Why 2SLS Results Need More Than a Coefficient
A 2SLS estimate is not fully reported by giving a coefficient and a significance marker. Readers also need to know what was estimated, how uncertainty was calculated, which observations contributed, and exactly what hypothesis a test addresses. These details matter because a standard error depends on the covariance estimator, and because a coefficient may be only one part of a scientifically meaningful linear contrast.
For a linear instrumental-variables model, let \(y_i\) be the outcome, \(D_i\) an endogenous regressor, \(X_i\) a vector of included exogenous regressors, and \(Z_i\) the excluded instrument vector. Write the structural equation as \(y_i=\beta_0+\beta_D D_i+X_i'\gamma+u_i\). 2SLS estimates the coefficient vector using the included exogenous regressors and excluded instruments as instruments for the regressors. Interpreting \(\hat{\beta}_D\) as a causal effect requires the relevant IV assumptions, including instrument relevance, instrument exogeneity, and exclusion, along with the model's identification conditions. A table can document the analysis; it cannot establish those assumptions by itself.
What to Put in the Table and Its Notes
A compact table can have columns for the regressor, 2SLS coefficient, standard error, and confidence interval. Use a clear row for each coefficient and label the outcome and model in the column heading or table caption. The accompanying note should specify the sample size, the instruments, included controls, any fixed effects, the covariance estimator, the confidence level, and the reference distribution used for inference. If the specification changes between columns, document those changes rather than relying on readers to infer them.
The standard error is the estimated standard deviation of the estimator under the stated covariance procedure; it is not the standard deviation of the observed outcome or a measure of the coefficient's practical importance. A confidence interval puts the estimate and its uncertainty on the coefficient's scale. For an estimate \(\hat{\beta}_j\) with standard error \(SE(\hat{\beta}_j)\), a large-sample 95% Wald interval is
This interval uses a normal approximation. If a justified finite-sample procedure uses a \(t\) reference distribution, report that choice and its degrees of freedom instead of silently using 1.96. For clustered inference, for example, the degrees of freedom may be tied to the number of independent clusters, subject to the particular variance estimator and inferential procedure. The interval has the stated repeated-sampling coverage under its assumptions; it is not a probability statement about the fixed parameter after the data have been observed.
A useful specification note distinguishes the sample size \(n\) from the number of independent clusters, if relevant. It also names the covariance estimator precisely: for example, heteroskedasticity-robust (HC1), cluster-robust by clinic, or HAC with the stated ordering and lag choice. “Robust standard errors” alone is ambiguous. The covariance estimator describes how uncertainty was estimated; it does not change the substantive model or repair invalid instruments.
Worked Example: Constructing a Coefficient Table
Suppose a hypothetical cross-sectional study estimates the effect of a program on annual earnings, in thousands of dollars, using 2SLS. The program indicator is endogenous; a policy eligibility rule is the excluded instrument. The model includes age and education as controls. There are \(n=840\) observations. The reported estimates and HC1 standard errors are: program \(0.42\) and \(0.18\), education \(0.11\) and \(0.05\), and age \(-0.008\) and \(0.006\). Assume the large-sample normal approximation is used for 95% intervals.
For the program coefficient, the interval is \(0.42\pm1.96(0.18)=0.42\pm0.3528\), or \([0.0672,0.7728]\). For education, it is \(0.11\pm1.96(0.05)=0.11\pm0.0980\), or \([0.0120,0.208]\). For age, it is \(-0.008\pm1.96(0.006)=-0.008\pm0.01176\), or \([-0.0198,0.00376]\). All coefficient units are thousands of dollars per unit of the corresponding regressor.
| Regressor | 2SLS coefficient | HC1 standard error | 95% confidence interval |
|---|---|---|---|
| Program | 0.420 | 0.180 | [0.0672, 0.773] |
| Education | 0.110 | 0.050 | [0.0120, 0.208] |
| Age | -0.00800 | 0.00600 | [-0.0198, 0.00376] |
A suitable note would state: “Outcome: annual earnings in thousands of dollars. 2SLS estimates; excluded instrument: policy eligibility rule. Controls: age and education. HC1 heteroskedasticity-robust covariance estimator. 95% Wald confidence intervals use the standard normal critical value 1.96. \(n=840\).” The program interval excludes zero, whereas the age interval includes zero. That describes the reported intervals; a causal interpretation of the program estimate still depends on the IV assumptions.
Reporting Linear Hypotheses
A hypothesis about one coefficient is a special case of a linear hypothesis about the coefficient vector \(\beta\). Write the restriction as \(H_0:R\beta=r\), where \(R\) specifies the linear combination or combinations and \(r\) gives their hypothesized values. For a single restriction, such as \(H_0:\beta_1+\beta_2=0\), the estimate of the contrast is \(\hat{\beta}_1+\hat{\beta}_2\). Its standard error must account for the covariance between the estimates:
For \(q\) restrictions, the Wald statistic is
where \(\widehat{V}\) is the covariance estimate for \(\hat{\beta}\). Under the null and the relevant large-sample conditions, \(W\) is compared with a \(\chi^2\) distribution with \(q\) degrees of freedom. A one-restriction test can equivalently be expressed as a signed Wald statistic compared with its stated reference distribution. In a table or its notes, report the restriction, test statistic or contrast and its standard error, and the reference distribution. A bare statement such as “the coefficients are jointly significant” does not identify which coefficients or restriction were tested.
