approximation

tauchen

Discretizes Gaussian linear AR(1) processes via Tauchen’s method

quantecon.markov.approximation.rouwenhorst(n, rho, sigma, mu=0.0)

Takes as inputs n, mu, sigma, rho. It will then construct a markov chain that estimates an AR(1) process of: \(y_t = \mu + \rho y_{t-1} + \varepsilon_t\) where \(\varepsilon_t\) is i.i.d. normal of mean 0, std dev of sigma

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

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

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

where \({p = q = \frac{(1 + \rho)}{2}}\)

Parameters:

nint

The number of points to approximate the distribution

rhofloat

Persistence parameter in AR(1) process, if you are approximating an AR(1) process then this is the autocorrelation across periods.

sigmafloat

The value of the standard deviation of the \(\varepsilon\) process

mufloat, optional(default=0.0)

The value \(\mu\) in the process. Note that the mean of this AR(1) process, \(y\), is simply \(\mu/(1 - \rho)\)

Returns:

mcMarkovChain

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

Notes

UserWarning: The API of rouwenhorst was changed from rouwenhorst(n, ybar, sigma, rho) to rouwenhorst(n, rho, sigma, mu=0.) in version 0.6.0.

quantecon.markov.approximation.std_norm_cdf(x)

quantecon.markov.approximation.tauchen(n, rho, sigma, mu=0.0, n_std=3)

Computes a Markov chain associated with a discretized version of the linear Gaussian AR(1) process

\[y_t = \mu + \rho y_{t-1} + \epsilon_t\]

using Tauchen’s method. Here \({\epsilon_t}\) is an i.i.d. Gaussian process with zero mean.

Parameters:

nscalar(int)

The number of states to use in the approximation

rhoscalar(float)

The autocorrelation coefficient, Persistence parameter in AR(1) process

sigmascalar(float)

The standard deviation of the random process

muscalar(float), optional(default=0.0)

The value \(\mu\) in the process. Note that the mean of this AR(1) process, \(y\), is simply \(\mu/(1 - \rho)\)

n_stdscalar(int), optional(default=3)

The number of standard deviations to approximate out to

Returns:

mcMarkovChain

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

Notes

UserWarning: The API of tauchen was changed from tauchen(rho, sigma_u, b=0., m=3, n=7) to tauchen(n, rho, sigma, mu=0., n_std=3) in version 0.6.0.