Hostname: page-component-76fb5796d-22dnz Total loading time: 0 Render date: 2024-04-25T11:58:41.438Z Has data issue: false hasContentIssue false
  • English
  • Français

Continuous Record Asymptotics in Systems of Stochastic Differential Equations

Published online by Cambridge University Press:  18 October 2010

Bent E. Sørensen
Bent E. Sørensen
Affiliation:
Brown University
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper considers estimation based on a set of T + 1 discrete observations, y(0), y(h), y(2h),…, y(Th) = y(N), where h is the sampling frequency and N is the span of the data. In contrast to the standard approach of driving N to infinity for a fixed sampling frequency, the current paper follows Phillips [35,36] and Perron [29] and examines the “dual” asymptotics implied by letting h tend to zero while the span N remains fixed.

We suggest a way of explicitly embedding discrete processes into continuous-time processes, and using this approach we generalize the results of the above-mentioned authors and derive continuous record asymptotics for vector first-order processes with positive roots in a neighborhood of one and we also consider the case of a scalar second-order process. We illustrate the method by two examples. The first example is a near unit root model with drift and trend.

Type
Articles
Information
Econometric Theory , Volume 8 , Issue 1 , March 1992 , pp. 28 - 51
Copyright
Copyright © Cambridge University Press 1992

