1 INTRODUCTION 1
2 STOCHASTIC PROCESSES 5
2.1 Review of Probability Theory . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Conditional Expectations . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4 Stationary Stochastic Processes . . . . . . . . . . . . . . . . . . . . . 12
2.5 Conditional Heteroskedasticity . . . . . . . . . . . . . . . . . . . . . . 16
2.6 Martingales and Random Walks . . . . . . . . . . . . . . . . . . . . . 18
2.A A Review of Measure Theory . . . . . . . . . . . . . . . . . . . . . . 19
2.B Convergence in Probability . . . . . . . . . . . . . . . . . . . . . . . . 29
2.B.1 Convergence in Distribution . . . . . . . . . . . . . . . . . . . 30
2.B.2 Propositions 2.2 and 2.3 for In¯nite Numbers of R.V.'s (Incom-
plete) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3 FORECASTING 33
3.1 Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1.1 De¯nitions and Properties of Projections . . . . . . . . . . . . 33
3.1.2 Linear Projections and Conditional Expectations . . . . . . . 35
3.2 Some Applications of Conditional Expectations and Projections . . . 37
3.2.1 Volatility Tests . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2.2 Parameterizing Expectations . . . . . . . . . . . . . . . . . . . 39
3.2.3 Noise Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.A Introduction to Hilbert Space . . . . . . . . . . . . . . . . . . . . . . 42
3.A.1 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.A.2 Hilbert Space . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4 ARMA AND VECTOR AUTOREGRESSION REPRESENTATIONS 51
4.1 Autocorrelation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2 The Lag Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.3 Moving Average Representation . . . . . . . . . . . . . . . . . . . . . 53
4.4 Autoregression Representation . . . . . . . . . . . . . . . . . . . . . . 55
iii
iv CONTENTS
4.4.1 Autoregression of Order One . . . . . . . . . . . . . . . . . . . 55
4.4.2 The p-th Order Autoregression . . . . . . . . . . . . . . . . . 57
4.5 ARMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.6 The Wold Representation . . . . . . . . . . . . . . . . . . . . . . . . 58
4.7 Fundamental Innovations . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.8 The Spectral Density . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.A Di®erence Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5 STOCHASTIC REGRESSORS IN LINEAR MODELS 67
5.1 The Conditional Gauss Markov Theorem . . . . . . . . . . . . . . . . 68
5.2 Unconditional Distributions of Test Statistics . . . . . . . . . . . . . 73
5.3 The Law of Large Numbers . . . . . . . . . . . . . . . . . . . . . . . 75
5.4 Convergence in Distribution and Central Limit Theorem . . . . . . . 76
5.5 Consistency and Asymptotic Distributions of OLS Estimators . . . . 80
5.6 Consistency and Asymptotic Distributions of IV Estimators . . . . . 82
5.7 Nonlinear Functions of Estimators . . . . . . . . . . . . . . . . . . . . 83
5.8 Remarks on Asymptotic Theory . . . . . . . . . . . . . . . . . . . . . 83
5.A Monte Carlo Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.A.1 Random Number Generators . . . . . . . . . . . . . . . . . . . 85
5.A.2 Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.A.3 Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.A.4 A Pitfall in Monte Carlo Simulations . . . . . . . . . . . . . . 90
5.A.5 An Example Program . . . . . . . . . . . . . . . . . . . . . . 92
6 ESTIMATION OF THE LONG-RUN COVARIANCE MATRIX 101
6.1 Serially Uncorrelated Variables . . . . . . . . . . . . . . . . . . . . . 102
