New Test for a Random Walk Detection Based on the Arcsine Law
In our work, we construct a new statistical test for a random walk detection, which is based on the arcsine law. Additionally, we consider a version of the unit root test for an autoregressive process of order 1, which is also related to the arcsine law.
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bstract In our work, we construct a new statistical test for a random walk detection, which is based on the arcsine law. Additionally, we consider a version of the unit root test for an autoregressive process of order 1, which is also related to the arcsine law. Furthermore, we conduct some simulation study in order to check the quality of the proposed test. Keywords Random walk · Arcsine law · Test for a random walk detection
1 Introduction Our objective is to introduce some proposal of a new test for a random walk detection. To the best of our knowledge, the main tools that have been applied in this context so far are the two celebrated tests—an Augmented Dickey–Fuller (ADF) test ([10]) and the Runs test ([13])—and through our work, we attempt to fill in a gap related to this field of investigations. The presented approach is a certain extension and a generalization of the research conducted in [2]. We also compare the quality of the proposed test with the efficiency and the power of the mentioned ADF and Runs test. The readers who are closely interested in the field of tests devoted to a random walk identification or to the existence of unit root are encouraged to refer to [6–9] and [11].
M. Dudzi´nski · K. Furma´nczyk (B) · A. Orłowski Institute of Information Technology, Warsaw University of Life Sciences, Warsaw, Poland e-mail: [email protected] M. Dudzi´nski e-mail: [email protected] A. Orłowski e-mail: [email protected]
© The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Switzerland AG 2020 O. Valenzuela et al. (eds.), Theory and Applications of Time Series Analysis, Contributions to Statistics, https://doi.org/10.1007/978-3-030-56219-9_4
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Our paper is organized as follows. In Sect. 1, we present a general idea leading to the construction of our test for a random walk identification, as well as we describe the construction of this test. In Sects. 2 and 3, we check the efficiency and the power of the introduced test, whereas we summarize our study in Sect. 4. The presented research and its results are an extension of the research and the results from [3].
1.1 Random Walk Random walk theory states that the price of financial instrument in the subsequent time point is the sum of its price in the previous time point and some random variable with a finite variance, i.e. it is modelled with the use of a stochastic process called a random walk. We say that a stochastic process S0 , S1 , S2 , . . . , Sn is a random walk, if the following relations hold: S0 = s0 , S1 = s0 + Y1 , S2 = s0 + Y1 + Y2 , .. . Sn = s0 + Y1 + Y2 + · · · + Yn , where Y1 , Y2 , . . . , Yn form an iid sequence of symmetric r.v.’s. In our considerations, we assume that s0 = 0. Then, St =
t
Yi , t = 1, 2, . . . , n.
i=1
1.2 Ordinary Random Walk Test Let n = |1 ≤ i ≤ n : Si > 0|. Then, obviously: n —the number of those among the sums S1 , . . . , Sn , which are positive, n /n—its frequency. From the first arcsine law ([4, 10]),
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