An Introduction to Econometrics: A Time Series Approach
PART I FOUNDATIONS
Chapter 1 Economics and Quantitative Economics
Chapter 2 Some Preliminaries
Chapter 3 An Introduction To Stationary And Non-stationary Random Variables
PART II ESTIMATION AND SIMULATION
Chapter 4 A Review Of Estimation And Model Building: The Bivariate Case
Chapter 5 Extending Estimation And Model Building To Several Regressors
Chapter 6 An Introduction To Nonstationary Univariate Time Series Models
Chapter 7 Developments Of Non-stationary Univariate Time Series Models
Chapter 8 Stationarity And Nonstationarity In Single Equation Regression Analysis
Chapter 9 Endogeneity And The Fully Modified OLS Estimator
PART III APPLICATIONS
Chapter 10 The Demand For Money
Chapter 11 The Term Structure of Interest Rates
Chapter 12 The Phillips Curve
Chapter 13 The Exchange Rate And Purchasing Power Parity
PART IV EXTENSIONS
Chapter 14 Multivariate Models And Cointegration
Chapter 15 Applications Of Multivariate Models Involving Cointegration
Chapter 16 Autoregressive Conditional Heteroscedasticity: Modelling Volatility
Detailed List of Contents
PART I FOUNDATIONS
Chapter 1 Economics and Quantitative Economics
1.1 Introduction
1.2 Defining Economics
1.3 Description, Construction And Models In Economics
1.4 The Scope Of Model Building In Quantitative Economics
1.4.1 An Historical Debate
1.4.2 Present Day Concerns
1.4.3 Stylisations Of Methodology
1.5 The Structure And Aims Of This Book
1.5.1 General Aims
1.5.2 Parts And Chapters
1.5.3 General Comments About The Structure Of This Book
1.5.4 Further Reading
1.6 Concluding Remarks
Review
Review Questions
Chapter 2 Some Preliminaries
2.1 Introduction
2.2 Distinguishing Characteristics Of The Data
2.2.1 The Series And Cross Section Data
2.2.2 Time Series Graphs
2.2.3 Frequency
2.2.4 Dimension Of A Variable
2.2.5 Some Examples Of Time Series Data
2.2.6 Non-experimental Data
2.2.7 Experimental Data
2.3 Lagging And Leading Time Series Data
2.3.1 Lagging Time Series Data
2.3.2 Leading Time Series Data
2.4 The Lag Operator
2.4.1 Definition Of The Lag Operator
2.4.2 The Lag Polynomial
2.4.3 Obtaining The Sum Of The Lag Coefficients
2.4.4 A Univariate Dynamic Model
2.5 Bivariate Relationships
2.5.1 A Deterministic Bivariate Model
2.5.2 A Stochastic Bivariate Model
2.5.3 Visual Representation of 2 Variables
2.5.4 Dynamic Bivariate Models
2.5.5 Autoregressive Distributed Lag (ADL) Models
2.5.6 The Distributed Lag Function
2.5.7 More Than One Conditioning Variable
2.5.8 Notation InMore Complex Models
2.6 Several Equations Together
2.7 Concluding Remarks
Review
Review Questions
Chapter 3 An Introduction To Stationary And Non-stationary Random Variables
3.1 Introduction
3.2 Time Series With A Varying Mean
3.2.1 Some Examples
3.3 Random Variables
3.3.1 The Expected Value Of A Random Variable
3.3.2 The Variance Of A Random Variable
