HSE electronic catalog. Textbook: A Guide to Modern Econometrics

The book introduces the reader to a wide range of topics in modern econometrics that are important for understanding and performing practical work. This book is a guide to alternative methods with a focus on lighting. specific issues, for example, when to use this method What are its advantages and disadvantages. The main focus of the book is not on calculations or formal proofs, but on explaining approaches to the problem and its practical solution.
The book is addressed to students, graduate students, teachers, as well as specialists in applied economics and econometrics. The content and style of presentation correspond to the standard curricula teaching these disciplines at the level of bachelor's degree (2nd, 3rd and 4th years of study) and master's degree (5th and 6th years of study) of higher educational institutions of an economic profile.

The economy must be economical.” Such a statement was once made by L. I. Brezhnev, the leader of the USSR during the stagnation - it is not known, however, whether he himself was the author. It seems to be a completely “banal truth” such as “oil must be oily”, “water must be liquid”, and the like. But truth cannot possibly cease to be truth, even if it is banal. There is a clear meaning in this phrase - the slogan. For something to be economical, it must first of all be measurable. It is necessary to come up with reference points and mechanisms for comparison. In short, we need to digitize (sounds almost like bewitching) the economy. And the economy is actually the whole life modern man- Now they even try to measure emotions. The hardest task! But there is nothing to do, and mathematicians, together with economists, not so long ago, some thirty years ago, began to build new science- econometrics. This science (and where there is mathematics, this is science) is carried by specialists in mathematical statistics and other related mathematical disciplines, while at the same time adapting these disciplines themselves to curb such a galloping horse as economics. To describe the complex of material, social, ethno-geographical, cultural and other connections and relations, united by one term " modern economy”, mathematicians mobilize all the methods achieved - from multivariate analysis to graph theory, and also have to invent new ones. So living economy pushes herself formal mathematics to perfection and enrich it.

Solev International Association is a large Russian consulting firm that specializes in arranging financing for investment projects and programs aimed at creating and/or modernizing industrial productions, mainly in the "old" areas of the economy, using schemes and methods of project financing for both private commercial projects and objects of public-private partnership.

Project finance considers transactions when full recourse to the borrower on different reasons impossible. And the only reasonable provision for the risks of investors and creditors is a thorough study of the project itself, in-depth marketing research, numerous examinations and competent forecasts, where without econometrics it’s almost like without hands, vision and hearing: it’s impossible to feel the future “cash flow”, it’s impossible to look, but even a deaf and blind person can smell at least a little with the help of smell, mental forecast and calculation.

