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Two-Sample IV: Efficient Two-Step Estimation and Tests for Overidentification and Weak-Instruments
Two-sample instrumental variables estimation was developed by Klevmarken1982, AngristKruegerJASA1992, \nocite{AngristKruegerJBES1995} ArellanoMeghirREStud1992 and InoueSolonWP2005\nocite{InoueSolonREStat2010}, among others. It applies to settings where an outcome $y$ and possibly endogenous explanatory variables $x$ are not observed in the same data set. Instead, in the standard two-sample setup we consider here, one has observations on $y$, instruments $z$ and covariates $c$ in one sample and observations on $x$, $z$ and $c$ in another sample.
The commonly used estimator in the linear setting we consider is the two-sample two-stage least squares (ts2sls) estimator as introduced by Klevmarken1982. This estimator is asymptotically normally distributed and efficient under standard sampling, homoskedasticity and homogeneity assumptions of the distributions of the two samples.
In the standard one-sample IV setting, when there are more instruments than endogenous variables and so the specification is overidentified, test results for overidentification are routinely reported. Standard tests are the Sargan test, which is valid under homoskedasticity, and the robust Hansen $J$-test, where robustness is to general forms of heteroskedasticity, including for example clustering. The $J$-test statistic is calculated using a two-step estimation procedure, where the two-step estimator is efficient under heteroskedasticity. Such a two-step estimator has not been considered for the two-sample setup and we develop it here, resulting in the two-sample $J$-test. A prime feature of our derivations is that we only need the summary statistics from two linear regressions, that of $y$ on $z$ and $c$ in sample 1 and $x$ on $z$ and $c$ in sample 2. These summary statistics are the vectors of estimated coefficients, their estimated variance matrices valid under homoskedasticity and the heteroskedasticity robust variance estimators. From these six objects we obtain the ts2sls estimator, their standard errors, including the robust ones as in PaciniWindEL2016, the efficient two-step estimator and the $J$-test statistic. We derive the (standard) limiting distributions fully, obtained simply from those of the least squares estimators in the two samples. This is done in Sections (ref) and (ref), with Section (ref) providing some simulation results.
The paper closest to our setup is \citet*{ZhaoetalStatSci2019}. They did consider the test for overidentifying restrictions, but in a homoskedastic setting, allowing for heterogeneity of the two samples. They did not fully establish the limiting distribution results, which we do also for their setting. Another related paper is \citet*{ChoietalJAE2017}, who propose the two-sample robust Anderson-Rubin statistic for weak-instrument robust inference, but do not consider the efficient two-step estimator and the associated two-sample $J$-test.
We further consider testing for weak instruments in Section (ref). For the one-variable, homoskedasticity and homogeneous samples case, we derive the relationship between the first-stage $F$-statistic and the relative bias of the ts2sls estimator of $\beta$ under weak-instrument asymptotics as in Staiger1997 and Stock2005. For the ts2sls estimator, the weak-instruments bias is towards $0$ and we find that the critical values of Stock2005 apply for the the relative bias results, where here the relative bias is relative to the true parameter value $\beta$.
Olea2013 propose the effective $F$-statistic for testing weak instruments for the standard one-sample 2sls estimator under heteroskedasticity. They provide critical value functions for the effective $F$-statistic for the Nagar bias of the 2sls estimator, relative to a benchmark bias. Windmeijer2025 generalizes the effective $F$-statistic to a general GMM setting, and we apply his results to derive the effective $F$-statistic as a test for weak instruments, related to the Nagar bias of the ts2sls estimator again relative to the true parameter value $\beta$.
Section (ref) presents estimation and test results for the study of marshall2019anti of the effects of education on political affiliation, a two-sample IV analysis wit cluster-robust inference.
For ease of exposition, we focus here an a single explanatory variable $x$, the details for multivariable $x$ are presented in the Appendix. We have the model specifications
where other exogenous explanatory variables that are observed in both data sets have been linearly partialled out. $z$ and $\pi_{x}$ are $k_{z}$-vectors.
The reduced form for $y$ is then given by
with the $k_{z}$-vector $\pi_{y}=\pi_{x}\beta$ and $v_{y}=u+v\beta$.
