Threshold Autoregression (TAR)

The threshold variable is a lagged, observable time series value that determines the active regime. Unlike Hidden Markov Models, the regime is directly observable and deterministic.

• Self-Exciting TAR (SETAR): The threshold variable is a lag of the dependent variable itself (e.g., y_{t-d}).
• Open-Loop TAR: The threshold variable is an exogenous series (e.g., VIX index, unemployment rate).
• Delay Parameter (d): Specifies which lag of the variable is used to determine the regime.

Each regime defined by the threshold has its own distinct autoregressive (AR) model with unique coefficients, capturing fundamentally different behaviors.

• Lower Regime: May model a mean-reverting process with negative serial correlation.
• Upper Regime: May model a trending, persistent process with high positive autocorrelation.
• Asymmetric Cycles: The model naturally captures the observation that financial markets fall faster (high negative drift) than they rise (low positive drift).

The threshold parameter (γ) is the critical value that partitions the state space. It is estimated simultaneously with the AR coefficients, typically via grid search.

• Methodology: The model is estimated for a sequence of candidate thresholds, and the value minimizing the residual sum of squares is selected.
• Trimming: A percentage of observations (e.g., 10-15%) is excluded from the extremes to ensure sufficient data in each regime.
• Confidence Intervals: Constructed using the likelihood profile or bootstrap methods to assess estimation uncertainty.

Before fitting a TAR model, a formal statistical test must reject the null hypothesis of a linear AR model to justify the added complexity.

• Tsay's F-test: Uses arranged autoregression and recursive least squares to detect threshold nonlinearity without specifying the threshold value a priori.
• Hansen's Supremum Test: Computes the maximum test statistic over a grid of possible thresholds and compares it to a bootstrapped null distribution.
• Issue: Standard likelihood ratio tests have non-standard distributions because the threshold is unidentified under the null hypothesis.

TAR models assume an instantaneous, discrete switch between regimes when the threshold is crossed. This contrasts with Smooth Transition Autoregression (STAR) models.

• TAR (Discrete): I(y_{t-d} > γ) — an indicator function creates a hard boundary.
• STAR (Continuous): Uses a logistic or exponential function for a gradual, weighted blend of regimes.
• Use Case: TAR is appropriate for modeling structural breaks or policy-triggered shifts; STAR is better for slowly evolving market sentiment transitions.

The basic two-regime TAR can be extended to multiple thresholds to capture more complex state-dependent dynamics.

• Three-Regime TAR: Models a 'neutral' or 'sideways' market between a lower and upper threshold, common in currency bands or mean-reverting arbitrage strategies.
• Band-TAR: A specific case where the middle regime is mean-reverting and the outer regimes are unit-root or trending processes.
• Estimation Complexity: The grid search dimension grows exponentially with the number of thresholds, requiring efficient optimization algorithms.

A time-series model where parameters switch between a finite number of regimes governed by an unobservable Markov chain. Unlike TAR, where the regime is determined by a known, observable threshold variable, the state in a Markov switching model is latent and inferred probabilistically. This captures structural breaks in financial data, such as shifts between bull and bear markets, where the driving force of the regime change is not directly measured.

A regime-switching model where the transition between regimes is continuous and smooth rather than abrupt. Governed by a logistic or exponential function of a transition variable, STAR models avoid the sharp discontinuity of TAR. This is ideal for modeling market dynamics where participants gradually shift sentiment, such as a slow transition from a low-volatility to a high-volatility environment, rather than an instantaneous break.

Statistical tests and algorithms designed to identify points in time where the underlying data-generating process has fundamentally changed. While TAR models regime shifts based on a variable crossing a threshold, structural break tests like the Chow test or Bai-Perron algorithm detect permanent shifts in model parameters without a predefined threshold variable. This is critical for identifying events like regulatory changes or black swan events that permanently alter market dynamics.

An extension of the Markov switching model where the probability of moving between regimes depends on observable exogenous variables, such as macroeconomic indicators or volatility indices. This bridges the gap between TAR's deterministic, observable threshold and the standard Markov model's fixed probabilities. For example, the probability of switching to a crisis regime might increase as the VIX index rises.

A deep learning architecture where a gating mechanism or mixture of experts activates different sub-networks based on the inferred market regime. This blends statistical regime detection with nonlinear function approximation. Unlike TAR's linear autoregressive components within each regime, these networks can model highly complex, nonlinear relationships within each state, learning both the regime boundaries and the within-regime dynamics simultaneously from data.

Algorithms that identify shifts in the statistical properties of a data stream in real-time, allowing trading systems to adapt immediately to new market conditions. Unlike TAR, which requires a predefined threshold variable, online changepoint detection methods like Bayesian online changepoint detection continuously monitor for any distributional shift, enabling immediate strategy adaptation without waiting for batch processing or relying on a single indicator.