TABLE OF CONTENTS
Preface to Russian edition 12
From scientific editor Russian edition 14
Preface 17
1. Introduction 20
1.1. About econometrics 20
1.2. Structure of this book 23
1.3. Examples and exercises 26
2. Introduction to linear regression model 29
2.1. Conventional method least squares as an algebraic tool 30
2.1.1. Ordinary least squares (OLS) 30
2.1.2. Simple (paired) linear regression model 34
2.1.3. Example: individual salary 36
2.1.4. Matrix notation 37
2.2. Linear model multiple regression 39
2.3. Properties of the OLS estimator for small samples 43
2.3.1. Gauss-Markov Assumptions 43
2.3.2. Properties of the least squares estimator 45
2.3.3. Example: individual salary (continued) 49
2.4. The quality of the “fit” of the data by the model (“goodness-of-fit”) 51
2.5. Examination statistical hypotheses 54
2.5.1. Simple ^-criterion 55
2.5.2. Example: individual salary (continued) 58
2.5.3. Testing a single linear constraint 59
2.5.4. Joint test for the significance of regression coefficients 60
2.5.5. Example: individual salary (continued) 63
2.5.6. General case linear constraints on regression coefficients 65
2.5.7. Size, power and p-values ​​of the test 67
2.6. Asymptotic Properties of LSM Estimates 69
2.6.1. Wealth 69
2.6.2. Asymptotic normality 73
2.7. Illustration: Financial Asset Pricing Model (CFAM) 75
2.7.1. CFAM as a regression model 76
2.7.2. Assessing and testing the CFAM 77
2.8. Multicollinearity 81
2.8.1. Example: individual salary (continued) 84
2.9. Forecasting 86
Exercise 87
3. Interpreting and comparing regression models 93
3.1. Linear Model Interpretation 93
3.2. Selection of multiple explanatory variables 99
3.2.1. Incorrect specification of the set of regressors 99
3.2.2. Choice of explanatory variables 101
3.2.3. Comparing non-nested models 107
3.3. Incorrectly specified functional form 110
3.3.1. Not linear models 111
3.3.2. Functional Form Testing 112
3.4. Example: explanation of house prices 113
3.5. Example: explanation of individual salary 120
3.5.1. Linear Models 121
3.5.2. Log Linear Models 125
3.5.3. Tender effects 130
3.5.4. Some Cautionary Notes 133
Exercises 134
4. Heteroskedasticity and autocorrelation 137
4.1. Implications for properties of the least squares estimator 138
4.2. Derivation of an alternative estimate 140
4.3. Heteroskedasticity 142
4.3.1. Introduction 142
4.3.2. Properties of Estimates and Hypothesis Testing 145
4.3.3. The case of unknown variances 146
4.3.4. Consistent Estimates of Standard Errors of OLS Estimates in the Presence of Heteroscedasticity 148
4.3.5. Model with two unknown variances 150
4.3.6. Multiplicative Heteroskedasticity 151
4.4. Testing for Heteroscedasticity 153
4.4.1. Testing the equality of two unknown variances 153
4.4.2. Testing for multiplicative heteroscedasticity.... 154
4.4.3. Breusch-Pagan test 155
4.4.4. White test 155
4.4.5. What test? 157
4.5. Example: explaining the demand for labor 157
4.6. Autocorrelation 164
4.6.1. First order autocorrelation 166
4.6.2. p value unknown 169
4.7. Testing for First Order Autocorrelation 170
4.7.1. Asymptotic tests 171
4.7.2. Durbin-Watson test 172
4.8. Example: demand for ice cream 174
4.9. Alternative autocorrelation structures 179
4.9.1. autocorrelation higher order 179
4.9.2. Remaining moving average 180
4.10. What to do when you find autocorrelation? 182
4.10.1. Wrong specification 183
4.10.2. Wealthy standard errors OLS estimates considering heteroscedasticity and autocorrelation 185
4.11. Example: risk premium in foreign exchange markets 188
4.11.1. Concepts and symbols 189
4.11.2. Risk premium tests in the one-month market 191
