188
Views
62
CrossRef citations to date
0
Altmetric
Original Articles

Basic structure of the asymptotic theory in dynamic nonlineaerco nometric models, part i: consistency and approximation concepts

&
Pages 125-216 | Published online: 21 Mar 2007

References

  • Amemiya , T. 1973 . Regression analysis when the dependent variable is truncated normal . Econornetrica , 41 : 997 – 1016 .
  • Amemiya , T. 1983 . “ Non-linear regression models ” . In Handbook of Econometrics , Edited by: Griliches , Z. and Intriligator , M. D. Vol. I , 333 – 389 . Amsterdam : North Holland .
  • Amemiya , T. 1985 . Advanced Econometrics , Cambridge, MA : Harvard University Press .
  • Andrews , D. W. K. 1984 . Non-strong mixing autoregressive processes . Journal of Applied Probabilit , 21 : 930 – 934 .
  • Andrews , D. W. K. 1987 . Consistency in nonlinear econometric models: a generic uniform law of large numbers . Econornetrica , 55 : 1465 – 1471 .
  • Andrews , D. W. K. 1988 . Laws of large numbers for dependent non-identically distributed random variables . Econometric Theory , 4 : 458 – 467 .
  • Andrews D. W. K. Asymptotics for semiparametric econometric models: I.Estimation Cowles Foundation for Research in Economics 1989a Working Paper No 908
  • Andrews D. W. K. Asymptotics for semiparametric econometric models: 11.Stochastic equicontinuity Cowles Foundation for Research in Economics 1989b Working Paper No 909
  • Andrews D. W. K. Asymptotics for semiparametric econometric models: 11.Stochastic equicontinuity Cowles Foundation for Research in Economics 1989b Working Paper No 909
  • Andrews , D. W. K. and Fair , R. 1988 . Inference in econometric models with structural change . Review of Economic Studies , 55 : 615 – 639 .
  • Athreya , K. B. and Pantula , S. G. 1986a . Mixing properties of harris chains and AR processes . Journal of Applied Probability , 23
  • Athreya , K. B. and Pantula , S. G. 1986b . A note on strong mixing of ARMA processes . Statistics and Probability Letters , 4 : 187 – 190 .
  • Brhadur , R. 1967 . Rates of convergence of estimates and test statistics . Amals of Mathematical Statistics , 38 : 303 – 324 .
  • Basawa , I. V. and Scott , D. J. 1983 . Asymptotic Optimal Inference for Non-ergodic Hodels , New York : Springer Verlag .
  • Bates C. Nonlinear parametric models with dependent observations University of Rochester, Department of Economlcs PhD dissertation 1984
  • Bates , B. C. and White , A. 1985 . A unified theory of consistent estimation for parametric models . Econometric Theory , 1 : 151 – 178 .
  • Berk , R. H. 1966 . Limiting behavior of posterior distributions when the model is incorrect . Annals of Mathematical Statistics , 37 : 51 – 58 .
  • Berk , R. H. 1970 . Consistency a posteriori . Annals of Mathematical Statistics , 41 : 894 – 906 .
  • Bickel , B. J. 1975 . One-step Huber estimates in the linear model . Journal of the American Statistical Association , 70 : 428 – 434 .
  • Bierens , H. J. 1981 . Robust Methods and Asymptotic Theory. Lecture Notes in Economics and Mathematical Systems 192 , Berlin : Springer Verlag .
  • Bierens , H. J. 1982a . A uniform weak law of large numbers under ⊘-mixing with applications to nonlinear least squares . Statlstica Neerlandica , 36 : 81 – 86 .
  • Bierens , H. J. 1982b . Consistent model specification tests . Journal of Econometrics , 20 : 105 – 134 .
  • Bierens , H. J. 1983 . Uniform consistency of kernel estimators of a regression function under generalized conditions . Journal of the American Statistical Association , 78 : 699 – 707 .
  • Bierens , H. J. 1984 . Model specification testing of time series regressions . Journal of Econometrics , 24 : 323 – 353 .
  • Bierens , H. J. 1987 . ARMAX model specification testing, with an application to unemployment in the Netherlands . Journal of Econometrics , 35 : 161 – 190 .
  • Billingsley , P. 1961 . Statistical Inference for Markov Processes , Chicago : University of Chicago Press .
  • Billingsley , P. 1968 . Convergence of Probability Measures , New York : Wiley .
  • Blum , J. R. and Rosenblatt , M. 1957 . A class of stationary processes and a central limit theorem . Duke Mathematical Journal , 24 : 73 – 78 .
