Part Ⅰ Introduction

1 Background and Overview

1.1 Background

1.2 Overview

2 Casting Models in Canonical Form

2.1 Notation

2.1.1 Log-Linear Model Representations

2.1.2 Nonlinear Model Representations

2.2 Linearization

2.2.1 Taylor Series Approximation

2.2.2 Log-Linear Approximations

2.2.3 Example Equations

3 DSGE Models：Three Examples

3.1 Model Ⅰ：A Real Business Cycle Model

3.1.1 Environment

3.1.2 The Nonlinear System

3.1.3 Log-Linearization

3.2 Model Ⅱ：Monopolistic Competition and Monetary Policy

3.2.1 Environment

3.2.2 The Nonlinear System

3.2.3 Log-Linearization

3.3 Model Ⅲ：Asset Pricing

3.3.1 Single-Asset Environment

3.3.2 Multi-Asset Environment

3.3.3 Alternative Preference Specifications

Part Ⅱ Model Solution Techniques

4 Linear Solution Techniques

4.1 Homogeneous Systems

4.2 Example Models

4.2.1 The Optimal Consumption Model

4.2.2 Asset Pricing with Linear Utility

4.2.3 Ramsey's Optimal Growth Model

4.3 Blanchard and Kahn's Method

4.4 Sims'Method

4.5 Klein's Method

4.6 An Undetermined Coefficients Approach

5 Nonlinear Solution Techniques

5.1 Projection Methods

5.1.1 Overview

5.1.2 Finite Element Methods

5.1.3 Orthogonal Polynomials

5.1.4 Implementation

5.1.5 Extension to the l-dimensional Case

5.1.6 Application to the Optimal Growth Model

5.2 Iteration Techniques：Value-Function and Policy-Function Iterations

5.2.1 Dynamic Programming

5.2.2 Value-Function Iterations

5.2.3 Policy-Function Iterations

5.3 Perturbation Techniques

5.3.1 Notation

5.3.2 Overview

5.3.3 Application to DSGE Models

5.3.4 Application to an Asset-Pricing Model

Part Ⅲ Data Preparation and Representation

6 Removing Trends and Isolating Cycles

6.1 Removing Trends

6.2 Isolating Cycles

6.2.1 Mathematical Background

6.2.2 Cramér Representations

6.2.3 Spectra

6.2.4 Using Filters to Isolate Cycles

6.2.5 The Hodrick-Prescott Filter

6.2.6 Seasonal Adjustment

6.2.7 Band Pass Filters

6.3 Spuriousness

7 Summarizing Time Series Behavior When All Variables Are Observable

7.1 Two Useful Reduced-Form Models

7.1.1 The ARMA Model

7.1.2 Allowing for Heteroskedastic Innovations

7.1.3 The VAR Model

7.2 Summary Statistics

7.2.1 Determining Lag Lengths

7.2.2 Characterizing the Precision of Measurements

7.3 Obtaining Theoretical Predictions of Summary Statistics

8 State-Space Representations

8.1 Introduction

8.1.1 ARMA Models

8.2 DSGE Models as State-Space Representations

8.3 Overview of Likelihood Evaluation and Filtering

8.4 The Kalman Filter

8.4.1 Background

8.4.2 The Sequential Algorithm

8.4.3 Smoothing

8.4.4 Serially Correlated Measurement Errors

8.5 Examples of Reduced-Form State-Space Representations

8.5.1 Time-Varying Parameters

8.5.2 Stochastic Volatility

8.5.3 Regime Switching

8.5.4 Dynamic Factor Models

Part Ⅳ Monte Carlo Methods

9 Monte Carlo Integration：The Basics

9.1 Motivation and Overview

9.2 Direct Monte Carlo Integration

9.2.1 Model Simulation

9.2.2 Posterior Inference via Direct Monte Carlo Integration

9.3 Importance Sampling

9.3.1 Achieving Efficiency：A First Pass

9.4 Efficient Importance Sampling

9.5 Markov Chain Monte Carlo Integration

9.5.1 The Gibbs Sampler

9.5.2 Metropolis-Hastings Algorithms

10 Likelihood Evaluation and Filtering in State-Space Representations Using Sequential Monte Carlo Methods

10.1 Background

10.2 Unadapted Filters

10.3 Conditionally Optimal Filters

10.4 Unconditional Optimality：The EIS Filter

10.4.1 Degenerate Transitions

10.4.2 Initializing the Importance Sampler

10.4.3 Example

10.5 Application to DSGE Models

10.5.1 Initializing the Importance Sampler

10.5.2 Initializing the Filtering Density

10.5.3 Application to the RBC Model

Part Ⅴ Empirical Methods

11 Calibration

11.1 Historical Origins and Philosophy

11.2 Implementation

11.3 The Welfare Cost of Business Cycles

11.4 Productivity Shocks and Business Cycle Fluctuations

11.5 The Equity Premium Puzzle

11.6 Critiques and Extensions

11.6.1 Critiques

11.6.2 Extensions

12 Matching Moments

12.1 Overview

12.2 Implementation

12.2.1 The Generalized Method of Moments

12.2.2 The Simulated Method of Moments

12.2.3 Indirect Inference

12.3 Implementation in DSGE Models

12.3.1 Analyzing Euler Equations

12.3.2 Analytical Calculations Based on Linearized Models

12.3.3 Simulations Involving Linearized Models

12.3.4 Simulations Involving Nonlinear Approximations

12.4 Empirical Application：Matching RBC Moments

13 Maximum Likelihood

13.1 Overview

13.2 Introduction and Historical Background

13.3 A Primer on Optimization Algorithms

13.3.1 Simplex Methods

13.3.2 Derivative-Based Methods

13.4 Ill-Behaved Likelihood Surfaces：Problems and Solutions

13.4.1 Problems

13.4.2 Solutions

13.5 Model Diagnostics and Parameter Stability

13.6 Empirical Application：Identifying Sources of Business Cycle Fluctuations

14 Bayesian Methods

14.1 Overview of Objectives

14.2 Preliminaries

14.3 Using Structural Models as Sources of Prior Information for Reduced-Form Analysis

14.4 Implementing Structural Models Directly

14.5 Model Comparison

14.6 Using an RBC Model as a Source of Prior Information for Forecasting

14.7 Estimating and Comparing Asset-Pricing Models

14.7.1 Estimates

14.7.2 Model Comparison

References

Index