scpi: A Package to Compute Synthetic Control Prediction Intervals With Multiple Treated Units and Staggered Adoption

Description

The package implements estimation, inference procedures, and produces plots for Synthetic Control (SC) methods using least squares, lasso, ridge, or simplex-type constraints. Uncertainty is quantified using prediction intervals according to Cattaneo et al. (2021) and Cattaneo et al. (2025).

Included functions are: scdata and scdataMulti for data preparation, scest for point estimation, scpi for inference procedures, and scplot and scplotMulti for plots.

print() and summary() methods are available for scest and scpi.

Companion Stata and Python packages are described in Cattaneo et al. (2025).

Related Stata, R, and Python packages useful for inference in SC designs are described in the following website:

https://nppackages.github.io/scpi/

For an introduction to synthetic control methods, see Abadie (2021) and references therein.

Author(s)

Matias Cattaneo, Princeton University. cattaneo@princeton.edu.

Yingjie Feng, Tsinghua University. fengyj@sem.tsinghua.edu.cn.

Filippo Palomba, Princeton University (maintainer). fpalomba@princeton.edu.

Rocio Titiunik, Princeton University. titiunik@princeton.edu.

References

Abadie A (2021). “Using Synthetic Controls: Feasibility, Data Requirements, and Methodological Aspects.” Journal of Economic Literature, 59(2), 391–425. ISSN 0022-0515, doi:10.1257/jel.20191450.

Cattaneo MD, Feng Y, Palomba F, Titiunik R (2025). “Uncertainty Quantification in Synthetic Controls with Staggered Treatment Adoption.” Review of Economics and Statistics, 1–46. ISSN 0034-6535, 1530-9142, doi:10.1162/rest_a_01588.

Cattaneo MD, Feng Y, Palomba F, Titiunik R (2025). “scpi: Uncertainty Quantification for Synthetic Control Methods.” Journal of Statistical Software, 113(2), 1–38. doi:10.18637/jss.v113.i01.

Cattaneo MD, Feng Y, Titiunik R (2021). “Prediction Intervals for Synthetic Control Methods.” Journal of the American Statistical Association, 116(536), 1865–1880. ISSN 0162-1459, doi:10.1080/01621459.2021.1979561.

See Also

Useful links:

• https://nppackages.github.io/scpi/

Coef Method for Synthetic Control Methods

Description

The coef method for synthetic control prediction fitted objects.

Usage

## S3 method for class 'scest'
coef(object, ...)

Arguments

Value

No return value, called to show scest constructed weights.

Author(s)

Matias Cattaneo, Princeton University. cattaneo@princeton.edu.

Yingjie Feng, Tsinghua University. fengyj@sem.tsinghua.edu.cn.

Filippo Palomba, Princeton University (maintainer). fpalomba@princeton.edu.

Rocio Titiunik, Princeton University. titiunik@princeton.edu.

See Also

scest for synthetic control prediction.

Supported methods: print.scest, summary.scest, coef.scest.

Coef Method for Synthetic Control Methods

Description

The coef method for synthetic control prediction fitted objects.

Usage

## S3 method for class 'scpi'
coef(object, ...)

Arguments

Value

No return value, called to show scpi constructed weights.

Author(s)

Matias Cattaneo, Princeton University. cattaneo@princeton.edu.

Yingjie Feng, Tsinghua University. fengyj@sem.tsinghua.edu.cn.

Filippo Palomba, Princeton University (maintainer). fpalomba@princeton.edu.

Rocio Titiunik, Princeton University. titiunik@princeton.edu.

See Also

scpi for synthetic control prediction.

Supported methods: print.scpi, summary.scpi, coef.scpi.

Summary Method for Synthetic Control

Description

The print method for synthetic control data objects.

Usage

## S3 method for class 'scdata'
print(x, ...)

Arguments

Value

No return value, called to print scdata results.

Author(s)

Matias Cattaneo, Princeton University. cattaneo@princeton.edu.

Yingjie Feng, Tsinghua University. fengyj@sem.tsinghua.edu.cn.

