSS_{U}}{%
RSS_{U}}\frac{n-k}{p}=\frac{R_{U}^{2}-R_{R}^{2}}{1-R_{U}^{2}}\frac{n-k}{p},%
\widehat{\beta }_{R}=\widehat{\beta }_{U}-\left( X^{\prime }X\right)
^{-1}\Gamma ^{\prime }\left( \Gamma \left( X^{\prime }X\right) ^{-1}\Gamma
^{\prime }\right) ^{-1}\left( \Gamma \widehat{\beta }-\Gamma \beta \right) $}

\frame{MLE} $\widehat{\beta }_{ML}=\widehat{\beta }_{OLS}=\left( X^{\prime
}X\right) ^{-1}X^{\prime }Y,$ $attains$ CLRB, $\widehat{\sigma }_{ML}^{2}=%
\frac{\widehat{e}^{\prime }\widehat{e}}{n}$ - biased, does not

$I\left( \beta ,\sigma ^{2}|X\right) ^{-1}=-E\left[ \frac{\partial ^{2}L}{%
\partial \theta \partial \theta ^{\prime }}|X\right] ^{-1}=\left[ 
\begin{array}{c}
\sigma ^{2}\left( X^{\prime }X\right) ^{-1} \\ 
0%
\end{array}%
\begin{array}{c}
0 \\ 
2\sigma ^{4}/n%
\end{array}%
\right] $, $\ \ \sqrt{n}\left( \widehat{\theta }_{ML}-\theta \right) \overset%
{d}{\rightarrow }N\left( 0,\lim \frac{1}{n}I_{n}^{-1}\right) $\\[0pt]
\underline{Tst}:$\frac{W}{n}=\left. h\left( \theta \right) ^{\prime }\left\{ 
\frac{\partial Q}{\partial \theta }^{\prime }\left[ \frac{\partial ^{2}Q}{%
\partial \theta \partial \theta ^{\prime }}\right] ^{-1}\frac{\partial Q}{%
\partial \theta }\right\} ^{-1}h\left( \theta \right) \right\vert _{\widehat{%
\theta }_{U}},\frac{LM}{n}=\left. \frac{\partial Q}{\partial \theta }%
^{\prime }\left[ \frac{\partial ^{2}Q}{\partial \theta \partial \theta
^{\prime }}\right] ^{-1}\frac{\partial Q}{\partial \theta }\right\vert _{%
\widehat{\theta }_{R}},\frac{LR}{n}=2\left( Q\left( \widehat{\theta }%
_{R}\right) -Q\left( \widehat{\theta }_{U}\right) \right) $

\underline{CNLR: $W=n\frac{RSS_{R}-RSS_{U}}{RSS_{U}},LM=n\frac{%
RSS_{R}-RSS_{U}}{RSS_{R}},LR=n\ln \frac{RSS_{R}}{RSS_{U}},LM\leq LR\leq
W,all\thicksim \chi ^{2}\left( rk\left( \Gamma \right) \right) $}

\frame{Deviations} \underline{Multicol}: $Var\left( \widehat{\beta }%
_{j}|X\right) =\sigma ^{2}\left( X_{j}^{\prime }M_{-j}X_{j}\right) ^{-1}=%
\frac{\sigma ^{2}}{\left( 1-R_{j}^{2}\right) \left( X^{\prime }M_{j}X\right) 
}$

\underline{Redund}: no bias \ \underline{Omit}:bias:$\left( X_{1}^{\prime
}X_{1}\right) ^{-1}X_{1}X_{2}\beta _{2}$, Both:Vtrue$\uparrow $,Vest?

\underline{Heter}:$V\left( X\right) ,Var\left( \widehat{\beta }|X\right)
=\sigma ^{2}\left( X^{\prime }X\right) ^{-1}X^{\prime }V\left( X\right)
X\left( X^{\prime }X\right) ^{-1},$ $\widehat{\beta }_{GLS}=\left( X^{\prime
}VX\right) ^{-1}X^{\prime }VY$ - BLUE

\underline{\underline{AR(1)}: $\varepsilon _{i}=\phi \varepsilon
_{i-1}+u_{i}\qquad Y_{i}-\phi Y_{i-1}=\left( X_{i}-\phi X_{i-1}\right) \beta
+u_{i}\qquad Y_{1}=\sqrt{1-\phi ^{2}}X_{1}\beta +u_{1}$}

