OLS Estimation of Markov switching VAR models: asymptotics and application to energy use
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OLS Estimation of Markov switching VAR models: asymptotics and application to energy use Maddalena Cavicchioli1 Received: 27 January 2020 / Accepted: 12 October 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We show that the ordinary least squares (OLS) estimates of population parameters for Markov switching vector autoregressive (MS VAR) models coincide with the maximum likelihood estimates. Then, we propose an algorithm in matrix form for the estimation of model parameters, and derive an explicit expression in closedform for the asymptotic covariance matrix of the OLS estimator of such models. The obtained characterization of the asymptotic variance is new to our knowledge. It is easier to program than the usual approach based on second derivatives, and more accurate. Our theorems generalize the classical results known for a linear VAR process, and complete those existing in the literature on the estimation of the asymptotic covariance matrix for multivariate stationary time series. Numerical simulations are provided to illustrate the obtained theoretical results. Finally, an application on energy use and economic growth in the Euro area gives some insights on the nonlinear nature of the corresponding time series, and reproduces the major stylized facts. Keywords Markov switching VAR model · OLS estimator · Asymptotic covariance matrix · Energy use · Economic growth JEL Classification C32 · C34 · C13
1 Introduction Economic time series prediction deals with the task of modelling the underlying data generation process using past observations and using the model to extrapolate the time series into the future. In the literature, two principal classes of models were
* Maddalena Cavicchioli [email protected] 1
Department of Economics “Marco Biagi”, University of Modena and Reggio Emilia, Viale Berengario 51, 41121 Modena, Italy
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studied for the purpose of forecasting i.e., the statistical time series models and the structural econometric models. Linear multivariate time series models, like vector autoregressive moving-average (VARMA) models, were among the first to be developed and widely used in statistics and economics to model simultaneously the dynamics of a set of endogenous variables. A leading class is formed by vector autoregressive (VAR) models, as special case of VARMA, pionereed by Sims (1980). More recently, Fernández-Villaverde et al. (2007) showed that linearized dynamic stochastic general equilibrium (DSGE) models imply that the variables of interest are generated by a VARMA process. For textbook discussions on several model specifications as well as estimation and inference procedures for linear time series see Hamilton (1994) and Lütkepohl (2007). Another important aspect of the modelling process is the evaluation of the fit of a selected model. Several authors were interested in goodness-of-fit tests for VARMA models i.e., tests which can be applied when no a priori information is available about what departu
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