Maddalena Cavicchioli

Determining the Number of Regimes in Markov Switching VAR and VMA Models

We give stable finite order VARMA(p*; q*) representations for M-state Markov switching second-order stationary time series whose autocovariances satisfy a certain matrix relation. The upper bounds for p* and q* are elementary functions of the dimension K of the process, the number M of regimes, the autoregressive and moving average orders of the initial model. If there is no cancellation, the bounds become equalities, and this solves the identification problem. Our class of time series include every M-state Markov switching multivariate moving average models and autoregressive models in which the regime variable is uncorrelated with the observable. Our results include, as particular cases, those obtained by Krolzig (1997), and improve the bounds given by Zhang and Stine (2001) and Francq and Zakoian (2001) for our classes of dynamic models. Data simulations and an application on foreign exchange rates complete the paper.

ANALYSIS OF THE LIKELIHOOD FUNCTION FOR MARKOV-SWITCHING VAR(CH) MODELS

In this work, we give simple matrix formulae for maximum likelihood estimates of parameters in a broad class of vector autoregressions subject to Markovian changes in regime. This allows us to determine explicitly the asymptotic variance–covariance matrix of the estimators, giving a concrete possibility for the use of the classical testing procedures. In the context of multivariate autoregressive conditional heteroskedastic models with changes in regime, we provide formulae for the analytic derivatives of the log likelihood. Then we prove the consistency of some maximum likelihood estimators and give some formulae for the asymptotic variance of the different estimators.

Acute triangulations of convex quadrilaterals

An acute triangulation of a polygon @C is a triangulation of @C into acute triangles. Let f(@C) denote the minimum number of triangles necessary for an acute triangulation of @C. We prove that the maximum value of f(Q) for all convex quadrilaterals Q is equal to 8. This solves a problem raised by Maehara (2001) in [4].

Asymptotic Fisher information matrix of Markov switching VARMA models

We study the Fisher information (FI) matrix of Markov switching vector autoregressive moving average (MS VARMA) models and derive an explicit expression in closed form for the asymptotic FI matrix of the underlying model. Our result is more general than the available one in the literature for linear VARMA models, which has been recently studied in Bao and Hua (2014), in two respects. First, we treat the variance of the error term in a more general setting rather than considering it as a nuisance parameter. Then, we consider non-trivial intercept in the MS VARMA model. Under general conditions, the asymptotic FI matrix can be used to derive the asymptotic covariance matrix of the Gaussian maximum likelihood estimator of the model parameters. Some examples and numerical applications illustrate the results.

Third and fourth moments of vector autoregressions with regime switching

ABSTRACT We derive matrix formulae in closed form for the unconditional third and fourth moments of a broad class of vector autoregressive time series with regime switching. First and second moments are well known. New measures of multivariate skewness and kurtosis are introduced and basic properties are investigated. The knowledge of series level, variation, co-movements, skewness, and kurtosis is useful to support model interpretation in real data application. Numerical examples complete the paper.

Estimation and asymptotic covariance matrix for stochastic volatility models

In this paper we compute the asymptotic variance-covariance matrix of the method of moments estimators for the canonical Stochastic Volatility model. Our procedure is based on a linearization of the initial process via the log-squared transformation of Breidt and Carriquiry (Modelling and prediction, honoring Seymour Geisel. Springer, Berlin, 1996). Knowledge of the asymptotic variance-covariance matrix of the method of moments estimators offers a concrete possibility for the use of the classical testing procedures. The resulting asymptotic standard errors are then compared with those proposed in the literature applying different parameter estimates. Applications on simulated data support our results. Finally, we present empirical applications on the daily returns of Euro-US dollar and Yen-US dollar exchange rates.

Markov Switching GARCH Models: Filtering, Approximations and Duality

This paper is devoted to show duality in the estimation of Markov Switching (MS) GARCH processes. It is well-known that MS GARCH models suffer of path dependence which makes the estimation step unfeasible with usual Maximum Likelihood procedure. However, by rewriting the model in a suitable state space representation, we are able to give a unique framework to reconcile the estimation obtained by filtering procedure with that coming from some auxiliary models proposed in the literature. Estimation on short-term interest rates shows the feasibility of the proposed approach.

Central Bank Independence, Financial Instability and Politics: New Evidence for OECD and Non-OECD Countries

This paper analyses the determinants of a new index of central bank independence, recently provided by Dincer and Eichengreen (2014), using a large database of economic, political and institutional variables. Our sample includes data for 31 OECD and 49 non-OECD economies and covers the period 1998-2010. To this aim, we implement factorial and regression analysis to synthesize information and overcome limitations such as omitted variables, multicollinearity and overfitting. The results confirm the role of the IMF loans program to guide all the economies in their choice of more independent central banks. Financial instability, recession and low inflation work in the opposite direction with governments relying extensively on central bank money to finance public expenditure and central banks’ political and operational autonomy is inevitably undermined. Finally, only for non-OECD economies, the degree of central bank independence responds to various measures of strength of political institutions and party political instability.

Validating Markov Switching VAR Through Spectral Representations

We develop a method to validate the use of Markov Switching models in modelling time series subject to structural changes. Particularly, we consider multivariate autoregressive models subject to Markov Switching and derive close-form formulae for the spectral density of such models, based on their autocovariance functions and stable representations. Within this framework, we check the capability of the model to capture the relative importance of high- and low-frequency variability of the series. Applications to U.S. macroeconomic and financial data illustrate the behaviour at different frequencies.

Business Cycle and Markov Switching Models with Distributed Lags: A Comparison between US and Euro Area

Business cycle models are often investigated by using reduced form time series models, other than (or in alternative to) structural highly grounded in economic theory models. Reduced form VARMA with fixed parameters play a key role in business cycle analysis, but it is often found that by their very nature they do not typically capture the changing phases and regimes which characterize the economy. ln this paper we show that well-known state space systems used to analyse business cycle in several empirical works can be comprised into a broad class of non linear models, the MSI-VARMA. These processes are M-state Markov switching VARMA models for which the intercept term depends not only on the actual regime but also on the last r regimes. We give stable finite order VARMA representations for these processes, where upper bounds for the stable VARMA orders are elementary functions of the parameters of the initial switching model. If there is no cancellation, the bounds become equalities, and this solves the identification problem. This result allows us to study US and European business cycles and to determine the number of regimes most appropriate for the description of the economic systems. Two regimes are confirmed for the US economy; the European business cycle exhibits, instead, strong nonlinearities and more regimes are necessary. This is taken into account when performing estimation and regime identification.