[QuantEcon]MATLAB混编FORTRAN语言

$$
y_{i} = X_i \beta_i + u_i, \quad i = 1, 2, \ldots, G ,
$$
where $y_i$ is a vector of the dependent variable,
$X_i$ is a matrix of the exogenous variables,
$\beta_i$ is the coefficient vector and
$u_i$ is a vector of the disturbance terms of the $i$th equation.

$$
\left[ \begin{array}{c}
      y_1 \\ y_2\\ \vdots\\ y_G
   \end{array} \right] =
   \left[ \begin{array}{cccc}
      X_1 & 0 & \cdots & 0\\
      0 & X_2 & \cdots & 0\\
      \vdots & \vdots & \ddots & \vdots\\
      0 & 0 & \cdots & X_G
   \end{array}\right]
   \left[ \begin{array}{c}
      \beta_1 \\ \beta_2 \\ \vdots\\ \beta_G
   \end{array} \right] +
   \left[ \begin{array}{c}
      u_1 \\ u_2 \\ \vdots\\ u_G
   \end{array} \right]
$$

$$
\E \left[ u_{it} \, u_{js} \right] = 0
   \; \forall \; t \neq s ,
$$

where $i$ and $j$ indicate the equation number
and $t$ and $s$ denote the observation number,
where the number of observations is the same for all equations.

However, we explicitly allow for contemporaneous correlation, i.e.,
$$
\E \left[ u_{it} \, u_{jt} \right] = \sigma_{ij} .
$$
thus, the covariance matrix of all disturbances is
$$
\E \left[ u \, u^\top \right] = \Omega = \Sigma \otimes I_T ,
$$

where $\Sigma = \left[ \sigma_{ij} \right]$ is the (contemporaneous)
disturbance covariance matrix,
$\otimes$ is the Kronecker product,
$I_T$ is an identity matrix of dimension $T$,
and $T$ is the number of observations in each equation.

If the whole system is treated as one single equation,
$\OHat$ in Equation~\ref{eq:cov-ols-wls-sur} is $\sHat^2 I_{G \cdot T}$,
where $\sHat^2$ is an estimator for the variance of all disturbances
$(\sigma^2 = \E [ u_{it}^2 ])$.
If the disturbance terms of the individual equations
are allowed to have different variances,
$\OHat$ in Equation~\ref{eq:cov-ols-wls-sur} is $\SHat \otimes I_T$,
where $\sHat_{ij} = 0 \; \forall \; i \neq j$ and
$\sHat_{ii} = \sHat_i^2$ is the estimated variance
of the disturbance term in the $i$th equation.

$$
\mathbb{E} \beta = \{ \alpha*\gamma \}
$$

$$
\ln m_t = \ln e \left( p_t, U_t \right)
= \alpha_0 + \sum_i \alpha_i \ln p_{it}
+ \frac{1}{2} \sum_i \sum_j \gamma_{ij}^* \ln p_{it} \ln p_{jt}
+ U_t \beta_0 \prod_i  p_{it}^{\beta_i},
$$
where
$m_t$ is total expenditure at time $t$,
$p_{it}$ is the price of good $i$ at time $t$,
$U_t$ is the utility level at time $t$,
and $\alpha$, $\beta$ and $\gamma^*$ are coefficients.

$$
x_{it} ( p_t , m_t )
= \frac{m_t}{p_{it}}
   \left( \alpha_i + \sum_j \gamma_{ij} \ln p_{jt}
   + \beta_i \ln \left( m_t / P_t \right) \right)
$$
where $x_{it}$ is the consumed quantity of good $i$,
$\gamma_{ij} = \frac{1}{2} ( \gamma_{ij}^{*} + \gamma_{ji}^{*} )$,
and $P_t$ is a translog price index:

$$
\ln P_{t} = \alpha_0 + \sum_i \alpha_i \ln p_{it}
+ \frac{1}{2} \sum_i \sum_j \gamma_{ij} \ln p_{it} \ln p_{jt}
$$

$$
s_{it} ( p_t , m_t )
= \alpha_i + \sum_j \gamma_{ij} \ln p_{jt}
+ \beta_i \ln \left( m_t / P_t \right)
$$

where $s_{it} = x_{it} \, p_{it} / m_t$ is the expenditure share
of good $i$.