Instrumental variables and the weak instrument problem

1. The problem: endogeneity and the failure of OLS

Consider the linear model written in the usual notation,

OLS relies on the exogeneity assumption that the regressors are uncorrelated with the error, \(E[x_i u_i]=0\). When this fails we say there is endogeneity, \(E[x_i u_i] e 0\). The consequence is immediate once we look at where OLS converges:

The lecture notes group the common causes of endogeneity into three recurring cases:

The canonical example in the notes is the returns to schooling regression of wages on years of education:

Natural ability raises wages but is not measured, so it sits inside the error: \(u_i=\delta_0 a_i+v_i\). Because ability is also positively correlated with schooling, the OLS bias \(\delta_0\,\mathrm{Cov}(s_i,a_i)/\mathrm{Var}(s_i)\) is positive, and OLS overestimates the true return to education.

2. The instrumental variables estimator

An instrumental variable is a way of isolating variation in the endogenous regressor that is unrelated to the error. We look for a variable \(z_i\) that satisfies two conditions:

Together these two conditions imply the population relationship \(\beta_0 = E[z_i x_i']^{-1}E[z_i y_i]\). Replacing the population moments with their sample averages gives the IV estimator,

In the simplest case of one endogenous regressor and one instrument, the two conditions reduce to clean covariance statements: exogeneity becomes \(\mathrm{Cov}(z_{1i},u_i)=0\) and identification becomes \(\mathrm{Cov}(z_{1i},x_{1i}) e 0\). The instrument must move with the regressor but not with the error.

The estimator is consistent under these assumptions. Writing it as the truth plus a noise term,

By the law of large numbers the first bracket converges to \(E[z_ix_i']^{-1}\) (which exists precisely because IV2 holds), while the second converges to \(E[z_iu_i]=0\) by IV1. So \(\hat\beta_{IV}\to^p\beta_0\), and under further regularity conditions \(\sqrt{N}(\hat\beta_{IV}-\beta_0)\) is asymptotically normal.

What makes a credible instrument? The notes use the returns-to-schooling literature: distance to college (Card, 1995), quarter of birth (Angrist & Krueger, 1991) and the Vietnam draft lottery (Angrist & Krueger, 1992) have all been proposed as instruments that may be uncorrelated with ability (IV1) but related to schooling (IV2). The Acemoglu & Robinson (2001) study of institutions and GDP uses settler mortality in the 17th-19th centuries as an instrument for the quality of institutions.

3. Two-stage least squares with more instruments than regressors

When there are more instruments than endogenous regressors, the model is over-identified and we use two-stage least squares (2SLS). The recipe is exactly what its name suggests:

Using the projection matrix \(P_Z = Z(Z'Z)^{-1}Z'\), the estimator can be written compactly as

The first stage replaces the endogenous regressor with the part of it that is explained by the (exogenous) instruments, stripping out the component correlated with the error. When the number of instruments equals the number of endogenous regressors, 2SLS collapses back to the simple IV estimator. Under the 2SLS assumptions the estimator is consistent and asymptotically normal with variance \(\sigma^2 Q_{XP_ZX}^{-1}\), where \(Q_{XP_ZX}=Q_{XZ}Q_{ZZ}^{-1}Q_{XZ}'\).

4. The weak instrument problem

The identification condition IV2/2SLS3 is not a yes/no matter in practice. It can hold technically while being almost false, and this is where the trouble starts.

To see how serious this is, compare the asymptotic variances of OLS and 2SLS in the single-regressor case. The notes show that the ratio is

where \(\rho=\mathrm{Corr}(z_{1i},x_{1i})\) is the correlation between instrument and regressor. As the instrument gets weaker, \(\rho\to 0\) and this ratio explodes. The price of using IV instead of OLS is paid in variance, and a weak instrument makes that price enormous.

The fully unidentified case is even starker. If \(E[z_ix_i]=0\) exactly, then \(N^{-1}Z'X\to^p 0\). Both \(N^{-1/2}Z'X\) and \(N^{-1/2}Z'u\) converge to correlated normal random variables, and the 2SLS estimator converges in distribution to

a ratio of two normals. This limit is not centred at \(\beta_0\), so 2SLS is inconsistent, and its variance is in fact infinite. A weak instrument sits on the spectrum between this disaster and the well-behaved strongly-identified case.

5. Testing the identification assumption: the first-stage F

Because weak instruments are so damaging, we test for them. With one endogenous regressor, one instrument and no controls, the first stage is

and identification is equivalent to \(\pi_{01} e 0\), because \(\mathrm{Cov}(x_{1i},z_{1i})=\pi_{01}\sigma_{z1}^2\). We simply estimate the first stage by OLS and test \(\pi_{01}=0\) with a t-test. With several instruments we run a joint F-test that all first-stage coefficients are zero.

6. A cautionary tale: quarter of birth

The lecture material closes with Angrist & Krueger's (1991) famous use of quarter of birth as an instrument for years of schooling. Because children must turn six by a fixed cut-off date to start school, and because compulsory schooling laws set a minimum leaving age, those born in different quarters end up with slightly different amounts of compulsory education. Quarter of birth is plausibly exogenous — it is hard to argue that the season you were born in directly affects your wage other than through schooling.

The problem is relevance. The effect of quarter of birth on schooling is tiny, so the instrument is weak. Bound, Baker & Jaeger (1995) showed that the first-stage F-statistic is small, and — strikingly — that replacing the real quarter-of-birth instrument with randomly simulated quarter-of-birth data produces very similar 2SLS estimates. That is the signature of a weak instrument: the procedure manufactures plausible-looking results from essentially no identifying variation. The lesson is that a credible exogeneity story is necessary but not sufficient; the instrument must also be strong.