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L52 Regimes and Nonlinear Models

  • Regimes = states (any two are not the same) (they are divided by thresholds)

Modelling Regimes

Threshold based regimes

  • Regime definition
\[ \begin{align} \text{Regime 1:} & y_{t-d} \lt \gamma \\ \text{Regime 2:} & y_{t-d} \gt \gamma \\ \end{align} \]
  • \(y_{t-d}\) is the lagged value of (another/same) variable
  • \(\gamma:\) threshold value
  • Example: TAR, SETAR

Smooth Transition regimes

  • 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
  • \(c:\) threshold value
  • Example: STAR model

Latent state based regimes

  • Markov transition matrix
\[ P = \begin{bmatrix} p_{11} & p_{12} \\ p_{21} & p_{22} \end{bmatrix} \]
  • \(p_{ij}:\) probability of transitioning from \(i\) to \(j\)
  • Example: Markov switching models like MSAR

Benefits of Regime Models

  • Improved forecasting
    • accuracy for systems with abrupt changes
    • we can capture the entire system properly if we differentiate the regimes by using different models
  • Interpretability:
    • clearly defined regimes
  • Flexibility:
    • can capture a wide variety of nonlinear behavior

Challenges of Regime Models

  • Threshold Selection
    • How to select \(\gamma\)?
    • Computationally intensive
  • Overfitting
    • Too many regimes \(\implies\) overly complex
    • generalize poorly
  • Data sufficiency
    • Each regime requires sufficient data points

TAR model

  • A class of nonlinear time series models
    • behavior switches between regimes
    • depending on whether threshold variable exceeds a predefined value
  • This structure is good for capturing regime shifts in TS data

Key Features

  • Regime Switching
    • Model assumes different dynamics in different regimes
    • Each regime is modelled by its own AR process
  • Threshold value: determines the regime (commonly, \(y_{t-d}\))
  • Piecewise linearity: overall nonlinear, it is linear within each regime
  • Flexibility: handles systems where relationship between variables change depending on the regime they are in

Model Formulation

  • Model formulation
\[ y_{t} = \begin{cases} \phi_{1,0} + \phi_{1,1} y_{t-1} + \dots \phi_{1,p} y_{t-p} + e_{t}, & \text{if } y_{t-d} \le \gamma \\ \phi_{2,0} + \phi_{2,1} y_{t-1} + \dots \phi_{2,p} y_{t-p} + e_{t}, & \text{if } y_{t-d} \le \gamma \\ \end{cases} \]
  • \(\phi_{i,j}:\) AR coefficient for regime \(i\) and lag \(j\)
  • \(y_{t-d}:\) threshold variable
  • \(\gamma:\) threshold value

Steps in Model Building

  1. Specify threshold variable \((y_{t-d})\)
  2. Determine lag structure (\(p\) in AR for each regime)
  3. Estimate threshold (\(\gamma\) using grid search, minimize residual variance or information criteria)
  4. Estimate parameters: (Fit the AR processes using methods like least squares, for each regime)
  5. Diagnostics
    • Residual autocorrelation, stationarity
    • Model accuracy
    • Hypothesis test for nonlinearity (e.g., Hansen's test)

Advantages of TAR Models

  • Interpretability
  • Flexibity
  • Stationarity in Regimes: Each regime can be stationary even if overall is not

Challenges

  • Threshold estimation
  • Overfitting
  • Boundary issues (sparse data near \(\gamma\) can make estimation difficult)

Applicaations

  • Economics
    • Business cycles
    • inflation dynamics with unemployment rates as \(\gamma\)
  • Climatology
    • Representing shift in climate variables
  • Engineering
    • operational thresholds, load-bearing structures or machinery