# Difference in differences

Difference in differences (sometimes 'Difference-in-Differences',[1] 'DID',[2] or 'DD'[3]) is a technique used in econometrics that measures the effect of a treatment at a given period in time. It is often used to measure the change induced by a particular treatment or event, though it may be subject to certain biases (mean reversion bias, etc.). In contrast to a within-subjects estimate of the treatment effect (that measures the difference in an outcome after and before treatment) or a between-subjects estimate of the treatment effect (that measures the difference in an outcome between the treatment and control groups), the DID estimator represents the difference between the pre-post, within-subjects differences of the treatment and control groups.

## Formal Definition

Consider the model

$y_{ist} ~=~ \gamma_s + \lambda_t + \delta D_{st} + \epsilon_{ist}$

where $y_{ist}$ is the dependent variable for individual $i$, given $s$ and $t$. The dimensions $s$ and $t$ may for example be state and time. $\gamma_s$ and $\lambda_t$ is then the vertical intercept for $s$ and $t$ respectively. $D_{st}$ is a dummy variable indicating treatment status, $\delta$ is the treatment effect, and $\epsilon_{ist}$ is an error term.

Let

$\overline{y}_{st} ~=~ \frac{1}{n} \sum_{i=1}^{n} y_{ist}$,

$\overline{\gamma}_s ~=~ \frac{1}{n}\sum_{i=1}^{n}\gamma_s ~=~ \gamma_s$,

$\overline{\lambda}_t ~=~ \frac{1}{n}\sum_{i=1}^{n}\lambda_t ~=~ \lambda_t$,

$\overline{D}_{st} ~=~ \frac{1}{n}\sum_{i=1}^{n} D_{st} ~=~ D_{st}$,

$\overline{\epsilon}_{st} ~=~ \frac{1}{n}\sum_{i=1}^{n}\epsilon_{ist}$,

and suppose for simplicity that $s=1,2$ and $t=1,2$. Then

$(\overline{y}_{11} - \overline{y}_{12}) - (\overline{y}_{21} - \overline{y}_{22})$

$= \left [ (\gamma_1 + \lambda_1 + \delta D_{11} + \overline{\epsilon}_{11}) - (\gamma_1 + \lambda_2 + \delta D_{12} + \overline{\epsilon}_{12}) \right ] - \left [ (\gamma_2 + \lambda_1 + \delta D_{21} + \overline{\epsilon}_{21}) - (\gamma_2 - \lambda_2 + \delta D_{22} + \overline{\epsilon}_{22}) \right ]$

$= \delta (D_{11} - D_{12}) + \delta(D_{22} - D_{21}) + \overline{\epsilon}_{11} - \overline{\epsilon}_{12} + \overline{\epsilon}_{22} - \overline{\epsilon}_{21}$.

The strict exogeneity assumption then implies that

$E \left [ (\overline{y}_{11} - \overline{y}_{12}) - (\overline{y}_{21} - \overline{y}_{22}) \right ] ~=~ \delta (D_{11} - D_{12}) + \delta(D_{22} - D_{21})$.

Without loss of generality, assume that $D_{22}=1$ and $D_{11}=D_{12}=D_{21}=0$, giving the DID estimator

$\hat{\delta} ~=~ (\overline{y}_{11} - \overline{y}_{12}) - (\overline{y}_{21} - \overline{y}_{22})$,

which can be interpreted as the treatment effect of the treatment indicated by $D_{st}$.

## Assumptions

Illustration of the parallel trend assumption

All the assumptions of the OLS model apply equally to DID. In addition, DID requires a parallel trend assumption. The parallel trend assumption says that $\lambda_2 - \lambda_1$ are the same in both $s=1$ and $s=2$. Given that the formal definition above accurately represents reality, this assumption automatically holds. However, a model with $\lambda_{st} ~:~ \lambda_{22} - \lambda_{21} \neq \lambda_{12} - \lambda_{11}$ may well be more realistic.

As illustrated to the right, the treatment effect is the difference between the observed value of y and what the value of y would have been with parallel trends, had there been no treatment. The Achilles' heel of DID is when something other than the treatment changes in one group but not the other at the same time as the treatment, implying a violation of the parallel trend assumption.

