Asymptotics for small switching probabilities

34 C . Gourieroux, J. Jasiak Economics Letters 70 2001 29 –41 Fig. 2. Autocorrelogram for switching mean. Cov Y ,Y 5 Cov E Y uZ , E Y uZ 1 ECov Y ,Y uZ f s d s dg s d s d t t 2h t t 2h t t 2h 5 Cov m 2 m Z , m 2 m Z fs d s d g 1 t 1 t 2h 2 5 m 2 m Cov Z ,Z , for h ± 0 s d s d 1 t t 2h VY 5 VE Y uZ 1 EVE Y uZ s d s d t t t 2 2 2 2 5 m 2 m 1 s 2 s VZ f s d g s d 1 1 t Therefore the autocorrelation function of Y is proportional to the autocorrelation function of Z, for 2 2 2 2 h 1, with a factor related to the ratio m 2 m s 2 s . 1 1

3. Asymptotics for small switching probabilities

In this section we study asymptotic properties of the empirical autocovariance function when the switching probabilities are small. We consider a Markov transition model, for a finite number of C . Gourieroux, J. Jasiak Economics Letters 70 2001 29 –41 35 Fig. 3. Autocorrelograms for switching variance. observations, which allows us to focus on the effect of infrequent switches on memory, and provides a tractable limit model. 3.1. The finite sample model and the limiting model j For a finite sample of size T, we consider a standard transition model, where p , j 5 0,1 denotes T the transition probability from state j to the complementary state between consecutive dates. These probabilities depend on T as well as on the corresponding distributions of durations of the two 1 regimes. To emphasize this dependence, we index by T the various variables, i.e. t , D , D . . . n,T j,T j,T 1 The duration variables are independent with geometric distributions of parameters p and p , T T 5 respectively. Under the assumption: 1 lim Tp 5 l , lim Tp 5 l A.1. T T 1 T → ` T → ` it is known that the finite sample process tends in distribution to a limiting process after an appropriate 1 change of time. More precisely the normalized durations D T and D T tend in distribution to j,T j,T 1 limiting duration variables D , D , which are independent with exponential distributions with j,` j,` respective parameters l , l . The normalized observation period becomes [0,1], whereas the limiting 1 5 See Granger and Hyung 1999 for a similar assumption. 36 C . Gourieroux, J. Jasiak Economics Letters 70 2001 29 –41 j j variables lim t T 5 t , lim N 5 N , lim A T 5 A are deduced from the limiting durations as n,T n,` T ` T ` T → ` T → ` T → ` they were in the finite sample model. Note that in the limiting model the number of switching dates N is finite, but stochastic. For ` instance in the special case l 5 l , the distribution of N is a Poisson distribution with a parameter 1 ` l 5 l . 1 3.2. Limiting behaviour of the autocorrelogram We study directly the asymptotic properties of the autocorrelogram or equivalently of the empirical ˆ autocovariance function. In fact g h tends in distribution to a limiting variable g h which is not T ` degenerate. This variable admits a mixture distribution, which takes into account the limiting number of switching dates. More precisely let us consider the first possible number of regimes and distinguish the cases of fixed and large lags h. i If there is only one limiting regime N 5 0, with probability p , say, we get: ` ˆ g h 5 0,;h, and also g h 5 0,;h T ` ii If there are two limiting regimes, N 5 1, with probability p , we get: ` 1 2 T 2 D 1 T 2 h 1,T 1 S D ˆ ] ]] ]]] g h 5 T 2 D 2 h 2 s d T 1,T T T T 1 2 D D h h 1,T 1,T S D S D ]] ] S ] D ]] 5 1 2 2 2 1 2 1 2 T T T T • If h is fixed, we deduce that: ˆ lim g h 5 1 2 D D 5 s d T 1,` 1,` T → ` The limiting pattern corresponds to a constant function with a stochastic level. 1 ˆ • If h is large proportional to T, h 5 aT, we get: lim g aT 5 1 2 D 2 a 2 1 2 a1 2 T T 1,` T → ` 2 D , or: 1,` 1 h h 2 ˆ S ] D S ] D g h | 1 2 D 2 2 1 2 1 2 D 6 s d T 1,` 1,` T T The limiting pattern corresponds to a piecewise linear function in the lag, featuring a rather slow decay which may be confused with a hyperbolic decay in practice. • iii If there are three limiting regimes, N 5 2, with probability p , we get: ` 2 1 1 ˆ lim g h 5 D 1 2 D s d T 1,` 1,` T → ` 1 1 1 2 ˆ lim g aT 5 D 2 a 2 1 2 a D s d s d s d T 1,` 1,` T → ` C . Gourieroux, J. Jasiak Economics Letters 70 2001 29 –41 37 And so on. It is important to note that the limiting distributions depend on the initial regime fixed at 0 in our computations. 3.3. Simulation To illustrate the limiting behaviour of the empirical autocovariance we have simulated a series of 1 length T 53000, with switching probabilities p 5 p 5 0.005. Despite the large number of observa- T T tions, the limiting distributions of the first and second order moments are not reduced to a point mass Figs, 4 and 5. The effect of state transitions on the pattern of the autocorrelogram and also on the estimated fractional degree can be observed by considering the joint distribution of estimated autocorrelations at two different lags Fig. 6. These limiting distributions may explain the large variation of the estimator of fractional coefficient for high frequency data, where the d parameter is estimated by rolling, since the estimated value is very sensitive to the subsample it is computed from. Fig. 4. Distribution of the mean. 38 C . Gourieroux, J. Jasiak Economics Letters 70 2001 29 –41 Fig. 5. Distribution of autocorrelations.

4. Concluding remarks