Convergence of the long memory Markov switching model to Brownian motion

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1 Convergence of the long memory Marov switching model to Brownian motion Changryong Bae Sungyunwan University Natércia Fortuna CEF.UP, Universidade do Porto Vladas Pipiras University of North Carolina February 5, 04 Abstract This technical note shows that the partial sum process of the Marov switching model converges to Brownian motion in the sense of finite-dimensional distributions. Definitions The Marov switching MS, in short model is defined as follows. Let s T {s T },...,T be a stationary Marov chain, taing the value 0 or, with the probability transition matrix p00 p P 00.. p p We shall tae below p 00 p 00 T and p p T as functions of T, which explains the use of the index T in the notation s T. Let µ 0, µ R. Consider the series X T µ s T + ϵ,,..., T,. {ϵ } is a sequence of i.i.d. random variables with mean 0 and variance σϵ, and its partial sum process S T t X T EXT, t [0, ]. The term µ s T in X T is not constant. It varies between two values µ 0 and µ. The switching mechanism is dictated by the Marov chain s T,,..., T. If one is in regime j, j 0,, one stays there with probability p jj and switches to the other regime with probability p jj. Suppose now that µ 0 µ.3 and, for j 0,, p jj p jj T c j T δ j,.4 JEL classification: C, C3, C4, C50. Keywords and phrases: Marov switching model, long memory, changes in mean.

2 0 < c j < and 0 < δ 0, δ <..5 Since p jj T, the Marov chain s T tends to stay in the same state as T. Under the assumptions.3.5, Diebold and Inoue 00, Proposition 3, showed that Var S T VarX T XT T CT d+,.6 d min{δ 0, δ } δ 0 δ 0,..7 The behavior.6 of the variance of the partial sum is consistent with that for long memory time series. Recall that a long memory time series X {X } Z is commonly defined as a secondorder stationary time series with autocovariance function CovX 0, X h ch d, as h,.8 c > 0 and d 0, /. Under.8, one can easily show that VarX X T CT d+, as T. The parameter d is nown as the long memory parameter. Convergence result We are interested here not only in the second-order properties of the partial sum process S T t but also in its convergence in distribution. For a long memory time series, a common limit of its partial sum process is fractional Brownian motion with parameter H d + /. Recall that B H {B H t} t R is fractional Brownian motion FBM if it is Gaussian, has stationary increments and is H-self-similar i.e. for any c > 0, the process B H ct and c H B H t have the same finitedimensional distributions. It is called standard if EB H, and if H /, then it is the usual Brownian motion. Does the partial sum process S T t of the Marov switching model converge to FBM with parameter H d + /? The answer to this question is negative, as will be shown below under additional assumptions. In fact, the limit turns out to be the usual Brownian motion. Consequences of this fact for estimation of the long memory parameter in the MS model are discussed in Bae, Fortuna and Pipiras 04. Before providing the convergence result, we state an auxiliary lemma which will be used in the proof of the main result. The result of the lemma is implicit in the proof of Proposition 3 in Diebold and Inoue 00. See Appendix for a proof. Lemma. The term µ s T in the Marov switching model. satisfies: for j, T, Cov µ s, µ T s µ Γ Tm m µ p 00 p p 00 p µ 0 µ λ m,. and µ µ0 µ, Γ j p 00 p p 00 p λ j λ j λ j λ j. λ p 00 + p..3 In the next result, we establish the convergence of the partial sum process to Brownian motion. See Appendix for a proof.

