1 Introduction
1.1 What Makes Time-Series Econometrics Unique?
1.2 Notation
1.3 Statistical Review
1.4 Specifying Time in Stata
1.5 Installing New Stata Commands
1.6 Exercises
2 ARMA(p,q) Processes
2.1 Introduction
2.1.1 Stationarity
2.1.2 A Purely Random Process
2.2 AR(1) Models
2.2.1 Estimating an AR(1) Model
2.2.2 Impulse Responses
2.2.3 Forecasting
2.3 AR(p) Models
2.3.1 Estimating an AR(p) Model
2.3.2 Impulse Responses
2.3.3 Forecasting
2.4 MA(1) Models
2.4.1 Estimation
2.4.2 Impulse Responses
2.4.3 Forecasting
2.5 MA(q) Models
2.5.1 Estimation
2.5.2 Impulse Responses
2.6 Non-zero ARMA Processes
2.6.1 Non-zero AR Processes
2.6.2 Non-zero MA Processes
2.6.3 Dealing with Non-zero Means
2.7 ARMA(p,q) Models
2.7.1 Estimation
2.8 Conclusion
3 Model Selection in ARMA(p,q) Processes
3.1 ACFs and PACFs
3.1.1 Theoretical ACF of an AR(1) Process
3.1.2 Theoretical ACF of an AR(p) Process
3.1.3 Theoretical ACF of an MA(1) Process
3.1.4 Theoretical ACF of an MA(q) Process
3.1.5 Theoretical PACFs
3.1.6 Summary: Theoretical ACFs and PACFs
3.2 Empirical ACFs and PACFs
3.2.1 Calculating Empirical ACFs
3.2.2 Calculating Empirical PACFs
3.3 Putting It All Together
3.4 Information Criteria
4 Stationarity and Invertibility
4.1 What Is Stationarity?
4.2 The Importance of Stationarity
4.3 Restrictions on AR coefficients Which Ensure Stationarity
4.3.1 Restrictions on AR(1) Coefficients
4.3.2 Restrictions on AR(2) Coefficients
4.3.3 Restrictions of AR(p) Coefficients
4.3.4 Characteristic and Inverse Characteristic Equations
4.3.5 Restrictions on ARIMA(p,q) Coefficients
4.4 The Connection Between AR and MA Processes
4.4.1 AR(1) to MA(∞)
4.4.2 AR(p) to MA(∞)
4.4.3 Invertibility: MA(1) to AR(∞)
4.5 What Are Unit Roots, and Why Are They Bad?
5 Non-stationarity and ARIMA(p,d,q) Processes
5.1 Differencing
5.1.1 Example of Differencing
5.2 The Random Walk
5.2.1 The Mean and Variance of the Random Walk
5.2.2 Taking the First Difference Makes it Stationary
5.3 The Random Walk with Drift
5.3.1 The Mean and Variance of the Random Walk with Drift
5.3.2 Taking the First Difference Makes it Stationary
5.4 Deterministic Trend
5.4.1 Mean and Variance
5.4.2 First Differencing Introduces an MA Unit Root
5.5 Random Walk with Drift vs Deterministic Trend
5.6 Differencing and Detrending Appropriately
5.6.1 Mistakenly Differencing (Overdifferencing)
5.6.2 Mistakenly Detrending
5.7 Replicating Granger and Newbold (1974)
5.8 Conclusion
6 Seasonal ARMA(p,q) Processes
6.1 Different Types of Seasonality
6.1.1 Deterministic Seasonality
6.1.2 Seasonal Differencing
6.1.3 Additive Seasonality
6.1.4 Multiplicative Seasonality
6.1.5 MA Seasonality
6.2 Identification
6.3 Invertibility and Stability
6.4 How Common are Seasonal Unit Roots?
6.5 Using De-seasonalized Data
6.6 Conclusion
7 Unit Root Tests
7.1 Introduction
7.2 Unit Root Tests
7.3 Dickey-Fuller Tests
7.3.1 A Random Walk vs a Zero-Mean AR(1) Process
7.3.2 A Random Walk vs an AR(1) Model with a Constant
