1.8.3 Threshold Models

Basic Prerequisites: Econometrics & Time Series (AR models, stationarity, ergodicity) and 01 · From Zero.

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1. Intuition & Practical Objective

Markov-switching (02 · MS) treats the regime as latent - you never see it. Threshold models flip this: the regime is observable (or nearly so), because it is decided by a lagged value of the observed series crossing a threshold. If returns (or some state variable like a drawdown measure) are below a level $\gamma$, you are in the low regime; above it, in the high regime. No hidden coin - the data itself flips the switch.

Two families (Tsay Ch 4, verified):

1. SETAR - Self-Exciting Threshold AR. The AR coefficients switch discontinuously when the delay-$d$ lag $x_{t-d}$ crosses $\gamma$:

$$x_t={\varphi }_0^{(j)}+\sum _i{\varphi }_i^{(j)}{x}_{t-i}+a_t^{(j)}\text{if }{\gamma }_{j-1}\le x_{t-d}<{\gamma }_j.$$

The regime is fully observed once $x_{t-d}$ is known. This gives threshold models a forecasting advantage Hamilton flags: when the threshold variable is observed, the next regime is known with certainty (SETAR is a single model at horizon $\le d$); Markov-switching always produces a mixture.

2. STAR - Smooth Transition AR. Replaces the hard cut with a smooth transition function $0\le F\le 1$ (logistic or exponential):

$$x_t=c_0+\sum _i{\varphi }_{0,i}{x}_{t-i}+F\left[\frac{x_{t-d}-\ell }{s}\right](c_1+\sum _i{\varphi }_{1,i}{x}_{t-i})+a_t,$$

so the conditional mean is a weighted combination of two linear models with weight $F$. The logistic transition $F(z)=1/(1+e^{-z})$ is $0$ for very negative $x$, $1$ for very positive, and $1/2$ at the location $\ell$ with slope controlled by $s$.

Practical objective: when you can name the switching variable (e.g., vol-level or drawdown state), threshold models estimate the threshold and the per-regime dynamics directly - cleaner than forcing a hidden Markov state. Tsay’s threshold-cointegration application (S&P 500 futures basis, Ch 8 §8.7) finds thresholds ${\hat{\gamma }}_1=-0.0226,{\hat{\gamma }}_2=0.0377$: outside the no-arbitrage band the ECM binds, inside it does not - a classic “regime = active vs inactive” story.

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2. Mathematical Ground Truth & Derivations

SETAR($2;d$) ergodicity (Tsay Ch 4, verified). For a TAR(1) with threshold variable $x_{t-1}$ (delay $d=1$), the two-regime process is ergodic if

$${\varphi }_1^{(1)}<1,{\varphi }_1^{(2)}<1,{\varphi }_1^{(1)}\cdot {\varphi }_1^{(2)}<1.$$

Note the product condition: one regime may have ${\varphi }_1>1$ as long as the other compensates - a key difference from a single linear AR, where $|{\varphi }_1|<1$ is required.

Estimation (least squares). For a fixed threshold $\gamma$, each regime is a linear regression on the observations whose $x_{t-d}$ falls on that side; total SSE is the sum over regimes. The threshold is then chosen by grid search over the observed values of $x_{t-d}$ (Tsay recommends candidate thresholds in the interior, e.g. the $8\%$–$92\%$ quantiles, to keep enough observations per regime):

$$\hat{\gamma }=\arg \underset{\gamma }{\min }[{\text{SSE}}_1(\gamma )+{\text{SSE}}_2(\gamma )],{\text{SSE}}_j=\sum _{t:x_{t-d}\in \text{regime }j}(x_t-{\hat{x}}_t{)}^2.$$

SETAR vs Markov-switching - the forecasting contrast (Tsay, verified). SETAR: if the horizon $\le d$ and $x_{t-d}$ is observed, the regime is known exactly, so the conditional mean is a single linear model. Only beyond horizon $d$ do you average over which regime might be entered. Markov-switching: the state is always latent, so the forecast is always a mixture. Latent vs observed state is a modeling choice with real forecasting consequences.

STAR as a smooth generalization. As the slope $s\to 0$, the logistic $F\to$ a step function and STAR collapses to SETAR. The smooth version avoids the discontinuity (better for estimation and for regimes that transition gradually) at the cost of the weakly-identified parameters $(\ell ,s)$.

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3. Computational Implementation - SETAR threshold recovered exactly, STAR smooths it

Part 1 - SETAR grid search. Simulate a 2-regime AR(1) with a hard threshold at $\gamma =0$ (low regime $\varphi =0.3$, high regime $\varphi =0.9$), then recover the threshold and both regimes by grid search. Stdlib only.

The grid search finds the threshold exactly at $0.000$ and recovers both regimes’ AR coefficients well - a sharp, unambiguous demonstration of how an observed-state regime model works.

Part 2 - STAR smooths the hard threshold. Fit a logistic-STAR to the same SETAR data and show it approximates the threshold with a smooth transition (midpoint $c\approx \gamma$) at nearly equal SSE.

The STAR midpoint lands at $0.00$ - right on the SETAR threshold - and reaches $1.035\times$ the SETAR SSE with a steep slope ($s=0.05$, nearly a step). STAR is the smooth generalization of SETAR; the “regime” is a weighted blend, not a hard cut.

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4. Failure Modes & First-Principles Breakdowns

1. Threshold undefined under $H_0$. Testing for a threshold is delicate: under the null of a single linear regime, the threshold $\gamma$ is not identified (the classic Davies problem), so a standard LR test statistic has no null distribution (Tsay Ch 4 notes this explicitly). Hypothesis tests must be simulated/bootstrap, not taken from ${\chi }^2$ tables.
2. Ergodicity vs one-regime stability. The two-regime process can be ergodic with one regime’s AR coefficient above $1$ (the product condition). Practitioners who impose $|\varphi |<1$ in each regime throw away valid nonlinear behavior; those who ignore it may simulate explosive paths. Understand which you are assuming.
3. Weak identification of the smooth parameters $(c,s)$. STAR’s location and slope are only weakly identified (a broad, flat likelihood), so fitted $(c,s)$ bounce between runs even when in-sample SSE is nearly identical - unlike SETAR’s sharply-identified threshold. Report the whole transition, not the point estimate.
4. Threshold-variable choice. The result depends on which lagged variable defines the regime and the delay $d$; misspecify the threshold variable and the “regimes” are artifacts. Cross-validate over candidate variables.

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5. References

• Tsay, Ruey S.: Analysis of Financial Time Series, 3rd ed.
• Tong, Howell: Non-Linear Time Series (1990) and Threshold Models in Non-linear Time Series Analysis (1983)
• Teräsvirta, Timo: Specification, Estimation, and Evaluation of Smooth Transition Autoregressive Models, JASA (1994)

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6. Connected Graph Bridges

• Back: 02 · Markov-Switching · Index Hub
• Forward: 05 · Failure Modes (the threshold-undefined-under-$H_0$ caveat)
• Contrast: 02 · Markov-Switching (latent state) and 04 · Hidden Markov Models (observed state vs hidden state modeling choice)
• Base: Econometrics & Time Series (AR stationarity/ergodicity)