1.8.5 Failure Modes & Real-World Practice

Basic Prerequisites: 02 · Markov-Switching Models and 04 · Hidden Markov Models.

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1. Intuition & Practical Objective

Regime models are beautifully estimable and empirically fragile in three specific ways. This page names them precisely and shows them in numbers, so a practitioner knows which failure mode to guard against. The objective is not cynicism - it is knowing exactly where the model’s identifiability and stability break so the output can be used safely.

The three failures, in one line each:

1. Label switching - the likelihood is identical under a permutation of the state labels, so two runs can return the same model with swapped meanings.
2. Overfitting regimes - adding states always raises the likelihood; the model will happily invent regimes that are clones of real ones.
3. Regime persistence & detection lag - the filter is deliberately conservative, so it is slow to switch; over a fast strategy’s horizon this lag is a real cost, and on non-regime data it oscillates and overreacts.

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2. Mathematical Ground Truth & Derivations

Where each failure lives.

• Label switching is a symmetry of the likelihood. For any parameter vector $\theta =(\mu ,\sigma ,P,\pi )$, the permuted vector ${\theta }^{\prime }$ (state $0\leftrightarrow 1$ everywhere) gives the same marginal likelihood because the sum over states is invariant:

$$\ln L(\theta )=\sum _{t=1}^T\ln [\sum _{j}f(y_t\mid s_t=j)\mathbb{P}[s_t=j\mid y_{1:t-1}]],$$

and the inner sum is unchanged when you rename the states. Hamilton (1989 §4.2) states this explicitly (“the decision of which state to call state 0 and which to call state 1 is arbitrary”) and fixes it by normalization, e.g. ${\mu }_1>{\mu }_0$.

• Overfitting is the usual bias–variance trade (ESL Ch 7). A $K$-state HMM has $2K$ mean/vol params plus $K(K-1)$ transition params. The likelihood is non-decreasing in $K$ (a $K$-state model can always mimic a $(K-1)$-state one by duplicating a state with $P_{ii}\to 1$). The correction is a complexity penalty:

$$\text{BIC}=-2\ln \hat{L}+K_{\text{params}}\ln T,$$

which, on genuine 2-state data, decisively rejects a spurious third state (verified below).

• Detection lag is Bayesian by construction. The filtered probability moves only when evidence accumulates; with overlapping regime densities a single return barely moves it. The expected “time to detect” is proportional to how separable the regimes are ($|{\mu }_1-{\mu }_0|/\sigma$). Persistence $P_{ii}$ then interacts: high persistence makes the filter confident (good) but makes false switches costly (a regime that actually changed is not believed for several periods).

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3. Computational Implementation - the failures in numbers

Experiment 1 - label switching. Run EM twice on the same data from different random starts; both converge to the same likelihood but with permuted labels. Stdlib only.

Run A labels the positive-mean state first ($0.0108$); run B finds the exact same model with the labels swapped ($-0.0133$ first). Same likelihood, swapped meaning. The fix is a constraint (${\mu }_1>{\mu }_0$) so the optimizer cannot cross the symmetry.

Experiment 2 - overfitting regimes. Fit 2- and 3-state HMMs to the same genuine 2-state data; the 3rd state clones an existing one, and BIC rejects it.

The third state buys only $+0.925$ of log-likelihood by splitting the bull state into two near-identical regimes ($0.0112$ and $0.0101$). BIC’s complexity penalty ($6$ extra params) dominates, correctly choosing 2 states. If you do not penalize, the model invents regimes.

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4. Failure Modes & First-Principles Breakdowns (numbered)

1. Label switching (identifiability). The likelihood is invariant to a permutation of the states; EM can return a swapped labeling with identical fit (verified: $|$diff$|=0.0002$). Fix with an ordering constraint on a parameter. This matters in production: a “bull→bear” trade based on run A is a “bear→bull” trade based on run B.
2. Overfitting the number of regimes. Likelihood rises monotonically in $K$; the third state clones a real one (verified). Use BIC (or cross-validated likelihood) to choose $K$ - never max-likelihood alone.
3. Regime persistence vs detection lag. High $P_{ii}$ makes the filter confident but slow to admit a genuine switch; low overlap $|{\mu }_1-{\mu }_0|/\sigma$ extends detection time. The filtered probability that you trade on is lagging by construction - cost that lag, don’t ignore it.
4. Weakly-identified parameters on rare regimes. A regime visited only a few times has its $\mu ,\sigma ,P$ estimated from a handful of effective observations (page 02’s $0.301$ vs $0.10$ transition), regardless of total sample size.
5. Regime detection on non-regime data. If returns are truly a single Gaussian (or the nonlinearity is in variance, not mean - Tsay Ch 3/Ch 4), the two-state filter still manufactures a story. Test for regime structure first (BDS, threshold tests under the correct null - see 03 · Threshold Models).

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5. References

• Hamilton (1989)
• ESL, Ch 7 (model selection, BIC, bias–variance) and Ch 8.5 (EM monotonicity). Verified: esl_ch6-10.md.
• Tsay
• Ang & Timmermann (2012)

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6. Connected Graph Bridges

• Back: 04 · Hidden Markov Models · Index Hub
• Forward: 06 · Advanced Extensions (Regime Allocation)
• Testing roots: Statistics & Inference (hypothesis testing, model selection) · Econometrics & Time Series
• Sibling: Backtesting Hygiene (multiple-testing: testing many regime specs inflates apparent skill) · Signal Processing & the Kalman Filter (filter lag in the continuous-state case)