Abstract
The econometric data to which autoregressive moving-average models are commonly applied are liable to contain elements from a limited range of frequencies. If the data do not cover the full Nyquist frequency range of [0,π] radians, then severe biases can occur in estimating their parameters. The recourse should be to reconstitute the underlying continuous data trajectory and to resample it at an appropriate lesser rate. The trajectory can be derived by associating sinc fuction kernels to the data points. This suggests a model for the underlying processes. The paper describes frequency-limited linear stochastic differential equations that conform to such a model, and it compares them with equations of a model that is assumed to be driven by a white-noise process of unbounded frequencies. The means of estimating models of both varieties are described.
Highlights
The paper considers the means by which a linear stochastic differential equation (LSDE) can be inferred from a series of observations taken at finite intervals in time
The paper represents a sequel to a previous paper (Pollock 2012), which described the hazards of estimating a discrete-time autoregressive moving-average (ARMA) model on the basis of data that have been sampled at a rate that exceeds the minimum rate that is needed in order to capture the information contained in the underlying continuous process
This section, which is followed by a brief summary of the paper, provides examples of LSDE models in which the forcing function consists of the increments of a Wiener process, and of continuous-time continuous-time ARMA (CARMA) models that are driven by frequency-limited white noise
Summary
The paper considers the means by which a linear stochastic differential equation (LSDE) can be inferred from a series of observations taken at finite intervals in time. The method of eliminating the non-observable derivatives by a process of substitution is thereby avoided, albeit that it continues to be the requisite method in the case of a multivariate process It is evident, from the partial-fraction decomposition of an improper rational function, which includes a quotient term, that an LSDE( p, p) model, which might accommodate flow variables, will have a discrete-time EDLM( p, p) counterpart. This section, which is followed by a brief summary of the paper, provides examples of LSDE models in which the forcing function consists of the increments of a Wiener process, and of continuous-time CARMA models that are driven by frequency-limited white noise. A more extensive treatment of Fourier analysis is given by Pollock (1999)
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