A Functional Central Limit Theorem for an ARMA(p, q) Process with Markov Switching

Transcription

1 Communications for Statistical Applications and Methods 2013, Vol 20, No 4, DOI: A Functional Central Limit Theorem for an ARMAp, q) Process with Markov Switching Oesook Lee 1,a a Department of Statistics, Ewha Womans University Abstract In this paper, we give a tractable sufficient condition for functional central limit theorem to hold in Markov switching ARMA p, q) model Keywords: Functional central limit theorem, Markov switching ARMA p, q) process, φ-mixing 1 Introduction Since the seminal work of Hamilton 1989), nonlinear time series models subject to Markov switching are widely used for modelling dynamics in different fields, especially in econometric studies Krolzig, 1997; Hamilton and Raj, 2002) Markov switching autoregressive moving averagemsarma)p, q) model, in which a hidden Markov process governs the behavior of an observable time series, exhibits structural breaks and local linearity When we consider a time series model as a data generating process, one of the important properties to show is the functional) central limit theorem Functional central limit theoremfclt) is applied for statistical inference in time series to establish the asymptotics of various statistics concerning, for example a test for stability such as CUCUM or MOSUM and unit root testing Probabilistic properties of MSARMAp, q) models have been studied in, eg, Francq and Zakoïan 2001), Yao and Attali 2000), Yang 2000), Lee 2005), and Stelzer 2009) The purpose of this paper is to find a sufficient condition under which FCLT holds for the partial sums processes of the given MSARMAp, q) models The typical approach to obtaining the FCLT for nonlinear time series models is to show that a specific dependence property such as various mixing condition, L p -NEDnear-epoch dependent), association or θ, L or ψ-weak dependence holds However, restrictive conditions such as distributional assumptions on errors and higher order moment are required in order to prove such dependence properties Ango Nze and Doukhan, 2004; De Jong and Davidson, 2000; Davidson, 2002; Dedecker et al, 2007; Doukhan and Wintenberger, 2007; Harrndorf, 1984) Our proof is based on Theorem 211 of Billingsley 1968) that is an extension of the results of Ibragimov 1962) The proof is short and relies on a second moment assumption This research was supported by Basic Science Research Program through the NRF funded by the Ministry of Education, Science and Technology2012R1A1A and 2013R1A1A ) 1 Professor, Department of Statistics, Ewha Womans University, Ewhayeodaegil 52, Seoul, Korea oslee@ewhaackr

2 340 Oesook Lee 2 Results We consider the following Markov switching autoregressive moving average MSARMA)p, q) model, where ARMA coefficients are allowed to change over time according to a Markov chain: x t = p ϕ i u t 1 )x t i + q θ j u t 1 )e t j + e t, t Z, 21) j=1 where p 1, q 0, {u t } is an irreducible aperiodic Markov chain on a finite state space E with n-step transition probability matrix P n) = p n) u,v) u,v E and the stationary distribution π, ϕ i u) and θ j u)i = 1, 2,, p, j = 1, 2,, q, u E) are constants and {e t } is a sequence of independent and identically distributed random variables with mean 0 and finite variance σ 2 We assume that {u t } and {e t } are independent Assuming without loss of generality that p = q and letting X t = x t, x t 1,, x t p+1 ) and ε t = e t, e t 1,, e t p+1 ), define p p matrix Φ t = Φu t ) = ϕ 1 u t ) ϕ 2 u t ) ϕ p 1 u t ) ϕ p u t ) and Θ t = Θu t ) = θ 1 u t ) θ 2 u t ) θ p 1 u t ) θ p u t ) Then X t = Φ t 1 X t 1 + Θ t 1 ε t 1 + ε t 22) and x t = c X t, c = 1, 0,, 0) Applying recursively the Equation 22), after m-step we have that m 1 X t = m j=1 Φ t jx t m + j=1 Φ t jφ t k + Θ t k )ε t k + m 1 j=1 Φ t jθ t m ε t m + ε t 23) assume 0 j=1 Φ t j = I) We make the assumptions A1) ρ := p α i < 1 with α i = sup u E Eϕ 2 i u 1 ) u 0 = u) ) 1 2 Note that the assumption A1) depends only on the autoregressive coefficients ϕ i u t ) of the Equation 21) Next theorem is for the existence of a strictly stationary solution to the Equation 21)