\textbf{ASYMPTOTICS}

\frame{Asy prop OLS}\underline{Ass.$Y_{i}=X_{i}\beta +\varepsilon _{i},$ $%
\left\{ X_{i},\varepsilon _{i}\right\} i.i.d.,$ $E\left[ X_{i}\varepsilon
_{i}\right] =0,$ $E\left\vert X_{i}\varepsilon _{i}\right\vert <\infty ,$ $%
E\left\vert X_{i}\right\vert ^{2}<\infty $}\underline{ $E\left[
X_{i}^{\prime }X_{i}\right] p.s.d.$} \ 
\end{center}

Consistent:$\widehat{\beta }=\beta +\left( \frac{1}{n}\sum X_{i}^{\prime
}X_{i}\right) ^{-1}\left( \frac{1}{n}\sum X_{i}^{\prime }\varepsilon
_{i}\right) \overset{p}{\rightarrow }\beta ,$ \ \underline{Ass: $%
same+E\left\vert X_{i}\varepsilon _{i}\right\vert ^{2}<\infty ,E\left[
X_{i}^{\prime }X_{i}\varepsilon _{i}^{2}\right] p.s.d.$}

\begin{center}
\underline{$\sqrt{n}\left( \widehat{\beta }-\beta \right) =\sqrt{n}\left( 
\frac{1}{n}\sum X_{i}^{\prime }X_{i}\right) ^{-1}\left( \frac{1}{n}\sum
X_{i}^{\prime }\varepsilon _{i}\right) \overset{d}{\rightarrow }N\left( 0,E%
\left[ X_{i}^{\prime }X_{i}\right] ^{-1}E\left[ X_{i}^{\prime
}X_{i}\varepsilon _{i}^{2}\right] E\left[ X_{i}^{\prime }X_{i}\right]
^{-1}\right) $}
\end{center}

\frame{i.i.d.}: $\widehat{\beta }\overset{A}{\sim }N\left( 0,\frac{1}{n}%
AV\left( \widehat{\beta }\right) \right) ,\frac{1}{n}\widehat{AV\left( 
\widehat{\beta }\right) }=\frac{1}{n}\left[ \frac{1}{n}\sum X_{i}^{\prime
}X_{i}\right] ^{-1}\left[ \frac{1}{n}\sum X_{i}^{\prime }X_{i}\varepsilon
_{i}^{2}\right] \left[ \frac{1}{n}\sum X_{i}^{\prime }X_{i}\right] ^{-1}$-- 
\textbf{White}

$t_{0}=\frac{\Gamma \widehat{\beta }-\Gamma \beta _{0}}{\sqrt{\Gamma \frac{1%
}{n}\widehat{AV\left( \widehat{\beta }\right) }\Gamma ^{\prime }}}\overset{d}%
{\rightarrow }N\left( 0,1\right) \qquad F_{0}=\left( \Gamma \widehat{\beta }%
-\Gamma \beta _{0}\right) ^{\prime }\left( \Gamma \frac{1}{n}\widehat{%
AV\left( \widehat{\beta }\right) }\Gamma ^{\prime }\right) ^{-1}\left(
\Gamma \widehat{\beta }-\Gamma \beta _{0}\right) \overset{d}{\rightarrow }%
\chi ^{2}\left( rk\left( \Gamma \right) \right) $

$W_{0}=\left( \gamma \left( \widehat{\beta }\right) -\gamma \left( \beta
\right) \right) ^{\prime }\left( \Gamma \left( \widehat{\beta }\right) \frac{%
1}{n}\widehat{AV\left( \widehat{\beta }\right) }\Gamma \left( \widehat{\beta 
}\right) ^{\prime }\right) ^{-1}\left( \gamma \left( \widehat{\beta }\right)
-\gamma \left( \beta \right) \right) \overset{d}{\rightarrow }\chi
^{2}\left( rk\left( \Gamma \right) \right) $ -- \textbf{Wald}

\frame{Asy WLS}:\underline{$Y_{i}=X_{i}\beta +\varepsilon _{i},$ $\left\{
X_{i},\varepsilon _{i}\right\} i.i.d.,$ $E\left[ X_{i}|\varepsilon _{i}%
\right] =0,$ $etc$} $\varepsilon _{i}^{2}=Z_{i}\alpha +u_{i},$ $\widehat{%
\varepsilon _{i}^{2}},$

