# ECONOMETRICS

[Italiano] AF: 76271 AR: 41960
 Official course title ECONOMETRIA Academic Year 2009/2010 Course code EM0004 University credits 6 Degree level Second Cycle Degree M.D.270 Educational sector code SECS-P/05 Semester/trimester 2° Periodo Course year 1 Where VENEZIA

# Contents

## Educational Goals

The course aims to handle some aspects of the econometric methods with respect to the regression models, both uni- and multi-equational like the models of simultaneous equations and the vector autoregressive models (VAR). Consequently, the purpose is to prepare the student to use the basic econometric tools for the measurement, interpretation and forecast of the economic and financial phenomena. The course is well equipped with econometric practice.

## Requirements

Elementary matrix algebra, Random variable theory, Statistical inference: Point estimation and testing.

## Contents

Introduction to linear regression
-specification of single equation multivariate model
-linearity and nonlinearity, endogenous and exogenous variables, observable and non observable variables, the unknown parameters
-ordinary least square: algebraic approach to calculate the unknown parameters
Reliability of estimated parameter values
-distribution of estimators
-properties of OLS estimators in finite sample
-hypothesis testing: testing linear constraints
-multicollinearity
Model validation: Analysis of Residuals
-autocorrelation and heteroscedasticity tests
-normality test
Modeling univariate time series
-static and dynamic models
-stochastic processes: moments, covariance and correlation functions
-strict and covariance stationarity
-MA, AR and ARMA processes
Specification strategies and selection of regression
-specification errors in the linear models: inclusion of irrelevant variables, exclusion of relevant variables
-general to specific strategy: motivations
Non stationary stochastic processes
-unit roots and integrated stochastic processes
-random walk process
-testing for unit roots: ADF test
Non stationary dynamic models
-dynamic analysis of ADL model: dynamic multiplier and long run coefficient
-Error Correction Mechanism (ECM) transformation
-some inferential aspects linked to unit roots: non standard statistic tests and spurious regression
ECM model for non stationary time series
-two-step Engle-Granger procedure
-cointegration test
-restriction for the Engle-Granger approach
Asymptotic analysis of regression models
-stochastic convergence
-the law of large numbers
-ergodic theorems: Khinchine, Tchebicheff, Markov
-asymptotic properties of OLS estimators
Bayesian statistical modeling
-sample, a priori, posterior and predictive distributions
-joint distribution in the sample and parameter space
-different a priori distributions and the convergence to the same posterior (learning by doing)
Relevant statistical principles
-some elements of decision theory
-not uniqueness of decision rules with respect to the parametric space and sample space
Identification problem
-Fisher information, Kullback-Leibler information
- information and identification
- weak and strong identification, global and local identification
Multivariate analysis of time series
- moments of a multivariate stochastic process
- stationarity and non stationarity in the multivariate stochastic processes
- multivariate ARMA processes
Econometric versus statistical models
- difference between econometric and statistical models
- simultaneity: parameters of interest and nuisance parameters
- structural, reduced and final forms of econometric models
- identification in the simultaneous equation models
Simultaneity: estimation methods for the structural parameters
- limited information methods
- full information methods
Reduction of econometric models
- conditional independence, reduction by conditioning
- sufficiency and ancillarity of a statistic and a parameter
- extended notion of ancillarity and conditional sufficient statistic
Exogeneity
- strict econometric exogeneity
- weak, strong and super exogeneity
- the use of the different exogeneity definitions for admissibility purposes
Prediction
- ex-post and ex-ante predictions
- static and dynamic predictions

References

Verbeek, M., 2004, A guide to modern econometrics, 2nd ed.,John Wiley & Sons Ltd.

1 Introduction 1
1.2 The Structure of this Book 3
1.3 Illustrations and Exercises 4

2 An Introduction to Linear Regression 7
2.1 Ordinary Least Squares as an Algebraic Tool 8
2.2 The Linear Regression Model 14
2.3 Small Sample Properties of the OLS Estimator 16
2.4 Goodness-of-fit 20
2.5 Hypothesis Testing 23
2.6 Asymptotic Properties of the OLS Estimator 32
2.7 Illustration: The Capital Asset Pricing Model 38
2.8 Multicollinearity 42
2.9 Prediction 44

3 Interpreting and Comparing Regression Models 51
3.1 Interpreting the Linear Model 51
3.2 Selecting the Set of Regressors 55
3.3 Misspecifying the Functional Form 62
3.4 Illustration: Explaining House Prices 65
3.5 Illustration: Explaining Individual Wages 68

4 Heteroskedasticity and Autocorrelation 79
4.1 Consequences for the OLS Estimator 79
4.2 Deriving an Alternative Estimator 81
4.3 Heteroskedasticity 82
4.4 Testing for Heteroskedasticity 90
4.5 Illustration: Explaining Labour Demand 92
4.6 Autocorrelation 97
4.7 Testing for First Order Autocorrelation 101
4.8 Illustration: The Demand for Ice Cream 103
4.9 Alternative Autocorrelation Patterns 106
4.10 What to do When you Find Autocorrelation? 108
4.11 Illustration: Risk Premia in Foreign Exchange Markets 112

5 Endogeneity, Instrumental Variables and GMM 121
5.1 A Review of the Properties of the OLS Estimator 122
5.2 Cases Where the OLS Estimator Cannot be Saved 125
5.3 The Instrumental Variables Estimator 131
5.4 Illustration: Estimating the Returns to Schooling 137
5.5 The Generalized Instrumental Variables Estimator 142

8 Univariate Time Series Models 255
8.1 Introduction 256
8.2 General ARMA Processes 261
8.3 Stationarity and Unit Roots 266
8.4 Testing for Unit Roots 268
8.5 Illustration: Long-run Purchasing Power Parity (Part 1) 276
8.6 Estimation of ARMA Models 279
8.7 Choosing a Model 281
8.8 Predicting with ARMA Models 288
8.9 Illustration: The Expectations Theory of the Term Structure 293
8.10 Autoregressive Conditional Heteroskedasticity 297

9 Multivariate Time Series Models 309
9.1 Dynamic Models with Stationary Variables 310
9.2 Models with Nonstationary Variables 313
9.3 Illustration: Long-run Purchasing Power Parity (Part 2) 319
9.4 Vector Autoregressive Models 321
9.5 Cointegration: the Multivariate Case 324
9.6 Illustration: Money Demand and Inflation 333
9.7 Concluding Remarks 339

A Vectors and Matrices 389
A.1 Terminology 389
A.2 Matrix Manipulations 390
A.3 Properties of Matrices and Vectors 391
A.4 Inverse Matrices 392
A.5 Idempotent Matrices 393
A.6 Eigenvalues and Eigenvectors 394
A.7 Differentiation 394
A.8 Some Least Squares Manipulations 395

B Statistical and Distribution Theory 397
B.1 Discrete Random Variables 397
B.2 Continuous Random Variables 398
B.3 Expectations and Moments 399
B.4 Multivariate Distributions 400
B.5 Conditional Distributions 401
B.6 The Normal Distribution 403
B.7 Related Distributions 405

## Assessment

The exam is shared in two parts:
- presentation of an econometric model (single equation multivariate regression) using economic or financial data;
- discussion on some topics of the course chosen by the student.

Conventional

Italian