The Joint Reduced-Form Limit Under Weak Instruments

Why the Joint Limit Matters

With a strong instrument, reduced-form coefficients become increasingly precise around their population values. Under a local-to-zero instrument sequence, however, the population slopes themselves shrink at the same rate as their sampling uncertainty. The limiting behavior is therefore described not by two separate approximations, but by a joint distribution that keeps both the local signal and the covariance between the estimates.

That covariance matters because the reduced-form outcome and endogenous-regressor equations share a source of variation. In particular, the outcome’s reduced-form error contains both the structural error and the first-stage error. If those errors are correlated, the covariance between the reduced-form estimates reflects that correlation.

Model and Reduced-Form Coefficients

Consider a scalar model with one instrument and no intercept or controls. The equations for observation $i$ in a sample of size $n$ are

x_i=\pi_n z_i+v_i,\qquad y_i=\beta x_i+u_i.

Here $y_i$ is the outcome, $x_i$ is the endogenous regressor, $z_i$ is the instrument, $\beta$ is the structural parameter, and $u_i$ and $v_i$ are the structural and first-stage errors. The coefficient $\pi_n$ is allowed to vary with sample size.

Definition: The reduced-form coefficient for $x$ is $\pi_n$, and the reduced-form coefficient for $y$ is $\beta\pi_n$. Their through-origin sample estimates are $$ \hat{\gamma}_{x,n}=\frac{\sum_{i=1}^n z_i x_i}{\sum_{i=1}^n z_i^2}, \qquad \hat{\gamma}_{y,n}=\frac{\sum_{i=1}^n z_i y_i}{\sum_{i=1}^n z_i^2}. $$

Use the following assumptions throughout. The triples $(z_i,u_i,v_i)$ are independent and identically distributed across observations, with a common distribution that does not depend on $n$. The instrument is independent of the error pair $(u_i,v_i)$, and

E[z_i]=0,\quad E[z_i^2]=Q>0,\quad E[u_i]=E[v_i]=0.

Write $\sigma_u^2=\operatorname{Var}(u_i)$, $\sigma_v^2=\operatorname{Var}(v_i)$, and $\sigma_{uv}=\operatorname{Cov}(u_i,v_i)$. Allow $\sigma_{uv}\ne 0$. Assume there is a $\delta>0$ for which $E|z_i|^{2+\delta}$, $E|u_i|^{2+\delta}$, and $E|v_i|^{2+\delta}$ are finite. Finally, take the local-to-zero sequence

\pi_n=\frac{c}{\sqrt n},

where $c$ is a fixed real number. Independence of $z_i$ from the errors and their zero means imply $E[y_i\mid z_i]=\beta\pi_n z_i$ and $E[x_i\mid z_i]=\pi_n z_i$. Thus the stated population reduced-form slopes are also the conditional mean slopes.

Worked Example: Writing the Two Reduced Forms

Suppose $x_i=(1.2/\sqrt n)z_i+v_i$ and $y_i=2x_i+u_i$. Substituting the first equation into the second gives

y_i=2\left(\frac{1.2}{\sqrt n}z_i+v_i\right)+u_i =\frac{2.4}{\sqrt n}z_i+(2v_i+u_i).

The coefficient of $z_i$ is $1.2/\sqrt n$ in the $x_i$ equation and $2.4/\sqrt n$ in the $y_i$ equation. The corresponding reduced-form errors are $v_i$ and $2v_i+u_i$, not $v_i$ and $u_i$. The latter distinction determines the covariance of the two coefficient estimates.

The Joint Reduced-Form Limit

The joint limit has a simple structure: the local sequence contributes a fixed shift to the limiting mean, while the error variances and covariance determine the limiting covariance matrix.

Theorem: Under the model and assumptions above, $$ \begin{pmatrix} \sqrt n\,\hat{\gamma}_{y,n}\\ \sqrt n\,\hat{\gamma}_{x,n} \end{pmatrix} \xrightarrow{d} N\left[ \begin{pmatrix}\beta c\\c\end{pmatrix}, \frac{1}{Q} \begin{pmatrix} \beta^2\sigma_v^2+2\beta\sigma_{uv}+\sigma_u^2 & \beta\sigma_v^2+\sigma_{uv}\\ \beta\sigma_v^2+\sigma_{uv} & \sigma_v^2 \end{pmatrix} \right]. $$

Equivalently, after centering at the population reduced-form coefficients,

\begin{pmatrix} \sqrt n\,(\hat{\gamma}_{y,n}-\beta\pi_n)\\ \sqrt n\,(\hat{\gamma}_{x,n}-\pi_n) \end{pmatrix} \xrightarrow{d} N\left[ \begin{pmatrix}0\\0\end{pmatrix}, \frac{1}{Q} \begin{pmatrix} \beta^2\sigma_v^2+2\beta\sigma_{uv}+\sigma_u^2 & \beta\sigma_v^2+\sigma_{uv}\\ \beta\sigma_v^2+\sigma_{uv} & \sigma_v^2 \end{pmatrix} \right].

