3.1 The Matching Framework

To establish the impossibility of nonparametric adjustment for unit-specific and time-specific unobserved confounders, it is useful to consider the 2FE estimator as a matching estimator (Imai and Kim Reference Imai and Kim2019). An intuitive explanation of this result is as follows. Although one could nonparametrically adjust for unit-specific (time-specific) unobserved confounders by matching a treated observation with control observations of the same unit (time period), no other observation shares the same unit and time indices. Thus, the 2FE estimator critically relies upon the linearity assumption for its simultaneous adjustment for the two types of unobserved confounders. The following proposition formalizes this argument.

Proposition 1 The Two-way Fixed Effects Regression Estimator as a Two-way Matching Estimator

The two-way fixed effects estimator defined in Equation (2) is equivalent to the following matching estimator,

$$ \begin{align*} \hat{\beta} & = \frac{1}{K}\left[\frac{1}{NT} \sum_{i=1}^N \sum_{t=1}^T \left\{X_{it} \left(Y_{it} - \widehat{Y_{it}(0)} \right) + (1-X_{it}) \left( \widehat{Y_{it}(1)} -Y_{it} \right) \right\}\right] \end{align*} $$

where for $x=0,1$ , the estimate of the potential outcome $Y_{it}(x)$ for unit i at time t under the treatment status $X_{it} = x$ is given by,

$$ \begin{align*} \widehat{Y_{it}(x)} & = \frac{1}{T-1}\sum_{t^{\prime} \neq t} Y_{it^{\prime}} + \frac{1}{N-1}\sum_{i^{\prime} \neq i} Y_{i^{\prime}t} - \frac{1}{(T-1)(N-1)}\sum_{i^{\prime} \neq i} \sum_{t^{\prime} \neq t} Y_{i^{\prime}t^{\prime}} \\ K & = \frac{1}{NT} \sum_{i=1}^N \sum_{t=1}^T \left\{X_{it}\left(\frac{\sum_{t^{\prime}\neq t} (1-X_{it^{\prime}})}{T-1} + \frac{\sum_{i^{\prime} \neq i} (1-X_{i^{\prime}t})}{N-1} - \frac{\sum_{i^{\prime} \neq i} \sum_{t^{\prime} \neq t} (1-X_{i^{\prime}t^{\prime}})}{(T-1)(N-1)}\right) \right.\nonumber \\ & \quad+ \left. (1-X_{it})\left( \frac{\sum_{t^{\prime} \neq t} X_{it^{\prime}}}{T-1} + \frac{\sum_{i^{\prime}\neq i} X_{i^{\prime}t}}{N-1} - \frac{\sum_{i^{\prime} \neq i} \sum_{t^{\prime} \neq t} X_{i^{\prime}t^{\prime}}}{(T-1)(N-1)}\right)\right\}. \end{align*} $$

Proof is given in Online Supplementary Information. The proposition shows that the estimated counterfactual outcome of a given observation, that is, $\widehat {Y_{it}(1-X_{it})}$ , is a function of three averages. First, the average of all the other observations from the same unit, that is, $\sum _{t^{\prime } \ne t} Y_{it^{\prime }}/(T-1)$ , and the average of all the other observations from the same time period, that is, $\sum _{i^{\prime } \ne i} Y_{i^{\prime } t}/(N-1)$ , are added together. We call them the within-unit matched set ${\mathcal {M}}_{it}$ and the within-time matched set ${\mathcal {N}}_{it}$ , respectively, and formally define them as,

(3) $$ \begin{align} {\mathcal{M}}_{it} & = \{(i^{\prime}, t^{\prime}): i^{\prime} = i, t^{\prime} \ne t\}, \quad \text{and} \quad {\mathcal{N}}_{it} \ = \ \{(i^{\prime}, t^{\prime}): i^{\prime} \ne i, t^{\prime} = t\}. \end{align} $$

