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Econometrics. Beginner course

Contents Foreword Preface to the first edition Preface to the third edition Preface to the sixth edition 1. Introduction 1.1. Models 1.2. Types of models 1.3. Data types 2. Paired regression model 2.1. Curve fitting 2.2. Least squares method (LSM) 2.3. Linear regression model with two variables 2.4. Gauss-Markov theorem. Estimation of error variance a2 2.5. Statistical properties of OLS estimates of regression parameters. Hypothesis testing b = bo- Confidence intervals for regression coefficients 2.6. Analysis of variation in the dependent variable in regression. Determination coefficient R2 2.7. Maximum likelihood estimation of regression coefficients Exercises 3. Multiple regression model 3.1. Main hypotheses 3.2. Least square method. Gauss-Markov theorem 3.3. Statistical properties of OLS estimates 3.4. Analysis of variation in the dependent variable in regression. R2 coefficients and adjusted R 3.5. Testing hypotheses. Confidence intervals and confidence regions Exercises 4. Various aspects of multiple regression 4.1. Multicollinearity 4.2. Dummy variables 4.3. Partial correlation 4.4. Model specification Exercises 5. Some generalizations of multiple regression 5.1. Stochastic regressors 5.2. Generalized least squares method 5.3. Accessible Generalized Least Squares Exercises 6. Heteroscedasticity and Time Correlation 6.1. Heteroscedasticity 6.2. Time correlation Exercises 7. Forecasting in regression models 7.1. Unconditional forecasting 7.2. Conditional forecasting 7.3. Forecasting in the presence of autoregressive errors Exercises 8. Instrumental variables 8.1. Consistency of estimates obtained using instrumental variables 8.2. Influence of measurement errors 8.3. Two-Step Least Squares Method 8.4. Hausman test Exercises 9. Systems of regression equations 3.1. Externally unrelated equations 9.1. Systems of simultaneous equations Exercises 10. Maximum likelihood method in regression models 10.1. Introduction 10.2. Mathematical apparatus 246 10.3. Maximum likelihood estimation of parameters of a multivariate normal distribution 10.4. Properties of maximum likelihood estimates 10.5. Maximum likelihood estimation in a linear model 10.6. Hypothesis testing in a linear model, I 10.7. Testing Hypotheses in a Linear Model, II 10.8. Nonlinear constraints Exercises 11. Time series 11.1. Distributed lag models 11.2. Dynamic models 11. 3. Unit roots and cointegration 11.4 Box-Jenkins models (ARIMA) 11.5. GARCH Models Exercises 12. Discrete Dependent Variables and Censored Samples 12.1. Binary and multiple choice models 12.2. Models with Reduced and Censored Samples Exercises 13. Panel Data 13.1 Introduction 13.2. Designations and basic models 13.3. Fixed effect model 13.4. Random effect model 13.5. Quality of fit 13.6. Model selection 13.7. Dynamic models 13.8. Binary choice models with panel data 13.9. Generalized method of moments Exercises 14. Pre-test: introduction 14.1. Introduction 14.2. Statement of problem 14.3. Main result 14.4. Pretest score 14.5. WALS score 14.6. Equivalence theorem 14.7. Pre-testing and the underestimation effect 14.8. "Understatement" effect. One auxiliary parameter 14.9. Model selection: from general to specific and from specific to general 10.14. "Understatement" effect. Two auxiliary parameters 14.11. Forecasting and preliminary testing 14.12. Generalizations 14.13. Other questions Exercises 15. Econometrics of financial markets 15.1. Introduction 15.2. Hypothesis of financial market efficiency 15.3. Optimization of the securities portfolio 15.4. Test for including new assets in an effective portfolio 15.5. Optimal portfolio in the presence of a risk-free asset 15.6. Models for valuation of financial assets Exercises 16. Econometrics perspectives 1.6.1. Introduction 16.2. What exactly does an econometrician do? 16.3. Econometrics and physics 16.4. Econometrics and mathematical statistics 16.5. Theory and practice 16.6. Econometric method 16.7. Weak link 16.8. Aggregation 16.9. How to use other works 16.10. Conclusion Appendix LA. Linear algebra 1. Vector space 2. Vector space LP 3. Linear dependence 4. Linear subspace 5. Basis. Dimension 6. Linear operators 7. Matrices 8. Operations with matrices 9. Matrix invariants: trace, determinant 10. Matrix rank 11. Inverse matrix 12. Systems of linear equations 13. Eigenvalues and vectors 14. Symmetric matrices 15. Positive definite matrices 16 Idempotent matrices 17. Block matrices 18. Kronecker product 19. Differentiation with respect to a vector argument Exercises Application MS. Probability theory and mathematical statistics 1. Random variables, random vectors 2. Conditional distributions 3. Some special distributions 4. Multivariate normal distribution 5. Law of large numbers. Central limit theorem 6 Basic concepts and problems of mathematical statistics 7. Estimation of parameters 8. Hypothesis testing Appendix EP. Overview of econometric packages 1. Origin of packages. Windows versions. Graphics 2. About some packages 3. Practical experience Appendix ST. Brief English-Russian dictionary of terms Appendix TA. Tables Literature Subject index

