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The Falsification Adaptive Set in Linear Models with Instrumental Variables that Violate the Exclusion or Conditional Exogeneity Restriction
Keywords:{ Instrumental variables, invalid instruments, falsification adaptive set}
JEL Codes:{ C26, C52}\\ { }
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Nicolas Apfel: [email removed]
Frank Windmeijer: [email removed], corresponding author
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We consider the classical linear IV model with a single endogenous variable and multiple correlated instruments. As in MastenPoirierEcta2021 (henceforth MP), if the baseline model is falsified, this is due to invalidity of instruments. MP introduced the falsification adaptive set (FAS), which includes “all parameter values consistent with the data and a model which is relaxed just enough to make it nonfalsified” (MP, p 1450) .
In order to construct the FAS, it is important to consider the two different ways in which instruments can be invalid. The first one is that an instrument is invalid because it has a direct or indirect effect on the outcome and hence should be included as a control in the model. When this is the case, we say that this instrument violates the exclusion assumption. The second one is that an instrument is invalid if it has a non-zero covariance with the error in the model which includes as controls invalid instruments that violate the exclusion assumption. We say that an instrument with such a non-zero covariance violates the conditional exogeneity assumption. The notions of indirect effect and conditional exogeneity are formalized in Section (ref).
MP derive the identified set for the parameter of interest $\beta$, the falsification frontier and the FAS under a natural relaxation of the exclusion restriction whilst maintaining the conditional exogeneity assumption. We therefore denote this FAS the FAS$_{excl}$. It is the closed interval with endpoints the minimum and maximum of the just-identified, and hence non-falsifiable, IV estimands in models where each relevant instrument is considered as the just-identifying instrument in turn, whilst all other instruments are included as controls. The FAS$_{excl}$ is thus an expanded set compared to the baseline point estimand to account for the uncertainty due to a violation of the exclusion assumption only.
In this paper, we study the consequences of violations of both exclusion and conditional exogeneity restrictions for the FAS. MP argue that, mathematically, the same technical analysis can be used to relax both the exclusion assumption and conditional exogeneity assumption. However, we show in Section (ref), where we consider the case where invalid instruments violate the exogeneity assumption only, that the natural way to relax the exogeneity restriction leads to a different identified set for $\beta$ and so a different FAS, termed FAS$_{exo}$. It is the closed interval with endpoints the minimum and maximum of the IV estimands in models where each relevant instrument is considered as the just-identifying instrument in turn, with all other instruments removed from the instrument set.
Although the identified sets for $\beta$ for a natural partial relaxation of the exclusion assumption only or exogeneity assumption only are both sharp identified sets, this does not imply that the FAS$_{excl}$ is better, in some metric, than the FAS$_{exo}$ when invalid instruments violate the exclusion assumption only. We illustrate this with a numerical example in Section (ref), where all instruments violate the exclusion assumption only, and where both FAS$_{excl}$ and FAS$_{exo}$ contain $\beta$, but the FAS$_{exo}$ is narrower than FAS$_{excl}$. We therefore consider the presence of a valid instrument, which is an instrument that satisfies both the exclusion assumption and conditional exogeneity assumption. Then if invalid instruments violate the exclusion assumption only, but there is at least one relevant valid instrument, \textit{FAS}$_{excl}$ is guaranteed to contain $\beta$, but \textit{FAS}$_{exo}$ is not, due to the instruments being correlated. Likewise, if invalid instruments violate the exogeneity assumption only, \textit{FAS}$_{exo}$ is guaranteed to contain $\beta$ if there is at least one valid and relevant instrument, but now \textit{FAS$_{excl}$} is not\textit{.}
In Section (ref) we then expand the analysis and consider the situation where an invalid instrument either violates the exclusion assumption or the conditional exogeneity assumption. We consider joint natural relaxations of the exclusion and conditional exogeneity assumptions for all patterns of instrument invalidity. For example, for the case with three instruments, we can relax the exclusion restriction for the first instrument and the conditional exogeneity restriction for the second and third instruments. This results in a sharp identified set for $\beta$ and the associated FAS for this particular relaxation of the baseline assumptions. For a general number of instruments $k_{z}$ we thus get a total of $2^{k_{z}}$ identified sets for $\beta$ and falsification adaptive sets, including FAS$_{excl}$ and FAS$_{exo}$. We propose the generalized FAS as the union of these $2^{k_{z}}$ falsification adaptive sets. This generalized FAS takes into account all of these possible violations of the exclusion and conditional exogeneity assumptions when the baseline model is falsified. If at least one of the instruments is valid and relevant then the generalized \textit{FAS} is guaranteed to contain $\beta$.
In Section (ref) we illustrate and compare the estimated FAS$_{excl}$, FAS$_{exo}$ and generalized FAS for one of the main examples in MP, the empirical analysis of roads and trade by Duranton2014.
We consider the linear model specifications as in HeckmanPinto2015,
with $Y$ and $X$ observable scalar random variables, and $\boldsymbol{Z}$ an observable $k_{z}$-vector of putative instruments. $A$, $E$ and $V$ are unobservable scalar random variables, with $\text{cov}\left(A,V\right)=\text{cov}\left(E,V\right)=0$ and $\text{cov}\left(\boldsymbol{Z},V\right)=\boldsymbol{0}$. $A$ is an unobserved confounder that is a common cause to both $Y$ and $X$ when $\rho\neq0$ and $\text{var}\left(A\right)>0$, and $X$ is then an endogenous regressor. $\beta$ is an unknown constant, and $\bar{\boldsymbol{\pi}}$ an unknown $k_{z}$-vector. Individual elements of $\boldsymbol{Z}$ are denoted $Z_{\ell}$, $\ell=1,\ldots,k_{z}$. For ease of exposition, a constant and other exogenous variables have been omitted as they can be partialled out in any application wlog.
The following sufficient variation assumption is maintained as in MP.
If the instruments are exogenous in the sense that $\text{cov}\left(\boldsymbol{Z},\left(A+E\right)\right)=\boldsymbol{0}$, and relevant in the sense that $\bar{\boldsymbol{\pi}}\neq\boldsymbol{0}$, the baseline model, then $\beta$ is point identified and the standard 2SLS estimator on a sample of observations $\left\{ Y_{i},X_{i},\boldsymbol{Z}_{i}^{\prime}\right\} _{i=1}^{n}$ is a consistent estimator of $\beta$. Exogeneity of all instruments can be falsified by for example the Sargan1958 and Hansen1982 tests for overidentifying restrictions. This is the type of falsification of the model that was considered by MP and for which they proposed the FAS. When the model is falsified and hence $\text{cov}\left(\boldsymbol{Z},\left(A+E\right)\right)\neq\boldsymbol{0}$, one needs to consider in what way invalid instruments violate this exogeneity condition.
The first violation we consider is that an instrument is invalid because it has a direct and/or indirect effect on the outcome $Y$ over and above the effect of $X$. This is a violation of the exclusion assumption.
We can rewrite specifications ((ref)) and ((ref)) to incorporate direct and indirect effects as
The second violation we consider is that an instrument is invalid because it does not satisfy the conditional exogeneity assumption $\text{cov}\left(Z_{\ell},\widetilde{A}+\widetilde{E}\right)=0$.
One way to conceptualize the violation of the conditional exogeneity assumption is by specifying the relationship between $\boldsymbol{Z}$, $\widetilde{A}$ and $\widetilde{E}$ as
with $\text{var}\left(\boldsymbol{W}\right)=\boldsymbol{\Sigma}_{w}$, $\text{cov}\left(\widetilde{A},\boldsymbol{W}\right)=\text{cov}\left(\widetilde{E},\boldsymbol{W}\right)=\boldsymbol{0}$ and $\text{cov}\left(\widetilde{A},\widetilde{E}\right)=0$. Then $\boldsymbol{\alpha}_{\widetilde{A}}=\text{var}\left(\widetilde{A}\right)\boldsymbol{\xi}_{\widetilde{A}}$ and $\boldsymbol{\alpha}_{\widetilde{E}}=\text{var}\left(\widetilde{E}\right)\boldsymbol{\xi}_{\widetilde{E}}$, and $\text{var}\left(\boldsymbol{Z}\right)=\boldsymbol{\Sigma}_{w}+\text{var}\left(\widetilde{A}\right)\boldsymbol{\xi}_{\widetilde{A}}\boldsymbol{\xi}_{\widetilde{A}}^{\prime}+\text{var}\left(\widetilde{E}\right)\boldsymbol{\xi}_{\widetilde{E}}\boldsymbol{\xi}_{\widetilde{E}}^{\prime}$.
