CONTINUOUS RECORD ASYMPTOTICS IN SYSTEMS OF STOCHASTIC DIFFERENTIAL-EQUATIONS

This paper considers estimation based on a set of T + 1 discrete obser vations, y(0),y(h),y(2h),...,y(Th) = y(N), where h is the sampling fre quency and N is the span of the data. In contrast to the standard appr oach of driving N to infinity for a fixed sampling frequency, the curr ent paper follows Phillips [35,36] and Perron [29] and examines the "d ual" asymptotics implied by letting h tend to zero while the span N re mains fixed. We suggest a way of explicitly embedding discrete process es into continuous-time processes, and using this approach we generali ze the results of the above-mentioned authors and derive continuous re cord asymptotics for vector first-order processes with positive roots in a neighborhood of one and we also consider the case of a scalar sec ond-order process. We illustrate the method by two examples. The first example is a near unit root model with drift and trend. We derive the continuous record approximation to the Dickey-Fuller rho(tau) triple- overdot test and to a recent test by Schmidt and Phillips [39] and tab ulate the distribution of the test statistics in the near continuous r ecord situation. The second example is a (near) I(2)-process. We prese nt a continuous record approximation to the least-squares estimator an d compare to recent results of Perron [32].