Overidentification in Shift-Share Designs

Overidentification in Shift-Share Designs

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 bartik1991 and blanchard1992yregional, 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 card2001immigrant, the consequence of trade liberalization for poverty topalova2010factor, the effect of import competition on labor markets david2013china, and the welfare implications of geographic sorting by education diamond2016determinants.

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 goldsmith2020bartik, attributes the exogeneity of the Bartik instrument to the exogeneity of the shares. The second, examined by adao2019shift and borusyak2022quasi, 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, goldsmith2020bartik necessitates the Bartik instrument to be exogenous conditional on the realization of the aggregate shocks, while adao2019shift and borusyak2022quasi 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 goldsmith2020bartik, our analysis implies that the entire vector of shares must be exogenous. As noted by goldsmith2020bartik, 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 adao2019shift and borusyak2022quasi, 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 goldsmith2020bartik.

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 chernozhuokov2022improved. Our tests utilize bootstrap based critical values and are robust to the presence of heteroskedasticity, clustering, and weighting.\footnote{In contrast, goldsmith2020bartik emply overidentification tests based on LIML (which requires homoskedasticity) and CHAO201415 (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 goldsmith2020bartik, adao2019shift. In such a setting, TSLS can readily be shown to estimate a weighted average of group specific treatment effects. As forcibly argued by kolesa2013estimation and blandhol2022tsls 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 goldsmith2020bartik, 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 adao2019shift and borusyak2022quasi, 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 goldsmith2020bartik or the aggregate shocks alone adao2019shift, borusyak2022quasi 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 HKM, RePEc:cup:etheor:v:38:y:2022:i:5:p:942-958_5 and may be of independent interest.

We highlight the empirical relevance of our analysis by revisiting the study by david2013china 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 goldsmith2020bartik as well as the identification framework of adao2019shift and borusyak2022quasi. 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 goldsmith2020bartik. Likewise, TSLS may not possess a causal interpretation in a heterogeneous effects model under the identification strategy of adao2019shift and borusyak2022quasi 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 goldsmith2020bartik as well borusyak2022quasi and adao2019shift. 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 david2013china. 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 goldsmith2020bartik and adao2019shift, 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 $Y_{i}\in\mathbb R$, $X_{i}\in\mathbb R$, $Z_i\in \mathbb R$, and $W_{i}\in\mathbb R^d$ 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

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

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

For instance, in canonical applications $\mathcal Z$ 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 $\mathcal Z$ 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

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 adao2019shift. The observable variables are the aggregate shocks $\mathcal Z$ 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 $\mathcal Z$. Hence, in such instances we would set $\mathcal G_n$ to include all aggregate variables, including $\mathcal Z$. Alternatively, when specialized to equations (ref) and (ref), the asymptotic framework of adao2019shift and borusyak2022quasi 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

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

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 $\mathcal Z$ 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 $\mathcal Z$. 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.\footnote{This setting rules out, for example, that $(W,\varepsilon)$ have a Bartik structure as in adao2019shift. 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

Equation (ref) highlights that the relevant exogeneity condition for this asymptotic framework depends on the realization of the aggregate shocks $\mathcal Z$ 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 $\mathcal Z$ and not others, we should in the interest of robustness demand that (ref) hold at all possible realizations of $\mathcal Z$. Provided the aggregate shocks $\mathcal Z$ exhibit sufficient variation, however, it then follows that condition (ref) can hold for all possible realizations of $\mathcal Z$ if and only if in fact $S_i$ itself is uncorrelated with the errors $\varepsilon_i$.

Our next simple proposition formalizes the preceding discussion.

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 goldsmith2020bartik, 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

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.

Identification Through Shocks

As an alternative identification strategy, adao2019shift and borusyak2022quasi advocate interpreting the exogeneity of the Bartik instrument $Z_i$ as originating from the exogeneity of the aggregate shocks $\mathcal Z$. 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 $n$.\footnote{As noted by adao2019shift, 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

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

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 $\{\mathcal Z_j\}_{j=1}^p$ 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 $\{\mathcal Z_j\}_{j=1}^p$ 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

Intuitively, if the aggregate shocks $\{\mathcal Z_j\}_{j=1}^p$ are exogenous in the sense that (ref) holds, then any suitable linear combination of the shocks, such as $Z_i = S_i^\prime \mathcal Z$, 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 $\varepsilon_i$.\footnote{Equivalently, note that since any suitably linear combination of $\mathcal Z$ 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 $\mathcal Z$, 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 $\mathcal Z$ is independent of all individual specific variables.

Proposition (ref) establishes that the exogeneity requirement needed to obtain identification through the aggregate shocks $\mathcal Z$ 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 $\mathcal Z$ 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

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.

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 chen2018overidentification). 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

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 adao2019shift and goldsmith2020bartik in imposing a linear first stage

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 imbens1994identification, 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

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

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 kolesa2013estimation, blandhol2022tsls. 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 $\mathcal Z$ 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).

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.\footnote{Under regularity conditions, Assumption (ref)(iii) is equivalent to $E[S_i|W_i]$ being linear in $W_i$.} blandhol2022tsls 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.

The first part of Proposition (ref) shows that the weights may be expressed as a quadratic form in the shocks $\mathcal Z$ (see (ref)). Intuitively, provided that the support of the shocks $\mathcal Z$ is sufficiently rich, it therefore follows that the weights can only be positive for all possible realizations of the shocks $\mathcal Z$ 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 $\mathcal Z$ contain a neighborhood of zero.\footnote{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.

Corollary (ref) implies that, whenever shares are correlated, TSLS necessarily lacks a causal interpretation under certain realizations of the shocks $\mathcal Z$. 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 goldsmith2020bartik, 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 $\mathcal Z$.

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 adao2019shift and borusyak2022quasi 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).

Assumption (ref) imposes the structure of our model and requires that the conditional mean of the instrument given the controls be linear.\footnote{Sufficient conditions for Assumption (ref)(ii) were introduced by adao2019shift; 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.

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 $\Lambda_i \text{Var}\{\mathcal Z|\mathcal G_n\}$ 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 $\Lambda_i\text{Var}\{\mathcal Z|\mathcal G_n\}$ 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.

Traditionally, the literature has assumed shocks to either be uncorrelated or clustered for asymptotic purposes adao2019shift,borusyak2022quasi. 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.

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 $\mathcal Z_t$ and variables $\{Y_{it},X_{it},Z_{it}, S_{it}\}_{i=1}^n$ satisfying

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

Crucially, the expectation in (ref) is taken over both the cross section and the time series -- e.g., over both $\mathcal Z_t$ and $(S_{it},\varepsilon_{it})$. In particular, the moment restriction in (ref) contrasts with goldsmith2020bartik and adao2019shift who instead consider conditional expectations given $\mathcal Z_t$ 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

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

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 $\mathcal Z_t$) 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

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

Crucially, the time series process $\{\mathcal Z_t^\prime \zeta_t\}_{t=1}^T$ has mean zero because of the moment restriction in (ref), while the panel process $\{\mathcal Z_t^\prime \nu_{it}\}_{i,t=1}^{n,T}$ 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:

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

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,

However, as we discussed in Section (ref), the conditional moment restriction in (ref) is exactly the identifying assumption made by goldsmith2020bartik 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:2step.

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

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 chernozhuokov2022improved. Specifically, for an estimator $\hat \theta_n$ of $\theta$ and $f_j(V_i,\hat \theta_n)$ denoting the $j^{th}$ coordinate of $f(V_i,\hat \theta_n)\in \mathbb R^q$ we set

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 $\{\psi_i\}_{i=1}^{b_n}\subset \mathbb R^q$ the test statistic approximately equals

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 adao2019shift (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

where $\hat \psi_{ij}$ denotes the $j^{th}$ coordinate of $\hat \psi_i \in \mathbb R^q$ 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.\footnote{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

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 $b_n$.

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

where $\mathcal G_n$ is understood to contain all aggregate variables, including $\mathcal Z$ goldsmith2020bartik. As our test statistic we therefore employ

where $S_{ij}$ denotes the $j^{th}$ coordinate of $S_i\in \mathbb R^p$, $\{\hat \varepsilon_i\}_{i=1}^n$ 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

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

provided the observations $\{Y_i,S_i,X_i\}_{i=1}^n$ are i.i.d.\ conditionally on $\mathcal G_n$.\footnote{ 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

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

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

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 $\hat c_{1-\alpha}$.

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

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

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 $g(\hat \varepsilon_i,W_i,S_i)\in \mathbb R^q$.

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

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

Under conditions similar to those employed by adao2019shift, it is then possible to show that under the null hypothesis the statistic $T_n$ satisfies the approximation\footnote{See the supplemental appendix for calculations supporting the claims in this section.}

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

and let $\hat {\mathcal E}\in \mathbb R^p$ 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

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

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

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 $\hat c_{1-\alpha}$.

