18. Difference-in-Differences

Difference-in-differences (DiD) identifies a treatment effect from panel (or repeated cross-section) data by comparing the change between before and after an intervention across a treatment group and a control group. The core idea: the change in the control group over time is a good estimate of the change the treatment group would have experienced had it not been treated.

Before discussing the DiD method, let's first introduce some basic terminology.

18.1.1 Relationship between cross-section and panel data - A cross-section of data has one dimension, the unit \(i\). - Panel data has both a unit \(i\) dimension and a time period \(t\) dimension. - A panel is a repeated cross-section, but the converse is not true, i.e. a repeated cross-section is not a panel. - In a panel we need that the same units are observed multiple times, which is not true in general for a repeated cross-section. - Sometimes this distinction is semantic. The repeated cross-section can be viewed as a panel, particularly in settings where data is observed or aggregated in groups. E.g., a setting of repeated cross-sections where data is aggregated by state could be viewed as a panel of states over time.

18.1.2 Balanced panel

18.2.1 Two periods and two groups - Two periods: \(t\in\{1,2\}\). - Before the intervention: \(t=1\). - After the intervention: \(t=2\). - \(N\) individuals: \(i\in\{1,2,\dots,N\}\). - Before the intervention: neither group is treated, i.e. \(D_{i1}=0\) for any individual \(i\) in both groups. - After the intervention: - the treatment group is treated by the intervention, i.e. \(D_{i2}=1\) for any individual \(i\) in the treatment group; - the control group is unaffected by the intervention, \(D_{i2}=0\) for any individual \(i\) in the control group.

18.2.2 Switching regression notation

Potential outcomes: $$\text{period 1: }Y_{i1}^0,\qquad\text{period 2: }Y_{i2}=\begin{cases}Y_{i2}^1&\text{if }D_{i2}=1\\[2pt]Y_{i2}^0&\text{if }D_{i2}=0\end{cases}$$

(Everyone is untreated in period 1, so only \(Y_{i1}^0\).) Equivalently, the observed outcome in switching-regression form is: $$Y_{it}=D_{it}Y_{it}^1+(1-D_{it})Y_{it}^0=D_{it}\left(Y_{it}^1-Y_{it}^0\right)+Y_{it}^0\tag{Table 2}$$

This switching-regression format is identical to before, but now we have the additional time structure. The change in individual \(i\)'s observed outcome over time is: $$Y_{i2}-Y_{i1}=D_{i2}\left(Y_{i2}^1-Y_{i2}^0\right)+Y_{i2}^0-Y_{i1}^0$$

which uses \(Y_{i1}=Y_{i1}^0\). Realized outcomes: - For the treatment group: \(Y_{i1}=Y_{i1}^0\) and \(Y_{i2}=Y_{i2}^1\). - For the control group: \(Y_{i1}=Y_{i1}^0\) and \(Y_{i2}=Y_{i2}^0\).

18.3.1 The two differences and DiD

First difference: the after-before difference for the treatment group $$\mathbb E[Y_{i2}-Y_{i1}\mid D_{i2}=1]=\mathbb E\!\left[Y_{i2}^1-Y_{i2}^0\mid D_{i2}=1\right]+\mathbb E\!\left[Y_{i2}^0-Y_{i1}^0\mid D_{i2}=1\right]$$

Second difference: the after-before difference for the control group $$\mathbb E[Y_{i2}-Y_{i1}\mid D_{i2}=0]=\mathbb E\!\left[Y_{i2}^0-Y_{i1}^0\mid D_{i2}=0\right]$$

Difference-in-Differences (DiD) is given by the difference in the two differences: $$\text{DiD}=\mathbb E[Y_{i2}-Y_{i1}\mid D_{i2}=1]-\mathbb E[Y_{i2}-Y_{i1}\mid D_{i2}=0]$$

To further characterize DiD, we need to make the common trend assumption.