We don't observe $y$, $x$ and $z$ in one sample, but $y$ and $z$ in sample 1, $x$ and $z$ in sample 2, the observations being $\left\{ y_{1i},z_{1i}^{\prime}\right\} _{i=1}^{n_{1}}$ and $\left\{ x_{2j},z_{2j}^{\prime}\right\} _{j=1}^{n_{2}}$. The samples are independent and we assume that $\lim_{n_{1}\rightarrow\infty,n_{2}\rightarrow\infty}\frac{n_{1}}{n_{2}}=\alpha$, with $\alpha>0$ and finite. As in ZhaoetalStatSci2019 we make the following structural assumption for the two samples.
We further assume that the instruments are relevant and valid.
As in ZhaoetalStatSci2019, under Assumption (ref) together with Assumption (ref), we obtain all our results for estimation and testing allowing for heterogenous samples. The homogeneity assumption of the samples is formulated as follows, see
For homogeneous samples, it follows that the first-stage relationship $x=z'\pi_{x}+v$ does not need to be structural, but can be a linear projection. It follows from Assumption (ref) that the linear projection parameters in the two samples are the same and equal to $\pi_{x}$. It follows then that $\pi_{y}=\pi_{x}\beta$ and from the linear projection that $\mathbb{E}\left(z_{2j}v_{x_{2},j}\right)=0$.
As mentioned, for the results obtained below we assume Assumptions (ref) and (ref). Some results simplify when the homogeneity Assumption (ref) holds, which we will highlight specifically. For example, for using the standard $F$-statistic as a test for weak-instruments we need the homogeneity assumption.
Let $y_{1}$ and $x_{2}$ be the $n_{1}$- and $n_{2}$-vectors $\left(y_{1i}\right)$ and $\left(x_{2j}\right)$ respectively, and $Z_{1}$ and $Z_{2}$ the $\left(n_{1}\times k_{z}\right)$ and $\left(n_{2}\times k_{z}\right)$ matrices of observations of the instruments in the two samples. The ts2sls estimator of $\beta$ is obtained as the OLS estimator for $\beta$ in the specification
resulting in \[ \widehat{\beta}_{ts2sls}=\left(\widehat{x}_{1}^{\prime}\widehat{x}_{1}\right)^{-1}\widehat{x}_{1}^{\prime}y_{1}=\left(\widehat{\pi}_{x,2}^{\prime}Z_{1}^{\prime}Z_{1}\widehat{\pi}_{x,2}\right)^{-1}\widehat{\pi}_{x,2}^{\prime}Z_{1}^{\prime}Z_{1}\widehat{\pi}_{y,1}, \] where $\widehat{\pi}_{x,2}$ and $\widehat{\pi}_{y,1}$ are the OLS estimators of $\pi_{x}$ and $\pi_{y}$ given by
Under Assumption 1 and standard regularity conditions for OLS estimators, the limiting distributions are given by
For the estimators of the variance of $\widehat{\pi}_{y,1}$ and $\widehat{\pi}_{x,2}$, we distinguish between those that are valid under conditional homoskedasticity and those that are robust to general forms of heteroskedasticity, including serial correlation and clustering. For example, the homoskedastic variance estimator for $\widehat{\pi}_{y,1}$ is given by \[ \widehat{V}_{\widehat{\pi}_{y,1}}=\widehat{\sigma}_{v_{y,1}}^{2}\left(Z_{1}^{\prime}Z_{1}\right)^{-1}, \] where $\widehat{\sigma}_{v_{y,1}}^{2}=\frac{1}{n}\widehat{v}_{y,1}^{T}\widehat{v}_{y,1}$, with $\widehat{v}_{y,1}=y_{1}-Z_{1}\widehat{\pi}_{y,1}$. $\widehat{V}_{\widehat{\pi}_{y,1}}$ is an estimator of $\Sigma_{\widehat{\pi}_{y,1}}/n_{1}$ in the sense that $n_{1}\widehat{V}\left(\widehat{\pi}_{y,1}\right)\stackrel{p}{\rightarrow}\Sigma_{\widehat{\pi}_{y,1}}$ if $\mathbb{E}\left(v_{y,1i}^{2}|z_{1i}\right)=\sigma_{v_{y_{1}}}^{2}$.
