This vignette demonstrates how to use the DiSCo package by way of the
empirical application in Gunsilius (2023),
which is based on Dube (2019). We
illustrate the use of the two main functions: 1) `DiSCo`

,
which estimates the raw distributional counterfactuals, computes
confidence intervals, and optionally performs a permutation test; and 2)
`DiSCoTEA`

, the “Treatment Effect Aggregator”, which takes in
the distributional counterfactuals and computes aggregate treatment
effects using a user-specified aggregation statistic.

We briefly review the main idea behind Distributional Synthetic
Controls. Denote \(Y_{jt,N}\) the
outcome of group \(j\) in time period
\(t\) in the absence of an
intervention. Also denote \(Y_{jt,I}\)
the outcome in the presence of an intervention at time \(t > T_0\). Denote the quantile function,
\[
F^{-1}(q):=\inf _{y \in \mathbb{R}}\{F(y) \geq q\}, \quad q \in(0,1),
\] where \(F(y)\) is the
corresponding cumulative distribution function. One unit \(j=1\) has received treatment, while the
other units \(j=2,\ldots,J\) have not.
Then the goal is to estimate the counterfactual quantile function \(F_{Y_{1 t}, N}^{-1}\) of the treated unit
had it not received treatment by an optimally weighted average of the
control units’ quantile functions, \[
F_{Y_{1 t}, N}^{-1}(q)=\sum_{j=2}^{J+1} \lambda_j^* F_{Y_{j t}}^{-1}(q)
\quad \text { for all } q \in(0,1)
\]

In practice, we do this by solving the following optimization problem:
\[
\vec{\lambda}_t^*=\underset{\vec{\lambda} \in
\Delta^J}{\operatorname{argmin}} \int_0^1\left|\sum_{j=2}^{J+1}
\lambda_j F_{Y_{j t}}^{-1}(q)-F_{Y_{1 t}}^{-1}(q)\right|^2 d q
\]which gives an optimal weight for each unit-period combination.
This problem can be solved by a simple weighted regression, which is
implemented in the `DiSCo_weights_reg`

function. To obtain
the overall optimal weights \(\vec{\lambda}^*\), we take the average of
the optimal weights across all periods.

To get coding, we load the data from Dube (2019), which is available in the package.

To learn more about the data, just type `?dube`

in the
console. We have already renamed the outcome, id, and time variables to
`y_col`

, `id_col`

, and `time_col`

,
respectively, which is required before passing the dataframe to the
DiSCo command. We also need to set the two following parameters:

which indicate the id of the treated unit and the time period of the intervention, respectively. We can now run the DiSCo command:

```
df <- copy(dube)
disco <- DiSCo(df, id_col.target, t0, G = 1000, num.cores = 1, permutation = TRUE, CI = TRUE, boots = 1000, graph = TRUE, simplex=TRUE, seed=1, q_max=0.9)
```

where we have chosen a grid (`G`

) of 1000 quantiles and
opted for parallel computation with 5 cores to speed up the permutation
test and confidence intervals (`CI`

is set to
`TRUE`

), which we calculate using 1000 resamples
(`boots`

). We also set the `seed`

explicitly in
the function call, which ensures reproducibility across the parallel
cores. We followed the “classical” approach of Abadie, Diamond, and Hainmueller (2010) by
restricting the weights to be between 0 and 1
(`simplex=TRUE`

). Finally, we restricted the data to the
0-90th quantile to account for the fact that the CPS data used in Dube (2019) is imputed for the top income
quantiles (more detail in the section below).

The returned `disco`

object contains a host of information
produced by the command. Typing `?DiSCoT`

in the console
pulls up the help files which lay out the precise structure of the
returned object. For now, we will just have a look at the top 10
estimated weights,

```
# retrieve the weights
weights <- disco$weights
# retrieve the control unit IDs
controls <- disco$control_ids
# store in a dataframe
weights_df <- data.frame(weights = weights, fips = controls)
# merge with state fips codes (built into the maps package)
state.fips <- as.data.table(maps::state.fips)
state.fips <- state.fips[!duplicated(state.fips$abb), c("fips", "abb")]
weights_df <- merge(weights_df, state.fips, by = "fips")
setorder(weights_df, -weights)
print(weights_df[1:10,])
```

When we ran the `DiSCo`

command, we set
`permutation`

to `TRUE`

, which runs the
permutation test described in the paper (see `?DiSCo_per`

for
more details). This will allow us to inspect the permutation inference
results below. Already, by setting `graph`

to
`TRUE`

, the function displayed a plot of the full
distribution of permutation tests. The black solid line shows the fit of
the “true” synthetic control. The fact that it does not diverge stronger
than the other lines in gray after treatment suggests that the treatment
had no effect.

