Do Markov-switching models capture nonlinearities in the data?: Tests using nonparametric methods

doi:10.1016/S0378-4754(03)00106-X

Mathematics and Computers in Simulation

Volume 64, Issues 3–4, 11 February 2004, Pages 401-407

https://doi.org/10.1016/S0378-4754(03)00106-X Get rights and content

Abstract

Markov-switching models have become popular alternatives to linear autoregressive models. Many papers which estimate nonlinear models make little attempt to demonstrate whether the nonlinearities they capture are of interest or if the models differ substantially from the linear option. By simulating the models and nonparametrically estimating functions of the simulated data, we can evaluate if and how the nonlinear and linear models differ.

Introduction

The purpose of this paper is to provide an example of how simulation methods, in combination with nonparametric density and regression estimation, may be used to evaluate the success (or failure) of nonlinear models in capturing important characteristics of the data. As an example, we look at Markov-switching (MS) models, popularized in the economics literature by Hamilton [7]. MS models have become a popular alternative to linear autoregressive models in part due to their intuitive appeal, in that they allow a latent state process to cause “switching” between different linear regimes. This corresponds to the popular conception of underlying states of the economy, for example high-growth or low-growth states in GDP and high-volatility and low-volatility states in the stock market.

Harding and Pagan [9] provided an early example of the use of simulation and nonparametric techniques applied to MS models while Clements and Galvao [4] and Pagan [11] apply nonparametric conditional mean estimation to a selection of nonlinear models. The idea of comparing nonparametric estimates of densities and models goes back to the late 1980s at least, and was used by Ait-Sahalia [1] and Pagan [10] in analyzing financial data.

In this paper, we consider a simple Markov-switching model estimated by Bodman [2] using Australian unemployment data. The example in this paper is illustrative of the problems which often arise in estimating and using Markov-switching models. Basically these are failures of the algorithm to reach the global optimum of a likelihood and an inability to fit important features of the data. Moreover, the fitted MS model is often indistinguishable from simpler alternatives. Breunig and Pagan [3] explore a number of other examples of MS models which share the same problems.

Section snippets

Markov-switching models

Consider the following example of a Markov-switching model which describes the movement of some stationary economic variable x_t as a function of its p most recent lagged values and a binary state variable, s_t, taking values of 0 or 1. $x_{t} =μ_{0} (1−s_{t})+μ_{1} s_{t} + ∑ l=1 p φ_{l} (x_{t} −μ_{0} (1−s_{t})−μ_{1} s_{t})+σv_{t}$ s_t evolves as a first-order Markov process with probability P₀₀ of remaining in state 0 next period conditional on being in state 0 this period while P₁₁ is defined analogously. Extensions of this simple model have been

Informal tests

The central idea of this paper is to simulate from the proposed model taking the parameter estimates, $θ ̂$ , as given. Using that simulated data, we can use nonparametric techniques to compute quantities, such as $f_{M} (x_{t} | θ ̂)$ , the implied density of x_t from the model, and $E_{M} (x_{t} |x_{t−j}; θ ̂)$ , the implied conditional mean from the model, and compare these to the respective quantities from the data. In order to minimize the degree of error in estimating these functions, we take a very large number of

Application

Bodman [2] fitted the two state MS model (1) to the first differences of the quarterly Australian unemployment rate, x_t, for the time period 1959:3 to 1997:3. The second column of Table 1 provides his estimates of the parameters.

Using these parameter estimates, we generated 63,000 observations from the implied model. We drop the first 3000 observations to remove the effect of starting values. Consequently, estimates of $f_{M} (x_{t} | θ ̂)$ and $E_{M} (x_{t} |x_{t−j}; θ ̂)$ are based upon sample sizes of 60,000. Fig. 1

Conclusions

The techniques outlined here have the advantage of being easy to apply, even for very complicated models. Clearly the choice of which conditional moments and which densities to examine depend upon the particular application. In the example presented here, the techniques uncovered a convergence problem with the algorithm that was used to estimate the model. Even after correcting the estimates, there is little evidence that the nonlinear model performs better than the linear alternative in ways

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Do Markov-switching models capture nonlinearities in the data?: Tests using nonparametric methods

Abstract

Introduction

Section snippets

Markov-switching models

Informal tests

Application

Conclusions

J. Monetary Econ.

J. Empirical Finance

Testing continuous-time models of the spot interest rate

Rev. Financial Stud.

Asymmetry and duration dependence in Australian GDP and unemployment

Econ. Record

Asymptotic null distribution of the likelihood ratio test in Markov switching models

Int. Econ. Rev.