4.11.3. Risk Premium Tests Using Overlapping Samples 195
Exercise 199
5. Endogeneity, instrumental variables, and the generalized method of moments (GMM) 202
5.1. Overview of properties of the least squares estimator 203
5.2. Cases when you can not use the OLS estimate 209
5.2.1. Autocorrelation of residuals and lagged dependent variable as a regressor 209
5.2.2. Example with measurement error 210
5.2.3. Simultaneity: The Keynesian Model 214
5.3. Instrumental variable estimation 217
5.3.1. Estimation with one endogenous regressor and one instrumental variable 218
5.3.2. Back to the Keynesian Model 222
5.3.3. Back to the Problem of Measurement Errors 224
5.3.4. Multiple endogenous regressors 225
5.4. Example: assessing returns to education 226
5.5. Generalized method of instrumental variables 234
5.5.1. Multiple endogenous regressors with an arbitrary number of instrumental variables 234
5.5.2. Two-Step Least Squares and back to the Keynesian model 240
5.6. Generalized method of moments 242
5.6.1. Example 243
5.6.2. Generalized method of moments 245
5.6.3. Some simple examples 248
5.7. Example: Estimating Intertemporal Financial Asset Pricing Models 250
5.8. Final remarks 255
Exercises 256
6. Maximum Likelihood Estimation and Specification Tests 259
6.1. Introduction to Maximum Likelihood Method 261
6.1.1. Some examples 261
6.1.2. General properties 266
6.1.3. Example (continued) 270
6.1.4. Normal linear regression model 271
6.2. Specification tests 273
6.2.1. Three Principles of Testing 273
6.2.2. Lagrange multiplier tests 276
6.2.3. Example (continued) 281
6.3. Tests in a Normal Linear Regression Model 283
6.3.1. Testing for significant non-included variables 283
6.3.2. Testing for heteroscedasticity 284
6.3.3. Testing for Autocorrelation 286
6.4. Quasi-Maximum Likelihood Method and Tests of Momentary Conditions 288
6.4.1. Quasi-maximum likelihood method 288
6.4.2. Moment Condition Tests 291
6.4.3. Testing the Normality Hypothesis 292
Exercises 293
7. Models with limited dependent variables 296
7.1. Binary Choice Models 297
7.1.1. Whether to apply linear regression? 297
7.1.2. Introduction to Binary Choice Models 298
7.1.3. Underlying Latent Model 300
7.1.4. Assessment 302
7.1.5. Quality of "fit" ("goodness-of-fit") data model 304
7.1.6. Example: Impact of Unemployment Benefits on Receipt 306
7.1.7. Specification Tests in Binary Choice Models 311
7.1.8. Relaxing Some Assumptions in Binary Choice Models 314
7.2. Multiple Response Models 316
7.2.1. Ordered Response Models 317
7.2.2. About normalization 319
7.2.3. Example: willingness to pay for natural areas not affected by human activities 320
7.2.4. Multinomial Models 324
7.3. Tobit models 329
7.3.1. Standard Tobit Model 329
7.3.2. Assessment 333
7.3.3. Example: Expenses for Alcohol and Tobacco (Part 1) 335
7.3.4. Specification tests for the tobit model 340
7.4. Generalizations of Tobit Models 343
7.4.1. Model Tobit II 344
7.4.2. Evaluation 348
7.4.3. Further generalizations 351
7.4.4. Example: Expenses for Alcohol and Tobacco (Part 2) 352
7.5. Bias due to selectivity 359
7.5.1. The nature of the selectivity problem 359
7.5.2. Semiparametric Estimation of a Constrained Model in Sampling 363
Exercises 365
8. Univariate time series models 370
8.1. Introduction 372
8.1.1. Some examples 372
8.1.2. Stationarity and autocorrelation function 375
8.2. General Processes autoregressive moving average (ARSS) 379
8.2.1. Formulation of APCC processes 379
8.2.2. Invertibility of polynomials from the shift operator 383
8.2.3. common roots 384
8.3. Stationarity and unit roots 385
8.4. Unit Root Testing 389
