On Spectral Representation of Varma Models with Change in Regime

We present various formulae in closed form for the spectral density of multivariate or univariate ARMA models subject to Markov switching, and describe some new properties of them. Many examples and numerical applications are proposed to illustrate the behaviour of the spectral density. This turns out to be useful in order to investigate various concepts of stationarity via spectral analysis.

where x t is a K-dimensional random vector with values in R K , (ξ t ) is an irreducible, aperiodic and ergodic Markov chain with finite space Ξ = {1, 2, . . . , d}, stationary transition probabilities denoted by p ij = pr(ξ t = j|ξ t−1 = i) and unconditional (or steady state) probabilities π i = pr(ξ t = i) for i ∈ Ξ. The a i (ξ t ) and b j (ξ t ) are K × K real random matrices. To allow for the possibility of change in variance, it is assumed that ϵ t = ϵ t (ξ t ) = σ(ξ t )η t , where σ(ξ t ) is a K × K real random matrix and (η t ) is a white noise with E(η t η ′ t ) = Ω, the covariance matrix Ω being nonsingular. In addition, assume that (η t ) is independent of (ξ t ). In order to investigate the stationarity properties of the process (x t ), Francq and Zakoïan (2001) proposed the following Markovian representation of (1): where: On Spectral Representation of Varma Models with Change in Regime Here M(m, n; R) denotes the vector space formed by all m × n real matrices. In order to avoid degeneracies in (2), we assume, without loss of generality, p ≥ 1. In the case of a purely autoregressive model (q = 0), it is understood that J and B t vanish, be the matrices obtained by replacing ξ t by i in Φ t , A t , B t , and Σ t , respectively. Then the following matrices are defined: Costa et al. (2005), Proposition 3.6, p.35, we see that ρ(P ) < 1 implies ρ(P * ) < 1, where ρ( · ) denotes the spectral radius. It can be easily checked that the converse is not true in general. See Remark 3.7, p.35, of the quoted book. In fact, for p 11 = p 22 = 0.5, Φ(1) = a(1) = 0.7 and Φ(2) = a(2) = 1.25, we get ρ(P ) = 1.02625 and ρ(P * ) = 0.975. The following theorem gives the necessary and sufficient conditions for the existence of second-order stationary Markov switching VARMA processes.
Theorem 1 (Francq and Zakoïan (2001)). Suppose that Then, for all t ∈ Z, the series converges in L 2 and the process (x t ), defined as the first component of (z t ), is the unique second-order stationary solution of (1). A necessary and sufficient condition for the existence of a second-order stationary solution of (1) is given by where Λ = (I K 2 (p+q) 2 · · · I K 2 (p+q) 2 ) ∈ M(K 2 (p + q) 2 , dK 2 (p + q) 2 ; R) and || · || denotes the matrix norm ||M || = ∑ i,j |m ij | for any matrix M = (m ij ). From now on, we assume ρ(P ) < 1, hence ρ(P * ) < 1. Once second-order stationarity is ensured, it can be useful to compute the autocovariance Γ x (k) of (x t ) for k ∈ Z. For any k ∈ Z, let us consider the matrix . . .
By Francq and Zakoïan (2001), p.349, for all k ≥ 0, we have In particular, the variance-covariance matrix of (x t ) is given by vec Γ x t+k ) ′ for any stationary process. As stated in Pataracchia (2011), p.13, we can compute the diagonal terms of the spectral density matrix F x (ω) of (x t ) by considering the summation where the frequency ω belongs to [−π, π]. This series converges as ρ(P * ) < 1. The following theorem gives a formula in closed form for the spectral density matrix of the process in (1).
Theorem 2 (Cavicchioli (2013)). Under the above notation, the spectral density matrix F x (ω) of the process (x t ) in (1) is given by and ReX(ω) denotes the real part of X(ω). In the univariate case with Ω = 1, we get the even real-valued function If an observed time series x 1 , . . . , x T can be described by an MS-VARMA(p, q) 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 x (ω).