$\widehat{\beta }_{FWLS}=\beta +\left( \frac{1}{n}\sum \frac{X_{i}^{\prime
}X_{i}}{\widehat{\varepsilon _{i}^{2}}}\right) ^{-1}\left( \frac{1}{n}\sum 
\frac{X_{i}^{\prime }\varepsilon _{i}}{\widehat{\varepsilon _{i}^{2}}}%
\right) \overset{p}{\rightarrow }\beta _{WLS}\overset{p}{\rightarrow }\beta $
asy more efficient then OLS\newpage 

\begin{center}
\textbf{EXTREMUM ESTIMATORS}
\end{center}

\frame{Extr.Est}:$\widehat{\theta }=\arg \max_{\theta \in \Theta
}Q_{n}\left( \theta \right) \qquad Lem:$\underline{$\Theta
compact,Q_{n}continuous$ $measurable$}$_{1}\Longrightarrow \exists \widehat{%
\theta }$

$Th1:\Theta Q_{n}same_{1},\underline{\exists Q_{0}\left( \theta \right)
:\exists !\theta _{0}=\arg \max_{\theta \in \Theta }Q_{0}\left( \theta
\right) ,Q_{n}\overset{p}{\rightrightarrows }Q_{0}}_{2}\Longrightarrow 
\widehat{\theta }\overset{p}{\rightarrow }\theta _{0}$

$Th2:\underline{\theta _{0}\in \Theta interior},Q_{n}$\underline{$concave$ }$%
meas,\exists Q_{0}\left( \theta \right) :\exists !\theta _{0}=\arg \max
Q_{0}\left( \theta \right) ,\underline{Q_{n}\overset{p}{\rightarrow }Q_{0}}%
\Longrightarrow \widehat{\theta }\overset{p}{\rightarrow }\theta _{0}$

\begin{center}
\frame{M-type}:$Q_{n}\left( \theta \right) =\frac{1}{n}\sum m\left(
W_{i},\theta \right) $ \ \ 
\end{center}

$Th3:X_{i}i.i.d.,\Theta compact,m\left( W,\theta \right) \in C^{0},E\left[
\sup_{\theta \in \Theta }\left\vert m\left( W_{i},\theta \right) \right\vert %
\right] <\infty \overset{ULLN}{\Longrightarrow }\widehat{\theta }\overset{p}{%
\rightarrow }\theta _{0}$

$Th4:i.i.d.,\theta _{0}\in \Theta int,m\left( W,\theta \right) concave,E%
\left[ \left\vert m\left( W_{i},\theta \right) \right\vert \right] <\infty 
\overset{LLN}{\Longrightarrow }\widehat{\theta }\overset{p}{\rightarrow }%
\theta _{0}$

$Th5:i.i.d.,\theta _{0}\in \Theta int,m\left( W,\theta \right) \in C^{2},%
\widehat{\theta }\overset{p}{\rightarrow }\theta _{0},E\left[ \sup
\left\vert \frac{\partial ^{2}m\left( W_{i},\theta \right) }{\partial \theta
\partial \theta ^{\prime }}\right\vert \right] <\infty ,0<E\left[ \frac{%
\partial ^{2}m\left( W_{i},\theta \right) }{\partial \theta \partial \theta
^{\prime }}\right] <\infty ,$

$\sum \frac{\partial m\left( W_{i},\theta \right) }{\sqrt{n}\partial \theta }%
\overset{d}{\rightarrow }N\left( 0,E\left[ \frac{\partial m\left(
W_{i},\theta \right) }{\partial \theta }\frac{\partial m\left( W_{i},\theta
\right) }{\partial \theta ^{\prime }}\right] p.s.d.\right) \overset{CLT}{%
\Longrightarrow }\sqrt{n}\left( \widehat{\theta }-\theta _{0}\right) \overset%
{d}{\rightarrow }N\left( 0,H_{0}^{-1}\Omega _{0}H_{0}^{-1}\right) $

\qquad \qquad \qquad \qquad \qquad \qquad\ \textbf{NONLINEAR LEAST SQUARES}

\frame{NLS}:\underline{Ass: $Y_{i}=g\left( X_{i},\beta _{0}\right)
+\varepsilon _{i},$ $\left\{ X_{i},\varepsilon _{i}\right\} i.i.d.,$ $E\left[
\varepsilon _{i}|X_{i}\right] =0,$ $\beta \neq \beta _{0}\Longrightarrow
g\left( X_{i},\beta \right) \neq g\left( X_{i},\beta _{0}\right) $}

$m\left( W_{i},\beta \right) =\left( Y_{i}-g\left( X_{i},\beta \right)
\right) ^{2},\frame{$Th1$}.Bcomp,g\left( X,\beta \right) \in C^{0},E\left[
\sup \left\vert \left\vert m\left( W_{i},\beta \right) \right\vert
\right\vert \right] <\infty \Longrightarrow \widehat{\beta }\overset{p}{%
\rightarrow }\beta _{0}$

