Non-Experimental Data: Natural Experiments and more on IV

1. Non-Experimental Data:Natural Experiments and more on IV

3. Non-Experimental Data • Refers to all data that has not been collected as part of experiment • Quality of analysis depends on how well one can deal with problems of: • Omitted variables • Reverse causality • Measurement error • selection • Or… how close one can get to experimental conditions

4. Natural/ ‘Quasi’ Experiments • Used to refer to situation that is not experimental but is ‘as if’ it was • Not a precise definition – saying your data is a ‘natural experiment’ makes it sound better • Refers to case where variation in X is ‘good variation’ (directly or indirectly via instrument) • A Famous Example: London, 1854

5. The Case of the Broad Street Pump • Regular cholera epidemics in 19th century London • Widely believed to be caused by ‘bad air’ • John Snow thought ‘bad water’ was cause • Experimental design would be to randomly give some people good water and some bad water • Ethical Problems with this

6. Soho Outbreak August/September 1854 • People closest to Broad Street Pump most likely to die • But breathe same air so does not resolve air vs. water hypothesis • Nearby workhouse had own well and few deaths • Nearby brewery had own well and no deaths (workers all drank beer)

7. Why is this a Natural experiment? • Variation in water supply ‘as if’ it had been randomly assigned – other factors (‘air’) held constant • Can then estimate treatment effect using difference in means • Or run regression of death on water source distance to pump, other factors • Strongly suggests water the cause • Woman died in Hampstead, niece in Islington

8. What’s that got to do with it? • Aunt liked taste of water from Broad Street pump • Had it delivered every day • Niece had visited her • Investigation of well found contamination by sewer • This is non-experimental data but analysed in a way that makes a very powerful case – no theory either

9. Methods for Analysing Data from Natural Experiments • If data is ‘as if’ it were experimental then can use all techniques described for experimental data • OLS (perhaps Snow case) • IV to get appropriate units of measurement • Will say more about IV than OLS • IV perhaps more common • If can use OLS not more to say • With IV there is more to say – weak instruments

10. Conditions for Instrument Validity • To be valid instrument: • Must be correlated with X - testable • Must be uncorrelated with ‘error’ – untestable – have to argue case for this assumption • These conditions guaranteed with instrument for experimental data • But more problematic for data from quasi-experiments

11. Bombs, Bones and Breakpoints:The Geography of Economic Activity Davis and Weinstein, AER, 2002 • Existence of agglomerations (e.g. cities) a puzzle • Land and labour costs higher so why don’t firms relocate to increase profits • Must be some compensatory productivity effect • Different hypotheses about this: • Locational fundamentals • Increasing returns (Krugman) – path-dependence

12. Testing these Hypotheses • Consider a temporary shock to city population • Locational fundamentals theory would predict no permanent effect • Increasing returns would suggest permanent effect • Would like to do experiment of randomly assigning shocks to city size • This is not going to happen

13. The Davis-Weinstein idea • Use US bombing of Japanese cities in WW2 • This is a ‘natural experiment’ not a true experiment because: • WW2 not caused by desire to test theories of economic geography • Pattern of US bombing not random • Sample is 303 Japanese cities, data is: • Population before and after bombing • Measures of destruction

14. Basic Equation • Δsi,47-40 is change in population just before and after war • Δsi,60-47 is change in population at later period • How to test hypotheses: • Locational fundamentals predicts β1=-1 • Increasing returns predicts β1=0

15. The IV approach • Δsi,47-40 might be influenced by both permanent and temporary factors • Only want part that is transitory shock caused by war damage • Instrument Δsi,47-40 by measures of death and destruction

16. The First-Stage: Correlation of Δsi,47-40with Z

17. Why Do We Need First-Stage? • Establishes instrument relevance – correlation of X and Z • Gives an idea of how strong this correlation is – ‘weak instrument’ problem • In this case reported first-stage not obviously that implicit in what follows • That would be bad practice