As an example, a robust variance estimator in a cross-section with conditionally heteroskedastic errors is given by \[ \widehat{V}_{r,\widehat{\pi}_{y,1}}=\left(Z_{1}^{\prime}Z_{1}\right)^{-1}\left(\sum_{i=1}^{n_{1}}\widehat{v}_{y,1}^{2}z_{i}z_{i}^{\prime}\right)\left(Z_{1}^{\prime}Z_{1}\right)^{-1}. \]
From the two OLS regressions, we collect the six objects \[ \left\{ \widehat{\pi}_{y,1},\widehat{V}_{\widehat{\pi}_{y,1}},\widehat{V}_{r,\widehat{\pi}_{y,1}},\widehat{\pi}_{x,2},\widehat{V}_{\widehat{\pi}_{x,2}},\widehat{V}_{r,\widehat{\pi}_{x,2}}\right\} . \] These six objects are the only ones needed to obtain the ts2sls estimator, its (robust) variance estimator, the efficient two-step estimator and two-sample Sargan and $J$-test statistics.
First, consider the one-step GMM-type estimator
From the relationship $\pi_{y}=\pi_{x}\beta$, it follows that
Relationship ((ref)) is a crucial observation that facilitates obtaining the limiting distribution of the ts2sls estimator from the limiting distributions of the OLS estimators as given in ((ref)) and ((ref)), and which was missing from the analysis in ZhaoetalStatSci2019. It follows from ((ref)) that \[ \sqrt{n_{1}}\left(\widehat{\beta}_{ts2sls}-\beta\right)=\left(\widehat{\pi}_{x,2}^{T}\widehat{V}_{\widehat{\pi}_{y,1}}^{-1}\widehat{\pi}_{x,2}\right)^{-1}\widehat{\pi}_{x,2}^{T}\widehat{V}_{\widehat{\pi}_{y,1}}^{-1}\sqrt{n_{1}}\left(\left(\widehat{\pi}_{y,1}-\pi_{y}\right)-\left(\widehat{\pi}_{x,2}-\pi_{x}\right)\beta\right) \] and so from the limiting distributions ((ref)) and ((ref)) and independence of the two samples, it follows that \[ \sqrt{n_{1}}\left(\widehat{\beta}_{ts2sls}-\beta\right)\stackrel{d}{\rightarrow}\mathcal{N}\left(0,\Sigma_{\widehat{\beta}_{ts2sls}}\right) \] where \[ \Sigma_{\widehat{\beta}_{ts2sls}}=C\left(\pi_{x},\Sigma_{\widehat{\pi}_{y,1}}^{-1}\right)^{T}\left(\Sigma_{\widehat{\pi}_{y,1}}+\beta^{2}\alpha\Sigma_{\widehat{\pi}_{x,2}}\right)C\left(\pi_{x},\Sigma_{\widehat{\pi}_{y,1}}^{-1}\right), \] with \[ C\left(\pi_{x},\Sigma_{\widehat{\pi}_{y,1}}^{-1}\right)\coloneqq\Sigma_{\widehat{\pi}_{y,1}}^{-1}\pi_{x}\left(\pi_{x}^{\prime}\Sigma_{\widehat{\pi}_{y,1}}^{-1}\pi_{x}\right)^{-1}. \]
It follows that, under homoskedasticity, the variance of $\widehat{\beta}_{ts2sls}$ can be estimated by \[ \widehat{V}_{\widehat{\beta}_{ts2sls}}=C\left(\widehat{\pi}_{x,2},\widehat{V}_{\widehat{\pi}_{y,1}}^{-1}\right)'\left(\widehat{V}_{\widehat{\pi}_{y,1}}+\widehat{\beta}_{ts2sls}^{2}\widehat{V}_{\widehat{\pi}_{x,2}}\right)C\left(\widehat{\pi}_{x,2},\widehat{V}_{\widehat{\pi}_{y,1}}^{-1}\right), \] whereas under heteroskedasticity, the robust variance estimator is given by \[ \widehat{V}_{r,\widehat{\beta}_{ts2sls}}=C\left(\widehat{\pi}_{x,2},\widehat{V}_{\widehat{\pi}_{y,1}}^{-1}\right)'\left(\widehat{V}_{r,\widehat{\pi}_{y,1}}+\widehat{\beta}_{ts2sls}^{2}\widehat{V}_{r,\widehat{\pi}_{x,2}}\right)C\left(\widehat{\pi}_{x,2},\widehat{V}_{\widehat{\pi}_{y,1}}^{-1}\right), \] which is the result of PaciniWindEL2016.
