Hostname: page-component-8448b6f56d-tj2md Total loading time: 0 Render date: 2024-04-16T17:40:33.381Z Has data issue: false hasContentIssue false
  • English
  • Français

Some Aspects of Asymptotic Theory with Applications to Time Series Models

Published online by Cambridge University Press:  11 February 2009

P. Jeganathan
P. Jeganathan
Affiliation:
University of Michigan
Get access Rights & Permissions [Opens in a new window]

Abstract

The primary purpose of this paper is to review a very few results on some basic elements of large sample theory in a restricted structural framework, as described in detail in the recent book by LeCam and Yang (1990, Asymptotics in Statistics: Some Basic Concepts. New York: Springer), and to illustrate how the asymptotic inference problems associated with a wide variety of time series regression models fit into such a structural framework. The models illustrated include many linear time series models, including cointegrated models and autoregressive models with unit roots that are of wide current interest. The general treatment also includes nonlinear models, including what have become known as ARCH models. The possibility of replacing the density of the error variables of such models by an estimate of it (adaptive estimation) based on the observations is also considered.

Under the framework in which the asymptotic problems are treated, only the approximating structure of the likelihood ratios of the observations, together with auxiliary estimates of the parameters, will be required. Such approximating structures are available under quite general assumptions, such as that the Fisher information of the common density of the error variables is finite and nonsingular, and the more specific assumptions, such as Gaussianity, are not required. In addition, the construction and the form of inference procedures will not involve any additional complications in the non-Gaussian situations because the approximating quadratic structure actually will reduce the problems to the situations similar to those involved in the Gaussian cases.

Type
Articles
Information
Econometric Theory , Volume 11 , Issue 5 , October 1995 , pp. 818 - 887
Copyright
Copyright © Cambridge University Press 1995