Name: Econometrics - Beginner course.

The textbook contains a systematic presentation of the fundamentals of econometrics and is written on the basis of lectures that the authors gave for a number of years at the Russian School of Economics and the Higher School of Economics. Linear regression models are studied in detail (least squares method, hypothesis testing, heteroscedasticity, autocorrelation of errors, model specification). Separate chapters are devoted to systems of simultaneous equations, the maximum likelihood method in regression models, models with discrete and limited dependent variables.
Three new chapters have been added to the sixth edition of the book. The chapter on "Panel Data" expands the book to a comprehensive list of topics traditionally included in modern basic econometrics courses. Chapters “Preliminary Testing” and “Econometrics of Financial Markets” have also been added, which will be useful for those interested in the theoretical and applied aspects of econometrics, respectively. The number of exercises has been significantly increased. Exercises with real data are included, available to the reader on the book's website.
For undergraduates, graduate students, teachers, as well as specialists in applied economics and finance.

Econometrics (along with microeconomics and macroeconomics) is one of the basic disciplines of modern economic education. What is econometrics? When dealing with a living, developing science, there is always a difficulty in trying to give a brief description of its subject and methods. Can we say that econometrics is the science of economic measurement, as its name suggests? Of course it is possible, but then the question arises, what is the meaning of the term “economic measurements”. This is similar to defining mathematics as the science of numbers. Therefore, without trying to develop this problem in more detail, we will cite statements from recognized authorities in economics and econometrics.