3 Theorem. Suppose that.3.5 hold and that Then, 3 min{δ 0, δ } + δ 0 δ <..4 S T t T /+d fdd σbt,.5 d is given in.7, fdd stands for the convergence of finite-dimensional distributions and B is a standard Brownian motion with and σ c 0c µ 0 µ c 3.6 c 0, if δ 0 < δ, c c, if δ < δ 0, c 0 + c, if δ 0 δ. The condition.4 in Theorem. was used to obtain a normal limit for the partial sum process S T. Note that, when δ δ 0 δ > 0, this condition becomes 0 < δ < /3.7 and hence see.7 d δ 0,. 6 More generally suppose δ 0 δ. Then.7 becomes d δ 0 δ. The condition d > 0 translates to δ 0 > δ /, thus δ / < δ 0 δ. Moreover,.4 becomes δ 0 + δ <, that is, δ 0 < δ /. Two cases need to be distinguished next. First, suppose that δ < δ / or δ < /3. Then, the two conditions δ / < δ 0 δ and δ 0 < δ / reduce to the first one, namely, δ / < δ 0 δ. In particular, d δ 0 δ δ and d 0,, 6 since δ < /3. Second, suppose that δ δ / or δ /3. Then, the two conditions δ / < δ 0 δ and δ 0 < δ / reduce to δ / < δ 0 δ / supposing δ < δ / or δ < /; otherwise, the intervals in the two conditions are disjoint. In particular, d δ 0 δ δ and d 0, 6 since δ /3. What happens when the condition.4 does not hold that is, for other values of δ 0, δ? When t, the convergence to a normal limit follows immediately from the results of Dobrushin 96. In fact, the asymptotic normality is expected for all finite-dimensional distributions. In the proof of Theorem., we used general convergence results for nonstationary Marov chains. In a nearly stationary case, the conditions of those results are expected to be too strong as discussed in the footnote on p. 76 of Dobrushin 956. But to the best of our nowledge, there are currently no results addressing this issue aside from the result corresponding to t. 3,

4 A Technical proofs Proof of Lemma.: Let ξ {s T 0} {s T }, so that µ s T µ ξ, prime denotes the transpose. Observe further that Cov µ s, µ T s Cov µ ξ Tm, µ ξ m µ Covξ, ξ m µ µ Γ m µ, A. Γ j Covξ 0, ξ j Cov{s T 0 {s T 0} j Cov {s T 0}, 0 {s T } j Cov {s T 0 }, {s T 0} j Cov {s T }, 0 {s T } j Ps T 0 0, s T j 0 PsT 0 0PsT j 0 PsT 0 0, st j PsT 0 0PsT j Ps T 0, st j 0 PsT 0 PsT j 0 PsT 0, st j PsT 0 PsT j Denoting by π π 0 π the invariant stationary probabilities of s T, and by p j nm the transition probability from state n to state m in j steps, we have π Γ j 0 p j 00 π 0 π 0 p j 0 π π p j 0 π 0 π p j π. A. To evaluate Γ j, we need expressions for π 0, π and p j nm. One can show that π 0 p p 00 p, π. p 00 p 00 p A.3 and that the probability transition matrix P in. is diagonalizable as π 0 π0 π P 0 : E π 0 0 λ 0 λ F, A.4 λ is given by.3. Note that and λ in A.4 are the eigenvalues of the matrix P and the matrices E and F are made of the right- and left-eigenvectors of P, satisfying F E. The latter fact implies that p 00 p P j π π λ j π0 π p + p 00 λ j p 00 λ j p λ j p 00 + p λ j π0 + π λ j π λ j π 0 λ j π + π 0 λ j p j 00 p j 0 p j 0 p j. A.5 By using A.3 and A.5, we obtain from A. that Γ j established by using A.. is given by. and. is finally Proof of Theorem.: It is enough to show that, for t,..., t n [0, ] and θ,..., θ n R, n S T t T /+d d N 0, σ n A.6 4

5 and that Since X T σ n lim T E n is a stationary series the relation A.7 follows from n S T t T /+d σ E Bt. A.7 ST t E T /+d σ t, A.8 as T. Write S T t µ s T Eµ s T + ϵ : S,T t + S,T t. Then, by the independence of s T and {ϵ }, A.9 and Hence, it is enough to show that Observe that ES,T t Var From Lemma. and by using ST t S,T t S,T t E T /+d E T /+d + E T /+d S,T t E σ ϵ 0. T +d µ s T T /+d µ Γ 0 + S,T t E σ t. A.0 T /+d Var µ ξ µ, jγ j + Γ j µ. j Covξ, ξ µ j λ j λ λ λ, j jλ j λ λ λ λ +, A. we get V 00 p 00 p p 00 p ES,T t V 00 µ 0 µ, λ λ + + λ+ λ λ. A. But by.3 and.4, λ p 00 + p c 0 T δ 0 c T δ. 5

