Memperoleh fungsi kemungkinan untuk IV-probit

Jadi saya memiliki model biner di mana adalah variabel laten yang tidak teramati dan yang diamati. menentukan dan adalah instrumen saya. Jadi singkatnya modelnya. Karena istilah kesalahan tidak independen tetapi, Saya menggunakan model IV-probit. $y_1^*$ $y_1 \in \{0,1\}$ $y_2$ $y_1$ $z_2$

\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}

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

Saya mengalami kesulitan memperoleh fungsi kemungkinan. Saya mengerti bahwa saya dapat menulis salah satu istilah kesalahan sebagai fungsi linear dari yang lain sehingga, dan bahwa harus digunakan untuk memaksakan 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

$\xi$

Saya telah mencari di manual Stata ( http://www.stata.com/manuals13/rivprobit.pdf ) untuk IV-probit dan mereka menyarankan menggunakan definisi kepadatan bersyarat untuk mendapatkan fungsi likelihood tetapi saya benar-benar tidak gunakan itu (dan ya saya berakhir dengan hasil yang salah ...). Upaya saya sejauh ini adalah,

\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}

$\begin{eqnarray} \mathcal{L}(y_1) &=& \prod_{i=1}^n \Pr(y_1=0 \mid y_2, \textbf{z} )^{1-y_1} \Pr(y_1=1 \mid y_2, \textbf{z} )^{y_1} \nonumber \\ &=& \prod_{i=1}^n \Pr(y_1^* \leq 0)^{1-y_1} \Big(\Pr(y_1^* > 0) f(y_2 \mid \textbf{z}) \Big)^{y_1} \nonumber \\ \text{[standardizing]} &=& \prod_{i=1}^n \Pr \Big( \frac{\xi}{\sqrt{1-\eta^2}} \leq - \frac{\delta_1 z_1 + \alpha_1 y_2 + \frac{\eta}{\tau^2}(y_2 - \textbf{z})}{\sqrt{1-\eta^2}}\Big)^{1-y_1} \\ &\cdot& \Big(\Pr \Big( \frac{\xi}{\sqrt{1-\eta^2}} < \frac{\delta_1 z_1 + \alpha_1 y_2 + \frac{\eta}{\tau^2}(y_2 - \textbf{z})}{\sqrt{1-\eta^2}}\Big) f(y_2 \mid \textbf{z}) \Big)^{y_1} \nonumber \\ &=& [1-\Phi(w)]^{1-y_i} \left[ \Phi(w)f(y_2 \mid \textbf{x}) \right]^{y_1} \end{eqnarray}$ Seperti yang saya katakan, saya belum menggunakan definisi untuk fungsi kerapatan sambungan seperti yang disebutkan di atas. Selain itu, saya berakhir dengan juga dinaikkan ke yang tampaknya salah. Adakah yang bisa memberi saya petunjuk tentang cara mendapatkan fungsi kemungkinan (log-) yang benar atau di mana saya salah?

f (y_{2} ∣ z)

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

y_{1}

$y_1$

maximum-likelihood econometrics probit

— Cederlöf
sumber

Ingat bahwa untuk variabel normal bivariat distribusi kondisional diberikan adalah

(\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}]) .

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

Dalam kasus ini, kita memiliki yang berarti mana (dan ini adalah kesalahan pertama Anda)

\begin{aligned} {kamu}_{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}

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

{kamu}_{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}}) .

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

Dengan demikian, kita dapat menulis ulang persamaan pertama

\begin{aligned} y_{1}^{*} & = δ_{1} z_{1} + α_{1} y_{2} + {kamu}_{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}

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

Sekarang, ingat bahwa fungsi kepadatan probabilitas bersyarat dari diberikan adalah $X=x$ $Y=y$

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

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

Dalam kasus ini, kita memiliki yang dapat diatur ulang ke ekspresi Anda

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) .