6.2 Serially Correlated Variables . . . . . . . . . . . . . . . . . . . . . . . 103
6.2.1 Unknown Order of Serial Correlation . . . . . . . . . . . . . . 103
6.2.2 Known Order of Serial Correlation . . . . . . . . . . . . . . . 108
7 TESTING LINEAR FORECASTING MODELS 112
7.1 Forward Exchange Rates . . . . . . . . . . . . . . . . . . . . . . . . . 113
7.2 The Euler Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
7.3 The Martingale Model of Consumption . . . . . . . . . . . . . . . . . 118
7.4 The Linearized Euler Equation . . . . . . . . . . . . . . . . . . . . . 119
7.5 Optimal Taxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
8 VECTOR AUTOREGRESSION TECHNIQUES 124
8.1 OLS Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
8.2 Granger Causality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
8.3 The Impulse Response Function . . . . . . . . . . . . . . . . . . . . . 129
8.4 Forecast error decomposition . . . . . . . . . . . . . . . . . . . . . . . 132
CONTENTS v
8.5 Structural VAR Models . . . . . . . . . . . . . . . . . . . . . . . . . . 133
8.6 Identi¯cation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
8.6.1 Short-Run Restrictions for Structural VAR . . . . . . . . . . . 136
8.6.2 Identi¯cation of block recursive systems . . . . . . . . . . . . . 138
8.6.3 Two-step ML estimation . . . . . . . . . . . . . . . . . . . . . 139
9 GENERALIZED METHOD OF MOMENTS 143
9.1 Asymptotic Properties of GMM Estimators . . . . . . . . . . . . . . . 143
9.1.1 Moment Restriction and GMM Estimators . . . . . . . . . . . 143
9.1.2 Asymptotic Distributions of GMM Estimators . . . . . . . . . 144
9.1.3 Optimal Choice of the Distance Matrix . . . . . . . . . . . . . 145
9.1.4 A Chi-Square Test for the Overidentifying Restrictions . . . . 146
9.2 Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
9.2.1 Ordinary Least Squares . . . . . . . . . . . . . . . . . . . . . 146
9.2.2 Linear Instrumental Variables Regressions . . . . . . . . . . . 147
9.2.3 Nonlinear Instrumental Variables Estimation . . . . . . . . . . 148
9.2.4 Linear GMM estimator . . . . . . . . . . . . . . . . . . . . . . 148
9.3 Important Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 149
9.3.1 Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
9.3.2 Identi¯cation . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
9.4 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
9.4.1 Sequential Estimation . . . . . . . . . . . . . . . . . . . . . . 151
9.4.2 GMM with Deterministic Trends . . . . . . . . . . . . . . . . 153
9.4.3 Minimum Distance Estimation . . . . . . . . . . . . . . . . . . 153
9.5 Hypothesis Testing and Speci¯cation Tests . . . . . . . . . . . . . . . 154
9.6 Numerical Optimization . . . . . . . . . . . . . . . . . . . . . . . . . 156
9.7 The Optimal Choice of Instrumental Variables . . . . . . . . . . . . . 158
9.8 Small Sample Properties . . . . . . . . . . . . . . . . . . . . . . . . . 158
9.A Asymptotic Theory for GMM . . . . . . . . . . . . . . . . . . . . . . 161
9.A.1 Asymptotic Properties of Extremum Estimators . . . . . . . . 162
9.A.2 Consistency of GMM Estimators . . . . . . . . . . . . . . . . 164
9.A.3 A Su±cient Condition for the Almost Sure Uniform Convergence165
9.A.4 Asymptotic Distributions of GMM Estimators . . . . . . . . . 170
9.B A Procedure for Hansen's J Test (GMM.EXP) . . . . . . . . . . . . . 174
10 EMPIRICAL APPLICATIONS OF GMM 181
10.1 Euler Equation Approach . . . . . . . . . . . . . . . . . . . . . . . . 181
10.2 Alternative Measures of IMRS . . . . . . . . . . . . . . . . . . . . . . 183