Filippo Palomba, Princeton University (maintainer). fpalomba@princeton.edu.

Rocio Titiunik, Princeton University. titiunik@princeton.edu.

See Also

scdata for synthetic control data preparation.

Supported methods: print.scdata, summary.scdata.

Summary Method for Synthetic Control

Description

The print method for synthetic control data objects.

Usage

## S3 method for class 'scdataMulti'
print(x, ...)

Arguments

Value

No return value, called to print scdataMulti results.

Author(s)

Matias Cattaneo, Princeton University. cattaneo@princeton.edu.

Yingjie Feng, Tsinghua University. fengyj@sem.tsinghua.edu.cn.

Filippo Palomba, Princeton University (maintainer). fpalomba@princeton.edu.

Rocio Titiunik, Princeton University. titiunik@princeton.edu.

See Also

scdataMulti for synthetic control data preparation.

Supported methods: print.scdataMulti, summary.scdataMulti.

Print Method for Synthetic Control Methods

Description

The print method for synthetic control prediction fitted objects.

Usage

## S3 method for class 'scest'
print(x, ...)

Arguments

Value

No return value, called to print scest results.

Author(s)

Matias Cattaneo, Princeton University. cattaneo@princeton.edu.

Yingjie Feng, Tsinghua University. fengyj@sem.tsinghua.edu.cn.

Filippo Palomba, Princeton University (maintainer). fpalomba@princeton.edu.

Rocio Titiunik, Princeton University. titiunik@princeton.edu.

See Also

scest for synthetic control prediction.

Supported methods: print.scest, summary.scest, coef.scest.

Print Method for Synthetic Control Inference

Description

The print method for synthetic control inference objects.

Usage

## S3 method for class 'scpi'
print(x, ...)

Arguments

Value

No return value, called to print scpi results.

Author(s)

Matias Cattaneo, Princeton University. cattaneo@princeton.edu.

Yingjie Feng, Tsinghua University. fengyj@sem.tsinghua.edu.cn.

Filippo Palomba, Princeton University (maintainer). fpalomba@princeton.edu.

Rocio Titiunik, Princeton University. titiunik@princeton.edu.

See Also

scpi for synthetic control inference

Supported methods: print.scpi, summary.scpi.

Data Preparation for scest or scpi for Point Estimation and Inference Procedures Using Synthetic Control Methods.

Description

The command prepares the data to be used by scest or scpi to implement estimation and inference procedures for Synthetic Control (SC) methods. It allows the user to specify the outcome variable, the features of the treated unit to be matched, and covariate-adjustment feature by feature. The names of the output matrices follow the terminology proposed in Cattaneo et al. (2021) and Cattaneo et al. (2025).

Companion Stata and Python packages are described in Cattaneo et al. (2025).

Companion commands are: scdataMulti for data preparation in the multiple treated units case with staggered adoption, scest for point estimation, scpi for inference procedures, scplot and scplotMulti for plots in the single and multiple treated unit(s) cases, respectively.

Related Stata, R, and Python packages useful for inference in SC designs are described in the following website:

https://nppackages.github.io/scpi/

For an introduction to synthetic control methods, see Abadie (2021) and references therein.

Usage

scdata(
  df,
  id.var,
  time.var,
  outcome.var,
  period.pre,
  period.post,
  unit.tr,
  unit.co,
  features = NULL,
  cov.adj = NULL,
  cointegrated.data = FALSE,
  anticipation = 0,
  constant = FALSE,
  verbose = TRUE
)

Arguments

Details

• cov.adj can be used in two ways. First, if only one feature is specified through the option features, cov.adj has to be a list with one (even unnamed) element (eg. cov.adj = list(c("constant","trend"))). Alternatively, if multiple features are specified, then the user has two possibilities:

  • provide a list with one element, then the same covariates are used for adjustment for each feature. For example, if there are two features specified and the user inputs cov.adj = list(c("constant","trend")), then a constant term and a linear trend are for adjustment for both features.