The 2FE estimator then adjusts for unit-specific and time-specific unobserved confounders by using observations that share the same unit or time as those in ${\mathcal {N}}_{it}$ and ${\mathcal {M}}_{it}$ , respectively, and subtracting their mean, that is, $\sum _{i^{\prime } \ne i} \sum _{t^{\prime } \ne t} Y_{i^{\prime } t^{\prime }}/(T-1)(N-1)$ , from this sum. We use ${\mathcal {A}}_{it}$ to denote this group of observations and call it the adjustment set for observation $(i,t)$ with the following definition,

(4) $$ \begin{align} {\mathcal{A}}_{it} & = \{(i^{\prime}, t^{\prime}): i^{\prime} \ne i, t^{\prime} \ne t, (i, t^{\prime}) \in {\mathcal{M}}_{it}, (i^{\prime}, t) \in {\mathcal{N}}_{it} \}. \end{align} $$

By construction, the number of observations in ${\mathcal {A}}_{it}$ equals the product of the number of observations in the within-unit and within-time matched sets, that is, $|{\mathcal {A}}_{it}| = |{\mathcal {M}}_{it}| \cdot |{\mathcal {N}}_{it}|$ .

Panel (a) of Figure 1 presents an example of the binary treatment matrix with five units and four time periods, that is, $N=5$ and $T = 4$ . In the figure, the red underlined $\color{red}{}\underline{1} $ entry represents a treated observation of interest, for which the counterfactual outcome $Y_{it}(0)$ needs to be estimated using other observations. This counterfactual quantity is estimated as the average of control observations from the same unit ${\mathcal {M}}_{it}$ (circles in the figure), plus the average of control observations from the same time period ${\mathcal {N}}_{it}$ (squares), minus the average of adjustment observations, ${\mathcal {A}}_{it}$ (triangles).

Figure 1

An Example of the Binary Treatment Matrix with Five Units and Four Time Periods. Panels (a) and (b) illustrate how observations (i, t) are used to estimate counterfactual outcomes for the two-way fixed effects estimator (Proposition 1) and the adjusted matching estimator (Proposition 2), respectively. In the figures, the red underlined $\color{red}{}\underline{1} $ entry (4,3) represents the treated observation, for which the counterfactual outcome $Y_{it}(0)$ needs to be estimated. Circles indicate the set of matched observations—(4,1), (4,2), (4,4) in Panel (a) and (4,1), (4,2) in Panel (b)—that are from the same unit, whereas squares indicate those—(1,3), (2,3), (3,3), (5,3) in Panel (a) and (2,3), (5,3) in Panel (b)—from the same time period. Finally, triangles represent the set of observations—(1,1), (1,2), (1,4), (2,1), (2,2), (2,4), (3,1), (3,2), (3,4), (5,1), (5,2), (5,4) in Panel (a) and (2,1), (2,2), (5,1), (5,2) in Panel (b)—that are used to make adjustment for unit and time effects. The shaded grey symbols represent the “mismatches” with the same treatment status, which are prevalent in the two-way fixed effects estimator. The matching estimator in Panel (b) is designed to eliminate the attenuation bias within unit and time, although the adjustment set may still include mismatches (shaded triangles).

Note that all of these three averages may include units with the same treatment status as the observation whose counterfactual outcome is being estimated. We refer to these observations as “mismatches” (shaded grey entries in the figure) because for the estimation of causal effects, an observation must be matched with another observation with the opposite treatment status. Therefore, mismatches imply the (partial) comparison of observations with the same treatment status, which generally leads to an attenuation bias. The 2FE estimator adjusts for this bias via the factor K, which is equal to the net proportion of proper matches between the observations of opposite treatment status. For example, for a treated observation with $X_{it}=1$ , we compute the proportion of matched control observations in the within-unit matched set, that is, $\sum _{t^{\prime } \ne t}{}(1-X_{it^{\prime }})/(T-1)$ , and the proportion of matched control observations in the within-time matched set, that is, $\sum _{i^{\prime } \ne i} (1-X_{i^{\prime } t})/(N-1)$ , and subtract from their sum the proportion of matched control observations in the adjustment set, that is, $\sum _{i^{\prime } \ne i} \sum _{t^{\prime } \ne t}(1-X_{i^{\prime } t^{\prime }})/(T-1)(N-1)$ .