Explicit matrix computations
(2.1) MS-AR(1). Let us consider a 2-state univariate AR(1) with Markov regime where ϵ t ∼ N (0, σ 2 ϵ ). By using Theorem 2, we explicitly compute now the spectral generating function f x (ω) for the process in (4). Similar computations were given in Pataracchia (2008) but our formula is much simpler than the one obtained in the quoted paper. In this case, we have K = 1, d = 2, p = 1, q = 0, Ω = 1, Σ = σ 2 ϵ , Φ(1) = a(1) and Φ(2) = a (2). The matrices P and S are given by Here φ P ( · ) denotes the characteristic polynomial of P , hence Since Here we have Then the spectral density function for the process in (4) has the following closed form where H and K (resp. Y (ω) and Z(ω)) are given by (5) (resp. (6)). It is an even real-valued function, so ω can be taken in [0, π]. Formula (7) is much simpler than the one obtained by Pataracchia (2008), Formula (12) in Section 3 and Appendix B. Formula (7) shows that the spectral density function of a 2-state MS-AR(1) presents itself as a sort of weighted average spectra of the underlying models. See also the pictures in the next section. For Model (4), local stationarity implies global stationarity. In fact, if |a(i)| < 1 for i = 1, 2, then The converse is not true in general, i.e., we can have global stationarity without local stationarity within each regime. For example, for p 11 = 0.2, p 22 = 0.4, a(1) = 1.2 and a(2) = 0.5, we get ρ(P ) ∼ 0.62 (and ρ(P * ) ∼ 0.76). Finally, ρ(P ) < 1 is a necessary and sufficient condition for global stationarity of the process in (4) where ϵ t ∼ N (0, σ 2 ϵ ). As before, we compute explicitly the spectral generating function f x (ω) for the process (x t ) in (8). In this case, The matrices P and S are given by where p ij and π i , i, j = 1, 2, are as in Subsection (2.1). The nonzero eigenvalues of the above 8 × 8 matrix P coincide with those of the 2 × 2 matrix P from Subsection (2.1). So these matrices have the same spectral radius. Furthermore, the previous discussion on stationarity of MS-AR(1) models maintains his validity also for MS-ARMA(1, 1). Now we can apply the formula in Theorem 2 where f = (2) ) .
Then we get It is interesting to note that the coefficients b(ξ t ) of the MA part in Model (8) do not appear in the expression of f x (ω) given in (9). The knowledge of an explicit formula for the spectral density function of an MS-ARMA(1, 1) model allows us to obtain the spectral representations for Markov Switching Generalized Autoregressive Conditional Heteroschedastic models (in short, MS-GARCH(1, 1)) and for Markov Switching Stochastic Volatility models (in short, MS-SV). In fact, it is well-known that these last processes admit MS-ARMA(1, 1) reprersentations via linearization. .
For every k ≥ 1, we have by induction and S 1 = p 11 I K(p+q) , R 1 = U 1 = I K(p+q) and T 1 = p 22 I K(p+q) . Setting On Spectral Representation of Varma Models with Change in Regime From (10) we have (2).
Equating the last two formulae and solving for y(ω) yield Here we have hence the Woodbury formula gives Reasoning as above, from (11), (12) and (13) we get respectively From (14) and (15) we obtain Substituting this formula into (14) gives y(ω). From (16) and (17) we get

Substituting this formula into (17) gives z(ω).
As an alternative method for computing F x (ω), let us consider X(ω) = (I 2K(p+q) e iω − P * ) −1 e iω from Theorem 2. Then we have By Woodbury formula we obtain Taking Ω = I K , we have Here we set By Woodbury formula we obtain