3 A Functional Central Limit Theorem for an ARMAp, q) Process with Markov Switching 341 Theorem 1 Suppose the assumption A1) holds Then there is a unique non-anticipative strictly stationary solution x t to 21) with Ex 2 t ) < Proof: Define X t = j=1 Φ t j) Φt k + Θ t k ) ε t k + ε t 24) Then according to Minkowski inequality, we have that Ec Xt ) 2) 1 2 = c Xt 2 c j=1 Φ t jφ t k + Θ t k )ε 2 t k + σ 2 Let β i := E π ϕ i u t ) + θ i u t )) 2 ) 1/2 < Apply the independence of {u t } and {e t } and Minkowski inequality and use the assumption A1) to obtain, after some tedious calculations, that c j=1 Φ t j Φ t k + Θ t k ) ε 2 t k Ck 1), k = 1, 2, 3,, where p C := C0) = σ β i, C1) = α 1 C ρc, C2) = α α 2) ρc, Cp) = α 1 Cp 1) + α 2 Cp 2) + + α p C ρc Moreover, In general, for 1 r p, m 0, Therefore, Cp + 1) = α 1 Cp) + α 2 Cp 1) + + α p C1) ρ 2 C, Cp + 2) = α 1 Cp + 1) + α 2 Cp) + + α p C2) ρ 2 C, Cmp + r) = α 1 Cmp + r 1) + α 2 Cmp + r 2) + + α p Cm 1)p + r) ρ m+1 C c j=1 Φ t jφ t k + Θ t k )ε 2 t k C ρ [ ) p ] +1 < 25) Hence we have that Ec Xt ) 2 < and xt = c Xt strictly stationary sequence satisfying 22) < ae One can easily verify that X t in 24) is a

4 342 Oesook Lee To prove the uniqueness, assume that ˆX t is any non-anticipative, strictly stationary solution to 22) with Ec ˆX t ) 2 < Obviously ˆX t satisfies 23) for every m 1 and thus we obtain that X t ˆX t = m j=1 Φ t j)x t m ˆX t m ) E x t ˆx t = E c ) Xt ˆX t = E c m j=1 Φ ) t j X t m ˆX t m p ) E x t m i+1 ˆx t m i+1 c m j=1 Φ t j) νi p xt m i+1 ˆx t m i+1 c 2 m j=1 Φ 2 t jν i, where ν i = 0,, 0, 1, 0,, 0), 0 except the i th entry which is 1 Adopt the similar method used to derive the inequality 25), then we have that c m j=1 Φ t jν i 2 Mρ [m 1)/p] 0 as m 0, where M = p E π ϕ i u t ) < Thus we have that E xt ˆx t = 0, and xt = ˆx t ae Take x t = xt to get the conclusion A stationary process x t ) t Z is called φ- mixing if φ n = sup { PE 2 E 1 ) PE 2 ) : E 1 F k, E 2 F k+n} 0 as n, where for s < t, F t s := σx s, x s+1,, x t ) Let [x] denote the largest integer not exceeding x and let D denote convergence in distribution Let B denote standard Brownian motion on [0, 1] Following theorem is our main result Theorem 2 Suppose the assumption A1) holds Then for a strictly stationary solution x t of 21) with Ex 2 t ) <, τ 2 = Varx 0 ) k< Covx 0, x t ) is convergent and as n, 1 [nξ] D x t τbξ), 0 ξ 1 n t=1 Proof: Note that v t := u t 1, e t, e t 1,, e t p ) is an aperiodic E R p+1 -valued Markov chain Since {u t } is a positive recurrent Markov chain with finite state space, {u t } is exponentially φ-mixing By independence of {u t } and {e t }, {v t } is also exponentially φ mixing with φ 1/2 n < By the above Theorem 1, and take x t = c x t,l = c l j=1 Φ t j) Φt k + Θ t k ) ε t k + c ε t, j=1 Φ t j) Φt k + Θ t k ) ε t k + c ε t

