Computer-Aided Introduction to Econometrics

Example
2.9 Three General Test Procedures
2.9.1 Likelihood Ratio Test (LR)
2.9.2 The Wald Test (W)
2.9.3 Lagrange Multiplier Test (LM)
2.9.4 Relationships and Properties of the Three General Testing Procedures
2.9.5 The Three General Testing Procedures in the MLRM Context
2.9.6 Example
2.10 Dummy Variables
2.10.1 Models with Changes in the Intercept
2.10.2 Models with Changes in some Slope Parameters
2.10.3 Models with Changes in all the Coefficients
2.10.4 Example
2.11 Forecasting
2.11.1 Point Prediction
2.11.2 Interval Prediction
2.11.3 Measures of the Accuracy of Forecast
2.11.4 Example

3 Dimension Reduction and Its Applications
3.1 Introduction
3.1.1 Real Data Sets
3.1.2 Theoretical Consideration
3.2 Average Outer Product of Gradients and its Estimation
3.2.1 The Simple Case
3.2.2 The Varying-coefficient Model
3.3 A Unified Estimation Method
3.3.1 The Simple Case
3.3.2 The Varying-coefficient Model
3.4 Number of E.D.R. Directions
3.5 The Algorithm
3.6 Simulation Results
3.7 Applications
3.8 Conclusions and Further Discussion
3.9 Appendix. Assumptions and Remarks

4 Univariate Time Series Modelling
4.1 Introduction
4.2 Linear Stationary Models for Time Series
4.2.1 White Noise Process
4.2.2 Moving Average Model
4.2.3 Autoregressive Model
4.2.4 Autoregressive Moving Average Model
4.3 Nonstationary Models for Time Series
4.3.1 Nonstationary in the Variance
4.3.2 Nonstationarity in the Mean
4.3.3 Testing for Unit Roots and Stationarity
4.4 Forecasting with ARIMA Models
4.4.1 The Optimal Forecast
4.4.2 Computation of Forecasts
4.4.3 Eventual Forecast Functions
4.5 ARIMA Model Building
4.5.1 Inference for the Moments of Stationary Processes
4.5.2 Identification of ARIMA Models
4.5.3 Parameter Estimation
4.5.4 Diagnostic Checking
4.5.5 Model Selection Criteria
4.5.6 Example: European Union G.D.P
4.6 Regression Models for Time Series
4.6.1 Cointegration
4.6.2 Error Correction Models

5 Multiplicative SARIMA models
5.1 Introduction
5.2 Modeling Seasonal Time Series
5.2.1 Seasonal ARIMA Models
5.2.2 Multiplicative SARIMA Models
5.2.3 The Expanded Model
5.3 Identification of Multiplicative SARIMA Models
5.4 Estimation of Multiplicative SARIMA Models
5.4.1 Maximum Likelihood Estimation
5.4.2 Setting the Multiplicative SARIMA Model
5.4.3 Setting the Expanded Model
5.4.4 The Conditional Sum of Squares
5.4.5 The Extended ACF
5.4.6 The Exact Likelihood

6 Auto Regressive Conditional Heteroscedastic Models
6.1 Introduction
6.2 ARCH(1) Model
6.2.1 Conditional and Unconditional Moments of the ARCH(1)
6.2.2 Estimation for ARCH(1) Process
6.3 ARCH(q) Model
6.4 Testing Heteroscedasticity and ARCH(1) Disturbances
6.4.1 The Breusch-Pagan Test
6.4.2 ARCH(1) Disturbance Test
6.5 ARCH(1) Regression Model
6.6 GARCH(p,q) Model
6.6.1 GARCH(1,1) Model
6.7 Extensions of ARCH Models
6.8 Two Examples of Spanish Financial Markets
6.8.1 Ibex35 Data
6.8.2 Exchange Rate US Dollar/Spanish Peseta Data (Continued)

7 Numerical Optimization Methods in Econometrics
7.1 Introduction
7.2 Solving a Nonlinear Equation
7.2.1 Termination of Iterative Methods
7.2.2 Newton-Raphson Method
7.3 Solving a System of Nonlinear Equations
7.3.1 Newton-Raphson Method for Systems
7.3.2 Example
7.3.3 Modified Newton-Raphson Method for Systems
7.3.4 Example
7.4 Minimization of a Function: One-dimensional Case
7.4.1 Minimum Bracketing
7.4.2 Example
7.4.3 Parabolic Interpolation
7.4.4 Example
7.4.5 Golden Section Search
7.4.6 Example
7.4.7 Brent's Method
7.4.8 Example
7.4.9 Brent's Method Using First Derivative of a Function
7.4.10 Example
7.5 Minimization of a Function: Multidimensional Case
7.5.1 Neider and Mead's Downhill Simplex Method (Amoeba)
7.5.2 Example
7.5.3 Conjugate Gradient Methods
7.5.4 Examples
7.5.5 Quasi-Newton Methods
7.5.6 Examples
7.5.7 Line Minimization
7.5.8 Examples
7.6 Auxiliary Routines for Numerical Optimization
7.6.1 Gradient
7.6.2 Examples
7.6.3 Jacobian
7.6.4 Examples
7.6.5 Hessian
7.6.6 Example
7.6.7 Restriction of a Function to a Line
7.6.8 Example
7.6.9 Derivative of a Restricted function
7.6.10 Example