Maddalena Cavicchioli

Transcription

1 SPECTRAL REPRESENTATION AND AUTOCOVARIANCE STRUCTURE OF MARKOV SWITCHING DSGE MODELS Maddalena Cavicchioli Abstract. In this paper we investigate the L 2 -structure of Markov switching Dynamic Stochastic General Equilibrium MS DSGE) models and derive conditions for strict and second order stationarity. Then we determine the autocovariance function of the process driven by a stationary MS DSGE model and give a stable VARMA representation of it. It turns out that the autocovariance structure of the process coincides with that of a standard VARMA. Finally, we propose a method to derive the spectral density matrix of MS DSGE models and describe some properties of this spectral representation. We apply the Riesz-Fischer theorem which defines the spectral representation as the Fourier transform of the autocovariance functions. Examples and numerical applications complete the paper. Keywords: Multivariate DSGE, State-Space models, Markov chains, changes in regime, autocovariance structure, spectral density function. JEL Classification: C01, C05, C32. Department of Economics Marco Biagi, Viale Berengario 51, 41121, Modena Italy). Office Phone No: <maddalena.cavicchioli@unimore.it> 1

2 2 1. Introduction Dynamic Stochastic General Equilibrium DSGE) models have recently gained a central role for analyzing the mechanisms of propagation of economic shocks. See, for example, Fernández Villaverde et al. 2007), Franchi and Vidotto 2013) and Franchi and Paruolo 2014). For a comprehensive review about the relationship between DSGE and VAR models, see Giacomini 2013), and references therein. However, empirical work has shown that DSGE models have failed to fit the data very well over a long period of time. In fact, the changes of parameters require the economists to re estimate them. This observation leads to Markov switching MS) DSGE models, that is, DSGE models in which the coefficient parameters are assumed to depend on the state of an unobserved Markov chain. Since the influential work of Hamilton 1989) 1990), Markov switching models are widely used to capture the business cycle regime shifts typically observed in economic data. See also Hamilton 1994), Chp.22, and Hamilton 2005) for more recent references. Markov switching VARMA models were studied by many authors as, for example, Hamilton 1989) 1990), Krolzig 1997) and Francq and Zakoïan 2001). For estimation, consistency, asymptotic variance and model selection of MS VARMA models see also Yang 2000), Zhang and Stine 2001) and Cavicchioli 2014a) 2014b). The L 2 structures and autocovariance functions of standard and Markov switching GARCH models were investigated in Francq and Zakoïan 2005). Markov switching DSGE models have been discussed by Billio and Monfort 1998). These authors proposed a combination of the partial Kalman filter and of sampling techniques in order to evaluate the likelihood function. Obviously, the computation of this function opens the way for maximum likelihood estimation methods and also for likelihood ratio tests. The first purpose of the present paper is to investigate the L 2 structure of the MS DSGE models. Precise definitions and assumptions are given in Section 2. We derive stationarity conditions and compute explicitly the autocovariance function of these nonlinear processes. This is useful for statistical model analysis. We find that, under existence conditions, the L 2 structure of the process driven by an MS DSGE model is that of a stable finite) VARMA model, and hence the former admits a stable VARMA representation. This is in particular the case for standard DSGE models. We are able to characterize the orders of such a stable VARMA in terms of the parameters defining the MS DSGE model. These results have statistical implications in terms

