Regimes & Nonlinear Models

Regimes are essentially distinct "states" of a system. In nonlinear modeling, we assume that any two states are not necessarily governed by the same rules, and they are typically divided by thresholds or transition mechanisms.

Modelling Regimes

1. Threshold-Based Regimes

Regimes are defined by a hard cutoff.
$$
\begin{align}
\text{Regime 1:} & \quad y_{t-d} \le \gamma \
\text{Regime 2:} & \quad y_{t-d} > \gamma \
\end{align}
$$
- \(y_{t-d}\): The lagged value of the same or another variable.
- \(\gamma\): The threshold value.
- Example: TAR, SETAR.

2. Smooth Transition Regimes

Instead of a hard switch, the system moves gradually between states using a transition function.
$\(G(z_{t-d}; \gamma, c) = \dfrac{1}{1+e^{ -\gamma(z_{t-d}-c) }}\)$
- \(z_{t-d}\): Threshold variable.
- \(\gamma\): Slope parameter (determines the speed of transition).
- \(c\): Threshold value (location of the transition).
- Example: STAR model.

3. Latent State-Based Regimes

Regimes are governed by an unobserved (latent) Markov process.
$\(P = \begin{bmatrix} p_{11} & p_{12} \\ p_{21} & p_{22} \end{bmatrix}\)$
- \(p_{ij}\): Probability of transitioning from state \(i\) to state \(j\).
- Example: Markov switching models like MSAR.

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Benefits & Challenges of Regime Models

Benefits	Challenges
Improved Forecasting: Better accuracy for systems with abrupt changes (e.g., market crashes).	Threshold Selection: Finding the "best" \(\gamma\) is computationally intensive.
Interpretability: Clearly defined states (e.g., "Expansion" vs "Recession").	Overfitting: Too many regimes make the model overly complex and poor at generalizing.
Flexibility: Can capture a wide variety of nonlinear behaviors that linear models miss.	Data Sufficiency: Each regime requires a sufficient number of data points for valid estimation.

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The TAR (Threshold Autoregressive) Model

The TAR model is a specific class of nonlinear models where behavior switches between regimes depending on whether a threshold variable exceeds a predefined value.

Key Features

• Regime Switching: The model assumes different dynamics in different states.
• AR Modeling: Each regime is modeled by its own specific Autoregressive (AR) process.
• Piecewise Linearity: While the overall system is nonlinear, it remains linear within each regime, aiding in estimation.
• Threshold Variable: Commonly a lagged value of the series itself (\(y_{t-d}\)).

Model Formulation

$$
y_{t} = \begin{cases}
\phi_{1,0} + \phi_{1,1} y_{t-1} + \dots \phi_{1,p} y_{t-p} + e_{t}, & \text{if } y_{t-d} \le \gamma \
\phi_{2,0} + \phi_{2,1} y_{t-1} + \dots \phi_{2,p} y_{t-p} + e_{t}, & \text{if } y_{t-d} > \gamma
\end{cases}
$$
- \(\phi_{i,j}\): AR coefficient for regime \(i\) and lag \(j\).
- \(\gamma\): The threshold value.

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Steps in Model Building

1. Specify Threshold Variable: Usually \(y_{t-d}\).
2. Determine Lag Structure: Choose the order \(p\) for the AR process in each regime.
3. Estimate Threshold (\(\gamma\)): Typically done using grid search to minimize residual variance or information criteria.
4. Estimate Parameters: Fit the AR processes for each regime (usually via Least Squares).
5. Diagnostics:
   • Check residual autocorrelation and stationarity.
   • Use hypothesis tests for nonlinearity, such as Hansen’s test.

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Applications

• Economics: Modeling business cycles (Expansion vs. Recession) or inflation dynamics using unemployment rates as the threshold \(\gamma\).
• Climatology: Representing sudden shifts in climate variables or weather patterns.
• Engineering: Identifying operational thresholds in load-bearing structures or machinery to predict failure.

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