3.2 The Impossibility of Nonparametric Adjustment

Given this result, it is natural to ask whether we can eliminate the mismatches and the adjustment set all together within the two-way fixed effects framework. We show below that this is generally impossible. In particular, although we can construct a weighted 2FE estimator that has fewer mismatches, this estimator in general still suffers from some mismatches and has an adjustment set.

To develop a weighted 2FE estimator with fewer mismatches, we begin by matching each observation only with other observations of the opposite treatment status to estimate the counterfactual outcome. That is, we use the following within-unit matched set ${\mathcal {M}}_{it}^\ast $ , which consists of the observations within the same unit but with the opposite treatment status,

(5) $$ \begin{align} {\mathcal{M}}_{it}^\ast \ = \ \{(i^{\prime}, t^{\prime}): i^{\prime} = i, X_{i^{\prime} t^{\prime}} = 1 - X_{it}\}. \end{align} $$

Similarly, we restrict the within-time matched set so that its observations belong to the same time period t but have the opposite treatment status,

(6) $$ \begin{align} {\mathcal{N}}_{it}^\ast & = \{(i^{\prime}, t^{\prime}): t^{\prime} = t, X_{i^{\prime} t^{\prime}} = 1-X_{it} \}. \end{align} $$

Then, using Equation (4), we can define the corresponding adjustment set ${\mathcal {A}}_{it}^\ast $ .

(7) $$ \begin{align} {\mathcal{A}}_{it}^\ast & = \{(i^{\prime}, t^{\prime}): i^{\prime} \ne i, t^{\prime} \ne t, (i, t^{\prime}) \in {\mathcal{M}}_{it}^\ast, (i^{\prime}, t) \in {\mathcal{N}}_{it}^\ast \}. \end{align} $$

The next proposition establishes that this two-way matching estimator, which eliminates mismatches within-unit and within-time dimension, can be written as a weighted 2FE estimator.

Proposition 2 The Two-way Matching Estimator with Fewer Mismatches as a Weighted Two-way Fixed Effects Regression Estimator

Assume that the treatment varies within each unit as well as within each time period, that is, $0 < \sum _{t=1}^T X_{it} < T$ for each i and $0 < \sum _{i=1}^N X_{it} < N$ for each t. Consider the following matching estimator,

$$ \begin{align*} \hat\beta^\ast & = \frac{1}{\sum_{i=1}^N \sum_{t=1}^T D_{it}} \sum_{i=1}^N \sum_{t=1}^T \frac{D_{it}}{K_{it}} \left\{X_{it}\left(Y_{it}- \widehat{Y_{it}(0)} \right) + (1-X_{it})\left(\widehat{Y_{it}(1)} - Y_{it}\right)\right\} \end{align*} $$

where $D_{it} = \mathbf {1}\{|{\mathcal {M}}_{it}^\ast |\cdot |{\mathcal {N}}_{it}^\ast |> 0\}$ , and for $x=0,1$ ,

$$ \begin{align*} \widehat{Y_{it}(x)} & = \frac{1}{|{\mathcal{M}}_{it}^\ast|}\sum_{(i,t^{\prime})\in {\mathcal{M}}_{it}^\ast}Y_{it^{\prime}} + \frac{1}{|{\mathcal{N}}_{it}^\ast|}\sum_{(i^{\prime},t)\in {\mathcal{N}}_{it}^\ast}Y_{i^{\prime}t} - \frac{1}{|{\mathcal{A}}_{it}^\ast|}\sum_{(i^{\prime}, t^{\prime})\in {\mathcal{A}}_{it}^\ast}Y_{i^{\prime}t^{\prime}} \\ K_{it} & = 1 + \frac{a_{it}}{|{\mathcal{A}}_{it}^\ast|} \end{align*} $$

and $a_{it} = |\{(i^{\prime }, t^{\prime }) \in {\mathcal {A}}_{it}^\ast : X_{i^{\prime } t^{\prime }} = X_{it} \}|$ . Then, this matching estimator is equivalent to the following weighted two-way fixed effects estimator,

