Nonlinear Model Extensions¶

Building on the basic Threshold Autoregressive (TAR) framework, we can extend these models to handle more specific behaviors, such as self-exciting dynamics, momentum-based triggers, and smooth transitions between states.

Extensions of the TAR Model¶

SETAR: The threshold variable is the time series itself ($y_{t-d}$). It "excites" its own regime changes.
MTAR: Uses the difference or momentum ($\Delta y_{t-d}$) as the threshold variable. Useful for capturing asymmetrical responses to price movements.
Multiple Thresholds: Models that allow for more than two regimes (e.g., Low, Medium, and High volatility states).
STAR: Transitions between regimes are smooth rather than abrupt, modeled using continuous transition functions.

The STAR Model (Smooth Transition Autoregressive)¶

Unlike the standard TAR model where the switch is instantaneous, the STAR model allows the system to move gradually from one state to another.

Key Features¶

Smooth Regime Transition: Captures the "gray area" between states.
Transition Function: Uses mathematical functions (Logistic or Exponential) to define the speed ($\gamma$) and location ($c$) of the transition.
Flexibility: Combines the simplicity of a linear AR model with the power to model nonlinear dynamics across different regimes.

Model Formulation¶

$$y_{t} = \phi_{0} + \phi_{1} y_{t-1} + \dots + \phi_{p} y_{t-p} + G(z_{t-d};\gamma,c)(\psi_{0}+ \psi_{1}y_{t-1}+\dots + \psi_{p}y_{t-p}) + e_{t}$$
* $\psi_{i}$ are the parameters for the second regime, weighted by the transition function $G$.

Transition Functions¶

1. Logistic Transition Function (LSTAR)¶

$$G(z_{t-d}; \gamma,c) = \dfrac{1}{1+e^{ -\gamma(z_{t-d}-c) }}$$
* Behavior: As $z_{t-d}$ moves from far below $c$ to far above $c$, $G$ moves smoothly from $0$ to $1$.
* Application: Modeling gradual shifts, such as the transition from a period of economic contraction to a period of steady expansion.

2. Exponential Transition Function (ESTAR)¶

$$G(z_{t-d}; \gamma,c) = 1 - e^{-\gamma(z_{t-d}-c)^2}$$
* Behavior: $G$ is $0$ when $z_{t-d} = c$ and moves toward $1$ as the variable moves away from $c$ in either direction.
* Application: Modeling oscillatory dynamics where the behavior is the same for large positive and large negative deviations (symmetry).

The SETAR Model (Self-Exciting TAR)¶

The SETAR model is a "self-referencing" system where the regime-switching dynamics are based entirely on the series' own historical values.

Model Formulation¶

\[ y_{t} = \begin{cases} \phi_{1,0} + \phi_{1,1} y_{t-1} + \dots \phi_{1,p} y_{t-p} + e_{t}, & \text{if } y_{t-d} \le \gamma \\ \phi_{2,0} + \phi_{2,1} y_{t-1} + \dots \phi_{2,p} y_{t-p} + e_{t}, & \text{if } y_{t-d} > \gamma \end{cases} \]

Model Building Steps¶

Determine Threshold Variable: Set $z_{t-d} = y_{t-d}$.
Lag Selection: Choose the number of regimes ($k=2$ or $k>2$) and the AR order $p$ for each.
Threshold Estimation: Use grid search or information criteria to find the optimal $\gamma$.
Nonlinearity Testing: Perform Hansen’s test to confirm that regime switching is statistically present (justifying the complex model over a simple linear AR).
Diagnostics: Check residuals for autocorrelation and perform out-of-sample forecasting to test the model's predictive power.

Specialized Extensions¶

CSETAR (Continuous SETAR): Uses continuous functions to smooth the transition within the self-exciting framework.
Seasonal SETAR: Accounts for seasonality by incorporating periodic thresholds, allowing the regime-switching logic to change depending on the time of year.
MTAR (Momentum TAR): Specifically tracks the speed of change. If the series is dropping quickly (high negative momentum), it might trigger a "crash" regime.