18.3.2 Common trend assumption

18.3.3 Graphical representation

Figure 3 (paraphrased): the horizontal axis is time (\(t=1,2\)), the vertical axis is the outcome. The treatment group rises from \(Y^{\text{treatment,before}}\) to \(Y^{\text{treatment,after}}\); the control group rises from \(Y^{\text{control,before}}\) to \(Y^{\text{control,after}}\). The common trend assumption: the treated group's counterfactual "if untreated" trajectory is parallel to the control group's actual trajectory. The DiD estimate = (treated after-before) − (control after-before), i.e. the vertical gap between the treated group's actual endpoint and its parallel counterfactual endpoint.

We could use both DiD and the first difference \(\mathbb E[Y_{i2}-Y_{i1}\mid D_{i2}=1]\) to identify the ATT \(\mathbb E[Y_{i2}^1-Y_{i2}^0\mid D_{i2}=1]\), but we are making different assumptions for those two methods.

• If we are using DiD to identify the ATT, then we are assuming the common trend, i.e. $$\mathbb E\!\left[Y_{i2}^0-Y_{i1}^0\mid D_{i2}=1\right]=\mathbb E\!\left[Y_{i2}^0-Y_{i1}^0\mid D_{i2}=0\right]$$
• If we are using the first difference to identify the ATT, then we are assuming: $$\begin{aligned}&\mathbb E[Y_{i2}-Y_{i1}\mid D_{i2}=1]=\mathbb E\!\left[Y_{i2}^1-Y_{i2}^0\mid D_{i2}=1\right]\\\Leftrightarrow{}&\mathbb E\!\left[Y_{i2}^1-Y_{i2}^0\mid D_{i2}=1\right]+\mathbb E\!\left[Y_{i2}^0-Y_{i1}^0\mid D_{i2}=1\right]=\mathbb E\!\left[Y_{i2}^1-Y_{i2}^0\mid D_{i2}=1\right]\\\Leftrightarrow{}&\mathbb E\!\left[Y_{i2}^0-Y_{i1}^0\mid D_{i2}=1\right]=0\end{aligned}$$ i.e. no time trend between the two periods.

Even though the common trend assumption may not hold in general, the DiD method is still better than the first difference method in the sense that the DiD method allows for the existence of a non-zero common time trend.

Define two dummy variables: $$\text{treat}_i=\begin{cases}1&\text{if }i\text{ is in treatment group}\\0&\text{if }i\text{ is in control group}\end{cases}\qquad\text{after}_t=\begin{cases}1&\text{if the observation is in }t=2\\0&\text{if the observation is in }t=1\end{cases}$$

Then the following regression specification $$Y_{it}=\beta_0+\beta_1\text{treat}_i+\beta_2\text{after}_t+\beta_3\,\text{treat}_i\times\text{after}_t+\epsilon_{it}$$ delivers the DiD estimate. Clearly \(\beta_3\) is the DiD estimator. The meaning of each coefficient is in the table below.

DiD can be estimated using non-parametric methods. However, here we discussed that DiD can also be estimated using regression models. There are two major advantages for doing regression to estimate DiD: - Doing regression is a convenient way to obtain standard errors in the estimation. However, in a non-parametric method, we can only obtain a point estimate of the DiD, and the standard error can only be obtained by non-parametric bootstrapping. - It is easy to add covariates to the regression, which can help justify the common trend assumption (i.e. common \(\beta_2\) for all individuals), and can potentially reduce the residual variance and increase the precision of the DiD \(\beta_3\). - Note that by adding covariates, we are assuming that the DiD \(\beta_3\) (i.e. ATT) are the same across all cells \(X=x\) specified by covariates \(X\).

