Overidentification in Shift-Share Designs
Jinyong Hahn
Guido Kuersteiner
University of Maryland
Andres Santos
Wavid Willigrod

This paper studies the testability of identifying restrictions commonly employed to assign a causal interpretation to two stage least squares (TSLS) estimators based on Bartik instruments.
For homogeneous effects models applied to short panels, our analysis yields testable implications previously noted in the literature for the two major available identification strategies.
We propose overidentification tests for these restrictions that remain valid in high dimensional regimes and are robust to heteroskedasticity and clustering.
We further show that homogeneous effect models in short panels, and their corresponding overidentification tests, are of central importance by establishing that:
(i) In heterogenous effects models, interpreting TSLS as a positively weighted average of treatment effects can impose implausible assumptions on the distribution of the data; and
(ii) Alternative identifying strategies relying on long panels can prove uninformative in short panel applications.
We highlight the empirical relevance of our results by examining the viability of Bartik instruments for identifying the effect of rising Chinese import competition on US local labor markets.

§ INTRODUCTION

Shift-share designs typically rely on linear combinations of unit specific variables  (the “shares") and aggregate level variables (the “shocks") as instruments to obtain identification.
Originally employed in the work of [Bartik, 1991] and [Blanchard and Katz, 1992], these instruments have become known in the literature as “Bartik" instruments.
Bartik instruments have proven to be remarkably versatile yielding insights into, among others, the impact of immigration on labor markets [Card, 2001], the consequence of trade liberalization for poverty [Topalova, 2010], the effect of import competition on labor markets [Autor et al., 2013], and the welfare implications of geographic sorting by education [Diamond, 2016].

In this paper, we examine the testable implications of identifying restrictions commonly employed to assign a causal interpretation to two stage least squares (TSLS) estimators based on Bartik instruments.
We largely focus our analysis on two recent complementary identification strategies.
The first, studied by [Goldsmith-Pinkham et al., 2020], attributes the exogeneity of the Bartik instrument to the exogeneity of the shares.
The second, examined by [Adao et al., 2019] and [Borusyak et al., 2022], instead attributes the exogeneity of the Bartik instrument to the exogeneity of aggregate shocks.
We systematically study the overidentifying content of both approaches by noting that their differing asymptotic frameworks either implicitly or explicitly require conditional moment restrictions to hold.
Specifically, [Goldsmith-Pinkham et al., 2020] necessitates the Bartik instrument to be exogenous conditional on the realization of the aggregate shocks, while [Adao et al., 2019] and [Borusyak et al., 2022] require the Bartik instrument to be exogenous conditional on the realization of the shares (among other variables).

The conditional moment restrictions implied by the different identification strategies yield previously noted overidentifying restrictions in homogeneous treatment effects models.
For instance, when applied to the framework of [Goldsmith-Pinkham et al., 2020], our analysis implies that the entire vector of shares must be exogenous.
As noted by [Goldsmith-Pinkham et al., 2020], the model is therefore overidentified because any deterministic linear combination of the shares yields a valid instrument.
In contrast, when applied to the framework of [Adao et al., 2019] and [Borusyak et al., 2022], our results imply that the conditional mean of the Bartik instrument given controls, shares, and residuals must be a linear function of the controls only.
As a result, the model is overidentified because any appropriately centered deterministic linear combination of the aggregate shocks yields a valid instrument.
Our impression is that the latter overidentifying restrictions have not received the same level of attention by practitioners as the testable implications of [Goldsmith-Pinkham et al., 2020].

Building on our overidentification analysis, we develop a framework for testing the overidentifying restrictions corresponding to both identification strategies.
In many of the applications that motivate us, the number of overidentifying restrictions can be “large" relative to the sample – e.g., in many applications the numbers of sectors exceeds the number of clusters.
We therefore focus on developing tests that remain valid in such high dimensional settings by employing the high dimensional central limit theorem of [Chernozhuokov et al., 2022].
Our tests utilize bootstrap based critical values and are robust to the presence of heteroskedasticity, clustering, and weighting.[In contrast, [Goldsmith-Pinkham et al., 2020] emply overidentification tests based on LIML (which requires homoskedasticity) and [Chao et al., 2014] (which precludes clustering). Neither test allows the number of restrictions to exceed the sample size.]

A natural way to proceed, for instance after rejecting the validity of the homogeneous effects model, is to consider weaker assumptions that still enable us to attribute a causal interpretation to TSLS.
To this end, we study a generalization of the model and an alternative identification strategy, but find both approaches to be potentially empirically limited in scope.
As a generalization of the model, we consider linear heterogeneous treatment effects models previously employed in the literature [Goldsmith-Pinkham et al., 2020, Adao et al., 2019].
In such a setting, TSLS can readily be shown to estimate a weighted average of group specific treatment effects.
As forcibly argued by [Kolesar, 2013] and [Blandhol et al., 2022] among others, however, the averaging weights should be positive in order to attribute a causal interpretation to TSLS.
Unfortunately, we find that in the case of Bartik instruments the weights can naturally be negative.
In particular, within the identification framework of [Goldsmith-Pinkham et al., 2020], we show that a necessary condition for the weights to be positive is that shares of different sectors be uncorrelated with each other – a requirement that a-fortiori fails when shares sum up to one.
Conversely, within the identification framework of [Adao et al., 2019] and [Borusyak et al., 2022],  we show that a necessary condition for the weights to be positive is that aggregate shocks to different sectors be (weakly) positively correlated with each other.

As an alternative identification strategy, we consider the possibility that the exogeneity of the instrument is not justified by the shares alone [Goldsmith-Pinkham et al., 2020] or the aggregate shocks alone [Adao et al., 2019, Borusyak et al., 2022] but by the shares and shocks together.
Formally, such an identification strategy corresponds to requiring that the Bartik instrument be uncorrelated with the error term when expectations are evaluated over both the time series distribution of aggregate shocks and the cross sectional distribution of shares.
While this identification strategy renders the model “just identified," it also necessitates a long panel in order to estimate the time series distribution of the aggregate shocks.
We show that as a result, the effective number of observations for computing TSLS standard errors is governed by the number of time periods in the panel.
Because the majority of empirical studies relying on Bartik instruments have relied on short panels, we expect TSLS to be statistically uninformative under this identification strategy.
Our asymptotic analysis relies on a novel simultaneous time series and cross sectional study of long panels.
These results significantly extend related work in [Hahn et al., 2019, Hahn et al., 2022] and may be of independent interest.

We highlight the empirical relevance of our analysis by revisiting the study by [Autor et al., 2013] on the impact of rising Chinese import competition on local US labor markets.
In this setting, our overidentification tests find evidence against the validity of the identification framework of [Goldsmith-Pinkham et al., 2020] as well as the identification framework of [Adao et al., 2019] and [Borusyak et al., 2022].
Moreover, since shares are empirically correlated across sectors, our analysis further suggests that TSLS may not have a causal interpretation in a heterogeneous effects model under the identification strategy of [Goldsmith-Pinkham et al., 2020].
Likewise, TSLS may not possess a causal interpretation in a heterogeneous effects model under the identification strategy of [Adao et al., 2019] and [Borusyak et al., 2022] when shocks within clusters are negatively correlated.
Finally, we note that applying the proposed alternative just identified long panel identification strategy is not viable in this application because there are only two time periods.

The rest of the paper is organized as follows.
Section <ref> characterizes the overidentifying restrictions implied by the identification strategies of [Goldsmith-Pinkham et al., 2020] as well [Borusyak et al., 2022] and [Adao et al., 2019].
Section <ref> examines the scope for attributing a causal interpretation to TSLS under heterogeneous effects models or long panel identification strategies.
Our overidentification tests are developed in Section <ref>, while Section <ref> illustrates the  relevance of our analysis by revisiting [Autor et al., 2013].
Section <ref> briefly concludes by providing recommendations for empirical practice.
A series of appendices contain proofs of our results and a Monte Carlo study.

§ OVERIDENTIFICATION

Following the main instrumental variables specifications of [Goldsmith-Pinkham et al., 2020] and [Adao et al., 2019], we begin by studying homogeneous treatment effects models in which the parameter of interest is common across all individuals.
In the next section, we will examine heterogenous treatment effects models instead.
For ease of exposition, we focus on cross sectional applications though note that an extension to short panel data settings is immediate; see, e.g., Remarks <ref> and <ref> below.

In what follows we let $Y_{i}\in\R$, $X_{i}\in\R$, $Z_i\in \R$, and $W_{i}\in\R^d$ denote the outcome variable, a scalar regressor of interest, a scalar instrument, and a vector of controls.
We will refer to each “$i$" as an individual, though note that in applications $i$ may represent, for example, a location.
The variables $(Y_i,X_i,W_i)$ are assumed to satisfy

\begin{equation}\label{model:eq1}
\end{equation}

where the restrictions on $\varepsilon_i$ will be stated shortly.
We will further require that the instrument $Z_{i}$ follow a Bartik structure in the sense
that it satisfies

\begin{equation*}%\label{model:eq2}
\end{equation*}

with $\Sh \in\R^p$ an aggregate variable and $S_{i}\in\R^p$ an individual specific variable.

For instance, in canonical applications $\Sh$ equals a vector of aggregate industry specific shocks and $S_i$ equals the share of each industry in the economy of location $i$.
For this reason, in what follows we refer to $\Sh$ as “shocks" and $S_i$ as “shares."

Other variables, such as $\varepsilon_i$, may have a Bartik structure as well though we only make the Bartik structure explicit for $Z_i$.
Finally, we complement equation (<ref>) with the first stage equation

\begin{equation}\label{model:eq3}
X_{i}=Z_{i} \lambda +W_{i}^{\prime}\gamma_{\mathtt{f}%
\end{equation}

Here, the $\mathtt{f}$ and $\mathtt{s}$ subscripts on $\gamma$ refer to the first and the second stage respectively.
While we let the first stage coefficients in (<ref>) be fixed for simplicity, it is worth emphasizing that all the results in this section continue section hold if the first stage coefficients are instead random as in [Adao et al., 2019].
The observable variables are the aggregate shocks $\Sh$ and the cross sectional variables $\left\{Y_{i},X_{i},W_{i},S_{i}\right\}_{i=1}^n$.

Different asymptotic frameworks either implicitly or explicitly condition on different variables when delivering asymptotic promises.
To reflect these differences, it will prove convenient to introduce the notation $\mathcal G_n$ to denote the relevant conditioning variables.
For instance, we will argue below that asymptotic approximations that rely on only the cross section being large implicitly condition on aggregate variables such as the shocks $\Sh$.
Hence, in such instances we would set $\mathcal G_n$ to include all aggregate variables, including $\Sh$.
Alternatively, when specialized to equations (<ref>) and (<ref>), the asymptotic framework of [Adao et al., 2019] and [Borusyak et al., 2022] explicitly conditions on the cross-sectional variables $\{W_i,S_i,\varepsilon_i\}_{i=1}^n$ so we would in that case simply set $\mathcal G_n$ to equal $\{W_i,S_i,\varepsilon_i\}_{i=1}^n$.

We will first examine conditions under which the two stage least squares (TSLS) estimand equals $\beta$ in the homogeneous effects model defined by (<ref>) and (<ref>).
To this end, it is helpful to define the “population" residualized instrument $\dot Z_{i,n}$ according to

\begin{equation}\label{model:eq4}
\dot Z_{i,n} \equiv Z_i - W_i^\prime \pi_n \hspace{0.5 in} \pi_{n}   \equiv\left(  \frac{1}{n}\sum_{i=1}^{n}E\left[  \left.  W_{i} W_{i}^{\prime}\right\vert \mathcal{G}_{n}\right]  \right)  ^{-1}\left( \frac{1}{n}\sum_{i=1}^{n}E\left[  \left.  W_{i}Z_{i}\right\vert \mathcal{G}_{n}\right]  \right),
\end{equation}

which allows us to define the TSLS estimand $\beta_{0,n}$ as the solution to the equation

\begin{equation}\label{model:eq5}
\frac{1}{n}\sum_{i=1}^{n}E\left[  \left.  \left(  Y_{i}-X_{i}\beta
_{0,n}\right)  \dot{Z}_{i,n}\right\vert \mathcal{G}_{n}\right]  =0.
\end{equation}

We note that in the notation we have let $\beta_{0,n}$ depend on $n$ to reflect that the TSLS estimand may depend on $n$ through the conditioning on $\mathcal G_n$.
Under an appropriate rank condition, plugging the outcome equation (<ref>) into the moment condition in (<ref>) readily yields that the estimand $\beta_{0,n}$ equals $\beta$ if and only if we have

\begin{equation}\label{model:eq6}
\frac{1}{n}\sum_{i=1}^n E\left[\varepsilon_i(Z_i-W_i^\prime \pi_n)|\mathcal G_n\right] = 0    .
\end{equation}

We next examine the implications of this exogeneity condition under the different choices of conditioning set $\mathcal G_n$ that correspond to different asymptotic approximations.