  • provide a list with as many elements as the number of features specified, then feature-specific covariate adjustment is implemented. For example, cov.adj = list('f1' = c("constant","trend"), 'f2' = c("trend")). In this case the name of each element of the list should be one (and only one) of the features specified. Note that if two (or more) features are specified and covariates adjustment has to be specified just for one of them, the user must still provide a list of the same length of the number of features, e.g., cov.adj = list('f1' = c("constant","trend"), 'f2' = NULL.

  This option allows the user to include feature-specific constant terms or time trends by simply including "constant" or "trend" in the corresponding element of the list.

  When outcome.var is not included in features, we automatically set \mathcal{R}=\emptyset, that is we do not perform covariate adjustment. This is because, in this setting it is natural to create the out-of-sample prediction matrix \mathbf{P} using the post-treatment outcomes of the donor units only.

• cointegrated.data allows the user to model the belief that \mathbf{A} and \mathbf{B} form a cointegrated system. In practice, this implies that when dealing with the pseudo-true residuals \mathbf{u}, the first-difference of \mathbf{B} are used rather than the levels.

Value

The command returns an object of class 'scdata' containing the following

Author(s)

Matias Cattaneo, Princeton University. cattaneo@princeton.edu.

Yingjie Feng, Tsinghua University. fengyj@sem.tsinghua.edu.cn.

Filippo Palomba, Princeton University (maintainer). fpalomba@princeton.edu.

Rocio Titiunik, Princeton University. titiunik@princeton.edu.

References

Abadie A (2021). “Using Synthetic Controls: Feasibility, Data Requirements, and Methodological Aspects.” Journal of Economic Literature, 59(2), 391–425. ISSN 0022-0515, doi:10.1257/jel.20191450.

Cattaneo MD, Feng Y, Palomba F, Titiunik R (2025). “Uncertainty Quantification in Synthetic Controls with Staggered Treatment Adoption.” Review of Economics and Statistics, 1–46. ISSN 0034-6535, 1530-9142, doi:10.1162/rest_a_01588.

Cattaneo MD, Feng Y, Palomba F, Titiunik R (2025). “scpi: Uncertainty Quantification for Synthetic Control Methods.” Journal of Statistical Software, 113(2), 1–38. doi:10.18637/jss.v113.i01.

Cattaneo MD, Feng Y, Titiunik R (2021). “Prediction Intervals for Synthetic Control Methods.” Journal of the American Statistical Association, 116(536), 1865–1880. ISSN 0162-1459, doi:10.1080/01621459.2021.1979561.

See Also

scdataMulti, scest, scpi, scplot, scplotMulti

Examples

data <- scpi_germany

df <- scdata(df = data, id.var = "country", time.var = "year",
             outcome.var = "gdp", period.pre = (1960:1990),
             period.post = (1991:2003), unit.tr = "West Germany",
             unit.co = setdiff(unique(data$country), "West Germany"),
             constant = TRUE, cointegrated.data = TRUE)

Data Preparation for scest or scpi for Point Estimation and Inference Procedures Using Synthetic Control Methods.

Description

The command prepares the data to be used by scest or scpi to implement estimation and inference procedures for Synthetic Control (SC) methods in the general case of multiple treated units and staggered adoption. It is a generalization of scdata, since this latter prepares the data in the particular case of a single treated unit.

The names of the output matrices follow the terminology proposed in Cattaneo et al. (2021) and Cattaneo et al. (2025).

Companion Stata and Python packages are described in Cattaneo et al. (2025).

Companion commands are: scdataMulti for data preparation in the multiple treated units case with staggered adoption, scest for point estimation, scpi for inference procedures, scplot and scplotMulti for plots in the single and multiple treated unit(s) cases, respectively.

Related Stata, R, and Python packages useful for inference in SC designs are described in the following website:

https://nppackages.github.io/scpi/

For an introduction to synthetic control methods, see Abadie (2021) and references therein.

