Derivando la función de probabilidad para IV-probit

Entonces tengo un modelo binario donde es la variable latente no observada y la observada. determina y es mi instrumento. En resumen, el modelo es. Dado que los términos de error no son independientes, Hago uso de un modelo probit IV. $y_1^*$ $y_1 \in \{0,1\}$ $y_2$ $y_1$ $z_2$

\begin{array}{rcl} y_{1}^{*} & = & δ_{1} z_{1} + α_{1} y_{2} + u_{1} \\ y_{2} & = & δ_{21} z_{1} + δ_{22} z_{2} + v_{2} = z δ + v_{2} \\ y_{1} & = & 1 [y^{*} > 0] \end{array}

$\begin{eqnarray} y_1^*&=& \delta_1 z_1 + \alpha_1 y_2 + u_1 \\ y_2 &=& \delta_{21} z_1 + \delta_{22}z_2 + v_2 = \textbf{z}\delta + v_2 \\ y_1 &=& \text{1}[y^*>0] \end{eqnarray}$

\begin{array}{rcl} (\begin{matrix} u_{1} \\ v_{2} \end{matrix}) \sim N (0, [\begin{matrix} 1 & η \\ η & τ^{2} \end{matrix}]) . \end{array}

$\begin{eqnarray} \begin{pmatrix} u_1 \\ v_2 \end{pmatrix} \sim \mathcal{N} \left(\textbf{0} \; , \begin{bmatrix} 1 &\eta \\ \eta &\tau^2 \end{bmatrix} \right). \nonumber \end{eqnarray}$

Tengo problemas para derivar la función de probabilidad. Entiendo que puedo escribir uno de los términos de error como una función lineal del otro, entonces, y que deben usarse para imponer un CDF normal.

\begin{array}{rcl} u_{1} = \frac{η}{τ^{2}} v_{2} + ξ, where ξ \sim N (0, 1 - η^{2}) . \end{array}

$\begin{eqnarray} u_{1} = \frac{\eta}{\tau^2}v_{2} + \xi, \qquad \text{where} \quad \xi \sim \mathcal{N}(0, 1-\eta^2). \end{eqnarray}$

ξ

$\xi$

He buscado en el manual de Stata ( http://www.stata.com/manuals13/rivprobit.pdf ) IV-probit y sugieren utilizar la definición de la densidad condicional para derivar la función de probabilidad, pero realmente no úsalo (y sí, termino con el resultado incorrecto ...). Mi intento hasta ahora es,

\begin{array}{rcl} f (y_{1}, y_{2} ∣ z) = f (y_{1} ∣ y_{2}, z) f (y_{2} ∣ z) \end{array}

$\begin{eqnarray} f(y_1, y_2 \mid \textbf{z}) = f(y_1 \mid y_2, \textbf{z}) f(y_2 \mid \textbf{z}) \end{eqnarray}$

\begin{array}{rcl} L (y_{1}) & = & \prod_{i = 1}^{n} Pr (y_{1} = 0 ∣ y_{2}, z)^{1 - y_{1}} Pr (y_{1} = 1 ∣ y_{2}, z)^{y_{1}} \\ = & \prod_{i = 1}^{n} Pr (y_{1}^{*} \leq 0)^{1 - y_{1}} (Pr (y_{1}^{*} > 0) f (y_{2} ∣ z))^{y_{1}} \\ [standardizing] & = & \prod_{i = 1}^{n} Pr (\frac{ξ}{\sqrt{1 - η^{2}}} \leq - \frac{δ_{1} z_{1} + α_{1} y_{2} + \frac{η}{τ^{2}} (y_{2} - z)}{\sqrt{1 - η^{2}}})^{1 - y_{1}} \\ \cdot & (Pr (\frac{ξ}{\sqrt{1 - η^{2}}} < \frac{δ_{1} z_{1} + α_{1} y_{2} + \frac{η}{τ^{2}} (y_{2} - z)}{\sqrt{1 - η^{2}}}) f (y_{2} ∣ z))^{y_{1}} \\ = & [1 - Φ (w)]^{1 - y_{i}} {[Φ (w) f (y_{2} ∣ x)]}^{y_{1}} \end{array}

