Instrumental Variables Models

1. Instrumental Variables Models Patsi Sinnott, PT, PhD March 2011

2. Outline • Causation Review • The IV approach • Examples • Testing the Instruments • Limitations

3. Causality Review • Randomized trial provides the structure for understanding causation • Does daily dark chocolate affect health? • Does rehab reduce long term costs of care in patients with THR/TKR?

4. Randomization Treatment Group (A) Outcome (Y) Random Sorting Recruit Participants Comparison Group (B) Outcome (Y)

5. Randomization • In OLS (yi = α + βxi + εi ) • The x’s explain the variation in y • ε = the random error • Randomization assumes a high probability that the two groups are similar

6. OLS Assumptions • Right hand side variables are measured without noise (i.e., considered fixed in repeated samples) • There is no correlation between the right hand side variables and the error term E(xiui)=0

7. However • Randomized trials may be • Unethical • Infeasible • Impractical • Not scientifically justified

8. Observational Studies • Natural Experiment • Administrative Data • Observable characteristics (e.g. age, gender, smoking status) can be included in the model, BUT……. • Treatment (sorting) is not randomly assigned • All characteristics are not likely to be “generally” evenly distributed

9. Potential violation of OLS assumptions • Explanatory variables are not fixed in repeated samples • Regressors are correlated with the error term (endogeneity)

10. Violations occur ….. • Measurement error in the explanatory variables • Autoregression with autocorrelated errors • Simultaneity (and reverse causation) • Omitted variables • Sample selection (in the population because of some unmeasured characteristic)

11. The IV approach • Assumptions for the instrument • Causes substantial variation in the problematic variable • Independent • No direct or indirect effect on the outcome

12. Instrumental Variable • Cause variation in the variable of interest • No direct or indirect effect on the outcome Instrumental variable Treatment Outcome Other variables

13. Example • Wage = fn(years of education)

14. Whiteboard • What’s missing (wage=fn (years of education)

15. Example • School performance = fn (class size)

16. Whiteboard • What’s missing: School performance = fn (class size)

17. Example • Mortality after AMI = fn (cardiac cath.)

18. Whiteboard • Mortality after AMI = fn (cardiac cath.) +

19. The IV approach • School performance = fn (class size) + parent involvement • Wage = fn (years of education) + ability • Mort. after AMI = fn (cardiac cath) + other var.

20. IV (Omitted variable) • Wage = fn (years of education) + ability • Instrument influence years of education but not ability (error term) • Distance to nearest college – related to years of college (not ability)

21. IV (Sample selection) • School performance = fn (class size) + parent involvement • Class size • Dummy variable (split class due to enrollment changes) • Captures effect of smaller classes (without other unmeasured characteristics of students/parents associated with smaller class size)

22. IV • Mortality after AMI = fn (cardiac cath.) + other var. • Instrument affects likelihood of cath. • No direct effect on mortality • Differential distance to nearest cath. hospital

23. Example • Does more intensive treatment of AMI in the elderly reduce mortality (McClellan; McNeil; Newhouse. JAMA. 9/21/94)

24. McClellan, McNeil and Newhouse • Preliminary results • 23% w/AMI received catheterization • Year 4 – 2/3rd of those who did NOT receive cath had died (compared to 30% of cath group)

25. Are the groups different?

26. Results • 37% points difference in mortality at 4 years in unadjusted • 28% points difference with observable characteristics (age, gender)at 4 years (OLS) • 7 percentage point difference in 1-day mort • 13 percentage point difference at 7 days • 18 percentage points at 30 days

27. IV for cath rate • Differential distance between nearest hospital and nearest catheterization facility/hospital • Pts w/AMI will go to nearest hospital • Distance from nearest hospital to nearest cath hospital will be independently predictive of catheterization for similar patients.

28. IV for cath rate • Zip code center to Zip code center • If nearest hospital = cath hospital then DD = 0 • Differential distance to cath hospital • ≤2.5 mi. and > 2.5 miles

29. Characteristics by DD

30. Results w/DD IV • That variation in IV causes variation in the treatment variable (cath) is satisfied • (26.2-19.5 = 6.75% point greater chance of getting cath within 90 days following AMI) • That DD affects outcomes only through likelihood of receiving treatment • reasonability

31. Results w/DD IV • Patient characteristics appear balanced • Hospital characteristics not

32. Multiple regression results • Patient characteristics • Hospital IVs including high volume hospital (1,0), rural residence (1,0) • DD IV

33. Multiple regression w/IV results * In percentage points, standard errors in parens.

34. Summary of results • Unadjusted – 37 % age pt effect on four year mortality • Adjusted w/out IV – 28 % age pt effect • Adjusted w/DD IV – 6.9 % age pt effect • Adjusted with DD, high volume hospital and rural IVs – approx 5 percentage pts. • The 5% age pt. difference at day 1

35. Inference • RCT provides the average effect for the population eligible for the study • Simple IV estimator • Effect of cath = ∆mortality/ ∆ cath rate = • =(58.06-58.02)/(26.2-19.5) = • -0.46/6.7 = -0.069 • That the 6.7% of patients who rec’d cath at nearby hospital had on average 6.9% percentage point additional chance of 4 year survival

36. Inference • Wide angle view of the effects of treatment • More general population than RCT • External validity • Marginal effect on a selection of the population: • Problematic for clinicians • Policy implications – incremental effects

37. Testing for strength of the instrument • Tells a good story • Validity of overidentification • Does the instrument have the expected sign and is significant • Compare to alternative instruments (if available)

38. Testing • Defend assumption that instrument is NOT an explanatory variable • Explain why instrument is not correlated with omitted explanatory variable

39. Testing • Test if errors are correlated with regressors • Hausman test • Test if instrument is uncorrelated with the error • Sargan test

40. Testing • Test if the correlation between the instrument and the troublesome variable is strong enough • F statistic, regressing troublesome variable on all instruments – to test the null that the slopes of all instruments equal zero • Staiger Stock test

41. Conclusion • Instrumental variables mimic randomization • But good instruments are hard to find. • An estimate of the marginal effect/influence on outcome

42. References

43. References

44. Other IV methods • Randomization • 2SLS