Variable Naming Convention: due to how scpi handles object internally, we kindly ask the users of the R version of the package to avoid including dots in the variable names. For example, "y.var" would generate issues with some parts of the code, whereas "yvar" or "y_var" would not.

Usage

scdataMulti(
  df,
  id.var,
  time.var,
  outcome.var,
  treatment.var,
  features = NULL,
  cov.adj = NULL,
  cointegrated.data = FALSE,
  post.est = NULL,
  units.est = NULL,
  donors.est = NULL,
  anticipation = 0,
  effect = "unit-time",
  constant = FALSE,
  verbose = TRUE,
  sparse.matrices = FALSE
)

Arguments

Details

• Covariate-adjustment. See the Details section in scdata for further information on how to specify covariate-adjustment feature-by-feature.

• Cointegration. cointegrated.data allows the user to model the belief that \mathbf{A} and \mathbf{B} form a cointegrated system. In practice, this implies that when dealing with the pseudo-true residuals \mathbf{u}, the first-difference of \mathbf{B} are used rather than the levels.

• Effect. effect allows the user to select between two causal quantities. The default option, effect = "unit-time", prepares the data for estimation of

  \tau_{ik},\quad k\geq, i=1,\ldots,N_1,

  whereas the option effect = "unit" prepares the data for estimation of

  \tau_{\cdot k}=\frac{1}{N_1} \sum_{i=1}^{N_1} \tau_{i k}

  which is the average effect on the treated unit across multiple post-treatment periods.

Value

The command returns an object of class 'scdataMulti' containing the following

Author(s)

Matias Cattaneo, Princeton University. cattaneo@princeton.edu.

Yingjie Feng, Tsinghua University. fengyj@sem.tsinghua.edu.cn.

Filippo Palomba, Princeton University (maintainer). fpalomba@princeton.edu.

Rocio Titiunik, Princeton University. titiunik@princeton.edu.

References

Abadie A (2021). “Using Synthetic Controls: Feasibility, Data Requirements, and Methodological Aspects.” Journal of Economic Literature, 59(2), 391–425. ISSN 0022-0515, doi:10.1257/jel.20191450.

Cattaneo MD, Feng Y, Palomba F, Titiunik R (2025). “Uncertainty Quantification in Synthetic Controls with Staggered Treatment Adoption.” Review of Economics and Statistics, 1–46. ISSN 0034-6535, 1530-9142, doi:10.1162/rest_a_01588.

Cattaneo MD, Feng Y, Palomba F, Titiunik R (2025). “scpi: Uncertainty Quantification for Synthetic Control Methods.” Journal of Statistical Software, 113(2), 1–38. doi:10.18637/jss.v113.i01.

Cattaneo MD, Feng Y, Titiunik R (2021). “Prediction Intervals for Synthetic Control Methods.” Journal of the American Statistical Association, 116(536), 1865–1880. ISSN 0162-1459, doi:10.1080/01621459.2021.1979561.

See Also

scdata, scest, scpi, scplot, scplotMulti

Examples

datager <- scpi_germany

datager$tr_id <- 0
datager$tr_id[(datager$country == "West Germany" & datager$year > 1990)] <- 1
datager$tr_id[(datager$country == "Italy" & datager$year > 1992)] <- 0

outcome.var <- "gdp"
id.var <- "country"
treatment.var <- "tr_id"
time.var <- "year"
df.unit <- scdataMulti(datager, id.var = id.var, outcome.var = outcome.var,
                       treatment.var = treatment.var,
                       time.var = time.var, features = list(c("gdp", "trade")),
               		    cointegrated.data = TRUE, constant = TRUE)

Prediction of Synthetic Control

Description

The command implements estimation procedures for Synthetic Control (SC) methods using least squares, lasso, ridge, or simplex-type constraints. For more information see Cattaneo et al. (2021) and Cattaneo et al. (2025).

Companion Stata and Python packages are described in Cattaneo et al. (2025).

Companion commands are: scdata and scdataMulti for data preparation in the single and multiple treated unit(s) cases, respectively, scpi for inference procedures, scplot and scplotMulti for plots in the single and multiple treated unit(s) cases, respectively.