It is possible to extend the simple 2-periods-2-groups model to multiple groups and multiple periods, i.e. consider the general regression: $$Y_{gt}=\beta\cdot D_{gt}+\alpha_g+\lambda_t+\varepsilon_{gt}$$ where \(g\in\{1,2,\dots,G\}\) is the index of groups, \(\alpha_g\) is the group dummy (fixed effect) and \(\lambda_t\) is the time period dummy. And \(D_{gt}=1\) if group \(g\) receives treatment in period \(t\) and \(0\) if not. - In this general regression setting, treatment \(D_{gt}\) can be continuous, and \(\beta\) could still identify ATT if we assume the treatment effect is constant. - Treatment \(D_{gt}\) can be implemented at different groups at different times, and \(\beta\) could still identify ATT if we assume the treatment effect is constant. - Note that we are still assuming the common trend, i.e. \(\lambda_t\) is the same across all groups.

18.6.1 The problem: correlation between residuals within group

Suppose we use the following regression (parametric) model to estimate DiD: $$Y_{gt}=\beta\cdot D_{gt}+\alpha_g+\lambda_t+\varepsilon_{gt}$$

For the estimate to be consistent, we need that \(\mathbb E[\varepsilon_{igt}\varepsilon_{jgt}]=0\) for \(i\ne j\). However, observations in each group \(g\) of each period \(t\) are likely to be correlated due to the existence of unobservables that are not included in the model, i.e. we might have $$\mathbb E[\varepsilon_{igt}\varepsilon_{jgt}]\ne0\quad(i\ne j)$$

18.6.2 Possible solutions - Imposing a parametric assumption: assume \(\varepsilon_{igt}=\upsilon_{gt}+\epsilon_{igt}\) where \(\epsilon\)'s are i.i.d. individual shocks and \(\upsilon_{gt}\) is the common shock to group \(g\). - Using cluster-robust standard errors which are robust to any correlation structure and heteroskedasticity. - Block bootstrapping, i.e. resampling with replacement for groups instead of individual observations and run the regression $$\overline Y_{gt}=\beta\cdot D_{gt}+\alpha_g+\lambda_t+\overline\varepsilon_{gt}$$ where \(\overline Y_{gt}\) and \(\overline\varepsilon_{gt}\) are the averages within each group \(g\).

The standard DiD model assumes the same common trend throughout all groups in the whole distribution of the outcome variable \(Y_{it}\). There are also models in which a common trend is assumed only at each quantile / each value of the distribution, such as quantile DiD and changes-in-changes (CiC).

The two models have the same idea of a quantile common trend but differ in the way of imposing that assumption: - Quantile DiD assumes a common trend in the outcome at a given quantile. - CiC assumes a common trend in the outcome at a given value of outcome.

Recall that the standard DiD assumes $$\mathbb E\!\left[Y_{i2}^0-Y_{i1}^0\mid D_{i2}=1\right]=\mathbb E\!\left[Y_{i2}^0-Y_{i1}^0\mid D_{i2}=0\right]$$ and identifies \(\mathbb E[Y_{i2}^0\mid D_{i2}=1]\) (the counterfactual expected outcome for the treated had they not been treated): $$\underbrace{\mathbb E\!\left[Y_{i2}^0\mid D_{i2}=1\right]}_{\text{counterfactual}}=\underbrace{\mathbb E\!\left[Y_{i1}^0\mid D_{i2}=1\right]}_{\text{observed data}}+\underbrace{\mathbb E\!\left[Y_{i2}^0-Y_{i1}^0\mid D_{i2}=0\right]}_{\text{observed data}}$$

Quantile DiD and CiC rely on a similar idea. For \(t=1,2\), let - \(F_t(Y)\) be the distribution of outcome \(Y\) in the treatment group; - \(G_t(Y)\) be the distribution of outcome \(Y\) in the control group.