Note that the sequential structure of the violations of the general exogeneity condition and the direction of the effects are important and cannot be captured by projection arguments. For example, in the linear projection $\left(A+E\right)=\boldsymbol{Z}'\boldsymbol{c}+r$, we have that $\text{cov}\left(\boldsymbol{Z},r\right)=0$, and $\boldsymbol{c}=\boldsymbol{\gamma}+\text{var}\left(\boldsymbol{Z}\right)\boldsymbol{\alpha}$, thus mixing the exclusion and conditional exogeneity violations. As $\text{var}\left(\boldsymbol{Z}\right)$ is a general variance matrix, with correlated instruments as illustrated in the empirical example in Section (ref), one cannot disentangle $\boldsymbol{\gamma}$ and $\boldsymbol{\alpha}$ from $\boldsymbol{c}$. In Section (ref) we use our setup to consider general violations $\boldsymbol{\gamma}+\boldsymbol{\alpha}$ under the restriction that $\gamma_{\ell}\text{\ensuremath{\alpha}}_{\ell}=0$, $\forall\text{\ensuremath{\ell}}.$
As discussed in MP, if the distribution of $\left(Y,X,\boldsymbol{Z}'\right)$ is such that the model is falsified then this could be due to misspecification of model ((ref)), which assumes homogeneous linear treatment effects, and/or instrument invalidity. As in MP, we maintain model ((ref)) and focus on failure of the instrument exclusion assumption or the instrument conditional exogeneity assumption as reasons for falsifying the baseline model.
Incorporating violations of the exclusion and conditional exogeneity assumptions, we get that \[ \text{cov}\left(\boldsymbol{Z},Y\right)-\text{cov}\left(\boldsymbol{Z},X\right)\beta=\text{var}\left(\boldsymbol{Z}\right)\boldsymbol{\gamma}+\boldsymbol{\alpha}, \] from which it follows that
MP derive the falsification adaptive set under the assumption that invalid instruments can violate the exclusion assumption only:
MP make the following partial exclusion assumption.
As MP (p 1453) argue, Assumption (ref) is a natural way to relax the exclusion restriction. As falsification is due to $\boldsymbol{\gamma}\neq0$, the natural relaxation is to consider a range of values for $\boldsymbol{\gamma}$, $-\boldsymbol{\delta}\leq\text{\ensuremath{\boldsymbol{\gamma}}}\leq\boldsymbol{\delta}$, componentwise.
Under Assumption (ref) it follows from ((ref)) that
Under Assumptions (ref), (ref) and (ref), Theorem 1 in MP then states that
is the identified set for $\beta$, where the inequalities are componentwise, and that the model is falsified if and only if $\mathcal{B}_{\gamma}\left(\boldsymbol{\delta}\right)$ is empty. The proof of Theorem 1 in MP is as follows. First, any value $\beta$ consistent with the model lies in $\mathcal{B}_{\gamma}\left(\boldsymbol{\delta}\right)$, which follows directly from Assumptions (ref) and (ref) and the resulting expression for $\boldsymbol{\gamma}$ in ((ref)). To show that $\mathcal{B}_{\gamma}\left(\boldsymbol{\delta}\right)$ is sharp, let $b\in\mathcal{B}_{\gamma}\left(\boldsymbol{\delta}\right)$ and define \[ \boldsymbol{\gamma}\left(b\right)\coloneqq\text{var}\left(\boldsymbol{Z}\right)^{-1}\left(\text{cov}\left(\boldsymbol{Z},Y\right)-\text{cov}\left(\boldsymbol{Z},X\right)b\right). \] Then $-\boldsymbol{\delta}\leq\boldsymbol{\gamma}\left(b\right)\leq\boldsymbol{\delta}$ by definition of $\mathcal{B}_{\gamma}\left(\boldsymbol{\delta}\right)$. From the expression of $\boldsymbol{\alpha}$ in ((ref)), we then get that \[ \boldsymbol{\alpha}\left(b\right)\coloneqq\text{cov}\left(\boldsymbol{Z},Y\right)-\text{cov}\left(\boldsymbol{Z},X\right)b-\text{var}\left(\boldsymbol{Z}\right)\boldsymbol{\gamma}\left(b\right)=0, \] and so Assumption (ref) holds and hence $\mathcal{B}_{\gamma}\left(\boldsymbol{\delta}\right)$ is sharp.
The falsification frontier (FF) is the minimal set of $\boldsymbol{\delta}$s which lead to a non-empty identified set. For a $\boldsymbol{\delta}\in FF$ this means that for any other $\boldsymbol{\delta}'<\boldsymbol{\delta}$, the identified set $\mathcal{B}\left(\boldsymbol{\delta}'\right)$ is empty and thus falsifies the model, where $\boldsymbol{\delta}'<\boldsymbol{\delta}$ means that $\delta'_{\ell}\leq\delta_{\ell}$ for all $\ell\in\left\{ 1,\ldots,k_{z}\right\} $ and $\delta'_{m}<\delta_{m}$ for some $m\in\left\{ 1,\ldots,k_{z}\right\} $, see Definition 1 in MP.
Let
then it follows that \[ \mathcal{B}_{\gamma}\left(\boldsymbol{\delta}\right)=\left\{ b\in\mathbb{R}:-\boldsymbol{\delta}\leq\left(\boldsymbol{\psi}-\boldsymbol{\pi}b\right)\leq\boldsymbol{\delta}\right\} . \]
As in MP, let $\mathcal{L}_{rel}$ denote the set of relevant instruments,
For model ((ref)), under Assumptions (ref), (ref) and (ref), Proposition 2 in MP specifies the falsification frontier as the set \[ FF_{excl}=\left\{ \boldsymbol{\delta}\left(b\right)\in\mathbb{R}_{\geq0}^{k_{z}}:\delta_{\ell}\left(b\right)=\left|\psi_{\ell}-b\pi_{\ell}\right|,\,\ell=1,\ldots,k_{z},\,b\in\left[\min_{\ell\in\mathcal{L}_{rel}}\frac{\psi_{\ell}}{\pi_{\ell}},\max_{\ell\in\mathcal{L}_{rel}}\frac{\psi_{\ell}}{\pi_{\ell}}\right]\right\} , \] where we have added the subscript “excl” to denote a relaxation of the exclusion assumption only and have defined $\boldsymbol{\delta}$ as a function of $b$. Note that for all $\boldsymbol{\delta}\left(b\right)\in FF_{excl}$, it follows that $\mathcal{B}_{\gamma}\left(\boldsymbol{\delta}\left(b\right)\right)=b$.
The falsification adaptive set, denoted FAS$_{excl}$, is then given in Theorem 2 of MP as
As MP point out in their Lemma 1, for $\ell\in\mathcal{L}_{rel}$, $\frac{\psi_{\ell}}{\pi_{\ell}}\left(=\beta+\frac{\gamma_{\ell}}{\pi_{\ell}}\right)$ is the IV/2SLS estimand in the just-identified model specification
where $\boldsymbol{Z}_{\{-\ell\}}=\boldsymbol{Z}\setminus\left\{ Z_{\ell}\right\} $, and using $Z_{\ell}$ as the excluded just-identifying instrument, see also WindmeijeretalJRSSB2021.
As just-identified models are not falsifiable, it follows that for $\ell\in\mathcal{L}_{rel}$, $\delta_{\ell}\left(\frac{\psi_{\ell}}{\pi_{\ell}}\right)=0$. From this, the results of the falsification frontier FF$_{excl}$ follow straightforwardly, as when moving $b$ from $\min_{j\in\mathcal{L}_{rel}}\frac{\psi_{j}}{\pi_{j}}$ to $\max_{j\in\mathcal{L}_{rel}}\frac{\psi_{j}}{\pi_{j}}$ there is at least one element in $\boldsymbol{\delta}\left(b\right)$ that decreases in value and at least one that increases in value.\footnote{For a numerical illustration, see Example 1 in Apfel2022.}
We can write the FAS$_{excl}$ alternatively as \[ FAS_{excl}=\left[\beta+\min_{\ell\in\mathcal{L}_{rel}}\frac{\gamma_{\ell}}{\pi_{\ell}},\beta+\max_{\ell\in\mathcal{L}_{rel}}\frac{\gamma_{\ell}}{\pi_{\ell}}\right]. \] It follows that the FAS$_{excl}$ contains $\beta$ if $0\in\left[\min_{\ell\in\mathcal{L}_{rel}}\frac{\gamma_{\ell}}{\pi_{\ell}},\max_{\ell\in\mathcal{L}_{rel}}\frac{\gamma_{\ell}}{\pi_{\ell}}\right]$.
