Two-way fixed effects instrumental variable regressions
in staggered DID-IV designs.^††thanks: I am grateful to Daiji Kawaguchi and Ryo Okui for their continued guidance and support. All errors are my own.

Our paper is related to the recent DID-IV literature (de Chaisemartin (2010); Hudson et al. (2017); de Chaisemartin and D’Haultfœuille (2018); Miyaji (2024)). In this literature, de Chaisemartin (2010) first formalizes $2\times 2$ DID-IV designs and shows that a Wald-DID estimand identifies the local average treatment effect on the treated (LATET) if the parallel trends assumptions in the treatment and the outcome, and a monotonicity assumption are satisfied. Hudson et al. (2017) also consider $2\times 2$ DID-IV designs with non-binary, ordered treatment settings. Build on the work in de Chaisemartin (2010), however, de Chaisemartin and D’Haultfœuille (2018) formalize $2\times 2$ DID-IV designs differently, and call them Fuzzy DID. Miyaji (2024) compares $2\times 2$ DID-IV to Fuzzy DID designs and points out the issues embedded in Fuzzy DID designs, and extends $2\times 2$ DID-IV design to multiple period settings with the staggered adoption of the instrument across units, which the author calls staggered DID-IV designs. Miyaji (2024) also provides a reliable estimation method in staggered DID-IV designs that is robust to treatment effect heterogeneity.

In this paper, we contribute to the literature by showing the properties of two-way fixed instrumental variable estimators in staggered DID-IV designs. In reality, when empirical researchers implicitly rely on the staggered DID-IV design, they commonly implement this design via TWFEIV regressions (e.g. Black et al. (2005), Lundborg et al. (2014), Meghir et al. (2018)). This paper presents the issues of the conventional approach, and provides the sufficient conditions for this estimand to attain its causal interpretation.

Our paper is also related to a recent DID literature on the causal interpretation of two-way fixed effects (TWFE) regressions and its dynamic specifications under heterogeneous treatment effects (Athey and Imbens (2022); Borusyak et al. (2021); de Chaisemartin and D’Haultfœuille (2020); Goodman-Bacon (2021); Imai and Kim (2021); Sun and Abraham (2021)).

Specifically, this paper is closely connected to Goodman-Bacon (2021), who derives the decomposition theorem for the TWFE estimator with settings of the staggered adoption of the treatment across units. In this paper, we establish the decompose theorem for the TWFEIV estimator with settings of the staggered adoption of the instrument across units, which is a natural generalization of their theorem 1.

This paper is also closely connected to de Chaisemartin and D’Haultfœuille (2020), who decompose the TWFE estimand and present the issue of using this estimand under DID designs: some weights assigned to the causal parameters in this estimand can be potentially negative. In their appendix, the authors also decompose the TWFEIV estimand and refer to the negative weight problem in this estimand. Specifically, they apply the decomposition theorem for the TWFE estimand to the numerator and denominator in the TWFEIV estimand respectively, and conclude that this estimand identifies the LATE as in Imbens and Angrist (1994) only if the effects of the instrument on the treatment and outcome are constant across groups and over time. However, their decomposition result for the TWFEIV estimand has some drawbacks. First, they do not formally state the target parameter and identifying assumptions in DID-IV designs. Second, their decomposition result is not based on the target parameter in DID-IV designs. Finally, the sufficient conditions for this estimand to be interpretable causal parameter are not well investigated.

In this paper, we investigate the causal interpretation of the TWFEIV estimand more clearly than that of de Chaisemartin and D’Haultfœuille (2020). Specifically, we first decompose the TWFEIV estimator into all possible $2\times 2$ Wald-DID estimators. We then formally introduce the target parameter and identifying assumptions in staggered DID-IV designs, built on the recent work in Miyaji (2024). This allows us to decompose the TWFEIV estimand into a weighted average of the target parameter in staggered DID-IV designs. Finally, we assess the causal interpretation of the TWFEIV estimand under a variety of restrictions on the effects of the instrument on the treatment and outcome, which clarifies the sufficient conditions for this estimand to attain its causal interpretation.

We note that this paper is distinct from the recent IV literature on the causal interpretation of two stage least square (TSLS) estimators with covariates under heterogeneous treatment effects (Słoczyński (2020), Blandhol et al. (2022)). These recent studies investigate the causal interpretation of the TSLS estimand with covariates under the random variation of the instrument conditional on covariates, and cast doubt on the LATE (or LATEs) interpretation of this estimand. In this literature, the identifying variations come from the assignment process of the instrument. In this paper, however, we investigate the causal interpretation of the TWFEIV estimand (where time and unit dummies can be viewed as covariates) under staggered DID-IV designs: our identifying variations mainly come from the parallel trends assumptions in the treatment and the outcome over time.

