Testing Stability in Functional Event Observations with an Application to IPO Performance

Abstract

Many sequentially observed functional data objects are available only at the times of certain events. For example, the trajectory of stock prices of companies after their initial public offering (IPO) can be observed when the offering occurs, and the resulting data may be affected by changing circumstances. It is of interest to investigate whether the mean behavior of such functions is stable over time, and if not, to estimate the times at which apparent changes occur. Since the frequency of events may fluctuates over time, we propose a change point analysis that has two steps. In the first step, we segment the series into segments in which the frequency of events is approximately homogeneous using a new binary segmentation procedure for event frequencies. After adjusting the observed curves in each segment based on the frequency of events, we proceed in the second step by developing a method to test for and estimate change points in the mean of the observed functional data objects. We establish the consistency and asymptotic distribution of the change point detector and estimator in both steps, and study their performance using Monte Carlo simulations. An application to IPO performance data illustrates the proposed methods.

Keywords:

• Change point analysis
• Functional data analysis
• IPO

Supplemental Materials

Online Supplement: In Section A, we provide the detailed proofs of all technical results. In Section B, we discuss the estimation of the long run variance function with uncorrelated errors. In Section C, we present the simulation results of the frequency change test. In Section D, we outline practical guidance on the implementation of our stability test. (PDF)

Computer Code: MATLAB code to perform the test described in the article. (zip file)

IPO dataset: Dataset used in the illustration of methods in Section 6. (.xlsx file)

Acknowledgments

We wish to thank the Editor (Professor Atsushi Inoue), the associate editor, and two anonymous referees whose comments and suggestions led to vast improvements of this work. We thank the Reading Academic Computing Cluster at the University of Reading for providing the high performance computer, which enables the successful execution of our highly computational simulations and the empirical application. We would like to thank Professor Marie Hušková for her helpful discussion at the 5^th International Workshop on Functional and Operatorial Statistics (Online, 2021).

Disclosure Statement

The authors report there are no competing interests to declare.

Funding

Notes

1 We consider the IPOs issued between December 1, 2015 to September 30, 2020. In order to fully observe the 60-day cumulative abnormal return curves, we include the price data up to December 31, 2020.

2 Under H₀, the unknown mean curve ${\mu }_t(u)$ on different dates becomes a common one $\mu (u)$ for all t.

3 It is also possible to suppress $y_t(u)$ from (1.1) in place of other conditions on the serial dependence in the data. In the literature, it is usually assumed that $\{y_t(u),1\le t\le N\}$ and $\{{ϵ}_{t,m}(u),1\le m\le n_t,1\le t\le N\}$ are independent or at least uncorrelated. Due to Assumption 3.1, we do not need conditions between $y_t(u)$ and ${ϵ}_{t,m}(u)$. Thus, we choose to include $y_t(u)$ to emphasize that time dependence is allowed between functional event observations at different dates.

4 We also provide a discussion of this choice in the Appendix D of the online supplement.

5 The sample period is chosen to avoid a prolonged period of IPO suspension in China. The China Securities Regulatory Commission (CSRC) suspended IPOs between July 4, 2015 and November 31, 2015 in order to slow a devastating stock market crash in 2015. Such suspension was lifted and the resumption of IPO was allowed in December 1, 2015.

6 We used a value-weighted portfolio of all stocks in China’s A-share market.

7 This definition allows us to use both high frequency intraday data as well as daily data. Due to the computational cost, we choose to use daily data in this study.

Additional information

Funding

Gregory Rice was supported by the Natural Sciences and Engineering Research Council of Canada under grant number RGPIN 50503-10477.