18.7.1 Quantile DiD

For any quantile \(\tau\in[0,1]\), denote the counterfactual outcome of the treated without treatment at quantile \(\tau\) by \(k^{\text{QDiD}}(\tau)\), then quantile DiD assumes $$k^{\text{QDiD}}(\tau)-F_1^{-1}(\tau)=G_2^{-1}(\tau)-G_1^{-1}(\tau)$$ and identifies the quantile treatment effect for the treated (QTE) by estimating the counterfactual outcome \(k^{\text{QDiD}}(\tau)\) without treatment at quantile \(\tau\) in the treatment group through $$\underbrace{k^{\text{QDiD}}(\tau)}_{\text{counterfactual}}=\underbrace{F_1^{-1}(\tau)}_{\text{observed data}}+\underbrace{G_2^{-1}(\tau)-G_1^{-1}(\tau)}_{\text{observed data}}$$

Summarized in three steps: 1. Fix a quantile of \(Y\) in the pre-intervention outcome distribution of the treatment group, i.e. \(F_1(Y)=\tau\). 2. The counterfactual post-intervention outcome \(k^{\text{QDiD}}(\tau)\) without treatment at quantile \(\tau\) in the treatment group is estimated by $$\underbrace{k^{\text{QDiD}}(\tau)}_{\text{counterfactual}}=\underbrace{F_1^{-1}(\tau)}_{\text{observed data}}+\underbrace{G_2^{-1}(\tau)-G_1^{-1}(\tau)}_{\text{observed data}}$$ 3. The QTE estimate at quantile \(\tau\) is $$F_2^{-1}(\tau)-k^{\text{QDiD}}(\tau)$$

Figure 4 (paraphrased): the horizontal axis is \(y\), the vertical axis the cumulative distribution \(F(y)\). The treatment distribution \(F_t(y)\) and control distribution \(G_t(y)\) each shift over time. At a fixed cumulative-probability level, read off the horizontal distances: \(\text{DF}\) is the horizontal shift of the treated quantile, \(\text{DG}\) that of the control quantile. In the graph $$k^{\text{QDiD}}(\tau)-F_1^{-1}(\tau)=G_2^{-1}(\tau)-G_1^{-1}(\tau)=DG$$ and $$DF-DG=F_2^{-1}(\tau)-k^{\text{QDiD}}(\tau)$$ i.e. the treated quantile's actual shift \(\text{DF}\) minus the counterfactual (common-trend) shift \(\text{DG}\) gives the QTE at that quantile.

18.7.2 CiC

The CiC model is fixing the outcome variable instead of the quantile. For any outcome \(y\) and corresponding quantiles \(G_1(y)\) and \(F_1(y)\), denote the counterfactual outcome of the treated without treatment at quantile \(F_1(y)\) by \(k^{\text{CiC}}(y)\), then the CiC assumes $$k^{\text{CiC}}(y)-y=G_2^{-1}\!\left(G_1(y)\right)-y$$ and identifies the treatment effect at level \(Y=y\) by estimating the counterfactual outcome: $$\begin{aligned}\underbrace{k^{\text{CiC}}(y)}_{\text{counterfactual}}&=\underbrace{y}_{\text{observed}}+\underbrace{G_2^{-1}\!\left(G_1(y)\right)-y}_{\text{observed}}\\&=\underbrace{G_2^{-1}\!\left(G_1(y)\right)}_{\text{observed}}\end{aligned}$$

Summarized in three steps: 1. Fix a level of outcome \(Y=y\). 2. The counterfactual post-intervention outcome \(k^{\text{CiC}}(y)\) without treatment at quantile \(F_1(y)\) in the treatment group is estimated by $$\underbrace{k^{\text{CiC}}(y)}_{\text{counterfactual}}=\underbrace{y}_{\text{observed}}+\underbrace{G_2^{-1}\!\left(G_1(y)\right)-y}_{\text{observed}}=\underbrace{G_2^{-1}\!\left(G_1(y)\right)}_{\text{observed}}$$ 3. The CiC estimate at level \(y\) is $$F_2^{-1}\!\left(F_1(y)\right)-k^{\text{CiC}}(y)$$

Figure 5 (paraphrased): similar to Figure 4, horizontal axis \(y\) and vertical axis \(F(y)\), but now the outcome value \(y\) is fixed rather than the quantile. \(\text{DF}\) and \(\text{DG}\) are the shifts of the treatment and control groups at that outcome value. In the graph $$k^{\text{CiC}}(y)-y=G_2^{-1}\!\left(G_1(y)\right)-y=DG$$ and $$DF-DG=F_2^{-1}\!\left(F_1(y)\right)-k^{\text{CiC}}(y)$$

18.8.1 Failure of common trend assumption

So far, we have established that the common trend assumption is crucial for the DiD to identify ATT. Unfortunately, the common trend assumption may not hold in some cases.