We have an i.i.d. sample of size $n$, $\left\{ Y_{i},X_{i},\boldsymbol{Z}_{i}^{\prime}\right\} _{i=1}^{n}$. The $n$-vectors $\left(Y_{i}\right)$ and $\left(X_{i}\right)$ are denoted $\boldsymbol{y}$ and $\boldsymbol{x}$ respectively, and here $\boldsymbol{Z}$ denotes the $n\times k_{z}$ matrix of observations on the instruments. The constant and other exogenous variables have been partialled out. MP suggest to estimate the set of relevant instruments by \[ \mathcal{\widehat{L}}_{rel}=\left\{ \ell\in\left\{ 1,\ldots,k_{z}\right\} :F_{\ell}\geq C_{n}\right\} , \] where $F_{\ell}$ is the first-stage $F$-statistic for model ((ref)), where $Z_{\ell}$ is considered as an instrument and $Z_{\left\{ -\ell\right\} }$ as controls. For all values of $\ell$, the first-stage model is therefore given by $\boldsymbol{x}=\boldsymbol{Z}\boldsymbol{\pi}+\boldsymbol{v}$ and so $F_{\ell}$ is the same as the Wald statistic for testing $H_{0}:\pi_{\ell}=0$ based on the OLS estimator of $\boldsymbol{\pi}$, denoted $\widehat{\boldsymbol{\pi}}$. The same first-stage hard thresholding was proposed in GuoetalJRSSB2018. Although $C_{n}\rightarrow\infty$ as $n\rightarrow\infty$ and $C_{n}=o\left(n\right)$ for consistent selection, MP choose $C_{n}=10$ as their default cutoff, or for the t-ratio, $\left|\frac{\widehat{\pi}_{\ell}}{se\left(\widehat{\pi}_{\ell}\right)}\right|\geq\sqrt{10}=3.16$.
Let $\widehat{\beta}_{\ell}$ be the IV estimator of $\beta_{\ell}$ in just-identified model specification ((ref)). Then FAS$_{excl}$ is estimated by \[ \widehat{FAS}_{excl}=\left[\min_{\ell\in\mathcal{\widehat{L}}_{rel}}\widehat{\beta}_{\ell},\max_{\ell\in\mathcal{\widehat{L}}_{rel}}\widehat{\beta}_{\ell}\right] \] and MP show that $\widehat{FAS}_{excl}$ is a consistent estimator of the $FAS_{excl}$ under the conditions of their Proposition 3.
We now make the assumption that invalid instruments can violate the exogeneity assumption only:
As $\boldsymbol{\gamma}=\boldsymbol{0}$, the exogeneity assumption is no longer conditional on the inclusion of instruments as controls.
MP, p 1453, argue that, mathematically, the same technical analysis can be used to relax the exogeneity assumption as above for the relaxation of the exclusion assumption. In MastenPoirierWP2020 (henceforth MP20) it is argued in Appendix G that a linear projection of $U\coloneqq A+E$ on $\boldsymbol{Z}$ results in
with $\text{cov}\left(\boldsymbol{Z},\dot{U}\right)=\boldsymbol{0}$ by construction. Then
which is the same type of specification as that of model ((ref)) and MP20 argue that the results above for the relaxation of the exclusion assumption apply here as well. MP20 stress that the key difference is the interpretation of the coefficients on $\boldsymbol{Z}$. We argue here that a violation of the exogeneity assumption only has its own natural relaxation, with implications for the identified set, falsification frontier and associated falsification adaptive set.
For the linear projection specification ((ref)) under Assumption (ref), we have that
If $\text{var}\left(\boldsymbol{Z}\right)$ is a diagonal matrix, then the distinction between a relaxation of the exclusion or (conditional) exogeneity assumption is immaterial. However, for a general $\text{var}\left(\boldsymbol{Z}\right)$, the two different relaxations need different treatment. The identified set $\mathcal{B}_{\gamma}\left(\boldsymbol{\delta}\right)$ as defined in ((ref)) for the natural relaxation of the exclusion restriction only, as detailed in Assumption (ref), and the associated FAS$_{excl}$ would apply to a relaxation of the exogeneity restriction given by $-\boldsymbol{\delta}\leq\text{var}\left(\boldsymbol{Z}\right){}^{-1}\boldsymbol{\alpha}\leq\boldsymbol{\delta}$, which is not a natural relaxation of the exogeneity restriction.
Like the natural partial exclusion Assumption (ref) of MP, we make the following natural partial exogeneity assumption.
Under Assumption (ref) and from ((ref)) we have
Together with Assumption (ref) we obtain the identified set for $\beta$ as detailed in the following proposition. This proposition is analogous to Theorem 1 in MP for the case where instruments can violate the exclusion restriction only.
The proof of Proposition (ref) is along the same lines as that of Theorem 1 in MP, as outlined in Section (ref). First it is clear that any value $\beta$ consistent with the model lies in $\mathcal{B}_{\alpha}\left(\boldsymbol{\eta}\right)$, which follows directly from Assumptions (ref) and (ref) and the resulting expression for $\boldsymbol{\alpha}$ in ((ref)). To show that $\mathcal{B}_{\alpha}\left(\boldsymbol{\eta}\right)$ is sharp, let $b\in\mathcal{B}_{\alpha}\left(\boldsymbol{\eta}\right)$ and define \[ \boldsymbol{\alpha}\left(b\right)\coloneqq\text{cov}\left(\boldsymbol{Z},Y\right)-\text{cov}\left(\boldsymbol{Z},X\right)b. \] Then $-\boldsymbol{\eta}\leq\boldsymbol{\alpha}\left(b\right)\leq\boldsymbol{\eta}$ by definition of $\mathcal{B}_{\alpha}\left(\boldsymbol{\eta}\right)$. From the expression of $\boldsymbol{\gamma}$ in ((ref)), we then get that \[ \boldsymbol{\gamma}\left(b\right)\coloneqq\text{var}\left(\boldsymbol{Z}\right)^{-1}\left(\text{cov}\left(\boldsymbol{Z},Y\right)-\text{cov}\left(\boldsymbol{Z},X\right)b-\boldsymbol{\alpha}\left(b\right)\right)=0, \] and so Assumption (ref) holds and hence $\mathcal{B}_{\alpha}\left(\boldsymbol{\eta}\right)$ is sharp.
Define the $k_{z}$-vectors $\boldsymbol{\pi}^{*}$, $\boldsymbol{\psi}^{*}$, $\boldsymbol{\alpha}^{*}$ and $\boldsymbol{\eta}^{*}$ with $\ell$-th elements given by
for $\ell=1,\ldots,k_{z}$. Then it follows that \[ \mathcal{B}_{\alpha}\left(\boldsymbol{\eta}\right)=\mathcal{B}_{\alpha^{*}}\left(\boldsymbol{\eta}^{*}\right)=\left\{ b\in\mathbb{R}:-\boldsymbol{\eta}^{*}\leq\left(\boldsymbol{\psi}^{*}-\boldsymbol{\pi}^{*}b\right)\leq\boldsymbol{\eta}^{*}\right\} . \] Let $\mathcal{L}_{rel}^{*}$ denote the set of relevant instruments
Under Assumptions (ref), (ref) and (ref), and following analogous arguments as in MP, the falsification frontier is then given as the set \[ FF_{exo}=\left\{ \boldsymbol{\eta}^{*}\left(b\right)\in\mathbb{R}_{\geq0}^{k_{z}}:\eta_{\ell}^{*}\left(b\right)=\left|\psi_{\ell}^{*}-b\pi_{\ell}^{*}\right|,\,\ell=1,\ldots,k_{z},\,b\in\left[\min_{\ell\in\mathcal{L}_{rel}^{*}}\frac{\psi_{\ell}^{*}}{\pi_{\ell}^{*}},\max_{\ell\in\mathcal{L}_{rel}^{*}}\frac{\psi_{\ell}^{*}}{\pi_{\ell}^{*}}\right]\right\} . \] The resulting falsification adaptive set is then
It follows that FAS$_{exo}$ contains $\beta$ if $0\in\left[\min_{\ell\in\mathcal{L}_{rel}^{*}}\frac{\alpha_{\ell}^{*}}{\pi_{\ell}^{*}},\max_{\ell\in\mathcal{L}_{rel}^{*}}\frac{\alpha_{\ell}^{*}}{\pi_{\ell}^{*}}\right]$.
