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L53 Nonlinear Model Extensions

TAR was basic, we will extend it further now

Extensions of the TAR Model

  • SETAR: threshold variable is the TS itself (\(y_{t-d}\))
  • MTAR: uses difference or momentum (\(\Delta y_{t-d}\)) as threshold
  • Multiple thresholds: more than two regimes
  • STAR: Transition between regimes are smooth

The STAR Model

  • Transition across regimes happen smoothly rather than abruptly as in TAR
  • Key features
    • Smooth regime transition
    • has a Transition Function (logistic, exponential \(\implies\) smoothness and location of transition)
    • Flexibility (simplicity of linear model, with ability to model nonlinear dynamics across 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 parameters for the second regime

Transition Functions

Logistic Transition Function

\[ G(z_{t-d}; \gamma,c) = \dfrac{1}{1+e^{ -\gamma(z_{t-d}-c) }} \]
  • Behavior
    • \(G \to 0\) \(z_{t-d}\) is far below \(c\) (not happening)
    • \(G \to 1\) \(z_{t-d}\) is far above \(c\) (transition is happening)
  • Example: Gradual shifts in economic expansions

Exponential Transition Function

\[ G(z_{t-d}; \gamma,c) = {1+e^{ -\gamma(z_{t-d}-c)^{2} }} \]

  • Behavior

    • \(G=0;\) \(z_{t-d} =c\)
    • \(G\to 1;\) \(z_{t-d}\) moves away in either side of \(c\)

  • Deviation from \(c\) is symmetric (Like a slide, go up and down)

  • Application: Oscillatory dynamics

Steps in Model Building

  • Specify the Threshold variable \(z_{t-d}\) = \(y_{t-d}\) or an external variable
  • Estimate linear model
    • Fit a linear AR as a benchmark to test for nonlinearity

  • Test for nonlinearity

    • LM test to justify using STAR

  • Select the transition function

    • choose between logistic or exponential based on application
  • Estimate parameters
    • \(\gamma\), \(c\) and AR coefficients
    • using MLE, Nonlinear least squares
  • Model dynamics
    • residual autocorrelation, heteroscedasticity
    • adequacy of smooth transition

SETAR

SE TAR models

  • DEFINITION: Threshold variable = \(y_{t-d}\)
  • self-referencing nature. Regime-switching dynamics based on its own history

Key Features

  • Regime-specific dynamics
    • based on the value of its own lags
  • Threshold Determination
  • Piecewise linearity
  • Flexibility

Model Formulation

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

  • Steps

    • Determine threshold values
    • Choose number of regimes
      • \(k=2\) or \(k\gt 2\)
    • Estimate the threshold
    • Estimate parameters
    • Test for nonlinearity
      • statistical test like Hansen's test to confirm presence of regime switching
    • Diagnostics
      • check residuals for autocorrelation
      • Perform out-of-sample forecasting

  • Advantages and challenges are the same

Extensions of SETAR Model

  • SETAR with multiple thresholds
  • MTAR
  • CSETAR (continuous SETAR) → Regime transition is smoothed using continuous functions
  • Seasonal SETAR: Accounts for seasonality by incorporating periodic thresholds