3 3 of regime number identification and parameter estimation. A second goal of the paper is to propose a tractable method to derive the spectral representation of MS DSGE models. The procedure simply relies on the Riesz Fischer theorem, which defines the spectral density function of a covariance stationary stochastic process as the Fourier transform of the autocovariance function. See, for example, Priestley 1981) for the spectral analysis of time series. Then we illustrate some properties of the spectral representation of MS DSGE models. Our methods and techniques relate with those developed by Francq and Zakoïan 2001) and 2005) for Markov Switching VARMA and GARCH models, respectively. The obtained results complete the analysis of Markov Swithing State Space models as in Billio and Monfort 1998) and Alencar et al. 2013). Furthermore, we refer to Francq and Zakoïan 2002) for the autocovariance structure of powers of switching regime ARMA processes. Stability properties of nonlinear dynamic models were characterized by Bask and De Luna 2002) in terms of smoothed Lyapounov exponents, which are a generalization of the classical ones for deterministic systems. We give analogous results for MS DSGE models by using a suitable Lyapounov exponent. Moreover, the spectral analysis of VAR models is studied in Ioannidis 2007) which is a linear subcase of our multivariate State Space. In an unpublished paper Bianchi 2012) computes analytical formulas for the first and second moments of MS DSGE models. Here we characterize the complete structure of the autocovariance for such models, and give a stable ARMA representations of them. Finally, there are numerous ongoing works on the topic of MS DSGE see, for instance, Foerster et al. 2014) and Giraitis et al. 2014)) so that an analytical characterization of these models is even more needed. The paper is organized as follows. In Section 2 we define the MS DSGE model and provide a representation of it, which is used to derive a sufficient condition for strict stationarity. Then a discussion of the second order stationarity properties of the model leads to necessary and sufficient conditions. In Section 3 we derive the autocovariance structure of the model and the stable VARMA representation of it. In Section 4 we study the spectral density matrix of the model and describe some properties of this spectral representation. Section 5 provides examples and numerical applications. Proofs are given in the Appendix. 2. Markov switching DSGE models Let us consider the following Markov-switching DSGE in short, MS DSGE)

4 4 model x t = A st x t 1 + B st w t 1) y t = C st x t 1 + D st w t 2) where x t is an n 1 vector of possibly unobserved state variables, y t is the k 1 vector of observable variables, and w t NID0, I m ) is an m 1 vector of economic shocks. The matrices A st R n n, B st R n m, C st R k n and D st R k m are real random matrices. The process s t ) is a homogeneous stationary irreducible and aperiodic Markov chain with finite state-space Ξ = {1,..., d}. Let πi) = Prs t = i) be the ergodic probabilities which are positive. Let pi, j) = Prs t = j s t 1 = i) be the stationary transition probabilities. Then the d d matrix P = pi, j)) is called the transition probability matrix. Assume that s t ) is independent of w t ). Note that Ex t ) = 0 and Ey t ) = 0. In order to investigate the stationarity properties of the process y t ), we use the following vectorial representation of the MS DSGE model: z t = Φ st z t 1 + Ψ st w t 3) where z t = xt y t ) Φ st = Ast ) 0 C st 0 Ψ st = Bst D st ) Then z t is p 1, Φ st is p p and Ψ st is p m, where p = n + k. Let Φ i and Ψ i be the matrices obtained by replacing s t by i in Φ st and Ψ st, for i = 1,..., d. Let us consider the following matrices: p1, 1){Φ 1 Φ 1 } p2, 1){Φ 1 Φ 1 } pd, 1){Φ 1 Φ 1 } p1, 2){Φ 2 Φ 2 } p2, 2){Φ 2 Φ 2 } pd, 2){Φ 2 Φ 2 } P Φ Φ =... p1, d){φ d Φ d } p2, d){φ d Φ d } pd, d){φ d Φ d } π1) {Ψ 1 Ψ 1 } π2) {Ψ 2 Ψ 2 } Π Ψ Ψ :=. πd) {Ψ d Ψ d } Then P Φ Φ is d p 2 ) d p 2 ) and Π Ψ Ψ is d p 2 ) m 2. Since {s t, w t )} is an ergodic process, {Φ st, w t )} and {Ψ st, w t )} are also strictly stationary

