Why Test the First Stage?
In instrumental variables analysis, the first stage describes how the excluded instruments predict the endogenous regressor after accounting for the included exogenous regressors. The first-stage F statistic gives a joint test of whether those excluded instruments add predictive information. It is therefore a useful diagnostic, but its meaning is precise and limited: it tests a particular zero-coefficient null, not every condition needed for credible IV estimation.
This tutorial considers one endogenous regressor and a linear first stage. Let \(D_i\) be the endogenous regressor for observation \(i\), let \(X_i\) be a vector of included exogenous regressors (including an intercept if the model has one), and let \(Z_i\) be a vector of excluded instruments. Write \(q\) for the number of excluded instruments and \(k_x\) for the number of coefficients on \(X_i\), counting the intercept when present.
The null concerns all \(q\) coefficients on \(Z_i\) jointly. The alternative is that at least one coefficient is nonzero. For this test, the instruments are not tested one at a time, and the included controls are present in both regressions. In particular, the overall F statistic for every regressor in the first stage is not generally the statistic of interest: it also tests coefficients on \(X_i\).
Formula and Assumptions
Let \(RSS_R\) be the residual sum of squares from the restricted regression of \(D_i\) on \(X_i\), and let \(RSS_U\) be the residual sum of squares from the unrestricted regression of \(D_i\) on both \(X_i\) and \(Z_i\). The unrestricted regression estimates \(k_x+q\) coefficients, so its residual degrees of freedom are \(n-k_x-q\). The statistic is
The numerator measures the reduction in residual variation when the excluded instruments are added, per restriction tested. The denominator estimates the first-stage error variance using the unrestricted residuals. Because the unrestricted model contains every regressor in the restricted model, \(RSS_R-RSS_U\) cannot be negative, apart from numerical rounding.
For the conventional finite-sample reference distribution \(F_{q,n-k_x-q}\) to be exact, assume the first-stage equation is correctly specified and linear, the regressor matrix has full column rank, and the errors are conditionally independent and normally distributed with mean zero and common variance given the regressors. In notation, conditional on \(X\) and \(Z\), \(v\sim N(0,\sigma^2 I)\). The usual homoskedastic F test also has a large-sample justification under suitable regularity conditions without normal errors. If errors are heteroskedastic or dependent, the conventional statistic and its reference distribution need not give valid inference; a suitable robust test is then required.
The model’s IV assumptions are a separate matter. The first-stage F test does not test whether \(Z_i\) is independent of unobserved causes of the outcome or excluded from the structural outcome equation. Nor does rejecting \(H_0\) prove that the instruments are sufficiently informative for reliable IV inference. It establishes evidence against zero conditional linear association under the test’s assumptions.
Why the Formula Has an F Distribution
The restricted and unrestricted regressions are nested: the restricted model is obtained by imposing the \(q\) restrictions \(\pi=0\). Under the null and the stated normal, homoskedastic model, the reduction in residual sum of squares, scaled by the error variance, has a chi-squared distribution with \(q\) degrees of freedom. The unrestricted residual sum of squares, scaled by the variance, has a chi-squared distribution with \(n-k_x-q\) degrees of freedom. These two quantities are independent under the normal linear model. Therefore their ratio after dividing each by its degrees of freedom has an F distribution:
The common variance cancels, yielding the statistic above. This also explains why the degrees of freedom in the numerator equal the number of excluded-instrument restrictions, while the denominator uses the unrestricted residual degrees of freedom.
Regress \(D_i\) on the included regressors \(X_i\) and record \(RSS_R\).
Regress \(D_i\) on the same \(X_i\) and the excluded instruments \(Z_i\); record \(RSS_U\).
Substitute the residual sums of squares, \(q\), and \(n-k_x-q\) into the formula. Compare the statistic with its \(F_{q,n-k_x-q}\) null distribution, or report its p-value.
Worked Calculations
Worked Example: One Excluded Instrument
Suppose a hypothetical cross-sectional sample has \(n=80\) observations. A first stage predicts a treatment variable \(D_i\) using an intercept and one included control, so \(k_x=2\), plus one excluded instrument, so \(q=1\). The restricted regression has \(RSS_R=160\), and the unrestricted regression has \(RSS_U=120\). These are hypothetical regression summary values from the same 80 observations.
The unrestricted residual degrees of freedom are \(80-2-1=77\). Thus
Under \(H_0:\pi=0\), compare this with an \(F_{1,77}\) distribution. The upper-tail p-value is below \(0.001\), so at \(\alpha=0.05\) the data provide evidence that the excluded instrument predicts \(D_i\) conditional on the included control. This conclusion is about the first-stage association; it does not establish instrument validity.
Worked Example: Joint Test of Two Excluded Instruments
In another hypothetical sample, \(n=120\). The first stage includes an intercept and two controls (\(k_x=3\)) and two excluded instruments (\(q=2\)). Suppose \(RSS_R=240\) and \(RSS_U=210\). The test concerns both excluded-instrument coefficients together.