7.3.3 A Random Walk with Drift vs a Deterministic Trend
7.3.4 Augmented Dickey-Fuller Tests
7.3.5 DF-GLS Tests
7.3.6 Choosing the Lag Length in DF-Type Tests
7.4 Phillips-Perron Tests
7.5 KPSS Tests
7.6 Nelson and Plosser
7.7 Testing for Seasonal Unit Roots
7.8 Conclusion and Further Readings
8 Structural Breaks
8.1 Structural Breaks and Unit Roots
8.2 Perron (1989): Tests for a Unit Root with a Known Structural Break
8.3 Zivot and Andrews' Test of a Break at an Unknown Date
8.3.1 Replicating Zivot and Andrews (1992) in Stata
8.3.2 The zandrews Command
8.4 Further Readings
9 ARCH, GARCH and Time-Varying Variance
9.1 Introduction
9.2 Conditional vs Unconditional Moments
9.3 ARCH Models
9.3.1 ARCH(1)
9.3.2 AR(1)-ARCH(1)
9.3.3 ARCH(2)
9.3.4 ARCH(q)
9.3.5 Example 1: Toyota Motor Company
9.3.6 Example 2: Ford Motor Company
9.4 GARCH Models
9.4.1 GARCH(1,1)
9.4.2 GARCH(p,q)
9.5 Variations on GARCH
9.5.1 GARCH-t
9.5.2 GARCH-M or GARCH-IN-MEAN
9.5.3 Asymmetric Responses in GARCH
9.5.4 I-GARCH or Integrated GARCH
9.6 Exercises
10 Vector Autoregressions I: Basics
10.1 Introduction
10.1.1 A History Lesson
10.2 A Simple VAR(1) and How to Estimate it
10.3 How Many Lags to Include?
10.4 Expressing VARs in Matrix Form
10.4.1 Any VAR(p) Can be Rewritten as a VAR(1)
10.5 Stability
10.5.1 Method 1
10.5.2 Method 2
10.5.3 Stata Command Varstable
10.6 Long-Run Levels: Including a Constant
10.7 Expressing a VAR as a VMA Process
10.8 Impulse Response Functions
10.8.1 IRFs as the Components of the MA Coefficients
10.9 Forecasting
10.10 Granger Causality
10.10.1 Replicating Sims (1972)
10.10.2 Indirect Causality
10.11 VAR Example: GNP and Unemployment
10.12 Exercises
11 Vector Autoregressions II: Extensions
11.1 Orthogonalized IRFs
11.1.1 Order Matters in OIRFs
11.1.2 Cholesky Decompositions and OIRFs
11.1.3 Why Order Matters for OIRFs
11.2 Forecast Error Variance Decompositions
11.3 Structural VARs
11.3.1 Reduced Form vs Structural Form
11.3.2 SVARs are Unidentified
11.3.3 The General Form of SVARs
11.3.4 Cholesky is an SVAR
11.3.5 Long-Run Restrictions: Blanchard and Quah (1989)
11.4 VARs with Integrated Variables
11.5 Conclusion
12 Cointegration and VECMs
12.1 Introduction
12.2 Cointegration
12.3 Error Correction Mechanism
12.3.1 The Effect of the Adjustment Parameter
12.4 Deriving the ECM
12.5 Engle and Granger's Residual-Based Tests of Cointegration
12.5.1 MacKinnon Critical Values for Engle-Granger Tests
12.5.2 Engle-Granger Approach
12.6 Multi-Equation Models and VECMs
12.6.1 Deriving the VECM from a Simple VAR(2)
12.6.2 Deriving the VECM(k-1) from a Reduced-form VAR(k)
12.6.3 Π = α β' is Not Uniquely Identified
12.6.4 Johansen's Tests and the Rank of Π
12.7 IRFs, OIRFs and Forecasting from VECMs
12.8 Lag-Length Selection
12.9 Cointegration Implies Granger Causality
12.9.1 Testing for Granger Causality
12.10 Conclusion
12.11 Exercises
13 Conclusion
A Tables of Critical Values
Bibliography
Index
