Empirical observation made in Econometrics and applied financial time series literature for long time horizons reveal that log-returns of various series of share prices, exchange-rates and interest rates depict unique stylized features. These features include: the frequency of large and small values is rather high suggesting that the data do not come from a normal but rather a heavy tailed distribution and that exceedances of high thresholds occur in clusters which indicates that there is dependence in the tails. It is also observed that the sample autocorrelations of data are small whereas the sample autocorrelation of the absolute and squared values is significantly different from zero even for large lags. This behavior suggests that there is some kind of long-range dependence in the data.
Various models have been proposed in order to describe these features. Among these models is the GARCH model which has been found appropriate in capturing volatility dynamics in financial time series particularly in modelling of stock market volatility as seen in  and derivative market volatility as utilized by  . GARCH (1, 1) in particular is often used in applications as it is believed to capture, despite its simplicity, variety of the empirically observed stylized features of the log-returns. However the log-return data cannot be modelled by one particular GARCH model over a long period of time  . They observe that in real financial time series the effect of non-stationarity of log-return series can be seen by considering the sample autocorrelation function of moving blocks of the same length as the estimates seem to differ from block to block. They suggest the use of change-point analysis of financial time series modelled by GARCH processes with parameters varying with time. The likelihood ratio scan method has been proposed by  for estimating multiple change points in piecewise stationary processes where they use a scan statistics to reduce the computationally infeasible global multiple change point estimation problem to a number of single change point detection problems in various local windows. The cumulative sum test is considered by  in determining volatility shifts in GARCH model against long range dependence. Cumulative sum test has also been used by  for change-point detection in copula ARMAGARCH Models. Markov switching GARCH model has been proposed by  where the volatility in each state is a convex combination of two different GARCH components with time varying weights making the model have a dynamic behavior to capture the variants of shocks. According to  change-point in the series could also be attributed to change in GARCH model order specification. The trio proposes an estimator based on the Manhattan distance of the sample autocorrelation of squared values. This paper aims at furthering the works of  by deriving the distributional convergence of the process used in deriving the estimator of change-point . Since is based on Manhattan distance of sample autocorrelation, the limit theory for sums of strictly stationary sequences is utilized. Conditions that ensure that partial sums of strictly stationary processes converge in distribution to an infinite variance stable distribution have provided by  . This is achieved by relating the regular variation condition and weak convergence of point processes. This was utilized by  in deriving the limit theory for the autocovariance function of linear processes which they later extended to bilinear processes in  . Limit theory for sample autocovariance of GARCH processes was also considered by  where they used weak convergence of point processes in combination with continuous mapping theorem. Point processes were also utilized by  in examining the convergence of the partial sum process of stationary regularly varying GARCH (1, 1) sequences for which the clusters of high thresholds excesses are broken down into asymptotically independent blocks which they established to be a stable Levy’s process. We utilize the point processes theory and restrict ourselves to qualitative results.
The paper is organized as follows. Section 2 outlines the GARCH model specification and change-point estimator with corresponding assumptions utilized. The weak convergence of point processes associated with the sequence is considered in Section 3. In Section 4, the asymptotic behavior of the change-point process is studied. Here the limiting distribution of is derived for a stationary GARCH sequence.
2. Change-Point Estimator
Let be a GARCH process of order given by the equation
By iterating the defining difference Equation (1) for the GARCH model can be further expressed as a stochastic differential equation as follows:
Let , , then ( ) satisfies the following stochastic differential equation
Specifically for the GARCH (1, 1) case with and Equation (2) reduces into a one-dimensional SDE
Assumption 1. (Strictly Stationary)
According to  the existence of a unique strictly stationary solution to (1) is the negativity of the top Lyapunov exponent. This however cannot be calculated explicitly but a sufficient condition for this is given by
Assumption 2. (Ergodic Process)
According to  standard ergodic theory yields that ( ) is an ergodic process. Thus its properties can be deduced from a single sufficiently large random sample of the sample.