Derivation of the Joint Limit

First substitute $x_i=\pi_nz_i+v_i$ into the outcome equation. The two sample slopes can then be written exactly as their population coefficients plus ratios of error-weighted instrument sums:

\hat{\gamma}_{x,n}-\pi_n = \frac{\sum_{i=1}^n z_i v_i}{\sum_{i=1}^n z_i^2}, \qquad \hat{\gamma}_{y,n}-\beta\pi_n = \frac{\sum_{i=1}^n z_i(\beta v_i+u_i)}{\sum_{i=1}^n z_i^2}.

Define the vector $\xi_i=z_i(\beta v_i+u_i,\ v_i)^\mathsf{T}$. Its mean is zero. Independence of $z_i$ and $(u_i,v_i)$ gives its covariance:

\operatorname{Var}(\xi_i) =Q \begin{pmatrix} \operatorname{Var}(\beta v_i+u_i)&\operatorname{Cov}(\beta v_i+u_i,v_i)\\ \operatorname{Cov}(\beta v_i+u_i,v_i)&\operatorname{Var}(v_i) \end{pmatrix} =Q \begin{pmatrix} \beta^2\sigma_v^2+2\beta\sigma_{uv}+\sigma_u^2& \beta\sigma_v^2+\sigma_{uv}\\ \beta\sigma_v^2+\sigma_{uv}&\sigma_v^2 \end{pmatrix}.

Proof: The moment assumptions and independence ensure that $\xi_i$ has finite second moments. The multivariate central limit theorem therefore gives $$ \frac{1}{\sqrt n}\sum_{i=1}^n \xi_i \xrightarrow{d}N\!\left(0,\operatorname{Var}(\xi_i)\right). $$ By the law of large numbers, $n^{-1}\sum_{i=1}^n z_i^2\xrightarrow{p}Q$. The exact slope decompositions show that the centered vector of slopes, multiplied by $\sqrt n$, is the vector sum above divided by $n^{-1}\sum_{i=1}^n z_i^2$. Since $Q>0$, Slutsky’s theorem gives a centered normal limit with covariance $\operatorname{Var}(\xi_i)/Q^2$. Substituting the displayed covariance yields the centered version of the theorem. Finally, $\sqrt n\,\pi_n=c$, so adding $(\beta c,c)^\mathsf{T}$ gives the stated uncentered limit. $\square$

The covariance calculation itself also shows why the structural error correlation cannot be discarded. The outcome reduced-form error is $\beta v_i+u_i$; its covariance with the regressor reduced-form error $v_i$ is $\beta\sigma_v^2+\sigma_{uv}$. In particular, this covariance is not generally zero even when $u_i$ and $v_i$ are uncorrelated.

Worked Example: Computing the Full Limit Covariance

Let $\beta=2$, $Q=4$, $\sigma_u^2=9$, $\sigma_v^2=4$, and $\sigma_{uv}=3$. These error moments are admissible: the determinant of their covariance matrix is $9(4)-3^2=27>0$. The variance of the outcome reduced-form error is

\operatorname{Var}(2v_i+u_i)=2^2(4)+2(2)(3)+9=16+12+9=37.

Its covariance with the regressor reduced-form error is

\operatorname{Cov}(2v_i+u_i,v_i)=2(4)+3=11.

Thus the limiting covariance matrix is

\frac14 \begin{pmatrix}37&11\\11&4\end{pmatrix} = \begin{pmatrix}9.25&2.75\\2.75&1\end{pmatrix}.

For $c=1.5$, the limiting mean is $(\beta c,c)^\mathsf{T}=(3,1.5)^\mathsf{T}$. The full limit is therefore a bivariate normal with that mean and the covariance matrix above. Its off-diagonal entry, $2.75$, records the dependence between the two reduced-form coefficient estimates.

