Partial R-Squared for Instrument Strength

Why Partial R-Squared Is Useful

A first-stage regression can explain substantial variation in an endogenous regressor because of included covariates, even when the excluded instruments add little explanatory power. The overall first-stage R-squared does not separate these contributions. Partial R-squared does: it measures the share of the variation left after controlling for included covariates that is explained by the excluded instruments.

This conditional measure complements the conventional first-stage F statistic. The partial R-squared describes the size of the instruments’ incremental fit on a bounded zero-to-one scale; the F statistic evaluates that fit relative to its sampling variation and degrees of freedom. For the conventional homoskedastic first-stage test, the two quantities are linked exactly.

The restricted model leaves the variation in \(D\) unexplained by \(X\). The unrestricted model asks how much of that remaining variation is explained when \(Z\) is added. Thus, partial R-squared is not generally the ordinary R-squared of the unrestricted regression. It is a measure of the incremental contribution of the excluded instruments, conditional on the included covariates.

Residualization and the F-Statistic Relationship

The same quantity can be understood by removing the linear contribution of \(X\) from both \(D\) and the columns of \(Z\), then regressing the residualized \(D\) on the residualized instruments. The resulting R-squared is the partial R-squared. This is the partial-regression interpretation: the comparison concerns what \(Z\) explains after the part associated with \(X\) has been set aside.

The conventional first-stage F test evaluates the null hypothesis that all excluded-instrument coefficients are zero, conditional on \(X\). Let \(n\) be the sample size and \(k\) the number of columns in \(X\), counting the intercept. The unrestricted regression has \(n-k-q\) residual degrees of freedom. Under the conventional homoskedastic OLS calculation,

The numerator is the reduction in residual sum of squares per excluded-instrument restriction. The denominator is the unrestricted residual mean square. Divide numerator and denominator by \(SSR_R\), and use the definition of partial R-squared:

This identity is exact for the conventional nested-model F statistic. It shows why partial R-squared and F are related but not interchangeable: F depends not only on the incremental fit, but also on the number of excluded instruments and the residual degrees of freedom. Conversely, if the F statistic and both degrees of freedom are known, rearrangement gives

For the usual finite-sample F reference distribution, the regression errors are assumed to be independent, normally distributed, and homoskedastic conditional on the regressors, with a correctly specified linear first-stage mean. The algebraic identity itself follows from the sums of squares; normality is needed for the exact conventional reference distribution, not for that arithmetic identity. If the test uses a heteroskedasticity-robust or otherwise modified covariance estimator, its reported statistic generally does not have this sums-of-squares identity.

Reading Partial R-Squared Alongside F

Partial R-squared is useful because it describes conditional explanatory power directly. A value near zero means adding the excluded instruments reduces the restricted residual sum of squares by only a small fraction. A larger value means a larger fraction of the variation left after fitting \(X\) is explained by \(Z\). The statistic is between zero and one when the unrestricted model includes the restricted model, both are fitted by least squares to the same observations, and \(SSR_R>0\).

The F statistic adds a dimension that partial R-squared alone cannot convey. For the same partial R-squared, increasing the number \(q\) of excluded-instrument restrictions changes the F statistic; changing the residual degrees of freedom changes it as well. Consequently, report the degrees of freedom with the F statistic and avoid treating one partial R-squared value as a universal threshold for strength.

Assumptions, Diagnostics, and Limitations

Before interpreting the calculation, check that the restricted and unrestricted regressions use the same observations, the same dependent variable, and the same included covariates; the unrestricted model should add the excluded instruments being assessed. The models must be nested for the stated sum-of-squares comparison. Subject to these conditions, adding regressors cannot increase the OLS residual sum of squares, so \(SSR_U\leq SSR_R\) and the computed partial R-squared lies between zero and one, apart from numerical rounding.

If software reports a negative value, a value above one, or a value inconsistent with the two residual sums of squares, first check whether the models use different samples or covariates, whether the reported statistic is a robust test rather than the conventional F test, and whether the correct residual sums of squares have been extracted. A large partial R-squared should also be checked for unusual leverage or influential observations: one or a few observations can materially affect a fitted regression. Such diagnostics help assess how representative the fit measure is; they do not change its definition.

Partial R-squared measures first-stage relevance in the observed sample, not instrument validity. A strong association between \(Z\) and \(D\) conditional on \(X\) does not show that \(Z\) is independent of the structural error or affects the outcome only through \(D\). Those are distinct requirements for an instrumental-variable interpretation. Nor does a small partial R-squared, by itself, prove that identification fails; it describes limited incremental fit, while the F statistic accounts for its restrictions and residual degrees of freedom.

The comparison with other summaries is therefore practical. Ordinary first-stage R-squared summarizes fit from all regressors together. Partial R-squared isolates the added fit from excluded instruments. The conventional first-stage F statistic tests their joint zero-coefficient null and combines incremental fit with degrees of freedom. Use the measure appropriate to the question, and state whether the F statistic is conventional or uses a robust procedure.