§.§ Conditioning on Shocks

Traditionally, empirical work employing shift-share designs has reported standard errors that are motivated by asymptotic approximations in which only the cross section becomes large (i.e. $n\to \infty$).
Probabilistic statements associated with these asymptotic approximations correspond to a thought experiment in which we only re-sample the individual specific variables $\{Y_i,X_i,W_i,S_i\}_{i=1}^n$.
For instance, in this context, the level of a confidence interval for $\beta$ refers to the probability that a randomly drawn sample $\{Y_i,X_i,W_i,S_i\}_{i=1}^n$ yields a confidence interval that indeed includes $\beta$.

Probabilistic statements based on re-sampling only individual specific variables a fortiori keep aggregate variables such as the shocks $\Sh$ fixed.
Consequently, in an asymptotic approximation in which only $n$ becomes large we are only able to identify the distribution of individual specific variables conditionally on the realizations of aggregate variables such as $\Sh$.
When evaluating the exogeneity condition on our instrument (i.e. restriction (<ref>)) under this asymptotic approximation,  we should therefore view all aggregate variables as belonging to the conditioning set  $\mathcal G_n$.

In order to illustrate the implications of these observations, we will for simplicity assume that the  variables $\{S_i,W_i,\varepsilon_i\}_{i=1}^n$ are i.i.d. and independent of all aggregate variables.[This setting rules out, for example, that $(W,\varepsilon)$ have a Bartik structure as in [Adao et al., 2019]. We note, however, that the main points made in this section carry over if we allow $(W,\varepsilon)$ to have a Bartik structure as well, though such an extension requires additional notation and assumptions. ]
Provided that the controls $W_i$ are uncorrelated with the error $\varepsilon_i$, it then follows from a bit of algebra that the exogeneity condition in (<ref>) is equivalent to

\begin{equation}\label{model:eq7}
\Sh^\prime E\left[S_i \varepsilon_i\right] = 0    .
\end{equation}

Equation (<ref>) highlights that the relevant exogeneity condition for this asymptotic framework depends on the realization of the aggregate shocks $\Sh$ and the correlation between the shares $S_i$ and the error $\varepsilon_i$.
Lacking a justification as to why (<ref>) should hold at the actual realization of $\Sh$ and not others, we should in the interest of robustness demand that (<ref>) hold at all possible realizations of $\Sh$.

Provided the aggregate shocks $\Sh$ exhibit sufficient variation, however, it then follows that condition (<ref>) can hold for all possible realizations of $\Sh$ if and only if in fact $S_i$ itself is uncorrelated with the errors $\varepsilon_i$.

Our next simple proposition formalizes the preceding discussion.

Suppose that $\mathcal Z\in \mathcal G_n$ and that  $\{S_i,W_i,\varepsilon_i\}_{i=1}^n$ are i.i.d. and independent of $\mathcal G_n$.
If $E[W\varepsilon] = 0$ and the support of $\mathcal Z\in \R^p$ has dimension $p$, then

\begin{equation}\label{prop:homoggssdisp}
\frac{1}{n}\sum_{i=1}^n E\left[\varepsilon_i(Z_i - W_i^\prime \pi_n)|\mathcal G_n \right] =0
\end{equation}

with probability one (over $\Sh$) if and only if $E[S_i \varepsilon_i ] = 0$.

Proposition <ref> establishes that the entire vector of shares $S_i$ must be a valid instrument in order for the scalar instrument $Z_i$ itself to be valid.
This conclusion echoes arguments in [Goldsmith-Pinkham et al., 2020], who similarly argue that the exogeneity of the instrument $Z_i$ should be understood in terms of the exogeneity of the shares $S_i$.
The principal implication of Proposition <ref> for our purposes is that the validity of the Bartik instrument renders the model overidentified.
For instance, it follows that we may construct an overidentification test by employing the sample moments

\begin{equation*}%\label{model:eq8}
\frac{1}{n}\sum_{i=1}^n S_i \hat \varepsilon_i ,
\end{equation*}

where $\{\hat \varepsilon_i\}_{i=1}^n$ denotes the fitted residuals from the estimated model.
One complication that arises in building overidentification tests from these moments is that the number of moments (i.e. the number of shares) is often too “large" for standard asymptotic approximations to remain accurate.
In Section <ref>, we address this challenge by developing overidentification tests that rely on high dimensional asymptotics instead.

While we have focused on a cross-sectional setting for ease of exposition, our conclusions readily extends to panel data applications.
Specifically, suppose there are $1\leq t \leq T$ time periods, each with an aggregate shock $\Sh_t$ and a cross-section $\{Y_{it},X_{it},S_{it},W_{it}\}_{i=1}^n$.
Under a short panel asymptotic approximation (i.e. $n\to\infty$ and $T$ fixed), conditions similar to those employed in Proposition <ref> readily imply that as the relevant exogeneity condition we should require that

\begin{equation*}%\label{rm:panelGSS1}
E[S_{it}\varepsilon_{it}] = 0 \quad \quad \text{ for } 1\leq t \leq T.
\end{equation*}

Thus, in this context, the time dimension yields additional overidentifying restrictions to those available in the cross sectional setting.

§.§ Identification Through Shocks

As an alternative identification strategy, [Adao et al., 2019] and [Borusyak et al., 2022] advocate interpreting the exogeneity of the Bartik instrument $Z_i$ as originating from the exogeneity of the aggregate shocks $\Sh$.
Within our context, the assumptions employed by their asymptotic approximations correspond to setting  $\mathcal G_n$ to equal $\{S_i,W_i,\varepsilon_i\}_{i=1}^n$ and letting the number of shocks $p$ increase with the sample size $n$.[As noted by [Adao et al., 2019], including $\{\varepsilon_i\}_{i=1}^n$ in the conditioning set is important for obtaining asymptotically valid standard errors in applications in which $\{\varepsilon_i\}_{i=1}^n$ possesses a Bartik structure.]
Under this asymptotic framework, the relevant exogeneity condition (i.e. restriction (<ref>)) is equivalent to

\begin{equation}\label{model:eq9}
\frac{1}{n}\sum_{i=1}^n \varepsilon_i\left(S_i^\prime E\left[\Sh|\mathcal G_n \right] - W_i^\prime \pi_n\right) = 0  .
\end{equation}

To gain intuition into this exogeneity requirement, it is instructive to consider the case in which there are no controls $W_i$.
Letting $\Sh_j$ denote the $j^{th}$ coordinate of the vector of aggregate shocks $\Sh\in \R^p$ and $S_{ij}$ the $j^{th}$ coordinate of the shares $S_i\in \R^p$, it then follows that the exogeneity condition in (<ref>) simplifies to the expression

\begin{equation}\label{model:eq10}
\sum_{j=1}^p \left(E[\mathcal Z_j|\mathcal G_n] \times  \left(\frac{1}{n}\sum_{i=1}^n \varepsilon_i S_{ij}\right)\right) = 0 .
\end{equation}

Moreover, since the moment restriction in (<ref>) must hold for any possible realization of the sample $\{S_i,\varepsilon_i\}_{i=1}^n$, the exogeneity condition in this context implies, under appropriate assumptions, that the aggregate shocks $\{\Sh_j\}_{j=1}^p$ must have mean zero and be mean independent of the individual specific variables $\{S_i,\varepsilon_i\}_{i=1}^n$.
However, if all the aggregate shock $\{\Sh_j\}_{j=1}^p$ are mean independent of the individual specific variables, then the instrument $Z_i = S_i^\prime \mathcal Z$ must itself be mean independent of $\{S_i,\varepsilon_i\}_{i=1}^n$ because

\begin{equation*}%\label{model:eq11}
E[Z_i|\{S_i,\varepsilon_i\}_{i=1}^n] = \sum_{j=1}^p E[\Sh_j|\mathcal G_n]S_{ij} = 0.
\end{equation*}

Intuitively, if the aggregate shocks $\{\Sh_j\}_{j=1}^p$ are exogenous in the sense that (<ref>) holds, then any suitable linear combination of the shocks, such as $Z_i = S_i^\prime \Sh$, must be mean independent of $\{S_i,\varepsilon_i\}_{i=1}^n$ as well.
In particular, we obtain the overidentifying restriction that $Z_i$ must be uncorrelated with any functions of the shares $S_i$ and the error $\varepsilon_i$.[Equivalently, note that since any suitably linear combination of $\Sh$ is also a valid instrument, it follows that there is a surplus of instruments and hence that the model is overidentified.]

While the inclusion of controls $W_i$ allows us to relax the requirements on the aggregate shocks $\Sh$, the model remains overidentified through its implications on the conditional mean of the instrument $Z_i$.
Our next proposition illustrates this conclusion in the simple setting in which $\Sh$ is independent of all individual specific variables.

Suppose $\mathcal G_n = \{S_i,W_i,\varepsilon_i\}_{i=1}^n$ and $\Sh$ is independent of $\mathcal G_n$.
If the support of $\{\varepsilon_i\}_{i=1}^n$ conditional on $\{S_i,W_i\}_{i=1}^n$ has dimension $n$, then

\begin{equation}\label{prop:homogakmdisp}
\frac{1}{n}\sum_{i=1}^n E[\varepsilon_i(Z_i - W_i^\prime \pi_n)|\mathcal G_n] = 0
\end{equation}

with probability one (over $\{S_i,W_i,\varepsilon_i\}_{i=1}^n$) if and only if $E[Z_i|\{S_i,W_i,\varepsilon_i\}_{i=1}^n] = W_i^\prime \pi_n$.

Proposition <ref> establishes that the exogeneity requirement needed to obtain identification through the aggregate shocks $\Sh$ effectively imposes that the conditional mean of the instrument $Z_i$ given $\{S_i,W_i,\varepsilon_i\}_{i=1}^n$ be equal to the linear projection of $Z_i$ onto $W_i$.
This conclusion can be shown to hold even if $\Sh$ is not independent of the individual specific variables $\{S_i,W_i,\varepsilon_i\}_{i=1}^n$ at the cost of additional notation and assumptions.
For our purposes, the principal implication of Proposition <ref> is that the residualized instrument $(Z_i - W_i^\prime \pi_n)$ must be uncorrelated with any function of the shares $S_i$, controls $W_i$, and errors $\varepsilon_i$.
This observation suggests employing the sample moments

\begin{equation*}%\label{model:eq12}
\frac{1}{n}\sum_{i=1}^n g(\hat \varepsilon_i,W_i,S_i)(Z_i - W_i^\prime \hat \pi_n)
\end{equation*}

to construct an overidentification test – here $g$ is an arbitrary vector valued function and $\{Z_i - W_i^\prime \hat \pi_n\}_{i=1}^n$ denotes the residuals from regressing the instrument on the controls.
In Section <ref> we will develop such an overidentification test while allowing the number of sample moments to potentially be high dimensional.

The overidentifying restrictions of Proposition <ref> generalize to panel data applications.
Specifically, suppose that at each time period $1\leq t \leq T$ we observe a  shock $\Sh_t \in \R^p$ and individual specific variables $\{Y_{it},X_{it},S_{it},W_{it}\}_{i=1}^n$.
In this context, the asymptotic framework in [Adao et al., 2019] and [Borusyak et al., 2022] corresponds to setting $\mathcal G_n$ to equal $\{S_{it},W_{it},\varepsilon_{it}\}_{i=1,t=1}^{n,T}$ and letting $T$ be fixed while $n$ and $p$ become large.
Under conditions similar to those imposed in Proposition <ref>, it follows that the exogeneity requirements imposed on the shocks $\{\Sh_t\}_{t=1}^T$ to obtain identification imply

\begin{equation*}%\label{rm:panelakm1}
E[Z_{it}|\mathcal G_n] = W_{it}^\prime \pi_n \hspace{0.5 in} \pi_n \equiv \left(\sum_{i=1}^n\sum_{t=1}^T W_{it}W_{it}^\prime \right)^{-1} \left(\sum_{i=1}^n\sum_{t=1}^T W_{it}E[Z_{it}|\mathcal G_n]\right).
\end{equation*}

As a result, the model is now overidentified by the restriction that the residualized instrument $(Z_{it}-W_{it}^\prime \pi_n)$ be uncorrelated with any function of $\{W_{it},S_{it},\varepsilon_{it}\}_{t=1}^T$.