$\begin{eqnarray} \mathcal{L}(y_1) &=& \prod_{i=1}^n \Pr(y_1=0 \mid y_2, \textbf{z} )^{1-y_1} \Pr(y_1=1 \mid y_2, \textbf{z} )^{y_1} \nonumber \\ &=& \prod_{i=1}^n \Pr(y_1^* \leq 0)^{1-y_1} \Big(\Pr(y_1^* > 0) f(y_2 \mid \textbf{z}) \Big)^{y_1} \nonumber \\ \text{[standardizing]} &=& \prod_{i=1}^n \Pr \Big( \frac{\xi}{\sqrt{1-\eta^2}} \leq - \frac{\delta_1 z_1 + \alpha_1 y_2 + \frac{\eta}{\tau^2}(y_2 - \textbf{z})}{\sqrt{1-\eta^2}}\Big)^{1-y_1} \\ &\cdot& \Big(\Pr \Big( \frac{\xi}{\sqrt{1-\eta^2}} < \frac{\delta_1 z_1 + \alpha_1 y_2 + \frac{\eta}{\tau^2}(y_2 - \textbf{z})}{\sqrt{1-\eta^2}}\Big) f(y_2 \mid \textbf{z}) \Big)^{y_1} \nonumber \\ &=& [1-\Phi(w)]^{1-y_i} \left[ \Phi(w)f(y_2 \mid \textbf{x}) \right]^{y_1} \end{eqnarray}$ Como dije, no he usado la definición de la función de densidad articular como se indicó anteriormente. Además, termino con también

f (y_{2} ∣ z)

$f(y_2 \mid \textbf{z})$ elevado a

y_{1}

$y_1$ que parece estar equivocado. ¿Alguien puede darme una pista sobre cómo derivar la función de probabilidad (log-) correcta o dónde me equivoqué?

maximum-likelihood econometrics probit

— Cederlöf
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Recuerde que para una variable normal bivariada la distribución condicional de dado es

(\begin{matrix} X \\ Y \end{matrix}) \sim N ([\begin{matrix} μ_{X} \\ μ_{Y} \end{matrix}], [\begin{matrix} σ_{X}^{2} & ρ σ_{X} σ_{Y} \\ ρ σ_{X} σ_{Y} & σ_{Y}^{2} \end{matrix}]),

$\begin{pmatrix}X \\ Y\end{pmatrix}\sim\mathcal{N}\left(\begin{bmatrix}\mu_X\\\mu_Y\end{bmatrix}, \begin{bmatrix}\sigma_X^2 & \rho\sigma_X\sigma_Y\\\rho\sigma_X\sigma_Y & \sigma_Y^2\end{bmatrix}\right),$

Y

$Y$

X

$X$

Y ∣ X \sim N (μ_{Y} + ρ σ_{Y} \frac{X - μ_{X}}{σ_{X}}, σ_{Y} [1 - ρ^{2}]) .

$Y\mid X \sim \mathcal{N}\left(\mu_Y+\rho\sigma_Y\frac{X-\mu_X}{\sigma_X},\sigma_Y\left[1-\rho^2\right]\right).$

En el presente caso, tenemos que significa que donde (y este fue su primer error)

\begin{aligned} u_{1} ∣ v_{2} & \sim N (0 + \frac{η}{1 \cdot τ} \cdot 1 \frac{v_{2} - 0}{τ}, 1 \cdot [1 - {(\frac{η}{1 \cdot τ})}^{2}]) \\ = N (\frac{η}{τ^{2}} v_{2}, 1 - \frac{η^{2}}{τ^{2}}), \end{aligned}

$\begin{align} u_1 \mid v_2 &\sim \mathcal{N}\left(0+\frac{\eta}{1\cdot\tau}\cdot1\frac{v_2-0}{\tau}, 1\cdot\left[1-\left(\frac{\eta}{1\cdot\tau}\right)^2\right] \right) \\ &= \mathcal{N}\left(\frac{\eta}{\tau^2}v_2, 1-\frac{\eta^2}{\tau^2} \right), \end{align}$

u_{1} = \frac{η}{τ^{2}} v_{2} + ξ

$u_1=\frac{\eta}{\tau^2}v_2+\xi$

ξ \sim N (0, 1 - \frac{η^{2}}{τ^{2}}) .

