A researcher uses an IV strategy where the first-stage F-statistic is 4.2. The IV estimate is large and statistically significant at p < 0.01. What is the most appropriate conclusion?
AThe results are reliable — statistical significance of the IV estimate confirms the instrument works
BThe large IV estimate itself validates the first stage, since a weak instrument would produce a near-zero estimate
CThe weak instrument undermines the IV estimate — even small violations of the exclusion restriction could be severely amplified
DAn F-statistic of 4.2 is borderline; the researcher should report it but the results are still trustworthy
A weak first stage (F < 10) is disqualifying regardless of the IV estimate's size or significance. With a weak instrument, IV estimates have severe finite-sample bias that can be worse than OLS, and the confidence intervals are unreliable. A large IV estimate with a weak instrument may simply reflect amplified noise or an exclusion restriction violation. Statistical significance of the IV estimate does not validate the instrument — it just means the noise is too small relative to the (possibly biased) signal. The F > 10 threshold exists precisely because finite-sample behavior is poor below it.
Question 2 Multiple Choice
The Wald estimator in instrumental variables is the ratio of:
AThe OLS coefficient on X divided by the first-stage coefficient on Z
BThe reduced-form coefficient on Z divided by the first-stage coefficient on Z
CThe first-stage coefficient on Z divided by the reduced-form coefficient on Z
DThe IV residual variance divided by the OLS residual variance
The Wald estimator is: (coefficient of Z in the y-on-Z regression) / (coefficient of Z in the X-on-Z regression). The numerator is the reduced-form: how much does y change when Z changes by one unit? The denominator is the first-stage: how much does X change when Z changes by one unit? Dividing removes the Z-to-X scaling, leaving the causal effect of X on y. This ratio logic makes IV transparent: the instrument affects y only through X, so the y-response is just the X-response scaled by the effect of X on y.
Question 3 True / False
A strong first-stage F-statistic (F > 10) is sufficient to validate an instrumental variables strategy.
TTrue
FFalse
Answer: False
A strong first stage is necessary but not sufficient. The instrument must also satisfy the exclusion restriction: Z must affect y only through X, not through any direct channel. This is a theoretical requirement that cannot be tested when you have exactly one instrument and one endogenous variable. A strong F-statistic tells you the instrument is relevant (Z strongly predicts X), but it says nothing about validity (Z ↛ y except through X). Both conditions are required for consistent IV estimation.
Question 4 True / False
When first-stage F is very small (near 1), IV estimates can actually be more biased than OLS estimates, even if the instrument is theoretically valid.
TTrue
FFalse
Answer: True
With a weak instrument, the IV estimator has poor finite-sample properties: it can have enormous variance and bias that far exceeds OLS bias from the endogeneity it was meant to correct. Intuitively, a weak instrument barely shifts X, so the IV estimate amplifies both signal and noise. Even a tiny violation of the exclusion restriction (Z has a small direct effect on y) gets amplified into a large bias when the first-stage coefficient is small. This is why weak instruments represent a complete failure of IV, not just reduced precision.
Question 5 Short Answer
Explain the logic of the Wald estimator: why does dividing the reduced-form coefficient by the first-stage coefficient recover the causal effect of X on y?
Think about your answer, then reveal below.
Model answer: The instrument Z affects y only through X (by the exclusion restriction). The reduced-form coefficient measures the total effect of Z on y, which is: (effect of Z on X) × (effect of X on y). The first-stage coefficient measures the effect of Z on X. Dividing removes the Z-to-X scaling: (effect of Z on y) / (effect of Z on X) = (effect of Z on X × effect of X on y) / (effect of Z on X) = effect of X on y. This is the causal effect we want — the instrument acts as a natural experiment, and the Wald ratio scales the outcome shift by the treatment shift.
The Wald estimator makes IV intuitive: it asks 'how much did y change per unit change in X that was induced by Z?' The first stage tells you how much X was induced; the reduced form tells you how much y moved in response. Their ratio is the causal effect. This logic breaks down with a weak first stage — if Z barely moves X, any noise in the reduced form gets magnified enormously.