Count and Duration Time Series with Equal Conditional Stochastic and Mean Orders

We consider a positive-valued time series whose conditional distribution has a time-varying mean, which may depend on exogenous variables. The main applications concern count or duration data. Under a contraction condition on the mean function, it is shown that stationarity and ergodicity hold when the mean and stochastic orders of the conditional distribution are the same. The latter condition holds for the exponential family parametrized by the mean, but also for many other distributions. We also provide conditions for the existence of marginal moments and for the geometric decay of the beta-mixing coefficients. We give conditions for consistency and asymptotic normality of several estimators of the conditional mean parameters which do not require fully specifying the conditional distribution. We compare Quasi-Maximum Likelihood Estimators (QMLEs) (in particuler the Poisson QMLE and the Exponential QMLE) and weighted least squares estimators. Simulation experiments and illustrations on series of stock market volumes and of greenhouse gas concentrations show that the multiplicative-error form of usual duration models deserves to be relaxed, as allowed in our approach.

Joint with Abdelhakim Aknouche.