3.3.3 Continuous Random Variables
3.4 Covariance, Autocovariance And Autocorrelation
3.4.1 Joint Events
3.4.2 Covariance And Autocovariance
3.4.3 Conditional Expectation
3.4.4 Autocovariances And Second Order Stationarity
3.4.5 Linear Combinations Of Random Variables
3.4.6 An Example Of A Non-stationary Time Series
3.4.7 Correlation And The Autocorrelation Function
3.4.8 The Variance Decomposition
3.4.9 Iterating Expectations
3.5 A Random Walk
3.5.1 The Coin Tossing Game
3.6 Estimation
3.6.1 Nonstationary Processes
3.6.2 Stationary Processes
3.6.3 Centered Moving Variance
3.7 Concluding Remarks
Review
Review Questions
PART II ESTIMATION AND SIMULATION
Chapter 4 A Review Of Estimation And Model Building: The Bivariate Case
4.1 Introduction
4.2 Statistical Background
4.2.1 Factorisation Of The Joint Density
4.2.2 The Conditional Expectation Function, CEF, As The Regression Function
4.2.3 Some Important Distributions
4.3 Estimation, Estimators and Estimates
4.3.1 The Analogy Principle And Instrumental Variables Estimation
4.3.2 The Least Squares Principle
4.4 Properties Of Estimators
4.4.1 Bias
4.4.2 Consistency
4.4.2a Examples
4.4.2b Speed Of Convergence
4.4.3 Asymptotic Bias
4.4.4 Efficiency
4.4.5 Linearity
4.5 Properties Of The OLS Estimators 
and 
4.5.1 Conditionally And Unconditionally Unbiased
4.5.2 Minimum Variance In The Class Of Linear Unbiased Estimators
4.5.2a The Variance Of
4.5.2b The Variance Of
4.5.2c The Unconditional Variances Of and
4.5.2d The Gauss-Markov Theorem
4.6 A Nonlinear CEF
4.7 Goodness Of Fit
4.7.1 Goodness Of Fit In The Population
4.7.2 Goodness Of Fit: In The Sample
4.7.3 %
As A Measure Of Goodness Of Fit
4.8 Estimation Of Dynamic Models
4.9 Structure And Regression
4.9.1 Weak Exogeneity And The Parameters Of Interest
4.9.2 Instrumental Variables Estimation
4.10 Tests And Associated Concepts
4.10.1 Significance Tests
4.10.2 The Alternative Hypothesis
4.10.3 Power
4.11 Summary Of OLS Estimators And An Empirical Example
4.11.1 Tabular Summary
4.11.2 Typical Computer Output
4.12 Concluding Remarks
Review
Review Questions
Appendices
A4.1 Maximum Likelihood Estimation
A4.1.1 A General Principle Of Estimation
A4.1.2 The Likelihood Function
A4.1.3 The Binomial Distribution Of Probabilities
A4.1.4 Maximum Likelihood Estimation: The Regression Model
A4.1.5 Estimation In Simultaneous Models
A4.1.6 Hypothesis Testing
A4.2 Computer Output
A4.2.1 MICROFIT
A4.2.1 TSP
A4.2.2 RATS
A4.2.3 PC GIVE
Chapter 5 Extending Estimation And Model Building To Several Regressors
5.1 Introduction
5.2 Extending The Bivariate Model: More Than Two Regressors
5.2.1 Multiple Regressors: The Basic Setup
5.2.2 Deriving The OLS Estimator
5.2.3 The Variance-Covariance Matrix of 
5.3 Generalised Least Squares, GLS
5.3.1 The GLS Estimator
5.3.2 The Variance-Covariance Matrix Of The GLS Estimator, Var(
)