The unrestricted residual degrees of freedom are \(120-3-2=115\). The calculation is
The reference distribution is \(F_{2,115}\); its upper-tail p-value is approximately \(0.00046\). At the 5% significance level, reject the joint null that both excluded-instrument coefficients are zero. The result does not say that both coefficients are individually nonzero: a joint test can reject when one instrument contributes most of the predictive information.
Worked Example: A Non-Significant First-Stage Test
Consider a third hypothetical sample with \(n=60\), an intercept and one included control (\(k_x=2\)), and one excluded instrument (\(q=1\)). Suppose the restricted and unrestricted residual sums of squares are \(90\) and \(85.5\), respectively. The unrestricted residual degrees of freedom are \(60-2-1=57\).
For \(F_{1,57}\), the upper-tail p-value is about \(0.089\). At \(\alpha=0.05\), do not reject \(H_0\). The appropriate conclusion is that this sample does not provide sufficient evidence, at that significance level, against zero conditional predictive contribution. It is not proof that the instrument has no relationship with \(D_i\), and it is not a direct estimate of how much weak-instrument bias remains.
What This Statistic Does—and Does Not—Tell You
The first-stage F statistic is useful because a weak relationship between the excluded instruments and the endogenous regressor can undermine the practical reliability of conventional IV estimates and inference. The test provides a compact, interpretable check of the joint zero-relevance null. However, a small p-value is not equivalent to a guarantee of strong identification: it reflects both the magnitude of predictive information and the sample size. With a large sample, even a modest conditional association can be statistically distinguishable from zero.
Keep the target of the test clear. The excluded-instrument test is a partial test: it asks whether adding \(Z_i\) improves the first stage after \(X_i\) is already included. An overall first-stage F statistic instead tests whether all slope coefficients in the first stage are zero, including those on \(X_i\). An individual t test addresses one coefficient and does not, by itself, test the joint null for multiple excluded instruments. These tests answer different questions.
| Quantity | Null or comparison | What it addresses |
|---|---|---|
| Excluded-instrument first-stage F | All coefficients on \(Z_i\) equal zero, conditional on \(X_i\) | Joint linear predictive contribution of the excluded instruments |
| Overall first-stage F | All tested slope coefficients in the first stage equal zero | Whether the tested regressors collectively predict \(D_i\) |
| Individual first-stage t test | One coefficient equals zero | Evidence about one coefficient, not the joint excluded-instrument null |
The earlier material on underidentification distinguishes a failure of the rank condition from weak identification. The first-stage F statistic should not be confused with a general test of every identification condition. In a one-endogenous-regressor setting, it is a relevant diagnostic of first-stage predictive strength, but the validity conditions for the instruments still require substantive justification and, where applicable, other diagnostics.
Diagnostics and Common Pitfalls
- Using the wrong F statistic. A software table may show an overall first-stage F. Check whether the statistic tests only the excluded instruments after controlling for \(X_i\). The target null is \(H_0:\pi=0\).
- Testing instruments one at a time. Individual t tests do not replace the joint test when \(q>1\). The joint restrictions must be tested together.
- Interpreting non-rejection as proof of irrelevance. A large p-value means the data did not provide sufficient evidence against the zero-coefficient null at the chosen level. It does not establish that the coefficients are exactly zero.
- Treating rejection as proof of instrument validity or adequate strength. Rejection says the instruments predict the endogenous regressor under the first-stage model. It does not test the exclusion restriction, independence from structural errors, or a universal threshold for reliable inference.
- Ignoring the error structure. If observations are clustered or errors are heteroskedastic, the conventional homoskedastic F reference distribution may be inappropriate. Use a test whose variance calculation matches the sampling design and error assumptions.
- Comparing regressions that differ in controls. The restricted and unrestricted regressions must use the same observations and the same included regressors; the instruments are the only added regressors in the unrestricted model.
As a practical check, verify that \(RSS_U\leq RSS_R\), that the denominator degrees of freedom are positive, and that the tested instrument set and included controls match the intended IV specification. Large changes in the first-stage sample due to missing data can invalidate a direct comparison of the two residual sums of squares.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- A first stage has \(n=100\), \(k_x=2\), \(q=2\), \(RSS_R=200\), and \(RSS_U=180\). Calculate the excluded-instrument F statistic and its numerator and denominator degrees of freedom.
- What is the null hypothesis when there are three excluded instruments?
- Why is the overall first-stage F statistic not necessarily the excluded-instrument first-stage F statistic?
- A test rejects the joint zero-coefficient null. Name one important IV condition this result does not establish.
- Why should a conventional homoskedastic first-stage F test be treated cautiously for clustered observations?
Solutions. (1) The residual degrees of freedom are \(100-2-2=96\), and \(F=((200-180)/2)/(180/96)=10/1.875=5.333\), with \(df=(2,96)\). (2) The null is that all three excluded-instrument coefficients are zero jointly, \(H_0:\pi_1=\pi_2=\pi_3=0\), conditional on the included regressors. (3) The overall test may also impose restrictions on included controls; the excluded-instrument test leaves their coefficients unrestricted and tests only the excluded instruments. (4) Rejection does not establish the exclusion restriction or instrument independence from structural errors. (5) Within-cluster dependence can make the conventional variance estimate and F reference distribution invalid; inference should account for the cluster structure.