$s\left( W_{i},\beta \right) =-\frac{\partial g\left( W_{i},\beta \right) }{%
\partial \beta }\left( Y_{i}-g\left( X_{i},\beta \right) \right) ,$ $H\left(
W_{i},\beta \right) =\frac{\partial g\left( W_{i},\beta \right) }{\partial
\beta }\frac{\partial g\left( W_{i},\beta \right) }{\partial \beta }-\frac{%
\partial ^{2}g\left( W_{i},\beta \right) }{\partial \beta \partial \beta
^{\prime }}\left( Y_{i}-g\left( X_{i},\beta \right) \right) $

\frame{$Th2$}$:i.i.d.,\beta _{0}\in Bint,m\left( W,\beta \right) \in C^{2},E%
\left[ \sup \left\vert H\left( W_{i},\beta \right) \right\vert \right]
<\infty ,0<E\left[ s\left( W_{i},\beta _{0}\right) ^{2}\right] <\infty ,$

$0<H_{0}=E\left[ H\left( W_{i},\beta _{0}\right) \right] <\infty ,\widehat{%
\beta }\overset{p}{\rightarrow }\beta _{0}\qquad \overset{CLT}{%
\Longrightarrow }\sqrt{n}\left( \widehat{\beta }-\beta _{0}\right) \overset{d%
}{\rightarrow }N\left( 0,H_{0}^{-1}\Omega _{0}H_{0}^{-1}\right) $

\qquad \qquad \qquad \qquad \qquad \qquad\ \ \ \ \textbf{MAXIMUM LIKELIHOOD}

\frame{ML}:\underline{Ass: $f\left( w_{i},\theta _{0}\right) ,$ $%
w_{i}i.i.d.,m\left( W_{i},\beta \right) =\ln f\left( w_{i},\theta \right) $}
, K-L: $\theta _{0}id\Longrightarrow E\left[ \ln f\left( w_{i},\theta
\right) \right] <E\left[ \ln f\left( w_{i},\theta _{0}\right) \right] $

$Th1:i.i.d.,\Theta compact,\theta _{0}id,\ln f\left( w_{i},\theta \right)
\in C^{0},E\left[ \sup_{\theta \in \Theta }\left\vert \left\vert \ln f\left(
w_{i},\theta \right) \right\vert \right\vert \right] <\infty \Longrightarrow 
\widehat{\theta }_{ML}\overset{p}{\rightarrow }\theta _{0}$

$Th2:i.i.d.,\theta _{0}\in \Theta int,\ln f\left( w_{i},\theta \right) >0\in
C^{2},\int \sup \left\vert \frac{\partial f\left( w_{i},\theta \right) }{%
\partial \theta }\right\vert dw_{i}<\infty ,\int \sup \left\vert \frac{%
\partial ^{2}f\left( w_{i},\theta \right) }{\partial \theta \partial \theta
^{\prime }}\right\vert dw_{i}<\infty ,$

$E\left[ \sup \left\vert \frac{\partial ^{2}\ln f\left( w_{i},\theta \right) 
}{\partial \theta \partial \theta ^{\prime }}\right\vert \right] <\infty
,0<i\left( \theta _{0}\right) =-E\left[ \left( \frac{\partial ^{2}\ln
f\left( w_{i},\theta \right) }{\partial \theta \partial \theta ^{\prime }}%
\right) ^{2}\right] <\infty ,\widehat{\theta }\overset{p}{\rightarrow }%
\theta _{0}$

\begin{center}
$\overset{CLT}{\Longrightarrow }\sqrt{n}\left( \widehat{\theta }_{ML}-\theta
_{0}\right) \overset{d}{\rightarrow }N\left( 0,i\left( \theta _{0}\right)
^{-1}\right) $
\end{center}

$\overline{\frame{TESTS}\frac{W}{n}=\left. h\left( \theta \right) ^{\prime
}\left\{ \frac{\partial h}{\partial \theta }^{\prime }\widehat{V}^{-1}\frac{%
\partial h}{\partial \theta }\right\} ^{-1}h\left( \theta \right)
\right\vert _{\widehat{\theta }_{U}},\frac{LM}{n}=\left. \frac{\partial Q_{n}%
}{\partial \theta }^{\prime }\left[ \frac{\partial ^{2}Q_{n}}{\partial
\theta \partial \theta ^{\prime }}\right] ^{-1}\frac{\partial Q_{n}}{%
\partial \theta }\right\vert _{\widehat{\theta }_{R}},\ \frac{LR}{n}=2\left(
Q_{n}\left( \widehat{\theta }_{R}\right) -Q_{n}\left( \widehat{\theta }%
_{U}\right) \right) }$

\begin{center}
all $\thicksim \chi ^{2}\left( rk\left( \Gamma \right) \right) \qquad $ex: $%
Q_{n}=\frac{1}{n}\sum \ln f\left( w_{i},\theta \right) $