6 Hence, as T, c is defined in.7, and λ λ c 0 T δ 0 + c T δ T min{δ0,δ}, c c 0 T δ c c { } 0 T δ T min{δ exp ct min{δ 0,δ } o, 0,δ } since δ 0, δ <. This shows that Now, V 00 p 00 p λ p 00 p λ p 00 p p 00 p T min{δ 0,δ }. c p 00 p p 00 p c 0T δ0 c T δ c T min{δ 0,δ } c 0c c T δ 0 δ, A.3 since δ 0 δ + min{δ 0, δ } δ 0 δ. Thus, by.4, Relations A. and A.4 now yield V 00 c 0c c 3 T min{δ 0,δ } δ 0 δ c 0c c 3 T d. A.4 ES,T T +d c 0c µ 0 µ t c 3 σ t, which establishes A.0. We now turn to the convergence A.6. In view of the decomposition A.9 and since, by the central limit theorem, as T, n 0 < σ,n <, it is enough to show that σ n is defined in A.7. Write n n f T t x S,T t T / d N 0, σ,n, S,T t T /+d d N 0, σ n, A.5 S,T t T /+d n T t f T t s T t, T /+d {t [T t ]}µ x Eµ s T t. Recall that s T t may be equal to 0 or. We shall apply next a central limit theorem for functionals of Marov chains. Let T : max Ps T t+ A s T t x Ps T t+ A s T t x x,x,a 6

7 be the so-called contraction coefficient for the Marov chain s T. Note first that T max Ps T t+ A s T t 0 Ps T t+ A s T t, A since x, x {0, } and the difference Ps T t+ A st t x Ps T t+ A st t x is zero whenever x x. Note next that the difference Ps T t+ A st t 0 Ps T t+ A st t is zero for A {0, }, and equals p 00 + p for A {0} and A {}. Hence, By.4, as T. Observe that T p 00 + p. α T : T c 0 T δ 0 + c T δ c T min{δ 0,δ }, sup t T sup f T t x C T : C x T /+d. A.6 By Theorem. in Sethuraman and Varadhan 005 see also Theorem in Dobrushin 956, the convergence A.5 follows if C T α 3 T T Since {t [T t ]} for large enough T, note that, for large enough T, Varf T t In view of A.3, this yields Varf T t t s T t n p 00 p T +d Varf T t s T t 0. A.7 s T t n T +d Varµ s T t n T +d µ Γ 0 µ. p 00 p µ 0 µ n µ 0 µ c 0 c c T d δ0 δ Combining this with the asymptotic form of α T given in A.6, we conclude that the condition A.7 holds if T +d T 3 min{δ 0,δ } T d+ δ 0 δ T 3 min{δ 0,δ }+ δ 0 δ 0. The convergence to 0 follows from the assumption.4. References Bae, C., Fortuna, N. and Pipiras, V. 04, Can Marov switching model generate long memory?, Preprint. Available at Diebold, F. X. and Inoue, A. 00, Long memory and regime switching, Journal of Econometrics 05, Dobrushin, R. L. 956, Central limit theorem for non-stationary Marov chains. I, Theory of Probability and its Applications, Dobrushin, R. L. 96, Limit theorems for a Marov chain of two states, in Select. Transl. Math. Statist. and Probability, Vol., Inst. Math. Statist. and Amer. Math. Soc., Providence, R.I., pp T. 7

8 Sethuraman, S. and Varadhan, S. 005, A martingale proof of Dobrushin s theorem for non-homogeneous Marov chains, Electronic Journal of Probability 0. Changryong Bae Natércia Fortuna Dept. of Statistics Faculdade de Economia Sungyunwan University Universidade do Porto 5-, Sungyunwan-ro, Jongno-gu Rua Dr. Roberto Frias Seoul, 0-745, Korea Porto, Portugal crbae@su.edu nfortuna@fep.up.pt Vladas Pipiras Dept. of Statistics and Operations Research UNC at Chapel Hill CB#360, Hanes Hall Chapel Hill, NC 7599, USA pipiras@ .unc.edu 8