8.4.1. Unit Root Testing in a First-Order Autoregressive Model 389
8.4.2. Unit Root Testing in Higher-Order Autoregressive Models 394
8.4.3. Example: quarterly disposable income 397
8.5. Example: long-term dynamic purchasing power parity (part 1) 400
8.6. Estimating APCC Models 405
8.6.1. Least Squares 406
8.6.2. Maximum likelihood method 407
8.7. Model 409 selection
8.7.1. Autocorrelation function 409
8.7.2. Partial autocorrelation function 411
8.7.3. Diagnostic check 413
8.7.4. Criteria for selecting model 413
8.7.5. Example: Quarterly Disposable Income Modeling 414
8.8. Forecasting with APCC Models 417
8.8.1. Optimal predictive function 418
8.8.2. Prediction Accuracy 421
8.9. Example: Expectancy Theory of Time Frame 424
8.10. Autoregressive conditional heteroscedasticity (ARHG) 430
8.10.1. AGC G-and AGC G-model 431
8.10.2. Estimation and forecasting 436
8.10.3. Example: Volatility in Daily Exchange Rates 438
8.11. What can be said about multidimensional models? 442
Exercise 443
9. Multivariate time series models 447
9.1. Dynamic Models with Stationary Variables 449
9.2. Models with non-stationary variables 453
9.2.1. Spurious regressions 453
9.2.2. Cointegration 456
9.2.3. Mechanisms of cointegration and correction of residuals 461
9.3. Example: Long-Term Dynamic Purchasing Power Parity (Part 2) 463
9.4. Autoregressive Vector Models 467
9.5. Cointegration: multivariate case 471
9.5.1. Cointegration in Vector Autoregressive Models 472
9.5.2. Example: Cointegration in a 2D Vector Autoregressive Model 475
9.5.3. Cointegration testing 476
9.5.4. Example: long-term dynamic purchasing power parity (part 3) 480
9.6. Example: Demand for Money and Inflation 483
9.7. Concluding remarks 492
Exercise 493
10. Models based on panel data 496
10.1. Benefits of panel data 497
10.1.1. Parameter Estimation Efficiency 499
10.1.2. Parameter identification 501
10.2. Static linear model 503
10.2.1. Fixed effects model 503
10.2.2. Models with random effects 507
10.2.3. Fixed effects or random? 511
10.2.4. Quality of data fit by model 514
10.2.5. Alternative estimates of the method of instrumental variables 516
10.2.6. Alternative structures of residues 519
10.2.7. Testing for heteroscedasticity and autocorrelation 521
10.3. Example: explanation of individual salary 524
10.4. Dynamic linear models 528
10.4.1. Panel data autoregression model 528
10.4.2. Dynamic models with exogenous variables 535
10.4.3. Unit Roots and Cointegration 537
10.5. Example: elasticity of demand for labor wages 539
10.6. Models with limited dependent variables 542
10.6.1. Binary Choice Models 543
10.6.2. Fixed effects logit model 545
10.6.3. Probit Model with Random Effects 547
10.6.4. Tobit models 549
10.6.5. Dynamics and problem initial conditions 550
10.7. Incomplete panel data and selectivity bias 553
10.7.1. Estimating with randomly missing data 555
10.7.2. Bias due to selectivity and some simple tests 557
10.7.3. Estimation with non-randomly missing data 561
Exercises 562
A. Vectors and matrices 567
A. 1. Terminology 567
A.2. Actions with matrices 568
A.Z. Properties of vectors and matrices 570
A.4. Inverse matrices 571
A.5. Idempotent matrices 572
A.6. Eigenvalues and eigenvectors 573
A.7. Differentiation 575
A.8. Some matrix actions associated with the method of least squares 575
B. Theory of statistics and the theory of distributions 578
8.1. Discrete random variables 578
8.2. Continuous random variables 579
8.3. Expected value and moments 581
8.4. Multivariate distributions 582
8.5. Conditional distributions 584
8.6. Normal distribution 586
8.7. Distributions related to normal distribution 589
Literature 592
Subject index 605.