5.3.3 OLS Or GLS?
5.4 Testing Hypotheses
5.4.1 Testing Principles: Wald, Likelihood Ratio, Lagrange-Multiplier
5.4.2 Extension To Multiple Hypotheses
5.5 Heteroscedasticity: Implications For OSL Estimation And Tests
5.5.1 Implications Of Heteroscedasticity
5.5.2 Tests for Heteroscedasticity
5.5.2a White's (1980) Test
5.5.2b The Goldfeld-Quandt Test
5.5.2c The Breusch-Pagan Test
5.5.3 Interpretation Of Significant Test Statistics For Heteroscedasticity
5.6 Misspecification: Diagnosis And Effects
5.6.1 Serial Correlation Of 
5.6.1a The Durbin-Watson, DW, Statistic
5.6.1b The Lagrange Multiplier (LM) Test For Serial Correlation
5.6.1c The Box-Pearce And Ljung-Box Tests
5.6.2 An Illustration Of The DW And LM tests
5.6.3 Interpretation Of Significant Test Statistics For Serial Correlation
5.6.4 The Newey-West Estimator Of The Variance(-Covariance) Matrix Of
5.7. Normality And The Jarque-Bera Test
5.7.1 Normality of
5.7.2 The Jarque-Bera Test
5.8 Functional Form And The RESET Test
5.8.1 Developing A Test For Nonlinearity
5.8.2 Ramsay's RESET Test
5.9 Stability Of The Regression Coefficients
5.9.1 Chow's (First) Test
5.9.2 Predictive/Forecast Failure Tests
5.9.2a Chow's (Second) Test: A Test For Predictive Failure
5.9.2b A Forecast (Deterioration) Test
5.9.3 Unknown Breakpoint(s)
5.10 Model Building And Evaluation
5.11 An Estimated Regression Model
5.11.1 The Basic Model
5.11.2 Estimation Of The Basic Model
5.11.3 Diagnostic And Misspecification Tests
5.11.3a Serial Correlation
5.11.3b Heteroscedasticity
5.11.4 Normality
5.11.5 Functional Form: The RESET Test
5.11.6 Chow Tests
5.11.6a Chow's First Test
5.11.6b Chow's Second Test: Predictive Failure
5.11.7 An Extended Model
5.12 Concluding Remarks
Review
Review Questions
Chapter 6 An Introduction To Nonstationary Univariate Time Series Models
6.1 Introduction
6.2 Non-deterministic Time Series
6.2.1 A Pure Random Walk
6.2.2 A Near Random Walk
6.2.3 A Random Walk With Drift
6.2.4 Unit Roots
6.2.5 A Near Random Walk With Drift
6.2.6 The Persistence Of Shocks
6.2.7 The Mean, Variance And Autocorrelations Of An AR(1) Process
6.2.8 Difference Stationary And Trend Stationary Series
6.3 Testing For A Unit Root
6.3.1
Test
6.3.2 
And 
Test Statistics
6.3.3 
And 
Test Statistics
6.3.4 The Empirical Power Of Some Dickey-Fuller Test Statistics
6.3.5 Distribution Of The Test Statistics On The Intercept And Trend
6.3.6 The Augmented Dickey-Fuller, ADF, Test
6.4 A Framework For Testing
6.4.1 Is The Data Series Trended?
6.4.2 The Data Is Not Obviously Trended And The Mean Under The Alternative Is Nonzero
6.4.3 The Data Is Not Obviously Trended And The Mean Under The Alternative Is Zero
6.4.4 Cumulation Of Type 1 Error
6.5 Concluding Remarks
Review
Review Questions
Chapter 7 Developments Of Non-stationary Univariate Time Series Models
7.1 Introduction
7.2 ARIMA Models
7.3 Pretesting, Power And Model Selection Strategies Using ADF Test Statistics
7.4 Other Tests
7.4.1 Dickey And Fuller's T(
- 1)
7.4.2 The Weighted Symmetric (WS) Estimator, Pantula et al (1994)
7.4.3 Phillips And Perron Versions Of The DF Tests
7.5 Structural And Reduced Form Univariate Time Series Models
7.5.1 Structural Univariate Time Series Models
7.5.2 Stationarity As The Null Hypothesis
7.6 Testing For 2 Unit Roots
7.7 Seasonality And Seasonal Integration
7.7.1 Integration In Seasonal Processes
7.7.2 Testing For A Unit Root In A Seasonal Process