For $\ell\in\mathcal{L}_{rel}^{*}$, the ratio \[ \beta_{\ell}^{*}:=\frac{\psi_{\ell}^{*}}{\pi_{\ell}^{*}}=\frac{\text{cov}\left(Z_{\ell},Y\right)}{\text{cov}\left(Z_{\ell},X\right)} \] is the IV estimand for $\beta$ in the specification $Y=X\beta+U$, using $Z_{\ell}$ as the just-identifying instrument for $X$, and treating the instruments $\boldsymbol{Z}_{\left\{ -\ell\right\} }$ as invalid and excluding them from the analysis.
The set of relevant instruments can here be estimated by \[ \mathcal{\widehat{L}}_{rel}^{*}=\left\{ \ell\in\left\{ 1,\ldots,k_{z}\right\} :F_{\ell}^{*}\geq C_{n}\right\} , \] where $F_{\ell}^{*}$ is the F-statistic for testing $H_{0}:\pi_{\ell}^{*}=0$ in the first-stage linear specification $\boldsymbol{x}=\boldsymbol{z}_{\ell}\pi_{\ell}^{*}+\boldsymbol{v}_{\ell}.$ Let $\widehat{\beta}_{\ell}^{*}=\frac{\boldsymbol{z}_{\ell}^{\prime}\boldsymbol{y}}{\boldsymbol{z}_{\ell}^{\prime}\boldsymbol{x}}$, where again the constant and other exogenous variables have been partialled out, then the consistent estimator of FAS$_{exo}$ is given by \[ \widehat{FAS}_{exo}=\left[\min_{\ell\in\mathcal{\widehat{L}}_{rel}^{*}}\widehat{\beta}_{\ell}^{*},\max_{\ell\in\mathcal{\widehat{L}}_{rel}^{*}}\widehat{\beta}_{\ell}^{*}\right]. \]
Although $\mathcal{B}_{\gamma}\left(\boldsymbol{\delta}\right)$ and $\mathcal{B}_{\alpha}\left(\boldsymbol{\eta}\right)$ are the sharp identified sets for the natural relaxations of the exclusion and exogeneity assumptions, Assumptions (ref) and (ref) respectively, this does not necessarily imply that the FAS$_{excl}$ is better, in some metric, than the FAS$_{exo}$ when instruments violate the exclusion restriction only, and vice versa. Consider the FAS$_{excl}$ associated with the natural relaxation $-\boldsymbol{\delta}\leq\boldsymbol{\gamma}\leq\boldsymbol{\delta}$ when instruments violate the exclusion restriction only. In comparison, the FAS$_{exo}$ in this case is then associated with a relaxation of the exogeneity condition $-\boldsymbol{\eta}\leq\text{var}\left(\boldsymbol{Z}\right)\boldsymbol{\gamma}\leq\boldsymbol{\eta}$. Alternatively, and as discussed in Section (ref), for the FAS$_{exo}$ associated with the natural relaxation $-\boldsymbol{\eta}\leq\boldsymbol{\alpha}\leq\boldsymbol{\eta}$ when instrument violate the exogeneity restriction only, its comparison is the FAS$_{excl}$ for a relaxation of the exclusion condition $-\boldsymbol{\delta}\leq\text{var}\left(\boldsymbol{Z}\right)^{-1}\boldsymbol{\alpha}\leq\boldsymbol{\delta}$.
The next simple numerical illustration with $k_{z}=2$ instruments considers a violation of the exclusion assumption only, finds that both FAS$_{excl}$ and FAS$_{exo}$ contain $\beta$, but FAS$_{exo}$ is a narrower interval.
It is clear that in this example both falsification adaptive sets include $\beta=\frac{1}{3}$. This follows as the two elements of both $\boldsymbol{\pi}$ and $\boldsymbol{\pi}^{*}$ have equal sign, whereas the two elements of both $\boldsymbol{\gamma}$ and $\boldsymbol{\alpha}\left(\gamma\right)$ have opposite signs. The width of the FAS$_{exo}$ is here $\frac{2}{3}$, whereas that of the FAS$_{excl}$ is wider and equal to $2$. This example therefore highlights that there is in general not a preference ordering for using FAS$_{excl}$ or FAS$_{exo}$, irrespective of the violation of instrument validity considered. We can, however, prefer one over the other if we consider the presence of a valid instrument and the requirement that a FAS should contain $\beta$ if at least one of the instruments is valid and relevant.
We define a valid instrument as follows.
When in Example (ref) we change the values of $\boldsymbol{\gamma}$ to $\boldsymbol{\gamma}=\left(0,1\right)'$ whilst keeping the values of $\beta$, $\text{var}\left(\boldsymbol{Z}\right)$ and $\text{cov}\left(\boldsymbol{Z},X\right)$ the same, we obtain $FAS_{excl}=\left[\frac{1}{3},\frac{4}{3}\right]$ and $FAS_{exo}=\left[\frac{2}{3},1\right]$. So now FAS$_{excl}$ is still wider than FAS$_{exo}$, but FAS$_{excl}$ contains $\beta$ whereas FAS$_{exo}$ does not. $Z_{1}$ is here a valid and relevant instrument, but due to the correlation of $Z_{1}$ and $Z_{2}$, only identifies $\beta$ when $Z_{2}$ is included in the model as a control. This is one of the estimands of FAS$_{excl}$, as detailed in equation ((ref)), but not of FAS$_{exo}$. Another way to look at it is that here $\boldsymbol{\alpha}\left(\gamma\right)=\text{var}\left(\boldsymbol{Z}\right)\boldsymbol{\gamma}=\left(0.5,1\right)'$, and so although there is a valid instrument, both elements of $\boldsymbol{\alpha}\left(\gamma\right)$ are non-zero due to the correlation of the instruments. Both elements are here positive, so $\beta\notin FAS_{exo}$. It is clear that \textit{FAS}$_{excl}$ is guaranteed to contain $\beta$ under Assumption (ref) that invalid instruments violate the exclusion assumption only if there is at least one valid and relevant instrument $Z_{\ell}$ with $\gamma_{\ell}=0$ and $\pi_{\ell}\neq0$. Similarly, $FAS_{exo}$ is guaranteed to contain $\beta$ under Assumption (ref) that invalid instruments violate the exogeneity assumption only if there is at least one valid and relevant instrument $Z_{\ell}$ with $\alpha_{\ell}=0$ and $\pi_{\ell}^{*}\neq0$.
A researcher who believes that invalid instruments violate the exclusion assumption only and that not all relevant instruments are invalid would then prefer the FAS$_{excl}$ as the appropriate set reflecting the model uncertainty that arises from a falsified baseline model. This could for example be the case when instruments are randomly assigned, but some could have a direct effect on the outcome. Or a researcher may believe the exclusion assumption to hold on economic grounds, but questions the exogeneity assumption of some of the instruments due to no or improper randomization, in which case the FAS$_{exo}$ would be the preferred set. This is because in both these cases $\beta$ is likely to be contained in the sets.
In general, however, a researcher may not know whether falsification is due to a violation of the exclusion or conditional exogeneity assumption. In the next section we consider the general case where some instruments can violate the exclusion assumption and some the conditional exogeneity assumption. As for the two cases discussed so far, each pattern of violations of the exclusion and conditional exogeneity assumptions results in a pattern-specific natural relaxation, identified set and FAS. An example of a pattern for $k_{z}=3$ is $\boldsymbol{\gamma}+\boldsymbol{\alpha}=\left(\alpha_{1},\gamma_{2},\alpha_{3}\right)'$, where $Z_{2}$ violates the exclusion assumption and $Z_{1}$ and $Z_{3}$ the conditional exogeneity assumption. Our proposed generalized FAS is then the union of all pattern-specific falsification adaptive sets. This generalized FAS therefore reflects the model uncertainty that arises from a falsified baseline model allowing for all possible combinations of instruments either violating the exclusion or conditional exogeneity assumption, an assumption we formalize in the next section. We also show that, given the definitions and assumptions made, the generalized FAS is guaranteed to contain $\beta$ if there is at least one valid and relevant instrument.
We now consider the cases where there can be a mixture of invalid instruments, with some invalid instruments violating the exclusion assumption and some the conditional exogeneity assumption.
