The Asymptotic Distribution of 2SLS

Why the Large-Sample Distribution Matters

The 2SLS coefficient is computed from a sample, so it varies across samples even when the population relationship is fixed. Its large-sample distribution describes that variation when the sample grows under suitable assumptions. The derivation also reveals what controls uncertainty: not just the strength of the relationship between instruments and regressors, but also the variability of the instrument-weighted errors.

This result concerns the 2SLS estimator itself. It does not say that a particular finite sample is approximately normal, nor does it make an invalid or weak instrument reliable. The population conditions must be considered before interpreting the approximation.

Model, Notation, and Assumptions

For observation \(i\), let \(y_i\) be a scalar outcome, \(x_i\) a \(k\)-vector of regressors, \(z_i\) an \(l\)-vector of instruments, and \(u_i\) the structural error. The linear model is \(y_i=x_i'\beta_0+u_i\), where \(\beta_0\) is the population coefficient vector. The instrument vector may include exogenous regressors as well as excluded instruments. Assume \(l\geq k\).

The asymptotic result below uses the following conditions. The observations \((y_i,x_i,z_i)\) are independent and identically distributed; \(E[z_i u_i]=0\); the entries of \(Q_{zz}\) and \(Q_{zx}\) are finite; and \(Q_{zz}\) is positive definite. The moment \(E[\|z_i u_i\|^2]\) is finite, so the instrument-weighted error has a finite covariance matrix. Finally, \(Q_{zx}\) has full column rank \(k\). This rank condition says that the instruments provide population variation in every regressor direction needed to identify \(\beta_0\).

These conditions imply that \(A\) is nonsingular: for every nonzero \(b\in\mathbb{R}^k\), \(Q_{zx}b\ne0\), and positive definiteness of \(Q_{zz}^{-1}\) gives \(b'A b=(Q_{zx}b)'Q_{zz}^{-1}(Q_{zx}b)>0\). In a sample, the corresponding matrices must be well-defined and invertible; under the population conditions, this occurs with probability approaching one.

From the Moment Representation to the Limit

Let \(X\) and \(Z\) be the \(n\times k\) and \(n\times l\) matrices with rows \(x_i'\) and \(z_i'\). The 2SLS estimator has the projection-matrix form \(\hat\beta_{2SLS}=(X'P_ZX)^{-1}X'P_Zy\), where \(P_Z=Z(Z'Z)^{-1}Z'\). Equivalently, using sample moment matrices, it is

Substitute \(y=X\beta_0+u\), with \(u\) the \(n\)-vector of errors. The part involving \(X\beta_0\) simplifies to \(\beta_0\). The remaining term yields the exact sample representation

where \(\hat Q_{zz}=Z'Z/n\), \(\hat Q_{zx}=Z'X/n\), \(\hat Q_{xz}=X'Z/n\), and \(\hat A=\hat Q_{xz}\hat Q_{zz}^{-1}\hat Q_{zx}\). Notice that \(Q_{zx}\) is \(l\times k\), whereas \(Q_{xz}\) is \(k\times l\); their order in this expression is required both by the derivation and by matrix dimensions.

By the law of large numbers, the sample moment matrices converge to their population counterparts. By the multivariate central limit theorem, \(n^{-1/2}\sum z_i u_i\) converges in distribution to a mean-zero normal vector with covariance \(\Omega\). Applying Slutsky’s theorem to the exact representation gives the result.

The matrix inside the normal distribution is the asymptotic covariance of the scaled estimation error. Consequently, the covariance of \(\hat\beta_{2SLS}\) is approximately that matrix divided by \(n\) for large \(n\). Its diagonal entries describe approximate variances for individual coefficients, and off-diagonal entries describe their joint sampling variation. The matrix is symmetric and positive semidefinite because it is a covariance matrix. No homoskedasticity assumption is needed for this form: \(\Omega\) retains the population variation in \(z_i u_i\).

Worked Calculations

Interpretation, Diagnostics, and Limitations

The formula separates two sources of precision. The matrices \(Q_{zx}\) and \(Q_{zz}\) describe how instrument variation relates to the regressors and how that variation is scaled. The matrix \(\Omega\) describes the covariance of the instrument-weighted structural errors. In general, changing either the instrument-regressor relationships or the error variation changes the asymptotic covariance.

In applications, inspect whether the sample analogue of the identifying matrix has full rank and whether it is poorly conditioned. First-stage relationships can help reveal whether the instruments provide little variation for some regressor direction. There is no universal sample diagnostic that verifies instrument validity: relevance can be assessed from observed data, while the exclusion and exogeneity assumptions require substantive justification. Weak instruments are especially consequential because the normal approximation can be poor even in samples that seem large.

The result assumes independent, identically distributed observations. Dependence across observations, such as clustering or serial correlation, changes the relevant limiting argument and covariance calculation. Those settings require methods tailored to their sampling structure. Also, asymptotic normality is a statement about behavior as \(n\) grows; it does not guarantee accurate finite-sample inference.

Compared with OLS, 2SLS uses instrument-driven variation to address endogeneity under valid-instrument assumptions. OLS can be more precise when its assumptions hold, but a difference in standard errors does not by itself establish which estimator is appropriate. 2SLS’s asymptotic distribution quantifies sampling variation conditional on its model and instrument assumptions; it cannot compensate for invalid instruments or identify a causal effect without the relevant assumptions.

A Short Derivation of the Covariance

The limiting random vector in the asymptotic linear representation is a fixed matrix multiplied by the limiting vector for \(n^{-1/2}\sum z_i u_i\). If a mean-zero vector has covariance \(\Omega\), then multiplying it by a matrix \(B\) gives covariance \(B\Omega B'\). Here \(B=A^{-1}Q_{xz}Q_{zz}^{-1}\). Since \(A\) and \(Q_{zz}\) are symmetric, \(B'=Q_{zz}^{-1}Q_{zx}A^{-1}\), which yields

This also explains why reversing \(Q_{xz}\) and \(Q_{zx}\) is not a harmless notational change: their dimensions differ, and only the displayed ordering gives a \(k\times k\) covariance matrix.

Common Mistakes

• Transposing the cross-moment matrix incorrectly. With \(z_i\) of dimension \(l\) and \(x_i\) of dimension \(k\), \(Q_{zx}=E[z_i x_i']\) is \(l\times k\), while \(Q_{xz}=Q_{zx}'\) is \(k\times l\). Check every product’s dimensions.
• Using the theorem without its rank condition. If \(A\) is singular, the inverse in the estimator’s representation and the stated covariance do not exist.
• Assuming the result makes weak instruments harmless. Full rank is a population identification condition; near-singularity can still undermine the normal approximation in realistic sample sizes.
• Calling the covariance of the scaled error the estimator’s covariance. The theorem gives the limiting covariance for \(\sqrt{n}(\hat\beta_{2SLS}-\beta_0)\); the estimator’s large-sample covariance is approximately that matrix divided by \(n\).