Endogeneity and instrumental variables

A regressor is endogenous when it is correlated with the error term: Cov(x, u) ≠ 0. Then the zero conditional mean assumption of lesson 2 fails, and OLS converges, not to the true slope, but to

There are three common sources, all met earlier:

• Omitted variable (lesson 7): something in u moves with x. In the survey, the regression of earnings on schooling alone has a slope of 1.81, against 1.3 once ability and town are held fixed (1.3): schooling is correlated with the unobserved ability that sits in the error.
• Measurement error in x (lesson 7): the recorded x contains noise that is also in the error of the equation, so the correlation of x with u is negative and the slope is attenuated.
• Simultaneity (lesson 12): x and y determine each other, as in price and quantity, so x is correlated with the error of the y equation.

Adding more observations cannot help, because the bias is in what OLS converges to. And because OLS makes the residuals orthogonal to x by construction, the residuals cannot show you the problem. Only an outside source of variation can: an instrument.

Suppose x is a mixture: some of its variation is tangled up with the error (ability, a shock to both), and some is not. An instrument z is a variable that moves x but is itself unrelated to the error. If you let only z-driven movement in x speak, you have isolated the clean part. Two conditions make z valid:

• Relevance: Cov(z, x) ≠ 0. The instrument actually moves the regressor. This can be tested, in the first stage.
• Exogeneity (the exclusion restriction): Cov(z, u) = 0. z affects y only through x, and is unrelated to the omitted factors. This cannot be tested when there is exactly one instrument per endogenous regressor; it has to be argued from the economics.

Examples of the kind of variable economists use: the distance from a person’s home to the nearest college, as an instrument for years of schooling (it shifts schooling, and should not otherwise affect earnings); rainfall, as an instrument for income in a poor agricultural economy; a randomly assigned offer of a programme, as an instrument for taking it up. In each case the argument for exogeneity is a story about the world, and critics attack it.

In the lab, y = 1 × x + u, and x is built from the instrument z (strength π) plus a first-stage error v that is correlated (ρ) with u. Each press draws 1,500 samples. Start with ρ = 0.6 and a strong instrument: the blue OLS estimates are centred well above 1 and the gold IV estimates are centred on 1, but wider. Then lower π towards zero and watch the IV distribution spread out and slide towards the OLS bias.

With a single instrument the IV estimator is the ratio of two covariances (the reduced-form effect of z on y over the first-stage effect of z on x). When z is a 0/1 indicator this becomes the Wald estimator:

The general method is two-stage least squares (2SLS), which handles several instruments, several endogenous regressors and extra exogenous controls:

• Stage 1: regress each endogenous regressor on all the exogenous variables (the instruments and the included controls). Keep the fitted values x̂, the part of x that the instruments explain.
• Stage 2: regress y on x̂ (and the same controls). The coefficient on x̂ is the 2SLS estimate.

With one instrument and one regressor, 2SLS equals the IV ratio exactly. Two cautions. Doing the second stage by hand gives the right coefficients but the wrong standard errors, because the residuals must be computed with the original x, not x̂; software does this correctly. And the equation must be identified: at least as many instruments as endogenous regressors (the order condition of lesson 12). With more instruments than endogenous regressors the model is over-identified and 2SLS uses them all optimally, and the surplus lets you test the instruments (below).

• Weak instruments. If z barely moves x, Cov(z, x) is near zero and the IV estimator, which divides by it, is erratic and biased towards OLS, and its standard errors cannot be trusted. The standard diagnostic is the first-stage F statistic for the excluded instruments: a rule of thumb says it should exceed about 10 (one instrument: t² above 10). The lab’s F readout shows what happens below it.
• Hausman test for endogeneity. Is x endogenous at all? Compare OLS with IV: if x is exogenous, both are consistent and should be close; if not, they differ. A simple regression-based version adds the first-stage residual to the OLS regression and tests its coefficient: significance means x is endogenous. In simultaneous-equations language, the Hausman specification test checks whether an endogenous variable is correlated with the error term.
• Sargan (over-identification) test. With more instruments than endogenous regressors, regress the 2SLS residuals on all the instruments: n × R² is chi-square with (instruments − endogenous regressors) degrees of freedom under the null that all the instruments are valid. A rejection says at least one instrument is correlated with the error. It cannot be run on a just-identified equation.

Remember what an instrument cannot do: it cannot create information. Without a credible instrument the honest statement is the one from lesson 1: the coefficient is a partial association, and a causal reading needs an argument about why the error is unrelated to x.

This lesson’s set is short because few papers ask about instruments; the exercises above cover the calculations. The simultaneous-equations version of the problem, and its identification conditions, is in lesson 12.