5 5 ergodic processes. Let denote any operator norm on the sets of matrices. Let, for a > 0, log + e a = max{log e a, 0}. It is clear that E log + e Φ st, E log + e Ψ st and E log + e w t are finite. Iterating 3) gives z t = i=0 Φ st Φ st 1 Φ st i+1 Ψ st i w t i 4) where we set Φ st Φ st 1 Φ st i+1 = I p if i = 0. From Brandt 1986) see also Bougerol and Picard 1992) and Francq and Zakoïan 2001)), the unique strictly stationary solution of 3) is given by 4), whenever the top Lyapunov exponent γ Φ = inf t N { E 1 t log e Φ st Φ st 1 Φ s1 } 5) is strictly negative. If Φ st = Φ for every t e.g., in the case of standard DSGE model), then γ Φ = E log e lim t + Φt 1 t = log e ρφ) < 0 if and only if ρφ) < 1, where ρ ) denotes the spectral radius. Then we have a sufficient condition for strict stationarity of MS DSGE models. Theorem 2.1. Suppose that γ Φ < 0 from 5). Then, for all t Z, the series z t ) in 4) converges a.s., and the process y t ), defined as the second component of z t ), is the unique strictly stationary solution of the MS DSGE model in 1) and 2). To verify that the process z t ) in 4) is well defined in L 2, it is sufficient to show that z t,i = Φ st Φ st 1 Φ st i+1 Ψ st i w t i 6) with z t,0 = Ψ st w t ) converges to zero in L 2 endowed with any matrix norm), at an exponential rate, as i goes to infinity. In fact, z t L 2 i=0 z t,i L 2. Here we are only interested in solutions of 1) and 2) which are nonanticipative. Recall that a process y t ) is called nonanticipative if, for all t Z, y t is measurable with respect to the σ field generated by {s τ, w τ : τ t}. We use arguments from Francq and Zakoïan 2001) 2005). From the independence between s t ) and w t ) and the fact that Ew t ) = 0, we have use 6)) vec Ez t,i z t,i) = Ez t,i z t,i ) = E[Φ st Φ st ) Φ st i+1 Φ st i+1 ) Ψ st Ψ st )] Ew t i w t i ) = Λ P i Φ Φ Π Ψ Ψ veci m )

6 6 where Λ = I p 2 I p 2) = e I p 2 and e = 1 1) R d. Now the following result can be proved as Theorem 2 from Francq and Zakoïan 2001). Theorem 2.2. Suppose that ρp Φ Φ ) < 1. Then, for all t Z, the series z t ) in 4) converges in L 2, and the process y t ), defined as the second component of z t ), is the unique nonanticipative second order stationary solution of the MS DSGE model in 1) and 2). A necessary and sufficient condition for the existence of such a solution is given by i=0 Λ P i Φ Φ Π Ψ Ψ veci m ) < + where denotes the matrix norm X = i,j x ij for any matrix X = x ij ). 3. Autocovariance structure and stable VARMA representations Once second order stationarity is ensured, it is useful to compute the autocovariance function of the process y t ) driven by an MS DSGE model. First, we study the autocovariance structure of the process z t ). For every h, define π1) Ez t z t h s t = 1) W h) = Π Ezt z t h ) := π2) Ez t z t h s t = 2). πd) Ez t z t h s t = d) Then W h) is d p) p. For h = 0, we shall prove in the Appendix the matrix formula: vec W 0) = P Φ Φ vec W 0) + Π Ψ Ψ veci m ) 7) Let P Φ be the d p) d p) matrix obtained by replacing Φ i Φ i by Φ i, for i = 1,..., d, in the definition of P Φ Φ. For h 1, we have the matrix relation see the Appendix) Iterating 8) gives W h) = P Φ W h 1) 8) W h) = P h Φ W 0) 9) for every h 0. From now on, we assume the condition ρp Φ Φ ) < 1. This implies that ρp Φ ) < 1; the converse is not true. The result follows

