Introductory Econometrics: Intuition, Proof, and Practice - Hardcover

Zax, Jeffrey

Hardcover

ISBN 10: 0804772622 ISBN 13: 9780804772624

Publisher: Stanford Economics and Finance, 2011

Synopsis

Introductory Econometrics: Intuition, Proof, and Practice attempts to distill econometrics into a form that preserves its essence, but that is acceptable―and even appealing―to the student's intellectual palate. This book insists on rigor when it is essential, but it emphasizes intuition and seizes upon entertainment wherever possible.

Introductory Econometrics is motivated by three beliefs. First, students are, perhaps despite themselves, interested in questions that only econometrics can answer. Second, through these answers, they can come to understand, appreciate, and even enjoy the enterprise of econometrics. Third, this text, which presents select innovations in presentation and practice, can provoke readers' interest and encourage the responsible and insightful application of econometric techniques.

In particular, author Jeffrey S. Zax gives readers many opportunities to practice proofs―which are challenging, but which he has found to improve student comprehension. Learning from proofs gives readers an organic understanding of the message behind the numbers, a message that will benefit them as they come across statistics in their daily lives.

An ideal core text for foundational econometrics courses, this book is appropriate for any student with a solid understanding of basic algebra―and a willingness to use that tool to investigate complicated issues.

About this title

About the Author

Jeffrey Zax is Professor of Economics at the University of Colorado at Boulder. His research interests include labor economics, public economics, and urban economics. Zax regularly teaches the Econometrics course. He was twice awarded the Stanford Calderwood Teaching Excellence Award.

Excerpt. © Reprinted by permission. All rights reserved.

Introductory Econometrics

Intuition, Proof, and PracticeBy Jeffrey S. Zax

Stanford University Press

Copyright © 2011 Board of Trustees of the Leland Stanford Junior University
All right reserved.
ISBN: 978-0-8047-7262-4

Contents

Contents.................................................................................................ixList of Tables and Figures...............................................................................xvPreface..................................................................................................xxiChapter 1 What Is a Regression?..........................................................................1Chapter 2 The Essential Tool.............................................................................35Chapter 3 Covariance and Correlation.....................................................................57Chapter 4 Fitting a Line.................................................................................88Chapter 5 From Sample to Population......................................................................136Chapter 6 Confidence Intervals and Hypothesis Tests......................................................191Chapter 7 Inference in Ordinary Least Squares............................................................232Chapter 8 What if the Disturbances Have Nonzero Expectations or Different Variances?.....................281Chapter 9 What if the Disturbances Are Correlated?.......................................................321Chapter 10 What if the Disturbances and the Explanatory Variables Are Related?...........................363Chapter 11 What if there is More Than One x?.............................................................411Chapter 12 Understanding and Interpreting Regression with Two x's........................................453Chapter 13 Making Regression More Flexible...............................................................501Chapter 14 More Than Two Explanatory Variables...........................................................541Chapter 15 Categorical Dependent Variables...............................................................582Epilogue.................................................................................................627Appendix.................................................................................................629References...............................................................................................637Index....................................................................................................639

Chapter One

What Is a Regression?

1.0 What We Need to Know When We Finish This Chapter 2 1.1 Why Are We Doing This? 3 1.2 Education and Earnings 5 1.3 What Does a Regression Look Like? 6 1.4 Where Do We Begin? 6 1.5 Where's the Explanation? 7 1.6 What Do We Look for in This Explanation? 9 1.7 How Do We Interpret the Explanation? 12 1.8 How Do We Evaluate the Explanation? 17 1.9 R² and the F-statistic 19 1.10 Have We Put This Together in a Responsible Way? 20 1.11 Do Regressions Always Look Like This? 25 1.12 How to Read This Book 28 1.13 Conclusion 28 Exercises 29

1.0 What We Need to Know When We Finish This Chapter

This chapter explains what a regression is and how to interpret it. Here are the essentials.

1. Section 1.4: The dependent or endogenous variable measures the behavior that we want to explain with regression analysis.

2. Section 1.5: The explanatory, independent, or exogenous variables measure things that we think might determine the behavior that we want to explain. We usually think of them as predetermined.

3. Section 1.5: The slope estimates the effect of a change in the explanatory variable on the value of the dependent variable.

4. Section 1.5: The t-statistic indicates whether the explanatory variable has a discernible association with the dependent variable. The association is discernible if the p-value associated with the t-statistic is .05 or less. In this case, we say that the slope is statistically significant. This generally corresponds to an absolute value of approximately two or greater for the t-statistic itself. If the t-statistic has a p-value that is greater than .05, the associated slope coefficient is insignificant. This means that the explanatory variable has no discernible effect.

5. Section 1.6: The intercept is usually uninteresting. It represents what everyone has in common, rather than characteristics that might cause individuals to be different.

6. Section 1.6: We usually interpret only the slopes that are statistically significant. We usually think of them as indicating the effect of their associated explanatory variables on the dependent variable ceteris paribus, or holding constant all other characteristics that are included in the regression.

7. Section 1.6: Continuous variables take on a wide range of values. Their slopes indicate the change that would be expected in the dependent variable if the value of the associated explanatory variable increased by one unit.

