Sometimes inference is straightforward, as when you use Census micro-data samples to study the American population. Often inference is more complex, however, especially with data that are clustered or grouped. Although inference issues are rarely very exciting, and often quite technical, the ultimate success of even a well-conceived and conceptually exciting project turns on the details of statistical inference.
This sometimes-dispiriting fact inspired the following econometrics haiku, penned by then-econometrics-Ph. As should be clear from the above discussion, the four research FAQs are part of a process of project development.
In other words, issues that arise once your research agenda has been set. Before turning to the nuts and bolts of empirical work, however, we begin with a more detailed explanation of why randomized trials give us our benchmark.
For instance, on the planet Earth, man had always assumed that he was more intelligent than dolphins because he had achieved so much— the wheel, New York, wars and so on— while all the dolphins had ever done was muck about in the water having a good time. But conversely, the dolphins had always believed that they were far more intelligent than man—for precisely the same reasons.
In fact there was only one species on the planet more intelligent than dolphins, and they spent a lot of their time in behavioral research laboratories running round inside wheels and conducting frighteningly elegant and subtle experiments on man. The Perry treatment group was randomly assigned to an intensive intervention that included preschool education and home visits. Dozens of academic studies cite or use the Perry …ndings see, e. Most importantly, the Perry project provided the intellectual basis for the massive Head Start pre-school program, begun in , which ultimately served and continues to serve millions of American children.
The Perry intervention seems to have done nothing for boys. To be concrete, consider a simple example: Do hospitals make people healthier? For our purposes, this question is allegorical, but it is surprisingly close to the sort of causal question health economists care about.
Some of these patients are admitted to the hospital. In fact, exposure to other sick patients by those who are themselves vulnerable might have a net negative impact on their health.
But will the data back this up? The natural approach for an empirically-minded person is to compare the health status of those who have been to the hospital to the health of those who have not. Error Hospital 2. Taken at face value, this result suggests that going to the hospital makes people sicker. Moreover, even after hospitalization people who have sought medical care are not as healthy, on average, as those who never get hospitalized in the …rst place, though they may well be better than they otherwise would have been.
The outcome of interest, a measure of health status, is denoted by yi. To address this question, we assume we can imagine what might have happened to someone who went to the hospital if they had not gone and vice versa. A naive comparison of averages by hospitalization status tells us something about potential outcomes, though not necessarily what we want to know.
Important references developing this idea are Rubin , , and Holland , who refers to a causal framework involving potential outcomes as the Rubin Causal Model.
The main thing, however, is that random assignment of di eliminates selection bias. This does not mean that randomized trials are problem-free, but in principle they solve the most important problem that arises in empirical research. How relevant is our hospitalization allegory? Experiments often reveal things that are not what they seem on the basis of naive comparisons alone.
A recent example from medicine is the evaluation of hormone replacement therapy HRT. This is a medical intervention that was recommended for middle-aged women to reduce menopausal symptoms. In contrast, the results of a recently completed randomized trial shows few bene…ts of HRT.
An iconic example from our own …eld of labor economics is the evaluation of government-subsidized training programs. The idea is to increase employment and earnings. Here too, selection bias is a natural concern since subsidized training programs are meant to serve men and women with low earnings potential. Not surprisingly, therefore, simple comparisons of program participants with non-participants often show lower earnings for the participants.
Randomized trials are not yet as common in social science as in medicine but they are becoming more prevalent. One area where the importance of random assignment is growing rapidly is education research Angrist, Congress mandates the use of rigorous experimental or quasi-experimental research designs for all federally-funded education studies. We can therefore expect to see many more randomized trials in education research in the years to come.
A key question in research on education production is which inputs produce the most learning given their costs. One of the most expensive inputs is class size - since smaller classes can only be had by hiring more teachers. The STAR experiment was meant to answer this question. Many studies of education production using non-experimental data suggest there is little or no link be- tween class size and student learning.
So perhaps school systems can save money by hiring fewer teachers with no consequent reduction in achievement. The observed relation between class size and student achieve- ment should not be taken at face value, however, since weaker students are often deliberately grouped into smaller classes.