  • Boente , G. and Fraiman , R. 1988 . On the asymptotic behavior of general maximum likelihood estimates for the nonregular case under nonstandard conditions . Biometrika , 75 : 45 – 56 .
  • Brown , L. D. and Purves , R. 1973 . Annals of of Statistics , 1 : 902 – 912 .
  • Bunke , H. and Schmidt , W. H. 1980 . Asymptotic results on .nonlinear approximation of regression functions and weighted least squares . Hathematische Operationsforschung und Statistik, Serie Statistik , 11 : 3 – 22 .
  • Burguete , J. F. , Gallant , A. R. and Souza , G. 1982 . On unification of the asymptotic theory of nonlinear econometric models . Econometric Reviews , 1 : 151 – 190 .
  • Caines , P. E. 1976 . Prediction error identification methods for stationary stochastlc processes . IEEE Transactions on Automatic Control , AC-21 : 500 – 506 .
  • Caines , P. E. 1978 . Stationary linear and non-linear cystern identification and predictor set completeness . IEEE Transactions on Automatic Control , AC-23 : 583 – 595 .
  • Caines P. E. Ljung L. Asymptotic normality and accuracy of prediction error estimators University of Toronto, Department of Electrical Engineering 1976 Research Report 7602
  • Chanda , K. C. 1974 . Strong mixing properties of linear stochastic processes . Journal of Applied Probability , 11 : 401 – 408 .
  • Chernick , M. R. 1981 . A limit theorem for the maximum of autoregressive processes with uniform marginal distributions . Annals of Probability , 9 : 145 – 149 .
  • Cramér , H. 1946 . Mathematical Methods of Statistics , Princeton : Princeton University Press .
  • Chernick , M. J. 1976 . Maximum likelihood estimation for dependent observations . Journal of the Royal Statistical Society , 38 : 45 – 53 .
  • Dahlhaus , R. and Pötscher , B. M. 1989 . Convergence results for maximum likelihood type estimators in multivariable ARMA models 11 . Journal of Multivariate Analysis , 30 : 241 – 244 .
  • Deistler , M. , Dunsmuir , W. and Hannan , E. J. 1978 . Vector linear time series models: Corrections and extensions . Advances in Applied Probability , 10 : 360 – 372 .
  • Deistler , M. and Pöscher , B. M. 1984 . The behaviour of the likelihood function for ARMA models . Advances in Applied Probability , 16 : 843 – 865 .
  • Dieudonnè , J. 1960 . Foundations of Modern Analysis , New York : Academic Press .
  • Domowitz , I. 1985 . New directions in non-linear estimation with dependent observations . Canadian Journal of Economics , 19 : 1 – 19 .
  • Domowitz , I. and White , H. 1982 . Misspecified models with dependent observations . Journal of Econometrics , 20 : 35 – 58 .
  • Donoho , D. L. and Liu , R. C. 1988 . Pathologies of some minimum distance estimators . Amals of Statistics , 16 : 587 – 608 .
  • Doob , J. L. 1934 . Probability and statistics . Transactions of the American Mathematical Society , 36 : 759 – 775 .
  • Dunsmuir , W. and Hannan , E. J. 1976 . Vector linear time series models . Advances in Applied Probability , 8 : 339 – 364 .
  • Dupačovà , J. and Wets , R. 1988 . Asymptotic behavior of statistical estimators and of optimal solutions of stochastic optimization problems . Annals of Statistics , 16 : 1517 – 1549 .
  • Epstein , L. G. and Denny , M. 1980 . Endogenous capital utilization in short-runproduction model: Theory. and empirical application . Journal of Econometrics , 12 : 189 – 207 .
  • Foutz , R. V. and Srivastava , R. C. 1977 . The performance of the likelihood ratio test when the model is incorrect . Annals of Statistics , 5 : 1183 – 1194 .
  • Foutz , R. V. and Srivastava , R. C. 1979 . Statistical inference for Markov processes when the model is incorrect . Advances in Applied Probability , 11 : 737 – 749 .
  • Freedman , D. A. and Diaconis , P. 1982 . On inconsistent M-estimators . Annals of Statistics , 10 : 454 – 461 .
  • Gallant , A. R. 1977 . Three stage least squares. estimation for a system of simultaneous nonlinear implicit equations . Journal of Econometrics , 5 : 71 – 88 .
  • Gallant , A. R. 1987 . Nonlinear Statistical Models , New York : Wiley .
  • Gallant , A. R. and White , H. 1988 . A Unified Theory of Estimation and Inference for Nonlinear Dynamic Models , New York : Basil Blackwell .
