October 22, 2013

Rouwenhorst (1995) Markov Transition Matrix

If you're going to approximate an AR(1) process using a finite state Markov-chain, many people will use the method given in either Tauchen (1986) or Tauchen and Hussey (1991). For a highly persistent process, however, Rouwenhorst (1995) outperforms both of these methods considerably. A rigorous comparison is given by Kopecky and Suen (2010).

Python code for Tauchen and Hussey is in a previous post. An implementation of Rowenhorst is given here (and shown below).

import numpy as np

def rouwen(rho, mu, step, num):
    Adapted from Lu Zhang and Karen Kopecky. Python by Ben Tengelsen.
    Construct transition probability matrix for discretizing an AR(1)
    process. This procedure is from Rouwenhorst (1995), which works
    well for very persistent processes.

    rho  - persistence (close to one)
    mu   - mean and the middle point of the discrete state space
    step - step size of the even-spaced grid
    num  - number of grid points on the discretized process

    dscSp  - discrete state space (num by 1 vector)
    transP - transition probability matrix over the grid

    # discrete state space
    dscSp = np.linspace(mu -(num-1)/2*step, mu +(num-1)/2*step, num).T

    # transition probability matrix
    q = p = (rho + 1)/2.
    transP = np.array([[p**2, p*(1-q), (1-q)**2], \
                    [2*p*(1-p), p*q+(1-p)*(1-q), 2*q*(1-q)], \
                    [(1-p)**2, (1-p)*q, q**2]]).T

    while transP.shape[0] <= num - 1:

        # see Rouwenhorst 1995
        len_P = transP.shape[0]
        transP = p*np.vstack((np.hstack((transP,np.zeros((len_P,1)))), np.zeros((1, len_P+1)))) \
        + (1-p)*np.vstack((np.hstack((np.zeros((len_P, 1)), transP)), np.zeros((1, len_P+1)))) \
        + (1-q)*np.vstack((np.zeros((1, len_P+1)), np.hstack((transP, np.zeros((len_P, 1)))))) \
        + q*np.vstack((np.zeros((1, len_P+1)), np.hstack((np.zeros((len_P, 1)), transP))))

        transP[1:-1] /= 2.

    # ensure columns sum to 1
    if np.max(np.abs(np.sum(transP, axis=1) - np.ones(transP.shape))) >= 1e-12:
        print('Problem in rouwen routine!')
        return None
        return transP.T, dscSp