1. Instrumental variables
Anant Nyshadham

2. Instrumental Variables
What is a natural experiment?
“situations where the forces of nature or government policy have conspired to produce an environment somewhat akin to a randomized experiment”
Angrist and Krueger (2001, p. 73)
Natural experiments can provide a useful source of exogenous variation in problematic regressors
But they require detailed institutional knowledge

3. Instrumental Variables and Natural Experiments
Some natural experiments in economics
Existing policy differences, or changes that affect some jurisdictions (or groups) but not others
Minimum wage rate
Excise taxes on consumer goods
Unemployment insurance, workers’ compensation
Unexpected “shocks” to the local economy
Coal prices and the Middle East oil embargo (1973)
Agricultural production and adverse weather events

4. Instrumental Variables and Natural Experiments
Some potential pitfalls
Not all policy differences/changes are exogenous
Political factors and past realizations of the response variable can affect existing policies or policy changes
Generalizability of causal effect estimates
Results may not generalize beyond the units under study
Heterogeneity in causal effects
Results may be sensitive to the natural experiment chosen in a specific study (L.A.T.E.)

5. Instrumental Variables and Natural Experiments
Some natural experiments used as IV which are of interest to development economists
Acemoglu Johnson & Robinson (2001): settler mortality
Paxson (1992): rainfall
Schultz & Tansel (1997): healthcare prices

6. True Model Suppose true model is: Do not observe V Can only estimate:
Y = a + bX + cV + e
a, b, and c are parameters to be estimated; e is error term
Do not observe V
Can only estimate:
Y = a + bX + e
What do we do to get b instead of b?

7. Methods Y = a + bX + η; η = cV + e Differencing/FE
Find groups with common V (assumption), but variation in X
Subtract off V to remove it from error term
Instrumental Variable
Find instrument Z; X = j + kZ + i
Predict portion of X which does not correlate with V
Use this portion in original estimating equation

8. IV Criteria and Assumptions
Step/Stage 1: X = j + kZ + I  X’ = k’Z
Step/Stage 2: Y = a + bX’ + η; recover true b
Criteria for Z
Z must sufficiently predict X: k>>0 or k<<0
Testable using estimate of k from first stage
Z must only impact Y through X
Cov(Z,η)=0; Cov(Z,V)=0 & Cov(Z,e)=0
Z does not belong original estimation equation
Assumption, untestable

9. An IV example: Angrist and Krueger (1991), J.L.E.
Returns to education (Y = wages)
Problem of omitted “ability bias”
Years of schooling vary by quarter of birth
Compulsory schooling laws, age-at-entry rules
Someone born in Q1 is a little older and will be able to drop out sooner than someone born in Q4
Q.O.B. can be treated as a useful source of exogeneity in schooling

10. Angrist and Krueger (1991), J.L.E.
People born in Q1 do obtain less schooling
But pay close attention to the scale of the y-axis
Mean difference between Q1 and Q4 is only 0.124, or 1.5 months
So...need large N since R2X,Z will be very small
A&K had over 300k for the cohort
Source: Angrist and Krueger (1991), Figure I

11. Angrist and Krueger (1991), J.L.E.
Final 2SLS model interacted QOB with year of birth (30), state of birth (150)
OLS: b = (s.e. = .0003)
2SLS: b = (s.e. = .0109)
Least squares estimate does not appear to be badly biased by omitted variables
But...replication effort identified some pitfalls in this analysis that are instructive

12. Bound, Jaeger, and Baker (1995), J.A.S.A.
Potential problems with QOB as an IV
Correlation between QOB and schooling is weak
Small Cov(X,Z) introduces finite-sample bias, which will be exacerbated with the inclusion of many IV’s
QOB may not be exogenous (correlated with unobservable determinants of wages, e.g. family income)
QOB may not satisfy exclusion restriction (e.g. age relative to peers changes social dynamics, competition, leadership skill etc.)

13. Bound, Jaeger, and Baker (1995), J.A.S.A.
Even if the instrument is “good,” matters can be made far worse with IV as opposed to LS
Weak correlation between IV and endogenous regressor can pose severe finite-sample bias
And…really large samples won’t help, especially if there is even weak endogeneity between IV and error
First-stage diagnostics provide a sense of how good an IV is in a given setting
F-test and partial-R2 on IV’s

14. Useful Diagnostic Tools for IV Models
Tests of instrument relevance
Weak IV’s → Large variance of bIV as well as potentially severe finite-sample bias
Tests of instrument exogeneity
Endogenous IV’s → Inconsistency of bIV that makes it no better (and probably worse) than bLS
Durbin-Wu-Hausman test
Endogeneity of the problem regressor(s)

15. Tests of Instrument Relevance
Diagnostics based on the F-test for the joint significance of the IV’s
Nelson and Startz (1990); Staiger and Stock (1997)
Bound, Jaeger, and Baker (1995)
Partial R-square for the IV’s
Shea (1997)
There is a growing econometric literature on the “weak instrument” problem

16. Tests of Instrument Exogeneity
Model must be overidentified, i.e., more IV’s than endogenous X’s
H0: All IV’s uncorrelated with structural error
Overidentification test:
1. Estimate structural model
2. Regress IV residuals on all exogenous variables
3. Compute NR2 and compare to chi-square
df = # IV’s – # endogenous X’s

17. Application: Adolescent Work and Delinquent Behavior
Prior research shows a positive correlation between teenage work and delinquency
Reasons to suspect serious endogeneity bias
2nd wave of the NLSY97 (N = 8,368)
Y = 1 if committed delinquent act (31.9%)
X = 1 if worked in a formal job (52.6%)
Z1 = 1 if child labor law allows 40+ hours (14.2%)
Z2 = 1 if no child labor restriction in place (39.6%)

