Introduction to Markov-Switching Models

by Eric · Published September 3, 2021 · Updated September 24, 2021

Introduction

Markov-switching models offer a powerful tool for capturing the real-world behavior of time series data. Today's blog provides an introduction to Markov-switching models including:

• What a regime switching model is and how it differs from a structural break model.
• When we should use the regime switching model.
• What a Markov-switching model is.
• What tools we use to estimate Markov-switching models.

What Is A Regime Switching Model?

Traditional time series models assume that one set of model parameters can be used to describe the behavior of the data over all time. This assumption isn't always valid for what we encounter in real-world data.

Real-world time series data may have different characteristics, such as means and variances, across different time periods. Regime-switching models:

• Characterize data as falling into different, recurring “regimes” or “states”.
• Allow the characteristics of time series data, including means, variances, and model parameters to change across regimes.
• Assume that at any given time period there is a probability that the series may be in any of the regimes and may transition to a different regime.

These components can allow regime change models to better capture the true behavior of real-world data than standard models.

How Are Regime Change Models Different Than Structural Break Models?

At first glance, it can be difficult to distinguish regime change models from structural break models. They both allow for changes in the underlying model of time series data. However, there are distinct differences:

Structural Break Models vs. Regime Change Models
Regime Change Models	Structural Break Models
Parameters vary across different regimes. Finite number of regimes. Regimes can be temporary and recurring. Used to model effects of cyclically occurring changes in the economy. Regime is unobserved and driven by stochastic process.	Parameters change at different times. Infinite number of structural changes. Non-recurring and permanent shifts. Usually used to model effects of permanent changes in economic structure.

In a way, we can think of structural change models as a very special case of regime change models, in which each possible "regime" occurs only once.

When Should You Use Regime Switching Models?

Regime switching models are most commonly used to model time series data that fluctuates between recurring "states". Put another way, if we are working data that seems to cycle between periods of behavior, we may want to consider a regime switching model.

Example Applications of Regime Change Models
Underlying cause	Description
Financial crises.	Many economic time series behave differently in times of financial stability than financial crisis.
Economic downturns.	The behavior of economic time series are characterized differently in periods of economic expansion versus economic recession.
Changes in tax policies.	Household behavior, such as income allocation between consumption and saving, changes depending on tax policy regimes.
Hyperinflation.	Economic fundamentals behave differently in periods of hyperinflation and "normal" rates of inflation.

The Markov-Switching Model

The Markov-switching model is a popular type of regime-switching model which assumes that unobserved states are determined by an underlying stochastic process known as a Markov-chain.

What is a Markov-chain?

A Markov-chain is a stochastic process used to describe how uncertain and unobserved outcomes occur. In the case of the Markov-switching model, it is used to describe how data falls into unobserved regimes. A Markov-chain has the property that future states are dependent only on present states (this is known as the Markov property).

A key characteristic of a Markov-chain is the transition probabilities. The transition probabilities describe the likelihood that the current regime stays the same or changes (i.e the probability that the regime transitions to another regime).

The Components of the Markov-Switching Model

The complete Markov-switching model includes:

• An assumed number of regimes.
• A dependent variable.
• Independent variables.
• Parameters relating the dependent variable to the independent variables for each regime.
• Transition probabilities.
• Statistical inferences on the model parameters and the determined states.

How Are Markov-Switching Models Estimated

Markov-switching models are usually estimated using:

• Maximum likelihood estimation.
• Bayesian estimation.

Maximum Likelihood Estimation of Markov-switching Models

Maximum likelihood estimation of Markov-switching models utilizes an iterative algorithm known as expectation-maximization. The expectation-maximization algorithm is data analysis for models where there is a latent (unobserved) variable in the model. This method was first proposed by John Hamilton in 1990.

The expectations-maximization algorithm broadly involves two steps:

1. Estimating the latent variable. This is known as the E-Step.
2. Estimating the parameters of the model given the value of the latent variable. This is known as the M-step.

In the context of the Markov-Switching model, this means we:

1. Use a filtering-smoothing algorithm, such as the Kalman smoother, to propose the path of the unobserved variable.
2. Use maximum likelihood, given the current regime, to estimate the model parameters, including the transition probabilities.
3. Repeat steps 1 & 2 using updated parameter estimates until convergence.

Bayesian Estimation of Markov-Switching Models

Bayesian estimation of Markov-switching models relies on drawing samples from a joint distribution of the parameters, states, and transition probabilities using a Markov Chain Monte Carlo method (MCMC). This method benefits from the fact that the likelihood function for the model doesn't have to be directly calculated.

A common tool used for Bayesian estimation of the Markov-switching models is the Gibbs sampler.

Conclusion

Congratulations! In today's blog, you learned the basics of the power Markov-switching model. After reading this blog, you should have a better understanding of:

• What a regime switching model is and how it differs from a structural break model.
• When to use a regime switching model.
• What a Markov-switching model is.
• What tools we use to estimate Markov-switching models.