We introduce the notation we use throughout this article. We consider a panel data setting with $T$ periods and $N$ units. For each $i\in\{1,\dots N\}$ and $t\in\{1,\dots,T\}$, let $Y_{i,t}$ denote the outcome and $D_{i,t}\in\{0,1\}$ denote the treatment status, and $Z_{i,t}\in\{0,1\}$ denote the instrument status. Let $D_{i}=(D_{i,1},\dots,D_{i,T})$ and $Z_{i}=(Z_{i,1},\dots,Z_{i,T})$ denote the path of the treatment and the path of the instrument for unit $i$, respectively. Throughout this article, we assume that $\{Y_{i,t},D_{i,t},Z_{i,t}\}_{t=1}^{T}$ are independent and identically distributed (i.i.d).

We make the following assumption about the assignment process of the instrument.

Assumption 1 requires that once units start exposed to the instrument, they remain exposed to that instrument afterward. In the DID literature, several recent papers impose this assumption on the adoption process of the treatment and sometimes call it the "staggered treatment adoption", see, e.g., Athey and Imbens (2022), Callaway and Sant’Anna (2021) and Sun and Abraham (2021).

Given Assumption 1, we can uniquely characterize the instrument path by the time period when unit $i$ is first exposed to the instrument, denoted by $E_{i}=\min\{t:Z_{i,t}=1\}$. If unit $i$ is not exposed to the instrument for all time periods, we define $E_{i}=\infty$. Based on the initial exposure period $E_{i}$, we can uniquely partition units into mutually exclusive and exhaustive cohorts $e$ for $e\in\{1,2,\dots,T,\infty\}$: all the units in cohort $e$ are first exposed to the instrument at time $E_{i}=e$. Hereafter, to ease the notation, we assume that the data contain $K$ cohorts $(K\leq T)$ where $e\in\{1,\dots,k,\dots,K\}$, and define $U$ as the never exposed cohort $E_{i}=\infty$.

Let $n_{e}$ be the relative sample share for cohort $e$ and let $\bar{Z}_{e}$ be the time share of the exposure to the instrument for cohort $e$:

$\displaystyle n_{e}\equiv\frac{\sum_{i}{\mathbf{1}\{E_{i}=e\}}}{N},\hskip 8.53581pt\bar{Z}_{e}\equiv\frac{\sum_{t}{\mathbf{1}\{t\geq e\}}}{T}.$

We also define $n_{ab}\equiv\displaystyle\frac{n_{a}}{n_{a}+n_{b}}$ to be the relative sample share between cohort $a$ and $b$.

In contrast to the staggered adoption of the instrument across units, we allow the general adoption process for the treatment: the treatment can potentially turn on/off repeatedly over time. de Chaisemartin and D’Haultfœuille (2020) and Imai and Kim (2021) consider the same setting in the recent DID literature.

The notations $PRE(a)$, $MID(a,b)$, and $POST(a)$ represent the corresponding time window, respectively: $PRE(a)\equiv[1,a)$, $MID(a,b)\equiv[a,b)$, and $POST(a)\equiv[a,T]$. Let $\bar{R}_{e}^{POST(a)}$ be the sample mean of the random variable $R_{i,t}$ in cohort $e$ during the time window $POST(a)$:

$\displaystyle\bar{R}_{e}^{POST(a)}\equiv\frac{1}{T-(a-1)}\sum_{a}^{T}\left[\frac{\sum_{i}R_{i,t}\mathbf{1}\{E_{i}=e\}}{\sum_{i}\mathbf{1}\{E_{i}=e\}}\right].$

We define $\bar{R}_{e}^{PRE(a)}$ and $\bar{R}_{e}^{MID(a,b)}$ analogously, representing the sample mean of the random variable $R_{i,t}$ in cohort $e$ during the time window $PRE(a)$ and $MID(a,b)$ respectively.