To guarantee the accuracy of the DID estimate, the composition of individuals of the two groups is assumed to remain unchanged over time. When using a DID model, various issues that may compromise the results, such as autocorrelation and Ashenfelter dips, must be considered and dealt with.

## Implementation

The DID method can be implemented according to the table below, where the lower right cell is the DID estimator.

$y_{st}$ $s=2$ $s=1$ Difference
$t=2$ $y_{22}$ $y_{12}$ $y_{12}-y_{22}$
$t=1$ $y_{21}$ $y_{11}$ $y_{11}-y_{21}$
Change $y_{21}-y_{22}$ $y_{11}-y_{12}$ $(y_{11}-y_{21})-(y_{12}-y_{22})$

Running a regression analysis gives the same result. Consider the OLS model

$y ~=~ \beta_0 + \beta_1 T + \beta_2 S + \beta_3 (T \cdot S) + \varepsilon$

where $T$ is a dummy variable for $t=2$, and $S$ is a dummy variable for $s=2$. The composite variable $(T \cdot S)$ is then a dummy variable indicating when $S=T=1$. Although it is not shown rigorously here, it turns out that the estimates in this model are

$\hat{\beta}_0 ~=~ (y ~|~ T=0,~ S=0)$

$\hat{\beta}_1 ~=~ (y ~|~ T=1,~ S=0) - (y ~|~ T=0,~ S=0)$

$\hat{\beta}_2 ~=~ (y ~|~ T=0,~ S=1) - (y ~|~ T=0,~ S=0)$

$\hat{\beta}_3 ~=~ [(y ~|~ T=1,~ S=1) - (y ~|~ T=0,~ S=1)] - [(y ~|~ T=1,~ S=0) - (y ~|~ T=0,~ S=0)]$,

which is equivalent to

$\hat{\beta}_3 ~=~ (y_{11} - y_{21}) - (y_{12} - y_{22})$.

But this is the expression for the treatment effect that was given in the formal definition and in the above table.

## Card & Krueger (1994) example

Consider one of the most famous DID studies, the Card and Krueger article on minimum wage in New Jersey, published in 1994.[4] Card and Krueger compared employment in the fast food sector in New Jersey and in Pennsylvania, in February 1992 and in November 1992, after New Jersey's minimum wage rose from $4.25 to$5.05 in April 1992. Observing a change in employment in New Jersey only, before and after the treatment, would fail to control for omitted variables such as weather and macroeconomic conditions of the region. By including Pennsylvania as a control in a difference-in-differences model, any bias caused by variables common to New Jersey and Pennsylvania are implicitly controlled for, even when these variables are unobserved. Assuming that New Jersey and Pennsylvania have parallel trends over time, Pennsylvania's change in employment can be interpreted as the change New Jersey would have experienced, had they not increased the minimum wage, and vice versa. The evidence suggested that the increased minimum wage did not induce an increase in unemployment in New Jersey, as standard economic theory would suggest. The table below shows Card & Krueger's estimates of the treatment effect on employment, measured as FTEs (or Full-time equivalents). Keeping in mind that the finding is controversial, Card and Krueger estimate that the \$0.80 minimum wage increase in New Jersey lead to a 2.76 FTE increase in employment.

New Jersey Pennsylvania Difference
February 20.44 23.33 -2.89
November 21.03 21.17 -0.14
Change 0.59 -2.16 2.76

## Critics

In 2004, the question "How Much Should We Trust Differences-in-Differences Estimates?" was asked in an article with the same name,[3] and apparently the answer is "not all that much." It is worth noting however that this is only the case if DiD is employed without considering the auto-correlation in calculating the standard errors. Most papers that employ Difference-in-Differences estimation use many years of data and focus on serially correlated outcomes but ignore that the resulting standard errors are inconsistent, leading to serious over-estimation of t-statistics and significance levels.[3] These conventional DID standard errors severely understate the standard deviation of the estimators: we find an "effect" significant at the 5 percent level for up to 45 percent of the placebo interventions.[3] To alleviate this problem two corrections based on asymptotic approximation of the variance-covariance matrix work well for moderate numbers of states and one correction that collapses the time series information into a "pre" and "post" period and explicitly takes into account the effective sample size works well even for small numbers of states.[3]