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About the Author
J. Edward Taylor is Professor of Agricultural and Resource Economics at the University of California, Davis. He has published more than 130 articles, book chapters, and books on topics ranging from international trade to ecotourism, immigration, and rural poverty. He has won research awards from the AAEA and teaching awards from UC Davis. He is listed in Who’s Who in Economics as one of the world’s most cited economists. A former editor of the American Journal of Agricultural Economics, he has worked on projects with the United Nations, the World Bank, and other agencies, as well as a number of foreign governments.
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Essentials of Applied Econometrics
By Aaron Smith, J. Edward Taylor
UNIVERSITY OF CALIFORNIA PRESSCopyright © 2017 The Regents of the University of California
All rights reserved.
Introduction to Econometrics
It is interesting that people try to find meaningful patterns in things that are essentially random.
— Data, Star Trek
Upon completing the work in this chapter, you will be able to:
* Define and describe the basics of econometrics
* Describe how to do an econometric study
Jaime Escalante was born in Bolivia in 1930. He immigrated to the United States in the 1960s, hoping for a better life. After teaching himself English and working his way through college, he became a teacher at Garfield High School in East Los Angeles. Jaime believed strongly that higher math was crucial for building a successful career, but most of the students at Garfield High, many of whom came from poor backgrounds, had very weak math skills. He worked tirelessly to transform these kids into math whizzes. Incredibly, more than a quarter of all the Mexican-American high school students who passed the AP calculus test in 1987 were taught by Jaime.
Hollywood made a movie of Jaime's story called "Stand and Deliver." If you haven't seen that movie, you've probably seen one of the other dozens with a similar plot. An inspiring and unconventional teacher gets thrown into an unfamiliar environment filled with struggling or troubled kids. The teacher figures out how to reach the kids, they perform well in school, and their lives change forever.
We all have stories of an inspiring teacher we once had. Or a terrible teacher we once had. Meanwhile, school boards everywhere struggle with the question of how to teach kids and turn them into economically productive adults. Do good teachers really make all the difference in our lives? Or do they merely leave us with happy memories? Not every school can have a Jaime Escalante. Is more funding for public schools the answer? Smaller class sizes? Better incentives for teachers? Technology?
Econometrics can provide answers to big questions like these.
WHAT IS ECONOMETRICS?
Humans have been trying to make sense of the world around them for as long as anyone knows. Data bombard our senses: movements in the night sky, the weather, migrations of prey, growth of crops, spread of pestilence. We have evolved to have an innate curiosity about these things, to seek patterns in the chaos (empirics), then explanations for the patterns (theories). Much of what we see around us is random, but some of it is not. Sometimes our lives have depended on getting this right: predicting where to find fish in the sea (and being smart enough to get off the sea when a brisk nor'easter wind starts to blow), figuring out the best time to plant a crop, or intervening to arrest the spread of a plague. A more complex world gives us ever more data we have to make sense of, from climate change to Google searches to the ups and downs of the economy.
Econometrics is about making sense of economic data (literally, it means "economy measurement"). Often, it is defined as the application of statistics to economic data, but it is more than that. To make sense of economic data, we usually need to understand something about the unseen processes that create these data. For example, we see differences in people's earnings and education (years of completed schooling). Econometric studies consistently find that there is a positive relationship between the two variables. Can we use people's schooling to predict their earnings? And if we increase people's schooling, can we say that their earnings will increase?
These are two different questions, and they get at the hardest part of econometrics — distilling causation from correlation. We may use an econometric model to learn that people with a college degree earn more than those without one. That is a predictive, or correlative, relationship. We don't know whether college graduates earn more because of useful things they learned in college — that is, whether college causes higher earnings. College graduates tend to have high IQ, and they might have earned a lot regardless of whether or not they went to college. Mark Twain (who was not educated beyond elementary school) once said: "I've never let my school interfere with my education." He might have had a point.
Often, an econometrician's goal is to determine whether some variable, X, causes an outcome, Y. But not all of econometrics is about causation. Sometimes we want to generate predictions and other times test a theory. Clearly defining the purpose of an econometrics research project is the first step toward getting credible and useful results. The second step is to formulate your research design and specify your econometric model, and the final step is to apply statistical theory to answer the question posed in step 1.
Most of your first econometrics course focuses on step 3, but don't forget steps 1 and 2! Throughout the book, we will remind you of these steps. Next, we discuss each of the three steps to put the rest of the book in context.
STEP 1: WHAT DO YOU WANT TO DO?
The first step in doing econometrics is to define the purpose of the modeling. It is easy to skip this step, but doing so means your analysis is unlikely to be useful.
Your purpose should be concrete and concise. "I want to build a model of the economy" is not enough. What part of the economy? What do you want to learn from such a model? Often, if you can state your purpose in the form of a question, you will see whether you have defined it adequately.