1. Introduction
1.1. Models
1.2. Model types
1.3. Data types
2. Paired regression model
2.1. Curve fitting
2.2. Least squares method (LSM)
2.3. Linear regression model with two variables
2.4. Gauss-Markov theorem. Error variance estimate a2
2.5. Statistical properties of OLS estimates of regression parameters. Hypothesis testing b = bo- Confidence intervals for regression coefficients
2.6. Analysis of variation in the dependent variable in regression. Determination coefficient R2
2.7. Maximum likelihood estimation of regression coefficients
Exercises
3. Multiple regression model
3.1. Main hypotheses
3.2. Least square method. Gauss-Markov theorem
3.3. Statistical properties of OLS estimators
3.4. Analysis of variation in the dependent variable in regression. R2 coefficients and adjusted R
3.5. Testing hypotheses. Confidence intervals and confidence regions
Exercises
4. Various aspects of multiple regression
4.1. Multicollinearity
4.2. Dummy variables
4.3. Partial correlation
4.4. Model Specification
Exercises
5. Some generalizations of multiple regression
5.1. Stochastic regressors
5.2. Generalized least squares
5.3. Available Generalized Least Squares
Exercises
6. Heteroscedasticity and correlation over time
6.1. Heteroscedasticity
6.2. Correlation over time
Exercises
7. Forecasting in regression models
7.1. Unconditional forecasting
7.2. Conditional Prediction
7.3. Forecasting in the presence of autoregressive errors
Exercises
8. Instrumental variables
8.1. Consistency of estimates obtained using instrumental variables
8.2. Impact of measurement errors
8.3. Two-Step Least Squares
8.4. Hausman test
Exercises
9. Systems of regression equations
3.1. Externally unrelated equations
9.1. Systems of simultaneous equations
Exercises
10. Maximum likelihood method in regression models
10.1. Introduction
10.2. Mathematical apparatus 246
10.3. Maximum likelihood estimation of parameters of a multivariate normal distribution
10.4. Properties of maximum likelihood estimators
10.5. Maximum likelihood estimation in a linear model
10.6. Hypothesis testing in a linear model, I
10.7. Hypothesis testing in a linear model, II
10.8. Nonlinear Constraints
Exercises
11. Time series
11.1. Distributed lag models
11.2. Dynamic models
11.3. Unit roots and cointegration
11.4 Box-Jenkins models (ARIMA)
11.5. GARCH models
Exercises
12. Discrete dependent variables and censored samples
12.1. Binary and multiple choice models
12.2. Models with trimmed and censored samples
Exercises
13. Panel data
13.1 Introduction
13.2. Designations and basic models
13.3. Fixed effect model
13.4. Random effect model
13.5. Quality of fit
13.6. Model selection
13.7. Dynamic models
13.8. Binary choice models with panel data
13.9. Generalized method of moments
Exercises
14. Pre-Test: Introduction
14.1. Introduction
14.2. Formulation of the problem
14.3. Main result
14.4. Pretest assessment
14.5. WALS assessment
14.6. Equivalence theorem
14.7. Pre-testing and the underreporting effect
14.8. "Understatement" effect. One auxiliary parameter
14.9. Model selection: from general to specific and from specific to general
14.10. "Understatement" effect. Two auxiliary parameters
14.11. Forecasting and preliminary testing
14.12. Generalizations
14.13. Other questions
Exercises
15. Econometrics of financial markets
15.1. Introduction
15.2. Financial Market Efficiency Hypothesis
15.3. Optimization of the securities portfolio
15.4. Test for the inclusion of new assets in an effective portfolio
15.5. Optimal portfolio in the presence of a risk-free asset
15.6. Financial asset valuation models
Exercises
16. Econometrics perspectives
1.6.1. Introduction
16.2. What exactly does an econometrician do?
16.3. Econometrics and physics
16.4. Econometrics and mathematical statistics
16.5. Theory and practice
16.6. Econometric method
16.7. Weak link
16.8. Aggregation
16.9. How to use other works
16.10. Conclusion
LA application. Linear algebra
1. Vector space
2. Vector space LP
3. Linear dependence
4. Linear subspace
5. Basis. Dimension
6. Linear operators
7. Matrices
8. Operations with matrices
9. Matrix invariants: trace, determinant
10. Matrix rank
11. Inverse matrix
12. Systems of linear equations
13. Eigenvalues and vectors
14. Symmetric matrices
15. Positive definite matrices
16. Idempotent matrices
17. Block matrices
18. Kronecker product
19. Differentiation by vector argument
Exercises
MS application. Theory of Probability and Mathematical Statistics
1. Random variables, random vectors
2. Conditional distributions
3. Some special distributions
4. Multivariate normal distribution
5. Law of large numbers. Central limit theorem
6 Basic concepts and tasks of mathematical statistics
7. Parameter estimation
8. Testing hypotheses
EP application. Overview of econometric packages
1. Origin of packets. Windows versions. Graphic arts
2. About some packages
3. Practical work experience
Application ST. Brief English-Russian dictionary of terms
TA application. Tables
Literature
Subject index

UDC 330.43(075.8)
BBK 65v6ya73

Magnus Y.R., Katyshev P.K., Peresetsky A.A.
Econometrics. Initial course: Textbook. — 8th ed., rev. - M.: , 2007. - 504 p.

ISBN 978-5-7749-0473-0

The chapter on “Panel Data” expands the book to a comprehensive list of topics traditionally included in modern basic econometrics courses. The chapters “Preliminary Testing” and “Econometrics of Financial Markets” will be useful for those interested in the theoretical and applied aspects of econometrics, respectively. The number of exercises has been significantly increased. Exercises with real data are included, available to the reader on the book's website.

For undergraduates, graduate students, teachers, as well as specialists in applied economics and finance.

Compared to the 1997 edition, the book includes three new chapters on maximum likelihood in regression models, time series, and models with discrete and bounded dependent variables. The number of examples from the Russian economy, tasks and exercises has been significantly increased.

For undergraduates, graduate students, teachers, as well as specialists in applied economics and finance.