[Adao et al., 2019] and [Borusyak et al., 2022] impose sufficient conditions for the overidentifying restrictions derived in Proposition <ref> to hold.
For instance, [Adao et al., 2019] require that the controls satisfy $W_i = \mathcal W^\prime S_i$, with $\mathcal W$ a $p\times d$ matrix of aggregate shocks, and impose that $E[\Sh|\{S_i,W_i,\varepsilon_i\}_{i=1}^n,\mathcal W] = \mathcal W\Gamma$ for some vector $\Gamma\in \R^d$.[More precisely, [Adao et al., 2019] impose that $W_i = \mathcal W^\prime S_i + U_i$ and require $U_i$ to be asymptotically negligible in a suitable sense; see their Assumptions 3(iii)(iv).]
Under this structure, it follows that $\pi_n$ equals $\Gamma$ and hence that

\begin{multline*}%\label{rm:akmrel1}
E[Z_i|\{S_i,W_i,\varepsilon_i\}_{i=1}^n] \\
= E[S_i^\prime E[\Sh | \{S_i,W_i,\varepsilon_i\}_{i=1}^n,\mathcal W]|\{S_i,W_i,\varepsilon_i\}_{i=1}^n] = E[S_i^\prime \mathcal W\Gamma | \{S_i,W_i,\varepsilon_i\}_{i=1}^n] = W_i^\prime \pi_n,
\end{multline*}

which implies the overidentifying restrictions of Proposition <ref> are indeed satisfied.

§ JUST IDENTIFICATION?

We have so far shown that the homogeneous effects model is overidentified in shift-share designs.
One possible way to proceed, for instance after statistically rejecting the model, is to weaken assumptions in a manner that renders the model potentially just identified  (e.g., in the sense of [Chen and Santos, 2018]).
In this section, we show that two natural relaxations to the model of Section <ref> are severely limited in the set of empirical contexts to which they may be successfully applied.
As a result, we conclude that the homogeneous effects model of Section <ref> is of central empirical importance in shift-share designs relying on Bartik instruments.

First, we examine the possibility of relaxing the model in Section <ref> to a linear heterogeneous effects model.
In this context, the TSLS estimand can be interpreted as a weighted average of causal effects for different population subgroups.
We find, however, that ensuring  the estimand equals a positively weighted average of causal effects requires strong, often unrealistic, assumptions on the distribution of the data.

Second, we examine the possibility of relaxing the model in Section <ref> by only requiring that the Bartik instrument be exogeneous when expectations are evaluated over both the shocks and the shares.
We show that, under this exogeneity requirement, identification crucially relies on the time series variation of the shocks and, as a result, that the standard errors of TSLS depend on the time dimension.
Because the majority of empirical studies relying on shift-share designs have employed short panels, we expect TSLS estimates to be highly imprecise under this identification strategy.

§.§ Heterogeneous Effects

We first re-examine the shift-share design of Section <ref> in the presence of heterogenous treatment effects.
To this end, we generalize the second stage equation by setting

\begin{equation}\label{model:eq13}
Y_i = X_i \beta_i + W_i^\prime \gamma_{\mathtt{s}} + \varepsilon_i;
\end{equation}

i.e. the “treatment effect" $\beta_i$ is allowed to depend on the individual but, for simplicity, we keep the coefficient for the controls fixed.
We complement the second stage by following [Adao et al., 2019] and [Goldsmith-Pinkham et al., 2020] in imposing a linear first stage

\begin{equation}\label{model:eq14}
X_i = \Sh^\prime \Lambda_i S_i + W_i^\prime \gamma_{\mathtt{f}} + \eta_i \hspace{0.5 in} \Lambda_i = \text{diag}\left\{\left(\lambda_{i1},\ldots, \lambda_{ip}\right)\right\},
\end{equation}

where “$\text{diag}\{(a_1,\ldots, a_p)\}$" denotes a diagonal matrix with diagonal entries $(a_1,\ldots, a_p)$ and again, for simplicity, we set the coefficient for the controls $W_i$ to be fixed.
In analogy to [Imbens and Angrist, 1994], we refer to $\Lambda_i$ as the “type" of the individual.

Within this context, we examine conditions under which the TSLS estimand retains a causal interpretation.
Formally, we remain interested in the parameter $\beta_{0,n}$ solving

\begin{equation*}%\label{model:eq15}
\frac{1}{n}\sum_{i=1}^n E\left[(Y_i - X_i \beta_{0,n})\dot Z_{i,n}\Big|\mathcal G_n\right] = 0,
\end{equation*}

but now study conditions under which $\beta_{0,n}$ equals a positively weighted average of the average treatment effects for different subgroups.
Under suitable exogeneity and rank conditions on the instrument $Z_i$, it is possible to show that the estimand $\beta_{0,n}$ satisfies

\begin{equation}\label{model:eq16}
\beta_{0,n} =  E\left[\left(\sum_{i=1}^n\omega_{i,n}E\left[\beta_i|\Lambda_i,W_i,\mathcal G_n\right]\right)\Bigg|\mathcal G_n\right] \hspace{0.3 in} \omega_{i,n} \equiv \frac{E\left[X_i\dot Z_{i,n}|\Lambda_i,W_i,\mathcal G_n\right]}{\sum_{i=1}^n E\left[X_i \dot Z_{i,n}|\mathcal G_n\right]};
\end{equation}

i.e. $\beta_{0,n}$ equals the expectation of a $\{\omega_{i,n}\}_{i=1}^n$-weighted average of the average treatment effects for subgroups determined by $\Lambda_i$ and $W_i$; see Lemma <ref> for a formal statement.
As forcibly argued in the literature, a minimal requirement for $\beta_{0,n}$ to possess a causal interpretation is that the weights $\{\omega_{i,n}\}_{i=1}^n$ be positive [Kolesar, 2013, Blandhol et al., 2022].
In what follows, we study the implications of requiring that such a positivity condition hold under different asymptotic frameworks – i.e. under different choices of $\mathcal G_n$.
For succinctness, we will refer to TSLS as having a causal interpretation whenever the weights $\{\omega_{i,n}\}_{i=1}^n$ are  positive with probability one.

§.§.§ Conditioning on Shocks

We begin by studying the conditions under which TSLS has a causal interpretation in an asymptotic setting in which only the cross section grows (i.e. $n \to \infty$).
Recall that in Section <ref> we showed that this setting implicitly conditions on the shocks $\Sh$ and requires the shares $S_i$ to be a valid instrument.

The analysis in this section relies on the following assumption and additional regularity conditions that we formally state in the Appendix; see Assumption <ref>.

(i) $\Sh$ and $\{Y_i,X_i,W_i,S_i\}_{i=1}^n$ satisfy equations (<ref>) and (<ref>);
(ii) $(\Sh,\Lambda_i,\beta_i,\varepsilon_i,\eta_i)\indep S_i$ conditionally on $W_i$;
(iii) $E[Z_i|W_i,\mathcal G_n] = W_i^\prime \pi_n$ with $\mathcal G_n = \{\Sh\}$.

Assumption <ref>(i) formally imposes the structure of the model, while Assumption <ref>(ii) requires the shares $S_i$ to be suitably exogenous.
In Assumption <ref>(iii) we further demand that the conditional mean of the instrument given the covariates be linear.[Under regularity conditions, Assumption <ref>(iii) is equivalent to $E[S_i|W_i]$ being linear in $W_i$.]
[Blandhol et al., 2022] showed in a related context that Assumption <ref>(iii) is a necessary condition for TSLS to have a causal interpretation.
We therefore impose Assumption <ref>(iii)  not because it is innocuous, but because it enables us to examine what additional conditions are necessary for TSLS to have a causal interpretation.
Assumption <ref>(iii) is of course testable, and practitioners may wish to examine its validity in assessing whether the TSLS estimator can be assigned a causal interpretation.

Our next proposition characterizes the weights $\{\omega_{i,n}\}_{i=1}^n$ and obtains necessary and sufficient conditions for them to be positive with probability one.

If Assumptions <ref> holds and $\mathcal G_n = \Sh$, then it follows that

\begin{equation}\label{prop:heterogssdisp}
\omega_{i,n} = \frac{\Sh^\prime (\Lambda_i {\rm Var}\{S_i|W_i\})\Sh}{\sum_{j=1}^n \Sh^\prime (\Lambda_j {\rm Var}\{S_j|W_j\})\Sh}.
\end{equation}

Moreover, if in addition Assumption <ref> holds, then for $n$ large $\omega_{i,n}$ is positive for all $i$ with probability one (over $\Lambda_i,W_i,\Sh$) if and only if the matrix $\Lambda_i {\rm Var}\{S_i|W_i\}$ is either positive semi-definite with probability one, or negative semi-definite with probability one.

The first part of Proposition <ref> shows that the weights may be expressed as a quadratic form in the shocks $\Sh$ (see (<ref>)).
Intuitively, provided that the support of the shocks $\Sh$ is sufficiently rich, it therefore follows that the weights can only be positive for all possible realizations of the shocks $\Sh$ if in fact the matrices $\Lambda_i\text{Var}\{S_i|W_i\}$ are positive semi-definite (or negative semi-definite) with probability one.
The second part of Proposition <ref> formalizes this intuition under the requirement that the support of the shocks $\Sh$ contain a neighborhood of zero.[This requirement is formally stated in Assumption <ref> in the Appendix.]

The necessary and sufficient conditions derived in Proposition <ref> for TSLS to have a causal interpretation are both highly restrictive and testable.
Our next corollary shows that, under mild conditions on the support of $\Lambda_i$, these conditions in fact necessarily fail whenever shares are correlated with each other.

Let the conditions of Proposition <ref> hold and suppose that the support of $\lambda_{ij}/\lambda_{ik}$ is unbounded for any $j\neq k$.
If for some $j\neq k$ we have that

\begin{equation}\label{cor:heteroimpdisp}
P\left({\rm Cov}\{S_{ij},S_{ik}|W_i\}\neq 0\right) > 0,
\end{equation}

then the weights $\{\omega_{i,n}\}_{i=1}^n$ are negative with positive probability (over $(\Lambda_i,W_i,\Sh)$).

Corollary <ref> implies that, whenever shares are correlated, TSLS necessarily lacks a causal interpretation under certain realizations of the shocks $\Sh$.
Because shares must be correlated whenever they sum up to one, we view Corollary <ref> as a  warning that TSLS based on the Bartik instrument can easily fail to have a causal interpretation under a heterogenous effects model.
However, it may be worth emphasizing that, as argued by [Goldsmith-Pinkham et al., 2020], empirical researchers may still be able to estimate causal parameters by carefully employing the shares as separate instruments instead of combining them into a single scalar $Z_i$ by employing the shocks $\Sh$.

Corollary <ref> yields testable implications beyond restricting the correlation between shares.
Notably, because $\text{Var}\{S_i|W_i\}$ must be diagonal and $\Lambda_i$ must have constant sign in order for TSLS to have a causal interpretation, it follows that the sign of the first stage must be constant under certain choices of instrument.
In particular, letting $\tilde {\Sh}_{ij} = f_j(W_i)\Sh_j$ for any positive $f_j$ and $\tilde {\Sh}_{i} = (\tilde {\Sh}_{i1},\ldots, \tilde {\Sh}_{ip})^\prime$, it follows that

\begin{equation}\label{rm:heterotest1}
\text{sign}\left\{E[X_i \dot Z_{i,n}|\Sh]\right\} = \text{sign}\left\{E[X_i (S_i - E[S_i|W_i])^\prime \tilde {\Sh}_{i}|\Sh ]\right\}.
\end{equation}

Since the conditional mean of the shares is linear in the controls in our setting, we obtain a simple diagnostic: Compare the sign of the covariances between $X_i$ and $\dot S_{i,n}^\prime \tilde {\Sh}_i$ for different choices of $\tilde {\Sh}_i$ – here, $\dot S_{i,n}$ denotes the residual from regressing $S_i$ on $W_i$.

§.§.§ Identification Through Shocks

We next examine the conditions under which TSLS retains a causal interpretation in an asymptotic framework in which identification is driven by the exogeneity of the aggregate shocks.
To this end, we follow [Adao et al., 2019] and [Borusyak et al., 2022] in augmenting the conditioning set $\mathcal G_n$ of Section <ref> to include all unobserved heterogeneity – i.e., we set $\mathcal G_n$ to equal $\{S_i,W_i,\Lambda_i,\beta_i,\varepsilon_i,\eta_i\}_{i=1}^n$.

Our analysis in this section relies on the following assumption and a set of regularity conditions that we formally state in the Appendix; see Assumption <ref>.

(i) $\Sh$ and $\{Y_i,X_i,W_i,S_i\}_{i=1}^n$ satisfy equations (<ref>) and (<ref>);
(ii) $E[Z_i|W_i,\mathcal G_n] = W_i^\prime \pi_n$ with $\mathcal G_n = \{S_i,W_i,\Lambda_i,\beta_i,\varepsilon_i,\eta_i\}_{i=1}^n$.