Foreword

Econometrics has developed rapidly over the past two decades, and the application of modern econometric methods has become increasingly standard practice in empirical work in many areas of economics. Common research topics include unit root tests; cointegration; estimation by the generalized method of moments; accounting for heteroscedasticity and autocorrelation of regression residuals; modeling of conditional heteroscedasticity; models based on panel data and models with limited dependent variables; endogenous regressors and problems associated with limitations in sampling ("sample selection problem"). In the same time software econometrics became more and more "friendly" for the user and more and more met modern requirements. As a result, users can implement rather complex methods without understanding the nature of the underlying theory, and therefore without understanding their potential shortcomings or dangers. At the same time, in many introductory textbooks of econometrics, it is unjustifiably great attention is given to the standard linear regression model under many strict assumptions. Needless to say, these assumptions are hardly satisfied in practice and are in fact unnecessary. On the other hand, more advanced econometrics textbooks are often overwhelmed with technical details, making it difficult for the average economist to grasp the basic ideas and extract necessary information. This book attempts to fill this gap.

The purpose of the book is to introduce the reader to a wide range of topics in modern econometrics, detailing issues that are important to understanding and doing empirical research. This book is more of a guide (rather than an overview) to alternative methods, so the presentation does not focus on formulas after (although necessary formulas are given), nor on formal proofs, and after describing the method and its practical justification, focuses on the development of its understanding. The book covers a wide range of topics that are not usually included in textbooks of this level. In particular, attention is paid to cointegration, the generalized method of moments, models with bounded dependent variables, and panel data models. As a result, the book discusses the further development of time series analysis, methods for analyzing spatial ("cross-sectional") and panel data. Dozens of full-scale empirical examples and illustrations are provided, taken from areas such as labor economics, finance, world economy, consumer behavior, environmental economics and macroeconomics. In addition, a number of exercises are of a specific empirical nature and require the use of real data.

The text presented is based on lecture notes given in the Applied Econometrics Master's degree programs in Economics at the Catholic University of Leuven and Tilburg University*^. The book is intended for an audience of economists and students of economics who would like to become familiar with modern econometric approaches and methods that are important for performing, understanding and evaluating empirical research. It meets the requirements for courses in applied econometrics at the master's or graduate level. In some higher educational institutions this book will meet the requirements for one or more courses at the master's level, provided that students have sufficient training in statistics. Some of the last chapters can be used in special courses covering specific topics such as panel data, models

Leuven (Belgium), Tilburg (Holland) (scientific ed. translation note).

with limited dependent variables or time series analysis. In addition, the book can serve as a guide for managers, research economists, and practitioners who wish to update or expand their knowledge of econometrics. The book uses elements of matrix algebra.

I am deeply indebted to Arie Kapteyn, Bertrand Melenberg, Theo Nijman and Arthur van Soest who contributed to my understanding of econometrics and shaped my thinking on many issues. . The fact that some of their ideas have come to fruition in this text is a tribute to their efforts. I must also thank the generations of students who helped me shape this text into its current form by commenting on previous versions and asking me questions they couldn't find answers to. The wide range of practical and empirical problems concerning econometrics offered to me by students and colleagues was an important stimulus for the completion of this book. My colleagues and friends read various parts of the manuscript and made corrections and comments. I am very grateful to Peter de Goeij, Ben Jacobsen, Wim Koevoets, Marco Lyrio, Konstantijn Maes, Wessel Marquering, Bertrand Bertrand Melenberg, Paulo Nunes, Anatoly Peresetsky, Max van de Sande Bakhuyzen, Erik Schokkaert, Arthur van Soest , Frederic Vermeulen, Kuo-chun Yeh, and many anonymous reviewers. Of course, I am personally responsible for any remaining errors. Special thanks go to Jef Flechet for his help with many empirical illustrations and his constructive comments on many previous versions. Finally, I want to thank my wife Marcella and my children Timo and Thalia for their patience and understanding throughout the time my soul was with this book when it should have been. them.

M.: 2008. - 616 p.

Marno Verbeek - professor of econometrics at the Center economic research University of Leuven (Belgium). He also works at the Center for Economic Research at Tilburg University (Netherlands).

The book introduces the reader to a wide range of topics in modern econometrics that are important for understanding and doing practical work. This book is a guide to alternative methods, focusing on specific issues such as when to use a given method, what are its advantages and disadvantages. The main focus of the book is not on calculations or formal proofs, but on explaining approaches to the problem and its practical solution. The book covers a wide range of topics, including those poorly covered in the domestic literature, such as time series regression analysis, cointegration, models with limited dependent variables, panel data analysis, and the generalized method of moments. Empirical examples are provided from areas such as labor economics, environmental economics, the world economy, finance, and macroeconomics. An overview is provided at the end of each chapter. key concepts explained on practical examples. For illustrations and exercises on the website on the Internet, the necessary data sets are presented on-line.

The book is addressed to students, graduate students, teachers, as well as specialists in applied economics and econometrics. The content and style of presentation corresponds to the standard curricula for teaching these disciplines at the undergraduate (2nd, 3rd and 4th years of study) and graduate (5th and 6th years of study) levels of higher educational institutions of economic profile.

