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What is a difference in differences estimator Difference in differences (DiD) is a tool to estimate treatment effects comparing the pre- and post-treatment differences in the outcome of a treatment and a control group. In general, we are interested in estimating the effect of a treatment $D_i$ (e.g. union status, medication, etc.) on an outcome $Y_i$ (e.g. ...

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The typical thing to do is visual inspection of the pre-treatment trends for the control and treatment group. This is particularly easy if you only have those two groups given a single binary treatment. Ideally the pre-treatment trends should look something like this: This graph was taken from a previous answer to the question why we need the common ...

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The typical way to estimate a difference in differences model with more than two time periods is your proposed solution b). Keeping your notation you would regress $$Y_{ist} = \alpha +\gamma_s (\text{Treatment}_s) + \lambda (\text{year dummy}_t) + \delta D_{st} + \epsilon_{ist}$$ where $D_t \equiv \text{Treatment}_s\cdot d_t$ is a dummy variable which equals ...

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A key assumption of difference-in-differences (DID) is that both groups have a common trend in the outcome variable before the treatment. This is important in order to make the argument that the change for the treated group is because of the treatment and not because the two groups were already different from each other to begin with. If you sample ...

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What you propose here is actually difference in difference in differences (DDD) instead of the usual difference in differences (see these lecture notes by Imbens and Wooldridge (2007) on the first two pages). This method can potentially account for the unobserved trends in wages of women across your two towns and the wage changes of both male and female ...

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Assuming you have the original data and not just the summary of the fits, the general solution to this problem is to fit a model with an interaction, i.e. to go back to the data and fit the model $$Y = \beta_0 + \beta_1 I(t>t_I) + \beta_2 (t-t_I) + \beta_3 I(t>t_I) (t-t_I)$$ where $I(t>t_I)$ is an indicator variable, i.e. =1 if $t>t_I$ and 0 ...