$$ \begin{align*} \hat\beta^\ast & = \operatorname*{\mathrm{argmin}}_{\beta} \sum_{i=1}^N \sum_{t=1}^T W_{it} \{(Y_{it} - \overline{Y}_i^\ast - \overline{Y}_t^\ast + \overline{Y}^\ast) - \beta(X_{it} - \overline{X}_i^\ast - \overline{X}_t^\ast + \overline{X}^\ast)\}^2 \end{align*} $$

where the asterisks indicate weighted averages, that is, $\overline {Y}_i^\ast = \sum _{t=1}^T W_{it} Y_{it}/\sum _{t=1}^T W_{it}$ , $\overline {Y}_t^\ast = \sum _{i=1}^N W_{it} Y_{it}/\sum _{i=1}^N W_{it}$ , $\overline {X}_i^\ast = \sum _{t=1}^T W_{it} X_{it}/\sum _{t=1}^T W_{it}$ , $\overline {X}_t^\ast = \sum _{i=1}^N W_{it} X_{it}/\sum _{i=1}^N W_{it}$ , $\overline {Y}^\ast = \sum _{i=1}^N \sum _{t=1}^T W_{it} {} Y_{it}/\sum _{i=1}^N \sum _{t=1}^T W_{it}$ , $\overline {X}^\ast = \sum _{i=1}^N \sum _{t=1}^T W_{it} X_{it}/\sum _{i=1}^N \sum _{t=1}^T W_{it}$ , and

$$ \begin{align*} W_{it} & = \sum_{i^{\prime}=1}^N \sum_{t^{\prime} = 1}^T w_{it}^{i^{\prime} t^{\prime}} \quad \text{and} \quad w_{it}^{i^{\prime} t^{\prime}} \ = \ \left\{ \begin{array}{cl} \frac{D_{i^{\prime} t^{\prime}}}{K_{i^{\prime} t^{\prime}}} & \text{if} \ (i,t) = (i^{\prime},t^{\prime}) \\ \frac{D_{i^{\prime} t^{\prime}}}{ K_{i^{\prime} t^{\prime}}\cdot|{\mathcal{M}}_{i^{\prime} t^{\prime}}^\ast|} & \text{if} \ (i,t) \in {\mathcal{M}}_{i^{\prime} t^{\prime}}^\ast \\ \frac{D_{i^{\prime} t^{\prime}}}{K_{i^{\prime} t^{\prime}}\cdot|{\mathcal{N}}_{i^{\prime} t^{\prime}}^\ast|} & \text{if} \ (i,t) \in {\mathcal{N}}_{i^{\prime} t^{\prime}}^\ast \\ \frac{D_{i^{\prime} t^{\prime}}(2X_{i t} - 1) (2X_{i^{\prime} t^{\prime}} - 1)}{K_{i^{\prime} t^{\prime}} \cdot |{\mathcal{A}}_{i^{\prime} t^{\prime}}^\ast |} & \text{if} \ (i,t) \in {\mathcal{A}}_{i^{\prime} t^{\prime}}^\ast \\ 0 & \text{otherwise.} \end{array} \right. \end{align*} $$

Proof is given in Online Supplementary Information. Unlike Proposition 1, the adjustment is done by deflating the estimated treatment effect for each treated observation $(i,t)$ by $1/K_{it}$ . This is because the attenuation bias from ${\mathcal {A}}_{it}^\ast $ (the “pooled” part) is subtracted from the sum of two estimates from ${\mathcal {M}}_{it}^\ast $ and ${\mathcal {N}}_{it}^\ast $ , inflating the estimated treatment effect for a given observation $(i,t)$ . In the example of Panel (b) of Figure 1, ${\mathcal {A}}_{it}^\ast $ contains two mismatches (shaded grey entries in triangles), that is, $a_{it} = 2$ , and hence the adjustment factor is $K_{it}=3/2=1 + 2/4$ . Note that such adjustment is not necessary (i.e., $K_{it} = 1$ ) when there are no mismatches in the adjustment set, that is, $a_{it} = 0$ .