Related Stata, R, and Python packages useful for inference in SC designs are described in the following website:

https://nppackages.github.io/scpi/

For an introduction to synthetic control methods, see Abadie (2021) and references therein.

Usage

scest(
  data,
  w.constr = NULL,
  V = "separate",
  V.mat = NULL,
  solver = "ECOS",
  plot = FALSE,
  plot.name = NULL,
  plot.path = NULL,
  save.data = NULL
)

Arguments

Details

Information is provided for the simple case in which N_1=1 if not specified otherwise.

• Estimation of Weights. w.constr specifies the constraint set on the weights. First, the element p allows the user to choose between imposing a constraint on either the L1 (p = "L1") or the L2 (p = "L2") norm of the weights and imposing no constraint on the norm (p = "no norm"). Second, Q specifies the value of the constraint on the norm of the weights. Third, lb sets the lower bound of each component of the vector of weights. Fourth, dir sets the direction of the constraint on the norm in case p = "L1" or p = "L2". If dir = "==", then

  ||\mathbf{w}||_p = Q,\:\:\: w_j \geq lb,\:\: j =1,\ldots,J

  If instead dir = "<=", then

  ||\mathbf{w}||_p \leq Q,\:\:\: w_j \geq lb,\:\: j =1,\ldots,J

  If instead dir = "NULL" no constraint on the norm of the weights is imposed.

  An alternative to specifying an ad-hoc constraint set on the weights would be choosing among some popular types of constraints. This can be done by including the element 'name' in the list w.constr. The following are available options:

  • If name == "simplex" (the default), then

    ||\mathbf{w}||_1 = 1,\:\:\: w_j \geq 0,\:\: j =1,\ldots,J.

  • If name == "lasso", then

    ||\mathbf{w}||_1 \leq Q,

    where Q is by default equal to 1 but it can be provided as an element of the list (eg. w.constr = list(name = "lasso", Q = 2)).

  • If name == "ridge", then

    ||\mathbf{w}||_2 \leq Q,

    where Q is a tuning parameter that is by default computed as

    (J+KM) \widehat{\sigma}_u^{2}/||\widehat{\mathbf{w}}_{OLS}||_{2}^{2}

    where J is the number of donors and KM is the total number of covariates used for adjustment. The user can provide Q as an element of the list (eg. w.constr = list(name = "ridge", Q = 1)).

  • If name == "ols", then the problem is unconstrained and the vector of weights is estimated via ordinary least squares.

  • If name == "L1-L2", then

    ||\mathbf{w}||_1 = 1,\:\:\: ||\mathbf{w}||_2 \leq Q, \:\:\: w_j \geq 0,\:\: j =1,\ldots,J.

    where Q is a tuning parameter computed as in the "ridge" case.

• Weighting Matrix.

  • if V <- "separate", then \mathbf{V} = \mathbf{I} and the minimized objective function is

    \sum_{i=1}^{N_1} \sum_{l=1}^{M} \sum_{t=1}^{T_{0}}\left(a_{t, l}^{i}-\mathbf{b}_{t, l}^{{i \prime }} \mathbf{w}^{i}-\mathbf{c}_{t, l}^{{i \prime}} \mathbf{r}_{l}^{i}\right)^{2},

    which optimizes the separate fit for each treated unit.

  • if V <- "pooled", then \mathbf{V} = \frac{1}{I}\mathbf{1}\mathbf{1}'\otimes \mathbf{I} and the minimized objective function is

    \sum_{l=1}^{M} \sum_{t=1}^{T_{0}}\left(\frac{1}{N_1^2} \sum_{i=1}^{N_1}\left(a_{t, l}^{i}-\mathbf{b}_{t, l}^{i \prime} \mathbf{w}^{i}-\mathbf{c}_{t, l}^{i\prime} \mathbf{r}_{l}^{i}\right)\right)^{2},

    which optimizes the pooled fit for the average of the treated units.