As in Assumptions (ref) and (ref) we maintain that $\gamma_{\ell}\alpha_{\ell}=0$, as otherwise a variable considered to be an instrument is itself an endogenous explanatory variable that needs to be instrumented in order to obtain point identification of $\beta$. Note that Assumption (ref) is a generalization that includes the previous two cases of violating the exclusion or exogeneity assumption only. In Section (ref) we present a further discussion, including a numerical example, on the issue of instruments that are themselves endogenous explanatory variables.
We now consider natural relaxations of the exclusion and conditional exogeneity assumptions for each possible pattern that $\boldsymbol{\gamma}+\boldsymbol{\alpha}$ can take with $\gamma_{\ell}\alpha_{\ell}=0$ for $\ell=1,\ldots,k_{z}$. There are $S=2^{k_{z}}$ such patterns. Denote these patterns by $\boldsymbol{\gamma}_{s}+\boldsymbol{\alpha}_{s}$, $s=1,\ldots,S$. Let $\mathcal{K}=\left\{ 1,\ldots,k_{z}\right\} $ and let $\mathcal{C}$ denote the collection of all $S$ possible subsets of $\mathcal{K}$, so $\mathcal{C}=\left\{ \emptyset,1,\ldots,\left\{ 1,2\right\} ,\ldots,\mathcal{K}\right\} $. Denote the subsets of $\mathcal{C}$ by $\mathcal{C}_{s}$, $s=1,\ldots,S$, and let $\mathcal{A}_{s}=\mathcal{K}\setminus\mathcal{C}_{s}$. Then we consider all possible patterns as follows. For $s=1,.\ldots,S$, for an invalid instrument $Z_{\ell}$, $\ell\in\mathcal{K}$, if $\ell\in\mathcal{C}_{s}$ it violates the exclusion restriction, $\gamma_{\ell}\neq0$, and if $\ell\in\mathcal{A}_{s}$ it violates the conditional exogeneity assumption, $\alpha_{\ell}\neq0$. To illustrate, for the $k_{z}=3$ example at the end of the previous section, with $\boldsymbol{\gamma}_{s}+\boldsymbol{\alpha}_{s}=\left(\alpha_{1},\gamma_{2},\alpha_{3}\right)'$, the associated sets are $\mathcal{C}_{s}=\left\{ 2\right\} $ and $\mathcal{A}_{s}=\left\{ 1,3\right\} $. Further, we obtain a violation of the exclusion assumption only, $\boldsymbol{\gamma}_{s}+\boldsymbol{\alpha}_{s}=\boldsymbol{\gamma}$, for $\mathcal{C}_{s}=\mathcal{K}$, $\mathcal{A}_{s}=\emptyset$, and a violation of the exogeneity assumption only, $\boldsymbol{\gamma}_{s}+\boldsymbol{\alpha}_{s}=\boldsymbol{\alpha}$, for $\mathcal{C}_{s}=\mathcal{\emptyset}$, $\mathcal{A}_{s}=\mathcal{K}$.
We can now make the following natural joint relaxation of the exclusion and conditional exogeneity assumption.
For each $s\in\left\{ 1,\ldots,S\right\} $ we can now obtain the identified set, falsification frontier and falsification adaptive set. For pattern $\left(\mathcal{C}_{s},\mathcal{A}_{s}\right)$ we have the model specification \[ Y=X\beta+\boldsymbol{Z}_{\mathcal{C}_{s}}^{\prime}\boldsymbol{\gamma}_{\mathcal{C}_{s}}+\widetilde{U};\,\,\,\text{cov}\left(\boldsymbol{Z}_{\mathcal{A}_{s}},\widetilde{U}\right)=\boldsymbol{\alpha}_{\mathcal{A}_{s}}. \] As a unifying framework, we consider linearly transformed instruments as follows. For $\ell\in\mathcal{C}_{s}$, let $\mathcal{C}_{s,-\ell}=\mathcal{C}_{s}\setminus\left\{ \ell\right\} $. Then, for $\ell\in\mathcal{C}_{s}$, we linearly partial out $\boldsymbol{Z}_{\mathcal{C}_{s,-\ell}}$ from $Z_{\ell}$, \[ Z_{\ell|\mathcal{C}_{s,-\ell}}=Z_{\ell}-\boldsymbol{Z}_{\mathcal{C}_{s,-\ell}}\left(\text{var}\left(\boldsymbol{Z}_{\mathcal{C}_{s,-\ell}}\right)\right)^{-1}\text{cov}\left(\boldsymbol{Z}_{\mathcal{C}_{s,-\ell}},Z_{\ell}\right). \] It then follows that \[ \frac{\text{cov}\left(Z_{\ell|\mathcal{C}_{s,-\ell}},Y\right)-\text{cov}\left(Z_{\ell|\mathcal{C}_{s,-\ell}},X\right)\beta}{\text{var}\left(Z_{\ell|\mathcal{C}_{s,-\ell}}\right)}=\gamma_{s,\ell},\,\,\,\forall\ell\in\mathcal{C}_{s}. \] Likewise, for $\ell\in\mathcal{A}_{s}$, we partial out $\boldsymbol{Z}_{\mathcal{C}_{s}}$ from $Z_{\ell}$, \[ Z_{\ell|\mathcal{C}_{s}}=Z_{\ell}-\boldsymbol{Z}_{\mathcal{C}_{s}}\left(\text{var}\left(\boldsymbol{Z}_{\mathcal{C}_{s}}\right)\right)^{-1}\text{cov}\left(\boldsymbol{Z}_{\mathcal{C}_{s}},Z_{\ell}\right), \] resulting in \[ \frac{\text{cov}\left(Z_{\ell|\mathcal{C}_{s}},Y\right)-\text{cov}\left(Z_{\ell|\mathcal{C}_{s}},X\right)\beta}{\text{var}\left(Z_{\ell|\mathcal{C}_{s}}\right)}=\frac{\alpha_{s,\ell}}{\text{var}\left(Z_{\ell|\mathcal{C}_{s}}\right)}\equiv\widetilde{\alpha}_{s,\ell},\,\,\,\forall\ell\in\mathcal{A}_{s}. \]
Then define $\widetilde{\boldsymbol{Z}}_{s}$ as the $k_{z}$-vector with $\ell$-th element either $Z_{\ell|\mathcal{C}_{s,-\ell}}$ if $\ell\in\mathcal{\mathcal{C}}_{s}$, or $Z_{\ell|\mathcal{C}_{s}}$ if $\ell\in\mathcal{A}_{s}$. For our example with $\boldsymbol{\gamma}_{s}+\boldsymbol{\alpha}_{s}=\left(\alpha_{1},\gamma_{2},\alpha_{3}\right)'$, we get $\widetilde{\boldsymbol{Z}}_{s}=\left(Z_{1|2},Z_{2},Z_{3|2}\right)'$. It then follows that \[ \left(\text{diag}\left(\text{var}\left(\widetilde{\boldsymbol{Z}}_{s}\right)\right)\right)^{-1}\left(\text{cov}\left(\widetilde{\boldsymbol{Z}}_{s},Y\right)-\text{cov}\left(\widetilde{\boldsymbol{Z}}_{s},X\right)\beta\right)=\boldsymbol{\gamma}_{s}+\widetilde{\boldsymbol{\alpha}}_{s}, \] where for a general square matrix $\boldsymbol{Q}$, $\text{diag}\left(\boldsymbol{Q}\right)$ is a diagonal matrix containing the diagonal elements of $\boldsymbol{Q}$, and $\widetilde{\boldsymbol{\alpha}}_{s}$ is the $k_{z}$-vector with $\ell$-th element either $\widetilde{\alpha}_{s,\ell}$ if $\ell\in\mathcal{A}_{s}$, or $0$ if $\ell\in\mathcal{C}_{s}$. Let $\widetilde{\boldsymbol{\omega}}$ denote the $k_{z}$-vector with $\ell$-th element either $\text{var}\left(\widetilde{Z}_{s,\ell}\right)^{-1}\omega_{s,\ell}$ if $\ell\in\mathcal{A}_{s}$, or $\omega_{s,\ell}$ if $\ell\in\mathcal{C}_{s}$, then it follows from Assumption (ref) that $-\widetilde{\boldsymbol{\omega}}_{s}\leq\boldsymbol{\gamma}_{s}+\widetilde{\boldsymbol{\alpha}}_{s}\leq\widetilde{\boldsymbol{\omega}}_{s}$. For each $s\in\left\{ 1,\ldots,S\right\} $ we then get the sharp identified set as detailed in the following proposition.