7.8 Structural Breaks
7.8.1 The Perron (1989) Approach To A Single Structural Break
7.8.2 Additive Outliers, Franses And Haldrup (1994)
7.8.3 Summary
7.9 Applications To Some Economic Time Series
7.9.1 U.K Consumer's Expenditure On Nondurables
7.9.2 U.K Unemployment Rate
7.9.3 U.S Unemployment Rate
7.9.4 Testing For Seasonal Nonstationarity: U.K Employees
7.10 'Nearly' Integrated And 'Nearly' Stationary Time Series
7.11 Concluding Remarks
Review
Review Questions
Chapter 8 Stationarity And Nonstationarity In Single Equation Regression Analysis
8.1 Introduction
8.2 Examining The Properties Of Estimators By Simulation
8.2.1 Xt Is Fixed In Repeated Samples
8.2.2 Xt Is A Stationary, Stochastic Variable
8.2.2a Xt Stationary: White Noise
8.2.2b Xt Stationary: An AR(1) Process
8.2.3 Xt Is A Nonstationary, Stochastic Varible
8.2.4 A Spurious Regression
8.2.5 The Distribution of R2
8.3 Cointegration
8.3.1 Cointegration: Basic Concepts
8.3.2 Cointegrating Versus Spurious Regressions
8.4 Testing For Non-Cointegration: The Engle-Granger Approach
8.4.1 The Engle-Granger (1987) Approach (The Bivariate Case)
8.4.2 Critical Values For The Test Statistic 
: Simulation
8.4.3 Mackinnon's Response Surface For Critical Values
8.4.4 More Than Two Variables
8.4.5 An Illustration Of The Testing Procedure
8.4.6 An Illustration Of A Spurious Regression
8.5 Links Between Cointegration And Error Correction Models
8.5.1 Granger's Representation Theorem
8.5.2 Cointegration And Error Correction: An Alternative Test Statistic For Cointegration
8.5.2a Known Cointegration Coefficients
8.5.2b Unknown Cointegration Coeffcients
8.6 Alternative Representations Of The Long-Run Relationship
8.6.1 The ADL Model And The ECM (Two Variable Case)
8.6.2 The Bewley Transformation
8.6.3 A Numerical Example
8.6.4 The More General ADL Model: Alternative Representations
8.7 Estimation, Inference And Simulation
8.7.1 A Comparison Of Alternative Ways Of Estimating The Cointegrating Coefficients
8.7.1a Simulation Set-Up
8.7.1a.i No Dynamics
8.7.1.a.ii Dynamics
8.7.2 Simulation Results
8.7.2a Estimating The Coefficients
8.7.2b Distribution Of The t Statistics
8.8 Concluding Remarks
Appendix
A8.1 MacKinnon's (1991) Critical Values For Cointegration Tests
Review
Review Questions
Chapter 9 Endogeneity And The Fully Modified OLS Estimator
9.1 Introduction
9.2 Distinguishing Variance Matrices
9.2.1 Conditional, Unconditional And Long-Run Variance Matrices
9.2.1a Autoregressive Process
9.2.1b First Order Moving Average Process
9.2.1c Decomposition Of The Long-Run Variance Matrix
9.2.2 A General Result
9.2.2a MA(1) Example
9.2.2b AR(1) Example
9.3 Endogeneity
9.3.1 Preliminaries
9.3.1a Contemporaneity
9.3.1b Weak Exogeneity
An Example Of The Failure Of Weak Exogeneity
9.3.2 The Regression (Or Conditional Expectation) Function And Weak Exogeneity
9.4 The Fully-Modified (Phillips-Hansen) OLS Estimator
9.4.1 Corrections For Bias And Enogeneity
9.4.1a A Bias Correction
9.4.1b An Endogeneity Correction
9.4.1c A Semi-Parametric Approach To Estimating The Corrections
9.4.2 Variations On A Theme: When OLS On The ADL Model Is Optimal
9.4.3 Examples Of FMOLS Estimation
9.4.3a The Consumption-Income Example
9.4.3b Long and Short Interest Rates
9.4.4 Simulation Findings
9.4.4a Simulation Results: Phillips And Hansen (1988)
9.4.4b Simulation Results: Hansen And Phillips (1990)
9.4.4c Simulation Results: Inder (1993)