  • Gänssler , P. 1983 . Empirical Process , Vol. 3 , California : Haywar . IMS Lecture Nutes-Monograph Series
  • Gorodetskii , V. V. 1977 . On the strong-mixing property for linear sequences . Theory of Probability and its Applications , 22 : 411 – 413 .
  • Gourieroux , C. , Monfort , A. and Trognon , A. 1985 . A general approach to serial correlation . Econometric Theory , 1 : 315 – 340 .
  • Haberman , S. 1989 . Concavity and Estimation . Annals of Statistics , 17 : 1631 – 1661 .
  • Hannan , E. J. 1971 . Nonlinear time series regression . Journal of Applied Probability , 8 : 767 – 780 .
  • Hannan , E. J. 1973 . The asymptotic theory of linear time series models . Journal of Applied Probability , 10 : 130 – 145 .
  • Hannan , E. J. and Deistler , M. 1988 . The Statistical Theory of Linear Systems , New York : wiley .
  • Hansen B. E. Strong laws for dependent heterogeneous processes University of Rochester 1990 Working Paper
  • Hansen , L. P. 1982 . Large sample properties of generalized method of moments estimators . Econometrica , 50 : 1029 – 1054 .
  • Hansen , L. P. 1985 . A method for calculating bounds on the asymptotic covariance matrices of generalized method of moments estimators . Journal of Econometrics , 30 : 203 – 238 .
  • Heijmans , R. H. D. and Magnus , J. R. 1986 . Consistent maximum likelihood estimation with dependent observations . Journal of Econometrics , 32 : 253 – 285 .
  • Heijmans , R. H. D. and Magnus , J. R. 1987 . Corrigenda . Journal of Econometrics , 35 : 395
  • Hoadley , B. 1971 . Asymptotic properties of maximum likelihood estimators for the independent not identically distributed case . Amals of Mathematical Statistics , 42 : 1977 – 1991 .
  • Huber , P. . The behavior of maximum likelihood estimates under nonstandard conditions . In Proceedings of the Fifth Berkeley Symposium in Mathematical Statistics and Probability . Vol. 1 , pp. 221 – 223 . Berkeley : University of California Press .
  • Huber , P. 1981 . Robust Statistics , New York : wiley .
  • Humak , K. M. S. 1983 . Statistische Methoden der Modellbildung II , Berlin : Akademie Verlag .
  • Ibragimov , I. A. 1962 . Some limit theorems for stationary processes . Theory of Probability and its Applications , 7 : 349 – 382 .
  • Ibragimov , I. A. and Linnik , Y. V. 1971 . Independent and Stationary Sequences of Random Variables , Wolters, Groningen : Noordhoff Publishing .
  • Jennrich , R. I. 1969 . Asymptotic properties of nonlinear least squares estimators . Amals of Mathematical Statistics , 40 : 633 – 643 .
  • Kabaila , P. 1983 . Parameter values of M A models minimizing the one step ahead prediction error when the true system is not in the model set . Journal of Applied Probability , 20 : 405 – 408 .
  • Kabaila , P. and Goodwin , G. C. 1980 . On the estimation of the parameters of an optimal interpolator when the class of interpolators is restricted . SIM Journal of Control and Optimisation , 18 : 121 – 144 .
  • Klimko , L. A. and Nelson , P. I. 1978 . On conditional least squares estimation for stochastic processes . Amais of Statistics , 6 : 629 – 642 .
  • Kumar , K. 1988 . Some recent developments in non-linear time series modelling , Monash University . Department of Econometrics
  • Läuter , H. 1987 . Note on the strong consistency of the least squares estimator in nonlinear regression , Edited by: Bunke , 0 and Jolivet , E. 153 – 175 . Berlin : Humboldt Universität . Seminarbericht No 89
  • Lecam , L. 1953 . On some asymptotic properties of maximum likelihood estimates and related Bayes' estimates , 277 – 328 . Berlin : University of California Press . University of California Publications in Statistics
  • Levine , D. A. 1983 . remark on serial correlation in maximum likelihood . Journal of Econometrics , 23 : 337 – 342 .
  • Ljung , L. 1976a . “ On consistency and identifiability ” . In Mathematical Programming Study , Vol. 5 , 169 – 190 . Amsterdam : North Holland .
  • Ljung , L. 1976b . “ On the consistency of prediction error identification methods ” . In System Identification, Advances and Case Studies , Edited by: Mehra , R. K. and Lainiotis , D. G. 121 – 164 . New York : Academic Press .
  • Ljung , L. 1978 . Convergence analysis of parametric identification methods . IEEE Transactions on Automatic Control , AC-23 : 770 – 783 .
  • Ljung , L. and Calnes , P. E. 1979 . Asymptotic normality of prediction error estimators for approximate models . Stochastics , 3 : 29 – 46 .
