tauchen: Discretize an AR(1) process into a Markov chain (Tauchen's...

In markovchain: Easy Handling Discrete Time Markov Chains

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

Description

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 an evenly spaced grid, following Tauchen (1986).

Usage

tauchen(alpha, sigma, rho, size, k = 3)

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.
k	A single positive number, the half-width of the grid in units of the process's unconditional standard deviation \sigma_y=\sigma/\sqrt{1-\rho^2}. The default, 3, follows Tauchen (1986)'s own recommendation and covers the great majority of the stationary distribution's mass for typical rho.

Details

The grid is n=\code{size} evenly spaced points y_1<\cdots<y_n spanning [\alpha-k\sigma_y,\ \alpha+k\sigma_y], with half-spacing w=(y_n-y_1)/(2(n-1)). Writing \Phi for the standard normal CDF, the transition probabilities from grid point y_i are

P_{i1} = \Phi\!\left(\frac{y_1-(1-\rho)\alpha-\rho y_i+w}{\sigma}\right),

P_{in} = 1-\Phi\!\left(\frac{y_n-(1-\rho)\alpha-\rho y_i-w}{\sigma}\right),

P_{ij} = \Phi\!\left(\frac{y_j-(1-\rho)\alpha-\rho y_i+w}{\sigma}\right) - \Phi\!\left(\frac{y_j-(1-\rho)\alpha-\rho y_i-w}{\sigma}\right), \quad 1<j<n,

i.e. the probability that y_t (a normal draw centred at the AR(1) conditional mean) lands in the half-open bin around y_j, with the two end bins extended to \pm\infty so that rows sum to exactly 1.

Tauchen's method is simple and fast (O(n^2) normal CDF evaluations) but the grid width is fixed by k regardless of size: for a coarse grid (small size) it under-resolves the bulk of the distribution, and for rho close to \pm1 the true unconditional variance is large and sensitive to k. See rouwenhorst for an alternative that tends to match the persistence of near-unit-root processes more accurately and needs no arbitrary grid-width parameter.

Value

A named list with two elements:

chain

The discretized markovchain object, with state names equal to the grid values of y formatted to 4 significant digits.

states

The numeric grid of y-values themselves, in the same order as chain's states. Returning the actual levels alongside the chain, rather than only generic state labels "1", "2", ..., is deliberate: the whole point of discretizing an AR(1) process is usually to do further numeric work with the levels (e.g. plugging them into a pricing formula), and re-deriving the grid from alpha, sigma, rho and k a second time by hand is both extra work and a place for an off-by-one or rounding mismatch to creep in.

References

Tauchen, G. (1986). Finite state markov-chain approximations to univariate and vector autoregressions. Economics Letters, 20(2), 177-181.

See Also

rouwenhorst

Examples

out <- tauchen(alpha = 0, sigma = 1, rho = 0.9, size = 5)
out$states
out$chain
# The chain's own stationary variance should be close to the AR(1)'s
# theoretical unconditional variance sigma^2 / (1 - rho^2).
pi <- as.numeric(steadyStates(out$chain))
sum(pi * (out$states - sum(pi * out$states))^2)
1 / (1 - 0.9^2)

markovchain documentation built on Oct. 10, 2026, 9:07 a.m.