Empirical Application: The China Syndrome

We illustrate the empirical relevance of our analysis by revisiting the seminal work of david2013china examining the impact of rising Chinese import competition on local US markets. We focus on the main specification of david2013china, in which

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.\footnote{Specifically, we employ the specification in column (6) of Table 3 in david2013china.} david2013china estimate the model in (ref) by TSLS based on a Bartik instrument $Z_{it} = \mathcal Z_t^\prime S_{it}$ in which $S_{it}$ and $\mathcal Z_t$ 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 david2013china 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.\

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 david2013china, 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.

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. goldsmith2020bartik 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 goldsmith2020bartik 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 adao2019shift, borusyak2022quasi. 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 borusyak2022quasi in clustering at the three level SIC code.

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 adao2019shift who similarly find that the manner in which they address the singularity of the shares design matrix affects their standard errors.\footnote{adao2019shift 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 david2013china, 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:

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

Proofs for Section (ref)

Proof of Proposition (ref). First note that since $\{S_i,W_i,\varepsilon_i\}_{i=1}^n$ are i.i.d.\ and independent of $\mathcal G_n$ and $\mathcal Z$ is measurable with respect to $\mathcal G_n$ it follows $E[W_i\varepsilon_i] = 0$ that

It is therefore immediate that $E[S_i\varepsilon_i] = 0$ is a sufficient condition for (ref) to hold with probability one over $\mathcal Z$. To see that this condition is also necessary, suppose by way of contradiction that $E[S_i\varepsilon_i]\neq 0$. By Lemma (ref), it then follows that there is a $z^*$ in the support of $\mathcal Z$ satisfying $(z^*)^\prime E[S_i\varepsilon_i]\neq 0$. Assuming without loss of generality that $(z^*)^\prime E[S_i\varepsilon_i] > 0$ and setting $B_\delta(z^*)\equiv \{z\in \mathbb R^p : \|z-z^*\|_2^2 < \delta\}$, we then obtain that $(E[S_i\varepsilon_i])^\prime z > 0$ for all $z\in B_\delta(z^*)$ provided that $\delta > 0$ is chosen to be sufficiently small. Hence, employing that $z^*$ is in the support of $\mathcal Z$ yields that

which implies that for (ref) to hold with probability one we must have $E[S_i\varepsilon_i] = 0$. \rule{2mm}{2mm}

Proof of Proposition (ref). First note that since $\mathcal G_n = \{S_i,W_i,\varepsilon_i\}_{i=1}^n$ and $\mathcal Z$ is independent of $\mathcal G_n$ by hypothesis, it follows from $Z_i = S_i^\prime \mathcal Z$ by definition that we have

Since $E[Z_i|\{S_i,W_i,\varepsilon_i\}_{i=1}^n] = S_i^\prime E[\mathcal Z]$, we can conclude that $E[Z_i|\{S_i,W_i,\varepsilon_i\}_{i=1}^n] = W_i^\prime \pi_n$ is a sufficient condition for (ref) to hold with probability one. In order to establish the converse direction, we next define the event $A_n$ to be given by

and note that $A_n$ is measurable with respect to $\{S_i,W_i\}$ by definition of $\pi_n$. For any realization of $\{S_i,W_i\}_{i=1}^n$ in the event $A_n$, Lemma (ref) allows us to conclude that there exists a $\{e_i^*\}_{i=1}^n$ in the support of $\{\varepsilon_i\}_{i=1}^n$ conditional on $\{S_i,W_i\}$ satisfying

Hence, defining $B_\delta(\{e_i^*\}_{i=1}^n) \equiv \{ \{e_i\}_{i=1}^n : \sum_i (e_i - e_i^*)^2 < \delta^2\}$ we obtain from $\{e_i^*\}_{i=1}^n$ being in the support of $\{\varepsilon_i\}_{i=1}^n$ conditional on $\{S_i,W_i\}_{i=1}^n$ that for $\delta$ sufficiently small

In particular, since the conclusion in (ref) applies to any $\{S_i,W_i\}_{i=1}^n$ in the event $A_n$, it follows that $A_n$ must have probability zero in order for condition (ref) to hold with probability one. Since $S_i^\prime E[\mathcal Z] = E[Z_i|\{S_i,W_i,\varepsilon_n\}_{i=1}^n]$, the lemma follows. \rule{2mm}{2mm}

{Proof.} Since the dimension of the linear span of $V$ equals $k$, it follows that there are vectors $\{v_j\}_{j=1}^k \subset \mathbb R^k$ and scalars $\{\alpha_j\}_{j=1}^k$ satisfying $x = \sum_j \alpha_j v_j$. In particular, since $x\neq 0$, we have $(\sum_j \alpha_j v_j)^\prime x = \|x\|_2^2 > 0$ and hence that $v_{j^*}^\prime x \neq 0$ for some $j^*$. \rule{2mm}{2mm}

Proofs for Section (ref)

The results in Section (ref) rely on the next set of regularity conditions. In the statement below, the notation $\text{supp}\{V\}$ refers to the support of a random variable $V$.

Assumption (ref)(i) allows for dependence across observations though it implies that the marginal distribution of $(\Lambda_i,W_i,\mathcal Z)$ is the same for all observations. Assumptions (ref)(ii)(iv) further impose support restrictions on $(\Lambda_i,W_i,\mathcal Z)$, while Assumption (ref)(iii) states our restrictions on the covariance matrix of $S_i \in \mathbb R^p$ conditionally on $W_i$ (denoted $\text{Var}\{S_i|W_i\})$. Assumption (ref)(iii) allows for $\text{Var}\{S_i|W_i\}$ to be rank-deficient to recognize that the shares may sum up to one. Finally, Assumptions (ref)(iv) simply demands that the “types" vector $(\lambda_{i1},\ldots,\lambda_{ip})$ be continuously distributed.

Proof of Proposition (ref). First note that Assumption (ref)(ii) implies $(\Lambda_i,\beta_i,\varepsilon_i,\eta_i)$ is independent of $(\mathcal Z,S_i)$ conditionally on $(\mathcal Z,W_i)$. Therefore, by Assumptions (ref)(i)(iii), we may apply Lemma (ref) with $\mathcal G_n = \{\mathcal Z\}$ to conclude that

where the second equality holds by $Z_i = S_i^\prime \mathcal Z$ and Assumption (ref)(iii), and the final equality holds by Assumption (ref)(ii). The claim of the proposition therefore follows from result (ref), the definition of $\omega_{i,n}$ in (ref), and Lemma (ref). \rule{2mm}{2mm}

Proof of Corollary (ref). Suppose that condition (ref) holds for some $j\neq k$ and define

Next, note that by Proposition (ref) a necessary condition for $\omega_{i,n}$ to be positive for all $1\leq i \leq n$ with probability one is that the corresponding $2\times 2$ matrix

be either positive semi-definite or negative semi-definite with probability one. Hence, setting $\bar \lambda_i = (\lambda_{ij} + \lambda_{ik})/2$ and noting that for any $2\times 2$ matrix $A$ and $a\in \mathbb R^2$ we have $a^\prime A a = a^\prime (A+A^\prime)a/2$, it follows that a necessary condition for $\omega_{i,n}$ to be positive for all $1 \leq i \leq n$ with probability one is that the matrices $\Omega_i$ defined by

be either positive semi-definite with probability one or negative semi-definite with probability one. However, in order for $\Omega_i$ to be positive semi-definite or negative-semidefinite with probability one we must have $\lambda_{ij}\lambda_{ik}\geq 0$ due to $\min\{\sigma_j^2(W_i),\sigma_k^2(W_i)\} > 0$ by Assumption (ref)(iii). Hence, since the determinant of $\Omega_i$ equals the product of its eigenvalues, a necessary condition for $\omega_{i,n}$ to be positive for all $i$ with probability one is

However, setting $\rho_{jk}(W_i) \equiv \sigma_{jk}(W_i)/\sigma_j(W_i)\sigma_k(W_i)$ we obtain by direct calculation that

where the first inequality follows from $\lambda_{ij}\lambda_{ik}\neq 0$ with probability one by Assumption (ref)(iv), and the final from Assumption (ref)(ii), the support of $\lambda_{ij}/\lambda_{ik}$ being unbounded, and $\rho^2_{jk}(W_i) \neq 0$ with positive probability due to condition (ref). \rule{2mm}{2mm}

{Proof.} First note that for any function $f$ of $(\Lambda_i,\beta_i,\varepsilon_i,\eta_i,W_i)$ it follows from the requirements that $(\Lambda_i,\beta_i,\varepsilon_i,\eta_i) \perp \!\!\! \perp (\mathcal Z,S_i)$ conditionally on $(\mathcal G_n,W_i)$ and $Z_i = \mathcal Z^\prime S_i$ that

where the final equality follows from $E[Z_i|W_i,\mathcal G_n] = W_i^\prime \pi_n$. Combining result (ref) with the specifications of the first and second stages in (ref) and (ref) then yields that