$\xi\sim\mathcal{N}\left(0,1-\frac{\eta^2}{\tau^2}\right).$

Así podemos reescribir la primera ecuación

\begin{aligned} y_{1}^{*} & = δ_{1} z_{1} + α_{1} y_{2} + u_{1} \\ = δ_{1} z_{1} + α_{1} y_{2} + \frac{η}{τ^{2}} v_{2} + ξ \\ = δ_{1} z_{1} + α_{1} y_{2} + \frac{η}{τ^{2}} (y_{2} - z δ) + ξ . \end{aligned}

$\begin{align} y_1^* &= \delta_1 z_1 + \alpha_1 y_2 + u_1 \\ &= \delta_1 z_1 + \alpha_1 y_2 + \frac{\eta}{\tau^2}v_2+\xi \\ &= \delta_1 z_1 + \alpha_1 y_2 + \frac{\eta}{\tau^2}(y_2-\textbf{z}\delta)+\xi. \end{align}$

Ahora, recuerde que la función de densidad de probabilidad condicional de dado es $X=x$ $Y=y$

f_{X} (x ∣ y) = \frac{f_{X Y} (x, y)}{f_{Y} (y)} .

$f_{X}(x \mid y)=\frac{f_{XY}(x,y)}{f_{Y}(y)}.$

En el presente caso, tenemos que se puede reorganizar a su expresión

f_{1} (y_{1} ∣ y_{2}, z) = \frac{f_{12} (y_{1}, y_{2} ∣ z)}{f_{2} (y_{2} ∣ z)},

$f_{1}(y_1 \mid y_2, \mathbf{z})=\frac{f_{12}(y_1,y_2 \mid \mathbf{z})}{f_{2}(y_2 \mid \mathbf{z})},$

f_{12} (y_{1}, y_{2} ∣ z) = f_{1} (y_{1} ∣ y_{2}, z) f_{2} (y_{2} ∣ z) .

$f_{12}(y_1, y_2 \mid \mathbf{z})= f_{1}(y_1 \mid y_2, \mathbf{z})f_{2}(y_2 \mid \mathbf{z}).$

Luego, podemos escribir la probabilidad en función de las densidades de los dos shocks independientes : $v_1,\xi_1$