7 7 from Proposition 3.6, p.35, of Costa et al. 2005). Thus the autocovariance structure of the process z t ) is given by Γ z h) = d πi) Ez t z t h s t = i) = e I p ) W h) i=1 for every h 0. Here e = 1 1) R d and W h) is given by 9), where vec W 0) = I d p 2 P Φ Φ ) 1 Π Ψ Ψ vec I m 10) by using 7). Of course, Γ z h) = [Γ z h)] for every h < 0. Since y t = f z t f = 0 k n I k ) R k p we have obtained the autocovariance structure of the process y t ) driven by an MS DSGE. More precisely, we have Theorem 3.1. Suppose that ρp Φ Φ ) < 1. With the above notations, the autocovariance function of the process y t ) defined by the MS DSGE model in 1) and 2) is given by Γ y h) = f e I p )P h Φ W 0) f for every h 0, where W 0) follows from 10). Furthermore, Γ y h) = [Γ y h)] for every h < 0. Now we give a stable VARMA representation for any second-order stationary process y t ) driven by an MS DSGE model. Theorem 3.2. Suppose that ρp Φ Φ ) < 1. Let n 0 be the algebraic multiplicity of the zero eigenvalue of P Φ. Then the process y t ) defined by the MS DSGE model in 1) and 2) admits a stable VARMAp, q ) representation, where p dn + k) n 0 and q dn + k) 1. We remark that the stable VARMA representation from Theorem 3.2 can be used, for example, to obtain the linear predictions of the observed process. Furthermore, from Theorem 3.2, we can estimate a lower bound for the number of regimes as follows. Calculate the sample covariances of the observations y t ), estimate p, q ) of a stable VARMAp, q ) representation from these, and then use the inequalities in Theorem 3.2 to bound d. As a corollary of Theorem 3.2 we have the following result.

8 8 Corollary 3.2. The number d of regimes of the Markov chain s t ) in 1) and 2) satisfies d max{p + n 0 )/n + k), q + 1)/n + k)}, where p and q are the orders of a stable VARMA representation of the process y t ). 4. Spectral representation In this section we apply the Riesz-Fischer theorem which defines the spectral density matrix as the Fourier transform of the autocovariance function F y ω) = h=0 Γ y h) e iωh + h=1 [Γ y h)] e iωh The multivariate spectral matrix describes the spectral density functions of each element of the state vector in the diagonal terms. The off-diagonal terms are defined to be cross spectral density functions and they are typically complex numbers. Here we are only interested to the diagonal terms. Therefore, we can compute them, without loss of generality, considering the summation F y ω) = h=0 Γ y h) e iωh + h=0 [ + = fe I p ) P h Φ e iωh + h=0 Γ y h) e iωh Γ y 0) h=0 P h Φ e iωh I dp ] W 0) f where the frequency ω belongs to [ π, π]. These series converge as ρp Φ ) < 1 by hypotheses. Then we have Theorem 4.1 Suppose that ρp Φ Φ ) < 1. Under the above notations, the spectral density matrix of the process y t ) driven by the MS DSGE in 1) and 2) is given by where F y ω) = fe I p )[ I dp + 2 Re Y ω)]w 0)f Y ω) = I dp P Φ e iω ) 1 is a dp) dp) complex matrix and ReY ω) denotes the real part of Y ω). In the univariate case k = 1), we get the scalar function f y ω) = {f [fe I p )[ I dp + 2 Re Y ω)]} I dp 2 P Φ Φ ) 1 Π Ψ Ψ veci m )