Letting $\mathcal Z_j$ denote the $j^{th}$ coordinate of $\mathcal Z$, $S_{ij}$ the $j^{th}$ coordinate of $S_i$, and recalling that $\Lambda_i = \text{diag}\{(\lambda_{i1},\ldots,\lambda_{ip})\}$ we may also employ (ref) and (ref) to obtain

which verifies that (ref) indeed holds. Finally, observe that $(\Lambda_i,\beta_i)$ being independent of $(\mathcal Z,S_i)$ conditionally on $(W_i,\mathcal G_n)$ further allows us to conclude that

where the final equality follows from result (ref) and $\Lambda_i \perp \!\!\! \perp (\mathcal Z,S_i)$ conditionally on $(W_i,\mathcal G_n)$. The claim that (ref) holds therefore follows from the definition of $\beta_{0,n}$, the rank condition $\sum_{i=1}^n E[X_i\dot Z_{i,n}|\mathcal G_n]\neq 0$ and results (ref), (ref), and (ref). \rule{2mm}{2mm}

Proof. For notational simplicity we first set $\Gamma_i \equiv \Lambda_i \text{Var}\{S_i|W_i\}$ and use “$\Gamma_i\geq 0$" and “$\Gamma_i\leq 0$" to denote that $\Gamma_i$ is positive semi-defintie and negative-semidefinite respectively. Then note that if either $P(\Gamma_i \geq 0) = 1$ for all $i$ or $P(\Gamma_i \leq 0) = 1$ for all $i$, then (ref) immediately holds. For the opposite direction, we next observe that

where the second equality follows from Assumption (ref)(i). However, since $a^n = o(1) $ for any $a\in [0,1)$ as $n$ tends to infinity, result (ref) implies that in order for the claim in (ref) to hold for $n$ sufficiently large we must have with probability one over $\mathcal Z$ that

By Assumption (ref)(ii), $\text{supp}\{(\Gamma_1,\mathcal Z)\} = \text{supp}\{\Gamma_1\}\times \text{supp}\{\mathcal Z\}$ and hence the distribution of $\Gamma_1$ is absolutely continuous with respect to the distribution of $\Gamma_1$ conditionally on $\mathcal Z$. Defining the sets $S_+$ and $S_{-}$ to be given by

it therefore follows from result (ref) holding with probability one that we must have

Next, set $\Gamma^* \equiv \Lambda^* \Sigma^*$ with $\Sigma^*$ any point in the support of ${\rm Var}\{S_1|W_1\}$ and $\Lambda^* = \text{diag}\{\lambda_{1}^*,\ldots, \lambda_p^*\}$ for any point $(\lambda_1^*,\ldots, \lambda_p^*)$ at which the density of $(\lambda_{11},\ldots, \lambda_{1p})$ is strictly positive. We next aim to show that $\Gamma^*$ is either positive semi-definite or negative semi-definite. We proceed by contradiction: Suppose that $\Gamma^*$ is neither positive semi-definite nor negative semi-definite. Then, by Lemma (ref), Assumption (ref)(iii), the density of $(\lambda_{11},\ldots,\lambda_{1p})$ being continuous and strictly positive at $(\lambda_1^*,\ldots, \lambda_p^*)$, and Assumption (ref)(v), there are $z_0 \in {\rm \text{supp}}\{\mathcal Z\}$ and $\Lambda_0,\tilde \Lambda_0 \in \text{supp}\{\Lambda_1\}$ satisfying

We may therefore select sufficiently small neighborhoods $N(z_0), N(\Lambda_0), N(\tilde \Lambda_0), N(\Sigma^*)$ of $z_0,\Lambda_0,\tilde \Lambda_0$, and $\Sigma^*$ such that the following inequalities are satisfied

Then note that since $\Lambda_0\Sigma^* \in \text{supp}\{\Lambda_1\text{Var}\{S_1|W_1\}\}$ by Assumption (ref)(ii), it follows

By identical arguments, but relying on $N(\tilde \Lambda_0)\times N(\Sigma^*)$ instead, similarly yield that

However, because $P(\mathcal Z \in N(z_0)) > 0$ due to $z_0 \in \text{supp}\{\mathcal Z\}$, results (ref) and (ref) together contradict (ref) and therefore $\Gamma^*$ must be either positive semi-definite or negative semi-definite as claimed. Thus, we have so far shown that

To conclude, note that by result (ref) and Lemma (ref) either $S_+$ or $S_{-}$ (or both) must contain $p$ linearly independent vectors. If $S_+$ contains $p$ linearly independent vectors $\{s_j\}_{j=1}^p$, then we obtain by the definition of $S_+$ in (ref) that

Next note that $z^\prime \Gamma_1 z = z^\prime(\Gamma_1 + \Gamma_1^\prime)z/2$ for any $z\in \mathbb R^p$ and use that $(\Gamma_1 + \Gamma_1^\prime)$ is diagonalizable and $s_j^\prime (\Gamma_1 + \Gamma_1^\prime)s_j = 0$ for all $1\leq j \leq p$ implies that all the eigenvalues of $(\Gamma_1+\Gamma_1^\prime)$ equal zero due to $\{s_j\}_{j=1}^p$ being linearly independent to conclude that

where the first inequality follows from $\text{Var}\{S_{1j}|W_i\}>0$ for all $1\leq j \leq p$ by Assumption (ref)(iii) and the second inequality by $\Gamma_1 = \text{diag}\{(\lambda_{11},\ldots, \lambda_{1p})\}\text{Var}\{S_1|W_1\}$. Hence, combining results (ref) and (ref) we obtain that if $S_+$ contains $p$ linearly independent vectors, then $P(\Gamma_1 = 0) = P(\Gamma_1 \leq 0)$, which together with result (ref) yields

Similarly, if $S_{-}$ instead contains $p$ linearly independent vectors, then it is possible to show that (ref) implies that $P(\Gamma_1\leq 0)=1$ and therefore the lemma follows. \rule{2mm}{2mm}

Proof. Let $\bar \Gamma = (\Gamma + \Gamma^\prime)/2$ and note that since $a^\prime \Gamma a = a^\prime \Gamma^\prime a$ for any $a \in \mathbb R^p$, we have

Next observe that $\bar \Gamma$ is a symmetric matrix and let $\{\pi_i\}_{i=1}^p$ denote its eigenvalues and $\{v_i\}_{i=1}^p$ its corresponding eigenvectors. If $\Gamma$ is neither positive semi-definite nor negative semi-definite, then result (ref) implies that $\bar \Gamma$ must have a strictly positive eigenvalue and a strictly negative eigenvalue. Assuming without loss of generality that $\pi_1 >0$ and $\pi_2 < 0$, let $\alpha >0$ and define the vectors $c_1$ and $c_2$ to be given by

Next note that $\|c_1\|^2 = \|c_2\|^2 = \alpha^2(1+ \pi_1/|\pi_2|)$ and that therefore we may set $\|c_1\| =\|c_2\| < \delta$ by selecting $\alpha$ to be sufficiently small. Moreover, by direct calculation we have

where the final equality follows from $\pi_2 < 0$. Next observe that since ${\rm rank}\{\Sigma^{1/2}\} = {\rm rank}\{\Sigma\} \geq p-1$ and $c_1,c_2\in \mathbb R^p$ are linearly independent, it follows that $\Sigma^{1/2} c_j \neq 0$ for some $j\in\{1,2\}$. Assuming without loss of generality that $\Sigma^{1/2} c_1 \neq 0$, we then set $z = c_1$ and note that results (ref), (ref), and $\Sigma^{1/2}z \neq 0$ imply that

Setting $x \equiv \Sigma z$ and letting $x_j$ and $z_j$ denote the $j^{th}$ coordinates of $x$ and $z$, then note

which implies that $z_{j} x_{j} > 0$ for some $1\leq j \leq p$. Assuming without loss of generality that $z_1 x_1 > 0$, then set $e_1 = (1,0,\ldots, 0)^\prime$ and for $\xi > 0$ define the matrices

noting that $\|D-H_1\| = \|D-H_2\| <\delta$ provided $\xi$ is chosen sufficiently small. In addition, $\Gamma \equiv D\Sigma$, $x \equiv \Sigma z$, result (ref), and $x_1z_1 > 0$ together yield that

Identical arguments imply $z^\prime(H_2 \Sigma)z < 0$, and hence the lemma follows. \rule{2mm}{2mm}

{\emph Proof.} By way of contradiction, suppose that ${\rm span}\{A_i\} \neq \mathbb R^p$ for $i \in \{1,2\}$. Then note that $A_i \subseteq V_i$ for some vector subspace $V_i \subset \mathbb R^p$ and that for each $i$ we may find a $v_i\in \mathbb R^p$ that is orthogonal to $V_i$ and satisfies $\|v_i\|^2 = 1$. Next define $v^*$ to equal

and note that $\langle v^*,v_i\rangle \neq 0$ for $i \in \{1,2\}$ due to $\|v_i\|^2 = 1$. Since $v_i$ is orthogonal to $V_i$ it follows that $\xi v^* \notin V_i$ and hence $\xi v^*\notin V_1\cup V_2$ for any $\xi > 0$. In particular, we obtain that $\xi v^*\notin A_1 \cup A_2$ due to $A_1\cup A_2 \subseteq V_1 \cup V_2$. However, for $\xi$ sufficiently small, $\xi v^*$ belongs to the support of $\mathcal Z$ by Assumption (ref)(v). Thus, selecting $N(\xi v^*)$ to be a neighborhood of $\xi v^*$ sufficiently small to ensure that $N(\xi v^*) \subseteq (A_1\cup A_2)^c$ we obtain

where the final inequality follows from $\xi v^*$ being in the support of $\mathcal Z$. Since (ref) contradicts the hypothesis that $P(\mathcal Z \in A_1 \cup A_2) =1$, the lemma follows. \rule{2mm}{2mm}

Proofs for Section (ref)

The results in Section (ref) rely on the following regularity conditions.