We consider a TWFEIV regression in multiple time period settings with the staggered adoption of the instrument across units:

		$\displaystyle Y_{i,t}=\phi_{i.}+\lambda_{t.}+\beta_{IV}D_{i,t}+v_{i,t},$		(3)
		$\displaystyle D_{i,t}=\gamma_{i.}+\zeta_{t.}+\pi Z_{i,t}+\eta_{i,t}.$		(4)

By substituting the first stage regression (4) into the structural equation (3), we obtain the reduced form regression:

$\displaystyle Y_{i,t}=\phi_{i.}+\lambda_{t.}+\alpha Z_{i,t}+v_{i,t}.$

(5)

The ratio between the first stage coefficient $\hat{\pi}$ and the reduced form coefficient $\hat{\alpha}$ yields the TWFEIV estimator $\hat{\beta}_{IV}$. By the Frisch-Waugh-Lovell theorem, the IV estimator $\hat{\beta}_{IV}$ is equal to the ratio between the coefficient from regressing $Y_{i,t}$ on the double-demeaning variable $\tilde{Z}_{i,t}$ and the coefficient from regressing $D_{i,t}$ on the same variable:

$\displaystyle\hat{\beta}_{IV}$

$\displaystyle=\frac{\frac{1}{NT}\sum_{i}\sum_{t}\tilde{Z}_{i,t}Y_{i,t}}{\frac{1}{NT}\sum_{i}\sum_{t}\tilde{Z}_{i,t}D_{i,t}},$

(6)

where $\tilde{Z}_{i,t}$ is the double demeaning variable defined below:

	$\displaystyle\tilde{Z}_{i,t}$	$\displaystyle=Z_{i,t}-\frac{1}{T}\sum_{t=1}^{T}Z_{i,t}-\frac{1}{N}\sum_{i=1}^{N}Z_{i,t}+\frac{1}{NT}\sum_{t=1}^{T}\sum_{i=1}^{N}Z_{i,t}$
		$\displaystyle\equiv(Z_{i,t}-\bar{Z}_{i})-(\bar{Z}_{t}-\bar{\bar{Z}}).$

Note that the TWFEIV regression runs the two-way fixed effects (TWFE) regression twice, as can be seen in equations (4) and (5). Because we assume the staggered assignment of the instrument across units, if we focus on the TWFE coefficient on $Z_{i,t}$ in the first stage or the reduced form regression, we can show that it is equal to a weighted average of all possible $2\times 2$ DID estimators of the treatment or the outcome from the decomposition result for the TWFE estimator shown by Goodman-Bacon (2021).

Consider the simple setting where we have only two periods and two cohorts: one cohort is not exposed to the instrument during the two periods ($E_{i}=U$), whereas the other cohort starts exposed to the instrument in the second period ($E_{i}=2$). In this setting, the TWFEIV estimator takes the following form, the so-called Wald-DID estimator (de Chaisemartin and D’Haultfœuille (2018), Miyaji (2024)):

$\displaystyle\hat{\beta}_{IV}=\frac{\bar{Y}_{2,2}-\bar{Y}_{2,1}-(\bar{Y}_{U,2}-\bar{Y}_{U,1})}{\bar{D}_{2,2}-\bar{D}_{2,1}-(\bar{D}_{U,2}-\bar{D}_{U,1})},$

where $\bar{R}_{a,t}$ is the sample mean of the random variable $R_{i,t}$ for cohort $E_{i}=a$ in time $t$. This estimator scales the DID estimator of the outcome by the DID estimator of the treatment between cohort $E_{i}=U$ and $E_{i}=2$.

The above observations bring us the intuition about how we can decompose the TWFEIV estimator with settings of the staggered adoption of the instrument across units; we expect that the TWFEIV estimator can be decomposed into a weighted average of all possible $2\times 2$ Wald-DID estimators (instead of DID-estimators).

To clarify this intuition, assume for now that we have only three cohorts, an early exposed cohort $k$, a middle exposed cohort $l$ $(k<l)$, and a never exposed cohort $U$ $(E_{i}=\infty)$. Figure 1 plots the simulated data for the time trends of the average treatment (first stage) and the average outcome (reduced form) in three cohorts.

From the data structure, we can construct the Wald-DID estimator in three ways. First, we can compare the evolution of the treatment and the outcome between exposed cohort $j=k,l$ and never exposed cohort $U$, exploiting the time window $POST(j)$ and $PRE(j)$, which we call an Unexposed/Exposed design:

	$\displaystyle\hat{\beta}_{IV,jU}^{2\times 2}$	$\displaystyle\equiv\frac{\left(\bar{y}_{j}^{POST(j)}-\bar{y}_{j}^{PRE(j)}\right)-\left(\bar{y}_{U}^{POST(j)}-\bar{y}_{U}^{PRE(j)}\right)}{\left(\bar{D}_{j}^{POST(j)}-\bar{D}_{j}^{PRE(j)}\right)-\left(\bar{D}_{U}^{POST(j)}-\bar{D}_{U}^{PRE(j)}\right)},\hskip 14.22636ptj=k,l,$		(7)
		$\displaystyle\equiv\frac{\hat{\beta}_{jU}^{2\times 2}}{\hat{D}_{jU}^{2\times 2}},\hskip 14.22636ptj=k,l.$