Consider the change-point test hypothesis to be investigated to be defined as:
Assumption 3. (Weight)
Let the weight be a measurable function that depends on the sample size n and change-point k. It is arbitrarily chosen such that it satisfies the condition that
Consider Assumption  , Assumption  and Assumption  to be satisfied. According to  the change-point estimator as hypothesized in (4) is based on the lower bound of the weighted Manhattan divergence measure of the sample autocorrelation function drawn for the process as
where and k denote sample autocorrelation function and the unknown change-point respectively which are estimated as:
Proof. The works of  are utilized here. Let be a k dimensional vector and be a dimensional vector. The autocovariance and autocorrelation functions can be expressed in terms of the inner product as
where and represents the standard deviation of X and Y respectively which represents an distance from the mean. Applying the Holder’s inequality in Theorem (7) to (8) and (9) yields
Following (10) we can define a sequence of autocorrelation functions where for fixed , and for fixed , to be such that we have two subsequences and
where and denote the autocorrelation of the sequence and for . A change-point
process quantifying the deviation between and using a divergence measure motivated by the weighted distance, with k denoting the change-point is proposed. Specifically, they assumed the case when resulting into a weighted Manhattan distance and by linearity and absolute value of inequalities of the expectation operator results into
The change-point estimator is processes is assumed to be the lower bound of the Manhattan divergence measure (11) where the weight is as specified in Assumption 3. The resultant process is as specified in (6). The change-point estimator of a change point is the point at which there is maximal sample evidence for a break in the sample autocorrelation function of the squared returns process. It is therefore estimated as the least value of k that maximizes the value of where is chosen as given in (7).
3. Point Process Theory
Point process techniques are utilized in obtaining the structure of limit variables and limit processes which occur in the theory of summation in time series analysis. The point process theory as developed by  is utilized. Consider the state space of the point process where . Let B be the collection of bounded Borel sets in . Let be a collection of bounded non-negative continuous functions on with bounded support and be a collection of bounded non-negative step functions on with bounded support. Write M for the collection of Radon counting measures on with null measure o. This means that if and only if μ is of the form , where , the points are distinct and
and is a Dirac measure at , that is for any . Let be the collection of measures μ such that , so that, . Define and let be the Borel set on .
Consider a strictly stationary sequence of random row vectors with values in , that is, . The characterization of the asymptotic behavior of the tails of the random variable X is examined through the regular variation condition.
Theorem 1. (Regular Variation Condition)
In light of  assume
1) there exist a number which is a unique solution of the equation
and there exist a positive constant such that
2) If for some , then
and the vector is jointly regularly varying such that
where denotes vague convergence on the Borel σ-field of the unit sphere S1 of , relative to the norm with
Proof. Following the works of  and  , assume ξ and η are independent non-negative random variables such that for some slowly varying function L and for some , then as .
Applying Theorem 1 yields
which completes proof.
Theorem 2. (Strongly Mixing Condition)
Let ( ) be a sequence of positive numbers such that
The sequence ( ) can be chosen as the -quantile of . Since is regularly varying, for slowly varying function . The condition (12) holds for ( ) if there exists a sequence of positive integers ( ) such that , as and
The condition (12) implies by the strong mixing condition of the stationary sequence ( ).
Assume that the joint regular variation in Theorem 1 and strongly mixing conditions in Theorem 2 are satisfied for a stationary sequence ( ), then, the statement can be made for the weak convergence of the sequence of point processes
where are independent and identically distributed as . It therefore follows that ( ) converges weakly if and only if does and they have the same limit N. N is identical in law to the point process where is a Poisson process with describing the radial part of the points and is a sequence of independent and identically distributed point processes with describing the spherical part and a joint distribution Q on .
Theorem 3. Assume that ( ) is a stationary sequence of random vectors for which all finite-dimensional distributions are jointly regularly varying index . To be specific, let be the -dimensional random row vector with values in the unit sphere , . Assume that the strongly mixing condition for ( ) and that
Then the limit
exists and is the extremal index of the sequence .
1) If , then
2) If , then
where is a Poisson process on with describing the radial part of the points and is a sequence of independent and identically distributed point processes with describing the spherical part and a joint distribution Q on , where Q is the weak limit of
Theorem 4. Utilizing the theory developed by  , let ( ) be a stationary GARCH (1, 1) process and assume that the jointly regularly varying and strongly mixing conditions hold. For fixed , set , then the conditions in the Theorem 2 above are met and hence
where and are as previously defined.
