Markov-switching regression (Hamilton 1989)

Fit a K-regime Markov-switching model with a switching mean and a switching variance, \(y_t \mid s_t = j \sim N(\mu_j, \sigma_j^2)\), by EM with the Hamilton filter and Kim smoother. This is the model MSwM::msmFit(lm(y ~ 1), k, sw = c(TRUE, TRUE)) fits, computed natively: the E- and M-steps reproduce MSwM's EM (including its transition-matrix update, which normalises the summed smoothed joint probabilities by the occupancy of the destination regime, so its fixed point can differ from the exact maximum likelihood estimate in the fourth decimal of the log-likelihood). Instead of MSwM's random starts the EM runs from two deterministic starts (regimes split by the quantiles of \(y\) and of \(|y - median(y)|\)) and keeps the higher log-likelihood. No external package is used.

Usage

morie_regime_switching(x, k_regimes = 2)

Arguments

x

Numeric univariate series.

k_regimes

Number of latent regimes. Default 2.

Value

Named list with \(mu, sigma, transition, smoothed_probabilities, loglik, n, k_regimes, method\). Regimes are ordered by increasing mu. transition uses MSwM's column convention: transition[j, i] is \(Pr(s_t = j \mid s_{t-1} = i)\) (columns sum to 1). smoothed_probabilities is n x K (observations 1..n). loglik is the maximised log-likelihood (MSwM's logLikel slot stores its negative).

References

Hamilton, J. D. (1989). A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57, 357-384.

Examples

set.seed(1)
morie_regime_switching(x = c(rnorm(40, 0, 0.5), rnorm(40, 2, 1.5)))$mu
#> [1] 0.04399582 2.16976670