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

Amemiya, T. (1985) Advanced Econometrics. Cambridge, MA: Harvard University Press.Google Scholar
Anderson, T.W. (1959) On asymptotic distributions of estimates of parameters of stochastic difference equation. Annals of Mathematical Statistics 30, 676–687.CrossRefGoogle Scholar
Anderson, T.W. (1971) The Statistical Analysis of Time Series. New York: Wiley.Google Scholar
Basawa, I.V. & Koul, H.L. (1979) Asymptotic tests of composite hypothesis for non-ergodic type stochastic processes. Stochastic Process and Applications 9, 291–305.CrossRefGoogle Scholar
Basawa, I.V. & Scott, D.J. (1983) Asymptotic Optimal Inference for Non-Ergodic Models. Lecture Notes in Statistics. New York: Springer.CrossRefGoogle Scholar
Beran, R. (1974) Asymptotically efficient adaptive rank estimates in location models. Annals of Statistics 2, 63–74.CrossRefGoogle Scholar
Bickel, P.J. (1982) On adaptive estimation. Annals of Statistics 10, 647–671.CrossRefGoogle Scholar
Brown, L.D. & Fox, M. (1974) Admissibility of procedures in two-dimensional location parameter problems. Annals of Statistics 2, 248–266.Google Scholar
Chan, N.H. (1988) On the parameter estimation for nearly non-stationary time series. Journal of the American Statistical Association 83, 857–862.CrossRefGoogle Scholar
Chan, N.H. & Wei, C.Z. (1987) Asymptotic inference for nearly nonstationary AR(1) process. Annals of Statistics 15, 1050–1063.CrossRefGoogle Scholar
Chan, N.H. & Wei, C.Z. (1988) Limiting distributions of least squares estimates of unstable autoregressive process. Annals of Statistics 16, 367–401.CrossRefGoogle Scholar
Davies, R.B. (1985) Asymptotic inference when the amount of information is random. In LeCam, L. & Olshen, R. (eds.), Proceedings of the Berkeley Conference in Honor of J. Neyman and J. Kiefer, vol. 2, pp. 841–864. Monterey, CA: Wadsworth.Google Scholar
Domowitz, I. & Muus, L. (1986) Inference in the Explosive First-Order-Linear Dynamic Regression Model. Technical report.Google Scholar
Eicker, F. (1967) Limit theorems for regressions with unequal and dependent errors. Proceedings of the 5th Berkeley Symposium on Mathematical Statistics and Probability 1, 59–82.Google Scholar
Fabian, V. & Hannan, J. (1987) Local asymptotic behavior of densities. Statistics and Decisions 5, 105–138.Google Scholar
Fuller, W.A., Hasza, D.P., & Goebel, J.J. (1981) Estimation of parameters for stochastic difference equations. Annals of Statistics 9, 531–543.CrossRefGoogle Scholar
Gallant, A.R. & White, H. (1988) A Unified Theory of Estimation and Inference for Nonlinear Dynamic Models. London: Basil Blackwell.Google Scholar
Ghosh, J.K. (1985) Efficiency of estimates–Part I. Sankhya, Series A 47, 310–325.Google Scholar
Hajek, J. (1970) A characterization of limiting distributions of regular estimates. Zeitschrift fur Wahrscheinlichkeitstheorie und Verwandte Gebiete 14, 323–330.CrossRefGoogle Scholar
Hájek, J. (1972) Local asymptotic minimax and admissibility in estimation. Proceedings of the 6th Berkeley Symposium on Mathematical Statistics and Probability 1, 175–194.Google Scholar
Hájek, J. & Šidak, Z. (1967) Theory of Rank Tests. New York: Academic Press.Google Scholar
Hall, P. & Heyde, C.C. (1980) Martingale Limit Theory and Its Application. New York: Academic Press.Google Scholar
Ibragimov, I.A. & Hasminskii, (1981) Statistical Estimation: Asymptotic Theory. New York: Springer.CrossRefGoogle Scholar
Inagaki, (1970) On the limiting distribution of a sequence of estimators with uniformity property. Annals of the Institute of Statistical Mathematics 22, 1–13.CrossRefGoogle Scholar
Jeganathan, P. (1980) An extension of a result of L. LeCam concerning asymptotic normality. Sankhya, Series A 42, 146–160.Google Scholar
Jeganathan, P. (1981) On a decomposition of the limit distribution of a sequence of estimators. Sankhya, Series A 43, 23–36.Google Scholar
Jeganathan, P. (1982) On the asymptotic theory of estimation when the limit of the log-likelihood is mixed normal. Sankhya, Series A 44, 173–212.Google Scholar
Jeganathan, P. (1983) Some asymptotic properties of risk functions when the limit of the experiment is mixed normal. Sankhya, Series A 45, 66–87.Google Scholar
Jeganathan, P. (1988) On the strong approximation of the distributions of estimators in linear stochastic models, I and II: Stationary and explosive AR models. Annals of Statistics 16 (3), 1283–1314.CrossRefGoogle Scholar
Jeganathan, P. (1989) A note on inequalities for probabilities of large deviations of estimators in non-linear regression models. Journal of Multivariate Analysis 30, 227–240.CrossRefGoogle Scholar
Jeganathan, P. (1991) On the asymptotic behavior of least squares estimators in AR time series with roots near the unit circle. Econometric Theory 7, 269–306.CrossRefGoogle Scholar
Kallianpur, G. (1980) Stochastic Filtering Theory. New York: Springer.CrossRefGoogle Scholar
Kaufmann, H. (1987) Regression models for nonstationary categorical time series: Asymptotic estimation theory. Annals of Statistics 15, 79–98.CrossRefGoogle Scholar
Klaassen, C.A.J. (1987) Consistent estimation of the influence function of locally asymptotically linear estimators. Annals of Statistics 15, 1548–1562.CrossRefGoogle Scholar
Koul, H. & Pflug, G. (1990) Adaptive estimation in explosive autoregression. Annals of Statistics 18, 939–960.Google Scholar
