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Linear Time Series Analysis

Chapter IX: Difference Equations and Lag Operators

Chapter IX — Difference Equations and Lag Operators

Chapter X: Linear Least Squares Projections (Regressions)

Chapter X — Linear Least Squares Projections (Regressions)

Linear Time Series (in the spirit of Chapter XI)

Introduction
Preliminary Concepts
The Cross Covariogram
A Mathematical Digression: Fourier and \(z\) Transforms
Evaluating the Inverse \(z\) Transform
Fourier Transform Pairs and the Uncertainty Principle
The Spectrum
The Cross Spectrum
Estimating Spectra, Cross Spectra, and Bispectra with the FFT
A Digression on Leading Indicators
Analysis of Some Filters: The Slutsky Effect and Kuznets’s Transformations
A Small Kit of \(h(e^{-i\omega})\)’s
Alternative Definitions of the Business Cycle
Index Models
Representation Theory
Linear Least Squares Prediction
Some Examples
Deriving the Moving Average Representation
Finding a Wold Representation: \(m\)th Order Moving Average
Finding a Wold Representation: An \(m\)th Order Moving Average, \(n\)th Order Autoregression
The Residue Theorem Behind Partial Fractions
Signal Extraction Problems
Predicting Geometric Distributed Leads
The Chain Rule of Forecasting
Some Applications to Rational Expectations Models
Vector Stochastic Difference Equations
A Compact Notation
Optimal Prediction: Compact Notation
Optimal Filtering Formula
The Relationship Between Wiener-Granger Causality and Econometric Exogeneity
Sims’s Application to Money and Income
Multivariate Prediction Formulas
The Effects of Filtering on One-Sided and Two-Sided Projections
Preservation of Orthogonality Conditions under Filtering
Theories Restricting One-Sided Projections
Seasonal Adjustment and Forecasting Geometric Distributed Leads
Seasonality and Approximation Errors
Aggregation over Time
Sims’s Formula Again
Effects of Errors in Variables
Bubbles
A Difficulty in Interpreting Vector Autoregressions
Exact Linear Rational Expectations Models
Money Demand in Hyperinflations: A Misspecified Regression and Sims’s Approximation Formula
Decomposing an Explosive Autoregression
Lucas’s Two Illustrations of the Quantity Theory and Whiteman’s Critique
Exercises
Solutions to Exercises

A Nonlinear Postscript

Nonlinear (Volterra) Moving-Average Representations
Volterra Moving-Average Models: Sample Paths and Higher-Order Spectral Analysis

Moving Spectra: Complex Demodulation

Complex Demodulation

Chapter XIV: Investment Under Uncertainty

Chapter XIV — Investment Under Uncertainty

Bibliography

Bibliography

Index

Index

Index

By Thomas J. Sargent