18. The IV Estimates

19. Why Are these other variables included? • Potential criticisms of instrument exogeneity • Government post-war reconstruction expenses correlated with destruction and had an effect on population growth • US bombing heavier of cities of strategic importance (perhaps they had higher growth rates) • Inclusion of the extra variables designed to head off these criticisms • Assumption is that of exogeneity conditional on the inclusion of these variables • Conclusion favours locational fundamentals view

20. An additional piece of supporting evidence…. • Always trying to build a strong evidence base – many potential ways to do this, not just estimating equations

21. The Problem of Weak Instruments • Say that instruments are ‘weak’ if correlation between X and Z low (after inclusion of other exogenous variables) • Rule of thumb - If F-statistic on instruments in first-stage less than 10 then may be problem (will explain this a bit later)

22. Why Do Weak Instruments Matter? • A whole range of problems tend to arise if instruments are weak • Asymptotic problems: • High asymptotic variance • Small departures from instrument exogeneity lead to big inconsistencies • Finite-Sample Problems: • Small-sample distribution may be very different from asymptotic one • May be large bias • Computed variance may be wrong • Distribution may be very different from normal

23. Asymptotic Problems I:Low precision • Class exercize asks you to show that asymptotic variance of IV estimator is larger the weaker the instruments • Intuition – variance in any estimator tends to be lower the bigger the variation in X – think of σ2(X’X)-1 • IV only uses variation in X that is associated with Z • As instruments get weaker using less and less variation in X

24. Asymptotic Problems II:Small Departures from Instrument Exogeneity Lead to Big Inconsistencies • Suppose true causal model is y=Xβ+Zγ+ε So possibly direct effect of Z on y. • Instrument exogeneity is γ=0. • Obviously want this to be zero but might hope that no big problem if ‘close to zero’ – a small deviation from exogeneity

25. But this will not be the case if instruments weak… consider just-identified case • If instruments weak then ΣZX small so ΣZX-1 large so γ multiplied by a large number

26. An Example: The Return to Education • Economists long-interested in whether investment in human capital a ‘good’ investment • Some theory shows that coefficient on s in regression: y=β0+β1s+β2x+ε Is measure of rate of return to education • OLS estimates around 8% - suggests very good investment • Might be liquidity constraints • Might be bias

27. Potential Sources of Bias • Most commonly mentioned is ‘ability bias’ • Ability correlated with earnings independent of education • Ability correlated with education • If ability omitted from ‘x’ variables then usual formula for omitted variables bias suggests upward bias in OLS estimate

28. Potential Solution • Find an instrument correlated with education but uncorrelated with ‘ability’ (or other excluded variables) • Angrist-Krueger “Does Compulsory Schooling Attendance Affect Schooling and Earnings” , QJE 1991, suggest using quarter of birth • Argue correlated with education because of school start age policies and school leaving laws (instrument relevance) • Don’t have to accept this – can test it

29. A graphical version of first-stage (correlation between education and Z)

30. In this case… • Their instrument is binary so IV estimator can be written in Wald form • And this leads to following expression for potential inconsistency: • Note denominator is difference in schooling for those born in first- and other quarters • Instrument will be ‘weak’ if this difference is small

31. Their Results

32. Interpretation (and Potential Criticism) • IV estimates not much below OLS estimates (higher in one case) • Suggests ‘ability bias’ no big deal • But instrument is weak • Being born in 1st quarter reduces education by 0.1 years • Means ‘γ’ will be multiplied by 10

33. But why should we have γ≠0 • Remember this would imply a direct effect of quarter of birth on earnings, not just one that works through the effect on education • Bound, Jaeger and Baker argued that evidence that quarter of birth correlated with: • Mental and physical health • Socioeconomic status of parents • Unlikely that any effects are large but don’t have to be when instruments are weak

34. An example: UK data Effect is small but significantly different from zero

35. A Back-of-the-Envelope Calculation • Being born in first quarter means 0.01 less likely to have a managerial/professional parent • Being a manager/professional raises log earnings by 0.64 • Correlation between earnings of children and parents 0.4 • Effect on earnings through this route 0.01*0.64*0.4=0.00256 i.e. ¼ of 1 per cent • Small but weak instrument causes effect on inconsistency of IV estimate to be multiplied by 10 – 0.0256 • Now large relative to OLS estimate of 0.08

