Discretize an AR(1) process into a Markov chain (Rouwenhorst's method)

Approximates the stationary first-order autoregressive process $$y_t = (1-\rho)\alpha + \rho y_{t-1} + \varepsilon_t, \qquad \varepsilon_t \overset{\mathrm{iid}}{\sim} \mathcal N(0,\sigma^2)$$ by a finite-state Markov chain, following Rouwenhorst (1995). Unlike tauchen, no grid-width parameter is needed and the method remains accurate for rho close to $\pm 1$.

Usage

rouwenhorst(alpha, sigma, rho, size)

Arguments

alpha: A single finite number: the unconditional mean of the process.
sigma: A single finite positive number: the standard deviation of the innovation $\varepsilon_t$.
rho: A single number in $(-1,1)$: the autocorrelation (persistence) of the process.
size: A single integer of at least 2: the number of grid points (states) of the discretized chain.

Value

A named list with two elements, chain and states, in the same form as returned by tauchen.

Details

The grid is $n=\code{size}$ evenly spaced points spanning $[\alpha-\psi,\ \alpha+\psi]$ with $\psi=\sigma_y\sqrt{n-1}$, where $\sigma_y=\sigma/\sqrt{1-\rho^2}$ is the process's unconditional standard deviation: this particular width (rather than a fixed multiple of $\sigma_y$ as in tauchen) is what the method needs in order to match the AR(1)'s variance and first-order autocorrelation exactly at every size, including for rho near $\pm1$. The transition matrix is built recursively. Let $\theta=(1+\rho)/2$ and, for two states, $$\Theta_2 = \begin{pmatrix}\theta & 1-\theta\\ 1-\theta & \theta\end{pmatrix}.$$ For $m$ states ($2<m\le n$), form the $m\times m$ matrix $$\Theta_m = \theta\begin{pmatrix}\Theta_{m-1} & 0\\ 0 & 0\end{pmatrix} + (1-\theta)\begin{pmatrix}0 & \Theta_{m-1}\\ 0 & 0\end{pmatrix} + (1-\theta)\begin{pmatrix}0 & 0\\ \Theta_{m-1} & 0\end{pmatrix} + \theta\begin{pmatrix}0 & 0\\ 0 & \Theta_{m-1}\end{pmatrix},$$ then divide every interior row (all but the first and last) by $2$ to restore row-stochasticity, since those rows receive contributions from two of the four corner blocks above.

Rouwenhorst's method reproduces the AR(1)'s unconditional variance and lag-1 autocorrelation $\rho$ exactly, for every size (Kopecky and Suen (2010) find it outperforms tauchen and several other methods across the persistence range typically seen in quarterly macroeconomic and actuarial time series, e.g. discretized short-rate or inflation processes for reserving and ALM work).

References

Rouwenhorst, K. G. (1995). Asset pricing implications of equilibrium business cycle models. In T. F. Cooley (Ed.), Frontiers of Business Cycle Research, 294-330. Princeton University Press.

Kopecky, K. A. and Suen, R. M. H. (2010). Finite state Markov-chain approximations to highly persistent processes. Review of Economic Dynamics, 13(3), 701-714.

Examples

out <- rouwenhorst(alpha = 0, sigma = 1, rho = 0.9, size = 5)
out$states
#>    -4.588    -2.294     0.000     2.294     4.588 
#> -4.588315 -2.294157  0.000000  2.294157  4.588315 
pi <- as.numeric(steadyStates(out$chain))
sum(pi * (out$states - sum(pi * out$states))^2) # matches 1/(1-rho^2) closely
#> [1] 5.263158
1 / (1 - 0.9^2)
#> [1] 5.263158