The proof follows the same arguments as those of Theorem 1 in MP, and Proposition (ref) in Section (ref). For each $s\in\left\{ 1,\ldots,S\right\} $, it is clear that any value $\beta$ consistent with the model lies in $\mathcal{B}_{s}\left(\widetilde{\boldsymbol{\omega}}_{s}\right)$. For any $b\in\mathcal{B}_{s}\left(\widetilde{\boldsymbol{\omega}}_{s}\right)$ define \[ \boldsymbol{\gamma}_{s}\left(b\right)+\widetilde{\boldsymbol{\alpha}}_{s}\left(b\right)\coloneqq\left(\text{diag}\left(\text{var}\left(\widetilde{\boldsymbol{Z}}_{s}\right)\right)\right)^{-1}\left(\text{cov}\left(\widetilde{\boldsymbol{Z}}_{s},Y\right)-\text{cov}\left(\widetilde{\boldsymbol{Z}}_{s},X\right)b\right), \] then $-\widetilde{\boldsymbol{\omega}}_{s}\leq\boldsymbol{\gamma}_{s}\left(b\right)+\widetilde{\boldsymbol{\alpha}}_{s}\left(b\right)\leq\widetilde{\boldsymbol{\omega}}_{s}$. If we specify for $\ell\in\mathcal{C}_{s}$, \[ \gamma_{s,\ell}\left(b\right):=\text{var}\left(\widetilde{\boldsymbol{Z}}_{s,\ell}\right)^{-1}\left(\text{cov}\left(\widetilde{\boldsymbol{Z}}_{s,\ell},Y\right)-\text{cov}\left(\widetilde{\boldsymbol{Z}}_{s,\ell},X\right)b\right), \] then $\widetilde{\alpha}_{s,l}\left(b\right)=0$. If we specify, for $\ell\in\mathcal{A}_{s}$, \[ \widetilde{\alpha}_{s,\ell}\left(b\right):=\text{var}\left(\widetilde{\boldsymbol{Z}}_{s,\ell}\right)^{-1}\left(\text{cov}\left(\widetilde{\boldsymbol{Z}}_{s,\ell},Y\right)-\text{cov}\left(\widetilde{\boldsymbol{Z}}_{s,\ell},X\right)b\right), \] then $\gamma_{s,\ell}\left(b\right)=0$, hence $\mathcal{B}_{s}\left(\widetilde{\boldsymbol{\omega}}_{s}\right)$ is sharp.
For $s=1\ldots,S$, define the $k_{z}$-vectors $\widetilde{\boldsymbol{\pi}}_{s}$ and $\widetilde{\boldsymbol{\psi}}_{s}$, with $\ell$-th elements given by
for $\ell=1,\ldots,k_{z}$. Then it follows that \[ \mathcal{B}_{s}\left(\widetilde{\boldsymbol{\omega}}\right)=\left\{ b\in\mathbb{R}:-\widetilde{\boldsymbol{\omega}}_{s}\leq\left(\widetilde{\boldsymbol{\psi}}_{s}-\widetilde{\boldsymbol{\pi}}_{s}b\right)\leq\widetilde{\boldsymbol{\omega}}_{s}\right\} . \] Let $\widetilde{\mathcal{L}}_{s.rel}$ denote the set of relevant instruments
Under Assumptions (ref), (ref) and (ref), the falsification frontier is then given by \[ FF_{s}=\left\{ \widetilde{\boldsymbol{\omega}}_{s}\left(b\right)\in\mathbb{R}_{\geq0}^{k_{z}}:\omega_{s,\ell}\left(b\right)=\left|\widetilde{\psi}_{s,\ell}-b\widetilde{\pi}_{s,\ell}\right|,\,\ell=1,\ldots,k_{z},\,b\in\left[\min_{\ell\in\widetilde{\mathcal{L}}_{s,rel}}\frac{\widetilde{\psi}_{s,\ell}}{\widetilde{\pi}_{s,\ell}},\max_{\ell\in\widetilde{\mathcal{L}}_{s,rel}}\frac{\widetilde{\psi}_{s,\ell}}{\widetilde{\pi}_{s,\ell}}\right]\right\} , \] and the resulting falsification adaptive set is
Here, \[ \widetilde{\beta}_{s,\ell}\coloneqq\frac{\widetilde{\psi}_{s,\ell}}{\widetilde{\pi}_{s,\ell}}=\frac{\text{cov}\left(\widetilde{Z}_{s,\ell},Y\right)}{\text{cov}\left(\widetilde{Z}_{s,\ell},X\right)} \] is the IV estimand for $\beta$ in the specification $Y=X\beta+U$, using the transformed instrument $\widetilde{Z}_{s,\ell}$ as the just-identifying instrument for $X$, but these imply different model specifications. For our $k_{z}=3$ example with $\boldsymbol{\gamma}+\boldsymbol{\alpha}=\left(\alpha_{1},\gamma_{2},\alpha_{3}\right)'$, we have that $\widetilde{\beta}_{1|2}=\frac{\text{cov}\left(Z_{1|2},Y\right)}{\text{cov}\left(Z_{1|2},X\right)}$ is the IV estimand for $\beta$ in the model with $Z_{2}$ included as a control and $Z_{3}$ excluded from the instrument set. It follows that for this example $\widetilde{\beta}_{1|2}=\beta+\frac{\widetilde{\alpha}_{1}}{\pi_{1|2}}$. For the $k_{z}=3$ case, if we consider violations of the exclusion assumption only, $\boldsymbol{\gamma}+\boldsymbol{\alpha}=\boldsymbol{\text{\ensuremath{\gamma}}}=\left(\gamma_{1},\gamma_{2},\gamma_{3}\right)'$, we have $\boldsymbol{Z}_{s}=\left(Z_{1|23},Z_{2|13},Z_{3|12}\right)'$, resulting in FAS$_{excl}$. For violations of the exogeneity assumption only, $\boldsymbol{\gamma}+\boldsymbol{\alpha}=\boldsymbol{\text{\ensuremath{\alpha}}}=\left(\alpha_{1},\alpha_{2},\alpha_{3}\right)'$, we have $\boldsymbol{Z}_{s}=\left(Z_{1},Z_{2},Z_{3}\right)'$, resulting in FAS$_{exo}$.
We now define the generalized FAS as follows.
The generalized FAS reflects the model uncertainty that arises from a falsified baseline model, considering all possible patterns of violations of the exclusion and conditional exogeneity assumptions that satisfy Assumption (ref). As we show next, if there is at least one valid and relevant instrument, then $\beta\in FAS$.
Although there are $S=2^{k_{z}}$ different patterns $\boldsymbol{\gamma}_{s}+\boldsymbol{\alpha}_{s}$, it is clear that there is overlap of the linearly transformed instruments when constructing the different FAS$_{s}$. There are a total of $J=k_{z}2^{k_{z-1}}$ transformed instruments. Let $\widetilde{\boldsymbol{Z}}$ denote the $J$-vector of transformed instruments. For example, for $k_{z}=3$, we have the following set of $J=12$ transformed instruments \[ \widetilde{\boldsymbol{Z}}=\left(Z_{1},Z_{1|2},Z_{1|3},Z_{1|23},Z_{2},Z_{2|1},Z_{2|3},Z_{2|13},Z_{3},Z_{3|1},Z_{3|2},Z_{3|12}\right)'. \]
Then define the $J$-vectors $\widetilde{\boldsymbol{\pi}}$ and $\widetilde{\boldsymbol{\psi}}$ with elements
for $j=1,\ldots,J$. Let \[ \mathcal{\widetilde{\mathcal{L}}}_{rel}=\left\{ j\in\left\{ 1,\ldots,J\right\} :\widetilde{\pi}_{j}\neq0\right\} . \]
When all transformed instruments are relevant, so $\mathcal{\widetilde{\mathcal{L}}}_{rel}=\left\{ 1,\ldots,J\right\} $, then it is straightforward to show that \[ FAS=\left[\min_{j}\frac{\widetilde{\psi}_{j}}{\widetilde{\pi}_{j}},\max_{j}\frac{\widetilde{\psi}_{j}}{\widetilde{\pi}_{j}}\right], \] as there are then always overlapping pattern-specific FAS$_{s}$, from the one containing $\min_{j}\frac{\widetilde{\psi}_{j}}{\widetilde{\pi}_{j}}$ to the one containing $\max_{j}\frac{\widetilde{\psi}_{j}}{\widetilde{\pi}_{j}}$. When not all transformed instruments are relevant, the generalized FAS could be a set of disjoint intervals. However, it is trivially the case that \[ FAS=\left[\min_{j\in\mathcal{\widetilde{\mathcal{L}}}_{rel}}\frac{\widetilde{\psi}_{j}}{\widetilde{\pi}_{j}},\max_{j\in\mathcal{\widetilde{\mathcal{L}}}_{rel}}\frac{\widetilde{\psi}_{j}}{\widetilde{\pi}_{j}}\right], \] if there is a set of overlapping FAS$_{s}$ that contain $\min_{j\in\mathcal{\widetilde{\mathcal{L}}}_{rel}}\frac{\widetilde{\psi}_{j}}{\widetilde{\pi}_{j}}$ and $\max_{j\in\mathcal{\widetilde{\mathcal{L}}}_{rel}}\frac{\widetilde{\psi}_{j}}{\widetilde{\pi}_{j}}$. We will illustrate this with estimated values in the empirical application in Section (ref).