Assumption <ref> imposes the structure of our model and requires that the conditional mean of the instrument given the controls be linear.[Sufficient conditions for Assumption <ref>(ii) were introduced by [Adao et al., 2019]; see Remark <ref>.]
As previously noted in Section <ref>, linearity has been shown to be a necessary condition for TSLS to preserve a causal interpretation in related contexts.
We reiterate that we therefore impose linearity to examine what additional conditions are needed for TSLS to have a causal interpretation, and not because we view linearity as an inoccuous assumption.
It is also worth emphasizing the connection between Assumption <ref>(ii) and Proposition <ref>, which established that linearity was a necessary condition for the instrument to be exogenous in a homogeneous effects model.
As a result, the overidentification test for the homogenous effects model that we develop in Section <ref> may be readily adapted to test Assumption <ref>(ii).

Our next proposition obtains a characterization of the weights $\{\omega_{i,n}\}_{i=1}^n$ and derives a necessary and sufficient condition for TSLS to have a causal interpretation.

If Assumption <ref> holds, and $\mathcal G_n = \{ S_i, W_i, \Lambda_i, \beta_i,\varepsilon_i,\eta_i\}_{i=1}^n$, then

\begin{equation}\label{prop:heteroakmdisp}
\omega_{i,n} = \frac{S_i^\prime (\Lambda_i {\rm Var}\{\Sh|\mathcal G_n\})S_i}{\sum_{j=1}^n S_j^\prime (\Lambda_j {\rm Var}\{\Sh|\mathcal G_n\})S_j}.
\end{equation}

Moreover, if in addition Assumption <ref> holds, then for $n$ large $\omega_{i,n}$ is positive for all $i$ with probability one (over $\mathcal G_n$) if and only if the distribution of $\Lambda_i{\rm Var}\{\Sh|\mathcal G_n\}$ satisfies

\begin{equation}\label{lm:heteroakmdisp2}
P\left(\sup_{s\in \R^p_+} s^\prime (\Lambda_i{\rm Var}\{\Sh|\mathcal G_n\}) s \leq 0\right) = 1  \text{ or } P\left(\inf_{s\in \R^p_+} s^\prime (\Lambda_i{\rm Var}\{\Sh|\mathcal G_n\}) s \geq 0\right) = 1.
\end{equation}

The first part of Proposition <ref> establishes that the weights $\{\omega_{i,n}\}_{i=1}^n$ equal a quadratic form in the shares $S_i$.

Intuitively, the expression for the weights as a quadratic form in the shares implies that the support of the random matrix $\Lambda_i \text{Var}\{\Sh|\mathcal G_n\}$ must be restricted in order for the weights to be positive for any possible realization of the shares.
The second part of Proposition <ref> formalizes this intuition under a requirement that the support of the shares is suitably rich; see Assumption <ref> for a formal statement.

Because, unlike the shocks, the shares are always positive, the necessary and sufficient condition for TSLS to have causal interpretation (i.e. (<ref>)) is weaker than requiring the random matrix $\Lambda_i\text{Var}\{\Sh|\mathcal G_n\}$ to be positive (or negative) semidefinite with probability one.
The conditions derived by Proposition <ref> are nonetheless restrictive and testable.
Our next corollary, for instance, employs Proposition <ref> to establish that TSLS fails to have a causal interpretation whenever shocks are negatively correlated.

Let the conditions of Proposition <ref> hold and suppose $P(\lambda_{ij}=0)=0$ for all $j$
and the support of $\lambda_{ij}/\lambda_{ik}$ is unbounded for any $j\neq k$.
If in addition

\begin{equation}\label{cor:heteroimpakmdisp}
P\left({\rm Cov}\{\Sh_j,\Sh_k|\mathcal G_n\} < 0\right) > 0
\end{equation}

for some $j\neq k$, then the weights $\{\omega_{i,n}\}_{i=1}^n$ are negative with positive probability.

Traditionally, the literature has assumed shocks to either be uncorrelated or clustered for asymptotic purposes [Adao et al., 2019, Borusyak et al., 2022].
Corollary <ref> highlights that the covariance structure of the shocks is important not only for delivering asymptotic approximations and computing standard errors, but also for assigning TSLS a causal interpretation.
Corollary <ref> additionally has important implications for applications in which standard errors are clustered.
In particular, since clustered standard errors reflect a concern that shocks within a cluster are correlated, Corollary <ref> implies that TSLS can only retain a causal interpretation if in fact all correlations within a cluster are positive conditionally on $\mathcal G_n$.
Empirical researchers clustering standard errors should therefore argue that all correlations within a cluster are positive or, alternatively, rely on a homogeneous effects model.

Because all entries of $\text{Var}\{\Sh|\mathcal G_n\}$ must be (weakly) positive and the sign of $\Lambda_i$ must be constant  for TSLS to possess a causal interpretation, it follows that

\begin{equation}\label{rm:heteroakmtest1}
\text{sign}\left\{\sum_{i=1}^n E[X_i\dot Z_{i,n}|\mathcal G_n]\right\} = \text{sign}\left\{\sum_{i=1}^n E\left[X_i(\Sh - E[\Sh|\mathcal G_n])^\prime f(S_i,W_i)|\mathcal G_n \right]\right\}
\end{equation}

for any positive $f(S_i,W_i)\in \R^p$ and $\mathcal G_n$ as in Corollary <ref>.
In parallel to Remark <ref>, we may interpret the restriction in (<ref>) as demanding that the sign of the first stage be constant under certain alternative choices of instrument.
Empirically evaluating this restriction can require an estimator for the conditional mean of the shocks.
For this purpose, we note that multiple such estimators have been proposed in the literature as they are also needed to compute the standard errors of the TSLS estimator when conditioning on $\mathcal G_n$ [Adao et al., 2019, Borusyak et al., 2022].

§.§ Long Panel

Our cross sectional analysis has so far conditioned on either the aggregate shocks or the shares to obtain identification.
Both approaches yield overidentifying restrictions for homogeneous effects models that carry over to short panel settings (i.e. $T$ fixed); see Remarks <ref> and <ref>.
In this section, we conclude by exploring the implications of employing unconditional moment restrictions for identification instead.

For illustrative purposes, we consider a homogeneous effects models with no controls and a scalar endogenous variable.
We assume that at each time period $1\leq t \leq T$ we observe an aggregate shock $\Sh_t$ and variables $\{Y_{it},X_{it},Z_{it}, S_{it}\}_{i=1}^n$ satisfying

\begin{equation*}%\label{eq:long1}
Y_{it} = X_{it}\beta + \varepsilon_{it} \hspace{0.5 in}  Z_{it} = S_{it}^\prime \Sh_t.
\end{equation*}

As an identifying assumption, we now employ the just identified moment restriction

\begin{equation}\label{eq:long2}
\frac{1}{nT} \sum_{i=1}^n\sum_{t=1}^T E[\varepsilon_{it} Z_{it}] = 0.
\end{equation}

Crucially, the expectation in (<ref>) is taken over both the cross section and the time series – e.g., over both $\Sh_t$ and $(S_{it},\varepsilon_{it})$.
In particular, the moment restriction in (<ref>) contrasts with [Goldsmith-Pinkham et al., 2020] and [Adao et al., 2019] who instead consider conditional expectations given $\Sh_t$ and $(S_{it},\varepsilon_{it})$ respectively; see Section <ref>.

The natural estimator for $\beta$ continues to be the same TSLS estimator that we have considered so far.
However, relying on the unconditional moment restriction in (<ref>) for identification now requires us to let both $n$ and $T$ grow so that we may approximate expectations over both the cross section and the time series.
As a result, when employing the just identified moment restriction in (<ref>) to obtain identification, we need to rely on “long panel" asymptotic approximations and their corresponding standard errors.
The main message of this section is that the long panel standard errors for TSLS decrease at a rate of $1/\sqrt T$.
Therefore, long panel standard errors, and hence the identification restriction in (<ref>), are unlikely to prove informative in applications in which $T$ is small.

In the rest of the section, we provide a summary of the long panel asymptotic properties of the TSLS estimator.
Due to the technical nature of the analysis, we relegate formal statements to the appendix and focus instead on providing intuition for the results.
To this end, we begin by noting that the TSLS estimator $\hat \beta_n$ satisfies

\begin{equation}\label{eq:long3}
\hat{\beta}_n  -\beta = \left(\frac{1}{nT} \sum_{t=1}^T\sum_{i=1}^n X_{it}Z_{it}\right)^{-1} \left(\frac{1}{nT} \sum_{i=1}^n \sum_{t=1}^T \Sh_t^\prime S_{it}\varepsilon_{it}\right).
\end{equation}

As the cross section and time series grow, the denominator in (<ref>) converge in probability to some constant ${\rm D}\neq 0$ under standard conditions; i.e., the denominator satisfies

\begin{equation}\label{eq:long4}
\frac{1}{nT} \sum_{t=1}^T\sum_{i=1}^n X_{it}Z_{it} \stackrel{p}{\rightarrow} {\rm D}.
\end{equation}

The asymptotic distribution of the TSLS estimator is therefore governed by the numerator in (<ref>).
In order to derive this asymptotic distribution, we let $\mathcal C_t$ denote all aggregate shocks at time $t$ (which includes $\Sh_t$) and assume that $\{S_{it}\varepsilon_{it}\}_{i=1}^n$ are i.i.d. across $i$ (but not $t$) conditionally on the aggregate shocks $\mathcal C_t$.
Defining the variables

\begin{equation*}%\label{eq:long5}
\zeta_t \equiv E\left[S_{it}\varepsilon_{it}|\mathcal C_t\right], \hspace{0.4 in}  \nu_{it} \equiv S_{it}\varepsilon_{it} - \zeta_t,
\end{equation*}

we then obtain a decomposition into a time series and a panel process by noting that

\begin{equation}\label{eq:long6}
\frac{1}{nT} \sum_{i=1}^n \sum_{t=1}^T \Sh_t^\prime S_{it} \varepsilon_{it} = \frac{1}{T}\sum_{t=1}^T \Sh_t^\prime \zeta_t + \frac{1}{nT}\sum_{t=1}^T \sum_{i=1}^n \Sh_t^\prime \nu_{it} .
\end{equation}

Crucially, the time series process $\{\Sh_t^\prime \zeta_t\}_{t=1}^T$ has mean zero because of the moment restriction in (<ref>), while the panel process $\{\Sh_t^\prime \nu_{it}\}_{i,t=1}^{n,T}$ has mean zero by construction.
Thus, under appropriate restrictions, we should expect both the time series and panel processes to be asymptotically normally distributed.

The preceding discussion is formalized in the appendix, where we establish:

Under the technical conditions presented in Appendix <ref> we have

\begin{equation}
\left(  \frac{1}{\sqrt{T}}\sum_{t=1}^{T}\mathcal{Z}_{t}^{\prime}\zeta
}\nu_{i,t}\right)  \stackrel{d}{\rightarrow} \left( \mathbb G_{\zeta}, \mathbb G_{\nu}\right) \label{vector-of-interest}%
\end{equation}

as $n,T \to \infty$, with $\mathbb G_{\zeta}$ and $\mathbb G_{\nu}$ independent Gaussian random variables.

The long panel asymptotic properties of the TSLS estimator immediately follow from Proposition <ref>.
For instance, combining Proposition <ref> with results (<ref>), (<ref>), and (<ref>) allows us to approximate the variance of $\hat \beta_n$ by the expression

\begin{equation}\label{eq:long7}
\text{Var}\left\{\hat \beta_n - \beta\right\} \approx \left(\frac{1}{{\rm D}}\right)^2\left\{ \frac{1}{T} \text{Var}\{\mathbb G_{\zeta}\} + \frac{1}{nT} \text{Var}\{\mathbb G_\nu\}\right\}.
\end{equation}

In particular, it follows that the long panel standard errors for TSLS decrease at a rate of $1/\sqrt T$.
Intuitively, this dependence on the time dimension results from the dependence of the moment restriction on the distribution of the time series.
One important exception to this phenomenon arises when the time series process has zero variance (i.e. $\text{Var}\{\mathbb G_{\zeta}\} = 0$).
This exception corresponds to the case in which $\zeta_t = 0$ or, equivalently,

\begin{equation}\label{eq:long8}
E[S_{it}\varepsilon_{it}|\mathcal C_t] = 0.
\end{equation}

However, as we discussed in Section <ref>, the conditional moment restriction in (<ref>) is exactly the identifying assumption made by [Goldsmith-Pinkham et al., 2020] that renders the model overidentified.
In summary, the long panel analysis suggests that practitioners should either report standard errors based on (<ref>) or rely on the short panel identification strategies discussed in Section <ref> and report the corresponding overidentification tests instead.

§ OVERIDENTIFICATION TESTS

We next build on our results by developing overidentification tests for the homogeneous effects model studied in Section <ref>.
While we focus on homogeneous effects models due to their demonstrated importance in shift-share designs, we note that it is also possible to test the overidentifying restrictions for heterogeneous effects models derived in Remarks <ref> and <ref> by adapting the insights in, e.g.,  [Bai et al., 2022].

