[QuantEcon]MATLAB混编FORTRAN语言

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

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

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

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

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
$$
\hat{θ}_{NLS} = arg \min_{θ}∑_{t=1}^{T}(y_t - Ψ_t'B'x_t)'(y_t - Ψ_t'B'x_t).
$$