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TABLE OF CONTENTS
Preface to the Russian edition 12
From the scientific editor of the Russian edition 14
Preface 17
1. Introduction 20
1.1. About econometrics 20
1.2. Structure of this book 23
1.3. Examples and exercises 26
2. Introduction to linear regression model 29
2.1. Ordinary least squares as an algebraic tool 30
2.1.1. Ordinary least squares (OLS) 30
2.1.2. Simple (paired) linear regression model 34
2.1.3. Example: individual salary 36
2.1.4. Matrix notation 37
2.2. Linear Multiple Regression Model 39
2.3. Properties of the OLS estimator for small samples 43
2.3.1. Gauss-Markov Assumptions 43
2.3.2. Properties of the least squares estimator 45
2.3.3. Example: individual salary (continued) 49
2.4. The quality of the “fit” of the data by the model (“goodness-of-fit”) 51
2.5. Testing statistical hypotheses 54
2.5.1. Simple ^-criterion 55
2.5.2. Example: individual salary (continued) 58
2.5.3. Testing a single linear constraint 59
2.5.4. Joint criterion for the significance of regression coefficients.. 60
2.5.5. Example: individual salary (continued) 63
2.5.6. General case of linear constraints on regression coefficients 65
2.5.7. Size, power and p-values ​​of the test 67
2.6. Asymptotic Properties of LSM Estimates 69
2.6.1. Wealth 69
2.6.2. Asymptotic normality 73
2.7. Illustration: Financial Asset Pricing Model (CFAM) 75
2.7.1. CFAM as a regression model 76
2.7.2. Assessing and testing the CFAM 77
2.8. Multicollinearity 81
2.8.1. Example: individual salary (continued) 84
2.9. Forecasting 86
Exercise 87
3. Interpreting and comparing regression models 93
3.1. Linear Model Interpretation 93
3.2. Selection of multiple explanatory variables 99
3.2.1. Incorrect specification of the set of regressors 99
3.2.2. Choice of explanatory variables 101
3.2.3. Comparing non-nested models 107
3.3. Incorrectly specified functional form 110
3.3.1. Nonlinear Models 111
3.3.2. Functional Form Testing 112
3.4. Example: explanation of house prices 113
3.5. Example: explanation of individual salary 120
3.5.1. Linear Models 121
3.5.2. Log Linear Models 125
3.5.3. Tender effects 130
3.5.4. Some Cautionary Notes 133
Exercises 134
4. Heteroskedasticity and autocorrelation 137
4.1. Implications for properties of the least squares estimator 138
4.2. Derivation of an alternative estimate 140
4.3. Heteroskedasticity 142
4.3.1. Introduction 142
4.3.2. Properties of Estimates and Hypothesis Testing 145
4.3.3. The case of unknown variances 146
4.3.4. Consistent Estimates of Standard Errors of OLS Estimates in the Presence of Heteroscedasticity 148
4.3.5. Model with two unknown variances 150
4.3.6. Multiplicative Heteroskedasticity 151
4.4. Testing for Heteroscedasticity 153
4.4.1. Testing the equality of two unknown variances 153
4.4.2. Testing for multiplicative heteroscedasticity.... 154
4.4.3. Breusch-Pagan test 155
4.4.4. White test 155
4.4.5. What test? 157
4.5. Example: explaining the demand for labor 157
4.6. Autocorrelation 164
4.6.1. First order autocorrelation 166
4.6.2. p value unknown 169
4.7. Testing for First Order Autocorrelation 170
4.7.1. Asymptotic tests 171
4.7.2. Durbin-Watson test 172
4.8. Example: demand for ice cream 174
4.9. Alternative autocorrelation structures 179
4.9.1. Higher order autocorrelation 179
4.9.2. Remaining moving average 180
4.10. What to do when you find autocorrelation? 182
4.10.1. Wrong specification 183
4.10.2. Consistent standard errors of least squares estimates, taking into account heteroscedasticity and autocorrelation 185
4.11. Example: risk premium in foreign exchange markets 188
4.11.1. Concepts and symbols 189