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

1.Basawa, I.V. & Rao, B.L.S. Prakasa. Statistical Inferencefor Stochastic Processes. London: Academic Press, 1980.Google Scholar
2.Bergstrom, A.R. (Ed.) Statistical Inference in Continuous Time Economic Models. Amsterdam: North-Holland Publishing Company, 1976.Google Scholar
3.Bergstrom, A.R.Gaussian estimation of structural parameters in higher order continuous time dynamic models. Econometrica 51 (1983): 117–152.CrossRefGoogle Scholar
4.Bergstrom, A.R. Continuous time stochastic models and issues of aggregation over time, pp. 1145–1212 in Griliches, Z. and Intriligator, M.D. (Eds.) Handbook of Econometrics. Amsterdam: North-Holland, 1984.CrossRefGoogle Scholar
5.Bhargava, A.On the theory of testing for unit roots in observed time series. Review ofEco-nomic Studies 53 (1986): 369–384.CrossRefGoogle Scholar
6.Billingsley, P. (1986): Convergence of Probability Measures. New York: Wiley, 1968.Google Scholar
7.Cavanagh, C. Roots Local to Unity. Harvard Institute of Economic Research, Discussion Paper No. 1259, 1986.Google Scholar
8.Chan, K.S. & Tong, H.. A note on embedding a discrete parameter ARMA model in a continuous parameter ARMA model. Journal of Time Series Analysis 8 (1987): 277–281.CrossRefGoogle Scholar
9.Chan, N.H.The parameter inference for nearly nonstationary time series. Journal of the American Statistical Association 83 (1988): 857–862.CrossRefGoogle Scholar
10.Chan, N.H. & Wei, C.Z.. Asymptotic inference for nearly nonstationary AR(1) processes. Annals of Statistics 15 (1987): 1050–1063.CrossRefGoogle Scholar
11.Chung, K.L. & Williams, R.J.. Introduction to Stochastic Integration. Boston: Birkhauser, 1983.CrossRefGoogle Scholar
12.Dickey, D.A. & Fuller, W.A.. Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association 74 (1979): 427–431.Google Scholar
13.Dickey, D.A. & Fuller, W.A.. Likelihood ratio statistics for autoregressive time series with a unit root. Econometrica 49 (1981): 1057–1072.CrossRefGoogle Scholar
14.Ethier, S.N. & Kurtz, T.G.. Markov Processes. Characterization and Convergence. New York: Wiley, 1986.CrossRefGoogle Scholar
15.Evans, G.B.A. & Savin, N.E.. Testing for unit roots: 1. Econometrica 49 (1981): 753–779.CrossRefGoogle Scholar
16.Evans, G.B.A. & Savin, N.E.. Testing for unit roots: 2. Econometrica 52 (1984): 1241–1269.CrossRefGoogle Scholar
17.Fuller, W.A.Introduction to Statistical Time Series. New York: Wiley, 1976.Google Scholar
18.Gantmacher, F.R.The Theory of Matrices. New York: Chelsea, 1959.Google Scholar
19.Gihman, I.I. & Skorohod, A.V.. The Theory of Stochastic Processes III. New York: Springer Verlag, 1979.CrossRefGoogle Scholar
20.Harrison, J.M., Pitbladdo, R. & Schaefer, S.M.. Continuous price processes in frictionless markets have infinite variation. Journal of Business 57 (1984): 353–365.CrossRefGoogle Scholar
21.He, S.W. & Wang, J.G.. On embedding a discrete-paramete r ARMA model in a continuous-parameter ARMA model. Journal of Time Series Analysis 10 (1989): 315–325.CrossRefGoogle Scholar
22.Jacod, J. & Shiryaev, A.N.. Limit Theorems for Stochastic Processes. New York: Springer-Verlag, 1987.CrossRefGoogle Scholar
23.Kolmogorov, A.N. & Fomin, S.V.. Introductory Real Analysis. New York: Dover Publications, 1975.Google Scholar
24.Liptser, R.S. & Shiryayev, A.N.. Statistics of Random Processes II. Applications. New York: Springer-Verlag, 1978.Google Scholar
25.Lo, A.W.Maximum likelihood estimation of generalized Ito processes with discretely sampled data. Econometric Theory 4 (1988): 231–248.CrossRefGoogle Scholar
26.Nabeya, S. & Tanaka, K.. A general approach to the limiting distribution for estimators in time series regression with nonstable autoregressive errors. Econometrica 58 (1990): 145–163.CrossRefGoogle Scholar
27.ØØsendal, B.Stochastic Differential Equations. Berlin: Springer-Verlag, 1985.CrossRefGoogle Scholar
28.Perron, P.The calculation of the limiting distribution of the least-squares estimator in a near-integrated model. Econometric Theory 5 (1989): 241–256.CrossRefGoogle Scholar
29.Perron, P. A continuous time approximation to the unstable first-order autoregressive process: The case without an intercept. Forthcoming: Econometrica.Google Scholar
30.Perron, P. A continuous time approximatio n to the stationary first-order autoregressive model. Forthcoming: Econometric Theory.Google Scholar
31.Perron, P. Test consistency with varying sampling frequency. Forthcoming: Econometric Theory.Google Scholar
32.Perron, P.The adequacy of limiting distributions in the AR(1) model with dependent errors. Mimeo: Princeton, 02, 1990.Google Scholar
33.Phadke, M.S. & Wu, S.M.. Modeling of continuous stochastic processes from discrete observations with application to sunspots data. Journal of the American Statistical Association 69 (1974): 325–329.CrossRefGoogle Scholar
34.Phillips, P.C.B.The problem of identification in finite parameter continuous time models. Journal of Econometrics 1 (1973): 351–362.CrossRefGoogle Scholar
35.Phillips, P.C.B.Time series regression with a unit root. Econometrica 55 (1987): 277–301.CrossRefGoogle Scholar
36.Phillips, P.C.B. Towards a unified asymptotic theory for autoregression. Biometrika 74 (1987): 535–547.CrossRefGoogle Scholar
37.Phillips, P.C.B.Regression theory for near integrated time series. Econometrica 56 (1988): 1021–1045.CrossRefGoogle Scholar
38.Protter, P.Stochastic Integration and Differential Equations. A New Approach. New York: Springer-Verlag, 1990.CrossRefGoogle Scholar
39.Schmidt, P. & Phillips, P.C.B.. Testing for a unit root in the presence of deterministic trends. Cowles Foundation Discussion Paper No. 933, 1989.Google Scholar
40.Schwert, G.W.Tests for unit roots: A Monte Carlo investigation. Journal of Business & Economic Statistics 1 (1989): 147–161.CrossRefGoogle Scholar
41.Sims, C.A., Stock, J.H. & Watson, M.W.. Inference in linear time series models with some unit roots. Econometrica 58 (1990): 113–145.CrossRefGoogle Scholar
42.White, J.S.The limiting distribution of the serial correlation coefficient in the explosive case. Annals of Mathematical Statistics 29 (1958): 1188–1197.CrossRefGoogle Scholar