For context, the y-axis is the squared Wasserstein distance between the counterfactual and observed quantile functions, \[ d_{t t}^2:=\int_0^1\left|F_{Y_{u t, N}}^{-1}(q)-F_{Y_{u t}}^{-1}(q)\right|^2 d q \]and, as in Abadie, Diamond, and Hainmueller (2010) we take the ratio of post- to pre-intervention Wasserstein distance to account for variation in the pre-treatment fit across placebo tests, \[ r_j=\frac{R_j\left(T_0+1, T\right)}{R_j\left(1, T_0\right)} \] and calculate the p-value for the permutation test as, \[ p=\frac{1}{J+1} \sum_{j=1}^{J+1} H\left(r_j-r_1\right), \]where \(H(x)\) is the Heaviside function which is 1 if \(x\geq 0\) and 0 otherwise. This p-value gives the probability of observing a placebo test with a larger ratio than the true treatment effect. It can be retrieved by calling

The p-value is larger than 0.05, which confirms the visual result from the plot above, while accounting for potential differences in pre-treatment fit across placebo units.

Finally, we can use the DiSCo Treatment Effect Aggregator
(`DiSCoTEA`

) function to aggregate the resulting
counterfactual quantile functions into various treatment effect
measures. For example, we can calculate the difference between the
counterfactual and observed quantile functions and cumulative
distribution functions (CDF) as follows.

Calling `summary`

on the returned object prints a table
summarizing the effects across the distribution. You can choose the
intervals of quantiles over which it aggregates using the
`samples`

parameter. If you calculated the permutation test
and confidence intervals in the `DiSCo`

function, these are
reported as well.

Setting `graph`

to `TRUE`

also prints a plot of
the distribution differences over time You can focus on specific years
using the `t_plot`

parameter. You can use the other
parameters of the `DiSCOTEA`

function to adjust the basic
appearance of the plots, or directly alter the ggplot object that is
stored in the returned `DiSCoT`

object.

Looking at the plot of the quantile and CDF differences, we can see two things: 1) the pre-treatment fit for the years 1998-2002 is decent but not perfect; 2) there do not appear to be any notable effects of the minimum wage inrease, asides from fluctuations in the function differences that are similar to those pre-treatment.

The package offers various ways to further test suspected effects. For example, we can focus on a specific part of the distribution, and construct a separate synthetic control for it. Mathematically, this comes down to,

\[ \vec{\lambda}_t^*=\underset{\vec{\lambda} \in \Delta^J}{\operatorname{argmin}} \int_{\text{q_min}}^{\text{q_max}}\left|\sum_{j=2}^{J+1} \lambda_j F_{Y_{j t}}^{-1}(q)-F_{Y_{1 t}}^{-1}(q)\right|^2 d q, \]

where \(\text{q_min} < \text{q_max}\) are the bounds of the quantile range we’re interested in. For example, following Dube (2019), we can focus on the lower end of the distribution, up until observations that earn 3.5 times the poverty income threshold. This corresponds to around the 0.65th quantile in our data:

We can simply run the following code,

`disco <- DiSCo(dube, id_col.target=id_col.target, t0=t0, G = 1000, num.cores = 1, permutation = TRUE, CI = TRUE, boots = 1000, graph = FALSE, q_min = 0, q_max=0.65, seed=1, simplex=TRUE)`

One could also try to deal with irregularity in the estimated
quantile functions is by using the `qmethod`

option in the
`DiSCo`

command, which allows for the use of alternative
quantile estimation methods that can account for non-smoothness and
extreme values. As above, we plot the results using the
`DiScoTEA`

function:

Again, we do not observe any subtantial changes in the quantile functions or CDFs. Also, the p-value of the permutation test is now larger, losing the marginal significance it had before. Overall, we cannot reject the null hypothesis of there being no effect of the treatment.

In this vignette, we have demonstrated the DiSCo package using the
data from the empirical application in Gunsilius
(2023). We estimated the difference in between the quantile
functions of the distributional synthetic control constructed with the
`DiSco`

command and the observed quantile function. To test
for the presence of treatment effects, we inspected the pre-treatment
distributional fit, carried out a permutation test, and re-estimated the
synthetic control for a restricted quantile range; all these robustness
tests are available in the DiSCo package, together with handy graphing
and aggregation tools.

Abadie, Alberto, Alexis Diamond, and Jens Hainmueller. 2010.
“Synthetic Control Methods for Comparative Case Studies:
Estimating the Effect of California’s Tobacco Control Program.”
*Journal of the American Statistical Association* 105 (490):
493–505.

Dube, Arindrajit. 2019. “Minimum Wages and the Distribution of
Family Incomes.” *American Economic Journal: Applied
Economics* 11 (4): 268–304.

Gunsilius, Florian F. 2023. “Distributional Synthetic
Controls.” *Econometrica* 91 (3): 1105–17.