9.5 Complications: Nearly Integrated Processes And Endogeneity
9.5.1 Xt Integrated/Nearly Integrated, No Endogeneity
9.5.2 Xt Integrated/Nearly Integrated And Endogenous
9.5.2a Contemporaneity
9.5.2b Failure Of Weak Exogeneity
9.5.2c Summary
9.5.3 Sensitivity To Changes In The Design Parameters: Slow Adjustment
9.6 Concluding Remarks
Review
Review Questions
PART III APPLICATIONS
Chapter 10 The Demand For Money
10.1 Introduction
10.2 The Demand For Money
10.2.1 A Definition Of Money
10.2.2 The Transactions Motive
10.2.3 The Precautionary Motive
10.2.4 The Speculative Motive
10.2.5 Bringing The Motives Together
10.2.6 Some Variations On A Theme: The Velocity Of Circulation
10.3 The Demand For Money During The German Hyperinflation
10.3.1 Historical Background
10.3.2 Cagan's Specification Of The Demand For Money Function: Background
10.3.3 Cagan's Demand For Money Function: Basic Specification
10.3.4 A Graphical Analysis Of The Data
10.3.5 Testing For Nonstationarity
10.3.6 Cointegration
10.3.7 Dynamic Models
10.4 The Demand For M1: A Study Using Recent U.S Data
10.4.1 Model Specification
10.4.2 Data Definitions
10.4.3 A Graphical Analysis Of The Data
10.4.4 Testing For Nonstationarity
10.4.5 Cointegration
10.4.6 Dynamic Models
10.4.7 Out Of Sample Performance
10.4.8 A Brief Comparison With Hoffman And Rasche (1991) And Baba, Hendry And Starr (1992)
10.5 Concluding Remarks
Review
Review Questions
Chapter 11 The Term Structure of Interest Rates
11.1 Introduction
11.2 Term Structure Of Interest Rates
11.2.1 Term To Maturity
11.2.2 The Discount Rate, The Interest Rate And Continuous Compounding
11.2.3 The Yield Curve
11.3 The Expectations Model Of The Term Structure
11.3.1 The Yield To Maturity And The Forward Rate
11.3.2 The Spread
11.3.3 Implications For Economic Policy
11.4 Assessing The Expectations Model
11.4.1 Three Implications Of The Expectations Model
11.4.2 The Data
11.4.3 A Graphical Analysis Of The Data: Yields
11.4.4 Unit Root Tests On The Yields
11.4.5 A Graphical Analysis Of The Data: Spreads
11.4.6 Unit Root Tests On The Spreads
11.4.7 Estimation Of The Spread Equations
11.4.8 Bivariate Regressions: The Perfect Foresight Spread
11.5 Other Studies And Other Methods Of Testing The Expectations Model
11.5.1 Methods And Results
11.5.2 Why Do Tests Of EH + REH Tend To Indicate Rejection?
11.6 Concluding Remarks
Review
Review Questions
Chapter 12 The Phillips Curve
12.1 Introduction
12.2 The Phillips Curve
12.2.1 Basic Ideas
12.2.2 Phillips' Original Estimates and Interpretation
12.2.3 The Phillips Curve: A Menu Of Choice?
12.2.4 The Phillips Curve In The United States: An Early View
12.2.5 A Graphical analysis Of Phillips' Data For 1861-1913
12.2.6 Testing For Nonstationarity
12.2.7 Re-estimation Of The Phillips Curve, 1981-1913
12.3 Is The Phillips Curve Misspecified?
12.3.1 Fisher (1926) And Phillips (1958)
12.3.2 Friedman's Model
12.3.3 Imperfect Competition
12.3.4 The Phillips, 'Fisher' And 'Friedman' Curves
12.3.5 Expectations And The Reformulation Of The Phillips Curve
12.3.6 A Supply Side Interpretation Of The Importance Of Inflation Expectations
12.4 Estimation Of The Expectations Augmented Phillips Curve (EAPC)
12.4.1 Timing Of Expectations
12.4.2 The Adaptive Expectations Hypothesis: Formulation
12.4.3 The AEH: Estimation
12.4.4 The Lucas/Sargent Critique Of The Identifying Assumption
