# High-Frequency Observations: Identifiability and Asymptotic Efficiency

# High-Frequency Observations: Identifiability and Asymptotic Efficiency

This chapter starts with a brief reminder about a number of concepts and results which pertain to classical statistical models, without specific reference to stochastic processes. It then introduces a general notion of identifiability for a parameter, in a semi-parametric setting. A parameter can be a number (or a vector), as in classical statistics; it can also be a random variable, such as the integrated volatility. The analysis is first conducted for Lévy processes, because in this case parameters are naturally non-random, and then extended to the more general situation of semimartingales. It also considers the problem of testing a hypothesis which is “random,” such as testing whether a discretely observed path is continuous or discontinuous: the null and alternative are not the usual disjoint subsets of a parameter space, but rather two disjoint subsets of the sample space, which leads to an ad hoc definition of the level, or asymptotic level, of a test in such a context. Finally, the chapter returns to the question of efficient estimation of a parameter, which is mainly analyzed from the viewpoint of “Fisher efficiency.”

*Keywords:*
parameter identifiability, high-frequency trading, financial data, stochastic processes, Lévy processes, semimartingales, Fisher efficiency

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