A randomized trial overcomes this problem by ensuring that we are comparing apples to apples, i. The study ran for four years, i. Schools with at least three classes in each grade could choose to participate in the experiment. Unfortunately, the STAR data fail to include any pre-treatment test scores, though it is possible to look at characteristics of children such as race and age. Table 2. The student Table 2. Free lunch. Age in 5. Attrition rate. Class size in kindergarten Percentile score in kindergarten The table shows means of variables by treatment status.
The P -value in the last column is for the F -test of equality of variable means across all three groups. All variables except attrition are for the …rst year a student is observed, The free lunch variable is the fraction receiving a free lunch.
The percentile score is the average percentile score on three Stanford Achievement Tests. The attrition rate is the proportion lost to follow up before completing third grade. Free lunch status is a good measure of family income, since only poor children qualify for a free school lunch. This suggests the random assignment worked as intended. The attrition rate was lower in small kindergarten classrooms.
This is potential a problem, at least in principle. The dependent variable is the Stanford Achievement Test percentile score. Robust standard errors that allow for correlated residuals within classes are shown in parentheses. The sample size is In many cases, such trials are impractical. Pupils who repeated or skipped a grade left the experiment. Students who entered an experimental school one grade later were added to the experiment and randomly assigned to one of the classes.
There was also some switching of children after the kindergarten year. We hope to …nd natural or quasi-experiments that mimic a randomized trial by changing the variable of interest while other factors are kept balanced.
Can we always …nd a convincing natural experiment? Of course not. Nevertheless, we take the position that a notional randomized trial is our benchmark. Not all researchers share this view, but many do. We heard it …rst from our teacher and thesis advisor, Orley Ashenfelter, a pioneering proponent of experiments and quasi-experimental research designs in social science. Here is Ashenfelter assessing the credibility of the observational studies linking schooling and income: How convincing is the evidence linking education and income?
Here is my answer: Pretty con- vincing. If I had to bet on what an ideal experiment would indicate, I bet that it would show that better educated workers earn more.
The quasi-experimental study of class size by Angrist and Lavy illustrates the manner in which non-experimental data can be analyzed in an experimental spirit. The Angrist and Lavy study relies on the fact that in Israel, class size is capped at Therefore, a child in a …fth grade cohort of 40 students ends up in a class of 40 while a child in …fth grade cohort of 41 students ends up in a class only half as large because the cohort is split.
This is in marked contrast with naive analyses, also reported by Angrist and Lavy, based on simple comparisons between those enrolled in larger and smaller classes. These comparisons show students in smaller classes doing worse on standardized tests. The hospital allegory of selection bias would therefore seem to apply to the class-size question as well.
With extremely well implemented randomized trial. In the hospital allegory, those who were treated had poorer health outcomes in the no-treatment state, while in the Angrist and Lavy study, students in smaller classes tend to have intrinsically lower test scores. The remainder of Table 2. Covariates play two roles in regression analyses of experimental data. First, the STAR experimental design used conditional random assignment. The comparison in column 1 of Table 2. We saw before that these individual characteristics are balanced across class types, i.
This is a point we expand on in Chapter 3. Although the control variables, Xi , are uncorrelated with di , they have substantial explanatory power for yi. Including these control variables therefore reduces the residual variance, which in turn lowers the standard error of the regression estimates.
The last column adds teacher characteristics. Regression plays an exceptionally important role in empirical economic research. Some regressions are simply descriptive tools, as in much of the research on earnings inequality. In some cases, regression can also be used to approximate experiments in the absence of random assignment. But before we can get into the important question of when a regression is likely to have a causal interpretation, it is useful to review a number of fundamental regression facts and properties.
These facts and properties are reliably true for any regression, regardless of your purpose in running it. I was still mostly interested in a career in special education, and had planned to go back to work as an orderly in a state mental hospital, my previous summer job. My research assistant duties included data collection and regression analysis, though I did not understand regression or even statistics at the time. The paper I was working on that summer Meltzer and Richard, , is an attempt to link the size of governments in democracies, measured as government expenditure over GDP, to income inequality.
