Threshold and Smooth Transition Models

A currency pair might drift almost like a random walk near its long-run fair value, but snap back forcefully once it strays too far — central bank intervention, or simple arbitrage pressure, kicks in only past a certain distance from equilibrium. A single linear model (one fixed set of coefficients everywhere) can't capture this: it either fits the calm, near-equilibrium behavior and misses the snap-back, or averages the two regimes together and describes neither well. Threshold and smooth transition models solve this by letting the model's coefficients themselves change depending on where the series currently sits — either abruptly, once a threshold is crossed (TAR), or gradually, phasing in over a transition zone (STAR).

An analogy: a rubber band with slack

Picture a rubber band tied loosely around a peg, with some slack. Move an object a little within the slack, and there's no pull back — it just drifts freely, like a random walk. But once the object moves far enough that the band pulls taut, a restoring force kicks in and pulls it back. A threshold model (TAR) is like a stiff rubber band that has zero tension right up to the point of going taut, then applies full tension instantly. A smooth transition model (STAR) is like a band made of a slightly stretchy material — the pull increases gradually as the object gets further out, rather than switching on all at once.

The mechanics, one symbol at a time

A two-regime threshold autoregressive (TAR) model switches abruptly between two AR processes depending on whether a "transition variable" $z_{t-d}$ (often a lag of the series itself) crosses a threshold $c$:

$$y_t = \begin{cases} \phi_1^{(1)} y_{t-1} + \varepsilon_t & \text{if } z_{t-d} \le c \\ \phi_1^{(2)} y_{t-1} + \varepsilon_t & \text{if } z_{t-d} > c \end{cases}.$$

In plain English: the series follows one set of dynamics (say, near-random-walk behavior, $\phi_1^{(1)}$ close to 1) when it's within the threshold, and a different set (say, strongly mean-reverting, $\phi_1^{(2)}$ well below 1) once it crosses the threshold — a hard, discontinuous switch. A smooth transition (STAR) model replaces the abrupt switch with a continuous blend:

$$y_t = \phi_1^{(1)} y_{t-1}\big(1-G(z_{t-d})\big) + \phi_1^{(2)} y_{t-1}\, G(z_{t-d}) + \varepsilon_t, \qquad G(z) = \frac{1}{1+e^{-\gamma(z-c)}},$$

where $G(z)$ is a logistic function sliding smoothly from 0 to 1 as $z_{t-d}$ crosses $c$, and $\gamma$ controls how sharp that transition is — a large $\gamma$ makes $G$ behave almost like the TAR model's hard switch, while a small $\gamma$ spreads the transition out over a wide zone, blending the two regimes' coefficients gradually rather than flipping between them.

Worked example 1: a TAR mean-reversion signal

Suppose a currency deviation series $y_t$ (percent from fair value) follows $\phi^{(1)} = 0.98$ (near unit root, essentially drifting) when $|y_{t-1}| \le 2\%$, and $\phi^{(2)} = 0.60$ (strong mean reversion) when $|y_{t-1}| > 2\%$. If today's deviation is $y_t = 1.5\%$ (inside the threshold), the forecast for tomorrow is $0.98 \times 1.5 = 1.47\%$ — barely any pull back, essentially persisting. If instead today's deviation is $y_t = 3.0\%$ (outside the threshold), the forecast is $0.60 \times 3.0 = 1.80\%$ — a much sharper pull back toward zero, a 40% reduction in one step, reflecting the regime switch into active mean-reversion dynamics.

Worked example 2: comparing TAR and STAR at the boundary

Using the same threshold $c=2\%$ but a STAR model with $\gamma = 5$, evaluate $G(z)$ at $z = 1.9\%$ (just inside) and $z=2.1\%$ (just outside): $G(1.9) = \frac{1}{1+e^{-5(1.9-2)}} = \frac{1}{1+e^{0.5}} \approx \frac{1}{1+1.65} \approx 0.377$, and $G(2.1) = \frac{1}{1+e^{-5(2.1-2)}} = \frac{1}{1+e^{-0.5}} \approx \frac{1}{1+0.61} \approx 0.622$. The blended coefficient at $z=1.9$ is $0.98(1-0.377) + 0.60(0.377) \approx 0.611 + 0.226 = 0.837$, and at $z=2.1$ it's $0.98(0.378) + 0.60(0.622) \approx 0.371+0.373=0.744$ — the effective mean-reversion strength shifts gradually from about 0.84 to 0.74 across this narrow zone, rather than jumping abruptly from 0.98 to 0.60 as the TAR model would at exactly $c=2\%$.

What this means in practice

Threshold and smooth transition models are the standard tool for capturing "band-limited" mean reversion — FX pairs kept within intervention bands, spreads that only correct once transaction costs are cleared, or entry/exit zones in pairs trading (see Entry and Exit Bands for Spreads) where reversion strength genuinely depends on how far a series has strayed, not just whether it has strayed at all.