In XUPUCHEN/lrem:

This package is solving linear rational expectation models, using the function "lre_auto", which contains two functions: lre_auto_bk, introduced by Blanchard and Kahn(1980), and lre_auto_klein, introduced by Klein(2000).

Examples

source("F:/ecomovement/lrem/R/lrem.R")

1, For the Blanchard and Kahn(1980) system as following: $$ x_{t+1} = Ax_t \Leftrightarrow \begin{bmatrix} x_{t+1}^1 \ x_{t+1}^2 \end{bmatrix} = \begin{bmatrix} A_{11} & A_{12} \ A_{21} & A_{22} \end{bmatrix} \begin{bmatrix} x_{t}^1 \ x_{t}^2 \end{bmatrix}, $$ We can define:

A <- matrix(c( 0.7000000, 0.000000, 1.2000000, 0.6363636, 1.909091, 0.1818182, 0.0000000, -1.000000, 1.0000000 ), byrow = TRUE, nrow = 3)
x0 <- 0.1

As there is no E in this system, we can use the function as following:

lre_auto (A, ,x0)

Then we define g and h as following, accroding to the system: \begin{align} x_t^2 &= g(x_t^1) \ x_{t+1}^1 &= h(x_t^1, x_t^2) = h(x_t). \end{align}

Therefore

g <- lre_auto (A, ,x0)$g
h <- lre_auto (A, ,x0)$h

and we set t=100

t <- 100

Here we can do the simulation:

simulate <- function(g, h, x0, t) {
n1 <- length(x0)  
n2 <- length(g(x0)) 
pre <- 1:n1 
npr <- (n1 + 1):(n1 + n2)
out <- matrix(0, t, n1 + n2)
out[1, pre] <- x0  
out[1, npr] <- g(x0) 
for (i in 1:(t - 1)) 
{ out[i + 1, pre] <- h(out[i, pre]) 
out[i + 1, npr] <- g(out[i + 1, pre]) } 
out
}

simulate(g, h, x0, t)
out<-simulate(g, h, x0, t)
plot(out[, 1])

2, For the Klein(2000) case: We use the matrix A and B in "http://www.nag.com/lapack-ex/node119.html" as A and E

A <- QZ::exAB2$A
E <- QZ::exAB2$B
ret <- QZ::qz(A, E)
abs(ret$ALPHA / ret$BETA)
ord <- abs(ret$ALPHA / ret$BETA) <= 1
ret2 <- QZ::qz.dtgsen(ret$S, ret$T, ret$Q, ret$Z, select = ord)
abs(ret2$ALPHA / ret2$BETA)

Therefore, we define

g <- lre_auto (A,E,x0)$g
h <- lre_auto (A,E,x0)$h

Then the simulation result becoms:

simulate <- function(g, h, x0, t) {
n1 <- length(x0)  
n2 <- length(g(x0)) 
pre <- 1:n1 
npr <- (n1 + 1):(n1 + n2)
out <- matrix(0, t, n1 + n2)
out[1, pre] <- x0  
out[1, npr] <- g(x0) 
for (i in 1:(t - 1)) 
{ out[i + 1, pre] <- h(out[i, pre]) 
out[i + 1, npr] <- g(out[i + 1, pre]) } 
out
}

simulate(g, h, x0, t)
out<-simulate(g, h, x0, t)
plot(out[, 1])

XUPUCHEN/lrem documentation built on May 4, 2019, 3:19 p.m.