Change point estimation for the telegraph process observed at discrete times
The telegraph process models a random motion with finite velocity and it is usually proposed as an alternative to diffusion models. The process describes the position of a particle moving on the real line, alternatively with constant velocity or . The changes of direction are governed by an homogeneous Poisson process with rate In this paper, we consider a change point estimation problem for the rate of the underlying Poisson process by means of least squares method. The consistency and the rate of convergence for the change point estimator are obtained and its asymptotic distribution is derived. Applications to real data are also presented.
Key words: discrete observations, change point problem, volatility regime switch, telegraph process.
The telegraph process describes a random motion with finite velocity and it is usually proposed as an alternative to classical diffusion models (see Goldstein, 1951 and Kac, 1974). The process defines the position of a particle initially located at the origin of the real line and moving alternatively with constant velocity or . The changes of direction are governed by an homogeneous Poisson process with rate The telegraph process or telegrapher’s process is defined as
where is the initial velocity taking values with equal probability and independently of the Poisson process . Many authors analyzed probabilistic properties of the process over the years (see for example Orsingher, 1990, 1995; Foong and Kanno, 1994; Stadje and Zacks, 2004; Zacks 2004). Di Crescenzo and Pellerey (2002) proposed the geometric telegraph process as a model to describe the dynamics of the price of risky assets where replaces the standard Brownian motion of the original Black-Scholes-Merton model. Conversely to the geometric Brownian motion, given that is of bounded variation, so is the geometric telegraph process. This seems a realistic way to model paths of assets in the financial markets. Mazza and Rulliere (2004) linked the process (1.1) and the ruin processes in the context of risk theory. Di Masi et al. (1994) proposed to model the volatility of financial markets in terms of the telegraph process. Ratanov (2004, 2005) proposed to model financial markets using a telegraph process with two intensities and two velocities . The telegraph process has also been used in ecology to model population dynamics (see Holmes et al., 1994) and the displacement of wild animals on the soil. In particular, this model is chosen because it preserves the property of animals to move at finite velocity and for a certain period along one direction (see e.g. Holmes, 1993, for an account).
For the telegraph process observed at equidistant discrete times , with , , and as , De Gregorio and Iacus (2006) proposed pseudo-maximum likelihood and implicit moment based estimators for the rate of the telegraph process. Under the additional condition as , Iacus and Yoshida (2007) studied the asymptotic properties of explicit moment type estimators and further propose a consistent, asymptotically gaussian and asymptotically efficient estimator based on the increments of the process.
In this paper we suppose that for a telegraph process occurs a switch of the rate from to at some time instant and the interest is in the estimation of the change point and both and .
The change point estimation theory has been employed widely by means of the likelihood function (see Csörgő and Horváth, 1997). Unfortunately, the likelihood function for the telegraph process is not known and the pseudo likelihood proposed in De Gregorio and Iacus (2006) is not easy to treat in this framework. We will then proceed using the alternative method based on least squares proposed in Bai (1994, 1997) and used in different contexts by many authors including Hsu (1977, 1979) for the i.i.d. case and Chen et al. (2005) for the mixing case. Our model is peculiar in itself for the properties of described in the above, because is a parameter related to the mean and the variance of the Poisson process and because the mesh plays a role in the definition of the rate of convergence of our estimators.
The paper is organized as follows. Section 2 describes the model, the observation scheme and the change point estimator. The consistency of change point estimator is discussed in Section 3 and distributional results are presented in Section 4. Finally, Section 5 contains an application of our method to real data: we will reanalyze the classical change point data sets of Dow-Jones weekly closing (see Hsu, 1977, 1979) and IBM stock prices (see Box and Jenkins, 1970).
2 Observation scheme and least squares estimator
We consider a telegraph process defined as in (1.1) and assume to observe its trajectory only in equidistant discrete times with , and . We assume that a rate of changes of direction shift occurs during the interval at an unknown time , . Therefore the changes of direction are governed by an inhomogeneous Poisson process with parameter where the positive values and the change point (or ) are unknown and to be estimated given the observations , , …, . In order to simplify the formulas we use the following notation: . The asymptotic framework is the following: and as .