Worked Example: Testing a Sum of Two Coefficients
In a hypothetical 2SLS model with an interaction, suppose the estimated main-effect coefficient is \(0.50\), the interaction coefficient is \(-0.20\), and their covariance matrix, using the selected robust estimator, is
The question is whether the sum of the two coefficients is zero: \(H_0:\beta_1+\beta_2=0\), against \(H_a:\beta_1+\beta_2\ne0\). The estimated sum is \(0.50+(-0.20)=0.30\). Its variance and standard error are
Using a normal reference, the 95% interval for the sum is \(0.30\pm1.96(0.1962)=[-0.0846,0.6846]\). The Wald statistic is \(z=0.30/0.1962=1.529\), so the equivalent one-restriction statistic is \(W=z^2=2.338\). The interval includes zero, and the two-sided test does not reject \(H_0\) at the 5% level. A report should name the sum being tested; separate tests of \(\beta_1=0\) and \(\beta_2=0\) do not answer this question.
Covariance Choices, Diagnostics, and Alternatives
The covariance estimator should follow the dependence and sampling structure of the data, not the result that produces the smallest standard error. A heteroskedasticity-robust estimator allows nonconstant error variance across observations while treating observations as independent. A cluster-robust estimator allows dependence within specified clusters and typically relies on independence across clusters. HAC estimators address certain forms of serial dependence and heteroskedasticity in ordered data. The chosen estimator must be appropriate for the design and its assumptions; listing it in a table is necessary for transparency but does not prove those assumptions hold.
Different valid covariance estimators can produce the same 2SLS point estimates but different standard errors, intervals, and tests. Finite-sample corrections and bootstrap procedures are alternative inferential choices and should be identified rather than mixed without explanation. The choice of estimator does not substitute for examining the design, the instrument assumptions, or the model specification. Instrument strength diagnostics are also a separate part of an IV analysis; coefficient-table precision alone does not establish strong identification.
Worked Example: How the Covariance Note Changes Interpretation
Suppose a hypothetical panel analysis estimates a program coefficient of \(0.42\) thousand dollars using \(n=840\) observations from 28 clinics. The HC1 standard error is \(0.18\), while the clinic-clustered standard error is \(0.24\). To illustrate a stated finite-cluster procedure, use a \(t\) reference with \(df=27\) and critical value 2.052 for the clustered interval.
The HC1 normal-based interval is \(0.42\pm1.96(0.18)=[0.0672,0.7728]\). The cluster-based interval is \(0.42\pm2.052(0.24)=0.42\pm0.49248=[-0.0725,0.9125]\). The latter includes zero. These are not conflicting point estimates: they are the same estimate paired with different uncertainty calculations and reference distributions. If within-clinic dependence is plausible and clinics are the independent sampling units, the HC1 interval may understate uncertainty. A transparent table should report which result is primary and why, and should show \(n=840\) and 28 clusters rather than giving only the observation count.
Before finalizing a table, check that every column uses the stated sample and model, that the displayed intervals can be reproduced from the coefficient, standard error, and critical value, and that the reported tests match their written restrictions. If the sample changes because of missing observations, report the sample size for each specification. If multiple outcomes or model columns appear, define all abbreviations and make clear which estimates are 2SLS.
- Pitfall: reporting only “robust SEs.” This does not say whether the estimator is heteroskedasticity-robust, clustered, or HAC. Name the covariance estimator and relevant clustering or HAC choices.
- Pitfall: using the wrong uncertainty for a sum. Adding two standard errors ignores their covariance. Use \(R\widehat{V}R'\) for a linear contrast.
- Pitfall: treating a confidence interval as a causal-validity check. A narrow interval quantifies uncertainty under the model and inference assumptions; it does not demonstrate instrument exogeneity or exclusion.
- Pitfall: omitting the sample size or hypothesis. Without \(n\), readers cannot tell what information supports the estimate; without a stated restriction, a test result is difficult to interpret.
Check Your Understanding
Use the stated normal approximation for the first two calculations. The solutions follow the questions.
- A 2SLS coefficient is \(0.30\), with standard error \(0.10\). Calculate its 95% Wald interval and state whether it includes zero.
- Two estimates have variances \(0.09\) and \(0.04\), and covariance \(-0.015\). Their estimated sum is \(0.45\). Calculate the standard error and 95% interval for the sum.
- A table says “robust SEs, \(n=500\)” for data from 20 clinics. Name two details needed to make its inference description clearer.
- In one sentence, explain why a 2SLS confidence interval does not by itself verify the instrument assumptions.
Solutions. (1) \(0.30\pm1.96(0.10)=0.30\pm0.196=[0.104,0.496]\), which excludes zero. (2) The variance of the sum is \(0.09+0.04+2(-0.015)=0.10\), so its standard error is \(\sqrt{0.10}=0.3162\). The interval is \(0.45\pm1.96(0.3162)=[-0.170,1.070]\), which includes zero. (3) State the specific covariance estimator and, if clustering is used, identify clinics as the clusters and report their number; also state the reference distribution and degrees of freedom. (4) The interval describes sampling uncertainty under the chosen model and inference assumptions, whereas instrument relevance, exogeneity, and exclusion are design and model assumptions not established by interval coverage.