§.§ The Approach

The identification strategies commonly employed for homogeneous effects models yield testable implications in the form of moment equality restrictions.
In this section, we present a general inference approach that is portable across the moment restrictions yielded by different identification strategies.

In what follows, we let $V_{i}\equiv (Y_{i},X_{i},W_{i},S_{i},\mathcal{Z})$ for notational simplicity.
The overidentifying moment restrictions we derived in Section <ref> have the structure

\begin{equation*}%\label{test:eq1}
E[\left.  f\left(  V_{i},\theta\right)  \right\vert \mathcal{G}_{n}]=0,
\end{equation*}

where $\theta$ represent some unknown parameter (e.g., $\beta$) and $f(V_{i},\theta)\in\mathbb{R}^{q}$ is a vector valued function.
In many applications, the number of moment equality restrictions is high dimensional in the sense that it is “large" relative to the sample size.
Because the distributional approximations implicit in chi-squared overidentification tests can be unreliable in high dimensional problems, we instead employ test statistics based on the high dimensional central limit theorem of [Chernozhuokov et al., 2022].
Specifically, for an estimator $\hat \theta_n$ of $\theta$ and $f_j(V_i,\hat \theta_n)$ denoting the $j^{th}$ coordinate of $f(V_i,\hat \theta_n)\in \R^q$ we set

\begin{equation}\label{test:eq2}
T_n \equiv \max_{1\leq j \leq q} \left|\sum_{i=1}^n f_j\left(V_i,\hat \theta_n\right)\right|
\end{equation}

as the test statistic for the null hypothesis that the moment restrictions hold.

The main assumption we impose to construct a test is that the sample moments be asymptotically linear; see Appendix <ref> for a formal statement of the assumptions and results for this section.
Intuitively, we require that for some collection of independent mean zero random variables $\{\psi_i\}_{i=1}^{b_n}\subset \R^q$ the test statistic approximately equals

\begin{equation*}%\label{test:eq3}
T_n \approx \max_{1\leq j \leq q} \left|\sum_{i=1}^{b_n} \psi_{ij}\right|,
\end{equation*}

where $\psi_{ij}$ denotes the $j^{th}$ coordinate of $\psi_i$.
We refer to the number $b_n$ of random variables $\psi_i$ as the “effective number of observations," and note that it need not equal the sample size.
Such a distinction is useful, for example, when the data is clustered (in which case $b_n$ equals the number of clusters) or when relying in the asymptotic framework in [Adao et al., 2019] (in which case $b_n$ equals the number of shocks $p$).

We obtain critical values for our test  by relying on a bootstrap approximation.
Specifically, given estimates $\{\hat \psi_i\}_{i=1}^{b_n}$ for $\{\psi_i\}_{i=1}^{b_n}$, we define a bootstrap statistic

\begin{equation*}%\label{test:eq4}
T_{n}^{\star}\equiv\max_{1\leq j \leq q}\left| \sum_{i=1}^{b_{n}}\omega_{i}\left(  \hat{\psi
}_{ij}-\frac{1}{b_{n}}\sum_{k=1}^{b_{n}}\hat{\psi}_{kj}\right)  \right|,
\end{equation*}

where $\hat \psi_{ij}$ denotes the $j^{th}$ coordinate of $\hat \psi_i \in \R^q$ and the random weights $\{\omega_i\}_{i=1}^{b_n}$ are drawn independently of the data $\{V_i\}_{i=1}^n$.
For instance, we may set the bootstrap weights $\{\omega_i\}_{i=1}^{b_n}$ to be i.i.d. and drawn from a standard normal or Rademacher distribution.[The Rademacher distirbution corresponds to setting $\omega_i$ to satisfy $P(\omega_i = 1) = P(\omega_i=-1) = 1/2$.]
The bootstrap critical value for a level $\alpha$ test is then given by the bootstrap quantile

\begin{equation*}%\label{test:eq5}
\hat{c}_{n}\equiv\inf\{c:P\left(  \left.  T_{n}^{\ast}\leq c\right\vert
\{V_{i}\}_{i=1}^{n}\right)  \geq1-\alpha\};
\end{equation*}

i.e. we employ the $1-\alpha$ quantile of $T_n^*$ conditional on the data $\{V_i\}_{i=1}^n$ but unconditionally on the bootstrap weights $\{\omega_i\}_{i=1}^n$.
The critical value $\hat c_n$ can as usual be obtained through simulation by employing multiple draws of the bootstrap weights $\{\omega_i\}_{i=1}^n$ to approximate the distribution of $T_n^*$ conditionally on the data.

Our next result establishes that a test that rejects whenever $T_n$ exceeds the critical value $\hat c_n$ has the correct asymptotic size.
We defer the statement of the relevant regularity conditions to Appendix <ref>, though note that they allow for the number of moment restrictions $q$ to grow together with the effective sample size $b_n$.

Under the regularity conditions discussed in Appendix <ref>, we have

$$\lim_{n\to \infty} P(T_{n}>\hat{c}_{n})=\alpha.$$

§.§ Conditioning on Shocks

The general approach discussed in the preceding section can readily be specialized to develop an overidentification test for application that implicitly condition on aggregate shocks in their analysis.
Recall that such applications require the restriction

\begin{equation*}%\label{test:gss1}
E\left[ S_i\varepsilon_i|\mathcal G_n\right] = 0,
\end{equation*}

where $\mathcal G_n$ is understood to contain all aggregate variables, including $\Sh$ [Goldsmith-Pinkham et al., 2020].
As our test statistic we therefore employ

\begin{equation}\label{test:gss2}
T_n = \max_{1\leq j \leq p} \left|\frac{1}{\hat \sigma_j} \sum_{i=1}^n S_{ij}\hat \varepsilon_i\right|,
\end{equation}

where $S_{ij}$ denotes the $j^{th}$ coordinate of $S_i\in \R^p$, $\{\hat \varepsilon_i\}_{i=1}^n$ are the residuals from the estimated model, and $\hat \sigma_j$ is an estimated weight that allows us to ensure that all $p$ moments are on a comparable scale.

In order to map this setting into the overidentification test of the preceding section, we simply set $f_j(V_i,\hat \theta_n) = S_{ij}\hat \varepsilon_i/\hat \sigma_j$ so that the test statistic in (<ref>) becomes a special case of (<ref>).
Setting $A_i \equiv (Z_i,W_i^\prime)^\prime$, $\sigma_j$ to be the probability limit of $\hat \sigma_j$, and defining

\begin{equation*}%\label{test:gss3}
\psi_{ij} = \frac{U_{ij}}{\sigma_j} \hspace{0.3 in} U_{ij} \equiv \left(S_{ij}\varepsilon_{i}-E[\left.  S_{ij}\left(  X_{i},W_{i}^{\prime}\right)  \right\vert \mathcal{G}_{n}]\left(  E[A_{i}\left(
X_{i},W_{i}^{\prime}\right)  |\mathcal{G}_{n}]\right)  ^{-1}A_{i}\varepsilon
\end{equation*}

it is then possible to show that, under the null hypothesis, $T_n$ approximately equals

\begin{equation*}%\label{test:gss4}
T_n \approx \max_{1\leq j \leq p} \left|\sum_{i=1}^n \psi_{ij}\right|
\end{equation*}

provided the observations $\{Y_i,S_i,X_i\}_{i=1}^n$ are i.i.d. conditionally on $\mathcal G_n$.[
See the supplemental appendix for calculations supporting the claims in this section.]
Our inference approach requires an estimator for the influence function $\psi_i$ and to this end we set

\begin{equation*}%\label{test:gss5}
\hat{\psi}_{ij}    \equiv \frac{\hat U_{ij}}{\hat\sigma_j} \hspace{0.3 in} \hat U_{ij} \equiv \left( S_{ij}\hat{\varepsilon}_{i}-\left(  \frac{1}{n}%
\sum_{k=1}^{n}S_{kj}\left(  X_{k},W_{k}^{\prime}\right)  \right)  \left(
\frac{1}{n}\sum_{k=1}^{n}A_{k}\left(  X_{k},W_{k}^{\prime}\right)  \right)
\end{equation*}

We also let $\hat \sigma_j^2$ be an estimator of the asymptotic variance of the $j^{th}$ moment by setting

\begin{equation*}%\label{test:gss6}
\hat \sigma_j^2 = \frac{1}{n}\sum_{i=1}^n \left(\hat U_{ij} - \frac{1}{n}\sum_{k=1}^n \hat U_{kj}\right)^2.
\end{equation*}

Following the approach in the preceding section, we obtain critical values for our test statistic by: (i) Drawing $b\in \{1,\ldots, B\}$ samples $\{\omega_i^{(b)}\}_{i=1}^n$ of standard normal random variables independent of the data; (ii) For each drawn sample $\{\omega_i^{(b)}\}_{i=1}^n$ computing

\begin{equation*}%\label{test:gss7}
T_n^{*(b)} \equiv \max_{1\leq j \leq p} \left|\sum_{i=1}^n \omega_i^{(b)} \left(\hat \psi_{ij} - \frac{1}{n}\sum_{k=1}^{n} \hat \psi_{kj}\right)\right|;
\end{equation*}

and (iii) For a level $\alpha$ test, letting $\hat c_{1-\alpha}$ denote the $1-\alpha$ quantile of $\{T_n^{*(b)}\}_{b=1}^B$.
Our overidentification test then rejects whenever $T_n$ exceeds the critical value $\hat c_{1-\alpha}$.

The proposed test can easily be adapted to account for clustered data.
Specifically, suppose observations are divided into clusters $\{c_1,\ldots, c_{|C|}\}\equiv C$ – e.g., $C$ may represent all states, $|C|$ the total number of states, and each cluster $c\in C$ a specific state.
While the test statistic remains the same, for our bootstrap implementation we now set $\hat \psi_{cj} \equiv \sum_{i\in c} \hat U_{ij}/\hat\sigma_j$ for $\hat \sigma_j^2$ the cluter robust sample variance of $\{\hat U_{ij}\}_{i=1}^n$, and let

\begin{equation*}%\label{rm:gsscluster1}
T_n^* \equiv \max_{1\leq j \leq p} \left|\sum_{c \in C} \omega_c \left(\hat \psi_{cj} - \frac{1}{|C|} \sum_{\tilde c \in C} \hat \psi_{\tilde c j}\right)\right|,
\end{equation*}

with $\{\omega_i\}_{c\in C}$ an i.i.d. sample of standard normal random variables drawn independently of the data – i.e., the bootstrap procedure is modified by simply employing the same draw of $\omega_c$ for all observations in the same cluster.
We note that this procedure maps into the notation of Section <ref> by letting $b_n$ equal the number of clusters.

The proposed test can easily be adapted to incorporate practical challenges encountered in applications.
Suppose, for instance, that the data has sample weights $\{w_i\}_{i=1}^n$ and is divided into clusters $\{c_1,\ldots, c_{|C|}\}\equiv C$ – e.g., $C$ may represent all states, $|C|$ the total number of states, and each cluster $c\in C$ a specific state.
In order to incorporate the sample weights into our test statistic, we simply set $T_n$ to equal

\begin{equation}\label{rm:gsscluster1}
T_n \equiv \max_{1\leq j \leq p} \left|\frac{1}{\hat \sigma_{j}} \sum_{i=1}^n w_{i} S_{ij}\hat \varepsilon_{i}^{w} \right|.
\end{equation}

where $\{\hat \varepsilon_i^w\}_{i=1}^n$ denotes the residuals from a weighted estimation of the model.
Similarly, to estimate the influence function of each cluster we set $\hat \psi_{cj} \equiv \sum_{i \in c} \hat U_{ij}/\hat\sigma_j$ with

\begin{equation}\label{rm:gsscluster2}
\hat U_{ij} \equiv \left( w_iS_{ij}\hat{\varepsilon}^w_{i}-\left(\sum_{i=1}^{n} w_iS_{ij}\left(  X_{i},W_{i}^{\prime}\right)  \right)  \left(
\sum_{i=1}^{n} w_iA_{i}\left(  X_{i},W_{i}^{\prime}\right)  \right)
^{-1}w_iA_{i}\hat{\varepsilon}_{i}^w \right).
\end{equation}

The appropriate critical value for a level $\alpha$ test is then given by the conditional $1-\alpha$ quantile (given the data) of the bootstrap statistic $T_n^*$ defined by

\begin{equation}\label{rm:gsscluster3}
T_n^* \equiv \max_{1\leq j \leq p} \left|\sum_{c \in C} \omega_c \left(\hat \psi_{cj} - \frac{1}{|C|} \sum_{\tilde c \in C} \hat \psi_{\tilde c j}\right)\right|,
\end{equation}

where $\{\omega_i\}_{c\in C}$ is an i.i.d. sample of standard normal random variables drawn independently of the data – i.e. we simply employ the same draw of $\omega_c$ for all observations in the same cluster.
Finally, we note that $\hat \sigma_j^2$ should also be modified to now equal the sample variance of $\{\hat \psi_{cj}\}_{c\in C}$ across clusters.