Examples and numerical applications
(3.1). We start with some 2-state univariate MS(2)-AR(1) models, considered in Pataracchia (2008), Section 3, pp.11-13, and report her interesting observations developed there to make the reading self-contained. In Figure  1 Figure 1(b). Such figures correspond to two MS(2)-AR(1) models with identical coefficients but different transition probability matrices. They show that the spectral density of a regime switching model presents itself as a sort of weighted average of the spectra of the underlying models. As the less volatile model becomes more likely (for instance in Figure 1(b)), the peak of the spectral density function decreases. In Figure 1(c) we show the spectral density of an MS(2)-AR(1) with p 11 = p 22 = 0.5, a(1) = −0.8 and a(2) = −0.2. When both a(1) and a(2) are positive (resp. negative), the spectral shape is similar to a typical spectral representation of a stable AR(1) with positive (resp. negative) coefficients. When the sign of the coefficients is opposite, the prevailing shape depends on which model dominates the other, in terms of both the underlying unconditional probabilities and the absolute values of its coefficients. See Figure 1 Figure 2. The global stationarity is met but not the local one. The simulations results, obtained in the quoted paper (see Tables 3 and 4, pp.362-363), are reported in Table 1 below. The first column contains the true values of the parameters. In the second column there are the parameter estimates which are obtained without stationarity constraints (wsc). The third (resp. fourth) column shows estimation under the global (resp. local) stationarity constraints, in short gsc (resp. lsc).
The estimations obtained without imposing any stationarity constraint are slightly better than those obtained under the global or local stationarity constraints. This is confirmed by the closest position of the spectral density to the true one. (3.3). Let us consider the following univariate MS(2)-AR(2) model On Spectral Representation of Varma Models with Change in Regime    .
The spectral radii of the matrices P and P * are given by We know that ρ(P ) < 1 implies ρ(P * ) < 1. The converse is not true in general. For example, if p 11 = 0.1, p 22 = 0.2 and a = 1.2, then ρ(P ) ∼ 1.05 and ρ(P * ) ∼ 0.95. However, if |a| < 1, then ρ(P ) and ρ(P * ) are both less than one. Spectra are shown in Figure 3. Both models are global stationarity. However, the former does not satisfy local stationary, while the latter does. Differences between the two spectra appear at the extremes of the cycle.
(3.4). Let us consider the following univariate MS(2)-AR(2) model  Figure 4. The former process is global stationary and its spectral density appears to be quite flat; on the contrary, the latter process which does not satisfy global stationary has a inverted-U shape spectral density function.  5, p.353: where η t ∼ IID(0, 1). Then we have For p 11 = 0.2, p 22 = 0.1, a 1 (1) = 1.8, a 2 (1) = −0.9 and a 1 (2) = −0.2, both regimes are stationary (in fact, ρ(Φ(1)) ∼ 0.95 and ρ(Φ(2)) = 0.2) but the resulting process is explosive (the explosion is exponential). See Figure  5 for the spectrum, which shows quite a striking peak. Because the probability of staying in the same regime is small, the local stationarity does not compensate the explosive nature of the regime changes as remarked in the quoted paper at p.353. In this case we have ρ(P ) ∼ 1.33 and ρ(P * ) ∼ 0.96. The spectral density could be then first step to immediatly evaluate the behavior of such an explosive process.
(3.6). Let us consider the following univariate MS(3)-ARMA(1, 2) model Thus ρ(P ) < 1 is only a sufficient secondorder stationarity condition in the general case. If p 12 = 0.8, p 23 = 0.8 and p 31 = 0.8, then ρ(P ) = 0.54 and ρ(P * ) = 0.4648. In this case, the stationary probabilities are π 1 = 0.3509, π 2 = 0.2807 and π 3 = 0.3684. The spectral density for the last model are shown in Figure 6. Let us compare the above spectrum with a similarly constructed spectrum for an MS(3)-ARMA(2, 1) model (see Figure 7). When the autoregressive prevails as in this case, the spectral density assume the common shape of autoregressive process. While in Figure 6 the shape was mostly similar to the spectral density of a moving average process.