Lecture 19:Instrumental Variables(Chapter 13.213.3)

AgendaReviewRandomizationInstrumental Variables (Chapter 13.2)Using Instrumental Variables (Chapter 13.2)

ReviewIf we relax the assumption that explanators are fixed across samples, we confront a number of serious complications.Stochastic explanators render the mathematics of finding unbiased estimators intractable.Instead, econometricians search for consistent estimators.

Review (cont.)The most problematic condition for OLS to be consistent is

Review (cont.)

Review (cont.)Asking that key variances be non-zero and bounded is often reasonable (except in certain time series cases).

Review (cont.)How reasonable an assumption is (i.e. Xi and ei are uncorrelated)?In practice, this assumption is often terrible.When an explanator is correlated with the error term, we call the explanator a troublesome variable.

Review (cont.)We have examined four complications that induce correlation between X and eOmitted Variables BiasMeasurement ErrorSimultaneous CausalityUsing Lagged Values of the Dependent Variable as Explanators, in the presence of serial correlation

Review (cont.)Omitting a variable creates a bias only if:X2 is an explanator of Y (so, when omitted, it becomes a component of the error term).X2 is correlated with X1 (so that X2 creates a correlation between X1 and the error term).

Review (cont.)Measurement error also induces a correlation between our included explanator and the error termInstead of observing Xi , we observe

Review (cont.)

Review (cont.)Mismeasuring X leads to ATTENUATION BIAS. The estimated coefficient is biased towards 0.The magnitude of the bias depends on the relative variances of X and v.

Review (cont.)Under simultaneous causality, X and Y are jointly determined.Because X and Y are determined simultaneously, X can adjust in response to shocks to Y (e).Thus X will be correlated with e

Review (cont.)Using lagged dependent variables as explanators is another potential source of correlation between an explanator and the error term.If there is first-order serial correlation in the error terms, then et-1 is correlated with etTherefore Yt-1 is correlated with et

RandomizationIn practice, correlation between X and e is endemic.Much of econometric work involves studying the process determining the explanators, to see how they might be correlated with e.

Randomization (cont.)The ideal X variable has been randomly assigned.If X has been randomly assigned, then it contains no information about e.

Randomization (cont.)If X has been randomly assigned, thenIt follows thatMoreover, if, then OLS is unbiased.

Randomization (cont.)Laboratory scientists typically work with randomly assigned explanators.In an experiment, the scientist randomly assigns different subjects to different treatments.

Randomization (cont.)Economists have been making increasing use of experiments.Some economists construct carefully controlled markets in computer laboratories (usually with undergraduate subjects).Other economists insert randomization into actual programs in the field, for example randomly assigning participants to different health care plans.

Randomization (cont.)See Chapter 15 for a discussion of controlled experiments.Most econometric research, however, studies field data.The econometrician observes variables as they are naturally determined, and attempts to draw inferences.

Randomization (cont.)Occasionally, variables are naturally determined by random chance. When nature assigns X randomly, econometricians can study a natural experiment.

Randomization (cont.)For example, Alvin Roth studies centralized computer matching programs to assign medical school graduates to hospital training programs. Graduates and hospitals each submit preference lists, and a computer sorts through the lists and creates matches.Game theory suggests key properties for such a matching program.

Randomization (cont.)Roth noticed that in the United Kingdom, each region of the country ran its own centralized matching program.In some of the regions, organizers had happened to adopt a matching program that possessed good game theoretic properties. Other regions did not adopt good designs.

Randomization (cont.)Roth speculated that the choice of program design was reasonably random, at least in the sense that program design was not systematically linked to any other properties of the region.Programs with good game theoretic properties were much more likely to be successful. Centralized programs with poor properties fell into disuse.

Randomization (cont.)In Roths study, advanced econometric methods were unnecessary.The data naturally provided the experiment he needed to observe.Of course, it is Roths duty to argue compellingly that the program rules really are uncorrelated with other aspects of the regions.

Randomization (cont.)Nature often provides a certain degree of randomization.For example, suppose we are studying the effect of military service on earnings.During the Vietnam War, the draft lottery was a largely random process.Thus, a study of Vietnam War veterans would be more random than a study of veterans from the more recent all-volunteer American army.

Randomization (cont.)However, service in Vietnam was determined by more than simple random chance.Some men voluntarily enlisted; others received draft deferments (or simply evaded the draft).The decision to volunteer, or to avoid being drafted, would undoubtedly be correlated with many other variables.

Randomization (cont.)Military service in Vietnam, then, consists of two elements:A clean random element determined by the draft lottery, that would be uncorrelated with e in an earnings regression; andAn unclean element, determined by active decision-making on the agents part, that could easily be correlated with e.

Randomization (cont.)Vietnam War military service is partially determined by an obviously random process.Nonetheless, a simple OLS estimate of the effect of military service on earnings would be inconsistent. Military service is also partially determined non-randomly.

Randomization (cont.)Fortunately, econometricians have discovered a method for separating out the random elements of explanators from the elements that may be correlated with e.Unfortunately, this method requires the data to include an instrumental variable with certain key properties.

