System classes
system_base-class: System base class
system_basic-class: Basic model's system class
system_deterministic_adjustment-class: Deterministic adjustment model's system class
system_directional-class: Directional system class
system_equilibrium-class: Equilibrium model's system class
system_stochastic_adjustment-class: Stochastic adjustment model's system class
demandDemand equation.
supplySupply equation.
correlated_shocksBoolean indicating whether the shock of the equations of the system are correlated.
sample_separationBoolean indicating whether the sample of the system is separated.
quantity_vectorA vector with the system's observed quantities.
price_vectorA vector with the system's observed prices.
rhoCorrelation coefficient of demand and supply shocks.
rho1$$\rho_{1} = \frac{1}{\sqrt{1 - \rho}}$$
rho2$$\rho_{2} = \rho\rho_{1}$$
lhLikelihood values for each observation.
gammaExcess demand coefficient.
delta$$\delta = \gamma + \alpha_{d} - \alpha_{s}$$
mu_P$$\mu_{P} = \mathrm{E}P$$
var_P$$V_{P} = \mathrm{Var}P$$
sigma_P$$\sigma_{P} = \sqrt{V_{P}}$$
h_P$$h_{P} = \frac{P - \mu_{P}}{\sigma_{P}}$$
lagged_price_vectorA vector with the system's observed prices lagged by one date.
mu_Q$$\mu_{Q} = \mathrm{E}Q$$
var_Q$$V_{Q} = \mathrm{Var}Q$$
sigma_Q$$\sigma_{Q} = \sqrt{V_{Q}}$$
h_Q$$h_{Q} = \frac{Q - \mu_{Q}}{\sigma_{Q}}$$
rho_QP$$\rho_{QP} = \frac{\mathrm{Cov}(Q,P)}{\sqrt{\mathrm{Var}Q\mathrm{Var}P}}$$
rho_1QP$$\rho_{1,QP} = \frac{1}{\sqrt{1 - \rho_{QP}^2}}$$
rho_2QP$$\rho_{2,QP} = \rho_{QP}\rho_{1,QP}$$
z_QP$$z_{QP} = \frac{h_{Q} - \rho_{QP}h_{P}}{\sqrt{1 - \rho_{QP}^2}}$$
z_PQ$$z_{PQ} = \frac{h_{P} - \rho_{PQ}h_{Q}}{\sqrt{1 - \rho_{PQ}^2}}$$
price_equationPrice equation.
zeta$$\zeta = \sqrt{1 - \rho_{DS}^2 - \rho_{DP}^2 - \rho_{SP}^2 + 2 \rho_DP \rho_DS \rho_SP}$$
zeta_DD$$\zeta_{DD} = 1 - \rho_{SP}^2$$
zeta_DS$$\zeta_{DS} = \rho_{DS} - \rho_{DP}\rho_{SP}$$
zeta_DP$$\zeta_{DP} = \rho_{DP} - \rho_{DS}\rho_{SP}$$
zeta_SS$$\zeta_{SS} = 1 - \rho_{DP}^2$$
zeta_SP$$\zeta_{SP} = \rho_{SP} - \rho_{DS}\rho_{DP}$$
zeta_PP$$\zeta_{PP} = 1 - \rho_{DS}^2$$
mu_D$$\mu_{D} = \mathrm{E}D$$
var_D$$V_{D} = \mathrm{Var}D$$
sigma_D$$\sigma_{D} = \sqrt{V_{D}}$$
mu_S$$\mu_{S} = \mathrm{E}S$$
var_S$$V_{S} = \mathrm{Var}S$$
sigma_S$$\sigma_{S} = \sqrt{V_{S}}$$
sigma_DP$$\sigma_{DP} = \mathrm{Cov}(D, P)$$
sigma_DS$$\sigma_{DS} = \mathrm{Cov}(D, S)$$
sigma_SP$$\sigma_{SP} = \mathrm{Cov}(S, P)$$
rho_DS$$\rho_{DS} = \frac{\mathrm{Cov}(D,S)}{\sqrt{\mathrm{Var}D\mathrm{Var}S}}$$
rho_DP$$\rho_{DP} = \frac{\mathrm{Cov}(D,P)}{\sqrt{\mathrm{Var}D\mathrm{Var}P}}$$
rho_SP$$\rho_{SP} = \frac{\mathrm{Cov}(S,P)}{\sqrt{\mathrm{Var}S\mathrm{Var}P}}$$
h_D$$h_{D} = \frac{D - \mu_{D}}{\sigma_{D}}$$
h_S$$h_{S} = \frac{S - \mu_{S}}{\sigma_{S}}$$
z_DP$$z_{DP} = \frac{h_{D} - \rho_{DP}h_{P}}{\sqrt{1 - \rho_{DP}^2}}$$
z_PD$$z_{PD} = \frac{h_{P} - \rho_{PD}h_{D}}{\sqrt{1 - \rho_{PD}^2}}$$
z_SP$$z_{SP} = \frac{h_{S} - \rho_{SP}h_{P}}{\sqrt{1 - \rho_{SP}^2}}$$
z_PS$$z_{PS} = \frac{h_{P} - \rho_{PS}h_{S}}{\sqrt{1 - \rho_{PS}^2}}$$
omega_D$$\omega_{D} = \frac{h_{D}\zeta_{DD} - h_{S}\zeta_{DS} - h_{P}\zeta_{DP}}{\zeta_{DD}}$$
omega_S$$\omega_{S} = \frac{h_{S}\zeta_{SS} - h_{S}\zeta_{SS} - h_{P}\zeta_{SP}}{\zeta_{SS}}$$
w_D$$w_{D} = - \frac{h_{D}^2 - 2 h_{D} h_{P} \rho_{DP} + h_{P}^2}{2\zeta_{SS}}$$
w_S$$w_{S} = - \frac{h_{S}^2 - 2 h_{S} h_{P} \rho_{SP} + h_{P}^2}{2\zeta_{DD}}$$
psi_D$$\psi_{D} = \phi\left(\frac{\omega_{D}}{\zeta}\right)$$
psi_S$$\psi_{S} = \phi\left(\frac{\omega_{S}}{\zeta}\right)$$
Psi_D$$\Psi_{D} = 1 - \Phi\left(\frac{\omega_{D}}{\zeta}\right)$$
Psi_S$$\Psi_{S} = 1 - \Phi\left(\frac{\omega_{S}}{\zeta}\right)$$
g_D$$g_{D} = \frac{\psi_{D}}{\Psi_{D}}$$
g_S$$g_{S} = \frac{\psi_{S}}{\Psi_{S}}$$
rho_dsrho_dpCorrelation of demand and price equations' shocks.
rho_spCorrelation of supply and price equations' shocks.
L_DLikelihood conditional on excess supply.
L_SLikelihood conditional on excess demand.
Classes with data and functionality describing systems of models.