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Stochastic difference equations with non-integral differences

Published online by Cambridge University Press:  01 July 2016

P. M. Robinson
P. M. Robinson*
Affiliation:
Harvard University
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Abstract

As an alternative to conventional discrete time models for stochastic processes that fluctuate within the sampling interval, we propose difference equations containing non-integral lags. We discuss the problems of stability, identification and estimation, for which an approximate model is needed. Least squares applied to an approximate Fourier-transformed model yields estimators of the coefficients that are consistent with respect to the true model under some conditions. The conditions are weak when the model contains predetermined variables that obey an “aliasing condition”; estimators of the lags as well as coefficients can then be found that are consistent, efficient and satisfy a central limit theorem. Optimal estimators for stochastic difference-differential equations are also available.

Keywords

NON-INTEGRAL DIFFERENCE EQUATIONSDIFFERENCE-DIFFERENTIAL EQUATIONSFOURIER METHODSSTATIONARY PROCESSESLIMIT THEOREMS
Type
Research Article
Information
Advances in Applied Probability , Volume 6 , Issue 3 , September 1974 , pp. 524 - 545
Copyright
Copyright © Applied Probability Trust 1974 

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References

Anderson, T. W. (1971) The Statistical Analysis of Time Series. Wiley, New York.Google Scholar
Ansoff, H. I. and Krumhansl, J. A. (1948) A general stability criterion for linear oscillating systems with constant time lag. Quart. Appl. Math. 6, 337–341.CrossRefGoogle Scholar
Bellman, R. and Cooke, K. L. (1963) Differential-Difference Equations. Academic Press, New York.Google Scholar
Bernoulli, J. (1728) Meditationes. Dechordibus vibrantibus…. Commentarii Academiae Scientiarum Imperialis Petropolitanae 3, 13–28.Google Scholar
Bochner, S. (1931) Allgemeine lineare differenzengleichungen mit asymptotisch konstanten koeffizienten. Math. Z. 33, 426–450.CrossRefGoogle Scholar
Box, G. E. P. and Jenkins, G. M. (1970) Time Series Analysis. Forecasting and Control. Holden-Day, San Francisco.Google Scholar
Brewer, K. R. W. (1973) Some consequences of temporal aggregation and systematic sampling for arma and armax models. Mimeographed.CrossRefGoogle Scholar
Burger, E. (1956) On the stability of certain economic systems. Econometrica 24, 488–493.CrossRefGoogle Scholar
Carmichael, R. D. (1933) Systems of linear difference equations and expansions in series of exponential functions. Trans. Amer. Math. Soc. 35, 1–28.CrossRefGoogle Scholar
Cebotarev, N. G. and Meiman, N. N. (1949) The Routh-Hurwitz problem for polynomials and entire functions. Trudy Mat. Inst. Steklov 26.Google Scholar
Dhrymes, P. J. (1971) Distributed Lags, Problems of Estimation and Formulation. Holden-Day, San Francisco.Google Scholar
Durbin, J. (1960) Estimation of parameters in time-series regression models. J. R. Statist. Soc. B 22, 139–153.Google Scholar
Engle, R. F. (1970) The inconsistency of distributed lag estimators due to misspecification by time aggregation. Mimeographed.Google Scholar
Frisch, R. and Holme, H. (1935) The characteristic solutions of a mixed difference and differential equation occurring in economic dynamics. Econometrica 3, 225–239.CrossRefGoogle Scholar
Granger, C. W. J. (1966) The typical spectral shape of an economic variable. Econometrica 34, 150–161.CrossRefGoogle Scholar
Griliches, Z. (1967) Distributed lags: a survey. Econometrica 35, 16–49.CrossRefGoogle Scholar
Hannan, E. J. (1970) Multiple Time Series. Wiley, New York.CrossRefGoogle Scholar
