The Threshold VAR forces an abrupt switch at a single cutoff. In practice, monetary policy does not flip overnight, credit conditions do not jump from loose to tight at a single spread level, and consumer sentiment drifts rather than snaps. The Smooth Transition Regression (STR), formalized by Granger and Terasvirta (1993) and extended by Terasvirta (1994), replaces the indicator function with a bounded, monotone transition function - typically logistic or exponential - that moves continuously from 0 to 1 as a transition variable crosses its location parameter.

The mechanism is a weighted average of two linear regressions. At every point in time, the model's effective coefficient vector is (1 - $G(s_t)) *$ beta_1 + $G(s_t) *$ beta_2, where G is the transition function evaluated at the transition variable $s_t$. When G is near 0 the model behaves like regime 1; when G is near 1 it behaves like regime 2; in between, both regimes contribute proportionally. Two parameters govern the transition: c (location - where the midpoint of the switch lies) and gamma (speed - how quickly G moves from 0 to 1). The Threshold VAR is the limiting case as gamma approaches infinity.

Estimation is by nonlinear least squares (NLS) or conditional maximum likelihood. The transition speed gamma is notoriously difficult to identify: the log-likelihood surface is often flat in the gamma direction, and different starting values can produce very different estimates. Terasvirta (1994) proposed a specification cycle: (1) test linearity using a third-order Taylor expansion of the transition function; (2) choose between logistic (LSTAR) and exponential (ESTAR) forms based on sequential F-tests; (3) estimate by NLS with a grid of starting values; (4) evaluate with misspecification tests.

The STR family is widely used in exchange-rate modeling (purchasing-power-parity adjustment speeds depend on the size of the deviation), interest-rate dynamics (adjustment toward equilibrium accelerates when rates are far from target), and output-gap studies (Phillips-curve slopes differ in high-gap versus low-gap regimes). The Bank of Canada and the European Central Bank have used LSTAR models in their forecasting suites.

The STR takes the same inputs as a linear regression or VAR - a dependent variable (or vector), a set of regressors including lagged values, and a transition variable $s_t$ that may be one of the regressors, a lagged endogenous variable, a time trend, or an exogenous indicator. The transition variable choice is the most consequential modeling decision, as it determines what drives the regime switch.

The core mechanical component is the transition function $G(s_t; gamma, c)$. The logistic form (LSTAR) produces asymmetric dynamics: behavior differs depending on whether the economy is above or below c. The exponential form (ESTAR) produces symmetric dynamics: behavior at both extremes (far above and far below c) differs from behavior near c. These are qualitatively different stories - the LSTAR says booms and recessions obey different rules; the ESTAR says large deviations from equilibrium (in either direction) trigger different dynamics than small deviations.

Estimation requires choosing starting values for gamma and c, then running NLS to minimize the residual sum of squares jointly over (beta_1, beta_2, gamma, c). The transition speed gamma is scale-dependent (its magnitude depends on the variance of $s_t)$, so practitioners often standardize $s_t$ before estimation or divide gamma by the sample standard deviation of $s_t$. Output includes the estimated transition function $G(s_t; gamma-hat, c-hat)$ evaluated at each observation, which provides a continuous regime-membership score between 0 and 1.

The European Central Bank uses LSTAR models to study asymmetric Phillips-curve dynamics: inflation responds more strongly to the output gap when the gap is positive (overheating) than when it is negative (slack). Terasvirta and Anderson (1992) documented this asymmetry for several OECD economies, and the finding has been replicated in more recent samples with time-varying natural rates.

Exchange-rate economists use ESTAR models to study purchasing-power-parity (PPP) adjustment. Michael, Nobay, and Peel (1997) showed that real exchange rates revert to PPP faster when deviations are large (in either direction), which is exactly the symmetric nonlinearity that ESTAR captures. This transaction-cost interpretation - small deviations are not worth arbitraging, large deviations trigger trade flows - motivates the exponential transition form.

The STR cannot handle latent regimes, multiple simultaneous transitions, or high-dimensional systems without severe curse-of-dimensionality problems. When the transition speed gamma is weakly identified (flat likelihood surface), the confidence interval for gamma can span the entire feasible range, making the nonlinearity essentially untestable. The model also assumes a single transition variable; if regime changes are driven by a combination of variables (e.g., output gap and credit spread jointly), the STR requires ad hoc dimension reduction.

Recent extensions include panel STAR models (Gonzalez, Terasvirta, and van Dijk 2005), smooth-transition GARCH for volatility dynamics, and multivariate STVAR systems (Hubrich and Terasvirta 2013). Bayesian estimation with priors on gamma helps regularize the transition speed estimate when the likelihood surface is flat.