Second, we can construct the Wald-DID estimator, leveraging variation in the timing of the initial exposure to the instrument between exposed cohorts. Consider an early exposed cohort $k$ and a middle exposed cohort $l$. Before period $l$, the early exposed cohort $k$ is already exposed to the instrument, while the middle exposed cohort $l$ is not yet exposed to the instrument. In this setting, we can view that the middle exposed cohort $l$ plays the role of the control group in both the first stage and the reduced form. From this observation, we can compare the evolution of the treatment and the outcome between the early exposed cohort $k$ and middle exposed cohort $l$, exploiting the time window $MID(k,l)$ and $PRE(k)$, which we call an Exposed/Not Yet Exposed design:

	$\displaystyle\hat{\beta}_{IV,kl}^{2\times 2,k}$	$\displaystyle\equiv\frac{\left(\bar{y}_{k}^{MID(k,l)}-\bar{y}_{k}^{PRE(k)}\right)-\left(\bar{y}_{l}^{MID(k,l)}-\bar{y}_{l}^{PRE(k)}\right)}{\left(\bar{D}_{k}^{MID(k,l)}-\bar{D}_{k}^{PRE(k)}\right)-\left(\bar{D}_{l}^{MID(k,l)}-\bar{D}_{l}^{PRE(k)}\right)}$		(8)
		$\displaystyle\equiv\frac{\hat{\beta}_{kl}^{2\times 2,k}}{\hat{D}_{kl}^{2\times 2,k}}.$

Finally, if we focus on the middle exposed cohort $l$, which changes the exposure status from being unexposed to being exposed at time $l$, we can regard the early exposed cohort $k$ as the control group after time $l$ because this cohort is already exposed to the instrument at time $l$. We can compare the evolution of the treatment and the outcome between early exposed cohort $k$ and middle exposed cohort $l$, exploiting the time window $MID(k,l)$ and $POST(l)$, which we call an Exposed/Exposed Shift design:

	$\displaystyle\hat{\beta}_{IV,kl}^{2\times 2,l}$	$\displaystyle\equiv\frac{\left(\bar{y}_{l}^{POST(l)}-\bar{y}_{l}^{MID(k,l)}\right)-\left(\bar{y}_{k}^{POST(l)}-\bar{y}_{k}^{MID(k,l)}\right)}{\left(\bar{D}_{l}^{POST(l)}-\bar{D}_{l}^{MID(k,l)}\right)-\left(\bar{D}_{k}^{POST(l)}-\bar{D}_{k}^{MID(k,l)}\right)}$		(9)
		$\displaystyle\equiv\frac{\hat{\beta}_{kl}^{2\times 2,l}}{\hat{D}_{kl}^{2\times 2,l}}.$

In each type of the DID-IV design, we have three sources of variation. First, each design exploits the subsample from all $NT$ observations. The Unexposed/Exposed DID-IV design in (7) uses two cohorts and all time periods, indicating that the relative sample share is $n_{k}+n_{u}$. The Exposed/Not Yet Exposed DID-IV design in (8) uses two cohorts but exploits only the time periods before period $l$, so the relative sample share is $(1-\bar{Z}_{l})(n_{k}+n_{l})$. The Exposed/Exposed Shift DID-IV design in (9) uses two cohorts but exploits only the time periods after period $k$, so the relative sample share is $\bar{Z}_{k}(n_{k}+n_{l})$.

Second, the variation in each type of the DID-IV design partly comes from the variation of the instrument in its subsample. It is equal to the variance of the double demeaning variable $\tilde{Z}_{i,t}$ in each design:

		$\displaystyle\hat{V}_{jU}^{Z}\equiv n_{jU}(1-n_{jU})\bar{Z}_{j}(1-\bar{Z}_{j}),\hskip 14.22636ptj=k,l,$		(10)
		$\displaystyle\hat{V}_{kl}^{Z,k}\equiv n_{kl}(1-n_{kl})\left(\frac{\bar{Z}_{k}-\bar{Z}_{l}}{1-\bar{Z}_{l}}\right)\left(\frac{1-\bar{Z}_{k}}{1-\bar{Z}_{l}}\right),$		(11)
		$\displaystyle\hat{V}_{kl}^{Z,l}\equiv n_{kl}(1-n_{kl})\left(\frac{\bar{Z}_{l}}{\bar{Z}_{k}}\right)\left(\frac{\bar{Z}_{k}-\bar{Z}_{l}}{\bar{Z}_{k}}\right),$		(12)

where the $\hat{V}_{jU}^{Z}$, $\hat{V}_{kl}^{Z,k}$ and $\hat{V}_{kl}^{Z,l}$ represent the variance of the double demeaning variable $\tilde{Z}_{i,t}$ in Unexposed/Exposed, Exposed/Not Yet Exposed, and Exposed/Exposed Shift DID-IV designs, respectively. In the staggered DID set up, Goodman-Bacon (2021) also describes the two variations, that is, the relative sample share and the variance of the double demeaning treatment variable in each type of the DID designs.