Instrumental Variables (Chapter 13.2)An Instrumental Variable is a variable that is correlated with X but uncorrelated with e.If Zi is an instrumental variable:E( Zi Xi ) 0E( Zi ei ) = 0

Instrumental Variables (cont.)For example, a Vietnam War lottery number is: Correlated with X (winning the lottery is highly correlated with military service), but Is uncorrelated with (the lottery number is not connected to cultural, psychological, or economic factors that might affect the decision to join the military, and that might also affect earnings).

Instrumental Variables (cont.)The econometrician can use an instrumental variable Z to estimate the effect on Y of only that part of X that is correlated with Z.Because Z is uncorrelated with e, any part of X that is correlated with Z must also be uncorrelated with e.

Instrumental Variables (cont.)An instrumental variable lets the econometrician find a part of X that behaves as though it had been randomly assigned.

Instrumental Variables (cont.)For example, lets revisit the question of how much mortality can be reduced by intensive cardiac treatment.

Instrumental Variables (cont.)If our observable control variables, such as DFemale, were the only differences between patients who received intensive treatment and those who did not, then DIntensivelyTreated would not tell us anything about e. OLS would be consistent.

Instrumental Variables (cont.)However, we reasonably believe that a doctors choice to perform intensive cardiac procedures are correlated with many other variables.

Instrumental Variables (cont.)Doctors might select patients to receive treatment based on their underlying health status.If health status is an unobservable determinant of mortality, then it is a component of e.OLS will give an inconsistent estimate of the benefits of intensive treatment.

Instrumental Variables (cont.)To eliminate Omitted Variables Bias, we need to find some determinant of a patients receiving intensive cardiac care that is unrelated to mortality.Is there any element of the cardiac care process that is reasonably random?

Instrumental Variables (cont.)Not every hospital is equipped to provide intensive cardiac care.Some patients live near cardiac care centers. When they have a heart attack, they are more likely to be transported to a center, and thus more likely to receive intensive treatment.

Instrumental Variables (cont.)McClellan, McNeil, and Newhouse argue that the distance from a patients home to the nearest hospital equipped for intensive cardiac care is a valid instrumental variable.They argue that geographic location is likely to be uncorrelated with unobserved traits that influence mortality.

Instrumental Variables (cont.)A major task for McClellan, McNeil, and Newhouse is to determine whether or not distance to a cardiac care center really is uncorrelated with mortality (except through the increased likelihood of receiving cardiac treatment).As a reader of such a study, you should ask yourself whether the authors have convinced you that their instrument is uncorrelated with e

Instrumental Variables (cont.)When McClellan, McNeil, and Newhouse use their instrumental variable, they find only a small benefit from intensive cardiac care for the marginal patient (some patients may still benefit quite a lot!).

Instrumental Variables (cont.)When the economist is worried about measurement error, a good choice of instrument is simply a different measure of the same variable.The new measure may have its own errors, but these errors are unlikely to be correlated with the mistakes in the first measure, or with any other component of e.

Instrumental Variables (cont.)For example, Ashenfelter and Rouse were studying the effect of education on earnings.Their data came from a survey of twins.They were concerned that individuals might mis-report their own years of schooling, leading to measurement error biases.

Instrumental Variables (cont.)However, Ashenfelter and Rouse had two separate measures for each individuals years of schooling.The survey asked each individual to list both his/her own years of schooling, and also the years of schooling for his/her twin.The twins report of an individuals schooling served as an instrumental variable for the individuals self-report.

Instrumental Variables (cont.)Another example: policy makers are greatly interested in the effects of tax rates on labor force participation (and other taxpayer behaviors).They would like to run regressions with an individuals tax rate as an explanator.However, an individual has some choice over his/her tax rate.

Instrumental Variables (cont.)Taxpayers who are close to the income threshold for a new tax bracket can choose to limit their taxable income.For example, they might take more of their pay in the form of untaxed benefits or deferred 401(k) compensation rather than pay higher taxes on the extra compensation.The ability and desire to adjust taxable income may well be correlated with e.

Instrumental Variables (cont.)When the government changes the tax rates, the individuals new tax rate is determined by two elements:The change in tax rates (which is uncorrelated with anything else about the individual), andThe individuals decisions about how to respond to the tax change (which could well be correlated with e).

Instrumental Variables (cont.)Public finance economists construct an instrumental variable that captures only the change in tax rates, not the change in behavior.They use the new tax tables to look up the tax rate individuals would face IF they did NOT change their behavior from before the tax change.

Instrumental Variables (cont.)The constructed tax rate is correlated with the tax rate the individuals face after the tax change.The constructed tax rate is uncorrelated with the behavioral adjustments individuals make in response to the tax rate.Such a constructed instrument is called a simulated instrumental variable.