  • if the user wants to provide their own weighting matrix, then it must use the option V.mat to input a v\times v positive-definite matrix, where v is the number of rows of \mathbf{B} (or \mathbf{C}) after potential missing values have been removed. In case the user wants to provide their own V, we suggest to check the appropriate dimension v by inspecting the output of either scdata or scdataMulti and check the dimensions of \mathbf{B} (and \mathbf{C}). Note that the weighting matrix could cause problems to the optimizer if not properly scaled. For example, if \mathbf{V} is diagonal we suggest to divide each of its entries by \|\mathrm{diag}(\mathbf{V})\|_1.

Value

The function returns an object of class 'scest' containing two lists. The first list is labeled 'data' and contains used data as returned by scdata and some other values.

The second list is labeled 'est.results' and contains estimation results.

Author(s)

Matias Cattaneo, Princeton University. cattaneo@princeton.edu.

Yingjie Feng, Tsinghua University. fengyj@sem.tsinghua.edu.cn.

Filippo Palomba, Princeton University (maintainer). fpalomba@princeton.edu.

Rocio Titiunik, Princeton University. titiunik@princeton.edu.

References

Abadie A (2021). “Using Synthetic Controls: Feasibility, Data Requirements, and Methodological Aspects.” Journal of Economic Literature, 59(2), 391–425. ISSN 0022-0515, doi:10.1257/jel.20191450.

Cattaneo MD, Feng Y, Palomba F, Titiunik R (2025). “Uncertainty Quantification in Synthetic Controls with Staggered Treatment Adoption.” Review of Economics and Statistics, 1–46. ISSN 0034-6535, 1530-9142, doi:10.1162/rest_a_01588.

Cattaneo MD, Feng Y, Palomba F, Titiunik R (2025). “scpi: Uncertainty Quantification for Synthetic Control Methods.” Journal of Statistical Software, 113(2), 1–38. doi:10.18637/jss.v113.i01.

Cattaneo MD, Feng Y, Titiunik R (2021). “Prediction Intervals for Synthetic Control Methods.” Journal of the American Statistical Association, 116(536), 1865–1880. ISSN 0162-1459, doi:10.1080/01621459.2021.1979561.

See Also

scdataMulti, scdata, scpi, scplot, scplotMulti

Examples

data <- scpi_germany

df <- scdata(df = data, id.var = "country", time.var = "year",
             outcome.var = "gdp", period.pre = (1960:1990),
             period.post = (1991:2003), unit.tr = "West Germany",
             unit.co = setdiff(unique(data$country), "West Germany"),
             constant = TRUE, cointegrated.data = TRUE)

result <- scest(df, w.constr = list(name = "simplex", Q = 1))
result <- scest(df, w.constr = list(lb = 0, dir = "==", p = "L1", Q = 1))

Prediction Intervals for Synthetic Control Methods

Description

The command implements estimation and inference procedures for Synthetic Control (SC) methods using least squares, lasso, ridge, or simplex-type constraints. Uncertainty is quantified using prediction intervals according to Cattaneo et al. (2021) and Cattaneo et al. (2025). scpi returns the estimated post-treatment series for the synthetic unit through the command scest and quantifies in-sample and out-of-sample uncertainty to provide confidence intervals for each point estimate.

Companion Stata and Python packages are described in Cattaneo et al. (2025).

Companion commands are: scdata and scdataMulti for data preparation in the single and multiple treated unit(s) cases, respectively, scest for point estimation, scplot and scplotMulti for plots in the single and multiple treated unit(s) cases, respectively.

Related Stata, R, and Python packages useful for inference in SC designs are described in the following website:

https://nppackages.github.io/scpi/

For an introduction to synthetic control methods, see Abadie (2021) and references therein.

