A Guide to Econometrics

Contents ... 6
Preface ... 11
Dedication ... 13
1 Introduction ... 14
	1.1  What is Econometrics? ... 14
	1.2  The Disturbance Term ... 15
	1.3  Estimates  and Estimators ... 17
	1.4  Good and Preferred Estimators ... 18
	General Notes ... 19
		1.2  The Disturbance Term ... 22
		1.3  Estimates and Estimators ... 22
		1.4  Good and Prefe rred Estimators ... 22
2 Criteria for Estimators ... 24
	2.1  Introduction ... 24
	2.2 Computational Cost ... 24
	2.3 Least Squares ... 25
	2.4 Highest  R2 ... 26
	2.5 Unbiasedness ... 27
	2.6 Efficiency ... 29
	2.7 Mean Square  Error ... 30
	2.8 Asymptotic Properties ... 31
	2.9 Maximum Likelihood ... 34
	2.10 Monte Carlo Studies ... 35
	2.11 Adding  Up ... 38
	General Notes ... 39
		2.2 Computational  Cost ... 39
		2.4 Highest R2 ... 39
		2.3 Least Squares ... 39
		2.5 Unbiasedness ... 41
3 The Classical Liner Regression Model ... 53
	3.1 Textbooks as Catalogs ... 53
	3.2 The Five Assumptions ... 54
	3.3 The OLS Estimator in the CLR Model ... 56
	General Notes ... 57
		3.1 Textbooks as Catalogs ... 57
		3.3 The OLS Estimator in the CLR Model ... 57
		3.2 The Five Assumptions ... 60
		3.3 The OLS Estimator in the CLR Model ... 61
4 Interval Estimation and Hypothesis Testing ... 64
	4.1 Introduction ... 64
	4.2 Testing a Single Hypothesis: The t Test ... 64
	4.3 Testing a Joint Hypothesis: the  F Test ... 65
	4.4 Interval Estimation for a Parameter Vector ... 67
	4.5 LR,W, and LM Statistics ... 69
	4.6 Bootstrapping ... 71
	General Notes ... 72
		4.1 Introduction ... 72
		4.2 Testing a Single Hypothesis: The t Test ... 75
		4.3 Testing a Joint Hypothesis: The F Test ... 75
		4.4 Interval Estimation fo r a Parameter Vector ... 77
		4.5 LR,W, and LM Statistics ... 77
		4.6 Bootstrapping ... 78
	Technical Notes ... 79
		4.1 Introduction ... 80
		4.3 Testing a Joint Hypothesis: The F Test ... 81
		4.5 LR, W, and LM Statistics ... 81
		4.6 Bootstrapping ... 82
5 Specification ... 84
	5.1 Introduction ... 84
	5.2 Three Methodologies ... 85
		5.2.1 Average Economic Regression (AER) ... 85
		5.2.2 Test,Test,Test (TTT) ... 86
		5.2.3 Fragility Analysis ... 86
	5.3 General Principles for Specification ... 88
	5.4 Misspecification Tests/D iagnostics ... 89
	5.5 R^2  Again ... 92
	General Notes ... 94
		5.1 Introduction ... 94
		5.2 Three Methodologies ... 94
		5.3 General Principles for Specification ... 98
		5.4 Misspecifica tion Tests/Diagnostics ... 100
		5.5 R^2 Again ... 102
	Technical Notes ... 102
		5.1 Introduction ... 102
		5.2 Three Methodologies ... 102
		5.4 Misspecification Tests/D iagnostics ... 103
6 Violating Assumption 1: Wrong Regressors, Nonliearities, Parameter Inconsistency ... 106
	6.1 Introduction ... 106
	6.2 Incorrect Set of Independent Variables ... 106
	6.3 Nonlinearity ... 108
		6.3.1 Transformations ... 108
		6.3.2 Computer-Assisted Numerical Techniques ... 109
	6.4 Changing Parameter Values ... 110
		6.4.1 Switching Regimes ... 111
		6.4.2 Parameters Determined by Other Variables ... 111
		6.4.3 Random Coefficients ... 112
	General Notes ... 113
		6.1 Introduction ... 113
		6.2 Incorrect Set of Independent Variables ... 113
		6.3 Nonlinearity ... 115
		6.4 Changing Parameter Values ... 118
	Technical Notes ... 119
		6.3 Nonlinearity ... 119
		6.4 Changing Parameter Values ... 121
7 Violating Assumption Two: Nonzero Expected Disturbance ... 122