9 9 If an observed times series y 1,..., y T can be described by an MS DSGE model, one good approach for estimating the population spectrum is to estimate the parameters of the model and substitute them into the expression of F y ω). Methods for the computation of the likelihood function, of filtered and smoothed values of s t ) for MS DSGE models can be found in Billio and Monfort 1998), Sections 3 5. The computation of the likelihood function obviously opens the way for maximum likelihood estimation methods and also for likelihood ratio tests, whereas the filtered and smoothed values of s t ) provide information about the regime at date t. The performance of the proposed approach is evaluated via Monte Carlo experiments see Section 6 of the quoted paper). Theorem 4.1 directly implies as a corollary the following result: Corollary 4.1 Suppose that P Φ is diagonalizable with P Φ = T DT 1 and D = dp λ j, where λ j are the eigenvalues of P Φ. Then the spectral density matrix is given by F y ω) = fe I p )T dp 1 λ 2 ) j 1 + λ 2 j 2λ T 1 W 0)f j cosω) Using Theorem 4.1 we can also obtain the following general result. Theorem 4.2 Let P Φ = T JT 1 be a Jordan normal form of P Φ with J = s r=1 J mr λ r ), where the m r m r Jordan block J mr λ r ) corresponds to the eigenvalue λ r of P Φ and m m r = dp. Then the spectral density matrix is given by F y ω) = fe I p )T [ I dp + s r=1{i mr e iω J mr λ r )) 1 e iω + I mr e iω J mr λ r )) 1 e iω }]T 1 W 0)f where e ±iω λ r ) 1 e ±iω λ r ) 2 e ±iω λ r ) mr I mr e ±iω J mr λ r )) 1 0 e ±iω λ r ) 1 e ±iω λ r ) mr 1) = e ±iω λ r ) 1

10 10 5. Examples 5.1). Consider the following 2-state Markov switching univariate DSGE model x t = a st x t 1 + σ st 1 Rs 1 t ) w t y t = σ st w t where w t NID0, 1), R st > 1, and s t {1, 2}. It generalizes the permanent income model, where a st = 1 and d = 1. See, for example, Franchi and Paruolo 2014). Here A st = a st, B st = σ st 1 Rs 1 t ), C st = 0 and D st = σ st, hence ) ) ast 0 1 R 1 s Φ st = Ψ 0 0 st = σ t st 1 Then P 1 O 2 P 2 O 2 P Φ Φ = O 2 O 2 O 2 O 2 P 3 O 2 P 4 O 2 O 2 O 2 O 2 O 2 where O 2 denotes the 2 2 null matrix and ) ) p1, 1)a 2 P 1 = p2, 2))a 2 P = ) ) 1 p1, 1))a 2 P 3 = 2 0 p2, 2)a 2 P = The spectral radius of P Φ Φ is given by where ρp Φ Φ ) = max{ 1 2 p1, 1)a2 1 + p2, 2)a 2 2 ± } = [p1, 1)a p2, 2)a 2 2] 2 + 4a 2 1a 2 2[1 p1, 1) p2, 2)] In Figure 1 we depict two spectral density functions in the time-invariant or linear solid line) and switching dashed line) cases. Both plots have a smooth flat) profile over frequencies, as one would expect. However, the one relative to the switching model is above the other since it is mixing normal and good times over the lifetime. Here parameters are chosen to be a = 0.8, σ = 2, R = 1.5 for the linear model and a 1 = 1, a 2 = 0.5, σ 1 = 1, σ 2 = 2,

11 11 Figure 1: Spectral density functions for the linear time-invariant) DSGE solid line) and the MS DSGE dashed line) models as illustrated in Example 5.1). R 1 = 1 and R 2 = 2 for the switching model with p = 0.1 and q = 0.2. Finally, in the latter case we have ρp Φ Φ ) = 0.5 and ρp Φ ) = ). The following model generalizes an example of standard time-invariant) DSGE discussed in Franchi and Paruolo 2014). Let n = 2, m = k = 1 and d = 2, and take A st = ) B st = ) 1 1 C st = ) D st = 1 if s t = 1 and A st = 1 ) B st = ) 1 4 C st = ) D st = 2 if s t = 2, with p = 0.1 and q = 0.2. Here ρp Φ Φ ) = 0.72 and ρp Φ ) = The spectral density function of this MS DSGE model is reported in Figure 2 along with spectra of the two underlying linear models. All the spectra show preponderance at high frequencies. From an empirical point of view, we note that business cycle bands correspond roughly to spectrum frequencies from 0.2 to See Onatski and Williams 2003). When the probability of being in one regime increases, the spectrum of the switching model moves toward