4.11.2. Risk premium tests in the one-month market 191
4.11.3. Risk Premium Tests Using Overlapping Samples 195
Exercise 199
5. Endogeneity, instrumental variables, and the generalized method of moments (GMM) 202
5.1. Overview of properties of the least squares estimator 203
5.2. Cases when you can not use the OLS estimate 209
5.2.1. Autocorrelation of residuals and lagged dependent variable as a regressor 209
5.2.2. Example with measurement error 210
5.2.3. Simultaneity: The Keynesian Model 214
5.3. Instrumental variable estimation 217
5.3.1. Estimation with one endogenous regressor and one instrumental variable 218
5.3.2. Back to the Keynesian Model 222
5.3.3. Back to the Problem of Measurement Errors 224
5.3.4. Multiple endogenous regressors 225
5.4. Example: assessing returns to education 226
5.5. Generalized method of instrumental variables 234
5.5.1. Multiple endogenous regressors with an arbitrary number of instrumental variables 234
5.5.2. Two-Step Least Squares and back to the Keynesian model 240
5.6. Generalized method of moments 242
5.6.1. Example 243
5.6.2. Generalized method of moments 245
5.6.3. Some simple examples 248
5.7. Example: Estimating Intertemporal Financial Asset Pricing Models 250
5.8. Concluding remarks 255
Exercises 256
6. Maximum Likelihood Estimation and Specification Tests 259
6.1. Introduction to Maximum Likelihood Method 261
6.1.1. Some examples 261
6.1.2. General properties 266
6.1.3. Example (continued) 270
6.1.4. Normal linear regression model 271
6.2. Specification tests 273
6.2.1. Three Principles of Testing 273
6.2.2. Lagrange multiplier tests 276
6.2.3. Example (continued) 281
6.3. Tests in a Normal Linear Regression Model 283
6.3.1. Testing for significant non-included variables 283
6.3.2. Testing for heteroscedasticity 284
6.3.3. Testing for Autocorrelation 286
6.4. Quasi-Maximum Likelihood Method and Tests of Momentary Conditions 288
6.4.1. Quasi-maximum likelihood method 288
6.4.2. Moment Condition Tests 291
6.4.3. Testing the Normality Hypothesis 292
Exercises 293
7. Models with limited dependent variables 296
7.1. Binary Choice Models 297
7.1.1. Should I use linear regression? 297
7.1.2. Introduction to Binary Choice Models 298
7.1.3. Underlying Latent Model 300
7.1.4. Assessment 302
7.1.5. Quality of "fit" ("goodness-of-fit") data model 304
7.1.6. Example: Impact of Unemployment Benefits on Receipt 306
7.1.7. Specification Tests in Binary Choice Models 311
7.1.8. Relaxing Some Assumptions in Binary Choice Models 314
7.2. Multiple Response Models 316
7.2.1. Ordered Response Models 317
7.2.2. About normalization 319
7.2.3. Example: willingness to pay for natural areas not affected by human activities 320
7.2.4. Multinomial Models 324
7.3. Tobit models 329
7.3.1. Standard Tobit Model 329
7.3.2. Assessment 333
7.3.3. Example: Expenses for Alcohol and Tobacco (Part 1) 335
7.3.4. Specification tests for the tobit model 340
7.4. Generalizations of Tobit Models 343
7.4.1. Model Tobit II 344
7.4.2. Evaluation 348
7.4.3. Further generalizations 351
7.4.4. Example: Expenses for Alcohol and Tobacco (Part 2) 352
7.5. Bias due to selectivity 359
7.5.1. The nature of the selectivity problem 359
7.5.2. Semiparametric Estimation of a Constrained Model in Sampling 363
Exercises 365
8. Univariate time series models 370
8.1. Introduction 372
8.1.1. Some examples 372
8.1.2. Stationarity and Autocorrelation 375
8.2. General Processes of Autoregressive-Moving Average (ARSS) 379
8.2.1. Formulation of APCC processes 379
8.2.2. Invertibility of polynomials from the shift operator 383
8.2.3. Common Roots 384
8.3. Stationarity and unit roots 385
8.4. Unit Root Testing 389
