Kim-Piger-Startz (2008) Estimation of Markov Regime-Switching Regression Models with Endogenous Switching

Summary

Relaxes the standard exogeneity assumption of Hamilton (1989) Markov-switching regression — that the latent state $S_t$ is uncorrelated with regression residuals $\varepsilon_t$ — and develops two bias-correction techniques. The first extends Maddala-Nelson (1975) endogenous switching to the serially-dependent Markov case under joint normality; the second uses instrumental variables (IV) treating the unobserved regime indicator as an endogenous dummy. Both methods are identified by the Markov serial dependence structure (lagged state is exogenous) or time-varying transition probabilities. Monte Carlo confirms good performance; applied to Turner-Startz-Nelson (1989) volatility-feedback model for equity returns.

Key Claims

• Endogeneity bias. When $S_t$ is correlated with $\varepsilon_t$, maximum likelihood (ML) estimation under the exogeneity assumption yields inconsistent estimates of regime-specific intercepts and slopes. Rewriting equation (1) shows that the endogenous state introduces predictability into $\varepsilon_t$ that mimics regime-switching in the intercept, making $\beta_0$ and the bias term unidentified without correction.
• Technique 1 — Joint normality / bias correction. Parameterize transition probabilities via probit on exogenous variables $z_t$ (time-varying-probability Markov-switching, TVP-MS) or as constants (fixed-transition-probability Markov-switching, FTP-MS). Assume joint normality between $\varepsilon_t$ and the latent state innovation. This gives explicit correction terms $\lambda(\cdot)$ analogous to inverse Mills ratios from the Maddala-Nelson bivariate probit, added as additional regressors in the likelihood. Estimation uses the Hamilton filter or expectation-maximization (EM) algorithm on the augmented model.
• Technique 2 — Instrumental variables. Interpret the regime dummy $S_t$ as an endogenous regressor and the regime-switching regression as a standard IV problem. Instruments come from: (a) time-varying transition probability drivers $z_t$ correlated with $S_t$ but exogenous to $\varepsilon_t$; (b) $S_{t-1}$ — the lagged (unobserved) state, which by the Markov property correlates with $S_t$ while being predetermined relative to $\varepsilon_t$. Lagged state enters via filtered regime probabilities from the Hamilton filter, making the IV feasible despite $S_t$ being latent.
• Identification in FTP case. Even without time-varying transition probabilities ($\gamma = 0$), identification holds if the Markov chain has serial persistence ($p_{11} \neq p_{10}$) and $S_{t-1}$ is exogenous. The lagged filtered probability $P(S_{t-1}=1 | I_{t-1})$ functions as a valid instrument.
• Tests for endogeneity. Both techniques yield Hausman-type tests: compare the endogeneity-robust estimator to the standard ML estimate. Rejection implies the exogeneity assumption is violated.
• Monte Carlo results. Both bias-correction methods substantially reduce coefficient bias relative to standard ML under endogenous switching. The joint-normality method is more efficient when the normality assumption holds; IV is more robust to distributional misspecification.
• Application: volatility feedback. Turner-Startz-Nelson (1989) model of equity returns where variance follows a two-state Markov process; the return equation is affected by contemporaneous variance. Correcting for endogeneity materially changes estimated slope coefficients.

Concepts Introduced or Extended

• Markov-Switching Model

Entities Mentioned

• Chang-Jin Kim

Quotes

"We show that failure of the exogeneity assumption leads to significant bias in the coefficients of a regime-switching regression when estimation methods based on the exogeneity assumption are used."

My Take

An important paper that plugs a gap in the Markov-switching literature: essentially every application since Hamilton (1989) implicitly assumes the regime is exogenous, but in business-cycle and finance applications this is often implausible. The joint-normality approach is elegant but parametric; the IV approach is more robust. The key practical insight is that the Markov serial dependence itself provides instruments — no external instruments are needed as long as the lagged state is predetermined. Closely related to the broader Hamilton filter literature and to Diebold-Lee-Weinbach (1994) TVP-MS models.

Published as Kim, Piger, and Startz (2008), "Estimation of Markov Regime-Switching Regression Models with Endogenous Switching," Journal of Econometrics 143(2): 567–582 (the citation of record); the working paper circulated July 2003. The file slug retains -2003.