Instantiation:
# create a new object lrem_obj <- new(gensys)
Fields:
Gamma_0 Input matrix $\Gamma_0$.Gamma_1 Input matrix $\Gamma_1$.Gamma_C Input matrix $\Gamma_C$.Psi Input matrix $\Psi$.Pi Input matrix $\Pi$.shocks_cov Covariance matrix of shocks.G_sol Solution matrix $G$.impact_sol Solution impact matrix $I$.C_sol Solution matrix $C$.F_state State-space matrix $F$.G_state State-space matrix $G$.Build:
lrem_obj$build(Gamma_0,Gamma_1,Gamma_C,Psi,Pi)
Gamma_0,Gamma_1,Gamma_C,Psi,Pi input matrices.Solve model:
lrem_obj$solve()
State-space representation:
lrem_obj$state_space()
Simulate:
lrem_obj$simulate(n_sim_periods,n_burnin)
n_sim_periods length of simulation output.n_burnin number of burnin periods.The Gensys system: \begin{equation} \Gamma_0 (\theta) \xi_t = \Gamma_C(\theta) + \Gamma_1 (\theta) \xi_{t-1} + \Psi (\theta) \epsilon_t + \Pi (\theta) \eta_t \end{equation} The solver returns a solution of the form: \begin{equation} \xi_t = C (\theta) + G (\theta) \xi_{t-1} + I (\theta) \epsilon_t \end{equation}
As an example, we solve the simple New-Keynesian model.
The full model is a system of 8 equations, six endogenous processes
\begin{align} y_t^g &= \mathbb{E}_t y_{t+1}^g -\eta(i_t - \mathbb{E}_t \pi_{t+1} - r_t^{n}) \\ \pi_t &= \beta \mathbb{E}_t \pi_{t+1} + \kappa y_t^g \\ i_t &= \phi_{\pi} \pi_t + \phi_{y} y_t^g + v_t \\ r_t^n &= \frac{1+\phi}{1 - \alpha + \eta(\phi+\alpha)} (\rho_a - 1) a_t \\ y_t &= a_t + (1- \alpha) n_t \\ y_t^g &= y_t - \frac{\eta(1+\phi)}{1 - \alpha + \eta(\phi+\alpha)} a_t \end{align}with two exogenous shocks, technology and monetary policy,
\begin{align} a_t &= \rho_a a_{t-1} + \varepsilon_{a,t} \\ v_t &= \rho_v v_{t-1} + \varepsilon_{v,t} \end{align}respectively.
rm(list=ls())
library(BMR)
#
alpha <- 0.33
vartheta <- 6
beta <- 0.99
theta <- 0.6667
eta <- 1
phi <- 1
phi_pi <- 1.5
phi_y <- 0.5/4
rho_a <- 0.90
rho_v <- 0.5
BigTheta <- (1-alpha)/(1-alpha+alpha*vartheta)
kappa <- (((1-theta)*(1-beta*theta))/(theta))*BigTheta*((1/eta)+((phi+alpha)/(1-alpha)))
psi <- (eta*(1+phi))/(1-alpha+eta*(phi + alpha))
sigma_T <- 1
sigma_M <- 0.25
G0_47 <- (1/eta)*psi*(rho_a - 1)
#Order: yg, y, pi, rn, i, n, a, v, yg_t+1, pi_t+1
Gamma0 <- rbind(c( -1, 0, 0, eta, -eta/4, 0, 0, 0, 1, eta/4),
c( kappa, 0, -1/4, 0, 0, 0, 0, 0, 0, beta/4),
c( phi_y, 0,phi_pi/4, 0, -1/4, 0, 0, 1, 0, 0),
c( 0, 0, 0, -1, 0, 0, G0_47, 0, 0, 0),
c( 0, -1, 0, 0, 0, 1-alpha, 1, 0, 0, 0),
c( -1, 1, 0, 0, 0, 0, -psi, 0, 0, 0),
c( 0, 0, 0, 0, 0, 0, 1, 0, 0, 0),
c( 0, 0, 0, 0, 0, 0, 0, 1, 0, 0),
c( 1, 0, 0, 0, 0, 0, 0, 0, 0, 0),
c( 0, 0, 1, 0, 0, 0, 0, 0, 0, 0))
Gamma1 <- rbind(c( 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),
c( 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),
c( 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),
c( 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),
c( 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),
c( 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),
c( 0, 0, 0, 0, 0, 0, rho_a, 0, 0, 0),
c( 0, 0, 0, 0, 0, 0, 0, rho_v, 0, 0),
c( 0, 0, 0, 0, 0, 0, 0, 0, 1, 0),
c( 0, 0, 0, 0, 0, 0, 0, 0, 0, 1))
#
C <- matrix(0,10,1)
Psi <- matrix(0,10,2)
Psi[7,1] <- 1
Psi[8,2] <- 1
Pi <- matrix(0,10,2)
Pi[9,1] <- 1
Pi[10,2] <- 1
#
Sigma <- rbind(c(sigma_T^2, 0),
c( 0, sigma_M^2))
#
dsge_obj <- new(gensys)
dsge_obj$build(Gamma0,Gamma1,C,Psi,Pi)
dsge_obj$solve()
dsge_obj$G_sol
dsge_obj$impact_sol
#
dsge_obj$shocks_cov = Sigma
dsge_obj$state_space()
dsge_obj$F_state
dsge_obj$G_state
#
var_names <- c("Output Gap","Output","Inflation","Natural Int",
"Nominal Int","Labour Supply","Technology","MonetaryPolicy")
dsge_irf <- IRF(dsge_obj,12,var_names=var_names)
dsge_sim <- dsge_obj$simulate(800,200)