For example, a well known violation of the common trend assumption is Ashenfelter's Dip. Ashenfelter (1978) notes that enrollment in a training program is more likely to happen if a temporary dip in earnings occurs just before the start of the program. So the individuals in the treatment group (i.e. participating in the training program) are more likely to catch up in their earnings (i.e. income grows faster) even without the training program since the dip is just temporary, which breaks the common trend assumption and thus the DiD estimator will overstate the impact of the training program.

18.8.2 Functional form dependence

Note that the common trend assumption $$\mathbb E\!\left[Y_{i2}^0-Y_{i1}^0\mid D_{i2}=1\right]=\mathbb E\!\left[Y_{i2}^0-Y_{i1}^0\mid D_{i2}=0\right]$$ depends on the functional form of \(Y_{it}^0\). For example, if the common trend assumption holds for \(Y_{it}^0=\) wages, then it won't hold for \(Y_{it}^0=\log\) wages. So, non-linear transformation of outcome variable may ruin the common trend assumption and thus fail the identification. And it is often hard to decide between common trend in levels and common trend in logs, because they mean very different stories.

18.8.3 Composition changes

The occurrence of treatment may affect the composition of the treatment and the control groups. If this happens, then the DiD fails to identify ATT because the treatment and control groups are different before and after the treatment. The solution to this problem is redefining the groups such that the composition of all groups doesn't change after the treatment.

When the common trend assumption does not hold, the following extensions could probably give us some reasonable estimate.

18.9.1 Time varying covariates and/or group specific time trends

Assume that the treatment effect on outcome (\(\beta\)) is the same across all groups. Then, the common trend assumption can be relaxed by including time varying covariates (\(X_{gt}\)) and/or group specific linear time trends (\(\mu_g\cdot t\)), i.e. consider the regression $$Y_{gt}=\beta\cdot D_{gt}+X_{gt}\cdot\pi+\mu_g\cdot t+\alpha_g+\lambda_t+\varepsilon_{gt}$$

Note that we need at least three periods to estimate the group specific time trends \(\mu_g\)'s. See Besley and Burgess (2004) for an example that studies the effect of labor market regulation on manufacturing performance in Indian states.

18.9.2 Difference-in-differences-in-differences (DDD)

Sometimes a third difference might work when we suspect violation of the common trend assumption.

For example, a state implements change in health care policy for people with age \(\ge65\). Then, \(age\ge65\) is the treatment group and \(age<65\) is the control group. Denote \(T\) as the treatment state and \(C\) as the control state.

• Possible \(\text{DiD}_1\): data on health in treatment state before and after the policy change, for people with age \(\ge65\) and $<65$, i.e. $$\begin{aligned}\text{ATT}=\text{DiD}_1&=\left(\mathbb E[\text{Health}\mid\text{after},age\ge65,T]-\mathbb E[\text{Health}\mid\text{before},age\ge65,T]\right)\\&\quad-\left(\mathbb E[\text{Health}\mid\text{after},age<65,T]-\mathbb E[\text{Health}\mid\text{before},age<65,T]\right)\end{aligned}\tag{18.1}$$

• This identification of ATT depends on the common trend in health between the old and the young, i.e. $$\begin{aligned}&\mathbb E[\text{Health}\mid\text{after but no treatment},age\ge65,T]-\mathbb E[\text{Health}\mid\text{before},age\ge65,T]\\=\,&\mathbb E[\text{Health}\mid\text{after},age<65,T]-\mathbb E[\text{Health}\mid\text{before},age<65,T]\end{aligned}$$ which is unlikely to hold under common sense.

