Sub-sequence incidence analysis within series of Bernoulli trials: application in characterisation of time series dynamics

ABSTRACT

This paper presents a new and widely applicable nonparametric approach to the characterisation of time series dynamics. The approach involves analysis of the incidence of occurrence of patterns in the direction of movement of the series, and may readily be applied to time series data measured on any scale. The paper includes derivations of analytic forms for two (infinite) families of distributions under the null hypothesis of random behaviour, and of a useful analytic form for the generation of the moments of these distributions. The distributions are asymptotically normal, so allowing for straightforward application of the approach presented in the paper too long series of high frequency and/or extended time period data. Areas of application in finance and accounting are suggested.

KEYWORDS:

• Bernoulli trials
• time series dynamics
• nonparametric test

JEL Classifications:

• C14
• C22
• C55

Acknowledgments

I am very grateful to two anonymous referees for the care which they took in review of this paper and for the expertise which they brought to bear. Also to Mark Tippett for his constructive criticism and his support in the development of this work. Any remaining errors remain my sole responsibility.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1 See Sivarajah et al. (2017) for an interesting review of the challenges and methods associated with big data.

2 In addition to presenting an approach for the distribution theory of runs based on FMCI, Fu and Koutras (1994) also provide a number of references to various combinatorial approaches in this area.

3 ‘Symmetry’ in Bernoulli trials meaning that the probability of one outcome equals the probability of the other (both being 0.5); and ‘overlap order’ being as defined in the third paragraph of Section 3.

4 See, for example, Knuth (1998) chapter 3.

5 Nomenclature regarding attributes, labels and ranking of ordinal scale classes follows Siegel and Castellan (1988) section 3.3.

6 This is true for the standard Wald–Wolfowitz runs test – against which the framework presented in this paper is compared in Section 6, and against which it is shown to compare favourably.

7 See Edwards, Magee, and Bassetti (2007) for an excellent overview of technical analysis.

8 See, for example, Hallin and Mélard (1988). It is not uncommon in literature for empirical analysis to be based on the assumption of normally distributed data, whilst at the same time reporting test statistics (e.g. Jarque–Bera statistics) which imply the assumption is violated – and without recognition or discussion of the impact of non-normality.

9 This is the case in the illustrative application set out in Section 6.3.

10 Note that the terminology ‘sub-sequence’ is dropped at this point in favour of the less cumbersome ‘sequence’.

11 Existence of mean and variance being satisfied (see sections on distributions and their moments), conduct of a z-test is against the null hypothesis as expanded to include the mutual independence of the time series under investigation.

12 Kendall, Stuart, and Ord (1987) Sections 3.7–3.11 gives a general treatment of factorial moments and associated generating functions.

13 See, for example, Billingsley (1995, 364), Ferguson (1996, 70) and Bradley (2007).

14 It is, of course, straightforward to ‘fool’ the standard runs test using a variety of synthetic series. In the set-up of Section 6.2, this may be achieved simply by adjusting parameter $c_2$ in expression (31). Here, any value of $c_2$ chosen from the integer range [6, 10] results in inability to reject the null of randomness under the standard runs test at a generally acceptable level of significance.

15 Referring to the cumulative probability distribution under the null hypothesis, as shown in Table 2.

16 All necessary variables obtained from Datastream.

17 In some cases the variable appears to cycle, then undergo an innovation (either cycling upwards/downwards, or via a jump) to a new level around which cycles recommence. This is highly suggestive of a concatenation of differing generating processes, rather than processes which are constant in terms of structure and parameters. Inspection also suggests that the variables’ dynamics appear to be largely firm-idiosyncratic: common patterns are hard to detect, even amongst firms in broadly similar industry categories.

18 Notice that in the sequence ↓↑↑↑↑↑↑↑↓ (for which l = 9 and p = 1), terms ‘↓’ at the beginning and the end demarcate a run of exactly seven ‘↑’ terms. In the sequence ↓↑↑↑↑↑↑↑ (for which l = 8 and p = 0), the end of the run of seven increases is not demarcated with a ‘↓’ term.