While econometric models with deterministic regressors have been around for more than half a century, the methods designed specifically to study such models have only appeared in the last decade. Short–Memory Linear Processes and Econometric Applications serves as a comprehensive source of asymptotic results for econometric models with deterministic regressors. The author provides a balanced presentation of both established and newly developed results in the field, highlighting regressors including linear trends, seasonally oscillating functions, and slowly varying functions as well as some specifications of spatial matrices in the theory of spatial models.
The book begins with central limit theorems (CLTs) for weighted sums of short–memory linear processes, which have proved to be most useful in modeling dependence over time. This discussion includes the analysis of certain operators in Lp spaces and their employment in the derivation of CLTs. Among the models considered are static linear models with slowly varying regressors, spatial models, time series autoregressions, and two nonlinear models, while the treatment of estimation procedures includes ordinary and nonlinear least squares, maximum likelihood, and method of moments. The book also contains an introduction to operators, probabilities, and linear models, Lp–approximable sequences of vectors, convergence in distribution of linear and quadratic forms, and strong convergence of least squares estimators.
Throughout the book, advanced high–quality results are included alongside new and updated research, approaches, and tools. Special attention has been paid to providing rigorous, detailed proofs with extensive cross–referencing, and all long proofs have been divided into easy–to–follow, logical parts. Methodological issues of the asymptotic theory in econometrics are highlighted and thoroughly illustrated.
Short–Memory Linear Processes and Econometric Applications is suitable for probability theory, time series, and econometric courses at the graduate and PhD level. The book also serves as an authoritative resource for econometricians, specialists working with probability, applied time series statisticians, and academics as well as for new researchers in these fields.
1 INTRODUCTION TO OPERATORS, PROBABILITIES AND THE LINEAR MODEL.
1.1 Linear Spaces.
1.2 Normed Spaces.
1.3 Linear Operators.
1.4 Hilbert Spaces.
1.5 Lp Spaces.
1.6 Conditioning on s–fields.
1.7 Matrix Algebra.
1.8 Convergence of Random Variables.
1.9 The Linear Model.
1.10 Normalization of Regressors.
1.11 General Framework in the case of K Regressors.
1.12 Introduction to L2–Approximability.
2 Lp–APPROXIMABLE SEQUENCES OF VECTORS.
2.1 Discretization, Interpolation and Haar Projector in Lp.
2.2 Convergence of Bilinear Forms.
2.3 The Trinity and Its Boundedness in lp.
2.4 Convergence of the Trinity on Lp–Generated Sequences.
2.5 Properties of Lp–Approximable Sequences.
2.6 Criterion of Lp–Approximability.
2.7 Examples and Counterexamples.
3 CONVERGENCE OF LINEAR AND QUADRATIC FORMS.
3.1 General Information.
3.2 Weak Laws of Large Numbers.
3.3 Central Limit Theorems for Martingale Differences.
3.4 Central Limit Theorems for Weighted Sums of Martingale Differences.
3.5 Central Limit Theorems for Weighted Sums of Linear Processes.
3.6 Lp–Approximable Sequences of Matrices.
3.7 Integral operators.
3.9 Convergence of Quadratic Forms of Random Variables.
4 REGRESSIONS WITH SLOWLY VARYING REGRESSORS.
4.1 Slowly Varying Functions.
4.2 Phillips Gallery 1.
4.3 Slowly Varying Functions with Remainder.
4.4 Results Based on Lp–Approximability.
4.5 Phillips Gallery 2.
4.6 Regression with Two Slowly Varying Regressors.
5 SPATIAL MODELS.
5.1 A Math Introduction to Purely Spatial Models.
5.2 Continuity of Nonlinear Matrix Functions.
5.3 Assumption on the Error Term and Implications.
5.4 Assumption on the Spatial Matrices and Implications.
5.5 Assumption on the Kernel and Implications.
5.6 Linear and Quadratic Forms Involving Segments of K.
5.7 The Roundabout Road.
5.8 Asymptotics of the OLS Estimator for Purely Spatial Model.
5.9 Method of Moments and Maximum Likelihood.
5.10 Two–Step Procedure.
5.11 Examples and Computer Simulation.
5.12 Mixed Spatial Model.
5.13 The Roundabout Road (Mixed Model).
5.14 Asymptotics of the OLS Estimator for Mixed Spatial Model.
6 CONVERGENCE ALMOST EVERYWHERE.
6.1 Theoretical Background.
6.2 Various Bounds on Martingale Transforms.
6.3 Marcinkiewicz Zygmund Theorems and Related Results.
6.4 Strong Consistency for Multiple Regression.
6.5 Some Algebra Related to Vector Autoregression.
6.6 Preliminary Analysis.
6.7 Strong Consistency for Vector Autoregression and Related Results.
7 NONLINEAR MODELS.
7.1 Asymptotic Normality of an Abstract Estimator.
7.2 Convergence of Some Deterministic and Stochastic Expressions.
7.3 Nonlinear Least Squares.
7.4 Binary Logit Models with Unbounded Explanatory Variables.
8 TOOLS FOR VECTOR AUTOREGRESSIONS.
8.1 Lp–Approximable Sequences of Matrix–Valued Functions.
8.2 T–Operator and Trinity.
8.3 Matrix Operations and Lp–Approximability.
8.5 Convergence and Bounds for Deterministic Trends.