Here are some examples.
Do Good Teachers Produce Better Student Outcomes?
To estimate whether good teachers improve life outcomes, we first need to measure teacher quality. In a 2014 study, Raj Chetty, John Friedman, and Jonah Rockoff constructed measures of how much an above-average teacher improves students' test scores over what they would have been with an average teacher. These are called "value-added" (VA) measures of teacher quality and were estimated using detailed data on elementary school records from a large urban school district. This research was deemed so important that it was presented in not one but two papers in the most prestigious journal in economics, the American Economic Review.
Chetty and his coauthors used econometrics with their VA measures to show that replacing an average teacher with a teacher whose VA is in the top 5% would increase students' earnings later in life by 2.8%. This might seem small, but the average 12-year-old in the United States can expect lifetime earnings of $522,000, so a 2.8% earnings bump is worth about $14,500 per student. Multiply that by 20 kids per classroom and an excellent teacher starts to look really valuable. It works the other way too — teachers with low VA potentially have large negative effects on lifetime earnings.
Does the Law of Demand Hold for Electricity?
In microeconomic theory, the law of demand predicts that when the price of a good rises, demand for the good falls. Does this theoretical prediction hold up in the real world? Is the own-price elasticity of demand really negative? How large is it? Finding a negative correlation between price and demand is consistent with economic theory; finding the opposite is not.
Katrina Jessoe and David Rapson asked this question using data on residential electricity consumption. They conducted an experiment in which they divided homes randomly into three groups. The first group faced electricity prices that jumped by 200–600% on certain days of the year. The second group faced the same price rises but also were given an electronic device that told them in real time how much electricity they were using. The third group was the control group: they experienced no change in their electricity prices.
Jessoe and Rapson used econometrics to estimate that consumers in the first group did not change their consumption significantly compared with the control group — they had a price elasticity of demand close to zero. However, the second group had a price elasticity of demand of ?0.14. Conclusion: the law of demand holds for electricity, but only if consumers know how much electricity they are using in real time. Without this knowledge, they don't know how much electricity is used when they run the air conditioner or switch on a light, so they don't respond to a price change.
Is It Possible to Forecast Stock Returns?
Lots of people think they can make money in the stock market. We often receive emails informing us of the next greatest stock tip. Business TV channels are full of people yelling about how to make money in the stock market. Every time the market crashes, there's a great story about the genius investor who saw it all coming and made money during the crash. But if it's so easy to make money in the stock market, why isn't everyone doing it?
Based on the theory of efficient financial markets, many economists cast a skeptical eye on claims that the stock market is highly predictable. If everyone knew the market was going to go up, then it would have already done so. However, economic theory also predicts that financial investments should have returns in proportion to risk. Riskier investments should have higher returns on average. So, if you can measure risk in the stock market, then you should be able to predict returns to some extent.
Ivo Welch and Amit Goyal took a large number of variables that people claimed could predict stock returns and used econometrics to test whether any of them actually could. They conclude that there is little, if any, statistical evidence of stock return predictability. So next time you hear a prognosticator claiming that the stock market is about to crash because it crashed the last seven times the president went skiing on a Tuesday ... or something. ..., change the channel!
STEP 2: FORMULATE YOUR RESEARCH DESIGN AND SPECIFY THE ECONOMETRIC MODEL
This step typically requires some economic theory, common sense, and a little cleverness. It is where you take your abstract objective from step 1 and convert it into an econometric model with data that can answer your questions.
Making a good choice about what data to collect and use determines whether you will be able to meet your objective. Let's look again at the three studies we highlighted above.
Do Good Teachers Produce Better Student Outcomes?
Microeconomic theory of the firm provides us with a theory of how a teacher might affect earnings. It's called human capital theory. Human capital theory predicts that workers are paid the marginal value product of their labor (MVPL). A firm will not hire a worker unless the additional value she produces (her MVPL) is at least as large as what the firm will have to pay the worker (i.e., the wage). Characteristics that raise workers' productivity, like intelligence, ability to concentrate and willingness to work hard, should be associated with higher wages. Having had a good teacher is one characteristic that may raise productivity.
One possible research design would be to build an econometric model of the determinants of test scores. If students in teacher A's class get better than average test scores, then teacher A must be a good teacher. There is a big problem with that approach. Classes with a lot of students from disadvantaged backgrounds will tend to get lower-than-average scores no matter how good the teacher is (unless the teacher is the subject of a Hollywood movie).
This is why Chetty and his coauthors developed their VA measures. Their method first predicts test scores for thousands of students based on variables such as last year's test score and family socioeconomic characteristics. Next, they look at how well each student does relative to the prediction. If the students in a class tend to do better than predicted, then Chetty and coauthors assign a high VA score to the teacher. They conduct a series of tests of their VA measure. For example, they look at what happens when an average teacher leaves a school and is replaced by a teacher with a higher VA. They find that test scores jump up from the previous year, which validates their method.