Checking UnderstandingSuppose you are studying the effect of price on the demand for cigarettes, using a cross-section of different states cigarette consumption and average price.You would like to regress

Checking Understanding (cont.)Because Pricei is endogenous, you need to instrument. Which of these variables would be suitable?Each states cigarette excise taxA measure of each states anti-smoking lawsEach states sales tax

Checking Understanding (cont.)Each states cigarette excise taxCigarette excise taxes are surely correlated with cigarette prices. However, they also reflect the level of anti-smoking sentiment in the state (MA has a tax of $1.51 per pack, NC has a tax of $0.05 per pack). Anti-smoking sentiment is an omitted determinant of consumption, so excise taxes are correlated with e. Excise taxes are not a valid instrument.

Checking Understanding (cont.)A measure of state anti-smoking lawsState anti-smoking laws might be correlated with price, but only through their effect on cigarette demand in the state. Such measures are an explanator of cigarette consumption; moreover, they are also a proxy for state anti-smoking sentiment. Anti-smoking laws are a component of e, and would make a terrible instrument.

Checking Understanding (cont.)Each states sales taxState sales taxes are correlated with cigarette prices. Higher sales taxes raise the prices of all goods.There is no reason to expect sales taxes to have any other effect on cigarette consumption, or to be correlated with any other determinant of consumption.State sales taxes are a reasonable instrument.

Using Instrumental Variables (Chapter 13.2)Instrumental variables are NOT the explanator of interest.We want to know the effect of military service on earnings, not just the effect of winning the draft lottery.We want to know the effect of intensive cardiac care on mortality, not just the effect of living near a cardiac care center.

Using Instrumental Variables (cont.)We do NOT simply use instrumental variables as proxies for the explanator of interest.Instead, we use IVs as a tool to tease out the random (or at least uncorrelated) component of X.

Using Instrumental Variables (cont.)Lets construct a consistent IV estimator for the case of measurement error.

DGP with E(Xiei ) 0

Using Instrumental VariablesIf Xi were uncorrelated with ei , we would want to weight more heavily observations with a high xi value.We know that Zi is correlated with the clean part of Xi , so now we want to weight more heavily observations with a high zi value.

Using Instrumental Variables (cont.)

Using Instrumental Variables (cont.)

Using Instrumental Variables (cont.)

Checking Understanding

Checking Understanding (cont.)

Using Instrumental VariablesWhat is the expectation of IV?

Using Instrumental Variables (cont.)What is the expectation of IV?

Using Instrumental Variables (cont.)What is the probability limit of IV?

Using Instrumental Variables (cont.)What is the probability limit of IV?

Using Instrumental Variables (cont.)What is the probability limit of IV?

Using Instrumental Variables (cont.)The asymptotic variance of b IV isThe greater the covariance between X and Z, the lower the asymptotic variance.

Using Instrumental Variables (cont.)In the multiple regression case, we may have more than one explanator that is correlated with the error term (i.e. more than one troublesome variable).However, an instrumental variable only needs to instrument for one of the troublesome variables.

Using Instrumental VariablesA variable Zi can instrument for a particular troublesome explanator, XRi, if:Cov( Zi,XRi ) 0Cov( Zi,ei ) = 0Zi must be correlated with the troublesome variable for which it instruments, but need not be correlated with all of the troublesome variables.

Using Instrumental Variables (cont.)To estimate a multiple regression consistently, we need at least one instrumental variable for each troublesome explanator.

Using Instrumental Variables (cont.)When we have just enough instruments for consistent estimation, we say the regression equation is exactly identified.When we have more than enough instruments, the regression equation is over identified.When we do not have enough instruments, the equation is under identified (and inconsistent).

Review (cont.)In practice, correlation between X and e is endemic.Much of econometric work involves studying the process determining the explanators, to see how they might be correlated with e

Review (cont.)The ideal X variable has been randomly assigned.If X has been randomly assigned, then it contains no information about eHowever, true randomization is relatively uncommon.

Review (cont.)Often, an explanator is partially determined in a way that is random, or at least uncorrelated with eHowever, the explanator is also influenced by omitted variables, or determined endogenously, or is in some other way correlated with e

Review (cont.)Fortunately, econometricians have discovered a method for separating out the random elements of explanators from the elements that may be correlated with e.Unfortunately, this method requires the data to include an instrumental variable with certain key properties.

Review (cont.)An Instrumental Variable is a variable that is correlated with X but uncorrelated with e.If Zi is an instrumental variable:E(ZiXi ) 0E(Ziei ) = 0

Review (cont.)If Xi were uncorrelated with ei , we would want to weight more heavily observations with a high xi value.We know that Zi is correlated with the clean part of Xi , so now we want to weight more heavily observations with a high zi value.

Review (cont.)

Review (cont.)What is the probability limit of IV?

Review (cont.)The asymptotic variance of b IV isThe greater the covariance between X and Z, the lower the asymptotic variance.

Review (cont.)To estimate a multiple regression consistently, we need at least one instrumental variable for each troublesome explanator.

Review (cont.)When we have just enough instruments for consistent estimation, we say the regression equation is exactly identified.When we have more than enough instruments, the regression equation is over identified.When we do not have enough instruments, the equation is under identified (and inconsistent).