L54 Markov Switching Models

• MSM
  • class of nonlinear TS models
  • capture regime-switching behavior
  • assuming transition between regime is governed by a hidden Markov process
• TAR and SETAR have regime changes depending on observable variables**
  • VS MSM, where regime switching depends on unobservable state variable

Key Features of MSM

• Regime switching
• Markov process
  • current regime depends only on previous regime¹
• Hidden state
  • regime is a latent (unobservable) variable
• Flexibility
  • suitable for capturing abrupt or smooth transition, dependent on state transition probabilities

Model Formulation

\[ y_{t} = \mu_{s_{t}} + \phi_{s_{t},1} y_{t-1} + \phi_{s_{t},2}y_{t-2} + \dots + \phi_{s_{t},p}y_{t-p} + e_{t} \]

• where
  • \(s_{t}\) is the latent state (regime) taking values \(1,2,\dots,k\)
  • \(\mu_{s_{t}}\) is a regime specific mean
  • \(\phi_{s_{t},i}\) are AR coefficients
  • \(e_{t}\) error terms with variance \(\sigma_{s_{t}}^{2}\)

Transition Probabilities

The regime-switching controlled by a state transition probability matrix (\(P\))

\[ P = \begin{bmatrix} p_{11} & \dots & p_{1k} \\ \vdots & \ddots & \vdots \\ p_{k1} & \dots & p_{kk} \\ \end{bmatrix} \]

• \(p_{ij} = P(s_{t}= j | s_{t-1}=i)\)
• Each row of \(P\) sums to \(1\)

Estimation

• Likelihood function
• Expectation-Maximization (EM) Algorithm
  • E-step = expected values of hidden states given params
  • M-step = updates parameters to maximize expected likelihood
• Filter and Smoother
  • Hamilton Filter = estimate probability of being in particular state at \(t\)
  • Kim Filter = smooths the probability to improve state estimation

Advantages of MSM

• Flexibility
• State dependence → improve model accuracy
• Latent Regime Models → infers unobservable states that drive the system

Challenges in MSM

• Computational Complexity (high dimensional likelihood function)
• Model selection (number of regimes, lag structure)
• Interpretability: regime probability must be carefully interpreted
  • Overfitting \(\impliedby\) too many parameters

Numerical Example

• \(n=300\)
• \(\mu = [1,-1]\)
• \(\phi = [0.8,0.5]\)
• \(\sigma= [0.5,1]\)

\[ P = \begin{bmatrix} 0.9 & 0.1 \\ 0.2 & 0.8 \end{bmatrix} \]

Estimation Interpretation

• mean \(\mu\), AR coefficients \(\phi\), variance \(\sigma^{2}\) for each regime
• transition probabilities \(p_{ij}\)
• Hidden states: smoothed probabilities indicate likelihood of TS being in each regime at a given time (Carefully address hidden states)

Output interpretation

• Parameter estimates
• Plot of smoothed probabilities
• Diagnostics

Other Possible Interesting Extensions

Bilinear Models

\[ y_{t} = \phi_{0} + \sum_{i=1}^{p} \phi_{i}y_{t-i} + \sum_{j=1}^{q} \sum_{k=1}^{p} \beta_{jk}y_{t-k} e_{t-j} + e_{t} \]

• Incorporates products of
  • past observations and noise
  • handles nonlinearities
• Uses: environmental data with interaction effects

Nonlinear ARX models

\[ y_{t} = f(y_{t-1},y_{t-2},\dots x_{t-1},x_{t-2},\dots) + e_{t} \]

• Nonlinear function of
  • past inputs and outputs
• \(f(\cdot)\) can be a polynomial, neural network or other nonlinear functions
• Applications: control systems, dynamic system modelling

Neutral-Network Based models

• Nonlinear Autoregressive Neural Network (NAR-NN)
  • use neural network to model the autoregressive relationship

\[ y_{t} = f_{NN}(y_{t-1},y_{t-2},\dots,e_{t}) \]

• Flexible, powerful for compl²ex nonlinear dependencies
• Long Short-Term Memory (LSTM) → handles sequential dependencies and long-term memory in nonlinear time series

Others

• ST-GARCH: Smooth Transition GARCH
  • Extends GARCH by allowing regime switching in variance equation
• Hidden Markov Models (HMMs)
  • switching dynamics in TS using discrete latent states
• Functional Time Series
  • high (infinite) dimensional
  • models temporal changes in curves/functions
• Wavelet Transform Models
  • decompose TS into different frequency components

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1. Markov Property: today's state is dependent only on yesterday's state ↩