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A Markov-switching regression model with non-Gaussian innovations: estimation and testing

  • Luca De Angelis EMAIL logo and Cinzia Viroli
Published/Copyright: March 1, 2017
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Studies in Nonlinear Dynamics & Econometrics
From the journal Studies in Nonlinear Dynamics & Econometrics Volume 21 Issue 2

Abstract

In this paper we propose a very general multivariate Markov-switching regression (MSR) model considering the normal inverse Gaussian (NIG) distribution as conditional form of financial returns and model innovations. It is indeed well-known that the Gaussian distribution is not able to capture many stylized facts of the return series such as skewness, excess kurtosis and heavy tails. Through a large simulation study and an empirical analysis of the US stock market, we show that a NIG-based MSR model allows to adequately account for both skewness and fat tails in the data and, according to model selection criteria, is the best overall model in the majority of the cases considered, even preferred over other popular distributional assumptions such as Student-t and GED. We develop an EM algorithm which allows the estimation of the model parameters in closed form. As a natural byproduct of the algorithm we also derive the scores of the model estimators that allow us to perform dynamic specification tests to check for autocorrelation and for the violation of the first-order Markov assumption.

Keywords: market regimes; Markov-switching; normal inverse Gaussian distribution; return time series analysis

Appendix: EM Algorithm

E step

In the E step we need to compute the conditional expectation of ui(t), uij(t) given the observations. Moreover, by putting (12) and (13) into (14), it turns out that also the conditional expectations of v and v−1 given the observations and the state of the latent variable s are required.

The conditional expectations of ui(t) and uij(t) given the returns rt and given the current parameter estimates can be computed according to the Baum-Welch recursions as follows:

E[ui(t)|rt]=Pr(st=i|rt)=li(t)mi(t)∑j = 1klj(t)mj(t)

and

E[uij(t)|rt]=Pr(st − 1=i, st=j|rt)=li(t−1)mj(t)pijf(rt|xt, zt, st=j)∑i = 1k∑j = 1kli(t−1)mj(t)pijf(rt|xt, zt, st=j)

The conditional expectation of v and v−1 can be computed following Protassov (2004) by

E[vt|rt, st=i]=ϕitα˜iKp˜ + 1(ϕitα˜i)Kp˜(ϕitα˜i), E[vt−1|rt, st=i]=α˜iϕitKp˜ − 1(ϕitα˜i)Kp˜(ϕitα˜i)

where p˜=−p+12,α˜i2=γ˜i2+β˜′iΣ˜iβ˜i and ϕit2=1+(rt−ξit)′Σ˜i−1(rt−ξit).

M step

Given the previous posterior distributions, the maximum likelihood for the model parameters can be obtained by evaluating the score function of (15) at zero. It is possible to show that all the parameter estimates have a closed form solution. More specifically with st =i we have,

πi=E[ui(1)|r1]=f(ui(1)|r1)pij=∑t =2TE[uij(t)|rt]∑h = 1k∑t = 2TE[uih(t)|rt]γ˜i=ni∑t = 1TE[ui(t)|rt]E[vt|rt, st=i]β˜i=∑t = 1TE[ui(t)|rt](rt−ξit)∑t = 1TE[ui(t)|rt]E[vt|rt, st=i]niΣ˜i=∑t = 1TE[ui(t)|rt] ((rt−ξit)(rt−ξit)′E[vt−1|rt, st=i])−∑t = 1TE[ui(t)|rt] (β˜i(rt−ξit)′)−∑t = 1TE[ui(t)|rt] ((rt−ξit)β˜′i)+∑t = 1TE[ui(t)|rt] (β˜iβ˜′iE[vt|rt, st=i])μi=∑t = 1TE[ui(t)|rt]E[vt−1|rt, st=i](rt−xtθi+ztψi)−β˜i∑t = 1TE[ui(t)|rt]∑t = 1TE[ui(t)|rt]E[vt−1|rt, st=i]

In the case p=n the EM estimate for θi is

θi=diag[∑t=1TE[ui(t)|rt]E[vt−1|rt, st=i]xtx′t]−1×[∑t = 1TE[ui(t)|rt]E[vt−1|rt, st=i](rt−μi+ztψi)x′t−β˜i∑t = 1TE[ui(t)|rt]x′t].

When n is a multiple of p, the estimate can be easily achieved in subsequent steps for each block of the matrix by considering the corresponding subset of xt in the previous formula.

For the remaining parameters we get

ψi=[∑t = 1TE[ui(t)|rt]E[vt−1|rt, st=i]ztz′t]−1×[∑t = 1TE[ui(t)|rt]E[vt−1|rt, st=i](rt−μi+xtθi)z′t−β˜i∑t = 1TE[ui(t)|rt]z′t]

where ni=∑t = 1TE[ui(t)|rt]. MLE of the model parameters can be recovered at each iteration by setting δi=|Σ˜i|1/2p,γi=γ˜i/δi,βi=Σ˜i−1β˜i and Δi=Σ˜i/δi2. Since the algorithm could be quite complicated, we recommend to initialize it with the starting values given by the moment estimates of the parameters, conditionally to reconstructed latent states given by the k-means method.

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Supplemental Material:

The online version of this article (DOI: 10.1515/snde-2015-0118) offers supplementary material, available to authorized users.

Published Online: 2017-3-1
Published in Print: 2017-4-1

©2017 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Forecast accuracy of a BVAR under alternative specifications of the zero lower bound
  3. A Markov-switching regression model with non-Gaussian innovations: estimation and testing
  4. Semi-global solutions to DSGE models: perturbation around a deterministic path
  5. Time elements and oscillatory fluctuations in the Keynesian macroeconomic system
  6. Macroeconomic (in)stability and endogenous market structure with productive government expenditure
https://doi.org/10.1515/snde-2015-0118
Keywords for this article
market regimes; Markov-switching; normal inverse Gaussian distribution; return time series analysis

Articles in the same Issue

  1. Frontmatter
  2. Forecast accuracy of a BVAR under alternative specifications of the zero lower bound
  3. A Markov-switching regression model with non-Gaussian innovations: estimation and testing
  4. Semi-global solutions to DSGE models: perturbation around a deterministic path
  5. Time elements and oscillatory fluctuations in the Keynesian macroeconomic system
  6. Macroeconomic (in)stability and endogenous market structure with productive government expenditure
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