The Impact of COVID-19 Crisis on Stock Markets’ Statistical Complexity

Table 1

Critical values for test (with a 0.05 probability).

Length of series	Gaussian noise	Fractional Gaussian noise	Brownian motion	Fractional Brownian motion	Geometric Brownian motion
100	0.544	0.544	0.575	0.576	0.579
250	0.419	0.419	0.489	0.489	0.494
500	0.286	0.286	0.408	0.408	0.412
1,000	0.156	0.156	0.334	0.334	0.336
2,000	0.070	0.070	0.280	0.280	0.281
5,000	0.025	0.025	0.243	0.243	0.244
7,500	0.016	0.016	0.236	0.235	0.236
10,000	0.012	0.012	0.232	0.232	0.232
20,000	0.006	0.006	0.228	0.228	0.228

Notes: Table 1 estimates the critical values for the proposed compound measure of complexity being equal to zero as a function of various types of series and different lengths for each of these types. The Brownian motion simulations are implemented in “somebm” R language package [22], while the method proposed by Paxson [23] to generate fractional Gaussian noise is implemented in “longmemo” R package [24, 25]. Despite their irregular and unpredictable character, the Gaussian/Brownian noises do not contain any nontrivial structure, so they cannot be viewed as “complex.” The Martin et al. [13] generalised (global) complexity measure, based on the Jenson–Shannon divergence, is implemented in Sippel et al. [18]. The complexity measure is normalised in the range [0, 1]. The figures are obtained via numerical simulation with 20,000 corresponding Gamma probability distributions and all the embedding dimensions between 2 and 6.