\textbf{SAMPLE SELECTION MODELS}
\end{center}

\frame{Qualit.Resp} $\Pr \left( y_{i}=1|x_{i}\right) =$: LPM: $x_{i}\beta
_{0},$ Probit:$\int_{-\infty }^{x_{i}\beta _{0}}\frac{-e^{-\frac{u^{2}}{2}}}{%
\sqrt{2\pi }}du,$ Logit: $\frac{e^{x_{i}\beta _{0}}}{1+e^{x_{i}\beta _{0}}},$
LogWeib: $e^{-e^{x_{i}\beta _{0}}}$

$E\left[ y_{i}|x_{i}\right] =F\left( x_{i}\beta _{0}\right) ,$ $NLS:\sum
\left( y_{i}-F\left( x_{i}\beta \right) \right) ^{2},$ $WNLS:$ $\sum \frac{%
\left( y_{i}-F\left( x_{i}\beta \right) \right) ^{2}}{F\left( x_{i}\beta
\right) \left( 1-F\left( x_{i}\beta \right) \right) },$

\underline{$ML:\prod F\left( x_{i}\beta \right) ^{y_{i}}\left( 1-F\left(
x_{i}\beta \right) \right) ^{1-y_{i}},$ MargEff: $f\left( \overline{x}\beta
\right) \beta ,$ $\delta $-met. Compare: Logit*0.5513}

\frame{Censored} $y_{i}=\max \left[ x_{i}\beta _{0}+\varepsilon _{i},0\right]
$: $\varepsilon _{i}|x_{i}\thicksim N\left( 0,\sigma _{0}^{2}\right) \qquad E%
\left[ y_{i}|x_{i}\right] =\left[ x_{i}\beta _{0}+\sigma _{0}\frac{\phi
\left( x_{i}\beta _{0}\right) }{\Phi \left( x_{i}\beta _{0}\right) }\right]
\Phi \left( x_{i}\beta _{0}\right) $

$Var\left[ y_{i}|x_{i}\right] =\sigma _{0}^{2}\left[ 1+\frac{x_{i}\beta
_{0}\phi \left( \frac{x_{i}\beta _{0}}{\sigma _{0}}\right) }{\sigma _{0}\Phi
\left( \frac{x_{i}\beta _{0}}{\sigma _{0}}\right) }-\left( \frac{\phi \left( 
\frac{x_{i}\beta _{0}}{\sigma _{0}}\right) }{\Phi \left( \frac{x_{i}\beta
_{0}}{\sigma _{0}}\right) }\right) ^{2}\right] \Phi \left( \frac{x_{i}\beta
_{0}}{\sigma _{0}}\right) ^{2}$

$NLS,$ $WNLS,$ $ML:\prod \left[ 1-\Phi \left( \frac{x_{i}\beta }{\sigma }%
\right) \right] ^{1-d_{i}}\left( \frac{1}{\sigma }\phi \left( \frac{%
y_{i}-x_{i}\beta }{\sigma }\right) \right) ^{d_{i}},Heck2S:probit\widehat{%
\gamma }\rightarrow x+\widehat{\lambda }\left( -x\widehat{\gamma }\right) $

MargEff:expected $\left( 1+\lambda ^{\prime }\left( -x\gamma _{0}\right)
\right) \beta ,$ realiz:$\beta \Phi \left( x\gamma _{0}\right) \qquad $

$\overline{\frame{Truncated}y_{i}=x_{i}\beta _{0}+u_{i},\varepsilon
_{i}|x_{i}\thicksim N\left( 0,\sigma _{0}^{2}\right) \qquad f\left[
y_{i}|x_{i}\right] =\frac{\phi \left( \frac{x_{i}\beta _{0}}{\sigma _{0}}%
\right) }{\sigma _{0}\Phi \left( \frac{x_{i}\beta _{0}}{\sigma _{0}}\right) }%
ML:\prod \frac{\frac{1}{\sigma }\phi \left( \frac{y_{i}-x_{i}\beta }{\sigma }%
\right) }{\Phi \left( \frac{x_{i}\beta }{\sigma }\right) }}$

\end{document}