8 Violating Assumption Three: Nonspherical Disturbances ... 125
	8.1 Introduction ... 125
	8.2 Consequences of Violation ... 126
	8.3 Heteroskedasticity ... 128
		8.3.1 The Eyeball Test ... 129
		8.3.2 The Goldfeld-Quandt Test ... 129
		8.3.3 The Breusch-Pagan Test ... 129
		8.3.4 The White Test ... 130
	8.4 Autocorrelated Disturbances ... 131
		8.4.1 Cochrane-Orcutt Iterative Least Squares ... 134
		8.4.2 Durbin\'s Two-Stage Method ... 134
		8.4.3 Hildreth-Lu Search Procedure ... 134
		8.4.4 Maximum Likelihood ... 134
	8.5 Generalized Method of Moments ... 135
	General Notes ... 136
		8.1 Introduction ... 136
		8.2 Consequences of Violation ... 136
		8.3 Heteroskedasticity ... 137
		8.4 Autocorrelated Disturbances ... 139
	Technical Notes ... 142
		8.1 Introduction ... 142
		8.2 Consequences of Violation ... 142
		8.3 Heteroskedasticity ... 144
		8.4 Autocorrelated Disturbances ... 145
		8.5 Generalized Method of Moments ... 147
9 Violating Assumption Four: Instrumental Variable Estimation ... 150
	9.1 Introduction ... 150
	9.2 The IV Estimator ... 154
	9.3 IV Issues ... 157
		9.3.1 How can we test if errors are correlated with regressors? ... 157
		9.3.2 How can we test if an instrument is uncorrelated with the error? ... 157
		9.3.3 How can we test if an instrument\'s correlation with the troublesome variable is strong enough? ... 158
		9.3.4 How should we interpret IV estimates? ... 158
	General Notes ... 159
		9.1 Introduction ... 159
		9.2 The IV Estimator ... 160
		9.3 IV Issues ... 162
	Technical Notes ... 164
		9.1 Introduction ... 164
		9.2 IV Estimation ... 164
		9.3 IV Issues ... 166
10 Violating Assumpton Four: Measurement Errors and Autoregression ... 170
	10.1 Errors in Variables ... 170
		10.1.1 Weighted Regression ... 171
		10.1.2 Instrumental Variables ... 172
		10.1.3 Linear Structural Relations ... 173
	10.2 Autoregression ... 173
	General Notes ... 176
		10.1 Errors in Variables ... 176
		10.2 Autoregression ... 179
	Technical Notes ... 180
		10.1 Errors in Variables ... 180
		10.2 Autoregression ... 181
11 Violating Assumption Four: Simultaneous Equations ... 184
	11.1 Introduction ... 184
	11.2 Identification ... 186
	11.3 Single-Equation Methods ... 189
		11.3.1 Ordinary Least Squares ... 190
		11.3.2 Indirect Least Squares ... 190
		11.3.3 The Instrumental Variable (IV) Technique ... 191
		11.3.4 Two-Stage Least Squares (2SLS) ... 191
		11.3.5 Limited Info rmation, Maximum Likelihood (LI/ML) ... 192
	11.4 Systems Methods ... 192
		11.4.1Three-Stage Least Squares (3SLS) ... 193
		11.4.2 Full Information,Maximum Likelihood (FUML) ... 194
	General Notes ... 194
		11.l Introduction ... 194
		11.2 Identification ... 196
		11.3 Single-Equation Methods ... 197
		11.4 Systems Methods ... 198
	Technical Notes ... 199
		11.1 Introduction ... 199
		11.2 Identification ... 200
		11.3 Single-Equation Methods ... 201
		11.4 Systems Methods ... 203
12 Violating Assumtion Five: Multicollinearity ... 205
	12.1 Introduction ... 205
	12.2 Consequences ... 206
	12.3 Detecting Multicollinearity ... 207
	12.4 What To Do ... 209
		12.4.1 Do Nothing ... 209
		12.4.2 Incorporate Additional Information ... 209
	General Notes ... 211
		12.2 Consequences ... 211
		12.3 Detecting Multicollinearity ... 212
		12.4 What to Do ... 212
	Technical Notes ... 215
13 Incorporating Extraneous Information ... 216
	13.1 Introduction ... 216
	13.2 Exact Restrictions ... 216