Unlike the staggered DID set up, however, each DID-IV design has an additional source of the variation; the effect of the instrument on the treatment in the first stage. This comes from the fact that each DID-IV design allows the noncompliance of receiving the treatment when units are exposed to the instrument. The amount of this variation is equal to the $2\times 2$ DID estimator of the treatment in each DID-IV design:

	$\displaystyle\hat{D}_{jU}^{2\times 2}\equiv\left(\bar{D}_{j}^{POST(j)}-\bar{D}_{j}^{PRE(j)}\right)-\left(\bar{D}_{U}^{POST(j)}-\bar{D}_{U}^{PRE(j)}\right)\hskip 14.22636ptj=k,l,$
	$\displaystyle\hat{D}_{kl}^{2\times 2,k}\equiv\left(\bar{D}_{k}^{MID(k,l)}-\bar{D}_{k}^{PRE(k)}\right)-\left(\bar{D}_{l}^{MID(k,l)}-\bar{D}_{l}^{PRE(k)}\right),$
	$\displaystyle\hat{D}_{kl}^{2\times 2,l}\equiv\left(\bar{D}_{l}^{POST(l)}-\bar{D}_{l}^{MID(k,l)}\right)-\left(\bar{D}_{k}^{POST(l)}-\bar{D}_{k}^{MID(k,l)}\right).$

Note that the denominator of the TWFEIV estimator $\hat{\beta}_{IV}$ in (6), which we denote $\hat{C}^{D,Z}$ hereafter, measures the covariance between the instrument $Z_{i,t}$ and the treatment $D_{i,t}$ in whole samples. By some calculations (see the proof of Theorem 1 below), one can show that $\hat{C}^{D,Z}$ is equal to a weighted average of all possible $2\times 2$ DID estimators of the treatment in each DID-IV design:

$\displaystyle\hat{C}^{D,Z}=\sum_{k\neq U}\hat{w}_{kU}\hat{D}_{kU}^{2\times 2}+\sum_{k\neq U}\sum_{l>k}[\hat{w}_{kl}^{k}\hat{D}_{kl}^{2\times 2,k}+\hat{w}_{kl}^{l}\hat{D}_{kl}^{2\times 2,l}],$

where the weights are:

	$\displaystyle\hat{w}_{kU}=(n_{k}+n_{u})^{2}\hat{V}_{kU}^{Z},$
	$\displaystyle\hat{w}_{kl}^{k}=((n_{k}+n_{l})(1-\bar{Z}_{l}))^{2}\hat{V}_{kl}^{Z,k},$
	$\displaystyle\hat{w}_{kl}^{l}=((n_{k}+n_{l})\bar{Z}_{k})^{2}\hat{V}_{kl}^{Z,l}.$

Hereafter, we refer to $\hat{w}_{kU}$, $\hat{w}_{kl}^{k}$, and $\hat{w}_{kl}^{l}$ as the first stage weights. This decomposition result for $\hat{C}^{D,Z}$ is almost identical to that of Goodman-Bacon (2021) for the TWFE estimator under staggered DID designs, but the slight difference here is that each weight is not scaled by the variance of the double demeaning variable $\tilde{Z}_{it}$ in whole samples.

We now present the decomposition theorem for the TWFEIV estimator under the staggered assignment of the instrument across units. Theorem 1 below is a generalization of the decomposition result for the TWFE estimator with settings of the staggered assignment of the treatment across units in Goodman-Bacon (2021).

Theorem 1 shows that when the assignment of the instrument is staggered across units, the TWFEIV estimator is a weighted average of all possible $2\times 2$ Wald-DID estimators. If there exist $K$ cohorts in the data, we have $K^{2}-K$ Wald-DID estimators, which come from either Exposed/Not Yet Exposed designs as in (8) or Exposed/Exposed shift designs as in (9). If the data contains a never exposed cohort $U$, we have additionally $K$ Wald-DID estimators, which come from Unexposed/Exposed designs as in (7). If both situations occur, the TWFEIV estimator equals a weighted average of $K^{2}$ Wald-DID estimators.