8.4.1. Unit Root Testing in a First-Order Autoregressive Model 389
8.4.2. Unit Root Testing in Higher-Order Autoregressive Models 394
8.4.3. Example: quarterly disposable income 397
8.5. Example: long-term dynamic purchasing power parity (part 1) 400
8.6. Estimating APCC Models 405
8.6.1. Least Squares 406
8.6.2. Maximum likelihood method 407
8.7. Model 409 selection
8.7.1. Autocorrelation function 409
8.7.2. Partial autocorrelation function 411
8.7.3. Diagnostic check 413
8.7.4. Criteria for selecting model 413
8.7.5. Example: Quarterly Disposable Income Modeling 414
8.8. Forecasting with APCC Models 417
8.8.1. Optimal predictive function 418
8.8.2. Prediction Accuracy 421
8.9. Example: Expectancy Theory of Time Frame 424
8.10. Autoregressive conditional heteroscedasticity (ARHG) 430
8.10.1. AGC G-and AGC G-model 431
8.10.2. Estimation and forecasting 436
8.10.3. Example: Volatility in Daily Exchange Rates 438
8.11. What can be said about multidimensional models? 442
Exercise 443
9. Multivariate time series models 447
9.1. Dynamic Models with Stationary Variables 449
9.2. Models with non-stationary variables 453
9.2.1. Spurious regressions 453
9.2.2. Cointegration 456
9.2.3. Mechanisms of cointegration and correction of residuals 461
9.3. Example: Long-Term Dynamic Purchasing Power Parity (Part 2) 463
9.4. Autoregressive Vector Models 467
9.5. Cointegration: multivariate case 471
9.5.1. Cointegration in Vector Autoregressive Models 472
9.5.2. Example: Cointegration in a 2D Vector Autoregressive Model 475
9.5.3. Cointegration testing 476
9.5.4. Example: long-term dynamic purchasing power parity (part 3) 480
9.6. Example: Demand for Money and Inflation 483
9.7. Concluding remarks 492
Exercise 493
10. Models based on panel data 496
10.1. Benefits of panel data 497
10.1.1. Parameter Estimation Efficiency 499
10.1.2. Parameter identification 501
10.2. Static linear model 503
10.2.1. Fixed effects model 503
10.2.2. Models with random effects 507
10.2.3. Fixed effects or random? 511
10.2.4. Quality of data fit by model 514
10.2.5. Alternative estimates of the method of instrumental variables 516
10.2.6. Alternative structures of residues 519
10.2.7. Testing for heteroscedasticity and autocorrelation 521
10.3. Example: explanation of individual salary 524
10.4. Dynamic linear models 528
10.4.1. Panel data autoregression model 528
10.4.2. Dynamic models with exogenous variables 535
10.4.3. Unit Roots and Cointegration 537
10.5. Example: wage elasticity of demand for labor 539
10.6. Models with limited dependent variables 542
10.6.1. Binary Choice Models 543
10.6.2. Fixed effects logit model 545
10.6.3. Probit Model with Random Effects 547
10.6.4. Tobit models 549
10.6.5. Dynamics and the problem of initial conditions 550
10.7. Incomplete panel data and selectivity bias 553
10.7.1. Estimating with randomly missing data 555
10.7.2. Selective bias and some simple tests 557
10.7.3. Estimation with non-randomly missing data 561
Exercises 562
A. Vectors and matrices 567
A. 1. Terminology 567
A.2. Actions with matrices 568
A.Z. Properties of vectors and matrices 570
A.4. Inverse matrices 571
A.5. Idempotent matrices 572
A.6. Eigenvalues ​​and eigenvectors 573
A.7. Differentiation 575
A.8. Some Matrix Operations Related to the Least Squares Method 575
B. Theory of statistics and the theory of distributions 578
8.1. Discrete random variables 578
8.2. Continuous random variables 579
8.3. Mathematical expectation and moments 581
8.4. Multivariate distributions 582
8.5. Conditional distributions 584
8.6. Normal distribution 586
8.7. Distributions related to the normal distribution 589
Literature 592
Index 605