Markov-Switching VAR

Python · R · Part 11 of the VAR arc — data-discovered regimes

Overview

Part 11 of the VAR arc: discrete regimes, discovered by the data. Part 5's time-varying-parameter VAR let the economy's dynamics drift smoothly. But some changes are not gradual — they are switches: the economy flips between a calm state and a turbulent one, between expansion and recession. The Markov-switching VAR (Hamilton 1989; Krolzig 1997) models exactly that — the VAR's parameters jump between a small number of regimes governed by a latent Markov chain the data infer. Where Part 10 imposed the Great-Moderation break to identify shocks, the MS-VAR is handed no dates at all: it discovers the regimes, and rediscovers 1984 on its own. Built from scratch with a Gibbs sampler whose centrepiece is the forward-filter–backward-sample of the hidden regime path.

y_t = c_{S_t} + B_1 y_{t-1} + \dots + B_p y_{t-p} + \varepsilon_t, \quad \varepsilon_t\sim\mathcal N(0,\Sigma_{S_t}), \qquad P_{jk}=\Pr(S_t=k\mid S_{t-1}=j)

An MSIH-VAR and its forward-filter–backward-sample Gibbs

An MSIH-VAR — switching intercept and covariance, constant autoregressive dynamics — where $S_t\in\{1,\dots,K\}$ is a latent Markov chain with transition matrix $P$ whose diagonal governs how persistent the regimes are. The from-scratch Gibbs sampler cycles three blocks: (1) the regime path $S_{1:T}$ by forward-filter–backward-sample — run the Hamilton filter forward (predict $\Pr(S_t\mid y_{1:t-1})$ through $P$ , update with each regime's VAR likelihood), then sample backwards $S_T$ then $S_t\mid S_{t+1}$ down to $t=1$ — the discrete-state analogue of the Carter–Kohn smoother; (2) the parameters — given $S$ , each regime's intercept and covariance are conjugate Normal / inverse-Wishart draws and the constant lag matrix a GLS draw across regimes; (3) the transition matrix $P$ — each row a Dirichlet draw from the counted transitions. Label-switching is fixed by ordering regimes by volatility (calm vs turbulent), on the same monetary system as Parts 4/10 (GDP growth, inflation, funds rate), $K=2$ , $p=2$ .

Results

The Great Moderation, rediscovered without a date. The turbulent (high-volatility) regime is active 40% of the time before 1984 but only 3% after — the model draws the Great-Moderation line on its own, precisely where Part 10 had to assume it, and the calm regime once entered essentially never leaves (bar 2008). Turbulent = volatile and recessionary: the turbulent regime carries 2–4× the shock volatility of the calm one (GDP-growth std 5.1 vs 2.4) and its smoothed probability spikes in the NBER recessions (mean 0.57 in recessions vs 0.13 in expansions) and across the high-inflation 1970s — the postwar economy's "stress" state. Persistence differs sharply: the transition matrix implies the calm regime lasts ~26 quarters on average and the turbulent one ~6 — long stable stretches punctuated by shorter bouts of turbulence. Switches, not drift: where Part 5 asked how much has the world drifted?, the MS-VAR answers which regime are we in right now?, delivering a probabilistic, dated chronology of postwar macro that a smooth-drift model cannot — and acting as an endogenous break detector that locates the Great Moderation with no prior information. Together, Parts 5, 10 and 11 span the ways a VAR can be non-constant: smooth drift, imposed regimes, and data-discovered regimes. Cross-checked in R (MSwM).

Notebooks

Python Markov-switching VAR from scratch — forward-filter–backward-sample Gibbs over the hidden regime path, recovering the postwar calm/turbulent chronology (Python) Open → R R cross-check — a Markov-switching model recovering the same regime chronology (MSwM) (R) Open →

Downloads

msvar.py The from-scratch MS-VAR module — Hamilton filter, forward-filter–backward-sample of the regime path, and the conjugate parameter / Dirichlet transition draws (new self-contained module) bvar_sv_data.csv The Part 4 monetary system — US quarterly GDP growth, GDP-deflator inflation, and the Fed funds rate, 1959–2019 (FRED-QD)

References

Hamilton, J. D. (1989). A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57(2), 357–384. — the Markov-switching model and the Hamilton filter
Krolzig, H.-M. (1997). Markov-Switching Vector Autoregressions. Springer. — the MSIH-VAR specification reproduced here
Chib, S. (1996). Calculating posterior distributions and modal estimates in Markov mixture models. Journal of Econometrics 75(1), 79–97. — the forward-filter–backward-sample of the regime path
Carter, C. K. & Kohn, R. (1994). On Gibbs sampling for state space models. Biometrika 81(3), 541–553. — the continuous-state smoother the FFBS is the discrete analogue of
Kim, C.-J. & Nelson, C. R. (1999). State-Space Models with Regime Switching. MIT Press. — Bayesian estimation of regime-switching state-space models
Sims, C. A. & Zha, T. (2006). Were there regime switches in U.S. monetary policy? American Economic Review 96(1), 54–81. — Markov-switching VARs for postwar US macro
Sanchez-Espigares, J. A. & Lopez-Moreno, A. (2021). MSwM: Fitting Markov Switching Models. R package. — the R cross-check implementation