On the Rising Prices in the Post-war Capitalist Economy

This is the first of two papers that provide an expository discussion of the basic structure of the asymptotic theory of M-estimators in dynamic nonlinear models and a revlew of the literature. In this paper we discuss consistency,uniform laws of large numbers and develop a new framework for laws of large numbers for dependent and heterogeneous processes, encompassing the theory of stochastically stable as well as near epoch dependent processes. This framework results in simplified catalogues of sufficient conditions for consistency. The second paper, Potscher and Prucha (1990b), deals with asymptotic distribution theory.

[1]  Dag Tjøstheim,et al.  Estimation in nonlinear time series models , 1986 .

[2]  L. C. A. Corsten,et al.  Statistische Methoden der Modellbildung III. , 1988 .

[3]  J. Dupacová,et al.  ASYMPTOTIC BEHAVIOR OF STATISTICAL ESTIMATORS AND OF OPTIMAL SOLUTIONS OF STOCHASTIC OPTIMIZATION PROBLEMS , 1988 .

[4]  C. Withers Conditions for linear processes to be strong-mixing , 1981 .

[5]  P. J. Huber The behavior of maximum likelihood estimates under nonstandard conditions , 1967 .

[6]  P. Caines Prediction error identification methods for stationary stochastic processes , 1976 .

[7]  L. Hansen Large Sample Properties of Generalized Method of Moments Estimators , 1982 .

[8]  H. Bierens Armax model specification testing, with an application to unemployment in the Netherlands , 1987 .

[9]  D. Pollard Empirical Processes: Theory and Applications , 1990 .

[10]  S. Rolewicz On intersections of multifunctions , 1980 .

[11]  Inference in Econometric Models with Structural Change , 1987 .

[12]  D. McLeish Dependent Central Limit Theorems and Invariance Principles , 1974 .

[13]  E. Hannan,et al.  Vector linear time series models: corrections and extensions , 1978, Advances in Applied Probability.

[14]  Benedikt M. Pötscher,et al.  A UNIFORM LAW OF LARGE NUMBERS FOR DEPENDENT AND HETEROGENEOUS DATA PROCESSES , 1989 .

[15]  M. B. Priestley,et al.  Non-linear and non-stationary time series analysis , 1990 .

[16]  Ian Domowitz New Directions in Non-linear Estimation with Dependent Observations , 1985 .

[17]  George G. Roussas,et al.  Extension to Markov processes of a result by A. Wald about the consistency of the maximum likelihood estimate , 1965 .

[18]  Patrick Billingsley,et al.  Statistical inference for Markov processes , 1961 .

[19]  P. Robinson Non-linear regression for multiple time-series , 1972, Journal of Applied Probability.

[20]  J. L. Doob,et al.  Probability and statistics , 1934 .

[21]  D. Pollard Convergence of stochastic processes , 1984 .

[22]  S. Silvey A Note on Maximum‐Likelihood in the Case of Dependent Random Variables , 1961 .

[23]  H. Bunke,et al.  Asymptotic results on nonlinear approximation of regression functions and weighted least squares , 1980 .

[24]  R. Berk,et al.  CONSISTENCY A POSTERIORI , 1970 .

[25]  I. Ibragimov,et al.  Independent and stationary sequences of random variables , 1971 .

[26]  Jan R. Magnus,et al.  Consistent maximum-likelihood estimation with dependent observations: The general (non-normal) case and the normal case , 1986 .

[27]  B. M. Pöscher Convergence results for maximum likelihood type estimators in multivariable ARMAmodels , 1987 .

[28]  H. Bierens Consistent model specification tests , 1982 .

[29]  W. Newey,et al.  Uniform Convergence in Probability and Stochastic Equicontinuity , 1991 .

[30]  D. McLeish A Maximal Inequality and Dependent Strong Laws , 1975 .

[31]  K. Chanda,et al.  Strong mixing properties of linear stochastic processes , 1974, Journal of Applied Probability.

[32]  Paul A. Ruud,et al.  Probit with Dependent Observations , 1988 .

[33]  Robert Serfling,et al.  Contributions to Central Limit Theory for Dependent Variables , 1968 .

[34]  D. McLeish Invariance principles for dependent variables , 1975 .

[35]  V. V. Gorodetskii,et al.  On the Strong Mixing Property for Linear Sequences , 1978 .

[36]  Alain Monfort,et al.  A General Approach to Serial Correlation , 1985, Econometric Theory.

[37]  Kuldeep Kumar,et al.  Some Recent Developments in Non-Linear Time Series Modelling , 1988 .

[38]  E. J. Hannan,et al.  Vector linear time series models , 1976, Advances in Applied Probability.