Assumptions (ref)(i)-(iii) impose conditions that simplify our analysis. In turn, Assumption (ref)(iv) demands that the support of the shares be sufficiently rich -- in its statement, $\|\cdot\|_1$ denotes the $\ell_1$ norm $\|s\|_1 = \sum_{i=1}^n |s^{(i)}|$ for any vector $s = (s^{(1)},\ldots, s^{(p)})^\prime \in \mathbb R^p$. We note that Assumption (ref)(iv) allows the shares to sum to less than one, as is sometimes the case in empirical applications.

Proof of Proposition (ref). First note that setting $\mathcal G_n = \{S_i,W_i,\Lambda_i,\beta_i,\varepsilon_i,\eta_i\}_{i=1}^n$ implies that $(\Lambda_i,\beta_i,\varepsilon_i,\eta_i)\perp \!\!\! \perp (\mathcal Z,S_i)$ conditionally on $(W_i,\mathcal G_n)$. Hence, Assumption (ref) yields

where the first equality follows from Lemma (ref) and $(\Lambda_i,W_i)\in\mathcal G_n$, the second equality holds by $Z_i = S_i^\prime \mathcal Z$ and Assumption (ref)(ii), and the third equality holds by direct calculation. The claim of the proposition therefore follows from result (ref), the definition of $\omega_{i,n}$ in (ref), and Lemma (ref). \rule{2mm}{2mm}

Proof of Corollary (ref). Suppose condition (ref) holds for some $j\neq k$ and define

Also note that Assumptions (ref)(i)(iii) and Proposition (ref) together imply that a necessary condition for $\omega_{i,n}$ to be positive for all $i$ with probability one is that

In what follows, we assume that $P(\Lambda_i\geq 0) =1$ and note that the case in which $P(\Lambda_i \leq 0) = 1$ follows by identical arguments. Next note that by Proposition (ref), a necessary condition for $\omega_{i,n}$ to be positive for all $i$ with probability one is that

be either positive for all $(s_j,s_k)\in \mathbb R_+$ with probability one over $\mathcal G_n$, or negative for all $(s_j,s_k)\in \mathbb R_+$ with probability one over $\mathcal G_n$. However, Assumption (ref)(i), $P(\lambda_{ij}=0) = 0$, and $P(\lambda_{ij}\geq 0)=1$ imply that setting $(s_j,s_k)=(1,0)$ yields a strictly positive number in (ref) with probability one. Hence, we can conclude that a necessary condition for $\omega_{i,n}$ to be positive for all $i$ with probability one is that we have

where in the second equality we employed Assumption (ref)(i) and in the third equality we profiled out $s_j$ by setting $s_j = -s_k\rho(\mathcal G_n)(\lambda_j+\lambda_k)/2\lambda_j$. However, note that

where the inequality holds due to condition (ref), Assumptions (ref)(i)(iii), and the support of $\lambda_j/\lambda_k$ being unbounded. Since result (ref) implies the necessary condition in (ref) cannot hold, the claim of the corollary follows. \rule{2mm}{2mm}

Proof. First note that since $S_i \in \mathbb R^p_+$ by Assumption (ref)(iv), it follows that condition (ref) implies (ref) holds. For the reverse direction, set $\mathcal W_n = \{W_i\}_{i=1}^n$ and $\Gamma_{i,n} = \Lambda_i\text{Var}\{\mathcal Z|\mathcal W_n\}$ for notational simplicity, and employ Assumptions (ref)(i) to obtain

where the second equality follows from Assumption (ref)(ii). For $n\geq 2$, result (ref) and the law of iterated expectations together imply that in order for the equality in (ref) to hold we must have with probability one over $(\Lambda_i,\mathcal W_n)$ that

Next note that Assumption (ref)(iii) implies that the distribution of $S_i$ is absolutely continuous with respect to the distribution of $S_i$ conditionally on $(\Lambda_i,\mathcal W_n)$. Therefore, letting $\mathbb M_p$ denote the set of $p\times p$ real matrices and defining the sets

we obtain from result (ref) holding with probability one over $(\Lambda_i,\mathcal W_n)$ that we have

Moreover, since the sign of $a^\prime G a$ equals the sign of $a^\prime G a/\|a\|_1^2$ for any matrix $G\in \mathbb M^p$ and vector $0\neq a\in \mathbb R^p$, Assumption (ref)(iv) allows us to conclude that

In particular, combining (ref) with (ref) yields that the distribution of $\Gamma_{i,n}$ satisfies

To conclude, note that by Lemma (ref) we have $P(\Lambda_i \geq 0) =1$ or $P(\Lambda_i \leq 0) = 1$. However, if $P(\Lambda_i \geq 0)=1$, then $\Gamma_{i,n} = \Lambda_i \text{Var}\{\mathcal Z|\mathcal W_n\}$ and Assumption (ref)(i) imply

where the final two inequalities hold by set inclusion. By result (ref) we thus obtain

where the final equality follows from (ref). Similarly, it is possible to show that if $P(\Lambda_i \leq 0) =1$, then $P(\sup_{a\in \mathbb R^p_+}a^\prime \Gamma_{i,n}a\leq 0) = 1$, and therefore the lemma follows. \rule{2mm}{2mm}

Proof. We will proceed by contradiction and instead suppose that in fact we have that

For notational simplicity, let $\mathcal W_n = \{W_i\}_{i=1}^n$, set $\Gamma_{i,n}\equiv \Lambda_i \text{Var}\{\mathcal Z|\mathcal W_n\}$, and define

Next note that if $\lambda_{ij} > 0$ for some $1\leq j \leq p$ (resp.\ $\lambda_{ij} < 0$ for some $1\leq j \leq p$) then $A_+(\Gamma_{i,n})$ (rep.\ $A_{-}(\Gamma_{i,n})$) has non-empty interior whenever $\text{Var}\{\mathcal Z_j|\mathcal W_n\} > 0$ for all $1\leq j \leq p$. Therefore, since $s \in A_+(\Gamma_{i,n})$ (resp.\ $A_{-}(\Gamma_{i,n})$) if and only if $s/\|s\|_1\in A_+(\Gamma_{i,n})$ (resp.\ $s/\|s\|_1\in A_{-}(\Gamma_{i,n})$), we obtain by Assumptions (ref)(i)(iii) that

Similarly, also note that display (ref) holding and Assumption (ref)(iii) together imply

To conclude, observe that for $n \geq 2$, display (ref) and Assumptions (ref)(i)(ii) yield

where the final inequality follows from results (ref) and (ref). Hence, we have arrived at a contradiction implying (ref) cannot hold. \rule{2mm}{2mm}

Proofs for Section (ref)

In this appendix we collect the technical results behind Proposition (ref). Section (ref) introduces the framework, Section (ref) provides an outline of the main argument, and Sections (ref) and (ref) establish the two key building blocks for the main result.

Preliminaries and Definitions

We consider a probability space $\left( \Xi\times\Psi,\mathcal{F}\times\mathcal{C},P_{u}\times P_{f}\right) $ where $\Xi$ and $\Psi$ are Polish spaces with their respective Borel $\sigma$-algebras $\mathcal{F}$ and $\mathcal{C}$; see, p.\ 270 in dudley1989real. An example of such spaces is $\Xi=\left( \mathbb{R}^{\infty}\times....\times\mathbb{R}^{\infty}\right)$; i.e.\ a finite number of products of $\mathbb{R}^{\infty}$. Let $\Omega\equiv\Xi\times\Psi,$ $\mathcal{X}\equiv \mathcal{F}\times\mathcal{C}$ and $P\equiv P_{u}\times P_{f}$ so that the probability space can be represented more compactly as $\left(\Omega,\mathcal{X},P\right)$. It is useful for later developments to impose further structure on $\left( \Xi,\mathcal{F},P_{u}\right) $. We assume that $\left\{ \left( \Xi_{i},\mathcal{F}_{i},P_{u,i}\right) \right\}_{i=1}^{\infty}$ is a sequence of probability spaces and define $\Xi \equiv \Xi_{1} \times \Xi_{2} \times....$, $\mathcal{F} \equiv \mathcal{F}_{1} \times \mathcal{F} _{2} \times...,$ and $P_{u} \equiv P_{u,1}\times P_{u,2}\times...$ such that $\left( \Xi,\mathcal{F},P_{u}\right) $ is an infinite dimensional product space. Let $\mathcal{F}_{t}^{i}\subset\mathcal{F}_i$ denote an array of filtrations, where ${\mathcal F}_{-\infty}^{i} =\left\{ \mathcal{\emptyset},\Xi_{i}\right\}$, ${\mathcal F}_{\infty}^{i}={\mathcal F}_{i}$, and $\mathcal{F}_{t}^{i} \subset \mathcal{F}_{t+1}^{i}$ for $i\leq n$.\footnote{We can allow for triangular arrays, but suppress it for simplicity of notations.} Similarly, let $\mathcal{C}_{t}\subset\mathcal{C}$ denote a triangular array of filtrations, where $\mathcal{C}_{-\infty} =\left\{ \mathcal{\emptyset},\Psi\right\}$, $\mathcal{C}_{\infty} = \mathcal{C}$, and $\mathcal{C}_{t} \subset \mathcal{C}_{t+1} $. In addition, define the filtrations $\mathcal{H}_{t}^{i}\equiv\mathcal{F}_{t}^{i}\times\mathcal{C}_{t}$. Then, $\mathcal{F}\times\mathcal{C}_{t}\subset\mathcal{X}$ for all $t$ and $\mathcal{H}_{t}^{i}$ can be embedded in $\mathcal{X}$ for each $i$ and all $t\leq T$ by padding up additional coordinates; see, e.g., page 140 in halmos1976measure. In addition, note that $\mathcal{H} _{t}^{i}\subset\mathcal{H}_{t+1}^{i}$ for all $i$, and $t$.

