$$
N(d) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^d \exp(-t^2/2) \;dt
$$

$$
x=\pi \,k-\frac{\pi }{4}\mathrm{}\mathrm{\ \ for\ all\ \ }\mathrm{}k\mathrm{}\mathrm{\ \ with\ \ }\mathrm{}k\in \mathbb{Z}\wedge 1\le k
$$

$$
\left[\begin{array}{l}
m\frac{{{d^2}x}}{{d{t^2}}} = T\left( t \right)\frac{{x\left( t \right)}}{r}\\
m\frac{{{d^2}y}}{{d{t^2}}} = T\left( t \right)\frac{{y\left( t \right)}}{r} - mg\\
x{\left( t \right)^2} + y{\left( t \right)^2} = {r^2}
\end{array}\right]
$$

$$
-\frac{\mathrm{sin}\left(x\right)}{2\,{\mathrm{sin}\left(\frac{x}{2}\right)}^2 -1}
$$

$$
\begin{array}{l}
\left(\begin{array}{ccc}
\frac{b-1}{{\left(a-1\right)}\,\sigma_2 } & 0 & 0\\
\frac{1}{\sigma_2 } & 0 & 0\\
0 & 0 & 0\\
0 & 0 & 1\\
0 & \frac{f-1}{\sigma_1 \,{\left(e-1\right)}} & 0\\
0 & \frac{1}{\sigma_1 } & 0
\end{array}\right)\\
\mathrm{}\\
\textrm{where}\\
\mathrm{}\\
\;\;\sigma_1 =\sqrt{\frac{{{\left(f-1\right)}}^2 }{{{\left(e-1\right)}}^2 }+1}\\
\mathrm{}\\
\;\;\sigma_2 =\sqrt{\frac{{{\left(b-1\right)}}^2 }{{{\left(a-1\right)}}^2 }+1}
\end{array}
$$

$$
\begin{array}{l}
\left(\begin{array}{ccc}
\frac{b-1}{{\left(a-1\right)}\,\sigma_2 } & 0 & 0\\
\frac{1}{\sigma_2 } & 0 & 0\\
0 & 0 & 0\\
0 & 0 & 1\\
0 & \frac{f-1}{\sigma_1 \,{\left(e-1\right)}} & 0\\
0 & \frac{1}{\sigma_1 } & 0
\end{array}\right)\\
\mathrm{}\\
\textrm{where}\\
\mathrm{}\\
\;\;\sigma_1 =\sqrt{\frac{{{\left(f-1\right)}}^2 }{{{\left(e-1\right)}}^2 }+1}\\
\mathrm{}\\
\;\;\sigma_2 =\sqrt{\frac{{{\left(b-1\right)}}^2 }{{{\left(a-1\right)}}^2 }+1}
\end{array}
$$

$$
\left(\begin{array}{ccc}
z & \frac{1}{{\mathrm{hy}}^2 }-2\,z & z\\
\frac{1}{{\mathrm{hx}}^2 }-2\,z & 4\,z-\frac{2}{{\mathrm{hx}}^2 }-\frac{2}{{\mathrm{hy}}^2 } & \frac{1}{{\mathrm{hy}}^2 }-2\,z\\
z & \frac{1}{{\mathrm{hy}}^2 }-2\,z & z
\end{array}\right)
$$

$$
d\,\frac{\partial^4 }{\partial x^4 }\;u\left(x,y\right)+2\,d\,\frac{\partial^2 }{\partial y^2 }\;\frac{\partial^2 }{\partial x^2 }\;u\left(x,y\right)+d\,\frac{\partial^4 }{\partial y^4 }\;u\left(x,y\right)
$$

$$
\left(\begin{array}{cccccc}
\frac{1}{360} & h & h & h & h & \frac{\partial^6 }{\partial x^6 }\;u\left(x,y\right)+5\,\frac{\partial^2 }{\partial y^2 }\;\frac{\partial^4 }{\partial x^4 }\;u\left(x,y\right)+5\,\frac{\partial^4 }{\partial y^4 }\;\frac{\partial^2 }{\partial x^2 }\;u\left(x,y\right)+\frac{\partial^6 }{\partial y^6 }\;u\left(x,y\right)
\end{array}\right)
$$