• Possible \(\text{DiD}_2\): data on health in treatment state before and after the policy change, for people with age \(\ge65\) in treatment state and neighbor state (control group), i.e. $$\begin{aligned}\text{ATT}=\text{DiD}_2&=\left(\mathbb E[\text{Health}\mid\text{after},age\ge65,T]-\mathbb E[\text{Health}\mid\text{before},age\ge65,T]\right)\\&\quad-\left(\mathbb E[\text{Health}\mid\text{after},age\ge65,C]-\mathbb E[\text{Health}\mid\text{before},age\ge65,C]\right)\end{aligned}\tag{18.2}$$

• This identification of ATT depends on the common trend in health between the two states, i.e. $$\begin{aligned}&\mathbb E[\text{Health}\mid\text{after but no treatment},age\ge65,T]-\mathbb E[\text{Health}\mid\text{before},age\ge65,T]\\=\,&\mathbb E[\text{Health}\mid\text{after},age\ge65,C]-\mathbb E[\text{Health}\mid\text{before},age\ge65,C]\end{aligned}$$ which is unlikely to hold due to the possible differences in the two states.

But we can reasonably assume that the difference in health trend between the old (without treatment) and the young is common between treatment state and control state, i.e. $$\begin{aligned}&\left(\mathbb E[\text{Health}\mid\text{after but no treatment},age\ge65,T]-\mathbb E[\text{Health}\mid\text{before},age\ge65,T]\right)\\&\quad-\left(\mathbb E[\text{Health}\mid\text{after},age<65,T]-\mathbb E[\text{Health}\mid\text{before},age<65,T]\right)\\=\,&\left(\mathbb E[\text{Health}\mid\text{after},age\ge65,C]-\mathbb E[\text{Health}\mid\text{before},age\ge65,C]\right)\\&\quad-\left(\mathbb E[\text{Health}\mid\text{after},age<65,C]-\mathbb E[\text{Health}\mid\text{before},age<65,C]\right)\end{aligned}$$

Then, the counterfactual \(\mathbb E[\text{Health}\mid\text{after but no treatment},age\ge65,\text{treatment state}]\) can be estimated by $$\begin{aligned}&\mathbb E[\text{Health}\mid\text{after but no treatment},age\ge65,T]\\=\,&\mathbb E[\text{Health}\mid\text{before},age\ge65,T]\\&+\left(\mathbb E[\text{Health}\mid\text{after},age<65,T]-\mathbb E[\text{Health}\mid\text{before},age<65,T]\right)\\&+\left(\mathbb E[\text{Health}\mid\text{after},age\ge65,C]-\mathbb E[\text{Health}\mid\text{before},age\ge65,C]\right)\\&-\left(\mathbb E[\text{Health}\mid\text{after},age<65,C]-\mathbb E[\text{Health}\mid\text{before},age<65,C]\right)\end{aligned}$$

And thus the ATT is identified by $$\begin{aligned}\text{ATT}&=\mathbb E[\text{Health}\mid\text{after},age\ge65,T]-\mathbb E[\text{Health}\mid\text{after but no treatment},age\ge65,T]\\&=\left(\mathbb E[\text{Health}\mid\text{after},age\ge65,T]-\mathbb E[\text{Health}\mid\text{before},age\ge65,T]\right)\\&\quad-\left(\mathbb E[\text{Health}\mid\text{after},age<65,T]-\mathbb E[\text{Health}\mid\text{before},age<65,T]\right)\\&\quad-\left(\mathbb E[\text{Health}\mid\text{after},age\ge65,C]-\mathbb E[\text{Health}\mid\text{before},age\ge65,C]\right)\\&\quad+\left(\mathbb E[\text{Health}\mid\text{after},age<65,C]-\mathbb E[\text{Health}\mid\text{before},age<65,C]\right)\end{aligned}\tag{18.3}$$