8. Section 1.6: Discrete variables, sometimes called categorical variables, indicate the presence or absence of a particular characteristic. Their slopes indicate the change that would occur in the dependent variable if an individual who did not have that characteristic were given it.

9. Section 1.7: Regression interpretation requires three steps. The first is to identify the discernible effects. The second is to understand their magnitudes. The third is to use this understanding to verify or modify the behavioral understanding that motivated the regression in the first place.

10. Section 1.7: Statistical significance is necessary in order to have interesting results, but not sufficient. Important slopes are those that are both statistically significant and substantively large. Slopes that are statistically significant but substantively small indicate that the effects of the associated explanatory variable can be reliably understood as unimportant.

11. Section 1.7: A proxy is a variable that is related to, but not exactly the variable we really want. We use proxies when the variables we really want aren't available. Sometimes this makes interpretation difficult.

12. Section 1.8: If the p-value associated with the F-statistic is .05 or less, the collective effect of the ensemble of explanatory variables on the dependent variable is statistically significant.

13. Section 1.8: Observations are the individual examples of the behavior under examination. All of the observations together constitute the sample on which the regression is based.

14. Section 1.8: The R², or coefficient of determination, represents the proportion of the variation in the dependent variable that is explained by the explanatory variables. The adjusted R² modifies the R² in order to take account of the numbers of explanatory variables and observations. However, neither measures statistical significance directly.

15. Section 1.9: F-statistics can be used to evaluate the contribution of a subset of explanatory variables, as well as the collective statistical significance of all explanatory variables. In both cases, the F-statistic is a transformation of R² values.

16. Section 1.10: Regression results are useful only to the extent that the choices of variables in the regression, variable construction, and sample design are appropriate.

17. Section 1.11: Regression results may be presented in one of several different formats. However, they all have to contain the same substantive information.

1.1 Why Are We Doing This?

The fundamental question that underlies most of science is, how does one thing affect another? This is the sort of question that we ask ourselves all the time. Whenever we wonder whether our grade will go up if we study more, whether we're more likely to get into graduate school if our grades are better, or whether we'll get a better job if we go to graduate school, we are asking questions that econometrics can answer with elegance and precision.

Of course, we probably think we have answers to these questions already. We almost surely do. However, they're casual and even sloppy. Moreover, our confidence in them is almost certainly exaggerated.

Econometrics is a collection of powerful statistical tools that are devoted to helping provide answers to the question of how one thing affects another. Econometrics not only teaches us how to answer questions like this more accurately but also helps us understand what is necessary in order to obtain an answer that we can legitimately treat as accurate.

We begin in this chapter with a primer on how to interpret regression results. This will allow us to read work based on regression and even to begin to perform our own analyses. We might think that this would be enough.

However, this chapter will not explain why the interpretations it presents are valid. That requires a much more thorough investigation. We prepare for this investigation in chapter 2. There, we review the summation sign, the most important mathematical tool for the purposes of this book.

We actually embark on this investigation in chapter 3, where we consider the precursors to regression: the covariance and the correlation. These are basic statistics that measure the association between two variables, without regard to causation. We might have seen them before. We return to them in detail because they are the mathematical building blocks from which regressions are constructed.

Our primary focus, however, will be on the fundamentals of regression analysis. Regression is the principal tool that economists use to assess the responsiveness of some outcome to changes in its determinants. We might have had an introduction to regression before as well. Here, we devote chapters 4, 5, and 7 through 14 to a thorough discussion.

Chapter 6 intervenes with a discussion of confidence intervals and hypothesis tests. This material is relevant to all of statistics, rather than specific to econometrics. We introduce it here to help us complete the link between the regression calculations of chapter 4 and the behavior that we hope they represent, discussed in chapter 5.

Chapter 15 discusses what we can do in a common situation where we would like to use regression, but where the available information isn't exactly appropriate for it. This discussion will introduce us to probit analysis, an important relative of regression. More generally, it will give us some insight as to how we might proceed when faced with other situations of this sort.

As we learn about regression, we will occasionally need concepts from basic statistics. Some of us may have already been exposed to them. For those of us in this category, chapters 3 and 6 may seem familiar, and perhaps even chapter 4. For those of us who haven't studied statistics before, this book introduces and reviews each of the relevant concepts when our discussion of regression requires them.

1.2 Education and Earnings

Few of us will be interested in econometrics purely for its theoretical beauty. In fact, this book is based on the premise that what will interest us most is how econometrics can help us organize the quantitative information that we observe all around us. Obviously, we'll need examples.

There are two ways to approach the selection of examples. Econometric analysis has probably been applied to virtually all aspects of human behavior. This means that there is something for everyone. Why not provide it?

Well, this strategy would involve a lot of examples. Most readers wouldn't need that many to get the hang of things, and they probably wouldn't be interested in a lot of them. In addition, they could make the book a lot bigger, which might make it seem intimidating.

The alternative is to focus principally on one example that may have relatively broad appeal and develop it throughout the book. That's the choice here. We will still sample a variety of applications over the course of the entire text. However, our running example returns, in a larger sense, to the question of section 1.1: Why are we doing this? Except now, let's talk about college, not this course.