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

Kemudian, kita dapat menulis kemungkinan sebagai fungsi dari kepadatan dari dua kejutan independen : $v_1,\xi_1$

\begin{aligned} L. (y_{1}, y_{2} ∣ z) & = \prod_{saya}^{n} f_{1} (y_{1 saya} ∣ y_{2 saya}, z_{saya}) f_{2} (y_{2 saya} ∣ z_{saya}) \\ = \prod_{saya}^{n} Pr {(y_{1 saya} = 1)}^{y_{1 saya}} Pr {(y_{1 saya} = 0)}^{1 - y_{1 saya}} f_{2} (y_{2 saya} ∣ z_{saya}) \\ = \prod_{saya}^{n} Pr {(y_{1 saya}^{*} > 0)}^{y_{1 saya}} Pr {(y_{1 saya}^{*} \leq 0)}^{1 - y_{1 saya}} f_{2} (y_{2 saya} ∣ z_{saya}) \\ = \prod_{saya}^{n} Pr {(δ_{1} z_{1 saya} + α_{1} y_{2 saya} + \frac{η}{τ^{2}} (y_{2 saya} - z_{saya} δ) + ξ_{saya} > 0)}^{y_{1 saya}} \\ Pr {(δ_{1} z_{1 saya} + α_{1} y_{2 saya} + \frac{η}{τ^{2}} (y_{2 saya} - z_{saya} δ) + ξ_{saya} \leq 0)}^{1 - y_{1 saya}} \\ f_{2} (y_{2 saya} ∣ z_{saya}) \\ = \prod_{saya}^{n} Pr {(ξ_{saya} > - [δ_{1} z_{1 saya} + α_{1} y_{2 saya} + \frac{η}{τ^{2}} (y_{2 saya} - z_{saya} δ)])}^{y_{1 saya}} \\ Pr {(ξ_{saya} \leq - [δ_{1} z_{1 saya} + α_{1} y_{2 saya} + \frac{η}{τ^{2}} (y_{2 saya} - z_{saya} δ)])}^{1 - y_{1 saya}} \\ f_{2} (y_{2 saya} ∣ z_{saya}) \\ = \prod_{saya}^{n} Pr {(\frac{ξ_{saya} - 0}{\sqrt{1 - \frac{η^{2}}{τ^{2}}}} > - \frac{δ_{1} z_{1 saya} + α_{1} y_{2 saya} + \frac{η}{τ^{2}} (y_{2 saya} - z_{saya} δ) + 0}{\sqrt{1 - \frac{η^{2}}{τ^{2}}}})}^{y_{1 saya}} \\ Pr {(\frac{ξ_{saya} - 0}{\sqrt{1 - \frac{η^{2}}{τ^{2}}}} \leq - \frac{δ_{1} z_{1 saya} + α_{1} y_{2 saya} + \frac{η}{τ^{2}} (y_{2 saya} - z_{saya} δ) + 0}{\sqrt{1 - \frac{η^{2}}{τ^{2}}}})}^{1 - y_{1 saya}} \\ f_{2} (y_{2 saya} ∣ z_{saya}) \\ = \prod_{saya}^{n} Pr {(\frac{ξ_{saya}}{\sqrt{1 - \frac{η^{2}}{τ^{2}}}} > - w_{saya})}^{y_{1 saya}} Pr {(\frac{ξ_{saya}}{\sqrt{1 - \frac{η^{2}}{τ^{2}}}} \leq - w_{saya})}^{1 - y_{1 saya}} f_{2} (y_{2 saya} ∣ z_{saya}) \\ = \prod_{saya}^{n} {[1 - Pr (\frac{ξ_{saya}}{\sqrt{1 - \frac{η^{2}}{τ^{2}}}} \leq - w_{saya})]}^{y_{1 saya}} Pr {(\frac{ξ_{saya}}{\sqrt{1 - \frac{η^{2}}{τ^{2}}}} \leq - w_{saya})}^{1 - y_{1 saya}} f_{2} (y_{2 saya} ∣ z_{saya}) \\ = \prod_{saya} {[1 - Φ (- w_{saya})]}^{y_{1 saya}} Φ (- w_{saya})^{1 - y_{1 saya}} φ (\frac{y_{2 saya} - z_{saya} δ}{τ}) \\ = \prod_{saya}^{n} Φ (w_{saya})^{y_{1 saya}} {[1 - Φ (w_{saya})]}^{1 - y_{1 saya}} φ (\frac{y_{2 saya} - z_{saya} 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}$ di mana dan adalah fungsi kerapatan kumulatif dan fungsi kerapatan probabilitas dari distribusi normal standar.

\begin{aligned} w_{saya} = \frac{δ_{1} z_{1 saya} + α_{1} y_{2 saya} + \frac{η}{τ^{2}} (y_{2 saya} - z_{saya} 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
sumber