Consider the following two DiD's, which are different from the \(\text{DiD}_1\) in (18.1) and \(\text{DiD}_2\) in (18.2): - Define \(\text{DiD}_a\) as the difference in health trend between the old and the young in the treatment state, i.e. $$\begin{aligned}\text{DiD}_a&=\left(\mathbb E[\text{Health}\mid\text{after},age\ge65,T]-\mathbb E[\text{Health}\mid\text{before},age\ge65,T]\right)\\&\quad-\left(\mathbb E[\text{Health}\mid\text{after},age<65,T]-\mathbb E[\text{Health}\mid\text{before},age<65,T]\right)\end{aligned}$$ - Define \(\text{DiD}_b\) as the difference in health trend between the old and the young in the control state, i.e. $$\begin{aligned}\text{DiD}_b&=\left(\mathbb E[\text{Health}\mid\text{after},age\ge65,C]-\mathbb E[\text{Health}\mid\text{before},age\ge65,C]\right)\\&\quad-\left(\mathbb E[\text{Health}\mid\text{after},age<65,C]-\mathbb E[\text{Health}\mid\text{before},age<65,C]\right)\end{aligned}$$

Define the difference-in-differences-in-differences (DiDiD) by $$\begin{aligned}\text{DiDiD}&=\text{DiD}_a-\text{DiD}_b\\&=\left(\mathbb E[\text{Health}\mid\text{after},age\ge65,T]-\mathbb E[\text{Health}\mid\text{before},age\ge65,T]\right)\\&\quad-\left(\mathbb E[\text{Health}\mid\text{after},age<65,T]-\mathbb E[\text{Health}\mid\text{before},age<65,T]\right)\\&\quad-\left(\mathbb E[\text{Health}\mid\text{after},age\ge65,C]-\mathbb E[\text{Health}\mid\text{before},age\ge65,C]\right)\\&\quad+\left(\mathbb E[\text{Health}\mid\text{after},age<65,C]-\mathbb E[\text{Health}\mid\text{before},age<65,C]\right)\\&=\text{ATT}\end{aligned}$$ where the last line is true by (18.3). So, the DiDiD identifies the ATT under the assumption of the same old-young contrast between the two states.

18.9.3 Difference-in-differences with instrument variables

Suppose the common trend assumption fails and the trends are group specific. Then, in the mis-specified regression $$Y_{gt}=\beta\cdot D_{gt}+\alpha_g+\lambda_t+\varepsilon_{gt}$$ the error term \(\varepsilon_{gt}\) will contain the group specific trend.

If the treatment \(D_{gt}\) is related to the group specific trend, then \(D_{gt}\) becomes an endogenous variable. To solve this endogeneity problem, we can construct an instrument variable that affects treatment but is arguably unrelated to group specific trends. Then, the IV estimate or TSLS estimate will give us a consistent estimate of \(\beta\). Given the assumption that the treatment effect is the same across all groups, the \(\beta\) is thus identified and estimated. See Haan et al. (2018) for example that study the effect of supply of schools on pupils' test scores.

18.9.4 Synthetic controls

• Panel data gives us multi-periods of one treated group and many untreated groups.
• Suppose that none of the untreated groups has the common trend with the treated group.
• Then, we can construct a weighted average of all the untreated groups that well matches the treated group prior to treatment period.
• And that weighted average of untreated groups is called synthetic control group.
• The DiD between the treatment group and the synthetic control group identifies the treatment effect.
• However, there are increasing concerns on the validity of this method, since the matching on pre-periods need not say anything about the post-periods.