18. Regression Model Ignoring Endogeneity
. reg pcrime work if nomiss==1 & wave==2
Source | SS df MS Number of obs =
F( 1, 8366) =
Model | Prob > F =
Residual | R-squared =
Adj R-squared =
Total | Root MSE =
pcrime | Coef. Std. Err t P>|t| [95% Conf. Interval]
work |
_cons |
Teenage workers significantly more delinquent
Modest effect but consistent with prior research

19. First-Stage Model State child labor laws affect probability of work
. reg work law40 nolaw if nomiss==1 & wave==2
Source | SS df MS Number of obs =
F( 2, 8365) =
Model | Prob > F =
Residual | R-squared =
Adj R-squared =
Total | Root MSE =
work | Coef. Std. Err t P>|t| [95% Conf. Interval]
law40 |
nolaw |
_cons |
State child labor laws affect probability of work
This is a really strong first stage (F, R2)

20. Two-Stage Least Squares Model
. ivreg pcrime (work = law40 nolaw) if nomiss==1 & wave==2
Instrumental variables (2SLS) regression
Source | SS df MS Number of obs =
F( 1, 8366) =
Model | Prob > F =
Residual | R-squared =
Adj R-squared =
Total | Root MSE =
pcrime | Coef. Std. Err t P>|t| [95% Conf. Interval]
work |
_cons |
Instrumented: work
Instruments: law40 nolaw

21. What Do the Models Suggest Thus Far?
Completely different conclusions!
OLS = Teenage work is criminogenic (b = +.026)
Delinquency risk increases by 8.5 percent (base = .305)
2SLS = Teenage work is prophylactic (b = –.074)
Delinquency risk decreases by 20.7 percent (base = .358)
Which model should we believe?
We still have some additional diagnostic work to do to evaluate the 2SLS model
Overidentification test

22. Overidentification Test from the Software
Tests of overidentifying restrictions:
Sargan N*R-sq test Chi-sq(1) P-value =
Basmann test Chi-sq(1) P-value =
IV’s jointly pass the exogeneity requirement
Notice that -overid- provides a global test, whereas the regression-based approach allows you to test the IV’s jointly as well as individually

23. So Where Do We Stand with the Work-Delinquency Question?
Are child labor laws correlated with work?
YES = first-stage F is large
Are child labor laws good IV’s?
YES = overidentification test is not rejected
Is teenage work endogenous?
YES = Hausman test is rejected
Prior research findings that teenage work is criminogenic are selection artifacts

24. Now…What Happens if I Throw in a Potentially Bogus Instrument?
Now there are three instrumental variables
Z1 = 1 if child labor law allows 40+ hours (14.2%)
Z2 = 1 if no child labor restriction in place (39.6%)
Z3 = 1 if high unemployment rate in county (20.1%)
A little more difficult to tell a convincing story that the unemployment rate is only related to delinquency through work experience
But let’s see what happens

25. First-Stage Model So far so good and consistent with expectation
. reg work law40 nolaw highun if nomiss==1 & wave==2
Source | SS df MS Number of obs =
F( 3, 8364) =
Model | Prob > F =
Residual | R-squared =
Adj R-squared =
Total | Root MSE =
work | Coef. Std. Err t P>|t| [95% Conf. Interval]
law40 |
nolaw |
highun |
_cons |
So far so good and consistent with expectation

26. Two-Stage Least Squares Model
. ivreg pcrime (work = law40 nolaw highun) if nomiss==1 & wave==2
Instrumental variables (2SLS) regression
Source | SS df MS Number of obs =
F( 1, 8366) =
Model | Prob > F =
Residual | R-squared =
Adj R-squared =
Total | Root MSE =
pcrime | Coef. Std. Err t P>|t| [95% Conf. Interval]
work |
_cons |
Instrumented: work
Instruments: law40 nolaw highun

27. Post-Hoc Diagnostics Overidentification gives cause for concern
Tests of overidentifying restrictions:
Sargan N*R-sq test Chi-sq(2) P-value =
Basmann test Chi-sq(2) P-value =
. ivendog
Tests of endogeneity of: work
H0: Regressor is exogenous
Wu-Hausman F test: F(1,8365) P-value =
Durbin-Wu-Hausman chi-sq test: Chi-sq(1) P-value =
Overidentification gives cause for concern
The p-value shouldn’t be anywhere near 0.05

28. Conclusion from Diagnostic Tests
2SLS “work effect” is similar
Without unemployment, b = –.074 (s.e. = .028)
With unemployment, b = –.066 (s.e. = .028)
But…the second model is invalidated because the unemployment rate is not exogenous
If affects criminality through other channels
We need to control for all other indirect pathways, or…
It should not be used as an IV at all

29. Closing Comments about Instrumental Variables Studies
In general, a lagged value of the endogenous regressor is not a good instrument
Traditional structural equation model uses lagged values of X and Y as instruments to break the simultaneity between the current values of X and Y X1 X2 Y1 Y2
These models impose the awfully strong assumption that lagged values of X and Y only affect the outcomes through current values

30. Rules for Good Practice with Instrumental Variables Models
IV models can be very informative, but it’s your job to convince your audience
Show the first-stage model diagnostics
Even the most clever IV might not be sufficiently strongly related to X to be a useful source of identification
Report test(s) of overidentifying restrictions
An invalid IV is often worse than no IV at all
Report LS endogeneity (DWH) test

31. Rules for Good Practice with Instrumental Variables Models
Most importantly, TELL A STORY about why a particular IV is a “good instrument”
Something to consider when thinking about whether a particular IV is “good”
Does the IV, for all intents and purposes, randomize the endogenous regressor?