Hannan, E. J. (1971) The identification problem for multiple equation systems with moving average errors. Econometrica 39, 751–765.CrossRefGoogle Scholar
Hannan, E. J. (1973) The asymptotic theory of linear time series models. J. Appl. Prob. 10, 130–145.CrossRefGoogle Scholar
Hannan, E. J. and Nicholls, D. F. (1972) The estimation of mixed regression, autoregression moving average, and distributed lag models. Econometrica 40, 529–547.CrossRefGoogle Scholar
Hannan, E. J. and Robinson, P. M. (1973) Lagged regression with unknown lags. J. R. Statist. Soc. B 35, 252–267.Google Scholar
Hayes, N. D. (1950) Roots of the transcendental equation associated with a certain difference-differential equation. J. London Math. Soc. 35, 226–232.CrossRefGoogle Scholar
James, R. W. and Belz, M. H. (1936) On a mixed difference and differential equation. Econometrica 4, 157–160.CrossRefGoogle Scholar
Jennrich, R. I. (1969) Asymptotic properties of non-linear least squares estimators. Ann. Math. Statist. 40, 633–643.CrossRefGoogle Scholar
Jorgenson, D. W. (1966) Rational distributed lag functions. Econometrica 32, 135–149.CrossRefGoogle Scholar
Kalecki, M. (1935) A macrodynamic theory of business cycles. Econometrica 3, 327–344.CrossRefGoogle Scholar
Kalecki, M. (1943) Studies in Economic Dynamics. Allen and Unwin, London.Google Scholar
Langer, R. E. (1931) On the zeros of exponential sums and integrals. Bull. Amer. Math. Soc. 37, 213–239.CrossRefGoogle Scholar
Mann, H. B. and Wald, A. (1943) On the statistical treatment of linear stochastic difference equations. Econometrica 11, 173–220.CrossRefGoogle Scholar
Marden, M. (1949) The geometry of zeros. Mathematical Surveys, Number III. Amer. Math. Soc., New York.Google Scholar
Martin, W. T. (1938) Linear difference equations with arbitrary real spans. Acta Math. 69, 57–98.CrossRefGoogle Scholar
Mundlak, Y. (1961) Aggregation over time in distributed lag models. Internat. Econ. Rev. 2, 154–163.CrossRefGoogle Scholar
Pincherle, S. (1926) Sur la résolution de l'équations fonctionelle Σh v (x+a v) =f(x) à coefficients constants. Acta Math. 48, 279–304. (First published in 1888.) CrossRefGoogle Scholar
Pinney, E. (1958) Ordinary Difference-Differential Equations. University of California Press, Berkeley.Google Scholar
Pontryagin, L. S. (1955) On the zeros of some elementary transcendental functions. Amer. Math. Soc. Transl. Ser. 2 1, 95–110. (First published in 1942.) Google Scholar
Raclis, R. (1930) Solution principale de l'équation linéaire aux différences finies. Acta Math. 55, 277–394.CrossRefGoogle Scholar
Ritt, J. F. (1927) A factorization theory for functions Σ a iexp (αi,x). Trans. Amer. Math. Soc. 29, 584–596.Google Scholar
Robinson, P. M. (1972) The estimation of continuous-time systems using discrete data. , Australian National University.Google Scholar
Robinson, P. M. (1973) The estimation of linear differential equations with constant coefficients. To appear.Google Scholar
Sargan, J. D. (1958) The instability of the Leontief dynamic model. Econometrica 26, 381–392.CrossRefGoogle Scholar
Sims, C. A. (1971) Discrete approximations to continuous time distributed lag models in econometrics. Econometrica 39, 545–563.CrossRefGoogle Scholar
Strodt, W. (1948) Linear difference equations and exponential polynomials. Trans. Amer. Math. Soc. 64, 439–466.CrossRefGoogle Scholar
Telser, L. G. (1967) Discrete samples and moving sums in stationary stochastic processes. J. Amer. Statist. Assoc. 62, 484–499.CrossRefGoogle Scholar
Walker, A. M. (1962) Large sample estimation of parameters for autoregressive processes with moving average residuals. Biometrika 49, 117–132.CrossRefGoogle Scholar
Yaglom, A. M. (1962) An Introduction to the Theory of Stationary Random Functions. Prentice-Hall, Englewood Cliffs, N. J. Google Scholar