Modeling record-breaking stock prices

We study the statistics of record-breaking events in daily stock prices of 366 stocks from the Standard and Poors 500 stock index. Both the record events in the daily stock prices themselves and the records in the daily returns are discussed. In both cases we try to describe the record statistics of the stock data with simple theoretical models. The daily returns are compared to i.i.d. RV's and the stock prices are modeled using a biased random walk, for which the record statistics are known. These models agree partly with the behavior of the stock data, but we also identify several interesting deviations. Most importantly, the number of records in the stocks appears to be systematically decreased in comparison with the random walk model. Considering the autoregressive AR(1) process, we can predict the record statistics of the daily stock prices more accurately. We also compare the stock data with simulations of the record statistics of the more complicated GARCH(1,1) model, which, in combination with the AR(1) model, gives the best agreement with the observational data. To better understand our findings, we discuss the survival and first-passage times of stock prices on certain intervals and analyze the correlations between the individual record events. After recapitulating some recent results for the record statistics of ensembles of N stocks, we also present some new observations for the weekly distributions of record events.Comment: 20 pages, 28 figure

Record occurrence and record values in daily and monthly temperatures

We analyze the occurrence and the values of record-breaking temperatures in daily and monthly temperature observations. Our aim is to better understand and quantify the statistics of temperature records in the context of global warming. Similar to earlier work we employ a simple mathematical model of independent and identically distributed random variables with a linearly growing expectation value. This model proved to be useful in predicting the increase (decrease) in upper (lower) temperature records in a warming climate. Using both station and re-analysis data from Europe and the United States we further investigate the statistics of temperature records and the validity of this model. The most important new contribution in this article is an analysis of the statistics of record values for our simple model and European reanalysis data. We estimate how much the mean values and the distributions of record temperatures are affected by the large scale warming trend. In this context we consider both the values of records that occur at a certain time and the values of records that have a certain record number in the series of record events. We compare the observational data both to simple analytical computations and numerical simulations. We find that it is more difficult to describe the values of record breaking temperatures within the framework of our linear drift model. Observations from the summer months fit well into the model with Gaussian random variables under the observed linear warming, in the sense that record breaking temperatures are more extreme in the summer. In winter however a significant asymmetry of the daily temperature distribution hides the effect of the slow warming trends. Therefore very extreme cold records are still possible in winter. This effect is even more pronounced if one considers only data from subpolar regions.Comment: 16 pages, 20 figures, revised version, published in Climate Dynamic

A New Method for Protecting Interrelated Time Series with Bayesian Prior Distributions and Synthetic Data

Organizations disseminate statistical summaries of administrative data via the Web for unrestricted public use. They balance the trade-off between confidentiality protection and inference quality. Recent developments in disclosure avoidance techniques include the incorporation of synthetic data, which capture the essential features of underlying data by releasing altered data generated from a posterior predictive distribution. The United States Census Bureau collects millions of interrelated time series micro-data that are hierarchical and contain many zeros and suppressions. Rule-based disclosure avoidance techniques often require the suppression of count data for small magnitudes and the modification of data based on a small number of entities. Motivated by this problem, we use zero-inflated extensions of Bayesian Generalized Linear Mixed Models (BGLMM) with privacy-preserving prior distributions to develop methods for protecting and releasing synthetic data from time series about thousands of small groups of entities without suppression based on the of magnitudes or number of entities. We find that as the prior distributions of the variance components in the BGLMM become more precise toward zero, confidentiality protection increases and inference quality deteriorates. We evaluate our methodology using a strict privacy measure, empirical differential privacy, and a newly defined risk measure, Probability of Range Identification (PoRI), which directly measures attribute disclosure risk. We illustrate our results with the U.S. Census Bureau’s Quarterly Workforce Indicators

Exact statistics of record increments of random walks and L\'evy flights

We study the statistics of increments in record values in a time series $\{x_0=0,x_1, x_2, \ldots, x_n\}$ generated by the positions of a random walk (discrete time, continuous space) of duration $n$ steps. For arbitrary jump length distribution, including L\'evy flights, we show that the distribution of the record increment becomes stationary, i.e., independent of $n$ for large $n$, and compute it explicitly for a wide class of jump distributions. In addition, we compute exactly the probability $Q(n)$ that the record increments decrease monotonically up to step $n$. Remarkably, $Q(n)$ is universal (i..e., independent of the jump distribution) for each $n$, decaying as $Q(n) \sim {\cal A}/\sqrt{n}$ for large $n$, with a universal amplitude ${\cal A} = e/\sqrt{\pi} = 1.53362\ldots$.Comment: 6 pages + 5 pages of supplemental material, 5 figures. Published versio

Why Do Shoppers Use Cash? Evidence from Shopping Diary Data

Recent studies find that cash remains a dominant payment choice for small-value transactions despite the prevalence of alternative methods of payment such as debit and credit cards. For policy makers an important question is whether consumers truly prefer using cash or merchants restrict card usage. Using unique shopping diary data, we estimate a payment choice model with individual unobserved heterogeneity (demandside factors) while controlling for merchants’ acceptance of cards (supply-side factors). Based on a policy simulation where we impose universal card acceptance among merchants, we find that overall cash usage would decrease by only 7.7 percentage points, implying that cash usage in small-value transactions is driven mainly by consumers’ preferences