Most income distributions have a long right tail, which means that average income tends to be way above the median. When inequality grows, more voters …nd themselves with below-average incomes. Annoyed by this, those with incomes between the median and the average may join those with incomes below the median in voting for …scal policies which - following Robin Hood - take from the rich and give to the poor. The size of government consequently increases. You might say this project marked the beginning of my interest in the social returns to education, a topic I went back to with more enthusiasm and understanding in Acemoglu and Angrist Today, I understand the Meltzer and Richard study as an attempt to use regression to uncover and quantify an interesting causal relation.
At the time, however, I was purely a regression mechanic. Sometimes I found the RA work depressing. The best part of the job was lunch with Alan Meltzer, a distinguished scholar and a patient and good-natured supervisor, who was happy to chat while we ate the contents of our brown-bags this did not take long as Allan ate little and I ate fast.
I remember asking Allan whether he found it satisfying to spend his days perusing regression output, which then came on reams of double-wide green-bar paper. Meltzer laughed and said there was nothing he would rather be doing.
Now, we too spend our days at least, the good ones happily perusing regression output, in the manner of our teachers and advisors in college and graduate school. This chapter explains why. Because the regressor of interest in the class size study discussed in Section 2. In most cases, however, regression is used with observational data. Without the bene…t of random assignment, regression estimates may or may not have a causal interpretation.
We return to the central question of what makes a regression causal later in this chapter. Setting aside the relatively abstract causality problem for the moment, we start with the mechanical properties of regression estimates. As applied econometricians, however, we believe we can summarize and interpret randomness in a useful way. On average, people with more schooling earn more than people with less schooling. The connection between schooling and average earnings has considerable predictive power, in spite of the enormous variation in individual circumstances that sometimes clouds this fact.
Of course, the fact that more educated people earn more than less educated people does not mean that schooling causes earnings to increase.
The question of whether the earnings-schooling relationship is causal is of enormous importance, and we will come back to it many times. This predictive power is compellingly summarized by the conditional expectation function CEF. The CEF for a dependent variable, yi given a k 1 vector of covariates, Xi with elements xki is the expectation, or population average of yi with Xi held …xed.
The population average can be thought of as the mean in an in…nitely large sample, or the average in a completely enumerated …nite population. Expectation is a population concept.
In practice, data usually come in the form of samples and rarely consist of an entire population. We therefore use samples to make inferences about the population.
This is always necessary but we postpone a discussion of the formal inference step taking us from sample to population until Section 3. The distribution of earnings is also plotted for a few key values: 4, 8, 12, and 16 years of schooling. The CEF in the …gure captures the fact that— the enormous variation individual circumstances notwithstanding— people with more schooling generally earn more, on average.
The average earnings gain associated with a year of schooling is typically about 10 percent. Figure 3. An important complement to the CEF is the law of iterated expectations. This law says that an unconditional expectation can be written as the population average of the CEF. The power of the law of iterated expectations comes from the way it breaks a random variable into two pieces.
Theorem 3. The CEF is a good summary of the relationship between yi and Xi for a number of reasons. First, we are used to thinking of averages as providing a representative value for a random variable. Let m Xi be any function of Xi. The last term is minimized at zero when m Xi is the CEF. In fact, the validity of linear regression as an empirical tool does not turn on linearity either.
In our world, this question or one like it is heard almost every day. The link between regression functions — i. To lay out these explanations precisely, it helps to be precise about the regression function we have in mind. At this point, we are not worried about causality. It bears emphasizing that this error term does not have a life of its own. In the multivariate case, i. The joint distribution is what you would observe if you had a complete enumeration of the population of interest or knew the stochastic process generating the data.
These reasons can be summarized by saying that you are interested in regression parameters if you are interested in the CEF. Then the population regression function is it. The classic 0 scenario is joint Normality, i. This is the scenario considered by Galton , father of regression, who was interested in the intergenerational link between Normally distributed traits such as height and intelligence. The Normal case is clearly of limited empirical relevance since regressors and dependent variables are often discrete, while Normal distributions are continuous.
Another linearity scenario arises when regression models are saturated. As reviewed in Section 3. Such models are inherently linear, a point we also discuss in Section 3.
