Multiple Time Series Analysis of Irregularly Spaced Data

Abstract

Let Y(t) be a d-dimensional real column vector stochastic process, where t is scalar. We may define Y(t) in continuous time (t can take all real values) or discrete time (t = 0, ±δ, ±2δ,...). Either way Y(t) is measured within a finite interval T = [1,T] of t-values, but the full vector is not observed for all real t ε T (continuous case) or all discrete t ε T (discrete case). An important special case is discrete equally-spaced observation of a continuous process (see for example Bergstrom [1], Phillips [14], Robinson [16], Sargan [21]), which we discuss only insofar as the identifiability problem typically raised may be resolved by irregular sampling (see Section 3). A related case, which we do not discuss, is skip-sampling, where the discrete process defined above is recorded at intervals of kδ units, for integer k > 1 (for example, Robinson [16]).