The algebraic equivalence result given in Proposition 2 clarifies the set of observations that are used to estimate the counterfactual for each unit and how the adjustments due to mismatches are reflected in the weighted two-way fixed effects estimator. Specifically, it shows that each observation $(i,t)$ is weighted differently according to the number of times it serves as a control unit. For example, if an observation $(i,t)$ has the treatment status opposite to another observation within-unit $(i^{\prime }, t^{\prime })$ , that is, $(i,t) \in {\mathcal {M}}_{i^{\prime } t^{\prime }}^\ast $ , then its overall weight $W_{it}$ is increased by $1/|{\mathcal {M}}_{i^{\prime } t^{\prime }}^\ast |$ along with other observations in the within-unit matched set. This contribution to the weight is then deflated by the adjustment factor $K_{i^{\prime } t^{\prime }}$ , correcting the attenuation bias due to mismatches (see the formula for computing $w_{it}^{i^{\prime } t^{\prime }}$ in the proposition).

Unfortunately, we cannot eliminate mismatches in ${\mathcal {A}}_{it}^\ast $ without additional restrictions on the matched sets, ${\mathcal {M}}_{it}^\ast $ and ${\mathcal {N}}_{it}^\ast $ (see Section 4.1). This point is illustrated by Panel (b) of Figure 1 where the adjustment set ${\mathcal {A}}_{it}^\ast $ (triangles) still includes the observations of the same treatment status. Therefore, even the weighted 2FE estimator, which has fewer mismatches than the standard 2FE estimator, suffers from some mismatches. The estimator also has an adjustment set whose observations belong to neither the same unit nor the same time period as the observation being matched with. This implies that it is impossible to simultaneously and nonparametrically adjust for unit-specific and time-specific unobserved confounders under the two-way fixed effects framework.

4.1 The Multi-period Difference-in-Differences Estimator

To establish the relations between the 2FE and DiD estimators, we begin by considering the following parallel trend assumption,

Assumption 1 Parallel Trend

For $i=1,2,\dots ,N$ and $t=2,\dots ,T$ ,

$$ \begin{align*} {\mathbb{E}}(Y_{it}(0) - Y_{i,t-1}(0) \mid X_{it} = 1, X_{i,t-1} = 0) & = {\mathbb{E}}(Y_{it}(0) - Y_{i,t-1}(0) \mid X_{it} = X_{i,t-1} = 0). \end{align*} $$

We emphasize that this assumption may not be credible in some settings (see, e.g., Bilinski and Hatfield Reference Bilinski and Hatfield2018; Kahn-Lang and Lang Reference Kahn-Lang and Lang2019; Rambachan and Roth Reference Rambachan and Roth2019). The goal of our analysis, however, is to shed new light on a popular justification of the 2FE estimator as the DiD estimator under the simplest setting.Footnote ¹ Under this parallel trend assumption, the estimand is the average treatment effect for the treated (ATT),

(8) $$ \begin{align} \tau \ = \ {\mathbb{E}}(Y_{it}(1) - Y_{it}(0) \mid X_{it} = 1, X_{i,t-1} = 0). \end{align} $$

To formulate a multi-period DiD estimator under the 2FE estimator framework, we follow the analytical strategy used in the previous section and define three sets of observations as illustrated in Figure 2—the within-unit matched set (represented by a circle), within-time matched set (represented by squares), and adjustment set (represented by triangles)—for a treated observation $(4, 3)$ (represented by the red underlined $\color{red}{}\underline{1} $ ). We next show that the DiD design eliminates mismatches from these three sets.