How Error Correlation Changes the Result

The formula contains $\sigma_{uv}$ in both the outcome variance and the cross-covariance. Its effect is not limited to the off-diagonal entry: because the outcome reduced-form error combines $u_i$ and $\beta v_i$, their covariance also changes that error’s variance. The next calculation isolates this point.

Worked Example: A Negative Structural Error Covariance

Suppose $\beta=1.5$, $Q=1$, $\sigma_u^2=4$, $\sigma_v^2=1$, and $\sigma_{uv}=-0.5$. The error covariance matrix is valid because its determinant is $4(1)-(-0.5)^2=3.75>0$. The outcome reduced-form error variance is

$$ (1.5)^2(1)+2(1.5)(-0.5)+4=2.25-1.5+4=4.75, $$

and the cross-covariance is

$$ 1.5(1)-0.5=1. $$

The limiting covariance matrix is therefore $\begin{pmatrix}4.75&1\\1&1\end{pmatrix}$. If the errors were incorrectly treated as uncorrelated, the cross-covariance would be reported as $1.5$, and the outcome variance as $6.25$. Both entries would be wrong.

The Local Shift and Covariance Structure

The sequence parameter $c$ appears in the mean of the uncentered joint limit, not in its covariance. For example, changing $c$ from $0$ to $2$ shifts the limit’s mean from $(0,0)^\mathsf{T}$ to $(2\beta,2)^\mathsf{T}$, while leaving the covariance matrix unchanged under the maintained assumptions. Thus the sequence describes how much signal remains on the sampling scale; the error distribution describes the joint sampling variation.

A useful check on the covariance formula is that it must be positive semidefinite. Let $\Sigma$ denote the covariance matrix in the theorem. Direct expansion gives

Result: The determinant and diagonal entries of the limiting covariance matrix satisfy $$ \det(\Sigma)=\frac{\sigma_u^2\sigma_v^2-\sigma_{uv}^2}{Q^2}\geq 0, \qquad \Sigma_{11}\geq0,\quad \Sigma_{22}\geq0. $$

To verify the determinant, expand the numerator before dividing by $Q^2$:

(\beta^2\sigma_v^2+2\beta\sigma_{uv}+\sigma_u^2)\sigma_v^2 -(\beta\sigma_v^2+\sigma_{uv})^2 =\sigma_u^2\sigma_v^2-\sigma_{uv}^2.

The right-hand side is nonnegative because the covariance matrix of $(u_i,v_i)^\mathsf{T}$ is positive semidefinite. The diagonal entries of $\Sigma$ are variances divided by $Q>0$, so they are nonnegative as well. For a symmetric $2$-by-$2$ matrix these conditions imply positive semidefiniteness. This check also shows that the determinant does not depend on $\beta$: transforming the error pair into $(\beta v_i+u_i,v_i)$ changes its individual variances and covariance, but preserves its determinant.

Common Mistake

Wrong reasoning: “The outcome equation is $y_i=\beta x_i+u_i$, so the reduced-form outcome error is just $u_i$. I can use $\sigma_u^2$ for its variance and ignore the covariance with the first-stage error.”
Correction: Substituting $x_i=\pi_nz_i+v_i$ gives $y_i=\beta\pi_nz_i+(\beta v_i+u_i)$. The reduced-form outcome error is $\beta v_i+u_i$. Its variance and covariance with $v_i$ must be calculated from that expression, retaining $\sigma_{uv}$.
Another common error: Centering both estimates at zero in the local sequence without accounting for their local means. For the uncentered scaled estimates, the correct mean is $(\beta c,c)^\mathsf{T}$; it is zero only when $c=0$.

The joint reduced-form limit is a key input for later weak-instrument analysis. Working with both coefficients together preserves their dependence and makes the local signal explicit. Treating the estimates as independent, or replacing the reduced-form error by the structural error alone, loses information already present in the model.

Check Your Understanding

Try these questions using the assumptions and notation above.

Under the local sequence $\pi_n=c/\sqrt n$, what is the limiting mean of $(\sqrt n\,\hat{\gamma}_{y,n},\sqrt n\,\hat{\gamma}_{x,n})^\mathsf{T}$?
Why does the reduced-form outcome error equal $\beta v_i+u_i$, rather than $u_i$?
Write the limiting cross-covariance of the two scaled reduced-form estimates and identify where structural error correlation enters.
If $\sigma_{uv}=0$, is the cross-covariance necessarily zero? Explain using its formula.
Why does changing $c$ affect the uncentered limiting mean but not the limiting covariance under the stated assumptions?