Lecture Notes for Econometrics 2002

Paul soderlind1
June 2002 (some typos corrected later)

Contents
1 Introduction 5
1.1 Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Maximum Likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 The Distribution of ˆ . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Diagnostic Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5 Testing Hypotheses about ˆ . . . . . . . . . . . . . . . . . . . . . . 9
A Practical Matters 10
B A CLT in Action 12
2 Univariate Time Series Analysis 16
2.1 Theoretical Background to Time Series Processes . . . . . . . . . . . 16
2.2 Estimation of Autocovariances . . . . . . . . . . . . . . . . . . . . . 17
2.3 White Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4 Moving Average . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.5 Autoregression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.6 ARMA Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.7 Non-stationary Processes . . . . . . . . . . . . . . . . . . . . . . . . 30
3 The Distribution of a Sample Average 38
3.1 Variance of a Sample Average . . . . . . . . . . . . . . . . . . . . . 38
3.2 The Newey-West Estimator . . . . . . . . . . . . . . . . . . . . . . . 42
3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4 Least Squares 45
4.1 Definition of the LS Estimator . . . . . . . . . . . . . . . . . . . . . 45

4.2 LS and R2  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.3 Finite Sample Properties of LS . . . . . . . . . . . . . . . . . . . . . 49
4.4 Consistency of LS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.5 Asymptotic Normality of LS . . . . . . . . . . . . . . . . . . . . . . 52
4.6 Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.7 Diagnostic Tests of Autocorrelation, Heteroskedasticity, and Normality 59
5 Instrumental Variable Method 65
5.1 Consistency of Least Squares or Not? . . . . . . . . . . . . . . . . . 65
5.2 Reason 1 for IV: Measurement Errors . . . . . . . . . . . . . . . . . 65
5.3 Reason 2 for IV: Simultaneous Equations Bias (and Inconsistency) . . 67
5.4 Definition of the IV Estimator—Consistency of IV . . . . . . . . . . 70
5.5 Hausman’s Specification Test . . . . . . . . . . . . . . . . . . . . . 76
5.6 Tests of Overidentifying Restrictions in 2SLS . . . . . . . . . . . . 77
6 Simulating the Finite Sample Properties 79
6.1 Monte Carlo Simulations in the Simplest Case . . . . . . . . . . . . . 79
6.2 Monte Carlo Simulations in More Complicated Cases . . . . . . . . 81
6.3 Bootstrapping in the Simplest Case . . . . . . . . . . . . . . . . . . . 83
6.4 Bootstrapping in More Complicated Cases . . . . . . . . . . . . . . 83
7 GMM 87
7.1 Method of Moments . . . . . . . . . . . . . . . . . . . . . . . . . . 87
7.2 Generalized Method of Moments . . . . . . . . . . . . . . . . . . . . 88
7.3 Moment Conditions in GMM . . . . . . . . . . . . . . . . . . . . . . 88
7.4 The Optimization Problem in GMM . . . . . . . . . . . . . . . . . . 91
7.5 Asymptotic Properties of GMM . . . . . . . . . . . . . . . . . . . . 95
7.6 Summary of GMM . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.7 Efficient GMM and Its Feasible Implementation . . . . . . . . . . . . 101
7.8 Testing in GMM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
7.9 GMM with Sub-Optimal Weighting Matrix . . . . . . . . . . . . . . 104
7.10 GMM without a Loss Function . . . . . . . . . . . . . . . . . . . . 106
7.11 Simulated Moments Estimator . . . . . . . . . . . . . . . . . . . . . 106
2
8 Examples and Applications of GMM 109
8.1 GMM and Classical Econometrics: Examples . . . . . . . . . . . . . 109
8.2 Identification of Systems of Simultaneous Equations . . . . . . . . . 113
8.3 Testing for Autocorrelation . . . . . . . . . . . . . . . . . . . . . . . 116
8.4 Estimating and Testing a Normal Distribution . . . . . . . . . . . . . 120
8.5 Testing the Implications of an RBC Model . . . . . . . . . . . . . . . 123
8.6 IV on a System of Equations . . . . . . . . . . . . . . . . . . . . . 125
11 Vector Autoregression (VAR) 127
11.1 Canonical Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
11.2 Moving Average Form and Stability . . . . . . . . . . . . . . . . . . 128
11.3 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
11.4 Granger Causality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
11.5 Forecasts Forecast Error Variance . . . . . . . . . . . . . . . . . . . 132
11.6 Forecast Error Variance Decompositions . . . . . . . . . . . . . . . 133
11.7 Structural VARs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
11.8 Cointegration, Common Trends, and Identification via Long-Run Restrictions144
12 Kalman filter 151
12.1 Conditional Expectations in a Multivariate Normal Distribution . . . . 151
12.2 Kalman Recursions . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
13 Outliers and Robust Estimators 158
13.1 Influential Observations and Standardized Residuals . . . . . . . . . . 158
13.2 Recursive Residuals . . . . . . . . . . . . . . . . . . . . . . . . . . 159
13.3 Robust Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
13.4 Multicollinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
14 Generalized Least Squares 164
14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
14.2 GLS as Maximum Likelihood . . . . . . . . . . . . . . . . . . . . . 165
14.3 GLS as a Transformed LS . . . . . . . . . . . . . . . . . . . . . . . 168
14.4 Feasible GLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

21 Some Statistics 170
21.1 Distributions and Moment Generating Functions . . . . . . . . . . . . 170
21.2 Joint and Conditional Distributions and Moments . . . . . . . . . . . 171
21.3 Convergence in Probability, Mean Square, and Distribution . . . . . . 174
21.4 Laws of Large Numbers and Central Limit Theorems . . . . . . . . . 176
21.5 Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
21.6 Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
21.7 Special Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 178
21.8 Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
22 Some Facts about Matrices 187
22.1 Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
22.2 Vector Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
22.3 Systems of Linear Equations and Matrix Inverses . . . . . . . . . . . 187
22.4 Complex matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
22.5 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . . . 190
22.6 Special Forms of Matrices . . . . . . . . . . . . . . . . . . . . . . . 191
22.7 Matrix Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . 193
22.8 Matrix Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
22.9 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
0 Reading List 204
0.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
0.2 Time Series Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 204
0.3 Distribution of Sample Averages . . . . . . . . . . . . . . . . . . . . 204
0.4 Asymptotic Properties of LS . . . . . . . . . . . . . . . . . . . . . . 205
0.5 Instrumental Variable Method . . . . . . . . . . . . . . . . . . . . . 205
0.6 Simulating the Finite Sample Properties . . . . . . . . . . . . . . . . 205
0.7 GMM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

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