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The model is fine but instead of standardizing the treatment years there is an easier way to incorporate different treatment times in difference in differences (DiD) models which would be to regress, $$y_{it} = \beta_0 + \beta_1 \text{treat}_i + \sum^T_{t=2} \beta_t \text{year}_t + \delta \text{policy}_{it} + \gamma C_{it} + \epsilon_{it}$$ where $\text{... 15 Yes, it makes sense and in this case the coefficient for the interaction of the post-treatment indicator and the treatment variable gives you the effect on the outcome that results from an increase in the treatment intensity. An example of this is the paper by Acemoglu, Autor and Lyle (2004), where they estimate the effect of World War II on female labor ... 15 A nice feature of difference-in-differences (DiD) is actually that you don't need panel data for it. Given that the treatment happens at some sort of level of aggregation (in your case cities), you only need to sample random individuals from the cities before and after the treatment. This allows you to estimate $$y_{ist} = A_g + B_t + \beta D_{st} + c X_{... 15 I will assume you have a thorough grasp of the two group/two period difference-in-differences (DD) design and you now want to extend your intuition of the method to the multi-group/multi-period case. Suppose we have multiple observations of i units (e.g., counties) across multiple t periods (e.g., years). In DD applications, the data is ‘aggregated up’ ... 10 There are two reasons for including covariates in a difference in differences regression: for identification of the treatment effect; to reduce the error variance (i.e. increase power of statistical tests). Suppose you want to know the effect of a job market program on employment in a city where this program was randomly assigned to unemployed individuals. ... 10 The difference in differences (DiD) model is actually a type of fixed effects because the differencing gets rid of the individual fixed effects.^1 Regarding the pros and cons, it really depends what you want to do. DiD is mainly for causal inference with observational data whereas the fixed effects model primary task is to get rid of the correlation ... 9 Linear DiD Methods You could stick with the linear probability model which you can easily estimate via least squares. Running a simple linear regression for your difference in differences analysis has several nice properties: the DiD coefficient is readily interpretable (which is not necessarily true for interaction terms in nonlinear models - see Ai and ... 9 What is typically done is that you plot the averages of the outcome variables for your treatment and control group over time. So the control group here are naturally all those who did not receive the treatment whilst the treatment group are those who receive any intensity of the treatment. That was done for instance in this presentation (slides 25 and 26, ... 9 In your setting you already control for aggregate time effects via the inclusion of time dummies (\text{month}_t), which are more flexible than a linear time trend. To probe for the robustness of their results, people typically include individual specific linear time trends. These help to rule out the possibility that treatment and control individuals were ... 8 RD is about comparing two groups that are very similar except for the treatment because the treatment depends discontinuously on some cutoff. For example, those with a test score of 1499 don't get to go college and those with 1501 do. The underlying ability of these two groups is probably similar enough, so the wage comparison for those on either side of ... 7 The fundamental problem with causal inference is that we never observe what would have happened to the treated if the treatment had not occured. If I give you a million dollars now, and next month I compare your consumption to what it is today then I will perhaps not get the treatment effect. Why? Perhaps your consumption next month would have changed anyway ... 7 You construct the policy dummy the way you first describe it, i.e. create a column of zeroes. Then for each firm you replace this with ones if a firm is in the treatment group AND it is in the post-treatment period. Something like this$$ \begin{array}{ccccc} \text{firm} & \text{time} & \text{treated} & \text{post} & \text{policy} \\ \hline ... 7 Essentially what happens is that the staggered DID can be interpreted as a weighted average of two-period DIDs. So if a group is treated in multiple periods and another is not it could happen that the treated group enters in as a control. This matters if there is treatment effect heterogeneity (for example the treatment has a larger effect in the 2nd active ... 6 A triple difference-in-difference is the correct specification for this problem. I'll present a conceptual explanation and then a mathematical one. Conceptually, the standard (double) difference-in-difference can also be thought of as estimating heterogeneous treatment effect. In this perspective, time is the "treatment", and we want to estimate how time ... 6 The difference in differences is what is called an interaction in statistics (as Dimitriy Masterov has already pointed out). You want to test whether the time effect is different when you intervene compared with when you don't. Your data is most naturally modelled as binomial, i.e., the number of top scores out of total people surveyed in each area at each ... 5 The book is correct, but it is easier to see if you insert the corresponding values of the dummies and check what happens to the regression equation. Let's go through the possible values together.$NJ_s = 0$is Pennsylvania$NJ_s = 1$is New Jersey$d_t = 0$is February$d_t = 1$is November So your baseline regression is$NJ_s = 0$and$d_t = 0$, i.e. the ... 5 For difference in differences you can compute the mean of the outcome$Y$for each group$g =${$C$control,$T$treated} in each period$t =${1 pre-treatment, 2 post-treatment}, $$(E[Y_{igt}|g=T, t=2] - E[Y_{igt}|g=T, t=1]) - (E[Y_{igt}|g=C, t=2] - E[Y_{igt}|g=C, t=1])$$ which you can do by hand with the corresponding if-statements in SPSS. If you need ... 5 You can should treat the interaction variable as a dummy and follow this advice from David Giles: If$Treat\cdot Post$switches from 0 to 1, the % impact on$Y$is$100 \cdot (\exp(\beta_4 - \frac{1}{2} \hat \sigma_{\beta_4}^2)-1).$5 The most common generalization is as follows (here I am following Bertrand et al. (2004), but for example Hansen (2007) also considers a similar approach): Let$Y_{ist}$be the outcome of interest for individual$i$in group$s$(such as a state) by time$t$(such as a year) and$I_{st}$be a dummy for whether the intervention has affected group$s$at time ... 5 Propensity score matching before a difference in differences analysis is a potent way to get around different parallel trends in the pre treatment period and has been used in several papers (e.g. Becker and Hvide, 2013; Ichino et al., 2007). So it definitely does make sense. The advantage over simply including country dummies in your difference in ... 5 The central assumption in DID estimation is that the trends in the outcome variable would have been parallel in the treated and control groups if there had been no treatment. A common way of checking if this assumption seems plausible is to see if the trends were parallel before the intervention. It isn't necessary to have random assignment for this ... 5 If including a constant term gives you coefficients that are not significant it means this variables are neutral with respect to their conditional effect and thus can be numerically either positive or negative without interpretation. If excluding a constant term gives you statistical significance this means that the variable in question has had a significant ... 5 Summary In general, the DiD analysis is mathematically identical to the interaction term from the repeated measures analysis. If any of that is confusing, or you'd like more explanation, or you want to know how to run these analyses, then keep reading! First, I think your understanding of a repeated measures ANOVA is ok, but your DiD formula is a little ... 5 Let$t=1,2,3$and$T=1$if treated, zero otherwise.$\mathbf{I}\$ is the indicator function. The expected value from the specification I suggested in the comments above is $$E[Y \vert X] = \beta_0 + \beta_1 T + \beta_2 \mathbf{I}_{t=2}+\beta_3 \mathbf{I}_{t=3}+\beta_4 \mathbf{I}_{t=2} \cdot T + \beta_5 \cdot \mathbf{I}_{t=3} \cdot T.$$ The DID for the third ...

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