“Econometrics allows for quantitative analysis of real economic phenomena based on modern developments in theory and observations related to the methods of drawing conclusions” (Samuelson).

“The main task of econometrics is to fill a priori economic reasoning with empirical content” (Klein).

“The goal of econometrics is the empirical derivation of economic laws. Econometrics complements theory by using real data to test and clarify postulated relationships” (Malenvaux).

This book is aimed primarily at students beginning to study econometrics for the first time, and has two purposes. First, we want to prepare the reader for applied research in economics. Secondly, we think that it will be useful to students who are going to further study the theory of econometrics in depth. No prior knowledge of econometrics is required. However, initial familiarity with courses in linear algebra, probability theory, and mathematical statistics is assumed (for example, Gelfand, 1971; Ilyin and Poznyak, 1984; Wentzel, 1964). We also assume that the reader has mastered mathematical analysis within the standard course of a technical university.

There are several excellent textbooks on econometrics in English. For example, the book (Greene, 1997) can rightfully be considered an “econometric encyclopedia” - it contains almost all sections of modern econometrics. The textbook (Goldberger, 1990) pays more attention to the formal mathematical side of econometrics. In our opinion, the book (Johnston and DiNardo, 1997) is very successful, modern and balanced in terms of theory and applications. Also noteworthy are the textbooks (Griffiths, Hill and Judge, 1993) and (Pindyck and Rubinfeld, 1991), which are aimed at readers without a strong mathematical background and provide a large number of examples and exercises. A good addition to standard textbooks is the book (Kennedy, 1998), which focuses on the substantive side of econometric analysis and contains a large number of interesting exercises. It is also necessary to mention the book (Hamilton, 1994), where the theory of time series is presented in great detail and at a high mathematical level, and the book (Stewart, 1991), which contains successful and compact sections on the theory of time series.

Therefore, it may be necessary to make some arguments in favor of writing a new book rather than simply translating one of the existing textbooks. Our book is based on the material of lectures that one of the authors (Ya. Magnus) gave as an initial econometrics course in the master program for students of the Russian Economic School (NES) in March-April 1993. Two other authors (P. Katyshev, A. .Peresetsky) conducted practical classes. The intensive 7-week course included the basics of econometrics. This was the first year of the existence of the Russian Economic School. In subsequent years, the authors collaborated in creating the syllabus for all three econometric courses for first-year students at NES. In the process of work, in particular, we compiled examples from the Russian economy, which we used instead of the traditionally considered examples from the economies of Western Europe and the USA. We eventually came to the conclusion that it would be desirable to have a textbook written specifically for Russian students, and we revised the course syllabus into a stand-alone book. This book is therefore the result of five years of experience teaching econometrics to Russian students.

Chapters 2-4 contain the classical theory of linear regression models. This material is the core of econometrics, and students should have a good grasp of it before moving on to the rest of the book. Chapter 2 examines the simplest model with two regressors, Chapter 3 is devoted to multivariate models. In some ways, Chapter 2 is redundant, but from a pedagogical point of view it is extremely useful to study two-variable regression models first. Then, for example, you can do without matrix algebra; in the two-dimensional case it is also easier to understand the graphical interpretation of regression. Chapter 4 contains several additional sections (the problem of multicollinearity, dummy variables, model specification), but its material can also be considered standard fundamentals of econometrics.

Chapters 5-9 explore some generalizations of the standard multiple regression model, such as stochastic regressors, generalized least squares, heteroscedasticity and autocorrelation of residuals, accessible generalized least squares, forecasting, and instrumental variables. The amazing thing about econometric theory is that at this level most of the theorems in the standard core theory (chapters 2-4) remain valid, at least approximately or asymptotically, when the conditions of the theorems are relaxed. We strongly recommend that the results of Chapters 5-9 be continually referenced with the main results presented in Chapters 2-4.

Chapter 10 contains the theory of systems of simultaneous equations, i.e. the case when the model contains more than one equation. The problems that an econometrician may encounter in practical work are considered.

The book includes several appendices, including an overview of econometric packages and a brief English-Russian dictionary of terms.

Our experience shows that the material in chapters 1-7 is sufficient for a 7-week course of 6 hours per week, and the material in chapters 1-10 is sufficient for a standard one-semester course. We have had good results with the following course structure: two two-hour lectures per week and one seminar (in smaller subgroups), but other course structures are also possible.