The weight assigned to each Wald-DID estimator consists of three parts: the relative sample share squared, the variance of the double demeaning variable $\tilde{Z}_{i,t}$, and the DID estimator of the treatment in each DID-IV design. The first part depends on the sample share of two cohorts and the timing of the initial exposure date. The second part reflects the variation of the instrument in the subsample, represented by (10)-(12), and depends on the relative sample share between two cohorts and the timing of the initial exposure date. Finally, the remaining part reflects variation in the evolution of the treatment between the two cohorts. Note that the weight is not guaranteed to be non-negative in finite sample settings: although the first and second parts are always non-negative, the DID estimator of the treatment can be potentially negative in the data.

Theorem 1 also shows that if we subset the data containing only two cohorts (cohorts $k$ and $l$), the TWFEIV estimator $\beta_{IV,kl}^{2\times 2}$ in the subsample can be written as:

$\displaystyle\beta_{IV,kl}^{2\times 2}=\frac{\hat{w}_{kl}^{k}\hat{D}_{kl}^{2\times 2,k}}{\hat{w}_{kl}^{k}\hat{D}_{kl}^{2\times 2,k}+\hat{w}_{kl}^{l}\hat{D}_{kl}^{2\times 2,l}}\beta_{IV,kl}^{2\times 2,k}+\frac{\hat{w}_{kl}^{l}\hat{D}_{kl}^{2\times 2,l}}{\hat{w}_{kl}^{k}\hat{D}_{kl}^{2\times 2,k}+\hat{w}_{kl}^{l}\hat{D}_{kl}^{2\times 2,l}}\beta_{IV,kl}^{2\times 2,l}.$

The TWFEIV estimator $\beta_{IV,kl}^{2\times 2}$ is a weighted average of the Wald-DID estimators which come from either Exposed/Not Yet Exposed design or Exposed/Exposed Shift design, and the weight assigned to each Wald-DID estimator reflects the first stage weight and the DID estimator of the treatment in each DID-IV design.

To make the DID-IV decomposition theorem concrete, we provide a simple numerical example. Suppose we have three cohorts with equal sample size, as shown in Figure 1. In this figure, we set an early exposed period $k$ and a middle exposed period $l$ such that $\bar{Z}_{k}=0.67$ and $\bar{Z}_{l}=0.21$. We assume that the effect of the instrument on the treatment is $0.15$ in cohort $k$ and $0.1$ in cohort $l$ over time. This means that the units in cohort $k$ are more induced to the treatment by the instrument than those in cohort $l$ and the effects are stable in both cohorts. The DID estimates of the treatment are $\{\hat{D}_{kU}^{2\times 2},\hat{D}_{lU}^{2\times 2},\hat{D}_{kl}^{2\times 2,k},\hat{D}_{kl}^{2\times 2,l}\}=\{0.15,0.1,0.15,0.1\}$. We also assume that the effect of the instrument on the outcome through treatment is $9$ in cohort $k$ and $10$ in cohort $l$ over time. The DID estimates of the outcome are $\{\hat{Y}_{kU}^{2\times 2},\hat{Y}_{lU}^{2\times 2},\hat{Y}_{kl}^{2\times 2,k},\hat{Y}_{kl}^{2\times 2,l}\}=\{9,10,9,10\}$. Dividing the DID estimate of the treatment by the DID estimate of the outcome yields the Wald-DID estimate: $\{\hat{\beta}_{kU}^{2\times 2},\hat{\beta}_{lU}^{2\times 2},\hat{\beta}_{kl}^{2\times 2,k},\hat{\beta}_{kl}^{2\times 2,l}\}=\{60,100,60,100\}$. The Wald-DID estimate is larger in cohort $l$ than that of cohort $k$, though as we already noted, the effect of the instrument on the treatment is larger in cohort $k$ than that of cohort $l$.