Kreiss, J. (1987a) On adaptive estimation in autoregressive models when there are nuisance parameters. Statistics and Decisions 5, 59–76.Google Scholar
Kreiss, J. (1987b) On adaptive estimation in stationary ARMA processes. Annals of Statistics 15, 112–133.CrossRefGoogle Scholar
LeCam, L. (1953) On some asymptotic properties of maximum likelihood estimates and related Bayes estimates. University of California Publications in Statistics 1, 125–142.Google Scholar
LeCam, L. (1956) On the asymptotic theory of estimation and testing hypothesis. Proceedings of the 3rd Berkeley Symposium on Mathematical Statistics and Probability 1, 129–156.Google Scholar
LeCam, L. (1960) Locally asymptotically normal families of distributions. University of California Publications in Statistics 3, 37–98.Google Scholar
LeCam, L. (1970) On the assumptions used to prove the asymptotic normality of maximum likelihood estimates. Annals of Mathematical Statistics 41, 802–828.CrossRefGoogle Scholar
LeCam, L. (1975) Distances between experiments. In Srivastava, J.N. (ed.), A Survey of Statistical Design and Linear Models, pp. 383–396. Amsterdam: North-Holland.Google Scholar
LeCam, L. (1986) Asymptotic Methods in Statistical Decision Theory. New York: Springer.Google Scholar
LeCam, L. & Yang, G.L. (1988) On the preservation of local asymptotic normality under information loss. Annals of Statistics 16, 483–520.Google Scholar
LeCam, L. & Yang, G.L. (1990) Asymptotics in Statistics: Some Basic Concepts. New York: Springer.CrossRefGoogle Scholar
Lehmann, E. (1959) Testing Statistical Hypotheses. New York: Wiley.Google Scholar
Levit, B.Ya. (1974) On the behavior of generalized Bayes' estimators for Markov observations. Theory of Probability and Applications 19, 342–346.Google Scholar
Loéve, M. (1978) Probability Theory, vol. II. New York: Springer.Google Scholar
Negal, R. (1988) Work in Progress. University of Michigan, Ann Arbor.Google Scholar
Park, J.Y. & Phillips, P.C.B. (1988) Statistical inference in regressions with integrated processes: Part I. Econometric Theory 4, 468–497.Google Scholar
Park, J.Y. & Phillips, P.C.B. (1989) Statistical inference in regressions with integrated processes: Part II. Econometric Theory 5, 95–131.CrossRefGoogle Scholar
Parthasarathy, K.R. (1977) Probability and Measure. New York: Macmillan.Google Scholar
Pfanzagl, J. & Wefelmeyer, W. (1982) Contributions to a General Asymptotic Statistical Theory. Lecture Notes in Statistics 31. New York: Springer.CrossRefGoogle Scholar
Phillips, P.C.B. (1987a) Time series regression with a unit root. Econometrica 55, 277–301.CrossRefGoogle Scholar
Phillips, P.C.B. (1987b) Towards a unified asymptotic theory for autoregression. Biometrica 74, 535–547.CrossRefGoogle Scholar
Phillips, P.C.B. (1988) Weak convergence to matrix stochastic integral ∫B dB'. Journal of Multivariate Analysis 24, 252–264.CrossRefGoogle Scholar
Phillips, P.C.B. (1989) Partially identified econometric models. Econometric Theory 5, 181–240.CrossRefGoogle Scholar
Rao, C.R. (1973) Linear Statistical Inference and Its Applications, 2nd ed.New York: Wiley.CrossRefGoogle Scholar
Roussas, G.G. (1972) Contiguity of Probability Measures. New York: Cambridge University Press.CrossRefGoogle Scholar
Schick, A. (1986) On asymptotically efficient estimation in semiparametric models. Annals of Statistics 14, 1139–1151.Google Scholar
Schick, A. (1987) A note on the construction of asymptotically linear estimators. Journal of Statistical Planning and Inference 16, 89–105.CrossRefGoogle Scholar
Schick, A. (1993) An Adaptive Estimator of the Autocorrelation Coefficient in a Semiparametric Regression Model with Autoregressive Errors. Preprint.Google Scholar
Seiders, A. & Dzhaparidze, K. (1987) A large deviation result for parameter estimators and its application to non–linear regression analysis. Annals of Statistics 15, 1031–1049.Google Scholar
Serfling, R.J. (1980) Approximation Theorems of Mathematical Statistics. New York: Wiley.CrossRefGoogle Scholar
Steigerwald, D. (1987) Adaptive Estimation in Time Series Regression Models. Preliminary draft, Department of Economics, Berkeley.Google Scholar
Stein, C. (1956) Efficient non-parametric testing and estimation. Proceedings of the 3rd Berkeley Symposium on Mathematical Statistics and Probability 1, 187–196.Google Scholar
Stone, C. (1975) Adaptive maximum likelihood estimation of a location parameter. Annals of Statistics 3, 269–284.CrossRefGoogle Scholar
Strasser, H. (1978) Global asymptotic properties of risk functions in estimation. Zeitschrift fur Wahrscheinlichkeitstheorie und Verwandte Gebiete 45, 35–48.CrossRefGoogle Scholar
Swensen, A.R. (1985) The asymptotic distribution of the likelihood ratio for autoregressive time series with a regression trend. Journal of Multivariate Analysis 16, 54–70.CrossRefGoogle Scholar
Venkataraman, K.N. (1968) Some limit theorems on a linear stochastic difference equation with a constant term and their statistical applications. Sankhya, Series A 30, 51–74.Google Scholar
Wald, A. (1943) Tests of statistical hypotheses concerning several parameters when the number of observations is large. Transactions of American Mathematical Society 54, 426–482.Google Scholar
White, H. (1984) Asymptotic Theory for Econometricians. New York: Academic Press.Google Scholar
White, J.S. (1958) The limiting distribution of the serial correlation in the explosive case. Annals of Mathematical Statistics 29, 1188–1197.CrossRefGoogle Scholar