We now consider the case of general heteroskedasticity and efficient two-step estimation. Let $\widehat{\beta}_{ts2sls}$ be the initial, one-step estimator as described above. Let \[ W_{n,r}\left(\widehat{\beta}_{ts2sls}\right)=\left(\widehat{V}_{r,\widehat{\pi}_{y,1}}+\widehat{\beta}_{ts2sls}^{2}\widehat{V}_{r,\widehat{\pi}_{x,2}}\right)^{-1}, \] then the efficient two-step estimator is given by \[ \widehat{\beta}_{2s}=\left(\widehat{\pi}_{x,2}^{\prime}W_{n,r}\left(\widehat{\beta}_{ts2sls}\right)\widehat{\pi}_{x,2}\right)^{-1}\widehat{\pi}_{x,2}^{\prime}W_{n,r}\left(\widehat{\beta}_{ts2sls}\right)\widehat{\pi}_{y,1}, \] with limiting distribution \[ \sqrt{n_{1}}\left(\widehat{\beta}_{2s}-\beta\right)\stackrel{d}{\rightarrow}\mathcal{N}\left(0,\left(\pi_{x}^{\prime}\left(\Sigma_{\widehat{\pi}_{y,1}}+\beta^{2}\alpha\Sigma_{\widehat{\pi}_{x,2}}\right)^{-1}\pi_{x}\right)^{-1}\right). \] An estimator for the variance of $\widehat{\beta}_{2s}$ is therefore given by \[ \widehat{V}_{r,\widehat{\beta}_{2s}}=\left(\widehat{\pi}_{x,2}^{\prime}W_{n,r}\left(\widehat{\beta}_{ts2sls}\right)\widehat{\pi}_{x,2}\right)^{-1}. \] Efficiency of this two-step GMM estimator is a standard result.
The test for overidentifying restrictions then follows. Under the null $H_{0}:\pi_{y}=\pi_{x}\beta$ and the standard assumptions we have that \[ J\left(\widehat{\beta}_{2s};\widehat{\beta}_{ts2sls}\right)=\left(\widehat{\pi}_{y,1}-\widehat{\pi}_{x,2}\widehat{\beta}_{2s}\right)^{T}W_{n,r}\left(\widehat{\beta}_{ts2sls}\right)\left(\widehat{\pi}_{y,1}-\widehat{\pi}_{x,2}\widehat{\beta}_{2s}\right)\stackrel{d}{\rightarrow}\chi_{k_{z}-1}^{2}. \] This is again a standard result, shown as follows. We have that
It follows that \[ W_{n,r}^{1/2}\left(\widehat{\beta}_{ts2sls}\right)\left(\widehat{\pi}_{y,1}-\widehat{\pi}_{x,2}\widehat{\beta}_{2s}\right) \]
As \[ W_{n,r}^{1/2}\left(\widehat{\beta}_{ts2sls}\right)\left(\widehat{\pi}_{y,1}-\widehat{\pi}_{x,2}\beta\right)\stackrel{d}{\rightarrow}\mathcal{N}\left(0,I_{k_{z}}\right) \] and $\left(I_{k_{z}}-A_{n}\right)$ is a symmetric idempotent matrix with rank equal to $tr$$\left(I_{k_{z}}-A_{n}\right)=k_{z}-1$, the result follows.