We now consider the convergence of point processes which are products of random variables, which forms the basis of the results on the weak convergence of sample autocovariance and autocorrelation for stationary processes.
Theorem 5 Let ( ) be a strictly stationary sequence such that satisfying the jointly regularly varying condition for some and further assume that Theorem 2 and Theorem 3 hold, then:
where the points and are as previously defined, and are point processes on meaning that points are not included in the point processes if or
We study the weak limit behaviour of the sample autocovariance and sample autocorrelation of a stationary sequence ( ). Construct from this process the strictly stationary n-dimensional processes , . Define the sample autocovariance function
and the corresponding sample autocorrelation function
Define the deterministic counterparts of the autocovariance and autocorrelation functions as follows
Theorem 6. Assume that ( ) is a strictly stationary sequence of random variables and that for a fixed , ( ) satisfies the regular variation condition and where the points and are as previously defined.
1) If , then
The vector is jointly stable in .
2) If and for
which implies that
4. Limit Theory of Change-Point Estimator
The following proposition is our main result on weak convergence for our proposed change-point process as specified in (6) for GARCH processes based on the point process theory. In addition to the previously stated Theorems are additional Theorems are utilized in the proof of the proposition, see Appendix.
Proposition 2 Let be a strictly stationary sequence of random variables irrespective of the distribution of initial value . Specifically, let be a GARCH (1, 1) process defined in the form of a stochastic differential Equation (3). For fixed , set . Assume that the regular variation conditions hold. Let be a sequence of constants such that the strongly mixing condition is satisfied, then where the points and are as defined in Theorem 2. Thus the conditions in Theorem 5 are met and hence there exists a sequence of bounded constants which converge in distribution to such that the following statements hold:
1) If , then
2) If and for
Proof. Consider the GARCH (1, 1) model in the context of a stochastic differential Equation (3) defined as , then the necessary and sufficient conditions for stationarity are and where the latter implies that
If we assume that the sample vector comes from a stationary model, then the initial values also have a stationary distribution. This means that the distribution of is stationary whatever the distribution of , given the latter is independent of and stationarity conditions. To show this consider two sequences and given the same stochastic differential equation recursion (2) but with initial conditions and Z where both vectors are independent of the future values . Further assume that has stationary distribution. By iteration of stochastic differential Equation (3) we have
Thus for any initial values Z we have the following recursion
Then for any and for GARCH (1, 1) model (3) the top Lyapunov exponent given by
If , and , then the right hand side is also finite. In addition given the stationary conditions previously stated then for some sufficiently small ε. Thus the left hand side of (22) decays to zero as . Thus we conclude that is stationary irrespective of the distribution of the initial values .
Now, consider the sample autocorrelation function as defined in (19), then the following statements hold,
From (23) and (24) it can be asserted that there exists constants and such that the autocorrelation functions and can be expressed in terms of the autocorrelation function as follows:
The change-point process (6) can be expressed in terms of (25) and (26) as
The weak limits of the process is characterized in terms of the limiting point processes for the sample autocovariance and autocorrelation functions through the application of the Continuous Mapping Theorem 12. To complete the proof we independently prove the convergence of and and apply Theorem 12.
Let , . In order to proof the results, we define several mappings
The set is bounded for any and thus the mappings are continuous with respect to the limit point processes N. Consequently by Continuous Mapping Theorem 12 we have that
The prove of the convergence of is examined for and .
For the case of , the point process results of Theorem 3 holds and a direct application of Theorem 5 yields:
For we commence with the sequence and establish the convergence of . We rewrite using the recurrence structure of the SDE (3) so that and
Now using this representation yields:
Assuming that the condition is satisfied, we first show that II converges in probability to zero by applying Karamata’s theorem (see  ) on the regular variation and tail behavior of a stationary distribution which yields the asymptotic equivalence.
Now examining I we have
We utilize the argument given in Theorem 12 where as . Therefore, we finally obtain that:
In the presence of a change-point k as hypothesized (4) it is evident that for all t but rather
Thus the convergence of and are respectively given by
Following (31), (32) and (33) it is concluded that .