The CEF-approximation problem, 3. These two theorems show us two more ways to view regression. Regression provides the best linear predictor for the dependent variable in the same way that the CEF is the best unrestricted predictor of the dependent variable. On the other hand, if we prefer to think about approximating E[yi jXi ], as opposed to predicting yi , the Regression-CEF theorem tells us that even if the CEF is nonlinear, regression provides the best linear approximation to it.
The regression-CEF theorem is our favorite way to motivate regression. The linear CEF theorem is for special cases only. The best linear predictor theorem is satisfyingly general, but it encourages an overly clinical view of empirical research.
The weights are given by the distribution of Xi , i. One of the estimation strategies used in this project regresses civilian earnings on a dummy for veteran status, along with personal characteristics and the variables used by the military to screen soldiers.
The earnings data come from the US Social Security system, but Social Security earnings records cannot be released to the public. Instead of individual earnings, Angrist worked with average earnings conditional on race, sex, test scores, education, and veteran status. An illustration of the grouped-data approach to regression appears below. This 0 variance depends on the microdata, in particular, the second-moments of Wi yi ; X0i , a point we elaborate on in the next section.
We therefore draw statistical inferences about these quantities using samples. Statistical inference is what much of traditional econometrics is about. Suppose the vector Wi is independently and identically distributed in yi ; X0i PN a sample of size N.
By the law of large numbers, this sample moment gets arbitrarily close to the corresponding population moment as the sample size grows. We might similarly consider higher-order moments of the elements of Wi , e. Following this principle, the method of moments estimator of replaces each expectation by a sum. Important pitfalls and problems with this asymptotic theory are covered in the last chapter.
Sometimes not very often we do too. Sample is limited to white men, age Derived from Stata regression output. Old- fashioned standard errors are the default reported. Robust standard errors are heteroscedasticity-consistent. Panel A uses individual-level data. Panel B uses earnings averaged by years of schooling. Before deriving this distribution, it helps to record the general asymptotic distribution theory that covers our needs. This basic theory can be stated mostly in words.
For the purposes of these statements, we assume the reader is familiar with the core terms and concepts of statistical theory e. For de…nitions of these terms and a formal mathematical statement of the theoretical propositions given below, see, e.
In other words, the probability that the sample mean is close to the population mean can be made as high as you like by taking a large enough sample. The covariance matrix is given by the variance of the underlying random variable.
In other words, in large enough samples, appropriately normalized sample moments are approximately Normally distributed. Formally, let aN be a statistic with a limiting distribution and let bN be a statistic with probability limit b. This allows us to replaces some sample moments by population moments i. Then aN bN and aN b have the same asymptotic distribution.
For example, the probability limit of any continuous function of a sample moment is the function evaluated at the corresponding population moment. Most scalar functions of this random variable are also asymptotically Normally distributed, with covariance matrix given by a quadratic form with the covariance matrix of the random variable on the inside and the gradient of the function evaluated at the probability limit of the random vari- able on the outside.
It remains only to …nd the covariance matrix of the asymptotic distribution from the gradient of this function. An easier and more instructive derivation uses the Slutsky and central limit theorems. This is as good a place as any to point out that these residuals are uncorrelated with the regressors by de…nition of.
We return to this important point in the discussion of causal regression models in Section 3. By the central limit theorem, this is asymptotically Normally distributed with mean zero and covariance matrix E[Xi X0i e2i ], since this fourth mo- ment is the covariance matrix of Xi ei.
These standard errors are said to be robust because, in large enough samples, they provide accurate hypothesis tests and con…dence intervals given minimal assumptions about the data and model. In particular, our derivation of the limiting distribution makes no assumptions other than those needed to ensure that basic statistical results like the central limit theorem go through. These are not, however, the standard errors that you get by default from packaged software. Our view of regression as an approximation to the CEF makes heteroskedasticity seem natural.
If the CEF is nonlinear and you use a linear model to approximate it, then the quality of …t between the regression line and the CEF will vary with Xi.
Hence, the residuals will be larger, on average, at values of Xi where the …t is poorer. Even if you are prepared to assumed that the conditional variance of yi given Xi is constant, the fact that the CEF is nonlinear means that E[ yi X0i 2 jXi ] will vary with Xi. Our favorite example in this context is the linear probability model LPM. A linear probability model is any regression where the dependent variable is zero-one, i. Suppose the regression model is saturated, so the CEF is linear.