Presumably, at least some of the answer to that question is that we believe college prepares us in an important way for adulthood. Part of that preparation is for jobs and careers. In other words, we probably believe that education has some important effect on our ability to support ourselves.

This is the example that we'll pursue throughout this book. In the rest of this chapter, we'll interpret a somewhat complicated regression that represents the idea that earnings are affected by several determinants, with education among them. In chapter 3, we'll return to the basics and simply ask whether there's an association between education and earnings. Starting in chapter 4, we'll assume that education affects earnings and ask: by how much? In chapter 10, we'll examine whether the assumption that education causes earnings is acceptable, and what can be done if it's not.

As we can see, we'll ask this question with increasing sophistication as we proceed through this book. The answers will demonstrate the power of econometric tools to address important quantitative questions. They will also serve as illustrations for applications to other questions. Finally, we can hope that they will confirm our commitment to higher education.

1.3 What Does a Regression Look Like?

Figure 1.1 is one way to present a regression.

Does that answer the question?

Superficially, yes. But what does it all mean?

This question can be answered on two different levels. In this chapter, we'll talk about how to interpret the information in figure 1.1. This should put us in a position to read and understand other work based on regression analysis. It should also allow us to interpret regressions of our own.

In the rest of the book, we'll talk about why the interpretations we offer here are valid. We'll also talk about the circumstances under which these interpretations may have to be modified or may even be untrustworthy. There will be a lot to say about these matters. But, for the moment, it will be enough to work through the mystery of what figure 1.1 could possibly be trying to reveal.

1.4 Where Do We Begin?

The first thing to understand about regression is the very first thing in figure 1.1. The word "earnings" identifies the dependent variable in the regression. The dependent variable is also referred to as the endogenous variable.

The dependent variable is the primary characteristic of the entities whose behavior we are trying to understand. These entities might be people, companies, governments, countries, or any other choice-making unit whose behavior might be interesting. In the case of figure 1.1, we might wonder if "earnings" implies that we're trying to understand company profits. However, "earnings" here refers to the payments that individuals get in return for their labor. So the entities of interest here are workers or individuals who might potentially be workers.

"Dependent" and "endogenous" indicate the role that "earnings" plays in the regression of figure 1.1. We want to explain how it gets determined. "Dependent" suggests that "earnings" depends on other things. "Endogenous" implies the same thing, though it may be less familiar. It means that the value of "earnings" is determined by other, related pieces of information.

The question of what it means to "explain" something statistically can actually be quite subtle. We will have some things to say about this in chapters 4 and 10. Initially, we can proceed as if we believe that the things that we use to "explain" earnings actually "cause" earnings.

1.5 Where's the Explanation?

Most of the rest of figure 1.1, from the equality sign to "(-.199)," presents our explanation of earnings. The equality sign indicates that we're going to represent this explanation in the form of an equation. On the right side of the equation, we're going to combine a number of things algebraically. Because of the equality, it looks as though the result of these mathematical operations will be "earnings." Actually, as we'll learn in chapter 4, it will be more accurate to call this result "predicted earnings."

The material to the right of the equality sign in figure 1.1 is organized into terms. The terms are separated by signs for either addition (+) or subtraction (-). Each term consists of a number followed by the sign for multiplication (x), a word or group of words, and a second number in parentheses below the first.

In each term, the word or group of words identifies an explanatory variable. An explanatory variable is a characteristic of the entities in question, which we think may cause, or help to create, the value that we observe for the dependent variable.

Explanatory variables are also referred to as independent variables. This indicates that they are not "dependent." For our present purposes, this means that they do not depend on the value of the dependent variable. Their values arise without regard to the value of the dependent variable.

Explanatory variables are also referred to as exogenous variables. This indicates that they are not "endogenous." Their values are assigned by economic, social, or natural processes that are not under study and not affected by the process that is. The variables listed in the terms to the right of the equality can be thought of as causing the dependent variable, but not the other way around. We often summarize this assumption as "causality runs in only one direction."

This same idea is sometimes conveyed by the assertion that the explanatory variables are predetermined. This means that their values are already known at the moment when the value of the dependent variable is determined. They have been established at an earlier point in time. The point is that, as a first approximation, behavior that occurs later, historically, can't influence behavior that preceded it.

This proposition is easy to accept in the case of the regression of figure 1.1. Earnings accrue during work. Work, or at least a career, typically starts a couple of decades into life. Racial or ethnic identity and sex are usually established long before then. Age accrues automatically, starting at birth. Schooling is usually over before earnings begin as well. Therefore, it would be hard to make an argument at this stage that the dependent variable, earnings, causes any of the explanatory variables.

(Continues...)

Excerpted from Introductory Econometricsby Jeffrey S. Zax Copyright © 2011 by Board of Trustees of the Leland Stanford Junior University. Excerpted by permission of Stanford University Press. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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Edition details

• Publisher: Stanford Economics and Finance
• Publication date: 2011
• Language: English
• ISBN 10: 0804772622
• ISBN 13: 9780804772624
• Binding: Hardcover
• Edition number: 1
• Number of pages: 672