§.§ Identification Through Shocks

We conclude our discussion of overidentification tests by specializing our approach to applications in which the exogeneity of the Bartik instrument originates from the exogeneity of the shocks.
Recall from Section <ref> that the asymptotic framework designed for such applications relies on setting $\mathcal G_n \equiv \{S_i,W_i,\varepsilon_i\}_{i=1}^n$, which yields the restriction

\begin{equation*}%\label{test:akm1}
E[Z_i|\{S_i,W_i,\varepsilon_i\}_{i=1}^n] = W_i^\prime \pi_n
\end{equation*}

for $\pi_n$  the population regression coefficient from regressing $\{Z_i\}_{i=1}^n$ on $\{W_i\}_{i=1}^n$ conditionally on $\mathcal G_n$ (as in (<ref>)).
As our overidentification test statistic we therefore employ

\begin{equation}\label{test:akm2}
T_n = \max_{1\leq j \leq q} \left|\frac{1}{\hat \sigma_j} \sum_{i=1}^n g_j(\hat \varepsilon_i,W_i,S_i)(Z_i - W_i^\prime \hat \pi_n)\right|,
\end{equation}

where $\{\hat \varepsilon_i\}_{i=1}^n$ are the residuals from the estimated model, $\hat \pi_n$ is the regression coefficient from regressing $\{Z_i\}_{i=1}^n$ on $\{W_i\}_{i=1}^n$, $\hat \sigma_j$ is again an estimated weight, and $g_j(\hat \varepsilon_i,W_i,S_i)$ denotes the $j^{th}$ coordinate of the vector valued function $g(\hat \varepsilon_i,W_i,S_i)\in \R^q$.

In order to obtain a suitable critical value, we first need a characterization of the influence function for each sample moment.
To this end, we define

\begin{align*}%\label{test:akm3}
\delta_j & \equiv \left(\sum_{i=1}^n W_iW_i^\prime\right)^{-1}\sum_{i=1}^n W_{i} g_j(\varepsilon_i,W_i,S_i) \notag \\
\kappa_j & \equiv \left(\sum_{i=1}^n E\left[S_i^\prime \mathcal E X_i|\mathcal G_n\right]\right)^{-1} \left(\sum_{i=1}^n E\left[S_i^\prime \mathcal E X_i|\mathcal G_n\right]\frac{\partial} {\partial \varepsilon}g_j(\varepsilon_i,W_i,S_i)\right)
\end{align*}

with $\mathcal E \equiv \Sh - E[\Sh|\mathcal G_n]$, and for $\sigma_j$ the probability limit of $\hat \sigma_j$ we let $\psi_{ij}$ be given by

\begin{equation*}%\label{test:akm4}
\psi_{ij} \equiv \frac{U_{ij}}{\sigma_j} \hspace{0.3 in} U_{ij} \equiv  \mathcal E_i \times \sum_{k=1}^n S_{ki}(g_j(\varepsilon_k,W_k,S_k) - W_k^\prime \delta_j - \varepsilon_k \kappa_j).
\end{equation*}

Under conditions similar to those employed by [Adao et al., 2019], it is then possible to show that under the null hypothesis the statistic $T_n$ satisfies the approximation[See the supplemental appendix for calculations supporting the claims in this section.]

\begin{equation*}%\label{test:akm5}
T_n \approx \max_{1\leq j \leq q} \left|\sum_{i=1}^p \psi_{ij}\right|.
\end{equation*}

We note that, because identification is driven by the exogeneity of the shocks, the effective number of observations equals $p$ and not $n$ – i.e. the sample $\{\psi_{i}\}$ is of size $p$.

Our bootstrap critical value also relies on an estimate of $\psi_i$, and to this end we define

\begin{align*}%\label{test:akm6}
\hat \delta_j & \equiv \left(\sum_{i=1}^n W_iW_i^\prime\right)^{-1}\sum_{i=1}^n W_i g_j(\hat \varepsilon_i,W_i,S_i)\\
\hat \kappa_j & \equiv \left(\sum_{i=1}^n (Z_i - W_i^\prime \hat \pi_n)X_i\right)^{-1}\left(\sum_{i=1}^n (Z_i - W_i^\prime \hat \pi_n)X_i \frac{\partial}{\partial \varepsilon} g_j(\hat \varepsilon_i,W_i,S_i)\right),
\end{align*}

and let $\hat {\mathcal E}\in \R^p$ denote a suitable estimator for $\mathcal E\equiv \mathcal Z - E[\mathcal Z|\mathcal G_n]$; see Remark <ref> for possible choices of $\hat {\mathcal E}$.
As our estimator for the influence function $\psi_{ij}$ we then set

\begin{equation*}%\label{test:akm7}
\hat \psi_{ij}\equiv \frac{\hat U_{ij}}{\hat \sigma_j} \hspace{0.3 in} \hat U_{ij} \equiv \hat {\mathcal E}_i \times \sum_{k=1}^n S_{ki}(g_j(\hat \varepsilon_k,W_i,S_i) - W_k^\prime \hat \delta_j - \hat \varepsilon_k \hat \kappa_j).
\end{equation*}

Finally, for the purposes of studentizing the sample moments we set $\hat \sigma_j^2$ to be given by

\begin{equation*}%\label{test:akm8}
\hat \sigma_j^2 \equiv \frac{1}{p}\sum_{i=1}^p \left(\hat U_{ij}-\frac{1}{p}\sum_{k=1}^p\hat U_{kj}\right)^2.
\end{equation*}

Given the estimates $\{\hat \psi_{i}\}$, we can obtain asymptotically valid critical values by following the same procedure as in the preceding section.
Specifically, we: (i) Draw $b\in \{1,\ldots, B\}$ samples $\{\omega_i^{(b)}\}_{i=1}^p$ of standard normal random variables independent of the data; (ii) For each drawn sample $\{\omega_i^{(b)}\}_{i=1}^p$ we compute the bootstrap statistic

\begin{equation*}%\label{test:akm9}
T_n^{*(b)} \equiv \max_{1\leq j \leq q} \left|\sum_{i=1}^p \omega_i^{(b)} \left(\hat \psi_{ij} - \frac{1}{p}\sum_{k=1}^{p} \hat \psi_{kj}\right)\right|;
\end{equation*}

and (iii) For a level $\alpha$ test, we let $\hat c_{1-\alpha}$ denote the $1-\alpha$ quantile of $\{T_n^{*(b)}\}_{b=1}^B$.
Our overidentification test then rejects whenever $T_n$ exceeds the critical value $\hat c_{1-\alpha}$.

Multiple estimates $\hat {\mathcal E}$ for $\mathcal E = \mathcal Z - E[\mathcal Z|\mathcal G_n]$ have been previously studied in the literature.
[Adao et al., 2019], for instance, propose the estimator

\begin{equation}\label{rm:akmtest1}
\hat {\mathcal E} \equiv \left(\sum_{i=1}^n S_{i}S_{i}^\prime\right)^{-1}\sum_{i=1}^n S_{it}(Z_{it}-W_{it}^\prime \hat \pi_n).
\end{equation}

Alternatively, in applications in which $W_i$ contains variables equal to $\mathcal Q^\prime S_i$ for some $p\times k$ observable matrix of aggregate variables $\mathcal Q$, [Borusyak et al., 2022] advocate setting

\begin{equation}\label{rm:akmtest2}
\hat {\mathcal E} \equiv \Sh - \mathcal Q(\mathcal Q^\prime \mathcal Q)^{-1}\mathcal Q^\prime \Sh
\end{equation}

In our empirical analysis of [Autor et al., 2013] we employ a ridge regression version of (<ref>) because: (i) The matrix $\sum_i S_{i}S_{i}^\prime$ is ill-conditioned in that application,[[Adao et al., 2019] instead address this ill-conditioning by dropping sectors from the shares vector.] which prevents the use of (<ref>); and (ii) We focus on the specification employed in [Autor et al., 2013], which does not contain controls with the structure required by (<ref>).

§ EMPIRICAL APPLICATION: THE CHINA SYNDROME

We illustrate the empirical relevance of our analysis by revisiting the seminal work of [Autor et al., 2013] examining the impact of rising Chinese import competition on local US markets.
We focus on the main specification of [Autor et al., 2013], in which

\begin{equation}\label{sec:china1}
Y_{it} = X_{it}\beta + W_{it}^\prime \gamma_{\mathtt s} + \varepsilon_{it}
\end{equation}

with $i$ indexing a commuting zone (CZ), $t$ indexing one of two time periods, $Y_{it}$ and $X_{it}$ denoting the change in employment manufacturing and a measure of import exposure, and $W_{it}$ denoting a set of CZ level characteristics.[Specifically, we employ the specification in column (6) of Table 3 in [Autor et al., 2013].]
[Autor et al., 2013] estimate the model in (<ref>) by TSLS based on a Bartik instrument $Z_{it} = \Sh_t^\prime S_{it}$ in which $S_{it}$ and $\Sh_t$ consist of a vector of lagged industry shares and a vector of Chinese imports growth to other countries for different industries.
Their analysis also employs four digit SIC codes to define sectors, leading to a total of $p = 397$ sectors.

The standard errors in [Autor et al., 2013] rely on an asymptotic approximation that implicitly conditions on the shocks.
As we argued in Section <ref>, such standard errors corresponds to an identification strategy in which shares are viewed as exogeneous; i.e.
\begin{equation}\label{sec:china2}
E[S_{it}\varepsilon_{it}]  = 0 \text{ for all } t.
\end{equation}

We therefore begin by examining the viability of a constant treatment effects model in this application by employing the overidentification test of Section <ref>.
Following [Autor et al., 2013], we conduct inference by clustering at the state level leading to only 48 effective observations – a number much smaller than the $p\times T = 794$ number of moment restrictions represented by (<ref>).
In a Monte Carlo study based on this dataset we found the finite sample rejection probabilities of our test to be below the nominal level when employing all 794 restrictions; see the Supplemental Appendix.
We therefore also test the validity of moment restrictions implied by (<ref>) by aggregating shares to higher level SIC codes and focusing on particular time periods.

Overidentification tests corresponding to conditioning on shocks

1cMoment Restrictions         $\#$ Moments         p-value
Four Digit SIC $\&$ all time periods            794                  0.127

Four Digit SIC $\&$ $t=1$                       397                  0.115

Four Digit SIC $\&$ $t=2$                       397                  0.098

Three Digit SIC $\&$ all time periods            272                  0.049

Three Digit SIC $\&$ $t=1$                       136                  0.009

Three Digit SIC $\&$ $t=2$                       136                  0.072

Two Digit SIC $\&$ all time periods            40                  0.005

Two Digit SIC $\&$ $t=1$                       20                  0.002

Two Digit SIC $\&$ $t=2$                       20                  0.366

Table reports p-values for overidentification tests of constant treatment effects models identified by implicitly conditioning on the realization of the shocks; see Section <ref>.
All tests are implemented by clustering at the state level and weighting commuting zones by their start of period population as in [Autor et al., 2013].

Table <ref> reports the p-values for our overidentification tests under different choices of moment restrictions.
Overall, our analysis yields evidence against the validity of the constant effects model under an identification strategy that implicitly conditions on shocks.
[Goldsmith-Pinkham et al., 2020] reach a similar conclusion, though we note that the overidentification tests they employ do not cluster at the state level and are not robust to a “large" number of moment restrictions.
Given the implausibility of the constant treatment effects model, it is natural to ask whether TSLS can be attributed a causal interpretation through a heterogeneous treatment effects model instead.
Unfortunately, Corollary <ref> establishes that such an interpretation necessitates that shares be uncorrelated across sectors – a requirement readily contradicted by the data, which exhibits correlations for some sectors in excess of 0.9.
We note, however, that as advocated by [Goldsmith-Pinkham et al., 2020] it may still be possible to proceed by employing the shares as instruments without combining them into a Bartik instrument.

We next examine the viability of a constant effects model under an identification strategy in which the exogeneity of the Bartik instrument is due to the exogeneity of the shocks [Adao et al., 2019, Borusyak et al., 2022].
To this end, we implement the overidentification test of Section <ref>, which requires us to select a vector of moments ($\{g_j\}$ in (<ref>)) and an estimator for $\mathcal E \equiv \mathcal Z- E[\mathcal Z|\mathcal G_n]$ (as in Remark <ref>).
For the former, we select a total of twenty moments and let the functions $\{g_j\}$ depend on $\varepsilon$ only.
In particular, we select one function $g_j$ to simply equal $\varepsilon^2$, and the other nineteen to equal the Logit pdf centered at equispaced points in the support of $\varepsilon$; see the Supplemental Appendix for details.
Finally, as discussed in Remark <ref>, we estimate $\mathcal E$ through Ridge regression and follow [Borusyak et al., 2022] in clustering at the three level SIC code.