12 12 Figure 2: Spectral density function of the MS DSGE model as described in Example 5.2) dashed line) along with the two underlying time-invariant spectra solid lines). the most probable underlying linear model. This is shown in Figure 3 with different probabilities: p = 0.8 and q = 0.2 top panel) and p = 0.1 and q = 0.8 bottom panel). The spectral radius of the matrix P Φ Φ is equal to 0.93 resp. 0.52) in the first resp. second) case. 6. Conclusion We characterize the autocovariance structure of a second-order stationary MS DSGE model, showing that it coincides with that of a stable finite) VARMA model. The orders of the VARMA representations are directly linked with the number of regimes of the unobserved Markov chain. Then we propose a simple procedure to compute the spectral representation of covariance stationary MS DSGE models. This representation is obtained as the Fourier transform of the autocovariance function by using the Riesz-Fischer theorem as in linear frameworks. Examples suggest that the switching recurrence plays an important role in the frequency decomposition of MS DSGE models. Finally, the introduced techniques can be easily applied to any MS DSGE of interest. For example, macroeconomic issues and policy evaluations can be analysed from a frequency-specific point of view. The statistical analysis based on such considerations might be an interesting topic for future research.

13 Figure 3: Spectral density functions of the MS DSGE models as described in Example 5.2) dashed line) along with the two underlying time-invariant spectra solid lines) with different probabilities: p = 0.8, q = 0.2 top panel) and p = 0.1, q = 0.8 bottom panel). 13

14 14 Acknowledgements. Work performed under the auspices of the Cournot Foundation, Mont-sous-Vaudrey, France, and financially supported by a research grant FAR 2014) given by the University of Modena and Reggio E., Italy. References [1] Alencar, A.P., Morettin, P.A., and Toloi, C. 2013) State space Markov switching models using wavelets, Studies in Nonlinear Dynamics and Econometrics 17 2), [2] Bask, M., and De Luna, X. 2002) Characterizing the degree of stability of non-linear dynamic models, Studies in Nonlinear Dynamics and Econometrics 6 1). [3] Bianchi, F. 2012) Methods for Markov-Switching Models, mimeo. [4] Billio, M., and Monfort, A. 1998) Switching State-Space models, Likelihood function, filtering and smoothing, Journal of Statistical Planning and Inference 68, [5] Bougerol, P., and Picard, N. 1992) Strict stationarity of generalized autoregressive processes, Annals of Probability 20, [6] Brandt, A. 1986) The stochastic equation Y n+1 = A n Y n + B n with stationary coefficients models, Advanced in Applied Probability 18, [7] Brockwell, P.J., and Davis, R.A. 1991) Time Series: Theory and Methods, Springer Verlag, Berlin-Heidelberg New York. [8] Cavicchioli, M. 2014a) Determining the number of regimes in Markov switching VAR and VMA models, Journal of Time Series Analysis 35 2), [9] Cavicchioli, M. 2014b) Analysis of the likelihood function for Markov switching VARCH) models, Journal of Time Series Analysis 35 6),

