On a family of test statistics for discretely observed diffusion processes

We consider parametric hypotheses testing for multidimensional ergodic diffusion processes observed at discrete time. We propose a family of test statistics, related to the so called $\phi$-divergence measures. By taking into account the quasi-likelihood approach developed for studying the stochastic differential equations, it is proved that the tests in this family are all asymptotically distribution free. In other words, our test statistics weakly converge to the chi squared distribution. Furthermore, our test statistic is compared with the quasi likelihood ratio test. In the case of contiguous alternatives, it is also possible to study in detail the power function of the tests. Although all the tests in this family are asymptotically equivalent, we show by Monte Carlo analysis that, in the small sample case, the performance of the test strictly depends on the choice of the function $\phi$. Furthermore, in this framework, the simulations show that there are not uniformly most powerful tests

Divergences Test Statistics for Discretely Observed Diffusion Processes

In this paper we propose the use of $\phi$-divergences as test statistics to verify simple hypotheses about a one-dimensional parametric diffusion process \de X_t = b(X_t, \theta)\de t + \sigma(X_t, \theta)\de W_t, from discrete observations $\{X_{t_i}, i=0, ..., n\}$ with $t_i = i\Delta_n$, $i=0, 1, >..., n$, under the asymptotic scheme $\Delta_n\to0$, $n\Delta_n\to\infty$ and $n\Delta_n^2\to 0$. The class of $\phi$-divergences is wide and includes several special members like Kullback-Leibler, R\'enyi, power and $\alpha$-divergences. We derive the asymptotic distribution of the test statistics based on $\phi$-divergences. The limiting law takes different forms depending on the regularity of $\phi$. These convergence differ from the classical results for independent and identically distributed random variables. Numerical analysis is used to show the small sample properties of the test statistics in terms of estimated level and power of the test

On penalized estimation for dynamical systems with small noise

We consider a dynamical system with small noise for which the drift is parametrized by a finite dimensional parameter. For this model we consider minimum distance estimation from continuous time observations under $l^p$-penalty imposed on the parameters in the spirit of the Lasso approach with the aim of simultaneous estimation and model selection. We study the consistency and the asymptotic distribution of these Lasso-type estimators for different values of $p$. For $p=1$ we also consider the adaptive version of the Lasso estimator and establish its oracle properties

Change point estimation for the telegraph process observed at discrete times

The telegraph process models a random motion with finite velocity and it is usually proposed as an alternative to diffusion models. The process describes the position of a particle moving on the real line, alternatively with constant velocity $+ v$ or $-v$. The changes of direction are governed by an homogeneous Poisson process with rate $\lambda >0.$ In this paper, we consider a change point estimation problem for the rate of the underlying Poisson process by means of least squares method. The consistency and the rate of convergence for the change point estimator are obtained and its asymptotic distribution is derived. Applications to real data are also presented

Clustering of discretely observed diffusion processes

In this paper a new dissimilarity measure to identify groups of assets dynamics is proposed. The underlying generating process is assumed to be a diffusion process solution of stochastic differential equations and observed at discrete time. The mesh of observations is not required to shrink to zero. As distance between two observed paths, the quadratic distance of the corresponding estimated Markov operators is considered. Analysis of both synthetic data and real financial data from NYSE/NASDAQ stocks, give evidence that this distance seems capable to catch differences in both the drift and diffusion coefficients contrary to other commonly used metrics

Empirical L2L^2-distance test statistics for ergodic diffusions

The aim of this paper is to introduce a new type of test statistic for simple null hypothesis on one-dimensional ergodic diffusion processes sampled at discrete times. We deal with a quasi-likelihood approach for stochastic differential equations (i.e. local gaussian approximation of the transition functions) and define a test statistic by means of the empirical $L^2$-distance between quasi-likelihoods. We prove that the introduced test statistic is asymptotically distribution free; namely it weakly converges to a $\chi^2$ random variable. Furthermore, we study the power under local alternatives of the parametric test. We show by the Monte Carlo analysis that, in the small sample case, the introduced test seems to perform better than other tests proposed in literature

Communicating corporate social responsibility to involve stakeholders. The case of employer branding for university students

La responsabilità sociale d'impresa (o Corporate Social Responsibility, CSR) è uno strumento efficace di marketing e la sua efficacia è mediata dalla modalità con cui le aziende comunicano le proprie attività di CSR agli stakeholder. La ricerca si propone di valutare l'effetto di cinque strategie di comunicazione della CSR, ordinate secondo un livello crescente di coinvolgimento degli stakeholder, sull'Employer Branding (EB) di un'azienda ipotetica. Attraverso la distribuzione di cinque varianti di un opuscolo, l’azienda è stata presentata a due campioni di studenti universitari (n=167; n=112) in cinque diversi scenari comunicativi della CSR. È stato somministrato un questionario con scale che misurano l’attrattività dell'azienda, il prestigio percepito dell'azienda, la disponibilità dei soggetti a entrare in contatto con l'azienda e l’impegno prospettico sul lavoro. È stata anche indagata la percezione della brand personality e della comunicazione della CSR. Le analisi confermano che tutti e cinque gli scenari sono caratterizzati da alti livelli delle dimensioni dell’EB, però lo specifico dialogo di CSR adottato può generare differenti percezioni della brand personality dell'azienda.Corporate Social Responsibility (CRS) is an effective marketing lever, and its effectiveness is mediated by the strategies companies use to communicate their CRS activities to stakeholders. The present research aims at assessing the effect of five CSR communicative strategies, ranked according to an increasing involvement level of stakeholders, on a fictional company’s Employer Branding (EB). The company was presented to two samples of university students (n=167; n=112) via the administration of five different versions of a brochure, corresponding to five different communicative scenarios of CSR. A self-report questionnaire was administrated, with scales measuring the company’s attractiveness, perceived prestige, intention to contact the company, and prospective engagement, as well as the company’s perceived brand personality and CSR communication. Analyses report high levels of the EB dimensions in all five communicative scenarios, which however produce different perceptions of the company’s brand personality