Usage

scpi(
  data,
  w.constr = NULL,
  V = "separate",
  V.mat = NULL,
  solver = "ECOS",
  P = NULL,
  u.missp = TRUE,
  u.sigma = "HC1",
  u.order = 1,
  u.lags = 0,
  u.design = NULL,
  u.alpha = 0.05,
  e.method = "all",
  e.order = 1,
  e.lags = 0,
  e.design = NULL,
  e.alpha = 0.05,
  sims = 200,
  rho = NULL,
  rho.max = 0.2,
  cores = 1,
  plot = FALSE,
  plot.name = NULL,
  w.bounds = NULL,
  e.bounds = NULL,
  force.joint.PI.optim = FALSE,
  save.data = NULL,
  verbose = TRUE
)

Arguments

Details

Information is provided for the simple case in which N_1=1 if not specified otherwise.

• Estimation of Weights. w.constr specifies the constraint set on the weights. First, the element p allows the user to choose between imposing a constraint on either the L1 (p = "L1") or the L2 (p = "L2") norm of the weights and imposing no constraint on the norm (p = "no norm"). Second, Q specifies the value of the constraint on the norm of the weights. Third, lb sets the lower bound of each component of the vector of weights. Fourth, dir sets the direction of the constraint on the norm in case p = "L1" or p = "L2". If dir = "==", then

  ||\mathbf{w}||_p = Q,\:\:\: w_j \geq lb,\:\: j =1,\ldots,J

  If instead dir = "<=", then

  ||\mathbf{w}||_p \leq Q,\:\:\: w_j \geq lb,\:\: j =1,\ldots,J

  If instead dir = "NULL" no constraint on the norm of the weights is imposed.

  An alternative to specifying an ad-hoc constraint set on the weights would be choosing among some popular types of constraints. This can be done by including the element 'name' in the list w.constr. The following are available options:

  • If name == "simplex" (the default), then

    ||\mathbf{w}||_1 = 1,\:\:\: w_j \geq 0,\:\: j =1,\ldots,J.

  • If name == "lasso", then

    ||\mathbf{w}||_1 \leq Q,

    where Q is by default equal to 1 but it can be provided as an element of the list (eg. w.constr = list(name = "lasso", Q = 2)).

  • If name == "ridge", then

    ||\mathbf{w}||_2 \leq Q,

    where Q is a tuning parameter that is by default computed as

    (J+KM) \widehat{\sigma}_u^{2}/||\widehat{\mathbf{w}}_{OLS}||_{2}^{2}

    where J is the number of donors and KM is the total number of covariates used for adjustment. The user can provide Q as an element of the list (eg. w.constr = list(name = "ridge", Q = 1)).

  • If name == "ols", then the problem is unconstrained and the vector of weights is estimated via ordinary least squares.

  • If name == "L1-L2", then

    ||\mathbf{w}||_1 = 1,\:\:\: ||\mathbf{w}||_2 \leq Q, \:\:\: w_j \geq 0,\:\: j =1,\ldots,J.

    where Q is a tuning parameter computed as in the "ridge" case.

• Weighting Matrix.

  • if V <- "separate", then \mathbf{V} = \mathbf{I} and the minimized objective function is

    \sum_{i=1}^{N_1} \sum_{l=1}^{M} \sum_{t=1}^{T_{0}}\left(a_{t, l}^{i}-\mathbf{b}_{t, l}^{{i \prime }} \mathbf{w}^{i}-\mathbf{c}_{t, l}^{{i \prime}} \mathbf{r}_{l}^{i}\right)^{2},

    which optimizes the separate fit for each treated unit.

  • if V <- "pooled", then \mathbf{V} = \mathbf{1}\mathbf{1}'\otimes \mathbf{I} and the minimized objective function is

    \sum_{l=1}^{M} \sum_{t=1}^{T_{0}}\left(\frac{1}{N_1^2} \sum_{i=1}^{N_1}\left(a_{t, l}^{i}-\mathbf{b}_{t, l}^{i \prime} \mathbf{w}^{i}-\mathbf{c}_{t, l}^{i\prime} \mathbf{r}_{l}^{i}\right)\right)^{2},

    which optimizes the pooled fit for the average of the treated units.

