In tatsuyaasami/lrem: Linear Rational Expectations Model

This source simulates a linear ratinal expectation model as following,

$$ E \mathbb{E}t x{t+1} = A x_{t} + B u_{t}, \ \ x_{t} \in \mathbb{R}^{n}, \ \ u_{t} \in \mathbb{R}^{m}. $$ $E$ need not be a reversible matrix. $x_{t} = (x^{1}{t}, \ x^{2}{t})^{t}$ and $x^{1}{t}$ is the predetermined vector, $x^{2}{t}$ is the un-predetermined vector. There must exist some z such that $\det(E z - A) \neq 0$.

We transform this system into the homogeneous system when the shock follows the AR process $u_{t + 1} = \Phi u_{t} + \epsilon_{t + 1}$

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

Example 1 (Hansen's RBS model)

We replicate the example in https://cdn.rawgit.com/rokko-ed17q1/final/64730073/lrem-hansen-rbc.html.

The parameters are

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

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

The simulation is following

library("lrem")
library("ggplot2")

policy <- lre_ar(A, E, B, Phi, 0, 1)

# Simulation period
steps <- 100

# Path of the growth rates
out <- simulate(policy$g, policy$h, 0, 1, steps) 
out0 <- matrix(0, nrow = 1, ncol = ncol(out))
out <- rbind(out0, out)
# 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]
}

result <- data.frame(out)
names(result) <- c("Z", "K", "N", "C")
result["t"] <- 0:(steps)

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

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

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

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

Example 2 (the canonical New Keynesian model)

We simulate the following New Keynesian model.

Phillips curve

$$ \pi_{t} = \beta \mathbb{E}{t} \pi{t + 1} + \kappa x_{t} + u^{S}_{t} $$

IS curve

$$ x_{t} = \mathbb{E}{t} x{t + 1} - \frac{1}{\sigma} (i_{t} - \mathbb{E}{t} \pi{t + 1}) + u^{D}_{t} $$

Monetary policy rule

$$ i_{t} = \alpha \pi_{t} + \iota x_{t} $$

Assume AR shock with $|\rho_{S}|, |\rho_{D}| < 1$ and $\mathbb{E}{t} \epsilon^{S}{t + 1} = \mathbb{E}{t} \epsilon^{D}{t + 1}$;

$$ u^{S}{t + 1} = \rho{S} u^{S}{t} + \epsilon^{S}{t + 1} \ u^{D}{t + 1} = \rho{D} u^{D}{t} + \epsilon^{D}{t + 1} $$

• $\pi_{t}$: log inflation rate
• $x_{t}$: output gap
• $i_{t}$: log gross nominal interest rate
• $\sigma$: intertemporal elastocity of substitution
• $\beta$: discount rate
• $\kappa$: slope pf the Phillips curve
• $\alpha$ and $\iota$: monetary policy parameters

Rewriting this system into the above matrix form, we obtain

$$ \begin{bmatrix} I & O \ O & E \end{bmatrix} \mathbb{E}t \begin{bmatrix} u{t + 1} \ \pi_{t + 1} \ x_{t + 1} \ i_{t + 1} \end{bmatrix} = \begin{bmatrix} \Phi & O \ B & A \end{bmatrix} \begin{bmatrix} u_{t} \ \pi_{t} \ x_{t} \ i_{t} \end{bmatrix} $$ where

$$ A = \begin{bmatrix} 1 & - \kappa & 0 \ 0 & 1 & 1 / \sigma \ \alpha & \iota & - 1 \end{bmatrix}, \ E = \begin{bmatrix} \beta & 0 & 0 \ 1 / \sigma & 1 & 0 \ 0 & 0 & 0 \end{bmatrix}, \ B = \begin{bmatrix} - 1 & 0 \ 0 & - 1 \ 0 & 0 \end{bmatrix}, \ \Phi = \begin{bmatrix} \rho_{S} & 0 \ 0 & \rho_{D} \end{bmatrix}, \ u_{t} = \begin{bmatrix} u^{S}{t} \ u^{D}{t} \end{bmatrix}. $$

The parameters are

alpha <- 2
iota <- 0
beta <- 0.99
sigma <- 1
kappa <- 0.132
rhoS <- 0.9
rhoD <- 0.9

The matrices are

A <- matrix(c(1, - kappa, 0,
              0, 1, 1 / sigma,
              alpha, iota, - 1), nrow = 3, byrow = TRUE)
E <- matrix(c(beta, 0, 0,
              1 / sigma, 1, 0,
              0, 0, 0), nrow = 3, byrow = TRUE)
B <- matrix(c(- 1, 0,
              0, - 1,
              0, 0), nrow = 3, byrow = TRUE)
Phi <- matrix(c(rhoS, 0,
              0, rhoD), nrow = 2, byrow = TRUE)

The values of these variables at the steady state are

rhoSbar <- 0
rhoDbar <- 0
pibar <- 0
xbar <- 0
ibar <- 0
ss <- c(rhoSbar, rhoDbar, pibar, xbar, ibar)

That's all of preparation of simulation.

# Initial shock is (us, ud) = (1, 1)
policy <- lre_ar(A, E, B, Phi, NULL, c(1, 1))
# Simulation period
steps <- 100

out <- simulate(policy$g, policy$h, NULL, c(1, 1), steps)
out0 <- matrix(ss, nrow = 1, ncol = ncol(out))
out <- rbind(out0, out)

result <- data.frame(out)
names(result) <- c("rhoS", "rhoD", "pi", "x", "i")
result["t"] <- 0:(steps)

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

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

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

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

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

tatsuyaasami/lrem documentation built on May 3, 2019, 11:50 p.m.