In order to simplify our analysis, we will assume that $\mathcal{F}_{t}^{i} =\sigma\left( ....,\eta_{i,t-1} ,\eta_{it}\right)$, where $\left( \eta_{it}\right) _{i=1,t=-\infty}^{\infty,\infty}$ is an array of some random variables on $\left( \Xi,\mathcal{F},P_{u}\right) $ where for each $i$, $\left( \eta_{it}\right) _{t=-\infty}^{\infty}$ is an array of random variables defined on $\left( \Xi_{i},\mathcal{F}_{i},P_{u,i}\right) $. The product space structure of $\left( \Xi,\mathcal{F},P_{u}\right) $ immediately implies independence of $\left( \eta_{it}\right) _{t=-\infty}^{\infty}$ and $\left( \eta_{jt}\right) _{t=-\infty }^{\infty}$ for any $i\neq j$. Likewise, we assume that $\mathcal{C}_{t} =\sigma\left( ...,v_{t-1},v_{t}\right)$, where $\left( v_{t}\right) _{t=-\infty}^{\infty}$ is a sequence of some random variables on $\left( \Psi,\mathcal{C} ,P_{f}\right) $.

We will introduce mixing measures as in mcleish1975; see also andrews1988 for triangular array versions of these measures. Recall $\mathcal{C}_{t}=\sigma\left( ...,v_{t-1},v_t\right) $ and define $\mathcal{C}_{t}^{t+m}\equiv \sigma\left( v_{t},v_{t+1},...,v_{t+m}\right) $ and $\mathcal{C} _{t}^{\infty}\equiv\sigma\left( v_{t},v_{t+1},...\right) $. Similarly, we let $$\mathcal{F}_{t}^{i,t+m} \equiv\sigma\left( \eta_{it} ,...,\eta_{i,t+m}\right) ~\text{for all }i\leq n. $$ Our asymptotic normality results employ $\alpha$-mixing coefficients, and we therefore define \[ \alpha_{f}\left( m\right) \equiv \sup_{t}\sup_{A\in\mathcal{C}_{t} ,B\in\mathcal{C}_{t+m}^{\infty}}\left\vert P\left( A\cap B\right) -P\left( A\right) P\left( B\right) \right\vert . \]

We will now impose the following restrictions on the measures $P_{u}$ and $P_{f}$. By Theorems 4.34 and A.46 in breiman1968probability, a regular conditional distribution on $\mathcal{X}$ given $\mathcal{C}\subset\mathcal{X}$ exists, and by Theorem 10.2.2 in dudley1989real, the regular conditional distribution is unique for $P$-almost all $\omega\in\Omega$. As in eaglson1975, let $\omega^{\prime}\in\Omega$ and consider the regular conditional (on $\mathcal{C}$) probability denoted by $Q_{\omega^{\prime}}\left( B,\mathcal{C}\right) =Q_{\omega^{\prime}}\left( B\right) $. It follows that for fixed $B\in\mathcal{X}$, $Q_{\omega^{\prime}}\left( B,\mathcal{C}\right) $ is a version of $P\left( \left. B\right\vert \mathcal{C}\right) $ and for fixed $\omega^{\prime}\in\Omega$, $Q_{\omega^{\prime}}\left( \cdot\right) $ is a probability measure on $\mathcal{X}$. Importantly, this means $Q_{\omega^{\prime}}\left( \cdot\right) $ is countably additive which ensures that the law of iterated expectations holds; see p.\ 270 in dudley1989real. We note that the results in dudley1989real are established for Polish spaces.

Consider the measure space $\left( \Omega,\mathcal{X},Q_{\omega^{\prime} }\right) $ with expectation $E_{\omega^{\prime}}$, which formalizes the idea of treating the aggregate variables $\varkappa_{t}$ (to be defined later) as constants through the choice of $\omega^{\prime}$. By a lemma in p.\ 558 of eaglson1975, the following holds:

For arbitrary $t$ and arbitrary $m\geq0$, we next define the filtration $\mathcal Y_t^{i,t+m}$ by

To establish marginal convergence as $T\rightarrow\infty$ for each $i\leq n$ fixed we follow the strategy of the proof of Theorem 2 in eaglson1975. This requires modifying the regularity conditions to the measure $Q_{\omega ^{\prime}}$. Define the conditional $L_{q}$ norm $\left\Vert y\right\Vert _{\left. q\right\vert \mathcal{C}}=\left( \int\left\vert y\right\vert ^{q}dP\left( \left. y\right\vert \mathcal{C}\right) \right) ^{1/q}$. By Lemma (ref) it follows that $\left\Vert y\right\Vert _{\left. q\right\vert \mathcal{C}}=\left( \int\left\vert y\right\vert ^{q} dQ_{\omega^{\prime}}\right) ^{1/q}$ $Q_{\omega^{\prime}}$ a.s. for $P$-almost all $\omega^{\prime}$. Similarly, define the conditional mixing coefficients

where $\alpha_{\left. \xi\right\vert \mathcal{C}}\left( m\right) $ are $\mathcal{C}$-measurable random variables.

Now, recall that our analysis is predicated on the moment restriction in (ref). In order to facilitate our asymptotic analysis, we impose the following:

It is important to note that the product structure $P_{u}=P_{u,1}\times P_{u,2}\times...$ implies that the variables $(u_{it})_{t=1}^{\infty}$ are independent over $i$. Finally, we define the variables \[ \xi_{it}\equiv(\mathcal{Z}_{t})^\prime \nu_{it}, \hspace{0.4 in} \varkappa_{t}\equiv(\mathcal{Z}_{t})^\prime \zeta_{t}, \] where $\zeta_t$ and $\nu_{it}$ are given by $\zeta_{t} \equiv E\left[\left. S_{it}\varepsilon_{it}\right| {\mathcal C}\right] = E\left[\left. S_{it}\varepsilon_{it}\right| f_t \right]$ and $\nu_{it} \equiv S_{it}\varepsilon_{it} - \zeta_{t}$.

Proof of Proposition (ref) (Outline)

Recall that Proposition (ref) derives an asymptotic distribution for the numerator (i.e.\ the score) of $(\hat \beta_n - \beta)$; see (ref). Given the introduced notation, we can decompose

We start the analysis by studying the conditional distribution given $\mathcal C$ of the term

By Condition (ref), $\xi_{it}$ are independent over $i$ conditional on $\mathcal{C}$, and we therefore expect that (ref) is asymptotically normal conditional on $\mathcal{C}$. Under additional regularity conditions, the limit distribution has a variance that does not depend on $\mathcal{C}$. Indeed, this is formally established in Theorem (ref) below. In particular, defining

we have that $\phi_{nT}\left( \left. \varsigma_{1}\right\vert \mathcal{C}\right) \rightarrow\phi\left( \varsigma_{1}\right) $ almost surely in $\mathcal C$, where $\phi\left( \varsigma_{1}\right) $ denotes the characteristic function of the limiting normal distribution.