36. Summary • Small deviations from instrument exogeneity lead to big inconsistencies in IV estimate if instruments are weak • Suspect this is often of great practical importance • Quite common to use ‘odd’ instrument – argue that ‘no reason to believe’ it is correlated with ε but show correlation with X

37. Finite Sample Problems • This is a very complicated topic • Exact results for special cases, approximations for more general cases • Hard to say anything that is definitely true but can give useful guidance • Will divide problems into 3 areas • Bias • Incorrect measurement of variance • Non-normal distribution • But really all different symptoms of same thing

38. Review and Reminder • If ask STATA to estimate equation by IV • Coefficients compute using formula given • Standard errors computed using formula for asymptotic variance (in class exercise) • T-statistics, confidence intervals and p-values computed using assumption that estimator is unbiased with variance as computed and normally distributed • All are asymptotic results

39. Difference between asymptotic and finite-sample distributions • This is normal case • Only in special cases e.g. linear regression model with normally distributed errors are small-sample and asymptotic distributions the same. • Difference likely to be bigger • The smaller the sample size • The weaker the instruments

40. Some Intuition for why Strength of Instruments is Important • Consider very strong instrument • Z can explain a lot of variation in X • Z very close to X-hat • Think of limiting case where correlation perfect – then X-hat=X • IV estimator identical to OLS estimator • Will have same distribution • If errors normal then this is same as asymptotic distribution

41. Now consider case of weak instrument… • Think of extreme case where true correlation between X and Z is useless • First-stage tries to find some correlation so estimate of coefficients will not normally be zero and will have some variation in X-hat • No reason to believe X-hat contains more ‘good’ variation than X itself • So central tendency is OLS estimate • But a lot more noise – so very big variance

42. A Simple Example • One endogenous variable, no exogenous variables, one instrument • All variables known to be mean zero so estimate equations without intercepts

43. Assumptions • Assume zi non-stochastic • Assume (εi,ui) have joint normal distribution with mean zero, variances σ2ε,σ 2u, and correlation coefficient ρ • If ρ=0 then no endogeneity problem and OLS estimator consistent • If ρ≠0 then endogeneity problem and OLS estimator is inconsistent

44. IV Estimator for this special case.. • Both numerator and denominator of final term are linear combinations of normal random variables so are also normally distributed • So deviation of IV estimator from β is ratio of two (correlated) normal random variables • Sounds simple but isn’t

45. Finite Sample Problems I: Bias • To address issue of bias want to take expectation of final term – would like it to be zero. • Problem – mean does not exist !!!! • Not just a curiosity in a statistics course • What does this mean?- ‘fat tails’ – sizeable probability of getting vary large outcome • This happens when Σzixi is small – more likely when instruments are weak

46. A Very Special Case: π=ρ=0 • X exogenous and Z useless in this case so would do OLS if knew this • But perhaps you don’t know it and are ‘playing safe’ by using IV • In this case numerator and denominator in: • are independent with mean zero • The IV estimator has a Cauchy distribution – this has no mean (or other moments)

47. Rules-of-Thumb • Mean of IV estimator exists if more than two over-identifying restrictions • Where mean exists: • Probably can use as measure of central tendency of IV estimator where mean does not exist • This is where rule-of-thumb on F-stat comes from

48. More on the F-statistic • This says mean of IV estimator is weighted average of true β and plim of OLS estimator • Lower is E(F) the higher the weight on the plim of OLS estimator • Can write F-statistic as (no exogenous variables, one instrument): • Where R2 comes from first-stage • F-statistic increasing in number of observations and strength of instruments

49. Finite Sample Problems IIMismeasurement of Variance • When mean of IV estimator does not exist, this is because tails of distribution is ‘too fat’ • Variance also does not exist (infinite) but STATA will give a finite answer • Hence computed asymptotic variance probably too low

50. Finite Sample Problems IIINon-Normal Distribution • Distribution of IV estimator often has fatter tails than the normal. • Implies extreme values more likely to occur then would expect given normal • Will reject hypotheses that coefficients are zero more often than one should • Distribution may not even be bell-shaped