We examine stock market responses during the COVID-19 pandemic period using fractional integration techniques. The evidence suggests that stock markets generally follow a synchronized movement before and the stages of the pandemic shocks. We find while mean reversion significantly declines, the degree of persistence and dependence has been increased in the majority of the stock market indices in whole sample analysis covering the period of August 02, 2019 and July 09, 2020. This outcome implies increasing integration and possibly declining benefits of diversification for the global stock portfolio management.

The objective of this paper is to examine the overreaction behavior of 20 commodity futures based on intraday data from November 20, 2019 to June 3, 2020 with a focus on the impact of the Covid-19 pandemic. A dynamic and non-parametric approach is applied on intraday data for four different frequencies (from 1 min to 1 h) and two different sub-periods (pre-Covid-19 pandemic and during Covid-19 pandemic) in order to detect overreaction behavior which is defined as a large change of prices followed by proportional price reversals. Our empirical findings show that the overreaction hypothesis is confirmed for the considered commodity futures. Furthermore, both the number and the amplitude of overreactions is higher during the Covid-19 pandemic. Our findings also indicate that soft and metal commodities show much less overreactions than precious metals and especially energy commodities. In particular, crude oil futures exhibit a different overreaction behavior compared to other commodities since it has a higher number of negative than positive overreactions during the Covid-19 pandemic. We also find that the data frequency is independent of the overreacting behavior in both periods as the results continuously improve when having more observations due to higher frequencies. Finally, we find that extreme overreactions during the Covid-19 pandemic provide a great potential for profitable trading returns, which can be exploited by traders.

BACKGROUND: The vast majority of microbiome research so far has focused on the structure of the microbiome at a single time-point. There have been several studies that measure the microbiome from a particular environment over time. A few models have been developed by extending time series models to accomodate specific features in microbiome data to address questions of stability and interactions of the microbime time series. Most research has observed the stability and mean reversion for some microbiomes. However, little has been done to study the mean reversion rates of these stable microbes and how sampling frequencies are related to such conclusions. In this paper, we begin to rectify this situation. We analyse two widely studied microbial time series data sets on four healthy individuals. We choose to study healthy individuals because we are interested in the baseline temporal dynamics of the microbiome.
RESULTS: For this analysis, we focus on the temporal dynamics of individual genera, absorbing all interactions in a stochastic term. We use a simple stochastic differential equation model to assess the following three questions. (1) Does the microbiome exhibit temporal continuity? (2) Does the microbiome have a stable state? (3) To better understand the temporal dynamics, how frequently should data be sampled in future studies? We find that a simple Ornstein-Uhlenbeck model which incorporates both temporal continuity and reversion to a stable state fits the data for almost every genus better than a Brownian motion model that contains only temporal continuity. The Ornstein-Uhlenbeck model also fits the data better than modelling separate time points as independent. Under the Ornstein-Uhlenbeck model, we calculate the variance of the estimated mean reversion rate (the speed with which each genus returns to its stable state). Based on this calculation, we are able to determine the optimal sample schemes for studying temporal dynamics.
CONCLUSIONS: There is evidence of temporal continuity for most genera; there is clear evidence of a stable state; and the optimal sampling frequency for studying temporal dynamics is in the range of one sample every 0.8-3.2 days.