For students

Problem solving is the key to learning mathematics, statistics, and also econometrics. Our teachers told us this when we were students, and we repeat it here. And that's true! Experimentation with data is essential for hands-on students. Remove a few observations from your data and see what happens to your estimates and why. Add explanatory variables and see how your estimates and predictions change. In general, experiment. A theory-oriented student must ask himself why this or that condition of the theorem is necessary. Why does the theorem cease to be true if you remove or change one of the conditions. Find counterexamples.

For teachers

It is important that all students have the required mathematical and statistical background at the start of the course. If this is not the case, then the course should begin with a review of the necessary concepts in linear algebra and mathematical statistics. Chapters 2-4 should be at the beginning of the course. There is some freedom in choosing further topics if time does not allow including the entire book in the course. If there is not enough time, you can postpone stochastic regressors (section 5.1) and tests for heteroscedasticity (but not the concept of heteroscedasticity itself) for the next course. Chapters 7-10 contain specialized but important sections that can be included in the course in varying degrees of detail, depending on the tastes of the instructor.

We will be grateful for any comments, messages about typos, unclear places, errors in this book.

Acknowledgments

We owe a huge debt to five generations of students of the Russian School of Economics, who, while studying the course, gave a lot of critical comments that we used when working on the book. Without them this book would never have been written.

We are grateful to NES graduates Vladislav Kargin and Alexey Onatsky, who prepared an example for the book on the apartment market in Moscow, as well as NES students Elena Paltseva and Gauhar Turmukhambetova, through whose efforts we managed to avoid many typos. We are also grateful to our colleague Alexander Slastnikov, who took on the responsibility of editing the manuscript. In the work on the manuscript, P. Katyshev and A. Peresetsky received financial support from the Russian Humanitarian Scientific Foundation, project 96-02-16011a.

Tilburg/Moscow, March 1997

6th ed., revised. and additional - M.: Delo, 2004. - 576 p.

Three new chapters have been added to the sixth edition of the book. The chapter on "Panel Data" expands the book to a comprehensive list of topics traditionally included in modern basic econometrics courses. Chapters “Preliminary Testing” and “Econometrics of Financial Markets” have also been added, which will be useful for those interested in the theoretical and applied aspects of econometrics, respectively. The number of exercises has been significantly increased. Exercises with real data are included, available to the reader on the book's website.

For undergraduates, graduate students, teachers, as well as specialists in applied economics and finance