Under Assumptions (ref) and (ref), leaving out the homogeneity assumption and the assumption of conditional homoskedasticity, we obtain the two-step estimator using the weight matrix based on the homoskedastic variance estimator \[ W_{n}\left(\widehat{\beta}_{ts2sls}\right)=\left(\widehat{V}_{\widehat{\pi}}+\widehat{\beta}_{ts2sls}^{2}\widehat{V}_{\widehat{\pi}_{x,2}}\right)^{-1}. \] Denoting this estimator $\widehat{\beta}_{2s,hom},$with the Sargan test statistic then given by \[ S\left(\widehat{\beta}_{2s,hom};\widehat{\beta}_{ts2sls}\right)=\left(\widehat{\pi}_{y,1}-\widehat{\pi}_{x,2}\widehat{\beta}_{2s,hom}\right)^{T}W_{n}\left(\widehat{\beta}_{ts2sls}\right)\left(\widehat{\pi}_{y,1}-\widehat{\pi}_{x,2}\widehat{\beta}_{2s,hom}\right)\stackrel{d}{\rightarrow}\chi_{k_{z}-1}^{2}. \]
Under homogeneity Assumption (ref), we don't need a two-step estimator, as then \[ S\left(\widehat{\beta}_{ts2sls}\right)=\left(\widehat{\pi}_{y,1}-\widehat{\pi}_{x,2}\widehat{\beta}_{ts2sls}\right)^{T}W_{n}\left(\widehat{\beta}_{ts2sls}\right)\left(\widehat{\pi}_{y,1}-\widehat{\pi}_{x,2}\widehat{\beta}_{ts2sls}\right)\stackrel{d}{\rightarrow}\chi_{k_{z}-1}^{2}. \] These are the results as in ZhaoetalStatSci2019.
We conduct a Monte Carlo study to simulate the performance of the proposed two-sample estimators and the two-sample over-identification test. The DGP follows a two-sample design: sample 1 ($n_{1}=500$) contains $(Y_{1},Z_{1},c_{1})$ and sample 2 ($n_{2}=1000$) contains $(X_{2},Z_{2},c_{2})$. We simulate $10{,}000$ replications with $k_{x}=2$ endogenous regressors and $k_{z}=3$ instruments. The structural parameters are $(\beta_{1},\beta_{2},\beta_{c},\beta_{0})=(0.3,-0.1,0.1,0.2)$. Two designs are considered: (i) conditional homoskedasticity and (ii) conditional heteroskedasticity. This follows the specifications of simulation study in the supplementary materials of PaciniWindEL2016.
Tables (ref) and (ref) summarize the main Monte Carlo findings. Across both designs, all estimators appear unbiased. Under homoskedasticity, homoskedastic and heteroskedastic standard errors closely track empirical standard deviations, coverage is near nominal, and over-identification tests are correctly sized. In contrast, under heteroskedasticity, homoskedastic standard errors substantially understate sampling variability, leading to pronounced over-rejection of Wald and Sargan tests. Heteroskedasticity-robust methods restore correct inference, and the robust two-step estimator typically achieves the smallest empirical dispersion while delivering well-calibrated Hansen over-identification tests. Overall, the results highlight the importance of robust weighting for valid inference and specification testing in two-sample settings.
Under conditional homoskedasticity and homogeneity Assumption (ref) we consider the standard first-stage $F$-statistic in sample 2, given by \[ F_{\widehat{\pi}_{x,2}}=\widehat{\pi}_{x,2}^{\prime}\widehat{V}_{\widehat{\pi}_{x,2}}^{-1}\widehat{\pi}_{x,2}/k_{z}. \] Under weak-instrument asymptotics, as in Staiger1997 and Stock2005, given by the representation \[ \pi_{x}=c/\sqrt{n_{2}}, \] we find that the same critical values can be used as tabulated by Stock2005 in relation to the weak-instruments test in terms of relative bias of the ts2sls estimator. Whereas in the one-sample case this relative bias is relative to the OLS estimator of $\beta$, in the two-sample case this relative bias is given by the proportional bias \[ PB_{n}=\left|\frac{\mathbb{E}\left[\widehat{\beta}_{ts2sls}\right]-\beta}{\beta}\right|. \] This is due to the fact that the weak-instruments bias of the ts2sls estimator is towards $0$. In the limit under the weak-instruments asymptotics we get \[ PB_{n}\rightarrow\left|\mathbb{E}\left[\frac{\left(\lambda+\xi_{2}\right)'\xi_{2}}{\left(\lambda+\xi_{2}\right)'\left(\lambda+\xi_{2}\right)}\right]\right| \] where $\lambda=\Sigma_{\widehat{\pi}_{x,2}}^{-1/2}c$ and $\xi_{2}\sim\mathcal{N}\left(0,I_{k_{z}}\right)$. This is the same expression as in Stock2005 for the 2sls bias relative to the OLS bias in the one-sample case.
