Sales figures and other business characteristics do not always develop smoothly over time but are also subject to disruptive influences from individual market-changing events. These include the entry or exit of market participants as well as market-regulating political measures. In the analysis, such shocks must be reflected in the underlying stochastic model. A comparable situation occurs, for example, in reliability theory. From there, the idea of a process with the "restarting property" will be adopted here. If the process is restarted at the time of such an event, the type of process is preserved, and the shock then results in a jump in the parameter space. This avoids a complete reboot, which would neglect the previous history. The counting process proposed here is motivated by the Bass model from product diffusion theory. However, the approach should not only be viewed in this context; it can also be framed economically in other ways. It can be used, for example, to forecast various opportunities and threats for a company. For the presentation we choose the formulation in discrete time.

Measuring dependence between two events, or equivalently between two binary random variables, amounts to expressing the dependence structure inherent in a 2x2 contingency table in a real number between -1 and 1. Countless such dependence measures exist, but there is little theoretical guidance on how they compare and on their advantages and shortcomings. Thus, practitioners might be overwhelmed by the problem of choosing a suitable measure. We provide a set of natural desirable properties that a proper dependence measure should fulfill. We show that Yule's Q and the little-known Cole coefficient are proper, while the most widely-used measures, the phi coefficient and all contingency coefficients, are improper. They have a severe attainability problem, that is, even under perfect dependence they can be very far away from -1 and 1, and often differ substantially from the proper measures in that they understate strength of dependence. The structural reason is that these are measures for equality of events rather than of dependence. We derive the (in some instances non-standard) limiting distributions of the measures and illustrate how asymptotically valid confidence intervals can be constructed. In a case study on drug consumption we demonstrate how misleading conclusions may arise from the use of improper dependence measures.

I develop an intuitive concept of perfect dependence between two variables of which at least one has a nominal scale that is attainable for all marginal distributions and propose a set of dependence measures that are 1 if and only if this perfect dependence is satisfied. The advantages of these dependence measures relative to classical dependence measures like contingency coefficients, Goodman-Kruskal's lambda and tau and the so-called uncertainty coefficient are twofold. Firstly, they are defined if one of the variables is real-valued and exhibits continuities. Secondly, they satisfy the property of attainability. That is, they can take all values in the interval [0,1] irrespective of the marginals involved. Both properties are not shared by the classical dependence measures which need two discrete marginal distributions and can in some situations yield values close to 0 even though the dependence is strong or even perfect.

Additionally, I provide a consistent estimator for one of the new dependence measures together with its asymptotic distribution under independence as well as in the general case. This allows to construct confidence intervals and an independence test, whose finite sample performance I subsequently examine in a simulation study. Finally, I illustrate the use of the new dependence measure in two applications on the dependence between the variables country and income or country and religion, respectively.

Several models for count time series have been developed during the last decades, often inspired by traditional autoregressive moving average (ARMA) models for real-valued time series, including integer-valued ARMA (INARMA) and integer-valued generalized autoregressive conditional heteroscedasticity (INGARCH) models. Both INARMA and INGARCH models exhibit an ARMA-like autocorrelation function (ACF). To achieve negative ACF values within the class of INGARCH models, log and softplus link functions are suggested in the literature, where the softplus approach leads to conditional linearity in good approximation (DOI: 10.5705/ss.202020.0353). However, the softplus approach is limited to the INGARCH family for unbounded counts, that is, it can neither be used for bounded counts, nor for count processes from the INARMA family.

In this talk, we present an alternative solution, named the Tobit approach, for achieving approximate linearity together with negative ACF values, which is more generally applicable than the softplus approach. The main part of the talk studies a Skellam–Tobit INGARCH model for unbounded counts in detail (DOI: 10.1111/sjos.12751), including stationarity, approximate computation of moments, maximum likelihood and censored least absolute deviations estimation for unknown parameters and corresponding simulations. The practical relevance of the Skellam–Tobit INGARCH model is illustrated by real-world data examples on lottery winners, yields from a chemical process, and air quality in Beijing. The talk concludes with a presentation of some work in progress, where the Tobit approach is used to define (unbounded or bounded) INARMA-type models. By contrast to existing INARMA models for count time series, the proposed Tobit INARMA models allow for negative ACF values while showing the typical ARMA properties in close approximation.