15 15 [10] Costa, O.L.V., Fragoso, M.D., and Marques, R.P. 2005) Discrete-Time Markov Jump Linear Systems, in: Probability and Its Applications Gani, J., Heyde, C.C., Jagers, P., and Kurtz, T.G., eds), Springer- Verlag, London-Berlin. [11] Fernández-Villaverde, J., Rubio-Ramirez, J., Sargent, T., and Watson, M. 2007) ABCs and Ds) of understanding VARs, American Economic Review 97, [12] Foerster, A., Rubio-Ramirez, J., Waggoner, D. F., and Zha, T. 2014) Perturbation methods for Markov-switching DSGE models, No. w20390 National Bureau of Economic Research. [13] Franchi, M., and Paruolo, P. 2014) Minimality of state space solutions of DSGE models and existence conditions for their VAR representation, Computational Economics, doi: /s [14] Franchi, M., and Vidotto, A. 2013) A check for finite order VAR representations of DSGE models, Economics Letters 120, [15] Francq, C., and Zakoïan, J.M. 2001) Stationarity of Multivariate Markov- Switching ARMA Models, Journal of Econometrics 102, [16] Francq, C., and Zakoïan, J. M. 2002) Autocovariance structure of powers of switching-regime ARMA processes, ESAIM: Probability and Statistics 6, [17] Francq, C., and Zakoïan, J.M. 2005) The L 2 structures of the standard and switching regime GARCH models, Stochastic Processes and Their Applications 115, [18] Giacomini, R. 2013) The relationship between DSGE and V AR models, Cemmap working paper, Centre for Microdata Methods and Practice, no. CWP21/13, [19] Giraitis, L., Kapetanios, G., Theodoridis, K., and Yates, T. 2014) Estimating time-varying DSGE models using minimum distance methods, mimeo. [20] Hamilton, J.D. 1989) A new approach to the economic analysis of nonstationary time series and the business cycle, Econometrica 57 2),

16 16 [21] Hamilton, J.D. 1990) Analysis of time series subject to changes in regime, Journal of Econometrics 45, [22] Hamilton, J.D. 1994) Time Series Analysis, Princeton Univ. Press, Princeton, N.J. [23] Hamilton, J.D. 2005) Regime Switching Models, in New Palgrave Dictionary of Economics. BROOKS, Chapter 9. [24] Ioannidis, E.E. 2007) Spectra of bivariate VARp) models, Journal of Statistical Planning and Inference 137 2), [25] Krolzig, H.M. 1997) Markov-Switching Vector Autoregressions: Modelling, Statistical Inference and Application to Business Cycle Analysis, Springer Verlag, Berlin-Heidelberg-New York. [26] Onatski, A., and Williams, N. 2003) Modeling model uncertainty, Journal of the European Economic Association 1 5), [27] Priestley, M.B. 1981) Spectral Analysis and Time Series, Academic Press, Orlando, Fl. [28] Yang, M. 2000) Some properties of vector autoregressive processes with Markov switching coefficients, Econometric Theory 16, [29] Zhang, J., and Stine, R.A. 2001) Autocovariance structure of Markov regime switching models and model selection, Journal of Time Series Analysis 22 1), Appendix Derivation of 7). For h = 0, we have πi) vec Ez t z t s t = i) = πi) Ez t z t s t = i) = πi) E[Φ i z t 1 + Ψ i w t ) Φ i z t 1 + Ψ i w t ) s t = i] = πi) Φ i Φ i ) Ez t 1 z t 1 s t = i) + πi) Φ i Ψ i ) Ez t 1 w t s t = i) + πi) Ψ i Φ i ) Ew t z t 1 s t = i) + πi) Ψ i Ψ i ) Ew t w t s t = i)

17 17 = πi) Φ i Φ i ) d Ez t 1 z t 1 s t = i, s t 1 = j) pj, i) πj) = + πi)ψ i Ψ i ) vec I m d pj, i) Φ i Φ i ) πj) Ez t 1 z t 1 s t 1 = j) + πi)ψ i Ψ i ) vec I m as Ez t 1 w t ) = Ew t z t 1 ) = 0. In fact, from 4) we have Ez t 1 w t ) = [EΦ st 1 Φ st 2 Φ st i Ψ st i 1 ) I m ] Ew t i 1 w t ) = 0 i=0 as s t ) is independent of w t ) and Ew t l w t ) = vec Ew t l w t) = 0 for every l 1. The proof for Ew t z t 1 ) = 0 is analogous. This implies 7). Derivation of 8). For h 1, we get πi) Ez t z t h s t = i) = πi) E[Φ i z t 1 + Ψ i w t )z t h s t = i) = πi) Φ i Ez t 1 z t h s t = i) + πi) Ψ i Ew t z t h s t = i) d = πi) Φ i Ez t 1 z t h s t = i, s t 1 = j) pj, i) πj) = d pj, i) Φ i πj) Ez t 1 z t h s t 1 = j) This implies 8). Proof of Theorem 3.2. For k 1 and n 1 the matrix P Φ is singular. In fact, we have P Φ = [ d Φ j ] P I p ) where P is the transition probability matrix and p = n + k. Then det P Φ = [ d det Φ j ] det P) dp hence det P Φ = 0 as det Φ j = 0. Let n 0 = n 0 P Φ ) denote the algebraic multiplicity of the zero eigenvalue of P Φ. Then n 0 P Φ ) dk 1. The