10.3 Habit Formation and Durability . . . . . . . . . . . . . . . . . . . . . 185
10.4 State-Nonseparable Preferences . . . . . . . . . . . . . . . . . . . . . 187
10.5 Time Aggregation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
10.6 Multiple-Goods Models . . . . . . . . . . . . . . . . . . . . . . . . . . 189
vi CONTENTS
10.7 Seasonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
10.8 Monetary Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
10.9 Calculating Standard Errors for Estimates of Standard Deviation, Cor-
relation, and Autocorrelation . . . . . . . . . . . . . . . . . . . . . . 193
10.10Real Business Cycle Models and GMM Estimation . . . . . . . . . . 194
10.11GMM and an ARCH Process . . . . . . . . . . . . . . . . . . . . . . 199
10.12Other Empirical Applications . . . . . . . . . . . . . . . . . . . . . . 202
11 UNIT ROOT NONSTATIONARY PROCESSES 210
11.1 De¯nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
11.2 Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
11.3 Tests for the Null of Di®erence Stationarity . . . . . . . . . . . . . . 214
11.3.1 Dickey-Fuller Tests . . . . . . . . . . . . . . . . . . . . . . . . 214
11.3.2 Said-Dickey Test . . . . . . . . . . . . . . . . . . . . . . . . . 216
11.3.3 Phillips-Perron Tests . . . . . . . . . . . . . . . . . . . . . . . 218
11.3.4 Park's J Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
11.4 Tests for the Null of Stationarity . . . . . . . . . . . . . . . . . . . . 220
11.5 Near Observational Equivalence . . . . . . . . . . . . . . . . . . . . . 221
11.6 Asymptotics for unit root tests . . . . . . . . . . . . . . . . . . . . . 222
11.6.1 DF test with serially uncorrelated disturbances . . . . . . . . 222
11.6.2 ADF test with serially correlated disturbances . . . . . . . . . 226
11.6.3 Phillips-Perron test . . . . . . . . . . . . . . . . . . . . . . . . 232
11.A Asymptotic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
11.A.1 Functional Central Limit Theorem . . . . . . . . . . . . . . . 239
11.B Procedures for Unit Root Tests . . . . . . . . . . . . . . . . . . . . . 239
11.B.1 Said-Dickey Test (ADF.EXP) . . . . . . . . . . . . . . . . . . 239
11.B.2 Park's J Test (JPQ.EXP) . . . . . . . . . . . . . . . . . . . . 240
11.B.3 Park's G Test (GPQ.EXP) . . . . . . . . . . . . . . . . . . . . 241
12 COINTEGRATING AND SPURIOUS REGRESSIONS 245
12.1 De¯nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
12.2 Exact Finite Sample Properties of Regression Estimators . . . . . . . 248
12.2.1 Spurious Regressions . . . . . . . . . . . . . . . . . . . . . . . 249
12.2.2 Cointegrating Regressions . . . . . . . . . . . . . . . . . . . . 253
12.3 Large Sample Properties . . . . . . . . . . . . . . . . . . . . . . . . . 254
12.3.1 Canonical Cointegrating Regression . . . . . . . . . . . . . . . 255
12.3.2 Estimation of Long-Run Covariance Parameters . . . . . . . . 257
12.4 Tests for the Null Hypothesis of No Cointegration . . . . . . . . . . . 259
12.5 Tests for the Null Hypothesis of Cointegration . . . . . . . . . . . . . 260
12.6 Generalized Method of Moments and Unit Roots . . . . . . . . . . . 261
12.A Procedures for Cointegration Tests . . . . . . . . . . . . . . . . . . . 263
12.A.1 Park's CCR and H Test (CCR.EXP) . . . . . . . . . . . . . . 263
CONTENTS vii
12.A.2 Park's I Test (IPQ.EXP) . . . . . . . . . . . . . . . . . . . . . 264
13 ECONOMIC MODELS AND COINTEGRATING REGRESSIONS247
13.1 The Permanent Income Hypothesis of Consumption . . . . . . . . . . 248