Now, let's consider the joint characteristic function of the vector in display (ref): $$ E\left[ \exp\left( \iota\varsigma_{2}\frac{1}{\sqrt{T}}\sum_{t=1} ^{T}\varkappa_{t}+\iota\varsigma_{1}\frac{1}{\sqrt{nT} }\sum_{t=1}^{T}\sum_{i=1}^{n}\xi_{it}\right) \right] =E\left[ \exp\left( \iota\varsigma_{2}\frac{1}{\sqrt{T}}\sum _{t=1}^{T}\varkappa_{t}\right) \phi_{nT}\left( \left. \varsigma_{1}\right\vert \mathcal{C}\right) \right] $$ where we note that $\varkappa_t$ is measurable with respect to $\mathcal{C}$. Because $\sum_{t=1}^T E[\varkappa_{t}]=0$ due to the moment restriction in (ref), a time series CLT should apply to the term

Indeed, the asymptotic normality of (ref) is formally established in Theorem (ref) in Appendix (ref) below. Letting $\varphi\left(\varsigma_{2}\right) $ denote the characteristic function of the limiting normal distribution of the term in (ref), we then obtain that

Because characteristic functions are bounded by one and $\phi_{nT}\left( \left. \varsigma_{1}\right\vert \mathcal{C} \right) \rightarrow\phi\left( \varsigma_{1}\right) $ almost surely in $\mathcal C$ we obtain from the dominated convergence theorem that

Similarly, since (ref) is asymptotically normally distributed and $\varphi\left(\varsigma_{2}\right)$ denotes the characteristic function of its limiting distribution, we also have that

To conclude, we see that (ref) and (ref) are jointly asymptotically normally distributed and independent under the conditions laid out in Theorems (ref) and (ref).

Unconditional Convergence

The main purpose of this section is to establish a time series central limit theorem for $$\frac{1}{\sqrt T}\sum_{t=1}^{T}\varkappa_{t},$$ which is formally established in Theorem (ref) below. For this purpose, it is convenient to assume that $\varkappa_{t}$ is $L_{2}$ near-epoch-dependent (NED). The concept of NED sequences was introduced by billingsley1968convergence. mcleish1975 or andrews1988 show that a NED process is also a mixingale. Therefore assumptions that impose NED type conditions lead to strong laws by showing that these processes also satisfy the requirements for strong laws of related mixingales; see Theorem 3 in mcleish1975 for the first result of this nature.

We impose the following condition that establishes the NED property for $\varkappa_{t}$, and imposes sufficient conditions for $\varkappa_{t}$ to satisfy the conditions for the SLLN in DEJONG1996 and the CLT in dejong1997. In the statement below, we say that a sequence $\delta_{m}$ is of size $-\lambda$ if $\delta_{m}=O\left( m^{-\lambda-\omega}\right) $ for some $\omega>0$; see p.\ 335 in dejong1997.

To establish the CLT we adopt the conditions given in dejong1997. The result is based on a blocking scheme that needs to be defined. Let $b_{T}$ and $l_{T}$ be positive, non-decreasing integer valued sequences that are the lengths of included and discarded blocks. Assume $b_{T}\geq l_{T}+1,$ $l_{T}\rightarrow\infty$, $l_{T}\geq1,$ $b_{T}\leq T$, $b_{T}/T\rightarrow0$ and $l_{T}/b_{T}\rightarrow0$. Let $r_{T}\equiv\left[ T/b_{T}\right] $ and define $\tilde{\varkappa}_{T,t}\equiv \varkappa_{t}/\sigma_{T,\varkappa}$, where where $\sigma_{T,\varkappa}^{2}$ is given by $$\sigma_{T,\varkappa}^{2}\equiv E\left[ \left( \sum_{t=1}^{T}\varkappa_{t}\right) ^{2}\right]. $$

The following condition is sufficient for obtaining the desired CLT.

Conditional Analysis

The primary purpose of this section is to establish the asymptotic normality of the term $$\frac{1}{\sqrt{nT}}\sum_{t=1}^{T}\sum_{i=1}^{n}\xi_{it}$$ conditional on $\mathcal{C}$, which is formally shown in Theorem (ref). The proof uses ideas similar to the development in HKM, which in turn relies on arguments in eaglson1975, to handle the conditioning step in the proof of the CLT; see also Lemma 2.9.5 in vandervaart:wellner:1996. However, the dependence structure of the panel is more complicated here than in HKM and requires a different approach to prove the CLT. HKM consider a scenario where conditional on $\mathcal{C}$, a cross-sectional average over i.i.d.\ draws is analyzed. Here, we need to extend these results to a panel setting with joint asymptotics as $N,T\rightarrow\infty$ and where we allow for general dependence and possible non-stationarity in the time series direction. We extend the notation from HKM to account for these extensions. The result of eaglson1975 formalizes the intuition that conditional on $\mathcal{C}$ the processes $\mathcal{Z}_{t}$ and $f_{t}$ can be treated as a fixed constant in deriving limiting results.

We assume that $\alpha_{\left. \xi\right\vert \mathcal{C}}\left( m\right) \rightarrow0$ almost surely as $m\rightarrow\infty.$ This is done in the next condition, which introduces conditional NED, by requiring $\alpha_{\xi|\mathcal C}$ be of size $r/(r-2)$.

We now turn to establishing our main result. The argument is based on the one-to-one mapping $h_T$ of the double index $\left( i,t\right) $ into a single index $s=h_T(i,t) \equiv \left( i-1\right) T+t$ for $i\leq n$ and $L\equiv nT$. We will let $(\bar{\iota}_{T}(s), \bar{t}_{T}(s))$ denote the $(i,t)$ that corresponds to $s$, i.e., $(\bar{\iota}_{T}(s), \bar{t}_{T}(s)) \equiv h_{T}^{-1}(s)$.\footnote{Note $\bar{\iota}_{T}(s)$ is the smallest integer larger than or equal to $s/T$, and $\bar{t}_{T}(s) = s - (\bar{\iota}_{T}(s)-1)\cdot T$.} With some abuse of notation, we then set \[ \left( nT\right) ^{-1/2}\sum_{i=1}^{n}\sum_{t=1}^{T}\xi_{it}=L^{-1/2} \sum_{s=1}^{L}\xi_{L,s}. \] The sum on the right hand side sums over components starting with $i=1$, $t=1,\ldots,T$ followed by $i=2,$ $t=1,\ldots,T$, and so on. Similarly, we construct the array of filtrations based on (ref), using coordinate identification rules of halmos1976measure (see p.\ 151 and p.\ 155) to organize coordinates, by defining

The construction guarantees that $\mathcal{K}_{L,s}\subset \mathcal{X}$ by padding missing coordinates with the trivial field $\mathcal{F}_{-\infty}^{j}=\left\{ \emptyset,\Xi_{j}\right\} $.

We introduce the binary operator $\ominus$ that maps two $\sigma$-fields generated by a sequence of random variables into a $\sigma$-field generated by the non-overlapping portion of members of the sequence. In particular, for $0<s<s^{\prime}$ and $\mathcal{F}_{-\infty}^{j,s^{\prime}},$ $\mathcal{F}_{-\infty}^{j,s}$ we define $\mathcal{F}_{-\infty}^{j,s^{\prime}}\ominus\mathcal{F}_{-\infty}^{j,s}\equiv\mathcal{F}_{s}^{j,s^{\prime}}$, where $\mathcal{F}_{t}^{i,t+m}=\sigma\left(\eta_{it},...,\eta_{i,t+m}\right).$ We further define $\ominus$ to have the properties $\left(\operatorname*{\scalebox{1.8}{$\times$}}_{j=1}^{\infty}\mathcal{A}_{nT}^{j}\right)\ominus\left(\operatorname*{\scalebox{1.8}{$\times$}}_{j=1}^{\infty}\mathcal{B}_{nT}^{j}\right)\equiv\operatorname*{\scalebox{1.8}{$\times$}}_{j=1}^{\infty}\left(\mathcal{A}_{nT}^{j}\ominus\mathcal{B}_{nT}^{j}\right)$ for $\mathcal{A}_{nT}^{j},\mathcal{B}_{nT}^{j}\subset\mathcal{F}_{-\infty}^{j,\infty}$, where we understand $\mathcal{F}_{-\infty}^{j,\infty}\ominus\mathcal{F}_{-\infty}^{j,\infty}\equiv\mathcal{F}_{-\infty}^{j}$, $\mathcal{F}_{-\infty}^{j}\ominus\mathcal{F}_{-\infty}^{j}\equiv\mathcal{F}_{-\infty}^{j}$ and ${\mathcal{C}}\ominus{\mathcal{C}}\equiv{\mathcal{C}}$. Using these properties we define the sigma fields $\mathcal{K}_{L,s}^{s^{\prime}}$ by $\mathcal{K}_{L,s}^{s^{\prime}}\equiv\mathcal{K}_{L,s^{\prime}}\ominus\mathcal{K}_{L,s}$.