\begin{aligned} L (y_{1}, y_{2} ∣ z) & = \prod_{i}^{n} f_{1} (y_{1 i} ∣ y_{2 i}, z_{i}) f_{2} (y_{2 i} ∣ z_{i}) \\ = \prod_{i}^{n} Pr {(y_{1 i} = 1)}^{y_{1 i}} Pr {(y_{1 i} = 0)}^{1 - y_{1 i}} f_{2} (y_{2 i} ∣ z_{i}) \\ = \prod_{i}^{n} Pr {(y_{1 i}^{*} > 0)}^{y_{1 i}} Pr {(y_{1 i}^{*} \leq 0)}^{1 - y_{1 i}} f_{2} (y_{2 i} ∣ z_{i}) \\ = \prod_{i}^{n} Pr {(δ_{1} z_{1 i} + α_{1} y_{2 i} + \frac{η}{τ^{2}} (y_{2 i} - z_{i} δ) + ξ_{i} > 0)}^{y_{1 i}} \\ Pr {(δ_{1} z_{1 i} + α_{1} y_{2 i} + \frac{η}{τ^{2}} (y_{2 i} - z_{i} δ) + ξ_{i} \leq 0)}^{1 - y_{1 i}} \\ f_{2} (y_{2 i} ∣ z_{i}) \\ = \prod_{i}^{n} Pr {(ξ_{i} > - [δ_{1} z_{1 i} + α_{1} y_{2 i} + \frac{η}{τ^{2}} (y_{2 i} - z_{i} δ)])}^{y_{1 i}} \\ Pr {(ξ_{i} \leq - [δ_{1} z_{1 i} + α_{1} y_{2 i} + \frac{η}{τ^{2}} (y_{2 i} - z_{i} δ)])}^{1 - y_{1 i}} \\ f_{2} (y_{2 i} ∣ z_{i}) \\ = \prod_{i}^{n} Pr {(\frac{ξ_{i} - 0}{\sqrt{1 - \frac{η^{2}}{τ^{2}}}} > - \frac{δ_{1} z_{1 i} + α_{1} y_{2 i} + \frac{η}{τ^{2}} (y_{2 i} - z_{i} δ) + 0}{\sqrt{1 - \frac{η^{2}}{τ^{2}}}})}^{y_{1 i}} \\ Pr {(\frac{ξ_{i} - 0}{\sqrt{1 - \frac{η^{2}}{τ^{2}}}} \leq - \frac{δ_{1} z_{1 i} + α_{1} y_{2 i} + \frac{η}{τ^{2}} (y_{2 i} - z_{i} δ) + 0}{\sqrt{1 - \frac{η^{2}}{τ^{2}}}})}^{1 - y_{1 i}} \\ f_{2} (y_{2 i} ∣ z_{i}) \\ = \prod_{i}^{n} Pr {(\frac{ξ_{i}}{\sqrt{1 - \frac{η^{2}}{τ^{2}}}} > - w_{i})}^{y_{1 i}} Pr {(\frac{ξ_{i}}{\sqrt{1 - \frac{η^{2}}{τ^{2}}}} \leq - w_{i})}^{1 - y_{1 i}} f_{2} (y_{2 i} ∣ z_{i}) \\ = \prod_{i}^{n} {[1 - Pr (\frac{ξ_{i}}{\sqrt{1 - \frac{η^{2}}{τ^{2}}}} \leq - w_{i})]}^{y_{1 i}} Pr {(\frac{ξ_{i}}{\sqrt{1 - \frac{η^{2}}{τ^{2}}}} \leq - w_{i})}^{1 - y_{1 i}} f_{2} (y_{2 i} ∣ z_{i}) \\ = \prod_{i} {[1 - Φ (- w_{i})]}^{y_{1 i}} Φ (- w_{i})^{1 - y_{1 i}} φ (\frac{y_{2 i} - z_{i} δ}{τ}) \\ = \prod_{i}^{n} Φ (w_{i})^{y_{1 i}} {[1 - Φ (w_{i})]}^{1 - y_{1 i}} φ (\frac{y_{2 i} - z_{i} δ}{τ}) \\ = Φ (w)^{y_{1}} {[1 - Φ (w)]}^{1 - y_{1}} φ (\frac{y_{2} - z δ}{τ}) \end{aligned}