For the $F$-statistic we have the weak-instruments asymptotics result that \[ F_{\widehat{\pi}_{x,2}}\stackrel{d}{\rightarrow}\chi_{k_{z}}^{2}\left(\lambda'\lambda\right)/k_{z}, \] which matches the one-sample case. Hence the same critical values for $F_{\widehat{\pi}_{x,2}}$ for a maximum proportional bias of say 10% apply. See the Appendix for further derivations.
Under general forms of heteroskedasticity we can use the results in Windmeijer2025 who generalized the use of the effective F-statistic as proposed by Olea2013 from testing weak instruments in terms of the Nagar bias of the 2sls estimator to a general class of linear GMM estimators. For the ts2sls estimator, the generalization of the effective F-statistic is given by
\[ \widehat{F}_{\text{eff}}\left(\widehat{V}_{\widehat{\pi}_{y,1}}\right)=\frac{\widehat{\pi}_{x,2}^{\prime}\widehat{V}_{y,1}^{-1}\widehat{\pi}_{x,2}}{\text{tr}\left(\widehat{V}_{r,\widehat{\pi}_{x,2}}\widehat{V}_{\widehat{\pi}_{y,1}}^{-1}\right)}. \]
As shown in the Appendix, the worst-case benchmark bias is again $-\beta$, and as also shown in the Appendix, the critical value function of Olea2013 applies directly.
We illustrate the practical relevance of the proposed two-sample estimators using the education and political affiliation study of marshall2019anti. The setting is canonical for two-sample IV: educational attainment and voting outcomes are observed in separate surveys over the same time period but share common instruments. This makes it a suitable environment to assess efficiency, inference, and specification testing in two-sample designs.
Outcome data are drawn from the National Annenberg Election Survey (NAES), covering U.S. presidential elections in 2000, 2004, and 2008, while educational attainment is obtained from the American Community Survey (ACS) for corresponding cohorts. The samples are linked by cohort and state of birth and the instruments are state-level compulsory schooling (drop-out) laws, which vary by cohort and state. The outcomes measure Democratic party self-identification, voting intention, and past voting behavior.
We re-estimate Marshall's baseline specifications using the ts2sls and efficient two-step estimators. Inference is conducted using both homoskedastic and heteroskedastic (cluster-robust) variance estimators, and over-identification tests are computed using the corresponding weighting matrices.
Estimation and test results are presented in Table (ref). Across all outcomes, we replicate the ts2sls estimates of marshall2019anti, who presented cluster-robust (cr) standard errors, where the clustering is by state. Whereas in our approach we simply use the cr standard errors from the two linear regression, using for example the “sandwich” routine in R, marshall2019anti calculates these separately and directly, based on the residuals $\widehat{u}=y_{1}-\widehat{x}_{1}\widehat{\beta}_{ts2sls}=y_{1}-Z_{1}\widehat{\pi}_{x,2}\widehat{\beta}_{ts2sls}$. These are clearly a type of reduced-form residuals. In our approach we use the least-squares residuals $\widehat{v}_{y,1}=y_{1}-Z_{1}\widehat{\pi}_{y,1}$, and, given the cr robust variance estimates of $\widehat{\pi}_{y,1}$and $\widehat{\pi}_{x,2}$, we don't need to perform any further direct calculation, or indeed access the data. It is easily seen the $\widehat{u}'\widehat{u}\geq\widehat{v}_{y,1}^{\prime}\widehat{v}_{y,1}$. The cluster-robust standard errors are also systematically smaller than Marshall's when constructed using our method and hence based on reduced-form least-squares residuals. The standard homoskedastic first-stage F statistics are large (around 40), while cr effective F-statistics are substantially smaller (around 8.5) and only marginally larger than the calculated critical values. The Sargan/Hansen test results indicate some specification problems for the “Intend” outcome.