Because $\mathcal{K}_{L,s}$ contains the coordinate $\mathcal{C}$ for all $L$ and $s$ the construction implies that when conditioning on $\mathcal{K}_{L,s} $, all $\varkappa_{t}$ and $\zeta_{t}$ are held fixed. We have the following Lemma relating mixing coefficients for $\left\{ \xi_{L,s},\mathcal{K} _{L,s}\right\} $ to mixing coefficients for $\left\{ \xi_{it} ,\mathcal{Y}_{-\infty}^{i,t}\right\} $.\footnote{Condition (ref) includes uniformity of the mixing and approximation coefficients, which is used here.}

Lemma (ref) is the basis for establishing a joint CLT as both $n,T\rightarrow\infty.$ For this, the earlier blocking definitions need to be adjusted. Let $L\equiv nT,$ and $n,T\rightarrow\infty$. Define $b_{L}$ and $l_{L}$ to be positive, non-decreasing integer valued sequences that are the length of included and discarded blocks. Assume $b_{L}\geq l_{L}+1,$ $l_{L} \rightarrow\infty,$ $l_{L}\geq1,$ $b_{L}\leq n$, $b_{L}/L\rightarrow0$ and $l_{L}/b_{L}\rightarrow0$. Let $r_{L}\equiv\left[ L/b_{L}\right] $ and define $\tilde{\xi}_{L,s}\equiv\xi_{L,s}/\sigma_{L,\xi}(\omega)$,\footnote{ Note that $\sigma_{L,\xi}$ in (ref) is a random variable. In order to emphasize it, we write $\tilde{\xi}_{L,s}\equiv\xi_{L,s}/\sigma_{L,\xi}(\omega)$. } where

We impose the following conditions that are modifications of the conditions for the case of marginal convergence when $T\rightarrow\infty.$

We now state the following conditional CLT for joint convergence.

Proofs for Section (ref)

In this appendix we provide a formal justification of the inference procedure proposed in Section (ref) by relying on the high dimensional central limit theorem of chernozhuokov2022improved. To this end, we impose the following assumptions.

Assumption (ref) demands that the vector of moments be asymptotically equivalent to a $q$-dimensional sample mean of random variables $\{\psi_{i}\}_{i=1}^{b_n}$. We note that Assumption (ref) effectively requires that the null hypothesis be true by requiring that the variables $\{\psi_i\}_{i=1}^{b_n}$ have mean zero. In turn, Assumptions (ref) and (ref) imposes moment restrictions on the variables $\{\psi_i\}_{i=1}^{b_n}$ that ensure that the high dimensional central limit theorem of chernozhuokov2022improved is applicable. Finally, Assumption (ref) demands a linearization requirement on our bootstrap statistic, while Assumption (ref) states requirements on the weights $\{\omega_i\}_{i=1}^{b_n}$ that we may employ. We note, in particular, that Assumption (ref) allows for the empirical bootstrap (through Assumption (ref)(i)) and the use of Standard Gaussian, Rademacher, or mammen:1993 weights (through Assumption (ref)(ii)).

Our next result encompasses Proposition (ref) as a special case. The first and second parts of the result provide conditions under which the level of the test is $1-\alpha$ unconditionally on $\mathcal G_n$ and conditionally on $\mathcal G_n$ respectively. We view the unconditional result as appropriate for the asymptotic framework in adao2019shift (in which elements of $\mathcal G_n$ are resampled), and the conditional result as more suitable for the asymptotic framework in goldsmith2020bartik (in which $\mathcal G_n$ is not resampled).

Proof. We begin by defining the conditional covariance matrix $\Sigma(\mathcal G_n)$ to be given by

and letting $\mathbb T_n \equiv \|\mathbb G_n\|_\infty$ for $\mathbb G_n\in \mathbb R^q$ a Gaussian vector satisfying $\mathbb G_n \sim N(0,\Sigma(\mathcal G_n))$. Further denote the linearized versions of $T_n$ and $T_n^*$ by letting $L_n$ and $L_n^*$ equal

and for notational convenience set $\delta_n(\mathcal G_n) \equiv (B_n^2(\mathcal G_n)\log^5(qb_n)/b_n)^{1/4}$. By Theorem 2.1 in chernozhuokov2022improved there then exists a $C_1$ not depending on $\mathcal G_n$ such that

Next note that, for any constant $\varrho > 0$, result (ref) allows us to conclude that

where the third inequality holds holds for some constant $C_{2}$ not depending on $\mathcal{G}_{n}$ by Lemma J.3 in chernozhuokov2022improved, and the final inequality holds for any $\eta > 0$ by result (ref), Assumption (ref), and setting $\varrho = \eta/\sqrt{\log(q)}$.

For $\mathcal D_n \equiv (\{Y_i,X_i,S_i\}_{i=1}^n,\mathcal Z,\mathcal G_n)$ and any constant $C_3 > 0$ next define the event

By Assumption (ref) and Lemmas 4.5 and 4.6 in chernozhuokov2022improved it then follows that we may select a constant $C_3$ not depending on $\mathcal G_n$ under which we have

Applying (ref) and the same arguments as in (ref) we obtain that if $\mathcal{E}(\mathcal{D}_{n})=1$, then

for any constant $\eta>0$. Next, plug in $t=\hat{c}_{n}b_{n}^{-1/2}$ into result (ref) and note that result (ref) then implies that whenever $\mathcal E(\mathcal D_n) = 1$ we must have

where the final equality is definitional.

To conclude, let $\mathcal A_n$ be a sigma field satisfying $\mathcal G_n \subseteq \mathcal A_n$ and note that for any $\varrho > 0$ we obtain from result (ref) and the law of iterated expectations that

Similarly, for any $\varrho > 0$ we may select $\eta > 0$ sufficiently small so as to ensure that

for some $\epsilon > 0$. Finally, observe that (ref) and the law of iterated expectations yield

Part (i) of the lemma therefore follows from $\varrho$ being arbitrary, results (ref), (ref), (ref), and setting $\mathcal A_n$ to be the trivial sigma field. Part (ii) of the lemma similarly follows from $\varrho$ being arbitrary, results (ref), (ref), (ref), and setting $\mathcal A_n = \mathcal G_n$. \rule{2mm}{2mm}

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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

We first examine the overidentification test introduced in Section (ref), which is designed for applications in which $\mathcal G_n$ denotes a set of aggregate shocks that include $\mathcal Z$. Recall that in Section (ref) we set $\psi_{ij} \equiv U_{ij}/\sigma_j$ and $\hat \psi_{ij} \equiv \hat U_{ij}/\hat \sigma_j$ with $U_{ij}$ and $\hat U_{ij}$ given by

where $A_i = (Z_i,W_i^\prime)^\prime$ and $\sigma_j^2$ and $\hat \sigma_j^2$ denote the population and sample variances

It will also prove helpful to define the variables $\{R_{ij}\}_{i=1}^n$ for $1\leq j \leq p$ according to

In particular, by definition of $T_n$ and standard manipulations it then follows that

In order to apply Lemma (ref)(ii), to justify the asymptotic validity of the overidentification test of Section (ref), we require suitable moment conditions and that

where probability statements are understood to be conditionally on $\mathcal G_n$ and requirements (ref) and (ref) to hold almost surely in $\mathcal G_n$. By relying on Lemma D.5 in chernozhukov2019inference, it is possible to show that requirement (ref) in fact implies that

where, again, probabilities are understood to be conditionally on $\mathcal G_n$ and (ref) to hold almost surely in $\mathcal G_n$. Moreover, by the triangle inequality and condition (ref) we have

where the final result follows from result (ref) and a standard maximal inequality; see, e.g., Lemma 2.2.2 in vandervaart:wellner:1996. Result (ref) together with (ref) and $\psi_{ij} \equiv U_{ij}/\sigma_j$ imply that Assumption (ref) holds with $b_n = n$, $q = p$, and $r_{1n}(\varrho,\mathcal G_n)$ satisfying $r_{1n}(\varrho,\mathcal G_n) = o_{as}(1)$ for any $\varrho > 0$.

In order to verify Assumption (ref), recall that $\mathcal D_n\equiv (\{Y_i,X_i,S_i\}_{i=1}^n,\mathcal Z,\mathcal G_n)$ and note that if $\{\omega_i\}_{i=1}^n$ are i.i.d.\ standard normal random variables independent of $\mathcal D_n$, then a standard maximal inequality yields that

Moreover, employing that $\hat \psi_{ij} \equiv \hat U_{ij}/\hat \sigma_j$ and $\psi_{ij} \equiv U_{ij}/\sigma_j$ we can bound (ref) by

where the final result holds by (ref) and a standard maximal inequality. By Markov's inequality and result (ref) it follows that Assumption (ref) holds for some $r_{2n}(\varrho,\mathcal D_n)$ satisfying $P(r_{2n}(\varrho,\mathcal D_n) > \epsilon|\mathcal D_n) = o_{as}(1)$ for any $\varrho > 0$ as required by Lemma (ref).