[39]  Tuan Pham,et al.  Some mixing properties of time series models , 1985 .

[40]  Herman J. Bierens,et al.  Uniform Consistency of Kernel Estimators of a Regression Function under Generalized Conditions , 1983 .

[41]  D. Pollard New Ways to Prove Central Limit Theorems , 1985, Econometric Theory.

[42]  R. Fortet,et al.  Sur le mélange d'un processus ARMA vectoriel , 1986 .

[43]  H. Bierens Model specification testing of time series regressions , 1984 .

[44]  K. Parthasarathy,et al.  Probability measures on metric spaces , 1967 .

[45]  Paul I. Nelson,et al.  On Conditional Least Squares Estimation for Stochastic Processes , 1978 .

[46]  Sastry G. Pantula,et al.  A note on strong mixing of ARMA processes , 1986 .

[47]  J. Wooldridge Asymptotic properties of econometric estimators , 1986 .

[48]  A. Shapiro Asymptotic Properties of Statistical Estimators in Stochastic Programming , 1989 .

[49]  Le Cam,et al.  On some asymptotic properties of maximum likelihood estimates and related Bayes' estimates , 1953 .

[50]  D. McLeish On the Invariance Principle for Nonstationary Mixingales , 1977 .

[51]  E. Hannan,et al.  The statistical theory of linear systems , 1989 .

[52]  J. Dieudonne Foundations of Modern Analysis , 1969 .

[53]  B. Hoadley Asymptotic Properties of Maximum Likelihood Estimators for the Independent Not Identically Distributed Case , 1971 .

[54]  Herman J. Bierens,et al.  A uniform weak law of large numbers under π‐mixing with application to nonlinear least squares estimation , 1982 .

[55]  L. Hansen A method for calculating bounds on the asymptotic covariance matrices of generalized method of moments estimators , 1985 .

[56]  L. Ljung,et al.  Prediction error estimators: Asymptotic normality and accuracy , 1976, 1976 IEEE Conference on Decision and Control including the 15th Symposium on Adaptive Processes.

[57]  W. Ploberger Slight Misspecifications of Linear Systems , 1982 .

[58]  Bruce E. Hansen,et al.  Strong Laws for Dependent Heterogeneous Processes , 1991, Econometric Theory.

[59]  W. Newey,et al.  Generalized method of moments specification testing , 1985 .

[60]  I. Prucha,et al.  On the computation of estimators in systems with implicity defined variables , 1988 .

[61]  A. Wald Note on the Consistency of the Maximum Likelihood Estimate , 1949 .

[62]  P. Billingsley,et al.  Convergence of Probability Measures , 1969 .

[63]  W. Newey,et al.  Asymptotic Equivalence of Closest Moments and GMM Estimators , 1988, Econometric Theory.

[64]  G. Goodwin,et al.  On the Estimation of the Parameter of an Optimal Interpolator When the Class of Interpolators is Restricted , 1980 .

[65]  L. Ljung On Consistency and Identifiability , 1976 .

[66]  S. Haberman Concavity and estimation , 1989 .

[67]  Benedikt M. Pötscher,et al.  A class of partially adaptive one-step m-estimators for the non-linear regression model with dependent observations , 1986 .

[68]  L. Ljung Convergence analysis of parametric identification methods , 1978 .

[69]  H. White,et al.  NONLINEAR REGRESSION ON CROSS-SECTION DATA , 1980 .

[70]  M. Crowder Maximum Likelihood Estimation for Dependent Observations , 1976 .

[71]  Lennart Ljung,et al.  On The Consistency of Prediction Error Identification Methods , 1976 .

[72]  R. Berk,et al.  Limiting Behavior of Posterior Distributions when the Model is Incorrect , 1966 .

[73]  R. R. Bahadur Rates of Convergence of Estimates and Test Statistics , 1967 .

[74]  David K. Levine,et al.  A remark on serial correlation in maximum likelihood , 1983 .

[75]  D. Andrews CONSISTENCY IN NONLINEAR ECONOMETRIC MODELS: A GENERIC UNIFORM LAW OF LARGE NUMBERS , 1987 .

[76]  PARAMETER VALUES OF ARMA MODELS MINIMISING THE ONE-STEP-AHEAD PREDICTION ERROR WHEN THE TRUE SYSTEM IS NOT IN THE MODEL SET , 1983 .

[77]  R. Jennrich Asymptotic Properties of Non-Linear Least Squares Estimators , 1969 .

[78]  R. H. Norden A Survey of Maximum Likelihood Estimation: Part 2 , 1973 .

[79]  M. Chernick A Limit Theorem for the Maximum of Autoregressive Processes with Uniform Marginal Distributions , 1981 .

