Discretize an AR(1) process into a Markov chain (Tauchen-Hussey 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 on Gauss-Hermite nodes, following Tauchen and Hussey (1991), with the choice of the standard deviation of the nodes proposed by Floden (2008) as an option.

Usage

tauchenHussey(
  alpha,
  sigma,
  rho,
  size,
  sigmaWeight = c("conditional", "floden", "unconditional")
)

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.
sigmaWeight: How the standard deviation $\hat\sigma$ of the normal distribution defining the nodes is chosen: "conditional" (the default, Tauchen and Hussey's original choice) uses $\hat\sigma = \sigma$; "unconditional" uses $\hat\sigma = \sigma_y = \sigma/\sqrt{1-\rho^2}$; "floden" uses Floden's compromise $\hat\sigma = w\sigma + (1-w)\sigma_y$ with $w = 1/2 + \rho/4$.

Value

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

Details

The nodes $y_1, \ldots, y_n$ and probability weights $\omega_1, \ldots, \omega_n$ are those of the $n$-point Gauss-Hermite rule for $N(\alpha, \hat\sigma^2)$, and $$P_{ij} \propto \omega_j \frac{\phi(y_j;\ (1-\rho)\alpha + \rho y_i,\ \sigma)}{\phi(y_j;\ \alpha,\ \hat\sigma)},$$ normalized by rows, with $\phi(\cdot; m, s)$ the normal density. It is fitFunction with the AR(1) transition density and quadrature = "gauss-hermite".

The method is accurate for moderate persistence, but with $\hat\sigma = \sigma$ the nodes are too concentrated when $\rho$ is close to one and the chain understates the unconditional variance; with $\hat\sigma = \sigma_y$ it overstates the persistence and understates the conditional variance. Floden (2008) shows that his weighted choice is a good compromise; for very persistent processes rouwenhorst is usually preferable (Kopecky and Suen 2010). The vignette reproduces the conditional and unconditional standard deviations of Floden's Table 1.

References

Tauchen, G. and Hussey, R. (1991). Quadrature-based methods for obtaining approximate solutions to nonlinear asset pricing models. Econometrica, 59(2), 371-396. doi:10.2307/2938261

Floden, M. (2008). A note on the accuracy of Markov-chain approximations to highly persistent AR(1) processes. Economics Letters, 99(3), 516-520.

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 <- tauchenHussey(alpha = 0, sigma = sqrt(0.03), rho = 0.95, size = 5,
                     sigmaWeight = "floden")
out$chain
#> Tauchen-Hussey AR(1) Approximation (rho = 0.95) 
#>  A  5 - dimensional discrete Markov Chain defined by the following states: 
#>  -0.7809, -0.3706, 0.0000, 0.3706, 0.7809 
#>  The transition matrix  (by rows)  is defined as follows: 
#>              -0.7809      -0.3706       0.0000       0.3706       0.7809
#> -0.7809 9.207167e-01 7.920494e-02 7.839379e-05 8.692134e-10 1.546272e-17
#> -0.3706 4.762712e-02 8.485696e-01 1.036614e-01 1.418607e-04 5.226711e-10
#> 0.0000  3.971920e-05 8.734415e-02 8.252323e-01 8.734415e-02 3.971920e-05
#> 0.3706  5.226711e-10 1.418607e-04 1.036614e-01 8.485696e-01 4.762712e-02
#> 0.7809  1.546272e-17 8.692134e-10 7.839379e-05 7.920494e-02 9.207167e-01
#> 
# unconditional standard deviation of the chain (Floden 2008, Table 1: 0.4792)
pi <- as.numeric(steadyStates(out$chain))
sqrt(sum(pi * (out$states - sum(pi * out$states))^2))
#> [1] 0.4792287