# Source code for quantecon.markov.approximation

"""
tauchen
-------
Discretizes Gaussian linear AR(1) processes via Tauchen's method

"""

from math import erfc, sqrt
from .core import MarkovChain

import numpy as np
from numba import njit

[docs]def rouwenhorst(n, ybar, sigma, rho):
r"""
Takes as inputs n, p, q, psi. It will then construct a markov chain
that estimates an AR(1) process of:
:math:y_t = \bar{y} + \rho y_{t-1} + \varepsilon_t
where :math:\varepsilon_t is i.i.d. normal of mean 0, std dev of sigma

The Rouwenhorst approximation uses the following recursive defintion
for approximating a distribution:

.. math::

\theta_2 =
\begin{bmatrix}
p     &  1 - p \\
1 - q &  q     \\
\end{bmatrix}

.. math::

\theta_{n+1} =
p
\begin{bmatrix}
\theta_n & 0   \\
0        & 0   \\
\end{bmatrix}
+ (1 - p)
\begin{bmatrix}
0  & \theta_n  \\
0  &  0        \\
\end{bmatrix}
+ q
\begin{bmatrix}
0        & 0   \\
\theta_n & 0   \\
\end{bmatrix}
+ (1 - q)
\begin{bmatrix}
0  &  0        \\
0  & \theta_n  \\
\end{bmatrix}

Parameters
----------
n : int
The number of points to approximate the distribution

ybar : float
The value :math:\bar{y} in the process.  Note that the mean of this
AR(1) process, :math:y, is simply :math:\bar{y}/(1 - \rho)

sigma : float
The value of the standard deviation of the :math:\varepsilon process

rho : float
By default this will be 0, but if you are approximating an AR(1)
process then this is the autocorrelation across periods

Returns
-------

mc : MarkovChain
An instance of the MarkovChain class that stores the transition
matrix and state values returned by the discretization method

"""

# Get the standard deviation of y
y_sd = sqrt(sigma**2 / (1 - rho**2))

# Given the moments of our process we can find the right values
# for p, q, psi because there are analytical solutions as shown in
# Gianluca Violante's notes on computational methods
p = (1 + rho) / 2
q = p
psi = y_sd * np.sqrt(n - 1)

# Find the states
ubar = psi
lbar = -ubar

bar = np.linspace(lbar, ubar, n)

def row_build_mat(n, p, q):
"""
This method uses the values of p and q to build the transition
matrix for the rouwenhorst method

"""

if n == 2:
theta = np.array([[p, 1 - p], [1 - q, q]])

elif n > 2:
p1 = np.zeros((n, n))
p2 = np.zeros((n, n))
p3 = np.zeros((n, n))
p4 = np.zeros((n, n))

new_mat = row_build_mat(n - 1, p, q)

p1[:n - 1, :n - 1] = p * new_mat
p2[:n - 1, 1:] = (1 - p) * new_mat
p3[1:, :-1] = (1 - q) * new_mat
p4[1:, 1:] = q * new_mat

theta = p1 + p2 + p3 + p4
theta[1:n - 1, :] = theta[1:n - 1, :] / 2

else:
raise ValueError("The number of states must be positive " +
"and greater than or equal to 2")

return theta

theta = row_build_mat(n, p, q)

bar += ybar / (1 - rho)

return MarkovChain(theta, bar)

[docs]def tauchen(rho, sigma_u, m=3, n=7):
r"""
Computes a Markov chain associated with a discretized version of
the linear Gaussian AR(1) process

.. math::

y_{t+1} = \rho y_t + u_{t+1}

using Tauchen's method. Here :math:{u_t} is an i.i.d. Gaussian process
with zero mean.

Parameters
----------
rho : scalar(float)
The autocorrelation coefficient
sigma_u : scalar(float)
The standard deviation of the random process
m : scalar(int), optional(default=3)
The number of standard deviations to approximate out to
n : scalar(int), optional(default=7)
The number of states to use in the approximation

Returns
-------

mc : MarkovChain
An instance of the MarkovChain class that stores the transition
matrix and state values returned by the discretization method

"""

# standard deviation of y_t
std_y = np.sqrt(sigma_u**2 / (1 - rho**2))

# top of discrete state space
x_max = m * std_y

# bottom of discrete state space
x_min = -x_max

# discretized state space
x = np.linspace(x_min, x_max, n)

step = (x_max - x_min) / (n - 1)
half_step = 0.5 * step
P = np.empty((n, n))

_fill_tauchen(x, P, n, rho, sigma_u, half_step)

mc = MarkovChain(P, state_values=x)
return mc

[docs]@njit
def std_norm_cdf(x):
return 0.5 * erfc(-x / sqrt(2))

@njit
def _fill_tauchen(x, P, n, rho, sigma, half_step):
for i in range(n):
P[i, 0] = std_norm_cdf((x[0] - rho * x[i] + half_step) / sigma)
P[i, n - 1] = 1 - \
std_norm_cdf((x[n - 1] - rho * x[i] - half_step) / sigma)
for j in range(1, n - 1):
z = x[j] - rho * x[i]
P[i, j] = (std_norm_cdf((z + half_step) / sigma) -
std_norm_cdf((z - half_step) / sigma))