In nuritovbek/lrem: Solving and simulating a class of LRE models

For the AR LRE, inputs are defined as follows: Matrices E and A in addition to a matrix B and $\Phi$ that satisfy the following system

$Ex_{t+1} = Ax_t + Bu_t \ u_{t+1} = \Phi u_t + e_t$ Note that the shock follows an AR process.

In addition the system also takes the following vector $x_{t} = (x^{1}{t}, \ x^{2}{t})^{t}$ where $x^{1}_{t}$ are predetermined variables, $x^{2}{t}$ are the nonpredetermined variables.

The function in lre_ar then transforms these inputs to a system that satisfies.

$$ \begin{bmatrix} I \ & E \end{bmatrix} \begin{bmatrix} u_{t + 1} \ x_{t + 1} \end{bmatrix} = \begin{bmatrix} \Phi \ B & A \end{bmatrix} \begin{bmatrix} u_{t} \ x_{t} \end{bmatrix} $$

Replicating Hansen's RBC model

Consider the following parameters, the specification of the model is ommitted but follows that specified in the course.

alpha = 0.33
beta = 0.99
delta = 0.023
chi = 1.75
rho = 0.95

q0 = (1 - beta + beta * delta) / alpha / beta
q1 = q0 ^ (1 / (1 - alpha))
q2 = q0 - delta

kbar = (1 - alpha) * q1 ^ (- alpha)
kbar = kbar / ((1 - alpha) * q0 + chi * q2)

cbar = q2 * kbar
nbar = q1 * kbar
zbar = 1

Where the matrices are

E = matrix(0, 3, 3)
A = matrix(0, 3, 3)
B = matrix(0, 3, 1)
Phi = matrix(rho, 1, 1)

E[1, 1] = alpha * (alpha - 1) * q0
E[1, 2] = alpha * q0
E[1, 3] = - (1 - delta + alpha * q0)
E[2, 1] = 1

A[1, 3] = E[1, 3]
A[2, 1] = - A[1, 3]
A[2, 2] = (1 - alpha) * q0
A[2, 3] = - q2
A[3, 1] = alpha
A[3, 2] = (- alpha - (1 - alpha) * nbar) / (1 - nbar)
A[3, 3] = -1

B[1, 1] = - alpha * q0 * rho
B[2, 1] = q0
B[3, 1] = 1

Proceeding with the simulation, the motion and decision rules are stored in the following

sim <- lrem::lre_ar(A, E, B, Phi, n = 1)
sim

Which are used in the simulation and standartised as follows

# Simulation period
steps <- 100

# Path of the growth rates
out <- lrem::simulate(sim[[1]], sim[[2]], x0 = c(0, 0), steps, e = c(1,0)) 
out[1:10,]

Converting these into levels as per the example

# Steady State
ss <- c(zbar, kbar, nbar, cbar)

# Convert percentage changes to levels
for (i in 1:ncol(out)) {
  out[, i] <- ss[i] * out[, i] + ss[i]
}

Plotting the simulation results using the ggplot2 package

library("ggplot2")

# Convert matrix to dataframe
result <- data.frame(out)
names(result) <- c("Z", "K", "N", "C")
result["t"] <- 0:(steps - 1)

# Plotting
ggplot(result) + geom_line(aes(x = t, y = K))

ggplot(result) + geom_line(aes(x = t, y = Z))

ggplot(result) + geom_line(aes(x = t, y = N))

ggplot(result) + geom_line(aes(x = t, y = C))

nuritovbek/lrem documentation built on May 4, 2019, 4:24 p.m.