18.10.1 Notation - Time runs from \(t=0,1,\dots,T\), and treatment is \(D_{it}\in\{0,1\}\). - \(E_i\) denotes individual \(i\)'s first treatment (event) date. - It is possible that \(E_i=\infty\), which means that the event never happens to individual \(i\). - In this set-up, treatment is absorbing, i.e. \(D_{it}=1\) if and only if \(t\ge E_i\). - \(E_i\) fully determines the sequence \(D_i\equiv(D_{i0},\dots,D_{iT})\) by $$E_i=e\quad\Leftrightarrow\quad D_i=\Big(0,\dots,0,\overset{t=e}{1},1,\dots,1\Big)$$ $$E_i=\infty\quad\Leftrightarrow\quad D_i=(0,\dots,0)$$ - \(Y_{it}(e)\) is the potential outcomes of individual \(i\) in period \(t\) with the counterfactual event date \(e\). - We can only observe \(Y_{it}=Y_{it}(E_i)\) for \(t=0,1,\dots,T\).

18.10.2 Baseline assumptions for an event study

• Common trends assumption: for any \(s
• No anticipation assumption: for all untreated periods \(s

18.10.3 Identification under the two assumptions

With the common trends assumption and the no anticipation assumption, we have that for any \(s

so the desired counterfactual \(\mathbb E[Y_{it}(\infty)\mid E_i=e]\) is identified. The second equality uses no anticipation \(Y_{is}(e)=Y_{is}(\infty)\).

Therefore, for \(t\ge e\), the average treatment effect on \(Y_{it}\) of being treated at time \(e\), denoted by \(\text{ATE}_t(e)\), can be identified as: $$\begin{aligned}\text{ATE}_t(e)&=\mathbb E\!\left[Y_{it}(e)-Y_{it}(\infty)\mid E_i=e\right]\\&=\mathbb E\!\left[Y_{it}(e)\mid E_i=e\right]-\mathbb E\!\left[Y_{it}(\infty)\mid E_i=e\right]\\&=\mathbb E\!\left[Y_{it}(e)\mid E_i=e\right]-\mathbb E\!\left[Y_{is}(e)\mid E_i=e\right]-\mathbb E\!\left[Y_{it}(\infty)-Y_{is}(\infty)\mid E_i=\infty\right]\\&=\mathbb E\!\left[Y_{it}\mid E_i=e\right]-\mathbb E\!\left[Y_{is}\mid E_i=e\right]-\mathbb E\!\left[Y_{it}-Y_{is}\mid E_i=\infty\right]\end{aligned}$$ where \(s

Note that there are multiple ways to non-parametrically estimate \(\text{ATE}_t(e)\) since we can choose any \(s

18.10.4 Event study as regression

We can also estimate the event effect by parametric methods. Following the same specification as in DiD, we can regress \(Y_{it}\) on unit dummies \(\mu_i\), period dummies \(\lambda_t\), and treatment \(D_{it}=\mathbf 1\{t\ge E_i\}\), i.e. $$Y_{it}=\mu_i+\beta\cdot D_{it}+\lambda_t+\varepsilon_{it}$$

Abraham and Sun (2018) shows that $$\beta=\sum_{e=0}^{T}\sum_{t=e}^{T}\omega_t(e)\,\text{ATE}_t(e)$$ where the weights \(\omega_t(e)\) are identified, sum up to 1, but can be negative if causal effects are heterogeneous at different \(e\)'s.

18.10.5 Adding leads and lags to the event date: dynamic specification

• We can instead regress \(Y_{it}\) on unit dummies \(\mu_i\), period dummies \(\lambda_t\), and leads and lags \(\{R_{it}^\ell\}_{\ell\in\mathcal L}\).
• \(R_{it}^\ell\equiv\mathbf 1\{\ell=t-E_i\}\) is the indicator for \(\ell\) periods since the event in \(E_i\).
• Note that \(R_{it}^\ell\) stands for leads when \(\ell<0\) and for lags when \(\ell\ge0\).
• Coefficients on \(\{R_{it}^\ell\}_{\ell\in\mathcal L}\) are the objects of interest:
• For \(\ell>0\), the coefficient on \(R_{it}^\ell\) is the dynamic treatment effects.
• For \(\ell<0\), the coefficient on \(R_{it}^\ell\) can be used to check for the validity of the design, i.e. the no anticipation assumption.