[80]  K. Parthasarathy PROBABILITY MEASURES IN A METRIC SPACE , 1967 .

[81]  Takeshi Amemiya,et al.  Regression Analysis when the Dependent Variable is Truncated Normal , 1973 .

[82]  A. Ronald Gallant,et al.  Three-stage least-squares estimation for a system of simultaneous, nonlinear, implicit equations , 1977 .

[83]  R. H. Norden A Survey of Maximum Likelihood Estimation , 1972 .

[84]  Ricardo Fraiman,et al.  On the asymptotic behaviour of general maximum likelihood estimates for the nonregular case under nonstandard conditions , 1988 .

[85]  H. White,et al.  Nonlinear Regression with Dependent Observations , 1984 .

[86]  Donald W. K. Andrews,et al.  Inference in Nonlinear Econometric Models with Structural Change , 1988 .

[87]  D. Andrews Asymptotics for Semiparametric Econometric Models: I. Estimation , 1990 .

[88]  A CLASS OF STATIONARY PROCESSES AND A CENTRAL LIMIT THEOREM. , 1957, Proceedings of the National Academy of Sciences of the United States of America.

[89]  E. Hannan The asymptotic theory of linear time-series models , 1973, Journal of Applied Probability.

[90]  R. C. Srivastava,et al.  Statistical inference for Markov processes when the model is incorrect , 1979, Advances in Applied Probability.

[91]  A. Ronald Gallant,et al.  On unification of the asymptotic theory of nonlinear econometric models , 1982 .

[92]  Note on the strong consistency of the least squares estimator in nonlinear regression , 1989 .

[93]  R. Rao Relations between Weak and Uniform Convergence of Measures with Applications , 1962 .

[94]  M. Obstfeld Speculative Attack and the External Constraint in a Maximizing Model of the Balance of Payments , 1984 .

[95]  E. Malinvaud The Consistency of Nonlinear Regressions , 1970 .

[96]  M. Priestley STATE‐DEPENDENT MODELS: A GENERAL APPROACH TO NON‐LINEAR TIME SERIES ANALYSIS , 1980 .

[97]  Asad Zaman Consistency Via Type 2 Inequalities: A Generalization of Wu's Theorem , 1989, Econometric Theory.

[98]  D. Donoho,et al.  Pathologies of some Minimum Distance Estimators , 1988 .

[99]  H. Kelejian,et al.  The Structure of Simultaneous Equation Estimators: A Generalization towards Nonnormal Disturbances , 1984 .

[100]  S. D. Silvey,et al.  The Lagrangian Multiplier Test , 1959 .

[101]  Changbao Wu,et al.  Asymptotic Theory of Nonlinear Least Squares Estimation , 1981 .

[102]  H. White,et al.  Misspecified models with dependent observations , 1982 .

[103]  A. Gallant,et al.  Nonlinear Statistical Models , 1988 .

[104]  H. White,et al.  A Unified Theory of Consistent Estimation for Parametric Models , 1985, Econometric Theory.

[105]  M. Perlman On the strong consistency of approximate maximum likelihood estimators , 1972 .

[106]  D. Freedman,et al.  On Inconsistent $M$-Estimators , 1982 .

[107]  H. White,et al.  A Unified Theory of Estimation and Inference for Nonlinear Dynamic Models , 1988 .

[108]  E. J. Hannan,et al.  Non-linear time series regression , 1971, Journal of Applied Probability.

[109]  D. Andrews Laws of Large Numbers for Dependent Non-Identically Distributed Random Variables , 1988, Econometric Theory.

[110]  P. Bickel One-Step Huber Estimates in the Linear Model , 1975 .

[111]  Noninvertibility and Pseudo-Maximum Likelihood Estimation of Misspecified ARMA Models , 1991, Econometric Theory.

[112]  I. Ibragimov,et al.  Some Limit Theorems for Stationary Processes , 1962 .

[113]  Manfred Deistler,et al.  The behaviour of the likelihood function for ARMA models , 1984 .

[114]  L. Brown,et al.  Measurable Selections of Extrema , 1973 .

[115]  Hung Man Tong,et al.  Threshold models in non-linear time series analysis. Lecture notes in statistics, No.21 , 1983 .

[116]  T. Amemiya Non-linear regression models , 1983 .

[117]  Donald W. K. Andrews NON-STRONG MIXING AUTOREGRESSIVE PROCESSES , 1984 .

[118]  Larry G. Epstein,et al.  Endogenous capital utilization in a short-run production model: Theory and an empiral application , 1980 .

[119]  Robert V. Foutz,et al.  The Performance of the Likelihood Ratio Test When the Model is Incorrect , 1977 .