18 18 Cayley-Hamilton theorem implies that there exists scalars α 1,..., α n0 such that ϕ PΦ P Φ ) = P M Φ α 1 P M 1 Φ α M n0 P n 0 Φ = 0 11) where ϕ PΦ λ) = λ M α 1 λ M 1 α M n0 λ n 0 is the characteristic polynomial of P Φ. From above, it has a zero ith term, i = 0,..., n 0 1. Equation 11) implies fe I p ) ) P M Φ α 1 P M 1 Φ α M n0 P n 0 Φ W 0) f = 0 hence Γ y M) M n 0 α j Γ y M j) = 0 By using the relation in the statement of Theorem 3.1, we get Γ y h + M) = f e I p ) P h+m Φ W 0) f Γ y h + M 1) = f e I p ) P h+m 1 Φ W 0) f. Γ y h + n 0 ) = f e I p ) P h+n 0 Φ W 0) f Multiplying the second,..., the last relation by α 1,..., α n0, respectively, yields Γ y h + M) M n 0 α j Γ y h + M j) = f e I p ) = f e I p ) P h Φ P h+m Φ P M Φ M n 0 M n 0 α j P h+m j Φ α j P M j Φ = f e I p ) P h Φ ϕ PΦ P Φ ) W 0) f = 0 ) ) W 0) f W 0) f for every h 0 use 11)). This formula can be rewritten in the equivalent form Γ y h) M n 0 α j Γ y h j) = 0

19 19 for every h M. This form characterizes the autocovariance function of a stable VARMAp, q ) process where p M n 0 = dn + k) n 0 and q M 1 = dn + k) 1. See, for example, Brockwell and Davis 1991). Proof of Theorem 4.1. From 2) we get [ + F y ω) = fe I p ) P Φ e iω ) h + as ρp Φ ) < 1. h=0 h=0 P Φ e iω ) h I dp ] W 0)f = fe I p ) [ I dp + I dp P Φ e iω ) 1 + I dp P Φ e iω ) 1] W 0)f Proof of Corollary 4.1. We have Then I dp P Φ e ±iω ) 1 = T T 1 T DT 1 e ±iω ) 1 = T I dp De ±iω ) 1 T 1 [ ] = T De ±iω ) h T 1 = T λ j e ±iω ) h F y ω) = fe I p )T = fe I p )T = T h=0 [ ] dp 1 T 1 1 λ j e ±iω [ I dp + dp dp Recall that λ j < 1 as ρp Φ ) < 1. dp h=0 1 1 λ j e + 1 iω 1 λ 2 j 1 + λ 2 j 2λ j cosω) 1 λ j e iω ) T 1 W 0)f T 1 )] T 1 W 0)f Proof of Theorem 4.2. The result follows from Theorem 4.1 and the relations I dp e ±iω P Φ ) 1 = T I dp e ±iω J) 1 T 1 = T [ s r=1 I mr e ±iω J mr λ r )) 1] T 1 where e ±iω λ r e ±iω λ r I mr e ±iω J mr λ r ) = e ±iω λ r e ±iω λ r