Let $Z_{\ell}$ be a valid instrument with $\gamma_{\ell}=\alpha_{\ell}=0$. Denote by $\widetilde{Z}_{v}$ its transformed version that appropriately takes into account the violations of the exclusion and conditional exogeneity assumptions of the other instruments. Under Assumption (ref) it follows that $\widetilde{Z}_{v}$ is an element of $\widetilde{\boldsymbol{Z}}$. If the transformed instrument $\widetilde{Z}_{v}$ is relevant, $\widetilde{\pi}_{v}\neq0$, then it follows that $\widetilde{\beta}_{v}=\frac{\widetilde{\psi}_{v}}{\widetilde{\pi}_{v}}=\text{\ensuremath{\beta}}$. As $\widetilde{\beta}_{v}\in FAS$, it follows that the generalized $FAS$ is guaranteed to contain $\beta$ if there is a valid instrument for which its correctly transformed version is relevant. We state this result formally in the following proposition.
For the $k_{z}=3$ example, let $\boldsymbol{\gamma}+\boldsymbol{\alpha}=\left(0,\gamma_{2},\alpha_{3}\right)'$, then $Z_{1|2}$ point-identifies $\beta$ if $\pi_{1|2}\neq0$. This corresponds to the point-identified model specification $Y=X\beta+Z_{2}\gamma_{2}+\widetilde{U}$, $X=Z_{1}\pi_{1|2}+Z_{2}\pi_{2|1}+V$.
We have again partialled out the constant and other exogenous variables. For the pattern-specific falsification adaptive sets, FAS$_{s}$, for each $s\in$$\left\{ 1,\ldots,S\right\} $, with $S=2^{k_{z}}$, for each $\ell\in\mathcal{C}_{s}$ the transformation of the $n$-vector of observations $\boldsymbol{z}_{\ell}$ is given by \[ \widetilde{\boldsymbol{z}}_{s,\ell}=\boldsymbol{M}_{Z_{\mathcal{C}_{s,-\ell}}}\boldsymbol{z}_{\ell}, \] where for a general full column rank matrix $\boldsymbol{A}$, $\boldsymbol{M}_{A}=\boldsymbol{I}_{n}-\boldsymbol{A}\left(\boldsymbol{A}'\boldsymbol{A}\right)^{-1}\boldsymbol{A}'$, with $\boldsymbol{I}_{n}$ is the $n$-dimensional identity matrix. For each $\ell\in\mathcal{A}_{s}$ the transformation of the $n$-vector of observations $\boldsymbol{z}_{\ell}$ is given by \[ \widetilde{\boldsymbol{z}}_{s,\ell}=\boldsymbol{M}_{Z_{\mathcal{C}_{s}}}\boldsymbol{z}_{\ell}. \] The $n\times k_{z}$ matrix of pattern-specific transformed instruments is then given by \[ \widetilde{\boldsymbol{Z}}_{s}=\left[\widetilde{\boldsymbol{z}}_{s,\ell}\right]. \]
For each $s$, the set of relevant instruments is estimated by
\[ \mathcal{\widehat{\widetilde{L}}}_{s,rel}=\left\{ \ell\in\left\{ 1,\ldots,k_{z}\right\} :\widetilde{F}_{s,\ell}\geq C_{n}\right\} , \] where $\widetilde{F}_{s,\ell}$ is the F-statistic for testing $H_{0}:\widetilde{\pi}_{s,\ell}=0$ in the first-stage linear specification $\boldsymbol{x}=\widetilde{\boldsymbol{z}}_{s,\ell}\widetilde{\pi}_{s,\ell}+\boldsymbol{v}_{s,\ell}.$ Let $\widehat{\widetilde{\beta}}_{s,\ell}=\frac{\widetilde{\boldsymbol{z}}_{s,\ell}^{\prime}\boldsymbol{y}}{\widetilde{\boldsymbol{z}}_{s,\ell}^{\prime}\boldsymbol{x}}$, then the consistent estimator of FAS$_{s}$ is given by \[ \widehat{FAS}_{s}=\left[\min_{\ell\in\mathcal{\widehat{\widetilde{L}}}_{s,rel}}\widehat{\widetilde{\beta}}_{\ell},\max_{\ell\in\mathcal{\widehat{\widetilde{L}}}_{s,rel}}\widehat{\widetilde{\beta}}_{s,\ell}\right]. \] The consistent estimator of the generalized falsification adaptive set is then obtained as \[ \widehat{FAS}=\cup_{s=1}^{S}\widehat{FAS}_{s}. \]
Let the $n\times J$ matrix of transformed instruments be \[ \widetilde{\boldsymbol{Z}}=\cup_{s=1}^{S}\widetilde{\boldsymbol{Z}}_{s}, \] with $J=k_{z}2^{k_{z}-1}$. Then let \[ \mathcal{\widehat{\widetilde{L}}}_{rel}=\left\{ j\in\left\{ 1,\ldots,J\right\} :\widetilde{F}_{j}\geq C_{n}\right\} , \] where $\widetilde{F}_{j}$ is the F-statistic for testing $H_{0}:\widetilde{\pi}_{j}=0$ in the first-stage linear specification $\boldsymbol{x}=\widetilde{\boldsymbol{z}}_{j}\widetilde{\pi}_{j}+\boldsymbol{v}_{j}$.
Let $\widehat{\widetilde{\beta}}_{j}=\frac{\widetilde{\boldsymbol{z}}_{j}^{\prime}\boldsymbol{y}}{\widetilde{\boldsymbol{z}}_{j}^{\prime}\boldsymbol{x}}$. If all transformed instruments are relevant, so $\mathcal{\widehat{\widetilde{L}}}_{rel}=\left\{ 1,\ldots,J\right\} $, then \[ \widehat{FAS}=\left[\min_{j\in\left\{ 1,\ldots,J\right\} }\widehat{\widetilde{\beta}}_{j},\max_{j\in\left\{ 1,\ldots,J\right\} }\widehat{\widetilde{\beta}}_{j}\right]. \] When not all transformed instruments are relevant, it is the case that \[ \widehat{FAS}=\left[\min_{j\in\mathcal{\widehat{\widetilde{L}}}_{rel}}\widehat{\widetilde{\beta}}_{j},\max_{j\in\mathcal{\widehat{\widetilde{L}}}_{rel}}\widehat{\widetilde{\beta}}_{j}\right] \] if there can be found a path of overlapping $\widehat{FAS}_{s}$ intervals from the $\widehat{FAS}_{s}$ that contains $\min_{j\in\mathcal{\widehat{\widetilde{L}}}_{rel}}\widehat{\widetilde{\beta}}_{j}$ to the $\widehat{FAS}_{s}$ that contains $\max_{j\in\mathcal{\widehat{\widetilde{L}}}_{rel}}\widehat{\widetilde{\beta}}_{j}$, as we demonstrate in the empirical example in Section (ref).
Assumption (ref) specified that $\gamma_{\ell}\alpha_{\ell}=0$, $\forall\ell$. Under this assumption we could generalize the analysis of MP for all patterns $\boldsymbol{\gamma}_{s}+\boldsymbol{\alpha}_{s}$, $s=1,\ldots,S$, with pattern-specific sharp identified sets for $\beta$, associated falsification frontiers and falsification adaptive sets, with the union of these falsification adaptive sets our proposed generalized FAS.
The generalized FAS is still a reflection of the model uncertainty that arises from a falsified baseline model when allowing for instruments that are themselves endogenous explanatory variables with $\gamma_{\ell}\alpha_{\ell}\neq0$. But now, in general, none of the just-identifying transformed instruments will point-identify $\beta$, even if for an instrument $Z_{j}$ we have that $\gamma_{j}=\alpha_{j}=0$.