$$
\left(\begin{array}{cccccc}
\frac{1}{360} & h & h & h & h & \frac{\partial^6 }{\partial x^6 }\;u\left(x,y\right)+5\,\frac{\partial^2 }{\partial y^2 }\;\frac{\partial^4 }{\partial x^4 }\;u\left(x,y\right)+5\,\frac{\partial^4 }{\partial y^4 }\;\frac{\partial^2 }{\partial x^2 }\;u\left(x,y\right)+\frac{\partial^6 }{\partial y^6 }\;u\left(x,y\right)
\end{array}\right)
$$

has got a wrong

$$\gamma_{t-x} = \phi_1 + \phi_2(t-x) + \phi_3(t-x)^2 + \epsilon_{t-x}, \quad \epsilon_{t-x} \sim N(0,\sigma^2) \quad \text{i.i.d.}.
$$

$$
\left[ \text{Pr}(y_i=j) = \frac{\exp \{x_{ij}'\beta_i\}}{\sum_{k=1}^p\exp\{x_{ik}'\beta_i\}} \right]
$$

$$
\left[ \Pr(\beta_{ik}^*) = \sum_{m=1}^M \pi_m \phi(z_i' \Delta \vert \mu_j, \Sigma_j) \right]
$$

$$\text｛hunman capital｝=\text｛N｝\times\text｛Education｝\tag｛1.0.3｝$$.

$$
G_t(s_t;γ,c)=\{G_{1,t}(s_{1,t};γ_{1},c_{1}),G_{2,t}(s_{2,t};γ_{2},c_{2}), …,G_{\tilde{n},t}(s_{\tilde{n},t};γ_{\tilde{n}},c_{\tilde{n}})\}.
$$

Each diagonal element $G_{i,t}^r$ is specified as a logistic cumulative density functions, i.e.
$$
G_{i,t}^r(s_{i,t}^r; γ_i^r, c_i^r) = ≤ft[1 + \exp\big\{-γ_i^r(s_{i,t}^r-c_i^r)\big\}\right]^{-1}
$$

for $i = 1,2, …, \tilde{n}$ and $r=0,1,…,m-1$, so that the first model is a Vector Logistic Smooth Transition AutoRegressive (VLSTAR) model. The ML estimator of θ is obtained by solving the optimization problem
$$
\hat{θ}_{ML} = arg \max_{θ}log L(θ)
$$

where log $L(θ)$ is the log-likelihood function of VLSTAR model, given by
$$
ll(y_t|I_t;θ)=-\frac{T\tilde{n}}{2}\ln(2π)-\frac{T}{2}\ln|Ω|-\frac{1}{2}∑_{t=1}^{T}(y_t-\tilde{G}_tB\,z_t)'Ω^{-1}(y_t-\tilde{G}_tB\,z_t)
$$

The NLS estimators of the VLSTAR model are obtained by solving the optimization problem
$$
\hat{θ}_{NLS} = arg \min_{θ}∑_{t=1}^{T}(y_t - Ψ_t'B'x_t)'(y_t - Ψ_t'B'x_t).
$$

$$
\begin{aligned}
VAR(y_t|y_{t-1},\ldots,y_1)&= VAR(E(y_t|\theta_t,y_{t-1},\ldots,y_1)|y_{t-1},\ldots,y_1)\\
&+ E(VAR(y_t|\theta_t,y_{t-1},\ldots,y_1)|y_{t-1},\ldots,y_1)\\
&= VAR(E(y_t|\theta_t)| y_{t-1},\ldots,y_1) + E(VAR(y_t|\theta_t)|y_{t-1},\ldots,y_1) .
\end{aligned}
$$

Ver ist das.
Schön ,


$$
\left[
\hat{\theta} = \arg\min_\theta \bar{g}(\theta)'\big[\hat{\Omega}(\theta)\big]^{-1}\bar{g}(\theta)
\right]
$$

Given some regularity conditions, the GMM estimator converges as $n$ goes to infinity to the following distribution:
$$
\left[
\sqrt{n}(\hat{\theta}-\theta_0) \stackrel{L}{\rightarrow} N(0,V),
\right]
$$
where

$$
\left[
V = E\left(\frac{\partial g(\theta_0,x_i)}{\partial\theta}\right)'\Omega(\theta_0)^{-1}E\left(\frac{\partial g(\theta_0,x_i)}{\partial\theta}\right)
\right]
$$
Inference can therefore be performed on $\hat{\theta}$ using the assumption that it is approximately distributed as $N(\theta_0,\hat{V}/n)$.  

c.d.f

$$
G(w(p))  = \frac{ \int_{\underline p}^p d \Gamma(x) }{\int_{\underline p}^{\bar p} d \Gamma(x) }
.
$$