L51 Nonlinear Time Series Models

Introduction

• Extension of traditional TS, account for systems where relationships cannot be adequately captured by linear models

Characteristics

Motivation: Why should be change to linear to nonlinear processes?

• Nonlinearity: not additive, depend on interactions or thresholds
• Complex dynamics: chaos, bifurcations, periodic behavior
• State Dependence: Based on the state of the current TS, the influence on future values will differ.
• Non-stationarity: Statistical properties (mean and variance) may change over time

Example of Nonlinear Processes²

• Threshold Models: TAR models: different dynamics above or below certain threshold limits
• Volatility models: GARCH
• Nonlinear Dynamical Systems: Deterministic chaos¹ (e.g. Lorenz attractor)
• Neural Networks: Recurrent Neural Networks (RNNs), Long Short-Term Memory (LSTM) and Transformer-based models
  • Polynomial and Rational Models

Threshold Models

• Class of Nonlinear TS models → Dynamics change based on whether the processes crosses certain thresholds
  • After a certain point (thresholds), the dynamics change
• Useful for capturing abrupt changes and regime shifts in the data

Key Concepts

• Regimes: TS operates under different "regimes", with distinct dynamics for each regimes
  • Regimes are subsets separated by thresholds
  • Distinct phases or states of a system, each governed by distinct dynamics
  • In Threshold models, regimes are defined based on crossing of thresholds by a variable (say, \(Y_{t-d}\))
• Threshold VariableA: The variable whose value determines the regime. (Often a lagged value of the TS)
  • Given, \(Y_{t}\) then \(Y_{t-1}\) or \(Y_{t-2}\) might be a threshold variable.
• Nonlinearity: exhibits piecewise linear behavior (making them interpretable)

Regime Characteristics

• Distinct Dynamics
  • Each regime has its own set of parameters and equations the describe the behavior of system
  • Economic growth rates might follow one pattern in expansion and different during recession³
• Transition Mechanism
  • Crossing a threshold
  • probabilistic switching mechanism
• Temporal persistence
  • persist for a period before transitioning (It has to be persistent inside a regime)
  • \(\implies\) clustering of similar states
• Nonlinearity
  • Overall system may appear nonlinear!
  • \(\impliedby\) abrupt or smooth transition between regimes

Types of Regime Transitions

• Abrupt transition (discrete switching)
  • Changes between regimes are instantaneous, when threshold condition is met
  • Example: TAR and SETAR
  • Uses:
    • Economic recession and recoveries,
    • sudden market crashes and booms
• Smooth Transitions
  • Changes between regimes → gradually over a range of values of the threshold variable
  • Example
    • STAR models
    • Logistic or exponential transition functions
  • Use cases
    • Gradual Policy shifts
    • transitioning between weather patterns
• Probabilistic Transitions
  • Neither abrupt nor smooth, but governed by probabilities (modelled as latent variables)
  • Example: MSAR models
  • Cases
    • Stock market volatility clustering
    • Hidden states in biological systems

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1. Complex dynamics, abruptions in the models ↩

2. Not Nonlinear "Time Series" models, just nonlinear models in general ↩

3. Expansion and recession are regimes ↩