5 A Functional Central Limit Theorem for an ARMAp, q) Process with Markov Switching 343 {x t,l } is stationary for each l For some measurable functions f and f l, we can write x t = f v t, v t 1, v t 2, ), and x t,l = f l v t, v t 1, v t 2,, v t l+1 ) Then we have that Thus, x t x t,l 2 = l=1 c C k=l+1 j=1 Φ ) t j Φt k + Θ t k ) ε t k 2 c j=1 Φ 2 t j) Φt k + Θ t k ) ε t k k=l+1 ρ k=l+1 [ ) ] p +1 x t x 2 t,l C Cp 2 ρ l=1 k=l+1 l=1 i=0 = Cp 2 ρ 1 ρ) 2 < [ ) ] p +1 ρ i+l Applying Theorem 211 in Billingsley 1968) yields the conclusion Remark 1 For x t in 21), define a process Y t = Au t )Y t 1 + η t, with Y t = x t 1,, x t p, e t,, e t q+1 ) for properly defined p + q) p + q) matrix Au t ) and p + q) 1 vector η t If we take W t = u t, Y t ), then W t is a Markov chain One way to prove the FCLT is to use the Markovian structure of the model to obtain mixing properties If W t is ϕ-irreducible weak Feller and top Lyapunov exponent of Au t ) is negative, then the process Y t is geometrically ergodic and β-mixing and the FCLT for x t is obtained Lee, 2005) The stationarity and geometric ergodicity for vector valued MSARMAp, q) process with a general state space parameter chain are examined by Stelzer 2009) Remark 2 It is known that if the largest modulus of the eigenvalues λ of the set Φu), u E is less than 1, then the FCLT holds for x t Davidson, 2002, p256) Following is a simple example, which shows that λ > 1, but the FCLT holds due to Theorem 2 Example 1 Consider a MSARMA3, 2) process given by x t = 3 ϕ i u t 1 )x t i + 2 θ j u t 1 )e t j + e t, 26) j=1

6 344 Oesook Lee where {u t } is a Markov chain with state space E = {1, 2, 3} and one step transition probability matrix P is given by P = Take ϕ 1 1), ϕ 2 1), ϕ 3 1)) = 01, 03, 001), ϕ 1 2), ϕ 2 2), ϕ 3 2)) = 11, 02, 02), ϕ 1 3), ϕ 2 3), ϕ 3 3)) = 02, 003, 02), and Φu) = ϕ 1 u) ϕ 2 u) ϕ 3 u) , u E For the above example, the largest modulus of the eigenvalues λ of the set Φ1), Φ2) and Φ3) is larger than 1 but 3 sup u E Eϕ2 i u 1) u 0 = u)) 1/2 < 1 Hence Theorem 1 and 2 guarantee the existence of a strictly stationary solution x t of 26) and the FCLT for {x t } References Ango Nze, P and Doukhan, P 2004) Weak dependence: Models and applications to econometrics, Econometric Theory, 20, Billingsley, P 1968) Convergence of Probability Measures, Wiley, New York Davidson, J 2002) Establishing conditions for functional central limit theorem in nonlinear and semiparametric time series processes, Journal of Econometrics, 106, Dedecker, J, Doukhan, P, Lang, G, Leon, J R, Louhichi, S and Prieur, C 2007) Weak dependence, Examples and Applications In Springer Lecture Notes in Statistics, 190 De Jong, R M and Davidson, J 2000) The functional central limit theorem and weak convergence to stochastic integrals I: weakly dependent processes, Econometric Theory, 16, Doukhan, P and Wintenberger, O 2007) An invariance principle for weakly dependent stationary general models, Probability and Mathematical Statistics, 27, Francq, C and Zakoïan, J M 2001) Stationarity of multivariate Markov-switching ARMA models, Journal of Econometrics, 102, Hamilton, J D 1989) A new approach to the economic analysis of nonstationary time series and business cycle, Econometrica, 57, Hamilton, J D and Raj, Beds) 2002) Advances in Markov- switching Models - Applications in Business Cycle Research and Finance, Physica-Verlag, Heidelberg Harrndorf, N 1984) A functional central limit theorem for weakly dependent sequences of random variables, The Annals of Probability, 12, Ibragimov, I A 1962) Some limit theorems for stationary processes, Theory of Probability and Its Applications, 7, Krolzig, H M 1997) Markov-Switching Vector Autoregressions: Lecture Notes in Economics and Mathematical Systems, 454, Springer, Berlin

7 A Functional Central Limit Theorem for an ARMAp, q) Process with Markov Switching 345 Lee, O 2005) Probabilistic properties of a nonlinear ARMA process with Markov switching, Communications in Statistics: Theory and Methods, 34, Stelzer, R 2009) On Markov-switching ARMA processes-stationarity, existence of moments and geometric ergodicity, Econometric Theory, 29, Yang, M 2000) Some properties of vector autoregressive processes with Markov switching coefficients, Econometric Theory, 16, Yao, J and Attali, J G 2000) On stability of nonlinear AR processes with Markov switching, Advances in Applied Probability, 32, Received June 12, 2013; Revised July 8, 2013; Accepted July 8, 2013