01:220:322 Chap.9 Instrumental Variables and Two-Stage Least Squares
Instrumental Variables and Two-Stage Least Squares
The endogeneity problem
An explanatory variable correlated with the error is endogenous, and OLS is then biased and inconsistent. The lectures offer two remedies: collect enough good controls, or find an instrumental variable.
Two conditions
An instrument must be exogenous, uncorrelated with the error, and relevant, correlated with the endogenous regressor.
Only relevance can be tested, by regressing the endogenous variable on the instrument and every exogenous control in a first stage. Exogeneity has to be argued from how the instrument was assigned, and random assignment alone does not guarantee it.
Estimation
The simple IV estimator is the covariance of the instrument with the outcome divided by its covariance with the regressor.
With controls or several instruments, two-stage least squares is run in Stata with ivregress and the robust option.
Running the two stages by hand gives wrong standard errors.
The price of IV
IV is consistent rather than unbiased, and its standard error is roughly the OLS standard error divided by the correlation between instrument and regressor, so it is usually much larger.
Weak instruments
An instrument with little first-stage explanatory power is weak.
The estimate is then biased towards OLS and its intervals are unreliable, and a small failure of exogeneity is magnified. A first-stage F above 10 is the common rule of thumb, with a stricter threshold for the robust statistic.
Tests of endogeneity and of overidentifying restrictions are covered as optional material.
Why the first stage includes the controls
Relevance is about the partial correlation between the instrument and the endogenous regressor after the exogenous controls are held fixed. Leaving a control out of the first stage can make an instrument look strong when it is merely proxying for that control.
The same controls must appear in the structural equation estimated by 2SLS.
Interpreting an IV estimate
Read the 2SLS coefficient exactly like an OLS slope, with the same units and the words holding the controls fixed, and then compare it with OLS. A gap in the direction the suspected bias predicts supports the story that motivated the instrument; a much larger standard error is expected.
What this chapter covers
- 01
Endogenous and exogenous regressors
- 02
Instrument relevance and exogeneity
- 03
Simple IV estimator
- 04
First stage with controls
- 05
Two-stage least squares in Stata
- 06
Weak instruments and the first-stage F
Worked example · free
Checking an instrument
- 2Relevance: distance to a station must move commuting time; test its coefficient in a first stage that includes all controls.
- 2Exogeneity: distance must be unrelated to unobserved drivers of satisfaction.
- 2Threat: people who live far from stations may choose quieter suburbs they enjoy, a direct link to satisfaction.
Key terms
- Instrumental Variable
- A variable that moves the endogenous regressor but is unrelated to the error term.
- First Stage
- The regression of the endogenous regressor on the instrument and all exogenous controls.
- Two-Stage Least Squares
- The IV method that uses the first-stage fitted values in place of the endogenous regressor.
- Weak Instrument
- An instrument with little explanatory power for the endogenous regressor.
- Exogeneity
- The condition that an instrument is uncorrelated with the error in the equation of interest.
- Relevance
- The condition that an instrument is correlated with the endogenous regressor after controls.
Instrumental Variables and Two-Stage Least Squares FAQ
Which instrument condition can the data test?
Relevance. Regress the endogenous variable on the instrument and all exogenous controls and test the instrument's coefficient. Exogeneity concerns the unobserved error and must be defended with an argument.
Why are IV standard errors so much larger than OLS ones?
IV uses only the variation in the regressor created by the instrument. When that variation is small, the estimate rests on little information, and its standard error scales roughly with one over their correlation.
What does a first-stage F below 10 tell me?
The instrument is weak by the usual rule of thumb. The IV estimate may be biased towards OLS and its confidence intervals may be misleading, so look for a stronger instrument.
Why does the IV estimate often differ a lot from OLS?
OLS absorbs the bias from the omitted factor that motivated the instrument, while IV removes it. IV is also far less precise, so part of any gap can be sampling noise; compare the two using the IV confidence interval.
Can I run two-stage least squares as two separate OLS regressions?
The point estimate matches, but the second-stage standard errors printed by OLS are wrong because they treat the fitted values as data. Use ivregress with the robust option instead.
Exam move
For every IV example, write the structural equation, the first stage with its controls, and one sentence defending exogeneity before looking at any estimate.
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