Smooth Transition Autoregression (STAR)

The defining feature of STAR is the smooth transition function G(s_t; γ, c). This function takes a value between 0 and 1, creating a continuum of states between the two extreme regimes.

• Logistic STAR (LSTAR): Uses a logistic function, suitable for modeling asymmetric cycles where expansion and contraction have different dynamics.
• Exponential STAR (ESTAR): Uses an exponential function, appropriate for symmetric adjustment where behavior depends on the magnitude, not the sign, of the deviation.
• The slope parameter γ controls the speed of transition; as γ → ∞, the model approaches a threshold model.

Regime determination in STAR models depends on an observable transition variable s_t, which governs the weighting between regimes.

• Self-exciting STAR: The transition variable is a lagged value of the dependent variable (e.g., y_{t-d}), capturing feedback-driven regime changes.
• Exogenous STAR: The transition variable is an external driver, such as a volatility index (VIX), interest rate spread, or macroeconomic indicator.
• The delay parameter d determines which lag of the transition variable influences the current regime, typically selected via information criteria.

The STAR model is expressed as a weighted average of two linear autoregressive processes:

y_t = (φ₁'x_t)(1 - G) + (φ₂'x_t)(G) + ε_t

• The model is linear in parameters conditional on the transition function parameters γ and c.
• This structure allows the autoregressive coefficients to evolve smoothly over time as the economy or market transitions between states.
• The location parameter c defines the threshold around which the transition is centered.

Parameter estimation in STAR models requires nonlinear optimization because the transition parameters γ and c enter the model nonlinearly.

• Concentrated grid search: Fix γ and c on a grid, estimate the linear AR parameters via OLS, then optimize over the grid.
• Numerical optimization: Use algorithms like BFGS or Newton-Raphson to jointly estimate all parameters.
• Starting values: Critical due to the potential for local optima; grid search provides robust initialization.
• Scaling: The transition variable is typically standardized to make γ scale-free.

Before fitting a STAR model, a linearity test determines whether nonlinear specification is warranted. The null hypothesis of linearity (γ = 0) presents a nuisance parameter problem.

• Luukkonen-Saikkonen-Teräsvirta (LST) test: Replaces the transition function with a Taylor series approximation, creating an auxiliary regression for an LM-type test.
• Model selection sequence: Test linearity → if rejected, choose between LSTAR and ESTAR via a sequence of nested hypothesis tests on the Taylor expansion coefficients.
• Delay selection: The test is repeated for different values of d; the one producing the strongest rejection is chosen.

A key implication of STAR models is that impulse responses are history-dependent—the effect of a shock depends on the current regime and the sign and magnitude of the shock.

• Generalized Impulse Response Functions (GIRF) : Simulate the model forward from a specific initial condition, averaging over bootstrap draws of future shocks.
• Shocks can push the system across the threshold, causing the dynamics to shift mid-response.
• This captures empirical phenomena like asymmetric business cycles where recoveries differ from contractions, or limit order book dynamics where market impact varies with liquidity state.

A nonlinear model where autoregressive coefficients change abruptly based on a lagged observable variable crossing a threshold. Unlike STAR, TAR imposes a hard boundary between regimes.

• Self-Exciting TAR (SETAR): The threshold variable is a lag of the dependent variable itself.
• Key Difference: TAR is a special case of STAR where the transition function is a step function (infinite speed).
• Application: Modeling asymmetric business cycles where contractions and expansions have distinct dynamics.

A regime-switching framework where the state is governed by an unobservable Markov chain with discrete states. Transitions are probabilistic, not deterministic.

• Contrast with STAR: Markov Switching treats the regime as a latent random variable; STAR uses an observable transition variable to determine the regime.
• Key Output: A Transition Probability Matrix defining the likelihood of switching between states.
• Strength: Capturing structural breaks without needing to specify the threshold variable a priori.

A doubly stochastic process where an underlying Markov chain emits observable data according to state-dependent probability distributions. The Baum-Welch Algorithm is used for parameter estimation.

• Inference: The Viterbi Algorithm decodes the most likely sequence of hidden regimes from observed returns.
• Finance Use: Inferring latent volatility regimes (low/high) from daily price data.
• Relation to STAR: Both model regime changes, but HMMs assume discrete, unobservable states, whereas STAR models a continuous transition driven by a specific variable.

An extension of the Markov Switching model where the transition probabilities are not constant but depend on exogenous information variables.

• Mechanism: A logistic function links observable variables (e.g., the yield spread) to the probability of switching regimes.
• Conceptual Bridge: TVTP introduces observable drivers into the Markov framework, creating a philosophical link to STAR's use of a transition variable.
• Advantage: Allows the persistence of a bull market to vary with macroeconomic conditions.

The mathematical core of the LSTAR model, defined as G(s_t; γ, c) = [1 + exp(-γ(s_t - c))]^-1. This function produces an S-shaped curve.

• γ (Gamma): The slope parameter determining the speed of transition. As γ → ∞, the function approaches a step function (TAR).
• c (Threshold): The location parameter where the transition is centered.
• Behavior: Allows the model to capture asymmetric adjustments where one regime has different dynamics than the other.

The defining mechanism of the ESTAR model, given by G(s_t; γ, c) = 1 - exp(-γ(s_t - c)^2). This function is symmetric and U-shaped.

• Symmetry: The function approaches 1 for large deviations of s_t from c, and 0 when s_t is near c.
• Application: Modeling mean-reverting behavior where the speed of adjustment increases with the size of the deviation from equilibrium.
• Real Exchange Rates: ESTAR is widely used to model nonlinear adjustment of exchange rates to purchasing power parity.