Short-Memory Linear Processes and Econometric Applications

Preface.

Acknowledgments.

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.

3.8 Classes.

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.

8.4 Resolvents.

8.5 Convergence and Bounds for Deterministic Trends.

REFERENCES.

Author Index.

Subject Index.