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Table of contents
Opening remarks 10
Preface to the first edition 13
Preface to the third edition 18
Preface to the sixth edition 23
1. Introduction 26
1.1. Models 26
1.2. Model types 28
1.3. Data types 30
2. Paired regression model 32
2.1. Curve fitting 32
2.2. Least squares method (LSM) 34
2.3. Linear regression model with two variables 38
2.4. Gauss-Markov theorem. Estimation of error variance a2 41
2.5. Statistical properties of OLS estimates of regression parameters. Hypothesis testing b = bo- Confidence intervals for regression coefficients 46
2.6. Analysis of variation in the dependent variable in regression. Determination coefficient R2 51
2.7. Maximum likelihood estimation of regression coefficients 55
Exercises 58
3. Multiple regression model 67
3.1. Main hypotheses 68
3.2. Least square method. Gauss-Markov theorem 69
3.3. Statistical properties of OLS estimators 72
3.4. Analysis of variation in the dependent variable in regression. R2 coefficients and adjusted R^, 74
3.5. Testing hypotheses. Confidence intervals and confidence regions 78"
Exercises 88
4. Various aspects of multiple regression 108
4.1. Multicollinearity 109;
4.2. Dummy variables 112
4.3. Partial correlation 118
4.4. Model 124 Specification
Exercises 135
5. Some generalizations of multiple regression 148
5.1. Stochastic regressors 149
5.2. Generalized least squares method.... 154
5.3. Available Generalized Least Squares 160
Exercises 163
6. Heteroscedasticity and correlation over time 167
6.1. Heteroscedasticity 168
6.2. Time correlation 184
Exercises 192
7. Forecasting in regression models 204
7.1. Unconditional forecasting 205
7.2. Conditional Forecasting 208
7.3. Forecasting in the presence of autoregressive errors 209
Exercises 211
8 . Instrumental Variables 212
8.1. Consistency of estimates obtained using instrumental variables 213
8.2. Impact of measurement errors 214
8.3. Two-step least squares method.... 215
8.4. Hausman test 217
Exercises 218
9. Systems of regression equations 220
3.1. Seemingly Unrelated Equations 221
9.1. Systems of simultaneous equations 224
Exercises 241
10. Maximum likelihood method in regression models 244
10.1. Introduction 245
10.2. Mathematical apparatus 246
10.3. Maximum likelihood estimation of the parameters of a multivariate normal distribution. . 248
10.4. Properties of maximum likelihood estimates. 249
10.5. Maximum likelihood estimation in a linear model 250
10.6. Hypothesis testing in a linear model, I 253
10.7. Hypothesis testing in a linear model, II 257
10.8. Nonlinear constraints 258
Exercises 260
11. Time series 264
11.1. Distributed lag models 266
11.2. Dynamic models 268
11.3. Unit roots and cointegration 276
11.4 Box-Jenkins Models (ARIMA) 28
11.5. GARCH models 3
Exercises 3J
12. Discrete dependent variables and censored samples 3
12.1. Binary and multiple choice models... 3!
12.2. Models with trimmed and censored samples 3.
Exercises 3;
13. Panel data 31
13.1 Introduction 3
13.2. Designations and basic models 3
13.3. Fixed effect model 3
13.4. Random effect model 31
13.5. Quality of fit Z1
13.6. Model selection 3"
13.7. Dynamic Models 3
13.8. Binary choice models with panel data 3
13.9. Generalized method of moments 3
Exercises 39
14. Pre-test: introduction 39
14.1. Introduction 3!
14.2. Statement of problem 40
14.3. Main result 40"
14.4. Pretest-assessment 4$
14.5. WALS score 40
14.6. Equivalence theorem 4
14.7. Pre-testing and the underestimation effect 407
14.8. "Understatement" effect. One auxiliary parameter 412
14.9. Model selection: from general to specific and from specific to general 415
14.10. "Understatement" effect. Two auxiliary parameters 419
11. Forecasting and preliminary testing 425
.12. Generalizations 429
13. Other questions 432
Exercises 434
15. Econometrics of financial markets 435
11.5.1. Introduction 436
15.2. Hypothesis of financial market efficiency. . . 438
15.3. Optimization of the securities portfolio 446
15.4. Test for the inclusion of new assets in an effective portfolio 450
15.5. Optimal portfolio in the presence of a risk-free asset 456
15.6. Financial asset valuation models 461
Exercises 471
16. Econometrics perspectives 472
1.6.1. Introduction 472
16.2. What exactly does an econometrician do? .... 473
16.3. Econometrics and Physics 474
16.4. Econometrics and mathematical statistics. . . 475
16.5. Theory and practice 476
16.6. Econometric method 477
16.7. Weak link 480
1.6.8. Aggregation 481
16.9. How to use other works 481
16.10. Conclusion 482
LA application. Linear Algebra 484
1. Vector space 484
2. Vector space LP 485
3. Linear dependence 485
4. Linear subspace 486
5. Basis. Dimension 486
6. Linear operators 487
7. Matrices 488
8. Operations with matrices 489
9. Matrix invariants: trace, determinant 492
10. Matrix rank 494
11. Inverse matrix 495
12. Systems of linear equations 496
13. Eigenvalues and vectors 496
14. Symmetric matrices 498
15. Positive definite matrices 500
16. Idempotent matrices 502
17. Block matrices 503
18. Kronecker product 504
19. Differentiation by vector argument. . 505
Exercises 507
MS application. Probability theory and mathematical statistics 509
1. Random variables, random vectors 509
2. Conditional distributions 516
3. Some special distributions 518
4. Multivariate normal distribution 524
5. Law of large numbers. Central limit theorem 528
6 Basic concepts and tasks of mathematical statistics 531
7. Parameter estimation 533
8. Hypothesis testing 539
EP application. Review of econometric packages 542
1. Origin of packets. Windows versions. Graphics 543
2. About some 544 packages
3. Practical work experience 546
Application ST. Brief English-Russian dictionary of terms 547
TA application. Tables 555
Literature 561
Subject index 570

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