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Monitoring the Intraday Volatility Pattern

  • Robertas Gabrys , Siegfried Hörmann EMAIL logo and Piotr Kokoszka

Abstract

A functional time series consists of curves, typically one curve per day. The most important parameter of such a series is the mean curve. We propose two methods of detecting a change in the mean function of a functional time series. The change is detected on line, as new functional observations arrive. The general methodology is motivated by, and applied to, the detection of a change in the mean intraday volatility pattern. The methodology is asymptotically justified by applying a new notion of weak dependence for functional time series. It is calibrated and validated by simulations based on real intraday volatility curves.

Appendix: Proofs of the results of Section 3

Before developing proofs of Theorems 3.1 and 3.2, we state and prove Lemma 5.1 which shows that under Assumption 3.1 the partial sums of the sequence can be approximated by the trajectories of a Wiener process.

Lemma 5.1Letbe a sequence in, satisfying Assumption 3.1 for some. Letbe given as in Section 2 and set. Then, the sequencecan be defined together with a standard Wiener processon a common probability space, such that for somewe have

Proof. We define

Then, we have

showing that the dependence structure (or approximability with m-dependent variables) for the scalar sequence is inherited from . Hence, by Theorem 1 in Berkes, Hörmann, and Schauer (2010), we can define on common space with two Wiener processes and such that for some small enough

[17]
[17]

The sequences and satisfy

with some . We have, by an application of Theorem 1.2.1 in Csörgö and Révész (1981), that for some large enough constant C

[18]
[18]

Furthermore, by the law of the iterated logarithm, it follows immediately that for some small enough

[19]
[19]

Combining eqs [1719] yields

with . ■

Proof of Theorem 3.1. We have

As by Assumption 3.2 , it is enough to study the asymptotics of . We assume without loss of generality that . Furthermore, since

we can assume in addition that . Next, we use

where , for some small , which will be chosen later. We show that

[20]
[20]

To this end, we notice that the triangular inequality and basic algebraic manipulations yield

Lemma 5.1 implies that the law of the iterated logarithm applies to , and thus with Assumption 3.3, we can conclude that

Under Assumption 3.3, the function is non-decreasing, and thus again in connection with Lemma 5.1, we conclude that

Hence, eq. [20] is established.

Next, we show that

[21]
[21]

Using the same arguments as above, with replaced by the Wiener process , one can show that eq. [21] follows from

[22]
[22]

By Lemma 5.1

Thus, [22] and consequently [21] will follow if

[23]
[23]

and

[24]
[24]

We assume now that in the definition of satisfies . Then, for , we have by Assumption 3.3

showing [23]. The argument for [24] is similar.

Relations [20] and [21] imply (still assuming ) that

And thus, what remains to show now is that

[25]
[25]

This follows from routine arguments for Wiener processes, showing [25] and completing the proof of Theorem 3.1. ■

Proof of Theorem 3.2. We use the same notation as for the proof of Theorem 3.1. By Assumption 3.4 it suffices to show that

[26]
[26]

Define . Then, we have

From Theorem 3.1, it follows that

while

Acknowledgments

The first version of this article was read by two referees who provided a large number of substantive and precise comments. We acknowledge the time and care of these experts dedicated to our work and thank them for the valuable advice. This research was partially supported by Communauté française de Belgique – Actions de Recherche Concertées (2010–2015) and the IAP research network grant nr. P7/06 of the Belgian government (Belgian Science Policy).

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Published Online: 2013-07-26

©2013 by Walter de Gruyter Berlin / Boston

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