Overidentification tests corresponding to identification through shocks

4cRidge Penalty
1e-3        1e-4          1e-5          1e-6
p-value             0.0012      0.0074        0.0368        0.065

Table reports p-values for overidentification tests of constant treatment effects models identified through the exogeneity of shocks; see Section <ref>.
All tests are implemented by clustering at the three digit SIC level and weighting commuting zones by their start of period population as in [Autor et al., 2013].

Table <ref> reports the p-values for our test corresponding to different choice of Ridge penalties.
Our results show some sensitivity of the p-value to the choice of Ridge penalty.
This conclusion echoes [Adao et al., 2019] who similarly find that the manner in which they address the singularity of the shares design matrix affects their standard errors.[[Adao et al., 2019] addressed the singularity of $\sum_i S_iS_i^\prime$ by dropping sectors from their analysis. As stated by the authors, their standard errors are affected by what sectors are dropped; see page 3 in https://github.com/kolesarm/ShiftShareSE/blob/master/doc/ShiftShareSE.pdf]
Overall, we interpret Table <ref> as providing evidence against the plausibility of the constant effects model under the identification strategy that attributes the exogeneity of the Bartik instruments to the shocks.
Our preferred choice of penalty for this application is 1e-5, yielding a p-value of 0.0368, which in a Monte Carlo study based on this dataset yielded finite sample rejection probabilities closest to a $5\%$ nominal level; see the Supplemental Appendix.
It is also worth emphasizing that clustering at the three digit SIC level has important implications for whether TSLS can be attributed a causal interpretation in a heterogeneous effects model.
For example, Corollary <ref> implies that such a causal interpretation necessitates all industries within the same 3-digit SIC code to be (weakly) positively correlated.
It is important to note, however, that the latter requirement is necessary but not sufficient for TSLS to have a causal interpretation; see Remark <ref> for additional restrictions.

Finally, we note that the “just identified" long panel identification strategy of Section <ref> is not viable in this application.
Specifically, because there are only two time periods in [Autor et al., 2013], it is not possible to learn the time series distribution of the shocks as required by such an approach.
In other words, it would be foolhardy to invoke an asymptotic approximation based on $T$ diverging to infinity when $T$ equals two.

§ SUMMARY AND RECOMMENDATIONS

In this paper, we examined the testable implications of identifying restrictions employed to assign a causal interpretation to TSLS based on Bartik instruments.
We developed specification tests for homogeneous effects models that are robust to heteroskedasticity, clustering, and weighting.
Because our tests are based on the high dimensional central limit theorem, we expect them to be more robust than their alternatives to the “high" degree of overidentification present in shift-share designs.
Finally, we argued that our overidentification tests are of central importance due to the potentially limited empirical scope of the natural alternatives to the homogeneous effects models.

Our analysis has a number of important implications for applied work:

* We recommend that practitioners report overidentification tests corresponding to the identification strategy of [Goldsmith-Pinkham et al., 2020] as well as that of [Adao et al., 2019] and [Borusyak et al., 2022].
These tests can provide evidence as to which identification strategy is more compatible with the data.

* Practitioners relying on the identification strategy of [Goldsmith-Pinkham et al., 2020] should be wary of assigning a causal interpretation to TSLS by relying on a heterogenous effects model.
Such an interpretation necessitates that shares be uncorrelated across sectors, which is empirically untenable.
On the other hand, as advocated by [Goldsmith-Pinkham et al., 2020], it may still be possible to employ the shares for causal inference by not combining them into a Bartik instrument.

* Practitioners should likewise be careful in assigning TSLS a causal interpretation by relying on a heterogeneous effects model under the identification strategy of [Adao et al., 2019] and [Borusyak et al., 2022].
A necessary condition for TSLS to be causal is that shocks to different sectors not be negatively correlated.
Reasoning through the correlation structure of shocks is thus important not only for the purposes of obtaining standard errors, but also for interpreting the TSLS estimand.

* In applications with long panels, it is possible to rely on an alternative identification strategy in which the randomness of the instrument is understood jointly over the cross sectional distribution of shares and the time series distribution of shocks.
Practitioners relying on this approach, however, need to employ different standard errors that are governed by the length of the panel.

This Appendix contains the proofs of all the results in the main text.

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§ PROOFS FOR SECTION <REF>

§ PROOFS FOR SECTION <REF>

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Supplemental Appendix

Jinyong Hahn
Guido Kuersteiner
University of Maryland
Andres Santos
Wavid Willigrod

This supplemental appendix includes: (i) Calculations that justify the asymptotic validity of the proposed overidentifiation tests; and (ii) A set of Monte Carlo experiments evaluating the finite sample performance of such tests.

§ INFLUENCE FUNCTIONS

In this section, we discuss how to verify the conditions of Lemma <ref>, and hence Proposition <ref>, in the context of the overidentification tests of Sections <ref> and <ref>.

§.§ Conditioning on Shocks

§.§ Identification Through Shocks

§ SIMULATION EVIDENCE

We next conduct a series of Monte Carlo simulations to evaluate the finite sample performance of the overidentification tests proposed in Section <ref>.
With the goal of informing the implementation of our tests in the empirical application of Section <ref>, we employ simulation designs based on the [Autor et al., 2013] dataset.
In particular, as in [Autor et al., 2013], our designs consist of short panels with $T = 2$ time periods, $n = 722$ commuting zones, and $p = 397$ sectors defined by four digit SIC codes.

In what follows, we incorporate the short panel structure into our notation by letting $Y_{it}$, $X_{it}$, $W_{it}$, and $Z_{it}$ respectively denote the outcome, regressor, controls, and instrument for commuting zone $i$ at time periods $t$.
We also note that in [Autor et al., 2013] both the regressor $X_{it}$ and instrument $Z_{it}$ have a Bartik structure and hence we now index shares and aggregate shocks by subscripts $x$ and $z$.
Concretely, we have

\begin{equation}\label{sec:mc1}
X_{it} = S_{xit}^\prime \Sh_{xt} \hspace{0.5 in} Z_{it} = S_{zit}^\prime \Sh_{zt}
\end{equation}

where $S_{xit}$ and $S_{zit}$ represent share vectors for commuting zone $i$ at time $t$ and $\Sh_{xt}$ and $\Sh_{zt}$ denote aggregate shocks at time $t$.
Finally, because [Autor et al., 2013] weight all observations by the start of period commuting zone population, in our simulations we employ the same weights throughout the analysis.

§.§ Conditioning on Shocks

We begin by examining the finite sample performance of the overidentification test proposed in Section <ref>, which recall was designed for applications that implicitly condition on the aggregate shocks – i.e. that employ asymptotic approximations based on only $n$ growing.
As discussed in Remark <ref>, implicitly conditioning on aggregate shocks in short panels yields the overidentifying moment restrictions

\begin{equation}\label{mc:gss1}
E[S_{zit}\varepsilon_{it}] = 0 \hspace{0.5 in} \text{ for } 1\leq t \leq T.
\end{equation}

Since $T =2$ and $p = 397$ in the context of [Autor et al., 2013], result (<ref>) represents a total of 794 possible moment restrictions.
Moreover, because [Autor et al., 2013] cluster observations at the state level, their effective number of observations is 48.

In designing our simulations, we aimed to reflect the clustering structure in [Autor et al., 2013] by employing a heteroskedastic version of the group shock model of [Moulton, 1986].
To this end, we let $c$ denote a cluster, which consists of the commuting zone time pairs $(i,t)$ for which $i$ belongs to the state represented by $c$, and let $C$ denote the collection of all clusters.
Employing the [Autor et al., 2013] dataset, we then estimate a model in which the errors $\varepsilon_{it}$ are assumed to have the structure

\begin{equation*}%\label{mc:gss2}
\varepsilon_{it} = \eta_c + \zeta_{it}
\end{equation*}

where $\eta_{c}$ are i.i.d. cluster level shocks and $\zeta_{it}$ are i.i.d. shocks and independent of $\eta_{c}$.
We further impose a parsimonious heteroskedasticity specification by supposing that

\begin{align*}%\label{mc:gss3}
E[\eta_c^2|\mathcal A_n] & = a_\eta + s_\eta (\sum_{(i,t)\in c} S^\prime_{zit}S_{zit}) \notag \\
E[\zeta_{it}^2|\mathcal A_n] & = a_\zeta + s_\zeta (S_{zit}^\prime S_{zit})
\end{align*}

for some constants $a_\eta,s_\eta,a_\zeta,$ and $s_\zeta$ and $\mathcal A_n \equiv \{S_{zit},S_{xit},W_{it}\}$.
In order to estimate this model, we employ the fitted residuals $\{\hat e_{it}\}$ from the weighted instrumental variable estimation in the main specification of [Autor et al., 2013] and let

\begin{align}\label{mc:gss4}
(\hat a_\eta,\hat s_\eta) & \equiv \arg\min_{a,s\in \mathbf R} \sum_{c\in C}     \sum_{(it)\neq (\tilde i,\tilde t) \in c} (\hat e_{it} \hat e_{\tilde i \tilde t} - a- s(\sum_{(i^\prime t^\prime)\in c} S_{zi^\prime t^\prime}^\prime S_{zi^\prime t^\prime}))^2 \notag \\
(\hat a_\zeta, \hat s_\zeta) & \equiv \arg\min_{a,s \in \mathbf R} \sum_{c\in C}\sum_{(i,t)\in c} (\hat e_{it}^2 - \hat \sigma_{\eta,c}^2 - a - s(S^\prime_{zit} S_{zit}))^2
\end{align}

\begin{equation*}%\label{mc:gss5}
\hat \sigma_{\eta,c}^2 \equiv \max\{\hat a_\eta + \hat s_\eta (\sum_{(i, t)\in c} S_{zi t}^\prime S_{zi t}),0\}.
\end{equation*}

Given these estimates, we generate our Monte Carlo samples as follows:

Step 1. We employ the same controls $\{W_{it}\}$, aggregate shocks $\mathcal{Z}_{zt}$ and $\mathcal{Z}_{xt}$, and regression weights as in the main specification of [Autor et al., 2013], which we keep fixed across all the simulations.

Step 2. For each $t\in \{1,2\}$ we draw $n$ observations $\{S_{xit}^{\ast
},S_{zit}^{\ast}\}_{i=1}^n$ with replacement from the original full sample set of shares $\{S_{xit},S_{zit}\}_{i=1}^n$.

Step 3. Given the sample $\{S_{xit}^*,S_{zit}^*\}$, we create a sample of instruments $\{Z_{it}^{\ast}\}$ and endogenous variables $\{X_{it}^\ast\}$ by setting $Z_{it}^{\ast}\equiv (S_{zit}^{\ast})^{\prime}\mathcal{Z}_{zt}$ and letting $X_{it}^{\ast}\equiv (S_{xit}^{\ast})^{\prime}\mathcal{Z}_{xt}$.

Step 4. For each commuting zone time pair $(i,t)$ and cluster $c$ we create the variances

\begin{align*}%\label{mc:gss6}
(\hat \sigma_{\zeta,it}^*)^2 & \equiv \max\{\hat a_\zeta + \hat s_\zeta (S_{zit}^*)^\prime S_{zit}^*,0\}   \\
(\hat \sigma_{\eta,c}^*)^2 & \equiv \max\{\hat a_\eta + \hat s_\eta \sum_{(i,t)\in c}(S^*_{zit})^\prime S_{zit}^*,0\}
\end{align*}

by employing the full sample estimates $\hat a_\eta,\hat s_\eta,\hat a_\zeta$ and $\hat s_\zeta$ from (<ref>).

Step 5. To create a sample of outcomes for our simulations, we draw $|C|$ i.i.d. standard normal variables $\{V_c\}_{c\in C}$, $n\times T$ i.i.d. standard normal variables $\{U_{it}\}$, and set

\begin{equation*}%\label{mc:gss7}
Y_{it}^* \equiv X_{it}^* \hat \beta + W_{it}^\prime \hat \gamma_{\mathtt{s}} + V_c \hat \sigma_{\eta,c}^* + U_{it} \hat \sigma^*_{\zeta,it},
\end{equation*}

where $(\hat \beta,\hat \gamma_{\mathtt{s}})$ denote the weighted instrumental variable estimators from the main specification in [Autor et al., 2013] and the “$c$" subscript is understood to refer to the state to which commuting zone $i$ belongs.