Figure 2

Illustration of how observations are used to estimate counterfactual outcomes for the DiD estimator (Equation (12)). The red underlined $\color{red}{}\underline{1} $ entry represents the treated observation $(4,3)$ , for which the counterfactual outcome $Y_{it}(0)$ needs to be estimated. Circle indicates the matched observation (4,2) within the same unit, ${\mathcal {M}}_{it}^{\textsf {DiD}}$ , whereas squares—(2,3) and (5,3)—indicate those from the same time period, ${\mathcal {N}}_{it}^{\textsf {DiD}}$ . Finally, triangles—(2,2) and (5,2)—represent the set of observations that are used to make adjustment for unit and time effects, ${\mathcal {A}}_{it}^{\textsf {DiD}}$ . Unlike the examples in Figure 1, ${\mathcal {A}}_{it}^{\textsf {DiD}}$ only contains control observations and hence no mismatches (i.e., shaded grey triangles) exist.

Formally, the within-unit matched set contains the observation of the same unit from the previous time period if it is under the control condition, and to be an empty set otherwise,

(9) $$ \begin{align} {\mathcal{M}}_{it}^{\textsf{DiD}} & = \{(i^{\prime}, t^{\prime}): i^{\prime} = i, t^{\prime} = t-1, X_{i^{\prime} t^{\prime}} = 0\}. \end{align} $$

Similarly, the within-time matched set is defined as a group of control observations in the same time period whose prior observations are also under the control condition,

(10) $$ \begin{align} {\mathcal{N}}_{it}^{\textsf{DiD}} & = \{(i^{\prime}, t^{\prime}): i^{\prime} \ne i, t^{\prime} = t, X_{i^{\prime} t^{\prime}} = X_{i^{\prime}, t^{\prime} - 1} = 0 \}. \end{align} $$

Finally, we define the adjustment set ${\mathcal {A}}_{it}^{\textsf {DiD}}$ , which contains the control observations in the previous period that share the same unit as those in ${\mathcal {N}}_{it}^{\textsf {DiD}}$ ,

(11) $$ \begin{align} {\mathcal{A}}_{it}^{\textsf{DiD}} & = \{(i^{\prime}, t^{\prime}): i^{\prime} \neq i, t^{\prime} = t-1, X_{i^{\prime} t^{\prime}} = X_{i^{\prime} t} = 0 \}. \end{align} $$

Thus, the number of observations in this adjustment set is the same as that in ${\mathcal {N}}_{it}^{\textsf {DiD}}$ . It is worth noting that all three sets only contain control observations, thereby eliminating all mismatches.

Using these matched and adjustment sets, we can define the multi-period DiD estimator as the average of two-time-period two-group DiD estimators applied whenever there is a change from the control condition to the treatment condition,

(12) $$ \begin{align} \hat\tau & = \frac{1}{\sum_{i=1}^N \sum_{t=1}^T D_{it}} \sum_{i=1}^N \sum_{t=1}^T D_{it} \left(Y_{it} - \widehat{Y_{it}(0)} \right) \end{align} $$

where $D_{i1}=0$ for all i, $D_{it} = X_{it} \cdot \mathbf {1}\{|{\mathcal {M}}_{it}^{\textsf {DiD}}| \cdot |{\mathcal {N}}_{it}^{\textsf {DiD}}|> 0 \}$ for $t> 1$ , and for $D_{it} = 1$ , we define,

(13) $$ \begin{align} \widehat{Y_{it}(0)} & = Y_{i,t-1} + \frac{1}{|{\mathcal{N}}_{it}^{\textsf{DiD}}|}\sum_{(i^{\prime},t)\in {\mathcal{N}}_{it}^{\textsf{DiD}}}Y_{i^{\prime}t} - \frac{1}{|{\mathcal{A}}_{it}^{\textsf{DiD}}|}\sum_{(i^{\prime}, t^{\prime})\in {\mathcal{A}}_{it}^{\textsf{DiD}}}Y_{i^{\prime}t^{\prime}} \end{align} $$

Thus, when the treatment status of a unit changes from the control condition at time $t-1$ to the treatment condition at time t (and there exists at least one unit $i^{\prime }$ whose treatment status does not change during the same time periods, that is, $D_{it}=1$ ), the counterfactual outcome for observation $(i,t)$ is estimated as follows. We subtract from $Y_{it}$ its own observed outcome of the previous period $Y_{i,t-1}$ as well as the average outcome difference between the same two time periods among the other units whose treatment status remains unchanged as the control condition.