The DID estimates of the treatment and the exposure timing determine the amount of the weight assigned to each Wald-DID estimate, holding the sample size equal across cohorts. In the above setting, the resulting weights are $\{\hat{w}_{IV,kU},\hat{w}_{IV,lU},\hat{w}_{IV,kl}^{k},\hat{w}_{IV,kl}^{l}\}=\{0.28,0.12,0.40,0.20\}$. In Unexposed/Exposed designs, we have $\hat{w}_{IV,kU}>\hat{w}_{IV,lU}$ for two reasons. First, the DID estimate of the treatment is larger in cohort $k$ than that of cohort $l$, that is, we have $\hat{D}_{kU}^{2\times 2}=0.15>0.1=\hat{D}_{lU}^{2\times 2}$. Second, the time period $k$ is closer to the middle in the whole period than the time period $l$, that is, we have $\bar{Z}_{k}(1-\bar{Z}_{k})=0.22>0.17=\bar{Z}_{l}(1-\bar{Z}_{l})$, which implies $\hat{w}_{kU}>\hat{w}_{lU}$ in the first stage weight. By the similar argument, we have $\hat{w}_{IV,kl}^{k}>\hat{w}_{IV,kl}^{l}$ between Exposed/Not Yet Exposed and Exposed/Exposed Shift designs: we have $\hat{D}_{kl}^{2\times 2,k}=0.15>0.1=\hat{D}_{kl}^{2\times 2,l}$ and $\hat{w}_{kl}^{k}>\hat{w}_{kl}^{l}$ in the first stage weight. If the DID estimates of the treatment are equal between the two designs, the exposure timing matters: we have $\hat{w}_{IV,kU}<\hat{w}_{IV,kl}^{k}$ and $\hat{w}_{IV,lU}<\hat{w}_{IV,kl}^{l}$. The DD estimates are the same in each comparison, that is, we have $\hat{D}_{kU}^{2\times 2}=\hat{D}_{kl}^{2\times 2,k}$ and $\hat{D}_{lU}^{2\times 2}=\hat{D}_{kl}^{2\times 2,l}$. However, the different initial exposure date yields different weights in the first stage, that is, we have $\hat{w}_{kU}<\hat{w}_{kl}^{k}$ and $\hat{w}_{lU}<\hat{w}_{kl}^{l}$, which make the difference above the two comparisons.

In this numerical example, the simple average of the Wald-DID estimates is $80$ and the weighted average is $100\times\frac{3}{5}+60\times\frac{2}{5}=84$ where the weight assigned to the Wald-DID estimate reflects the relative amount of the DID estimate of the treatment. The TWFEIV estimate, however, is $\hat{\beta}_{IV}=60\times(0.28+0.40)+100\times(0.12+0.20)=72.8$ because it assigns more weights on the smaller Wald-DID estimate.

Theorem 1 is a decomposition result for the TWFEIV estimator and not for the estimand. Related to the work in this paper, de Chaisemartin and D’Haultfœuille (2020) decompose the TWFE estimand and present the issue regarding the use of this estimand under DID designs: some weights assigned to the causal parameters in this estimand can be potentially negative. In their appendix, the authors also decompose the TWFEIV estimand, and refer to the negative weight problem in this estimand. Specifically, they apply their decomposition theorem for the TWFE estimand to the numerator and the denominator of the TWFEIV estimand respectively, and conclude that this estimand identifies the local average treatment effect as in Imbens and Angrist (1994) only if the effects of the instrument on the treatment and the outcome are homogeneous across groups and over time. In fact, the population coefficients on the instrument in the first stage and the reduced form regressions take the form of the TWFE estimand and their decomposition theorem for the TWFE estimand is also applicable to the analysis of the TWFEIV estimand. However, the way of their decomposition for the TWFEIV estimand has some drawbacks. First, they do not formally state the target parameter and identifying assumptions in DID-IV designs. Second, their decomposition for the TWFEIV estimand is not based on the target parameter in DID-IV designs. Finally, the sufficient conditions for this estimand to have its causal interpretation are not well explored.

In the following section, we explore the causal interpretation of the TWFEIV estimand under staggered DID-IV designs. In section 3, we first define the target parameter and identifying assumptions in staggered DID-IV designs. In section 4, based on the decomposition theorem for the TWFEIV estimator, we then provide the causal interpretation of the TWFEIV estimand under staggered DID-IV designs. Finally, we investigate the sufficient conditions for this estimand to attain its causal interpretation under staggered DID-IV designs.

First, we introduce the potential outcomes framework. Let $Y_{i,t}(d,z)$ denote the potential outcome in period $t$ when unit $i$ receives the treatment path $d\in\mathcal{S}(D)$ and the instrument path $z\in\mathcal{S}(Z)$. Similarly, let $D_{i,t}(z)$ denote the potential treatment status in period $t$ when unit $i$ receives the instrument path $z\in\mathcal{S}(Z)$.

Assumption 1 allows us to rewrite $D_{i,t}(z)$ by the initial adoption date $E_{i}=e$. Let $D_{i,t}^{e}$ denote the potential treatment status in period $t$ if unit $i$ is first exposed to the instrument in period $e$. Let $D_{i,t}^{\infty}$ denote the potential treatment status in period $t$ if unit $i$ is never exposed to the instrument. Hereafter, we call $D_{i,t}^{\infty}$ the "never exposed treatment". Since the adoption date of the instrument uniquely pins down one’s instrument path, we can write the observed treatment status $D_{i,t}$ for unit $i$ at time $t$ as

$\displaystyle D_{i,t}=D_{i,t}^{\infty}+\sum_{1\leq e\leq T}(D_{i,t}^{e}-D_{i,t}^{\infty})\cdot\mathbf{1}\{E_{i}=e\}.$

We define $D_{i,t}-D_{i,t}^{\infty}$ to be the effect of the instrument on the treatment for unit $i$ at time $t$, which is the difference between the observed treatment status $D_{i,t}$ to the never exposed treatment status $D_{i,t}^{\infty}$. Hereafter, we refer to $D_{i,t}-D_{i,t}^{\infty}$ as the individual exposed effect in the first stage. In the DID literature, Callaway and Sant’Anna (2021) and Sun and Abraham (2021) define the effect of the treatment on the outcome in the same fashion.

