The Annals of Statistics

Continuous Time Regressions with Discrete Data

P. M. Robinson

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A general continuous time distributed lag model is considered. The problem is that of estimating the parameters of the kernel when, as is often the case, the available data consist not of a continuous record but of discrete observations recorded at regular intervals of time. Fourier transformation of the model and insertion of the computable, discrete Fourier transforms of the variables produce an approximate model which is of non-linear regression type and is relatively easy to handle. Estimators are proposed and their asymptotic properties established, assuming principally that the variables are stationary and ergodic and that an "aliasing" condition on the independent variable is satisfied. The results of the paper imply a theory for the estimation of rather general continuous time systems, involving the operations of differentiation, integration and translation through time.

Article information

Ann. Statist., Volume 3, Number 3 (1975), 688-697.

First available in Project Euclid: 12 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62E20: Asymptotic distribution theory
Secondary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 42A68 62M15: Spectral analysis

Stationary processes non-linear regressions Fourier methods asymptotic theory


Robinson, P. M. Continuous Time Regressions with Discrete Data. Ann. Statist. 3 (1975), no. 3, 688--697. doi:10.1214/aos/1176343131.

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