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 $c$:

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:

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.