Does the Law of Demand Hold for Electricity?
Microeconomic theory posits that the price and quantity in a market are determined at the point where supply equals demand. When some exogenous shock (like a new invention) shifts the supply outward, the price drops to convince consumers to buy more. When demand increases, the price rises to convince suppliers to produce more. When testing the law of demand, an econometrician wants to see how much consumers respond when a supplier changes the price. The problem is that often in the real world the price is high because consumers really like the product.
When the weather gets hot, consumers turn on their air conditioners and their electricity consumption goes up. This weather-induced increase in demand could cause electricity prices to go up. It would generate a positive correlation between consumption and price. Should we conclude, then, that the law of demand is false? Of course not, because the positive correlation comes from high demand causing high prices, not high prices causing more demand.
One way to think about whether you have a good research design is to imagine what experiment you would run if you could. Jessoe and Rapson went one better and actually ran it. They convinced an electric utility to let them raise prices for a random set of customers on hot days and keep prices the same for other customers. Because they were controlling prices themselves, and because they randomly assigned who got the high prices rather than cherry picking receptive customers, they could be confident they were really measuring consumer responses. This is an example of experimental economics.
Even if you can't run the experiment, thinking about how you would conduct that experiment can help you figure out whether you have data that can answer your research question.
Is It Possible to Forecast Stock Returns?
Finance theory says that stock returns should be higher when investors are more risk averse. (A higher return is needed to incentivize these investors to take on more risk and invest.) Researchers have proposed many measures of risk – often referred to as systematic risk factors. For example, if you have a high chance of losing your job, you are likely to be more risk averse than otherwise. You would only want to put your money in something risky like stocks if the price were low enough and the future expected gains high enough to make it worth the risk.
Welch and Goyal used this theory and the research from past studies to come up with a list of predictors to test. Without theoretical guidance on which predictors to consider, the possibilities would be endless.
Even with a theoretically motivated group of predictors, demonstrating whether a predictor works is hard because we don't really know how the predictor was chosen. A previous researcher may have engaged in data snooping, which means they searched repeatedly until they found a variable that correlated significantly with stock returns and then made up an economic story about why it measures risk aversion. If you search hard enough, you will find a significant looking but meaningless correlation. Welch and Goyal account for this possibility using what is called an out-of-sample test. They fit econometric models using data prior to 1965, and then they see whether the predictors that perform well prior to 1965 continue to work after 1965.
In each of these studies, the researchers applied some economic theory, some common sense, and a little cleverness. The best sources of theoretical insights are the intermediate theory courses that most likely were a prerequisite for this econometrics course. Throughout this book, we will refer to what we learned in our theory courses as a useful resource in building econometric models.
The researchers each thought about mathematical form of their models. Does Y increase linearly with X? Is this relationship likely to be quadratic instead of linear? Logarithmic? Which control variables should be included in the equation? We can use econometrics to test whether the relationship between X and Y is linear or nonlinear, and what mathematical form is best to predict an outcome of interest. For example, we might find that there is a significant relationship between X and Y that is evident using a nonlinear model but not a linear one. In short, we need both economic theory and mathematics — plus a little experimentation with functional forms — to come up with the model we want to estimate.
STEP 3: APPLY STATISTICAL THEORY
Statistical theory helps us fill the gap between the numbers we compute from our data and the broader world. There is a gap because no econometric model can perfectly predict every data point and because we usually only observe data from a sample of the population.
Suppose you are trying to predict earnings of individuals for a population of 100,000 people. Now, imagine you get really lucky: someone hands you data on earnings and years of education for the whole population. That's right, 100,000 people. (It sounds like a lot of data, but Aaron and Ed work with samples a lot larger than this sometimes, and it's puny compared to what Google works with!) We said "really lucky" because we almost never have data on whole populations.
Excerpted from Essentials of Applied Econometrics by Aaron Smith, J. Edward Taylor. Copyright © 2017 The Regents of the University of California. Excerpted by permission of UNIVERSITY OF CALIFORNIA PRESS.
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Table of ContentsPreface
About the Authors
1 Introduction to Econometrics
2 Simple Regression
3 Multiple Regression
4 Generalizing from a Sample
5 Properties of Our Estimators
6 Hypothesis Testing and Confidence Intervals
7 Predicting in a Nonlinear World
8 Best of BLUE I: Cross-Section Data and Heteroskedasticity (Assumption CR2)
9 Best of Blue II: Correlated Errors (Assumption CR3)
10 Sample Selection Bias (Assumption CR1)
11 Identifying Causation
12 Instrumental Variables: A Solution to the Endogeneity Problem
Appendix: Critical Values for Commonly Used Tests in Econometrics