DiSCo {DiSCos}

Distributional Synthetic Controls

Description

This function implements the distributional synthetic controls (DiSCo) method from Gunsilius (2023). as well as the alternative mixture of distributions approach.

Usage

DiSCo(
  df,
  id_col.target,
  t0,
  M = 1000,
  G = 1000,
  num.cores = 1,
  permutation = FALSE,
  q_min = 0,
  q_max = 1,
  CI = FALSE,
  boots = 500,
  replace = TRUE,
  uniform = FALSE,
  cl = 0.95,
  graph = FALSE,
  qmethod = NULL,
  qtype = 7,
  seed = NULL,
  simplex = FALSE,
  mixture = FALSE,
  grid.cat = NULL
)

Arguments

df

Data frame or data table containing the distributional data for the target and control units. The data table should contain the following columns:

• y_col A numeric vector containing the outcome variable for each unit. Units can be individuals, states, etc., but they should be nested within a larger unit (e.g. individuals or counties within a state)

• id_col A numeric vector containing the aggregate IDs of the units. This could be, for example, the state if the units are counties or individuals

• time_col A vector containing the time period of the observation for each unit. This should be a monotonically increasing integer.

id_col.target

Variable indicating the name of the target unit, as specified in the id_col column of the data table. This variable can be any type, as long as it is the same type as the id_col column of the data table.

t0

Integer indicating period of treatment.

M

Integer indicating the number of control quantiles to use in the DiSCo method. Default is 1000.

G

Integer indicating the number of grid points for the grid on which the estimated functions are evaluated. Default is 1000.

num.cores

Integer, number of cores to use for parallel computation. Default is 1. If the permutation or CI arguments are set to TRUE, this can be slow and it is recommended to set this to 4 or more, if possible. If you get an error in "all cores" or similar, try setting num.cores=1 to see the precise error value.

permutation

Logical, indicating whether to use the permutation method for computing the optimal weights. Default is FALSE.

q_min

Numeric, minimum quantile to use. Set this together with q_max to restrict the range of quantiles used to construct the synthetic control. Default is 0 (all quantiles). Currently NOT implemented for the mixture approach.

q_max

Numeric, maximum quantile to use. Set this together with q_min to restrict the range of quantiles used to construct the synthetic control. Default is 1 (all quantiles). Currently NOT implemented for the mixture approach.

CI

Logical, indicating whether to compute confidence intervals for the counterfactual quantiles. Default is FALSE. The confidence intervals are computed using the bootstrap procedure described in Van Dijcke et al. (2024).

boots

Integer, number of bootstrap samples to use for computing confidence intervals. Default is 500.

replace

Logical, indicating whether to sample with replacement when computing the bootstrap samples. Default is TRUE.

uniform

Logical, indicating whether to construct uniform bootstrap confidence intervals. Default is FALSE If FALSE, the confidence intervals are pointwise.

cl

Numeric, confidence level for the (two-sided) confidence intervals.

graph

Logical, indicating whether to plot the permutation graph as in Figure 3 of the paper. Default is FALSE.

qmethod

Character, indicating the method to use for computing the quantiles of the target distribution. The default is NULL, which uses the quantile function from the stats package. Other options are "qkden" (based on smoothed kernel density function) and "extreme" (based on parametric extreme value distributions). Both are substantially slower than the default method but may be useful for fat-tailed distributions with few data points at the upper quantiles. Alternatively, one could use the q_max option to restrict the range of quantiles used.

qtype

Integer, indicating the type of quantile to compute when using quantile in the qmethod argument. The default 7. See the documentation for the quantile function for more information.

seed

Integer, seed for the random number generator. This needs to be set explicitly in the function call, since it will invoke RNGkind which will set the seed for each core when using parallel processes. Default is NULL, which does not set a seed.

simplex

Logical, indicating whether to use to constrain the optimal weights to the unit simplex. Default is FALSE, which only constrains the weights to sum up to 1 but allows them to be negative.

mixture

Logical, indicating whether to use the mixture of distributions approach instead. See Section 4.3. in Gunsilius (2023). This approach minimizes the distance between the CDFs instead of the quantile functions, and is preferred for categorical variables. When working with such variables, one should also provide a list of support points in the grid.cat parameter. When that is provided, this parameter is automatically set to TRUE. Default is FALSE.

grid.cat

List, containing the discrete support points for a discrete grid to be used with the mixture of distributions approach. This is useful for constructing synthetic distributions for categorical variables. Default is NULL, which uses a continuous grid based on the other parameters.

Details

This function is called for every time period in the DiSCo function. It implements the DiSCo method for a single time period, as well as the mixture of distributions approach. The corresponding results for each time period can be accessed in the results.periods list of the output of the DiSCo function. The DiSCo function returns the average weight for each unit across all periods, calculated as a uniform mean, as well as the counterfactual target distribution produced as the weighted average of the control distributions for each period, using these averaged weights.

Value

A list containing the following elements:

• results.periods A list containing, for each time period, the elements described in the return argument of DiSCo_iter, as well as the following additional elements:

  • DiSco

    • quantile The counterfactual quantiles for the target unit.

    • weights The optimal weights for the target unit.

    • cdf The counterfactual CDF for the target unit.

• weights A numeric vector containing the synthetic control weights for the control units, averaged over time. When mixture is TRUE, these are the weights for the mixture of distributions, otherwise they are the weights for the quantile-based approach.

• CI A list containing the confidence intervals for the counterfactual quantiles and CDFs, if CI is TRUE. Each element contains two named subelements called upper, lower, se which are the upper and lower confidence bands and the standard error of the estimate, respectively. They are G x T matrices where G is the specified number of grid points and T is the number of time periods. The elements are:

  • cdf The bootstrapped CDF

  • quantile The bootstrapped quantile

  • quantile_diff The bootstrapped quantile difference

  • cdf_diff The bootstrapped CDF difference

  • bootmat A list containing the raw bootstrapped samples for the counterfactual quantiles and CDFs, if CI is TRUE. These are not meant to be accessed directly, but are used by DiSCoTEA to compute aggregated standard errors. Advanced users may wish to access these directly for further analysis. The element names should be self-explanatory. #'

  • control_ids A list containing the control unit IDs used for each time period, which can be used to identify the weights associated with each control as the returned weights have the same order as the control IDs.

  • perm A permut object containing the results of the permutation method, if permutation is TRUE. Call summary on this object to print the overall results of the permutation test. #'

  • evgrid A numeric vector containing the grid points on which the quantiles were evaluated.

  • params A list containing the parameters used in the function call.

References

Gunsilius FF (2023). “Distributional synthetic controls.” Econometrica, 91(3), 1105–1117.

Van Dijcke D, Gunsilius F, Wright AL (2024). “Return to Office and the Tenure Distribution.” Working Paper 2024-56, University of Chicago, Becker Friedman Institute for Economics.()