We conclude that LPM residuals are necessarily heteroskedastic unless the only regressor is a constant. These points of principle notwithstanding, as an empirical matter, heteroskedasticity may matter little. In the micro-data schooling regression depicted in Figure 3. These terms originate in an experimentalist tradition that uses regression to model discrete treatment-type variables.
This language is now used more widely in many …elds, however, including applied econometrics. For readers unfamiliar with these terms, this section provides a brief review. Saturated regression models are regression models with discrete explanatory variables, where the model includes a separate parameter for all possible values taken on by the explanatory variables.
For example, when working with a single explanatory variable indicating whether a worker is a college graduate, the model is saturated by including a single dummy for college graduates and a constant. We can also saturate when the regressor takes on many values. This is an important special case of the regression-CEF theorem. If there are two explanatory variables, say one dummy indicating college graduates and one dummy indicating sex, the model is saturated by including these two dummies, their product, and a constant.
This is not the only saturated parameterization; any set of indicators dummies that can be used to identify each value taken on by the covariates produces a saturated model. For example, an alternative saturated model includes dummies for male college graduates, male dropouts, female college graduates, and female dropouts, but no intercept.
Let x1i indicate college graduates and x2i indicate women. Note that there is a natural hierarchy of modeling strategies with saturated models at the top. On the other hand, saturated models generate a lot of interaction terms, many of which may be uninteresting or imprecise. You might therefore sensibly choose to omit some or all of these. Equation 3. This is a good approximation if the returns to college are similar for men and women.
Consequently, the results of estimating 3. For example, this is true for linear probability models and other limited dependent variable models e. This under- standing, however, does not help us with the deeper question of when regression has a causal interpretation.
In particular, we might think of schooling decisions as being made in a series of episodes where the decision-maker might realistically go one way or another, even if certain choices are more likely than others. For example, in the middle of junior year, restless and unhappy, Angrist glumly considered his options: dropping out of high school and hopefully getting a job, staying in school but taking easy classes that lead to a quick and dirty high school diploma, or plowing on in an academic track that leads to college.
Although the consequences of such choices are usually unknown in advance, the idea of alternative paths leading to alternative outcomes for a given individual seems uncontroversial. As we discussed in Chapter 2, experiments ensure that the causal variable of interest is independent of potential outcomes so that the groups being compared are truly comparable. Here, we would like to generalize this notion to causal variables that take on more than two values, and to more complicated situations where we must hold a variety of "control variables" …xed for causal inferences to be valid.
This leads to the conditional independence assumption CIA , a core assumption that provides the sometimes implicit justi…cation for the causal interpretation of regression. This assumption is sometimes called selection-on-observables because the covariates to be held …xed are assumed to be known and observed e. The big question, therefore, is what these control variables are, or should be. For now, we just do the econometric thing and call the covariates "Xi ".
As far as the schooling problem goes, it seems natural to imagine that Xi is a vector that includes measures of ability and family background. For starters, think of schooling as a binary decision, like whether Angrist goes to college.
Denote this by a dummy variable, ci. This is what we would measure if we could go back in time and nudge i onto the road not taken.
We therefore hope to measure the average of y1i y0i , or the average for some group, such as those who went to college. The CIA asserts that conditional on observed characteristics, Xi , selection bias disappears. In this example, the CIA says, fy0i ,y1i g q ci jXi : 3. In an observational study, the CIA means that si can be said to be "as good as randomly assigned," conditional on Xi. But given the CIA, conditional-on-Xi comparisons of average earnings across schooling levels have a causal interpretation.
Given the CIA, however, high school graduation is independent of potential earnings conditional on Xi , so the selection-bias vanishes. In practice, there are many details to worry about when implementing a matching strategy. We …ll in some of the technical details on the mechanics of matching in Section 3. Here we note that a global drawback of the matching approach is that it is not "automatic," rather it requires two steps, matching and averaging.
Estimating the standard errors of the resulting estimates may not be straightforward, either. Two routes can be traced from the CIA to regression.