12.4.5 Estimation Results: Adaptive Expectations Augmented Phillips Curve
12.4.6 Rational Expectations (RE): General Principles
12.4.7 Implementing Rational Expectations
12.4.8 Estimating Results With (Weakly) Rational Expectations
12.5 The Phillips Correlation
12.5.1 Granger Causation Tests
12.5.2 Estimation And Hypothesis Tests
12.5.3 Practical Problems
12.5.4 Granger-Causation Tests: Wage/Price Inflation And Unemployment, U.K
12.5.5 Granger-Causation Tests: Wage/Price Inflation And Unemployment, U.S
12.6 Concluding Remarks
Review
Review Questions
Chapter 13 The Exchange Rate And Purchasing Power Parity
13.1 Introduction
13.2 Purchasing Power Parity
13.2.1 Complications For PPP
13.2.2 Short Run And Long-Run Considerations
13.2.3 The Nominal Exchange Rate, Et, And The Real Exchange Rate, REt
13.2.4 A Strategy For Testing PPP
13.3 Assessing The Evidence For PPP
13.3.1 The Nature Of The Evidence
13.3.2 Measuring The Real Exchange Rate
13.3.3 Visual Impression Of The Data
13.3.4 Dickey-Fuller Unit Root Tests
13.4 The Real Exchange Rate: Some More Considerations And Tests
13.4.1 An Example Of The Persistence Of Shocks
13.4.2 Pooling Observations: A Panel Unit Root Test
13.4.3 Estimating The Speed Of Response To A Shcok To The Real Exchange Rate
13.5 Simple Tests For Non-cointegration
13.5.1 Relaxing The (1, -1) Cointegrating Vector
13.5.2 OLSEG Estimation Of The Cointegrating Rgeressions
13.5.3 An Illustration Of The Modified ADF Test Statistic
13.5.4 (Very) 'Weak' Form PPP
13.6 Other Models Of The Exchange Rate
13.6.1 The Flexible Price Monetary Model
13.6.2 An Illustration Of The FPMM With U.S:U.K Quarterly Data
13.7 Concluding Remarks
Review
Review Questions
PART IV EXTENSIONS
Chapter 14 Multivariate Models And Cointegration
14.1 Introduction
14.2 Some Basic Concepts
14.2.1 The Var
14.2.2 Stability And Stationarity In The VAR
14.2.3 Stability And Roots In The Univariate Model
14.2.4 Eigenvalues And Roots: The Multivariate Model
14.2.5 What To Do If There Is A Unit Root
14.3 Simple Multivariate (Vector) Error Correction Models
14.3.1 A Bivariate Model
14.3.2 The Eigenvalues Of P And the Existence Of Cointegrating Vectors
14.3.3 More Than One Cointegrating Vector
14.3.4 Longer Lags
14.3.5 The Multivariate Model: The Existence Of A Unit Root And Reduced Rank of P
14.4 Testing For Cointegration
14.4.1 Establishing A Firm Base For Inference On The Cointegrating Rank
14.4.2 Estimation Of The UVAR And Test Statistics For Testing The Cointegrating Rank (Optional)
14.4.3 Hypothesis Tests On The Cointegrating Rank
14.4.4 An Alternative Method Of Selecting The Cointegrating Rank
14.4.5
Intercepts And Trends In The Var For The Trace
and l max Statistics
14.4.6 Separating I(1) And I(0) Variables
14.5 Identification
14.5.1 Structural And Reduced Form Error Correction Models
14.5.2 Identification Of The Cointegrating Vector
14.5.3 Testing Overidentifyinhg Restrictions On The Cointegrating Vectors
14.5.4 Identification Of The Short-Run Structure
14.6 Concluding Remarks
Review
Review Questions
Chapter 15 Applications Of Multivariate Models Involving Cointegration
15.1 Introduction
15.2 Purchasing Power Parity And Uncovered Interest Parity, Johansen And Juselius (1992)
15.2.1 An Outline Of PPP And UIP
15.2.2 Generic Identification Of The Cointegrating Relationships
15.2.3 Estimating The Cointegrating Rank