Identification Through Shocks

We next discuss the overidentification test of Section (ref), which is designed for applications in which identification is driven by exogeneity of the shocks $\mathcal Z$. Recall that in the corresponding asymptotic framework, originally developed by adao2019shift, we set $\mathcal G_n = \{S_i,W_i,\varepsilon_i\}_{i=1}^n$. Following the notation in Section (ref), we further set

where $\mathcal E \equiv \mathcal Z - E[\mathcal Z|\mathcal G_n]$ and we note that $\delta_j$ and $\kappa_j$ depend on $n$ (through $\mathcal G_n$), but we suppress the dependence from the notation. As estimators for $\delta_j$ and $\kappa_j$ we employ

where $\hat \pi_n$ denotes the coefficient from regressing $\{Z_i\}_{i=1}^n$ on $\{W_i\}_{i=1}^n$. In addition let

for $\hat {\mathcal E}_i$ an estimator of $\mathcal E_i$ (see Remark (ref)), and recall that $\psi_{ij} \equiv U_{ij}/\sigma_j$ and $\hat \psi_{ij} \equiv \hat U_{ij}/\hat \sigma_j$, where $\sigma_j$ and $\hat \sigma_j$ respectively denote the population and finite sample variances

It will also prove convenient to define the variables $\{R_{ij}\}_{i=1}^n$ for $1\leq j \leq q$ according to

The asymptotic validity of the overidentification test of Section (ref) may be justified by employing Lemma (ref)(i). In order to appeal to Lemma (ref)(i), first note that if $\{\mathcal E_i\}_{i=1}^p$ are (uniformly) Sub-Gaussian almost surely in $\mathcal G_n$, then Assumption (ref) can be verified by setting $B_n(\mathcal G_n) = KC_n$ for $K$ large enough and $C_n$ given by

In turn, Assumptions (ref) and (ref) can be verified under the key requirements

where the convergence in probability statement should be understood as jointly over all the data (rather than conditionally on $\mathcal G_n$). In particular, under the condition that

for some $0<c<1$, it is possible to argue by relying on Lemma D.5 in chernozhukov2019inference that requirement (ref) in fact implies that

Moreover, by applying Lemma D.3 in chernozhukov2019inference and relying on the rate condition in (ref) it is also possible to obtain the rate bounds

Combining results (ref) and (ref) with the same arguments employed in Section (ref), it is then straightforward to show that conditions (ref) and (ref) imply Assumptions (ref) and (ref) hold with $b_n = p$ and $r_{1n}(\varrho,\mathcal G_n) \vee r_{2n}(\varrho,\mathcal G_n) = o_P(1).$ To conclude verifying the main requirements of Lemma (ref)(i) we note that the condition $B_n^2(\mathcal G_n) \log^5(qp)/p = o_P(1)$ is implied by requirement (ref) (up to logs).

Condition (ref) is more challenging to verify than its analogue in Section (ref) (i.e.\ (ref)) because there are $n$ terms $\{R_{ij}\}$ but $p$ terms $\{U_{ij}\}$. Fortunately, as we next outline, it is possible to establish that (ref) holds by building on the assumptions and arguments in adao2019shift. To this end, we start with the decomposition

It is also helpful to note that since $Z_i = S_i^\prime \mathcal Z$, $E[Z_i|\mathcal G_n] = W_i^\prime \pi_n$ under the null hypothesis, and $E[Z_i|\mathcal G_n] = S_i^\prime E[\mathcal Z|\mathcal G_n]$ due to $S_i\in \mathcal G_n$, it follows that

Next, note that (ref), the definition of $\delta_j$ in (ref), the equality $S_k^\prime \mathcal E = \sum_{i=1}^p S_{ki} \mathcal E_i$, and some algebra allows us to express term (ref) as being equal to

We analyze the term in (ref) through a linearization argument. To this end, we set

and, following adao2019shift, we let $n_k \equiv \sum_{i=1}^n S_{ki}$ and set $r_n = (\sum_{k=1}^p n_k^2 )^{-1}$. Then note that by result (ref) and a standard Taylor expansion we obtain that

where $\tilde \varepsilon_i$ is some intermediate value between $\hat \varepsilon_i$ and $\varepsilon_i$. If the covariates $W_i$ are bounded almost surely, then a maximal inequality (applied conditionally on $\mathcal G_n$) yields

where the final equality follows by definition of $r_n$. We can therefore conclude that

where the probability is understood to be over the entire data. Similarly, adapting the arguments in the proof of Proposition 3 in adao2019shift (see in particular the proof of their display (A.4)) and employing a maximal inequality for degenerate U-statistics (see, e.g., equation (3.5) in gine2000exponential) it is possible to establish that

Moreover, the arguments in adao2019shift can additionally be used to conclude that

while sup norm bound on the quadratic term in (ref) and the definition of $M_{2q}$ imply

Thus, since result (ref) implies that $|\hat \beta_n - \beta| = O_P((n\sqrt{r_n})^{-1})$, our analysis so far yields

Finally, using that $\|\hat \pi_n - \pi_n\|\vee |\hat \beta - \beta| = O_P((n\sqrt{r_n})^{-1})$ and relying on the mean value theorem allows to bound in probability the term in (ref) by

To simplify our bounds, we suppose that $\|\hat \gamma - \gamma_{\mathtt{s}}\|$ has the same rate of convergence as $|\hat \beta - \beta|$ so that $\|\hat \gamma - \gamma_{\mathtt{s}}\| = O_P((n\sqrt{r_n})^{-1})$. Combining our analysis of the terms in (ref)-(ref) together with the definition of $R_{ij}$ and $U_{ij}$ we can then conclude that

Thus, finally setting $\underline{\sigma} \equiv \min_{1\leq j \leq q} \sigma_j$ we obtain that (ref) is implied by the condition

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 david2013china dataset. In particular, as in david2013china, 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 david2013china 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

where $S_{xit}$ and $S_{zit}$ represent share vectors for commuting zone $i$ at time $t$ and $\mathcal Z_{xt}$ and $\mathcal Z_{zt}$ denote aggregate shocks at time $t$. Finally, because david2013china 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

Since $T =2$ and $p = 397$ in the context of david2013china, result (ref) represents a total of 794 possible moment restrictions. Moreover, because david2013china 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 david2013china by employing a heteroskedastic version of the group shock model of moulton1986random. 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 david2013china dataset, we then estimate a model in which the errors $\varepsilon_{it}$ are assumed to have the structure

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

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 david2013china and let

where

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 david2013china, which we keep fixed across all the simulations. \rule{2mm}{2mm}

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 $\{S_{xit},S_{zit}\}_{i=1}^n$. \rule{2mm}{2mm}

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 $X_{it}^{\ast}\equiv (S_{xit}^{\ast})^{\prime}\mathcal{Z}_{xt}$. \rule{2mm}{2mm}

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

by employing the full sample estimates $\hat a_\eta,\hat s_\eta,\hat a_\zeta$ and $\hat s_\zeta$ from (ref). \rule{2mm}{2mm}

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

where $(\hat \beta,\hat \gamma_{\mathtt{s}})$ denote the weighted instrumental variable estimators from the main specification in david2013china and the “$c$" subscript is understood to refer to the state to which commuting zone $i$ belongs. \rule{2mm}{2mm}

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 david2013china. All critical values were obtained by following the procedure in Section (ref) with one thousand bootstrap draws.

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

where

In order to ensure that the null hypothesis holds in our simulation design, we rely on a model proposed by adao2019shift as a sufficient condition for (ref). Specifically, we suppose that for some $p\times d$ matrix of shock $\mathcal Z_{wt}$ and $d\times 1$ vector $\Gamma$ we have

To estimate this model in the original david2013china dataset, we first compute a ridge regression of the coordinates of $W_{it}\in \mathbb R^{d}$ on $S_{zit}$ by setting

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

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

We further aim to reflect the clustering structure in david2013china. To this end, we follow adao2019shift and borusyak2022quasi who in re-examining the empirical analysis of david2013china cluster shocks at the three digit SIC code. As in Section (ref), we estimate a common shock model in which $\mathcal Z_{zt}$ satisfies

where $\mathcal Z_{ztj}$ denotes the $j^{th}$ coordinate of $\mathcal Z_{zt}$, $\eta_c$ 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

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

Finally, in order to reflect the strength of the instrument in david2013china in our simulation design, we run the following regression on the aggregate shocks

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

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

Step 1. We first create controls $\hat W_{it} \equiv \hat\mathcal Z_{wt}^\prime S_{zit}$, 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 david2013china and keep all of them fixed across simulation designs. \rule{2mm}{2mm}

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

where the “$c$" subscript refers to the three digit SIC code to which sector $j$ belongs. As our instrument we then employ $Z_{it}^* \equiv S_{zt}^\prime \mathcal Z^*_{zt}$. Note that, because $\hat E[\mathcal Z_{zt}|\mathcal G_n] \equiv \hat \mathcal Z_{wt} \hat \Gamma$, the shocks $\mathcal Z^*_{zt}$ have the structure required by model (ref). \rule{2mm}{2mm}

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

As our regressor, we then employ $X_{it}^* \equiv S_{xit}^\prime \mathcal Z_{xt}^*$. \rule{2mm}{2mm}

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

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 david2013china. \rule{2mm}{2mm}

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

In the main specification of david2013china, there are no control variables with the structure required by the estimator for $\mathcal E_t^*$ proposed by borusyak2022quasi. We therefore instead adapt the estimator of $\mathcal E^*_t$ advocated by adao2019shift by employing

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 adao2019shift who use ordinary regression (i.e.\ $\lambda = 0$), but instead drop sectors from the regression to address the ill conditioning of the design matrix.\footnote{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.\footnote{Analogously, as noted by adao2019shift, 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

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 david2013china. 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.

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