By repeating Steps 1-5 we generate one thousand samples $\{Y_{it}^*,X_{it}^*,Z_{it}^*,W_{it}\}$ on which we evaluate the finite sample properties of our test.
We note that the number of moment restrictions ($p\times T =794$) in (<ref>) far exceeds the number of clusters in the simulations ($48$ states).
Since any linear combination of the moment restrictions in (<ref>) is also a valid moment restriction, we also examine the performance of our test when adding restrictions across time periods and/or different levels of SIC codes.
Table <ref> reports the finite sample rejection probabilities for tests based on different choices of moments and significance levels.
The final column of Table <ref> additionally reports the p-value obtained when the test is implemented in the data of [Autor et al., 2013].
All critical values were obtained by following the procedure in Section <ref> with one thousand bootstrap draws.

Rejection Probabilities

3cSignificance Level

Moment Restrictions                                $\#$ Moments         $1\%$       $5\%$         $10\%$      p-value
Four Digit SIC $\&$ all time periods        794                  0.000       0.004          0.013     0.1274

Four Digit SIC $\&$ time aggregated         397                  0.000       0.008          0.031      0.168

Three Digit SIC $\&$ all time periods       272                  0.000       0.007          0.029     0.0488

Three Digit SIC $\&$ time aggregated        136                  0.000       0.007         0.025     0.0748

Two Digit SIC $\&$ all time periods         40                   0.001       0.030         0.072      0.0054

Two Digit SIC $\&$ time aggregated          20                   0.004       0.034         0.095     0.0018

Finite sample rejection probabilities for overidentification tests of the validity of the moment restrictions imposed when asymptotic approximation implicitly condition on aggregate shocks.

Overall we find that the test is able to control size across all specifications.
However, for larger values of the number of moments, the finite sample rejection probability of the test is significantly below its nominal level.
The test performs best when aggregating across time periods and two digit SIC codes.
For this specification, which consists of twenty moments, the finite sample rejection probabilities of the test are close to the nominal levels.
Because the design only has 48 clusters, we view this specification as still employing a large number of moments relative to the sample size.

§.§ Identification Through Shocks

We conclude by examining the finite sample performance of the overidentification test proposed in Section <ref>, which was designed for applications in which the exogeneity of the instrument is due to the exogeneity of the aggregate shocks.
Recall that in these applications $\mathcal G_n = \{S_{zit},S_{xit},W_{it},\varepsilon_{it}\}$ and the overidentifying restriction is given by

\begin{equation}\label{mc:akm1}
E[Z_{it}|\mathcal G_n] = W_{it}^\prime \pi_n
\end{equation}

\begin{equation*}%\label{mc:akm2}
\pi_n = (\sum_{t=1}^T\sum_{i=1}^n W_{it}W_{it}^\prime)^{-1}\sum_{t=1}^T\sum_{i=1}^n W_{it} (S_{zit}^\prime E[Z_{it}|\mathcal G_n]).
\end{equation*}

In order to ensure that the null hypothesis holds in our simulation design, we rely on a model proposed by [Adao et al., 2019] as a sufficient condition for (<ref>).
Specifically, we suppose that for some $p\times d$ matrix of shock $\Sh_{wt}$ and $d\times 1$ vector $\Gamma$ we have

\begin{equation}\label{mc:akm3}
E[\Sh_{zt}|\mathcal G_n] = \Sh_{wt} \Gamma \hspace{0.5 in} W_{it} = \Sh_{wt}^\prime S_{zit}.
\end{equation}

To estimate this model in the original [Autor et al., 2013] dataset, we first compute a ridge regression of the coordinates of $W_{it}\in \R^{d}$ on $S_{zit}$ by setting

\begin{equation*}%\label{mc:akm4}
\hat \delta_{jt} \equiv (\sum_{i=1}^n S_{zit}S_{zit}^\prime +  \lambda I_p)^{-1}\sum_{i=1}^n S_{zit} W_{itj},
\end{equation*}

for each $j$ and $t$, where $W_{itj}$ denotes the $j^{th}$ coordinate of the vector $W_{it}\in \R^{d}$, $I_p$ is a $p\times p$ identity matrix, and we set the penalty $\lambda$ to equal $0.1$.
Given these estimates, we let $\hat \Sh_{wt} \equiv [\hat \delta_{1t},\ldots, \hat \delta_{dt}]$ and estimate $\Gamma$ through the regression

\begin{equation*}%\label{mc:akm5}
\hat \Gamma \equiv \arg\min_{g \in \R^{d}} \sum_{t=1}^T \|\Sh_{zt} - \hat \Sh_{wt}g\|^2.
\end{equation*}

In what follows, it will be helpful to define $\hat E[\Sh_{zt}|\mathcal G_n] \equiv \hat \Sh_{wt}\hat \Gamma$ and $\hat \nu_t \equiv \Sh_{zt}-\hat \Sh_{wt}\hat \Gamma$.

We further aim to reflect the clustering structure in [Autor et al., 2013].
To this end, we follow [Adao et al., 2019] and [Borusyak et al., 2022] who in re-examining the empirical analysis of [Autor et al., 2013] cluster shocks at the three digit SIC code.
As in Section <ref>, we estimate a common shock model in which $\Sh_{zt}$ satisfies

\begin{equation*}%\label{mc:akm6}
\Sh_{ztj} = \eta_c + \zeta_{tj}
\end{equation*}

where $\Sh_{ztj}$ denotes the $j^{th}$ coordinate of $\Sh_{zt}$, $\eta_c$ are i.i.d. cluster level shocks, and $\zeta_{tj}$ are i.i.d. shocks independent of $\eta_c$.
We estimate the variance of these shocks by setting

\begin{align*}%\label{mc:akm7}
\hat \sigma_\eta^2 & \equiv \frac{1}{|C|}\sum_{c\in C}\frac{1}{n_c(n_c-1)}\sum_{(t,j)\neq (\tilde t,\tilde j)\in c} \hat \nu_{tj}\hat \nu_{\tilde t \tilde j}\notag \\
\hat \sigma_\zeta^2 & \equiv \frac{1}{|C|}\sum_{c\in C} \frac{1}{n_c} \sum_{(s,t)\in c} \hat \nu_{tj}^2 - \hat \sigma_\eta^2,
\end{align*}

where $n_c$ denotes the number of observations in cluster $c$ and $\nu_{tj}$ denotes the $j^{th}$ coordinate of the vector $\hat \nu_t \in \R^p$.

Finally, in order to reflect the strength of the instrument in [Autor et al., 2013] in our simulation design, we run the following regression on the aggregate shocks

\begin{equation*}%\label{mc:akm8}
(\hat \alpha,\hat \kappa ) \equiv \arg\min_{a,k\in \R} \sum_{t=1}^T\sum_{j=1}^p (\Sh_{xtj} - a - k \Sh_{ztj})^2
\end{equation*}

and let $\hat \sigma^2_\xi$ denote the sample variance of the residuals $\hat \xi_{tj} \equiv \Sh_{xtj} - \hat \alpha - \hat \kappa \Sh_{ztj}$.

Given these estimates, we generate our Monte Carlo samples as follows:

Step 1. We first create controls $\hat W_{it} \equiv \hat\Sh_{wt}^\prime S_{zit}$, which we note have the structure required by model (<ref>).
We combine the controls $\{\hat W_{it}\}$ with the shares $\{S_{zit},S_{xit}\}$ in [Autor et al., 2013] and keep all of them fixed across simulation designs.

Step 2. To generate our instrument, we first draw $|C|$ i.i.d. standard normal variables $\{V_c\}_{c\in C}$, $p\times T$ i.i.d. standard normal variables $\{U_{ztj}\}$, and set

\begin{equation*}%\label{mc:akm8p5}
\Sh_{ztj}^* = \hat E[\Sh_{ztj}|\mathcal G_n] + V_c \hat \sigma_\eta + U_{ztj} \hat \sigma_\zeta,
\end{equation*}

where the “$c$" subscript refers to the three digit SIC code to which sector $j$ belongs.
As our instrument we then employ $Z_{it}^* \equiv S_{zt}^\prime \Sh^*_{zt}$.
Note that, because $\hat E[\Sh_{zt}|\mathcal G_n] \equiv \hat \Sh_{wt} \hat \Gamma$, the shocks $\Sh^*_{zt}$ have the structure required by model (<ref>).

Step 3. Similarly, in order to generate aggregate shocks for our regressor, we draw $p\times T$ i.i.d. standard normal random variables $\{U_{xtj}\}$ and let

\begin{equation*}%\label{mc:akm9}
\Sh^*_{xtj} = \hat \alpha + \hat \kappa \Sh_{ztj}^* + V_{xtj} \hat \sigma_\xi.
\end{equation*}

As our regressor, we then employ $X_{it}^* \equiv S_{xit}^\prime \Sh_{xt}^*$.

Step 4. Finally, we generate a sample of outcomes $Y_{it}^*$ by simply setting $Y_{it}^*$ to equal

\begin{equation*}%\label{mc:akm10}
Y_{it}^* = X_{it}^* \hat \beta + \hat W_{it}^\prime \hat \gamma_{\mathtt s} + \hat e_{it},
\end{equation*}

where $(\hat \beta,\hat \gamma_{\mathtt s})$ and $\{\hat e_{it}\}$ denote the estimators and residuals obtained from replacing $W_{it}$ with $\hat W_{it}$ in the main specification of [Autor et al., 2013].

By repeating Steps 1-4 we generate one thousand samples $\{Y_{it}^*,X_{it}^*,Z_{it}^*,\hat W_{it}\}$ on which we evaluate the performance of the overidentification test proposed in Section <ref>.
In order to implement the test, we need to select the moments to employ (i.e. the functions $g_j$ in (<ref>)) and an estimator $\hat {\mathcal E}^*_t$ for the demeaned shock

\begin{equation*}%\label{mc:akm11}
\mathcal E^*_t \equiv \Sh^*_{zt}-E[\Sh^*_{zt}|\mathcal G_n].
\end{equation*}

In the main specification of [Autor et al., 2013], there are no control variables with the structure required by the estimator for $\mathcal E_t^*$ proposed by [Borusyak et al., 2022].
We therefore instead adapt the estimator of $\mathcal E^*_t$ advocated by [Adao et al., 2019] by employing

\begin{equation}\label{mc:akm12}
\hat {\mathcal E}_t^* = (\sum_{i=1}^n S_{zit} S_{zit}^\prime + \lambda I_p)^{-1}\sum_{i=1}^n S_{zit}(Z_{it}^*-\hat W_{it}^\prime\hat \pi_n^*)
\end{equation}

for $\hat \pi_n^*$ the coefficient obtained from a weighted regression of $\{Z_{it}^*\}$ of $\{\hat W_{it}\}$.
We introduce ridge regression in (<ref>) because the design matrix is ill-conditioned.
In this regard, our estimator differs from that in [Adao et al., 2019] who use ordinary regression (i.e. $\lambda = 0$), but instead drop sectors from the regression to address the ill conditioning of the design matrix.[See page 3 in https://github.com/kolesarm/ShiftShareSE/blob/master/doc/ShiftShareSE.pdf]
The p-values of the test can depend on $\lambda$, and we employ the simulations to inform the choice of penalty $\lambda$ for our application.[Analogously, as noted by [Adao et al., 2019], dropping sectors to ensure that the design matrix is well conditioned similarly affects the resulting standard errors.]

Finally, for our moments we select the square of the residual and moments based on the pdf of the Logit distribution, which may be interpreted as different kernel estimators.
Specifically, we employ a total of twenty moments by setting

\begin{equation*}%\label{mc:akm13}
g_j(\varepsilon,w,s) = \left\{\begin{array}{cl} \varepsilon^2 & \text{ if } j = 1\\
\frac{\exp(\varepsilon -a_j)}{(\exp(\varepsilon-a_j)+1)^2} & \text{ if } 2\leq j \leq 20
\end{array}\right.
\end{equation*}

with $a_2,\ldots, a_{20} = -2.25, -2, \ldots, 2, 2.25$.
Table <ref> reports the finite sample rejection probabilities of the resulting test for different choices of the ridge parameter $\lambda$.
The final column of Table <ref> additionally reports the p-value obtained when the test is implemented in the data of [Autor et al., 2013].
The results are based on one thousand simulations with the bootstrap implementation relying on one thousand replications.
Overall, we find that the rejection probability is close to the nominal level of the test provided that the ridge parameter is sufficiently small.

Rejection Probabilities

3cSignificance Level

Ridge Parameter                   $1\%$       $5\%$         $10\%$                p-value
$\lambda = 1{\rm e-3}$                  0.029       0.103         0.177        0.0012

$\lambda = 1{\rm e-4}$                  0.011       0.061         0.127         0.0074

$\lambda = 1{\rm e-5}$                  0.008       0.049         0.114         0.0368

$\lambda = 1{\rm e-6}$                  0.008       0.047         0.100          0.0650

Finite sample rejection probabilities for overidentification tests of the validity of the moment restrictions imposed when identification is driven by the exogeneity of the aggregate shocks.

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