As an illustration, for the numerical Example (ref), with $\beta=\frac{1}{3}$, we keep $\boldsymbol{\gamma}=\left(\gamma_{1},\gamma_{2}\right)'=\left(-1,1\right)'$, but change $\boldsymbol{\alpha}=\boldsymbol{0}$ to $\boldsymbol{\alpha}=\left(0,-0.25\right)'$. We still have that $\boldsymbol{\pi}=\left(1,1\right)'$ and $\boldsymbol{\pi}^{*}=\left(1.5,1.5\right)'$, but now $\boldsymbol{\psi}=\left(-0.5,1\right)'$ and $\boldsymbol{\psi}^{*}=\left(0,0.75\right)$. We therefore get that $FAS_{excl}\equiv FAS_{\gamma_{1},\gamma_{2}}=[-0.5,1]$ and $FAS_{exo}\equiv FAS_{\alpha_{1},\alpha_{2}}=[0,0.5]$. As before, for this particular example, both these falsification adaptive sets contain $\beta$, and the FAS$_{exo}$ is the narrower interval. For the other patterns, we get $FAS_{\gamma_{1},\alpha_{2}}=[0,1]$ and $FAS_{\alpha_{1},\gamma_{2}}=[-0.5,0.5]$. The generalized $FAS=[-0.5,1]$.
We next set $\gamma_{1}=0$. Whilst previously with $\boldsymbol{\alpha}=0$, FAS$_{excl}$ and hence the generalized FAS were guaranteed to contain $\beta$, we now get with $\boldsymbol{\alpha}=\left(0,-0.25\right)'$ that $\boldsymbol{\psi}=\left(0.5,1\right)'$ and $\boldsymbol{\psi}^{*}=\left(1,1.25\right)$. Therefore, $FAS_{excl}=\left[\frac{3}{6},1\right]$, $FAS_{exo}=\left[\frac{4}{6},\frac{5}{6}\right]$, $FAS_{\gamma_{1},\alpha_{2}}=[\frac{4}{6},1]$ and $FAS_{\alpha_{1},\gamma_{2}}=[\frac{3}{6},\frac{5}{6}]$. The generalized is thus $FAS=[0.5,1]$ and does not contain $\beta$. Because $Z_{2}$ is here an endogenous explanatory variable, and because of the correlation between $Z_{1}$ and $Z_{2}$, the fact that $\alpha_{1}=\gamma_{1}=0$ does not lead to a just-identifying transformed instrument that point-identifies $\beta$. To achieve point identification one will need to further find a relevant and valid instrument for $Z_{2}$.
One of the main examples in MP is the empirical analysis of roads and trade by Duranton2014. The outcome variable is a measure of how much a city exports. The one considered in MP is called the “propensity to export weight”. The treatment variable is the log number of kilometers of interstate highway within a city in 2007. Duranton2014 estimate the causal effect of within-city highways on the propensity to export weight using instrumental variables. There are three potential instruments: $Z_{1}=$ Plan is the log number of kilometers of highway in the city according to a planned highway construction map, approved by the federal government in 1947; $Z_{2}=$ Railroads is the log number of kilometers of railroads in the city in 1898; and $Z_{3}=$ Exploration is a measure of the quantity of historical exploration routes that passed through the city. For a fuller description see Duranton2014 and MP.
We will focus here on the model specification and estimation results as displayed in column 2 of Table 5 in Duranton2014 and in column 2 of Table I in MP. This model specification includes the additional control variables “log employment” and “Market access (export)”. Estimation results for the two additional control variables have been omitted. The first column in Table (ref) replicates the 2SLS estimation results using all 3 instruments. This specification is falsified by the $J$-statistic, which has a p-value of 0.043. The next columns give the estimation results for all twelve just-identified model specifications. Using the cutoff of $10$ for the first-stage F-statistic, as in MP, the instruments $Z_{2|1,3}$ and $Z_{2|1}$ are found to be not relevant. The resulting estimates of the falsification adaptive sets are given by $\widehat{FAS}{}_{excl}=\left[-0.32,0.28\right]$ as in MP, $\widehat{FAS}{}_{exo}=\left[0.13,1.09\right]$ and $\widehat{FAS}=\left[-0.61,1.18\right]$.
The $\widehat{FAS}$ has quite a wide range of values here, as it correctly takes into account possible violations of the exclusion and conditional exogeneity assumptions. If for example $\boldsymbol{\gamma}+\boldsymbol{\alpha}=\left(\alpha_{1},0,\gamma_{3}\right)'$ with $\gamma_{3}\neq0$ and $\alpha_{1}\neq0$, then $Z_{2|3}$ is a valid instrument. It is found to be a relevant instrument, and the associated IV estimate is given by $1.18$, which is the largest coefficient estimate of all just-identified specifications. $Z_{2|1,3}$ and $Z_{2|1}$ are found to be not relevant, with their F-statistics less than $10$. In contrast, $Z_{2}$ as well as $Z_{2|3}$ are found to be relevant. This can be explained by the strong correlation of the instruments $Z_{1}$ and $Z_{2}$. The sample partial correlation coefficients are here given by $\widehat{\rho}_{12}=0.57$, $\widehat{\rho}_{13}=0.34$ and $\widehat{\rho}_{23}=0.11$.
Note that here $\widehat{FAS}=\left[\min_{j\in\mathcal{\widehat{\widetilde{L}}}_{rel}}\widehat{\widetilde{\beta}}_{j},\max_{j\in\mathcal{\widehat{\widetilde{L}}}_{rel}}\widehat{\widetilde{\beta}}_{j}\right]$, as the FAS$_{s}$ for $\boldsymbol{\gamma}_{s}+\boldsymbol{\alpha}_{s}=\left(\alpha_{1},\alpha_{2},\gamma_{3}\right)'$, with $\boldsymbol{Z}_{s}=\left(Z_{1|3},Z_{2|3},Z_{3}\right)$ is given by $\left[0.40,1.18\right],$and the the FAS$_{s}$ for $\boldsymbol{\gamma}_{s}+\boldsymbol{\alpha}_{s}=\left(\gamma_{1},\alpha_{2},\alpha_{3}\right)'$, with $\boldsymbol{Z}_{s}=\left(Z_{1},Z_{2|1},Z_{3|1}\right)$ is given by $\left[-0.61,0.55\right]$, and so these two falsification adaptive sets are overlapping and contain $\min_{j\in\mathcal{\widehat{\widetilde{L}}}_{rel}}\widehat{\widetilde{\beta}}_{j}$ and $\max_{j\in\mathcal{\widehat{\widetilde{L}}}_{rel}}\widehat{\widetilde{\beta}}_{j}$.
As a final observation, note that the 2SLS estimator and the test for overidentifying restrictions are the same when using any combination of three instruments from the twelve transformed just-identifying ones, as long as all indices $\left\{ 1,2,3\right\} $ are involved, as in for example $\left\{ Z_{1|3},Z_{2|3},Z_{3}\right\} $. The 2SLS weights for the linear combination of the just-identified estimates for $\left\{ Z_{1},Z_{2},Z_{3}\right\} $ and $\left\{ Z_{1|2,3},Z_{2|1,3},Z_{3|1,2}\right\} $ are here given by $w=\left\{ 0.757,0.126,0.117\right\} $.\footnote{For further details on the 2SLS weights, see Apfel2022.}
We propose a generalization of the falsification adaptive set of MastenPoirierEcta2021 for the classical linear model with an endogenous variable, estimated by the method of instrumental variables with multiple correlated instruments. It is the union of all pattern-specific falsification adaptive sets, where a pattern is defined by any $\boldsymbol{\gamma}_{s}+\boldsymbol{\alpha_{s}}$, with $\gamma_{s,\ell}\text{\ensuremath{\alpha_{s,\ell}}}=0$, and where non-zero $\gamma_{s,\ell}$ values reflect violations of the exclusion assumption and non-zero $\alpha_{s,\ell}$ values reflect violations of the conditional exogeneity assumption. It reflects the model uncertainty when the baseline model is falsified, taking into account possible violations of both the exogeneity and exclusion assumptions. Under the assumption that an invalid instrument can violate either the exclusion or exogeneity assumption, the generalized FAS is guaranteed to contain $\beta$ if there is at least one valid and relevant instrument, and we recommend researchers to report estimates of this set when their baseline model is falsified.
We would like to thank three anonymous referees for their detailed comments and suggestions that helped to improve the paper substantially.