QE 6: IVs

Question 1

What are the three requirements of a valid instrument?

Answer 1

(1) Relevance: the instrument is a useful predictor of the endogenous variable, even after controlling for other x-variables.

(2) Exogeneity: Z is uncorrelated with the error term u; there is no indirect effect of Z on Y through other determinants.

(3) Exclusion: Z has no direct effect on Y, and does not appear in the model equation.

Question 2

Does the relationship between Z and X need to be causally interpretable, or just associational?

Answer 2

No, an associational relationship is sufficient.

Question 3

What is two-stage least squares?

Answer 3

Regress X on Z, and let the fitted values of X from this regression be X*. Regress Y on X*.

Question 4

How can we test for relevant instruments?

Answer 4

Perform an F test on the first-stage regression. We usually require F > 10.

Question 5

What is a ‘weak instrument’? Why are they a problem?

Answer 5

A weak instrument is one that is relevant to the endogenous variable, but with a weak association: F may be below 10. Asymptotic normality and standard inference will then fail to hold.

Question 6

How can we test for exogeneity?

Answer 6

We require at least two instruments. We first obtain the residuals from the 2SLS regression, and then regress the residuals û on the X. With only one Z_i, cov(Z,û) by construction, so this test will fail. Otherwise, we are implicitly comparing the estimates that would be generated by each instrument individually. If these are similar, we are unlikely to reject the null of exogeneity.

Question 7

Why can instrumental variables help us in RCTs where compliance is imperfect?

Answer 7

If random assignment fails, allocation to treatment is often a good instrument for actual treatment. It will be:

Question 8

What is the LATE? What possibly undesirable properties does it have?

Answer 8

If treatment effects are heterogenous, IV estimates the local average treatment effect (LATE). It is a weighted average of treatment effects, according to how responsive people are to the instrument.

The LATE therefore depends on the instrument used. Different instruments may therefore generate quite different estimates of the same causal effect.