15.2.4 Interpreting The Unrestricted Cointegrating Vectors
15.3 Wage Differentials In The U.S, Dickey And Rossana (1994)
15.3.1 Estimating The Cointegrating Rank
15.3.2 Cointegration Of Real Wages
15.3.3 Identification Of The Cointegrating Vectors
15.4 The IS/LM Model, Johansen And Juselius (1994)
15.4.1 Identifying The Short-Run Structure
15.4.2 The Simultaneous Structure
15.5 The Demand For Money In The U.K, Hendry And Mizon (1993)
15.5.1 Estimating The Cointegrating Rank
15.5.2 Unrestricted Estimates Of The Cointegrating Vectors And Adjustment Coefficients
15.5.3 Identification Of The Cointegrating Vectors
15.5.4 Estimating A SECM
15.5.5 Identification Of The Short-Run Structure
15.6 Weak Exogeneity: When Is It Valid To Model The Patial System?
15.6.1 Containing The Number Of Variables In The VAR
15.6.2 Closed Or Open Systems?
15.6.3 Joint, Conditional And Marginal Models
15.6.4 Hypothesis Testing And Weak Exogeneity
15.6.5 Examples Of Testing For Weak Exogeneity
15.7 An Extended Illustration: Urbain's (1995) Study Of The Demand For Imports In Belgium
15.7.1 Estimating The Cointegrating Rank
15.7.2 Identification Of The Cointegrating Vectors
15.7.3 Testing Restrictions
15.7.4 The Parsimonious VAR, PVAR, And SECM
15.8 Revisiting The Demand For Money In The U.S
15.8.1 A Multivariate Approach: Choosing The Lag Length
15.8.2 A Multivariate Approach: Estimating The Cointegrating Rank By The Johansen Method
15.8.3 A Multivariate Approach: Estimating The Cointegrating Rank By The Schwarz Information Criterion (SIC)
15.8.4 Robustness Of Specification
15.8.5 A Comparison With The OLS Results
15.8.6 A Structural Error Correction Model And Parsimonious Encompassing
15.8.7 An Estimated SECM For Money, Income And The Interest Rate
15.9 Concluding Remarks
Review
Review Questions
Chapter 16 Autoregressive Conditional Heteroscedasticity: Modelling Volatility
16.1 Introduction
16.2 Basic Concepts
16.2.1 Conditional And Unconditional Variances: A Crucial Distinction
16.2.2 ARCH(q)
16.2.3 GARCH(p, q)
16.2.4 What Does Data With An ARCH Effect Look Like?
16.3 Stationarity And Persistence In Some Standard Models
16.3.1 ARCH(q)
16.3.2 GARCH(p, q)
16.3.3 IGARCH(1, 1)
16.3.4 Nonnegativity Constraints In GARCH Models
16.4 Estimation
16.4.1 Specification
16.4.2 A Nonlinear Estimator 'Beats' The Linear OLS Estimator
16.5 Testing For ARCH/GARCH Effects
16.5.1 LM Test For ARCH Effects
16.5.2 GARCH(p, q)
16.6 Variations On An ARCH/GARCH Theme
16.6.1 ABSGARCH, EGARCH
16.6.1a ABSGARCH
16.6.1b EGARCH
16.6.2 ARCH-M, GARCH-M, ABSARCH-M, EGARCH-M
16.7 The Importance Of Asymmetry In ARCH Models
16.7.1 The News Impact Curve
16.7.2 Examples Of The News Impact Curve
16.7.3 Asymmetry In More Detail
16.7.3a The AGARCH and GJR Asymmetric Models
16.7.4 Tests For Asymmetry
16.8 Examples
16.8.1 The U.S Inflation Rate
16.8.2 The U.K Savings Ratio
16.8.3 ARCH-M Applied To Excess Returns
16.8.4 Testing For Asymmetry In The Returns For Standard and Poor's 500 Index For The U.S
16.9 Concluding Remarks
Appendix
A1.1 The Likelihood Function For the ARCH Model
A1.2 Non-normality
A1.3 Properties Of The Maximum Likelihood Estimators In GARCH Models
A1.4 Practical ARCH/GARCH
Review
Review Questions
Appendix Statistical Tables
A.1 The Normal Distribution
A.2 The t distribution
A.3 The
Distribution
A.4 The F Distribution
References
Index