approximation¶
tauchen¶
Discretizes Gaussian linear AR(1) processes via Tauchen’s method
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quantecon.markov.approximation.rouwenhorst(n, ybar, sigma, rho)[source]¶ Takes as inputs n, p, q, psi. It will then construct a markov chain that estimates an AR(1) process of: \(y_t = \bar{y} + \rho y_{t-1} + \varepsilon_t\) 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:
\[\begin{split}\theta_2 = \begin{bmatrix} p & 1 - p \\ 1 - q & q \\ \end{bmatrix}\end{split}\]\[\begin{split}\theta_{n+1} = p \begin{bmatrix} \theta_n & 0 \\ 0 & 0 \\ \end{bmatrix} + (1 - p) \begin{bmatrix} 0 & \theta_n \\ 0 & 0 \\ \end{bmatrix} + q \begin{bmatrix} 0 & 0 \\ \theta_n & 0 \\ \end{bmatrix} + (1 - q) \begin{bmatrix} 0 & 0 \\ 0 & \theta_n \\ \end{bmatrix}\end{split}\]Parameters: - n : int
The number of points to approximate the distribution
- ybar : float
The value \(\bar{y}\) in the process. Note that the mean of this AR(1) process, \(y\), is simply \(\bar{y}/(1 - \rho)\)
- sigma : float
The value of the standard deviation of the \(\varepsilon\) 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
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quantecon.markov.approximation.tauchen(rho, sigma_u, b=0.0, m=3, n=7)[source]¶ Computes a Markov chain associated with a discretized version of the linear Gaussian AR(1) process
\[y_{t+1} = b + \rho y_t + u_{t+1}\]using Tauchen’s method. Here \({u_t}\) is an i.i.d. Gaussian process with zero mean.
Parameters: - b : scalar(float)
The constant term of {y_t}
- 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