  • Malinvaud , E. 1970 . The consistency of nonlinear regressions . Amals of Mathematical Statistics , 41 : 956 – 969 .
  • McLeish , D. L. 1974 . Dependent central limit theorems and invariance principles . Annals of Probability , 2 : 620 – 628 .
  • McLeish , D. L. 1975a . A maximal inequality and dependent strong laws . Annals of Probability , 3 : 829 – 839 .
  • McLeish , D. L. 1975b . Invariance principles for dependent variables . Zeitschrift für Vahrscheinlichkeitstheorie und Verwandte Gebiete , 32 : 165 – 178 .
  • McLeish , D. L. 1977 . On the invariance principle for nonstationary mixingales . Annals of Probability , 5 : 616 – 621 .
  • Mokkadem , A. 1986 . Ssr le mélange d'un processus ARMA vectoriel . Comptes Rendus de 1'Académie Sciences , 303 : 519 – 521 .
  • Mokkadem , A. 1987 . Conditions suffisantes de mélange géométrique pour des autorégressifs polynomiaux; application aux processus ARMA et aux processus bilinéaires . Comptes Rendus de1'Académie Sciences , 305 : 477 – 480 .
  • Newey , W. K. 1985 . Generalized method of moments specification testing . Journal of Econometrics , 29 : 229 – 256 .
  • Newey , W. K. March 1987 . Expository notes on uniform laws of large numbers , March , Princeton University . Department of Economics
  • Newey , W. K. 1988 . Asymptotic equivalence of closest moments and GMM estimators . Econometric Theory , 4 March : 336 – 340 .
  • Newey , W. K. 1989 . Uniform convergence in probability and stochastic equicontinuity , Princeton University . Department of Economics
  • Norden , R. H. 1972 . A survey of maximum likelihood estimation,Part I . International Statistical Review , 40 : 329 – 354 .
  • Norden , R. H. 1973 . A survey of maximum likelihood estimation,Part II . International Statistical Review , 41 : 39 – 58 .
  • Parthasarathy , K. R. 1967 . Probability Measures on Metric Spaces , New York : Academic Press .
  • . On the strong consistency of approximate maximum likelihood estimators . Proceedings of the Sixth Berkeley Symposium in Mathematical Statistics and Probability . Vol. 1 , pp. 263 – 281 . Berkeley : University of California Press .
  • Pham , D. T. and Tran , L. T. 1985 . Some mixing properties of time series . Stochastic Processes and their Applications , 19 : 297 – 304 .
  • Poirier , D. J. and Ruud , P. A. 1988 . Probit kith dependent observations . Review of Economic Studies , 55 : 593 – 614 .
  • Ploberger , W. 1982a . “ Slight misspecification of linear systems ” . In Operations Research in Progress , Edited by: Feichtinger , G. and Kall , P. 413 – 424 . Dordrecht : D.Reidel Publishing Company .
  • Ploberger W. On the prediction error estimation of linear rational models I University of Technology Vienna, Department of Econometrics and OperationsResearch 1982b Research Report No.13
  • Pollard , P. 1984 . Convergence of stochastic Processes , New York : Springer Verlag .
  • Pollard , P. 1985 . New ways to Drove central limit theorems . Econometric Theory , 1 : 295 – 314 .
  • Pollard , P. 1988 . Bracketing methods in statistics and econometrics , Yale University . Department of Statistics
  • Pollard , P. . Empirical Processes: Theory and Applications . Proceedings of the Sixth Berkeley Symposium in Mathematical Statistics and Probability . Vol. 2 ,
  • Pötscher , B. M. 1987 . Convergence results for maximum likelihood type estimators in multivariable ARMA models . Journal of Multivariate Analysis , 21 : 29 – 52 .
  • Pötscher B. M. Nonivertibility and pseudo maximum likelihood estimation of misspecified ARMA models University of Maryland, Department of Economics 1990 Working Paper, Forthcoming in Econometric Theory
  • Pötscher , B. M. and Prucha , I. R. 1986a . A class of partially adaptive one-step M-estimators for the nonlinear regression mode1 with dependent . Journal of Econometrics , 32 : 219 – 251 .
  • Pötscher B. M. Prucha I. R. Consistency in nonlinear econometrics: A generic uniform law of large numbers and some comments on recent results University of Maryland, Department of Economics 1986b Working Paper 86-9
  • Pötscher , B. M. and Prucha , I. R. 1988 . Approximation concepts and weak laws of large numbers , University of Maryland . Department of Economics, Manuscript in preparation
  • Pötscher , B. M. and Prucha , I. R. 1989a . A uniform law of large numbers for dependent and heterogeneous data processes . Econornetrica , 57 : 675 – 683 .
