Observe that all these GARCH dcfinitions require E~~~3; t~?~~a; c 1. The strong GARCH definition has been adopted by, e.g., Engle (1982) and Bollerslev (1986). "f'he most popular distributions are normal and t distributions. The second definition has been adopted by, e.g., Weiss (1986). Evidently a strong GARCH process will also be semi-strong GARCH. On the othcr hand a semi-strong GARCFí proccss with time-varying higher order conditional moments of the rescaled innovations~,-e,, h, is not strong GARCH (sce, e.g., Example 3). Finally the requirements for weak GARCH are met both by strong and semi-strong GARCH processes. Observe that the weak GARCH definition is quite general and captures the characterizing features of the other GARCH formulations. As we will prove below, it is still possible to obtain strongly consistent estimators of the GARCH parameters in this general formulation.

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Integer-valued auto-regressive (INAR) processes have been introduced to model non-negative integer-valued phenomena that evolve over time. The distribution of an INAR(p) process is essentially described by two parameters: a vector of auto-regression coefficients and a probability distribution on the non-negative integers, called an immigration or innovation distribution. Traditionally, parametric models are considered where the innovation distribution is assumed to belong to a parametric family. The paper instead considers a more realistic semiparametric INAR(p) model where there are essentially no restrictions on the innovation distribution. We provide an (semiparametrically) efficient estimator of both the auto-regression parameters and the innovation distribution.

We propose a simple jump-diffusion model for an exchange rate target zone. The model captures most stylized facts from the existing target zone models while remaining analytically tractable. The model is based on a modi®ed two-limit version of the COX, INGERSOLL and ROSS (1985) model. In the model the exchange rate is kept within the band because the variance decreases as the exchange rate approaches the upper or lower limits of the band. We also consider an extension of the model with parity adjustments, which are modeled as Poisson jumps. Estimation of the model is by GMM based on conditional moments. We derive prices of currency options in our model, assuming that realignment jump risk is idiosyncratic. Throughout, we apply the theory to EMS exchange rate data. We show that, after the EMS crisis of 1993, currencies remain in an implicit target zone which is narrower than the of®cially announced target zones.