	13.3 Stochastic Restrictions ... 217
	13.4 Pre-Test Estimators ... 217
	13.5 Extraneous Information and MSE ... 219
	General Notes ... 220
		13.1 Introduction ... 220
		13.2 Exact Restrictions ... 221
		13.3 Stochastic Restrictions ... 222
		13.4 Pre-test Estimators ... 223
		13.5 Extraneous Information and MSE ... 223
	Technical Notes ... 224
		13.3 Stochastic Restrictions ... 224
		13.5 Extraneous Information and MSE ... 225
14 The Bayesian Approach ... 226
	14.1 Introduction ... 226
	14.2 What is a Bayesian Analysis? ... 226
	14.3 Advantages of the Bayesian Approach ... 229
	14.4 Overcoming Practitioners\' Complaints ... 230
		14.4.1 Choosing a Prior ... 230
		14.4.2 Finding and Using the Posterior ... 232
		14.4.3 Convincing Others ... 232
	General Notes ... 233
		14.1 Introduction ... 233
		14.2 What is a Bayesian Analysis? ... 233
		14.3 Advantages of the Bayesian Approach ... 236
		14.4 Overcoming Practitioners\' Complaints ... 237
	Technical Notes ... 239
		14.1 Introduction ... 239
		14.2 What is a Bayesian Analysis? ... 239
		14.3 Advantages of the Bayesian Approach ... 241
		14.4 Overcoming Practitioners\' Complaints ... 243
15 Dummy Variables ... 245
	15.1 Introduction ... 245
	15.2 Interpretation ... 246
	15.3 Adding Another Qualitative Variable ... 247
	15.4 Interacting with Quantitative Variables ... 248
	15.5 Observation-Specific Dummies ... 249
	General Notes ... 250
		15.1 Introduction ... 250
		1 5.4 Interacting with Quantitative Variables ... 251
		15.5 Observation-Specific Dummies ... 252
	Technical Notes ... 253
16 Qualitative Dependent Variables ... 254
	16.1 Dichotomous Dependent Variables ... 254
	16.2 Polychotomous Dependent Variables ... 257
	16.3 Ordered Logit/Probit ... 258
	16.4 Count Data ... 259
	General Notes ... 259
		16.1 Dichotomous Dependent Variables ... 259
		16.3 Ordered Logit/Probit ... 266
		16.4 Count Data ... 266
	Technical Notes ... 267
		16.1 Dichotomous Dependent Variables ... 267
		16.2 Polychotomous Dependent Variables ... 269
		1 6.3 Ordered Logit/Probit ... 271
		16.4 Count Data ... 272
17 Limited Dependent Variables ... 275
	17.1  Introduction ... 275
	17.2 The Tobit Model ... 276
	17.3 Sample Selection ... 278
	17.4 Duration Models ... 280
	General Notes ... 282
		17.1 Introduction ... 282
		17.2 The Tobit Model ... 282
		17.3 Sample Selection ... 283
		17.4 Duration Models ... 286
	Technical Notes ... 286
		17.1 Introduction ... 286
		17.2 The Tobit Model ... 287
		17.3 Sample Selection ... 288
		17.4 Duration Models ... 289
18 Panel Data ... 294
	18.1 Introduction ... 294
	18.2 Allowing for Different Intercepts ... 295
	18.3 Fixed Versus Random Effects ... 297
	18.4 Short Run Versus Long Run ... 299
	18.5 Long, Narrow Panels ... 300
	General Notes ... 301
		18.1 Introduction ... 301
		18.2 Allowing for Different Intercepts ... 302
		18.3 Fixed Versus Random Effects ... 303
		18.4 Short Run Versus Long Run ... 304
		18.5 Long,Narrow Panels ... 305
	Technical Notes ... 305
		18.2 Allowing for Different Intercepts ... 305
		18.3 Fixed versus Random Effects ... 305
		18.5 Long, Narrow Panels ... 308
19 Time Series Econometrics ... 309
	19.1 Introduction ... 309
	19.2 ARIMA Models ... 310
	19.3 VARs ... 311
	19.4 Error Correction Models ... 312
	19.5 Testing for Unit Roots ... 314
	19.6 Cointegration ... 315
	General Notes ... 317
		19.1 Introduction ... 317
		19.2 ARIMA Models ... 317