The telegraph process is not Markovian and, as mentioned in the Introduction, it is not possible to derive the explicit likelihood function of the observations ’s, therefore we can not apply the statistical methods based on the likelihood function. To work out our estimation problem, we shall follow the approach developed in Bai (1994), which involves least squares-type estimators. The same point of view has been applied by Chen et al., 2005, in a context of financial time series. For our model, the time increment plays an active role in the study of the asymptotics of our estimators so the proofs, although in some cases along the lines of Bai (1994) require some technical, but crucial, adjustments.
In order to obtain our estimator we introduce some basic notations. Let
where is the increment between two consecutive observations. We indicate the mean value of with , . We observe that the random variables are independent and identically distributed because depend by the increments . Iacus and Yoshida (2007) proved that the estimators
are consistent, gaussian and asymptotically efficient estimators of and respectively. We will use these properties in the following without necessarily mentioning them.
We assume that the change occurs exactly at time , therefore , , where represents the integer-valued function. The change point estimator is obtained as follows
We indicate the sum of the squares of residuals in the following manner
and , are respectively the least squares estimators of and This gives the two estimators
By setting , , simple algebra leads to
Therefore, formula (2.3) implies that
Our first result concerns the asymptotic distribution of under the null hypothesis that . This permits us to test if a shift has taken place during the interval .
Under , i.e. we have that for as the following result hods
where is a Brownian bridge.
Let , . Then, and We introduce the following function
with We note that
Since the Lindeberg condition is true
Now, by applying Donsker’s theorem (invariance principle) we are able to write that
with and representing respectively a standard Brownian motion and a Brownian bridge. Let , we can write
We observe that
It is easy to see (by Chebyshev inequality) that
By the law of large number while . Therefore from (2.13) follows that
The same convergence result of the Theorem 2.1 follows when we consider
where is any consistent estimator for .
From Theorem 2.1 we derive immediately that for
The last asymptotic results are useful to test if doesn’t exist a change point. In particular it is possible to obtain the asymptotic critical values for the distribution (2.15) by means of the same arguments used in Csörgő and Horváth (1997), pag. 25.
3 The consistency properties of the estimator
We shall study the consistency and the rate of convergence of the change point estimator (2.7). It is convenient to note that the rate of convergence is particularly important not only to describe how fast the estimator converges to the true value, but also to get the limiting distribution. The next Theorem represents our first result on the consistency.
The estimator satisfies
By the same arguments of Bai (1994), Section 3 and by using the formulas (10)-(14) therein, we have that
where is a constant depending only on . Let , given that
we obtain that
where and . By applying Hajék-Renyi inequality for martingales we have that
Choosing and observing that , for some (see e.g. Bai, 1994), we have that
By means of the law of iterated logarithm we obtain immediately the following rate of convergence which improve the previous result. We have that
Theorem 3.1 implies that, under the additional hypothesis , we have also consistency, i.e. in probability for any .
We are able to improve the rate of convergence of .
We have the following result
We use the same framework of the proof of the Proposition 3 in Bai (1994), Section 4, therefore we omit the details.
We choose a such that . Since is consistent for , for every , when is large. In order to prove (3.7) it is sufficient to show that is small when and are large, where . We are interested to study the behavior of for , . We define for any the set . Then we have that
for every Thus we study the behavior of . It is possible to prove that
By observing that , the Hajék-Renyi inequality yields
We prove that tends to zero when and are large. Thus we consider only or more precisely those values of such that . For , we have
where . When and are large the last three terms are negligible. Analogously we derive the proof of .
4 Asymptotic distributions
We want to study in this Section the asymptotic distribution of under our limiting framework for small variations of the rate of change of the direction. The case equal to a constant is less interesting because when is large the estimate of is quite precise.
We note that implies . By adding the condition
and to define a two-sided Brownian motion in the following manner
where are two independent Brownian motions. Now we present the following convergence in distribution result.
Under assumption (4.1), for as , we have that
where is a two-sided Brownian motion and is any consistent estimator for or .
The proof follows the same steps in Bai (1994), Theorem 1, hence we only sketch the parts of the proof that differ. We consider only because of symmetry. Let and
with We note that
similarly for the second term and for the third term we apply the invariance principle (2.12). Now we explicit the limiting distribution for
For simplicity we shall assume that and are integers. We observe that
where in the last step we have used the invariance principle (2.12). Analogously we show that