One assumes that fi s is both linear in s and the same for everyone except for an additive error term, in which case linear regression is a natural tool to estimate the features of fi s. At this point, we want to focus on the conditions required for regression to have a causal interpretation and not on the details of the regression-matching analog.
Again, s is written without an i subscript to index individuals, because equation 3. In this case, however, the only individual-speci…c and random part of fi s is a mean-zero error component, i, which captures unobserved factors that determine potential earnings.
Substituting the observed value si for s in equation 3. Importantly, because equation 3. Because is de…ned by the regression of i on Xi ;the residual vi and Xi are uncorrelated by construction.
It bears emphasizing once again that the key assumption here is that the observable characteristics, Xi , are the only reason why i and si equivalently, fi s and si are correlated. This is the selection-on-observables assumption for regression models discussed over a quarter century ago by Barnow, Cain, and Goldberger It remains the basis of most empirical work in Economics. This important formula is often motivated by the notion that a longer regression, i.
To make this discussion concrete, suppose the set of relevant control variables in the schooling regression can be boiled down to a combination of family background, intelligence and motivation. In practice, ability is hard to measure. What are the consequences of leaving ability out of regression 3.
Cov yi ;si This formula is easy to derive: plug the long regression into the short regression formula, V si : Not surprisingly, the OVB formula is closely related to the regression anatomy formula, 3.
On the other hand, as a matter of economic theory, the direction of the correlation between schooling and ability is not entirely clear. This is not a foregone conclusion, however: Mick Jagger dropped out of the London School of Economics and Bill Gates dropped out of Harvard, perhaps because the opportunity cost of schooling for these high-ability guys was high of course, they may also be a couple of very lucky college dropouts. The omitted variables bias formula tells us that these reductions are a result of the fact that the additional controls are positively correlated with both wages and schooling.
Standard errors are shown in parentheses. The sample is restricted to men and weighted by NLSY sampling weights. Although simple, the OVB formula is one of the most important things to know about regression. And the regression you want usually has a causal interpretation. The best-case scenario is random assignment of si , conditional on Xi , in some sort of possibly natural experiment. An example is the study of a mandatory re-training program for unemployed workers by Black, et al.
The authors of this study were interested in whether the re-training program succeeded in raising earnings later on. They exploit the fact that eligibility for the training program they study was determined on the basis of personal characteristics and past unemployment and job histories. Workers were divided up into groups on the basis of these characteristics.
While some of these groups of workers were ineligible for training, those in other groups were required to take training if they did not take 12 A large empirical literature investigates the consequences of omitting ability variables from schooling equations.
Key early references include Griliches and Mason , Taubman , Griliches , and Chamberlain When some of the mandatory training groups contained more workers than training slots, training opportunities were distributed by lottery.
Hence, training requirements were randomly assigned conditional on the covariates used to assign workers to groups. The Education Maintenance Allowance, which pays British high school students in certain areas to attend school, is one such policy experiment Dearden, et al, A second type of study that favors the CIA exploits detailed institutional knowledge regarding the process that determines si.
Since voluntary military service is not randomly assigned, we can never know for sure. The motivation for a control strategy in this case is the fact that the military screens soldier-applicants primarily on the basis of observable covariates like age, schooling, and test scores. The CIA in Angrist amounts to the claim that after conditioning on all these observed characteris- tics veterans and nonveterans are comparable. The core methods in today's econometric toolkit are linear regression for statistical control, instrumental variables.
Angrist at Booksamillion. The core methods in today's econometric toolkit are linear. Companion, by Joshua D. Keywords: gn Mostly harmless econometrics: an empiricist's companion When you are looking for a reference work in econometrics that will be on your 'frequently used' bookshelf for the next years to come, please.
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This includes political. Angrist and J. University Press, MHE. Mostly harmless econometrics: an empiricist's companion - harvard Shipping options ». Paperback On Its Way. Mostly harmless econometrics : an empiricist's companion - worldcat APA 6th ed. Angrist, J. Mostly harmless econometrics: An empiricist's companion.
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An Empiricist's Companion Joshua D. Mostly harmless econometrics: an empiricist's companion. Angrist and Jorn-Steffen Pischke. New Jersey: Princeton University Press, pp.
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