  • Pötscher B. M. Prucha I. R. On the formulation of uniform laws of large numbers: A truncation approach University of Maryland, Department of Economics 1989b Working Paper 89-27
  • Pötscher B. M. Prucha I. R. Generic uniform convergence and equicontinuity concepts for random functions: An exploration of the basic structure University of Maryland, Department of Economics 1990a Working Paper 90-14
  • Pötscher B. M. Prucha I. R. Basic structure of the asymptotic theory indynamic nonlinear econometric models, part 11: Asymptotic normality University of Maryland, Department of Economics 1990b Working Paper 90-15
  • Priestley , M. B. 1980 . State-dependent models: A general approach to time seriesanalysis . Journal of Time Series Analysis , 1 : 47 – 71 .
  • Priestley , M. B. 1988 . Non-linear and Non-stationary Time Series Analysis , London : Academic Press .
  • Prucha , I. R. and Kelejian , H. H. 1984 . The structure of simultaneous equation estimators: A generalization towards nonnormal disturbances . Econometrica , 52 : 721 – 736 .
  • Prucha , I. R. and Nadiri , M. I. 1988 . On the computation of estimators in systems with implicitly defined variables . Economic Letters , 26 : 141 – 145 .
  • Prucha I. R. Nadiri M. I. Endogenous capital utilization and productivitymeasurement in dynamic factor demand models: Theory and an application tothe U.S. electrical machinery industry University of Maryland, Department of Economics 1990 Working Paper 90-19
  • Ranga Rao , R. 1962 . Relations between weak and uniform convergence of measures with applications . Annals of Mathematical Statistics , 33 : 659 – 680 .
  • Robinson , P. M. 1972 . Non-linear regression for multiple time series . Journal of Applied Probability , 9 : 758 – 768 .
  • Roussas , G. F. 1965 . Extension to Markov processes of results by A. Wald about the consistency of the maximum likelihood estimate . Zeitschrift fur Wahrscheinlichkeitstheorie und Verwandte Gebiete , 4 : 69 – 73 .
  • Royden , H. L. 1968 . Real Analysis , New York : Macmillan .
  • Schmidt , W. H. . Asymptotic results for inadequate nonlinear regressions . Seminarbericht . Edited by: Bunke , O. and Jolivet , E. Vol. 89 , pp. 103 – 129 . Berlin : Humboldt Universität .
  • Serfling , R. J. 1968 . Contributions to central limit theory for dependent variables . Annals of Mathematical Statistics , 39 : 1158 – 1175 .
  • Shapiro , A. 1989 . Asymptotic properties of statistical estimators in stochastic programming . Annals of Statistics , 17 : 841 – 858 .
  • Silvey , S. D. 1959 . The Lagrangian multiplier test . Annals of Mathematical Statistics , 30 : 389 – 407 .
  • Silvey , S. D. 1961 . A note on maximum likelihood in the case of dependent random variables . Journal of the Royal Statistical Society , 23 : 444 – 452 .
  • Tj⊝stheim , D. 1986 . Estimation in nonlinear tine series models . Stochastic Processes and their Applications , 21 : 251 – 273 .
  • Tong , H. 1983 . “ Threshold models in nonlinear time series analysis ” . In Lecture Notes in Statistics , Vol. 21 , New York : Springer Veriag .
  • Wald , A. 1949 . Note on the consistency of the maximum likelihood estimate . Annals of Mathematical Statistics , 20 : 595 – 600 .
  • White , H. 1980 . Nonlinear regression on cross-section data . Econometrica , 48 : 721 – 746 .
  • White , H. and Domowitz , I. 1984 . Nonlinear regression with dependent observations . Econometrica , 52 : 143 – 161 .
  • Willard , S. 1970 . General Topology , Reading, MA : Addison-Wesley .
  • Withers , C. S. 1981 . Conditions for linear processes to be strong-mixing . Zeitschrift für Vahrscheinlichkeitstheorie und verwandte Gebiete , 57 : 477 – 480 .
  • Wooldrldge J. Asymptotic properties of econometric estimators University of California San Diego, Department of Economics PhD dissertation 1986
  • Wu , C. F. 1981 . Asymptotic theory of nonlinear least squares estimation . Annals of Statistics , 9 : 501 – 513 .
  • Zaman A. Necessary and sufficient conditions for consistency of M-estimates Columbia University, Department of Economics 1985
  • Zaman , A. 1989 . Consistency via type 2 inequalities: A generalization of Wu's theorem . Econometric Theory , 5 : 272 – 286 .

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.