2SLS Versus OLS Standard Errors

Why the Same Coefficient Does Not Mean the Same Standard Error

Two-stage least squares (2SLS) produces coefficients by using instruments to isolate variation in endogenous regressors. Its uncertainty calculation must reflect that instrument-based construction. A coefficient may match the coefficient from an ordinary least squares (OLS) regression on fitted regressors, but the standard OLS variance formula does not follow merely from that equality.

The key distinction is between the coefficient calculation and its sampling variability. OLS treats the regressors as the source of the fitted variation. 2SLS instead weights the outcome and regressors through the instrument space. That distinction changes the variance formula, even when the estimated coefficient is identical.

For a structural equation \(y=X\beta+u\), the OLS formula is conventionally paired with a homoskedastic error model and exogenous regressors: \(E[u\mid X]=0\) and \(\operatorname{Var}(u\mid X)=\sigma^2 I\). The IV model allows some columns of \(X\) to be correlated with \(u\), so OLS exogeneity fails for those columns. Instead, IV relies on valid instruments: in particular, the instruments satisfy the required orthogonality conditions with the structural error, and the instruments provide enough independent variation to identify the coefficients. Those conditions do not make endogenous regressors exogenous.

For the conventional homoskedastic 2SLS variance, assume the structural equation is correctly specified, the instruments are valid and relevant, and the errors have common variance \(\sigma^2\) with no cross-observation covariance. In a sample or moment-based variance calculation, homoskedasticity makes the covariance of the instrument-weighted error score proportional to \(Z'Z\). The formula below is the conventional variance approximation under these conditions; it is not a claim that OLS exogeneity holds for \(X\).

Deriving the Conventional Homoskedastic Formula

Substitute \(y=X\beta+u\) into the 2SLS estimator. Its estimation error is

The random error enters through \(Z'u\), the vector of instrument-error cross-products. Under homoskedasticity and no cross-observation error covariance, the covariance of this score, conditional on the instruments, is \(\sigma^2 Z'Z\). Treating the design matrices in the variance calculation as fixed, or using their sample counterparts in the usual moment-based approximation, gives

The simplification follows because the middle factors reduce to \(X'Z(Z'Z)^{-1}Z'X=X'P_ZX\), and the inverse matrix is symmetric. In practice, \(\sigma^2\) is unknown. A conventional estimate uses the structural residuals \(\hat{u}=y-X\hat{\beta}_{2SLS}\), not the residuals from regressing \(y\) on fitted regressors:

The \(n-k\) denominator is a common degrees-of-freedom convention for this estimate; software and finite-sample conventions may differ. The central point for this comparison is the matrix \(X'P_ZX\), not \(X'X\). The next sections show what that difference means in calculations and in two-stage software workflows.

Why Naive Second-Stage OLS Standard Errors Can Mislead

In the first stage, each endogenous regressor is projected onto the instruments and included exogenous regressors. Let \(\hat{X}=P_ZX\) denote the resulting fitted-regressor matrix, with the included exogenous columns represented consistently in \(Z\). Regressing \(y\) on \(\hat{X}\) reproduces the 2SLS coefficient, because \(X'P_ZX=\hat{X}'\hat{X}\) and \(X'P_Zy=\hat{X}'y\). But a software command that treats \(\hat{X}\) as observed data and applies the ordinary second-stage OLS variance calculation may use the wrong residual variance. The fitted regressors are estimated from the same sample, and second-stage residuals \(y-\hat{X}\hat{\beta}_{2SLS}\) are not the structural residuals \(y-X\hat{\beta}_{2SLS}\).

Diagnostics, Alternatives, and Limits

A conventional homoskedastic standard error can be inappropriate even when the coefficient calculation is correct. If error variance differs across observations, the covariance of \(Z'u\) is not \(\sigma^2 Z'Z\); the simplifying cancellation in the derivation does not apply. The heteroskedasticity-robust 2SLS variance replaces that homoskedastic score covariance with a residual-weighted “sandwich” calculation. If observations are dependent within clusters or over time, the variance calculation must also reflect that dependence. These are different uncertainty estimators, not different 2SLS coefficient estimators.

The first diagnostic is substantive and design-based: ask whether a common error variance and independence across observations are credible. Residual plots can reveal patterns such as spread increasing with fitted values, but a flat plot cannot prove homoskedasticity. Also check that the instrumented design is informative. When instruments predict the endogenous regressors only weakly, \(X'P_ZX\) can be small in relevant directions, making conventional standard errors large or making ordinary approximations unreliable. A small standard error is not evidence that instruments are valid or strong.

The main alternatives answer different needs. OLS with its classical standard error is appropriate for an exogenous-regressor model under its error assumptions; it is not a substitute for IV inference when regressors are endogenous. Conventional homoskedastic 2SLS uses the common-variance restriction. Heteroskedasticity-robust 2SLS relaxes that restriction but still requires the relevant instrument and sampling assumptions. Cluster-robust or other dependence-aware procedures address additional forms of dependence. No standard error formula repairs an invalid instrument, and conventional formulas should not be treated as a guarantee of reliable finite-sample inference with weak instruments.

A standard error describes estimated sampling uncertainty for a specified estimator under a specified variance model. It does not measure bias from invalid instruments, and comparing an OLS standard error with a 2SLS standard error does not rank the methods without considering their estimands and assumptions. Under endogeneity, OLS can appear more precise partly because it uses variation that 2SLS excludes precisely because that variation may be confounded.

Common Mistakes

• Using the OLS formula because the output contains an OLS coefficient. A coefficient equality between second-stage OLS on fitted regressors and 2SLS does not establish equality of their standard-error calculations.
• Using second-stage residuals as structural residuals. The conventional 2SLS variance estimate uses \(y-X\hat{\beta}_{2SLS}\), not \(y-\hat{X}\hat{\beta}_{2SLS}\).
• Interpreting a smaller OLS standard error as proof of greater precision for the causal effect. The estimators can target different quantities when regressors are endogenous.
• Assuming robust standard errors validate the instrument. Robustness changes the variance calculation; it does not establish instrument relevance or validity.