$\begin{align} \mathcal{L}(y_1,y_2\mid \mathbf{z}) &= \prod_i^n f_{1}(y_{1i} \mid y_{2i}, \mathbf{z}_i)f_{2}(y_{2i} \mid \mathbf{z}_i) \\ &= \prod_i^n \Pr\left(y_{1i}=1\right)^{y_{1i}}\Pr\left(y_{1i}=0\right)^{1-y_{1i}}f_{2}(y_{2i} \mid \mathbf{z}_i) \\ &= \prod_i^n \Pr\left(y_{1i}^*>0\right)^{y_{1i}}\Pr\left(y_{1i}^*\leq0\right)^{1-y_{1i}}f_{2}(y_{2i} \mid \mathbf{z}_i) \\ &= \prod_i^n \Pr\left(\delta_1 z_{1i} + \alpha_1 y_{2i} + \frac{\eta}{\tau^2}(y_{2i}-\textbf{z}_{i}\delta)+\xi_i>0\right)^{y_{1i}}\\ &\qquad\quad \Pr\left(\delta_1 z_{1i} + \alpha_1 y_{2i} + \frac{\eta}{\tau^2}(y_{2i}-\textbf{z}_i\delta)+\xi_i\leq0\right)^{1-y_{1i}}\\ &\qquad\quad f_{2}(y_{2i} \mid \mathbf{z}_i) \\ &= \prod_i^n \Pr\left(\xi_i>-\left[\delta_1 z_{1i} + \alpha_1 y_{2i} + \frac{\eta}{\tau^2}(y_{2i}-\textbf{z}_i\delta)\right]\right)^{y_{1i}}\\ &\qquad\quad \Pr\left(\xi_i\leq-\left[\delta_1 z_{1i} + \alpha_1 y_{2i} + \frac{\eta}{\tau^2}(y_{2i}-\textbf{z}_i\delta)\right]\right)^{1-y_{1i}}\\ &\qquad\quad f_{2}(y_{2i} \mid \mathbf{z}_i) \\ &= \prod_i^n \Pr\left(\frac{\xi_i-0}{\sqrt{1-\frac{\eta^2}{\tau^2}}}>-\frac{\delta_1 z_{1i} + \alpha_1 y_{2i} + \frac{\eta}{\tau^2}(y_{2i}-\textbf{z}_i\delta)+0}{\sqrt{1-\frac{\eta^2}{\tau^2}}}\right)^{y_{1i}}\\ &\qquad\quad \Pr\left(\frac{\xi_i-0}{\sqrt{1-\frac{\eta^2}{\tau^2}}}\leq-\frac{\delta_1 z_{1i} + \alpha_1 y_{2i} + \frac{\eta}{\tau^2}(y_{2i}-\textbf{z}_i\delta)+0}{\sqrt{1-\frac{\eta^2}{\tau^2}}}\right)^{1-y_{1i}}\\ &\qquad\quad f_{2}(y_{2i} \mid \mathbf{z}_i) \\ &= \prod_i^n \Pr\left(\frac{\xi_i}{\sqrt{1-\frac{\eta^2}{\tau^2}}}>-w_i\right)^{y_{1i}} \Pr\left(\frac{\xi_i}{\sqrt{1-\frac{\eta^2}{\tau^2}}}\leq-w_i\right)^{1-y_{1i}} f_{2}(y_{2i} \mid \mathbf{z}_i) \\ &= \prod_i^n \left[1-\Pr\left(\frac{\xi_i}{\sqrt{1-\frac{\eta^2}{\tau^2}}}\leq-w_i\right)\right]^{y_{1i}} \Pr\left(\frac{\xi_i}{\sqrt{1-\frac{\eta^2}{\tau^2}}}\leq-w_i\right)^{1-y_{1i}} f_{2}(y_{2i} \mid \mathbf{z}_i) \\ &= \prod_i \left[1-\Phi(-w_i)\right]^{y_{1i}} \Phi(-w_i)^{1-y_{1i}} \varphi\left(\frac{y_{2i}-\mathbf{z}_i\delta}{\tau}\right) \\ &= \prod_i^n \Phi(w_i)^{y_{1i}} \left[1-\Phi(w_i)\right]^{1-y_{1i}} \varphi\left(\frac{y_{2i}-\mathbf{z}_i\delta}{\tau}\right) \\ &= \Phi(w)^{y_{1}} \left[1-\Phi(w)\right]^{1-y_{1}} \varphi\left(\frac{y_{2}-\mathbf{z}\delta}{\tau}\right) \\ \end{align}$ donde y son la función de densidad acumulativa y la función de densidad de probabilidad de la distribución normal estándar.

\begin{aligned} w_{i} = \frac{δ_{1} z_{1 i} + α_{1} y_{2 i} + \frac{η}{τ^{2}} (y_{2 i} - z_{i} δ)}{\sqrt{1 - \frac{η^{2}}{τ^{2}}}} . \end{aligned}

$\begin{align} w_i = \frac{\delta_1 z_{1i} + \alpha_1 y_{2i} + \frac{\eta}{\tau^2}(y_{2i}-\textbf{z}_i\delta)}{\sqrt{1-\frac{\eta^2}{\tau^2}}}. \end{align}$

Φ (z)

$\Phi(z)$

φ (z)

$\varphi(z)$

— Fredrik P
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