Convergence of is determined in a similar manner where
Consequently for arbitrary lags we have
In the presence of a change-point k the convergence of and are respectively given by
Now we consider the sequence and establish the convergence of as follows:
Equation (35) follows directly from Equation (27). In a similar way to Equation (28), .
Now examining III and following the results obtained in Equation (29) we have that III converges as follows
Thus we have that
Similarly it can be shown that the convergence of and are respectively given by
Next we consider the sequence and establish the convergence of as follows:
Now examining VI we have
where is a constant and since is strongly mixing with geometric rate, thus there exist a and a constant K such that and .
Now examining V we have
By applying Karamata Theorem  to (36)
Examining VII we have
Since then for XI we have
Thus we have that
By extending to arbitrary lags the convergence of is given by
Consequently the convergence of is given by
We have been able to examine the limiting behavior of for two cases. In the first case, when , the variance of is infinite and thus has a random limit without any normalization. When , the
process has a finite variance but infinite fourth moment and converges to an -stable distribution. By Theorem 8 convergence of implies that the sequence is bounded with .
We now examine the convergence of . Consider , we can express as follows:
By the Bolzano-Weierstrass theorem, a bounded sequence has always a convergent subsequence. This is further confirmed through the invariance property of subsequences in Theorem 10 which states that if converges, then every subsequence say, and converges. By linearity rule of sequences as prescribed in Theorem 11, converges. This further implies that and are bounded since every convergent sequence is bounded. The subsequences and
are also bounded with and , thus their absolute difference is also bounded as . Further assume that we are considering only significant sample autocorrelation coefficients where , then is also bounded. Applying the quotient property of subsequences, then is also convergent.
Consider the proposed change-point process as defined in (6), then we can derive the limit of as follows:
Thus applying Theorem 5 to 37 we have
From (38) above, the sequence converges in distribution to as follows
By application of Continuous Mapping Theorem 12, we have the limiting behavior of the proposed change-point process for the three cases , and as follows.
for and by application of Theorem 5 (i):
for and by application of Theorem 5 (ii):
which completes proof.
The asymptotic behavior of the change-point process is established on the basis of examining the asymptotic behavior of the sample autocovariance and sample autocorrelation functions. The limits of the suitably normalized sample autocovariance and sample autocorrelation functions are expressed in terms of the limiting point processes. The limit distributions are the difference of ratios of the infinite variance stable vectors or functions of such vectors. As a result, determination of the quantiles from the limit distributions is difficult. The limits are also generally random as a result of the infinite variance. Future work will be aimed at identifying the limit distributions so as to make the results directly applicable for hypothesis testing purposes.
The authors thank the Pan-African University Institute of Basic Sciences, Technology and Innovation (PAUSTI) for funding this research.
Theorem 7. (Holder’s Inequality)
Let I be a finite or countable index set. Given , if and , where then and
Theorem 8. (Convergent sequences are bounded)
Let be a convergent sequence. Then the sequence is bounded and the limit is unique.
Theorem 9. (Bolzano-Weierstrass)
Let be a sequence of real numbers that is bounded. Then there exists a subsequence that converges.
Theorem 10. (Invariance property of subsequences)
If is a convergent sequence, then every subsequence of that sequence converges to the same limit.
Theorem 11. (Algebra on Sequences)
If the sequences converges to L and converges to M then the following hold:
3) for and
Theorem 12. (Continuous Mapping)
Let a function be continuous in every point of a set C such that . Then if then .
Theorem 13. (Algebra on Series)
Let and be two absolutely convergent series. Then:
1) the sum of the two series is again absolutely convergent. Its limit is the sum of the limit of the two series.
2) the difference of the two series is again absolutely convergent. Its limit is the difference of the limit of the two series.
3) the product of the two series is again absolutely convergent. Its limit is the product of the limit of the two series.
Theorem 14. Let be a strictly stationary sequence. Define the partial sums of the sequence by .
1) if then
where has a stable distribution
2) if and for all , then
where S is the distributional limit of
as , μ is the measure in section 2.1 which has a stable distribution.
For every , the mapping from M in section 2.1 into is defined by
and is almost surely continuous with respect to the point process N. Thus by continuous mapping theorem
As , .
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