Testing When Ordinary Regression Can Stand In for a Structural Model

When economists use schooling to explain wages, they must decide whether ordinary regression is adequate or whether endogeneity requires instrumental-variable methods. The relevant question here is mean exogeneity: after conditioning on the regressors, does the structural error have an average of zero? If so, the ordinary conditional mean equals the structural relationship. If not, interpreting the regression structurally can be misleading.

In Testing Exogeneity in Nonparametric Instrumental Variable Models, Xiaojun Song and Jichao Yuan develop a test for this condition without imposing a fixed functional form on the structural relationship. Their contribution combines an indicator-based test with a bootstrap correction designed to reproduce the uncertainty created by estimating the ordinary regression.

Avoiding the difficult structural estimate

Nonparametric instrumental-variable models allow an unknown structural function rather than a predetermined linear or polynomial equation. But recovering that function involves an ill-posed inverse problem, which can produce slow convergence. Directly comparing an estimated structural function with an ordinary regression estimate can therefore weaken a test’s ability to detect endogeneity.

The authors instead use a conditional-moment restriction involving the ordinary regression residual: the outcome minus its conditional mean given the regressors. They ask whether that residual also has conditional mean zero given the instruments.

This is equivalent to mean exogeneity under maintained instrument validity and a completeness condition. Completeness means, roughly, that the instruments contain enough information to distinguish different functions of the regressors through conditional expectations. Without it, the moment restriction need not identify every structural departure from exogeneity. The procedure does not independently establish instrument validity.

To construct the test, the authors estimate the ordinary regression with leave-one-out kernel smoothing. They then average the fitted residuals after weighting them by indicators selecting observations whose instrument values fall below specified thresholds. Two statistics summarize the resulting process: Kolmogorov–Smirnov uses the largest absolute departure, while Cramér–von Mises aggregates squared departures. Both are evaluated at observed instrument vectors.

Why a simple bootstrap fails

A central finding concerns calibration: deciding how large a statistic must be before rejecting exogeneity. Simply multiplying fitted residuals by independent random weights does not generally reproduce the test’s limiting distribution. Estimating the regression and regressor density changes the process’s first-order behavior, so a bootstrap that ignores those effects generally has the wrong covariance.

The proposed remedy corrects each indicator by subtracting its estimated conditional expectation given the regressors. This correction enters the bootstrap process, not the observed statistic. The fitted residuals and conditional-distribution estimates remain fixed across bootstrap draws; each draw requires only fresh multipliers and weighted sums.

Under the paper’s regularity conditions, the authors establish asymptotic validity and consistency against fixed alternatives when completeness holds. They also characterize behavior under departures that shrink at the inverse square root of sample size. The theory concerns mean exogeneity and does not require the stronger condition that the structural error average to zero after jointly conditioning on regressors and excluded instruments.

Simulated performance and schooling evidence

The simulations examine three designs, sample sizes of 200, 400 and 800, and several strengths of endogeneity. One design deliberately satisfies mean exogeneity without satisfying the stronger joint-conditioning restriction.

At the baseline bandwidth and a nominal 5% significance level, the proposed tests reject between 4.5% and 5.8% of samples when the null holds. Detection rates generally increase with sample size and departure strength. The new tests outperform the implemented Blundell–Horowitz and Breunig comparisons at most reported alternatives, but the authors explicitly reject any claim of universal power superiority. Larger smoothing bandwidths can substantially reduce power, particularly in smaller samples.

The empirical illustration uses young-men wage data previously analyzed by Card. Schooling is the potentially endogenous regressor, experience is the control, and the excluded instrument is either proximity to a four-year college or mother’s education. Both proposed tests reject mean exogeneity at the 1% level in both specifications, with reported baseline p-values between 0.001 and 0.003.

Those findings require caution. The application uses discrete variables, whereas the sufficient theory assumes continuous supports. Instrument validity and completeness need separate justification, especially for the binary proximity instrument. Only experience is controlled for, and the two samples overlap. The results therefore provide evidence against mean exogeneity within the maintained specifications—not an independently validated estimate of schooling’s causal wage effect.