4.2 Equivalence to the Weighted Two-way Fixed Effects Estimator with Some Negative Regression Weights

It is well known that the standard nonparametric DiD estimator is numerically equivalent to the 2FE estimator in the simplest setting, in which there are only two time periods and the treatment is administered only to one group of units in the second time period. Unfortunately, we show that this equivalence result does not generalize to the current multi-period DiD design, in which the number of time periods may exceed two and different units may switch in and out of the treatment condition multiple times and at different points in time.Footnote ² Instead, the following theorem establishes that the general multi-period DiD estimator given in Equation (12) is equivalent to a weighted two-way fixed effects regression estimator.

Theorem 1 Difference-in-Differences Estimator as a Weighted Two-way Fixed Effects Estimator

Assume that there is at least one treated and control unit, that is, $0 < \sum _{i=1}^N \sum _{t=1}^T X_{it} < NT$ , and that there is at least one unit with $D_{it} =1$ , that is, $0 < \sum _{i=1}^N \sum _{t=1}^T D_{it}$ . The difference-in-differences estimator $\hat \tau $ , defined in Equation (12), is equivalent to the following weighted two-way fixed effects regression estimator,

$$ \begin{align*} \hat\tau \ = \ \hat\beta_{{\textsf{WFE2}}} & = \operatorname*{\mathrm{argmin}}_\beta \sum_{i=1}^N \sum_{t=1}^T W_{it} \{(Y_{it} - \overline{Y}_i^\ast - \overline{Y}_t^\ast + \overline{Y}^\ast) - \beta (X_{it} - \overline{X}_i^\ast - \overline{X}_t^\ast + \overline{X}^\ast)\}^2 \end{align*} $$

where the asterisks indicate weighted averages, and the weights are given by,

$$ \begin{align*} W_{it} & = \sum_{i^{\prime}=1}^N \sum_{t^{\prime} = 1}^T D_{i^{\prime} t^{\prime}} \cdot w_{it}^{i^{\prime} t^{\prime}} \quad \text{and} \quad w_{it}^{i^{\prime} t^{\prime}} \ = \ \left\{ \begin{array}{cl} 1 & \text{if} \ (i,t) = (i^{\prime},t^{\prime}) \\ 1/|{\mathcal{M}}_{i^{\prime} t^{\prime}}^{\textsf{DiD}}| & \text{if} \ (i,t) \in {\mathcal{M}}_{i^{\prime} t^{\prime}}^{\textsf{DiD}} \\ 1/|{\mathcal{N}}_{i^{\prime} t^{\prime}}^{\textsf{DiD}}| & \text{if} \ (i,t) \in {\mathcal{N}}_{i^{\prime} t^{\prime}}^{\textsf{DiD}} \\ (2X_{i t} - 1) (2X_{i^{\prime} t^{\prime}} - 1)/|{\mathcal{A}}_{i^{\prime} t^{\prime}}^{\textsf{DiD}}| & \text{if} \ (i,t) \in {\mathcal{A}}_{i^{\prime} t^{\prime}}^{\textsf{DiD}} \\ 0 & \text{otherwise.} \end{array} \right. \end{align*} $$

Proof is in Appendix A. Theorem 1 shows that the DiD estimator can be obtained by calculating the weighted linear two-way fixed effects regression estimator.

Theorem 1 has two important implications. First, in contrast to a common belief held among applied researchers, the (unweighted) 2FE estimator is not in general equivalent to the multi-period DiD estimator. Second, although the multi-period DiD estimator can be shown to be equivalent to the weighted 2FE estimator, some control observations will have negative regression weights. This occurs when they frequently enter into the adjustment set, ${\mathcal {A}}_{i^{\prime } t^{\prime }}^{\textsf {DiD}}$ , for multiple treated observations (i.e., $(2X_{i t} - 1) (2X_{i^{\prime } t^{\prime }} - 1) = -1$ ). Since the regression weights should generally positive, the results of this section shows that the justification of the 2FE estimator as the DiD estimator is not warranted unless the linearity assumption is imposed.