		19.3 VARs ... 318
		19.5 Testing fo r Unit Roots ... 320
		19.6 Cointegration ... 322
	Technical Notes ... 327
		19.1 Introduction ... 327
		19.2 ARIMA Models ... 327
		19.3 VARs ... 333
		19.4 Error Correction Models ... 335
		19.5 Testing for Unit Roots ... 336
		19.6 Cointegration ... 340
20 Forecasting ... 344
	20.1 Introduction ... 344
	20.2 Causal Forecasting/Econometric Models ... 345
	20.3 Time Series Analysis ... 346
	20.4 Forecasting Accuracy ... 347
	General Notes ... 348
		20.1 Introduction ... 348
		20.2 Causal Forecasting/Econometric  Models ... 350
		20.3 Time Series Analysis ... 352
		20.4 Forecasting Accuracy ... 353
	Technical Notes ... 355
		20.l Introduction ... 355
		20.2 Causal Forecasting/E conometric Models ... 356
		20.4 Forecasting Accuracy ... 356
21 Robust Estimation ... 358
	21.1 Introduction ... 358
	21.2 Outliers and Influential Observations ... 359
	21.3 Guarding Against Influential Observations ... 360
	21.4 Artificial Neural Networks ... 362
	21.5 Nonparametric Estimation ... 363
	General Notes ... 365
		21.1 Introduction ... 365
		21.2 Outliers and Influential Observations ... 365
		21.3 Guarding against Influential Observations ... 366
		21.4 Artificial Neural Networks ... 367
		21.5 Nonparametric Estimation ... 368
	Technical Notes ... 369
		21.3 Guarding against Influential Observation ... 369
		21.4 Artificial Neural Networks ... 370
		21.5 Nonparametric Estimation ... 370
22 Applied Econometrics ... 374
	22.1 Introduction ... 374
	22.2 The Ten Commandments of Applied Econometrics ... 375
	22.3 Getting the Wrong Sign ... 381
	22.4 Common Mistakes ... 384
	22.5 What do Practitioners Need to Know? ... 386
	General Notes ... 387
		22.1 Introduction ... 387
		22.2 The Ten Commandments of Applied Econometrics ... 388
		22.3 Getting the Wrong Sign ... 391
		22.4 Common Mistakes ... 392
		22.5 What do Practitioners Need to Know? ... 392
	Technical Notes ... 396
		22.2 The Ten Commandments of Applied Econometrics ... 396
		22.5 What do Practitioners Need to Know? ... 396
23 Computational Considerations ... 398
	23.1 Introduction ... 398
	23.2 Optimizing via a Computer Search ... 399
	23.3 Estimating Integrals via Simulation ... 401
	23.4 Drawing Observations from Awkward Distributions ... 403
	General Notes ... 405
		23.1 Introduction ... 405
		23.2 Optimizing via a Computer Search ... 405
		23.3 Estimating Integrals via Simulation ... 408
		23.4 Drawing Observations from Awkward Distributions ... 409
	Technical Notes ... 410
		23.2 Optimizing via a Computer Search ... 410
		23.3 Estimating Integrals via Simulation ... 413
		23.4 Drawing Observations from Awkward Distributions ... 414
Appendix A Sampling Distributions ... 416
	I An Example ... 416
	2 Implications for Studying Econometrics ... 417
	3 Calculating Sampling Distributions ... 418
Appendix B All About Variance ... 420
	1 Definition ... 420
	2 Estimation ... 421
	3 Well-Known Formulas ... 421
	4 More-General Formulas ... 421
	5 Examples of the More-General Formulas ... 421
	6 Cramer-Rao Lower Bound ... 423
Appendix C A Primer on Asymptotics ... 425
	1 Convergence in Probability ... 425
	2 Convergence in Distribution ... 427
	3 Asymptotic Distributions ... 427
	Notes ... 428
Appendix D Exercises ... 430
Appendix E Answers to Even-Numbered Questions ... 492
Glossary ... 516
Bibliography ... 524
Name Index ... 576
Subject Index ... 586