Continuous Record Asymptotics in Systems of Stochastic Differential Equations

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Abstract
This paper considers estimation based on a set of T + 1 discrete observations, y(0), y(h), y(2h),…, y(Th) = y(N), where h is the sampling frequency and N is the span of the data. In contrast to the standard approach of driving N to infinity for a fixed sampling frequency, the current paper follows Phillips [35,36] and Perron [29] and examines the “dual” asymptotics implied by letting h tend to zero while the span N remains fixed.We suggest a way of explicitly embedding discrete processes into continuous-time processes, and using this approach we generalize the results of the above-mentioned authors and derive continuous record asymptotics for vector first-order processes with positive roots in a neighborhood of one and we also consider the case of a scalar second-order process. We illustrate the method by two examples. The first example is a near unit root model with drift and trend.