13.2 Present Value Models of Asset Prices . . . . . . . . . . . . . . . . . . 251
13.3 Applications to Money Demand Functions . . . . . . . . . . . . . . . 253
13.4 The Cointegration Approach to Estimating Preference Parameters . . 253
13.4.1 The Time Separable Addilog Utility Function . . . . . . . . . 255
13.4.2 The Time Nonseparable Addilog Utility Function . . . . . . . 259
13.4.3 Engel's Law and Cointegration . . . . . . . . . . . . . . . . . 264
13.5 The Cointegration-Euler Equation Approach . . . . . . . . . . . . . . 267
13.5.1 The Economy . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
13.5.2 The 2-Step Estimation Method . . . . . . . . . . . . . . . . . 274
13.5.3 Measuring Intertemporal Substitution: The Role of Durable
Goods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
13.6 Purchasing Power Parity . . . . . . . . . . . . . . . . . . . . . . . . . 276
14 ESTIMATION AND TESTING OF LINEAR RATIONAL EXPEC-
TATIONS MODELS 284
14.1 The Nonlinear Restrictions . . . . . . . . . . . . . . . . . . . . . . . . 284
14.1.1 Stationary dt . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
14.1.2 Di®erence Stationary dt . . . . . . . . . . . . . . . . . . . . . 287
14.2 Econometric Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
14.2.1 Stationary dt . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
14.2.2 Di®erence Stationary dt . . . . . . . . . . . . . . . . . . . . . 289
15 VECTOR AUTOREGRESSIONS WITH UNIT ROOT NONSTA-
TIONARY PROCESSES 296
15.1 Identi¯cation on Structural VAR Models . . . . . . . . . . . . . . . . 297
15.1.1 Long-Run Restrictions for Structural VAR Models . . . . . . . 297
15.1.2 Short-run and Long-Run Restrictions for Structural VAR Models298
15.2 Vector Error Correction Model . . . . . . . . . . . . . . . . . . . . . . 301
15.2.1 The model and Long-run Restrictions . . . . . . . . . . . . . . 301
15.2.2 Identi¯cation of Permanent Shocks . . . . . . . . . . . . . . . 303
15.2.3 Impulse Response Functions . . . . . . . . . . . . . . . . . . . 305
15.2.4 Forecast-Error Variance Decomposition . . . . . . . . . . . . . 307
15.2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
15.3 Structural Vector Error Correction Models . . . . . . . . . . . . . . . 309
15.4 An Exchange Rate Model with Sticky Prices . . . . . . . . . . . . . . 311
15.5 The Instrumental Variables Methods . . . . . . . . . . . . . . . . . . 318
15.6 Tests for the Number of Cointegrating Vectors . . . . . . . . . . . . . 322
15.7 How Should an Estimation Method be Chosen? . . . . . . . . . . . . 324
viii CONTENTS
15.7.1 Are Short-Run Dynamics of Interest? . . . . . . . . . . . . . . 325
15.7.2 The Number of the Cointegrating Vectors . . . . . . . . . . . 325
15.7.3 Small Sample Properties . . . . . . . . . . . . . . . . . . . . . 326
15.A Estimation of the Model . . . . . . . . . . . . . . . . . . . . . . . . . 327
15.B Monte Carlo Integration . . . . . . . . . . . . . . . . . . . . . . . . . 332
15.C Johansen's Maximum Likelihood Estimation and Cointegration Rank
Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
16 PANEL AND CROSS-SECTIONAL DATA 343
16.1 Generalized Method of Moments . . . . . . . . . . . . . . . . . . . . . 343
16.2 Tests of Risk Sharing . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
16.3 Decreasing Relative Risk Aversion and Risk Sharing . . . . . . . . . . 347
16.4 Euler Equation Approach . . . . . . . . . . . . . . . . . . . . . . . . 349
16.5 Panel Unit Root Tests . . . . . . . . . . . . . . . . . . . . . . . . . . 350
16.6 Cointegration and Panel Data . . . . . . . . . . . . . . . . . . . . . . 352