Next, we introduce the group variable which describes the type of unit $i$ at time $t$, based on the reaction of potential treatment choices at time $t$ to the instrument path $z$. Let $G_{i,e,t}\equiv(D_{i,t}^{\infty},D_{i,t}^{e})(t\geq e)$ be the group variable at time $t$ for unit $i$ and the initial exposure date $e$. Specifically, the first element $D_{i,t}^{\infty}$ represents the treatment status at time $t$ if unit $i$ is never exposed to the instrument $E_{i}=\infty$ and the second element $D_{i,t}^{e}$ represents the treatment status at time $t$ if unit $i$ starts exposed to the instrument at $E_{i}=e$. Following to the terminology in Imbens and Angrist (1994), we define $G_{i,e,t}=(0,0)\equiv NT_{e,t}$ to be the never-takers, $G_{i,e,t}=(1,1)\equiv AT_{e,t}$ to be the always-takers, $G_{i,e,t}=(0,1)\equiv CM_{e,t}$ to be the compliers and $G_{i,e,t}=(1,0)\equiv DF_{e,t}$ to be the defiers at time $t$ and the initial exposure date $e$.

Finally, we make a no carryover assumption on potential outcomes $Y_{i,t}(d,z)$.

This assumption requires that potential outcomes $Y_{i,t}(d,z)$ depend only on the current treatment status $d_{t}$ and the instrument path $z$. In the DID literature, several recent papers impose this assumption with settings of a non-staggered treatment; see, e.g., de Chaisemartin and D’Haultfœuille (2020) and Imai and Kim (2021). Although it can be possible to weaken this assumption by introducing the treatment path $d$ in potential outcomes $Y_{i,t}(d,z)$, this requires the cumbersome notation and complicates the definition of our target parameter, thus is beyond the scope of this paper.

Henceforth, we keep Assumption 1 and 2. In the next section, we define the target parameter in staggered DID-IV designs.

Our target parameter in staggered DID-IV designs is the cohort specific local average treatment effect on the treated (CLATT) defined below.

This parameter measures the treatment effects at a given relative period $l$ from the initial instrument adoption date $E_{i}=e$, for those who belong to cohort $e$, and are the compliers $CM_{e,e+l}$, that is, who are induced to treatment by instrument at time $e+l$. Each CLATT_e,l can potentially vary across cohorts and over time, as it depends on cohort $e$, relative period $l$, and the compliers $CM_{e,e+l}$.

In this section, we state the identifying assumptions in staggered DID-IV designs based on Miyaji (2024).

Assumption 3 requires that the path of the instrument does not directly affect the potential outcome for all time periods and its effects are only through treatment. Given Assumption 2 and Assumption 3, we can write the potential outcome $Y_{i,t}(d,z)$ as $Y_{i,t}(d_{t})=D_{i,t}Y_{i,t}(1)+(1-D_{i,t})Y_{i,t}(0)$.

Here, we introduce the potential outcomes at time $t$ if unit $i$ is assigned to the instrument path $z\in\mathcal{S}(Z)$:

$\displaystyle Y_{i,t}(D_{i,t}(z))\equiv D_{i,t}(z)Y_{i,t}(1)+(1-D_{i,t}(z))Y_{i,t}(0).$

Since the exposure timing $E_{i}$ completely determines the path of the instrument, we can write the potential outcomes for cohort $e$ and cohort $\infty$ as $Y_{i,t}(D_{i,t}^{e})$ and $Y_{i,t}(D_{i,t}^{\infty})$, respectively. The potential outcome $Y_{i,t}(D_{i,t}^{e})$ represents the outcome status at time $t$ if unit $i$ is first exposed to the instrument at time $e$ and the potential outcome $Y_{i,t}(D_{i,t}^{\infty})$ represents the outcome status at time $t$ if unit $i$ is never exposed to the instrument. Hereafter, we refer to $Y_{i,t}(D_{i,t}^{\infty})$ as the "never exposed outcome".