  • if the user wants to provide their own weighting matrix, then it must use the option V.mat to input a v\times v positive-definite matrix, where v is the number of rows of \mathbf{B} (or \mathbf{C}) after potential missing values have been removed. In case the user wants to provide their own V, we suggest to check the appropriate dimension v by inspecting the output of either scdata or scdataMulti and check the dimensions of \mathbf{B} (and \mathbf{C}). Note that the weighting matrix could cause problems to the optimizer if not properly scaled. For example, if \mathbf{V} is diagonal we suggest to divide each of its entries by \|\mathrm{diag}(\mathbf{V})\|_1.

• Regularization. rho is estimated through the formula

  \varrho = \sqrt{d_0\log(d)\log(T_0)}\mathcal{C}T_0^{-1/2}

  where d is the dimension of \widehat{\boldsymbol{\beta}} and d_0 denote the number of nonzeros in \widehat{\boldsymbol{\beta}} \mathcal{C} = \widehat{\sigma}_u / \min_j \widehat{\sigma}_{b_j} if rho = 'type-1' and \mathcal{C} = \max_{j}\widehat{\sigma}_{b_j}\widehat{\sigma}_{u} / \min_j \widehat{\sigma}_{b_j}^2 if rho = 'type-2', rho = 'type-2' is the default option from version 3.0.0 onwards, while previously 'type-1' was the default option. rho defines a new sparse weight vector as

  \widehat{w}^\star_j = \mathbf{1}(\widehat{w}_j\geq \varrho)

• In-sample uncertainty. To quantify in-sample uncertainty it is necessary to model the pseudo-residuals \mathbf{u}. First of all, estimation of the first moment of \mathbf{u} can be controlled through the option u.missp. When u.missp = FALSE, then \mathbf{E}[u\: |\: \mathbf{D}_u]=0. If instead u.missp = TRUE, then \mathbf{E}[\mathbf{u}\: |\: \mathbf{D}_u] is estimated using a linear regression of \widehat{\mathbf{u}} on \mathbf{D}_u. The default set of variables in \mathbf{D}_u is composed of \mathbf{B}, \mathbf{C} and, if required, it is augmented with lags (u.lags) and polynomials (u.order) of \mathbf{B}. The option u.design allows the user to provide an ad-hoc set of variables to form \mathbf{D}_u. Regarding the second moment of \mathbf{u}, different estimators can be chosen: HC0, HC1, HC2, HC3, and HC4 using the option u.sigma.

• Out-of-sample uncertainty. To quantify out-of-sample uncertainty it is necessary to model the out-of-sample residuals \mathbf{e} and estimate relevant moments. By default, the design matrix used during estimation \mathbf{D}_e is composed of the blocks in \mathbf{B} and \mathbf{C} corresponding to the outcome variable. Moreover, if required by the user, \mathbf{D}_e is augmented with lags (e.lags) and polynomials (e.order) of \mathbf{B}. The option e.design allows the user to provide an ad-hoc set of variables to form \mathbf{D}_e. Finally, the option e.method allows the user to select one of three estimation methods: "gaussian" relies on conditional sub-Gaussian bounds; "ls" estimates conditional bounds using a location-scale model; "qreg" uses conditional quantile regression of the residuals \mathbf{e} on \mathbf{D}_e.

• Residual Estimation Over-fitting. To estimate conditional moments of \mathbf{u} and e_t we rely on two design matrices, \mathbf{D}_u and \mathbf{D}_e (see above). Let d_u and d_e be the number of columns in \mathbf{D}_u and \mathbf{D}_e, respectively. Assuming no missing values and balanced features, the number of observation used to estimate moments of \mathbf{u} is N_1\cdot T_0\cdot M, whilst for moments of e_t is T_0. Our rule of thumb to avoid over-fitting is to check if N_1\cdot T_0\cdot M \geq d_u + 10 or T_0 \geq d_e + 10. If the former condition is not satisfied we automatically set u.order = u.lags = 0, if instead the latter is not met we automatically set e.order = e.lags = 0.

Value

The function returns an object of class 'scpi' containing three lists. The first list is labeled 'data' and contains used data as returned by scdata and some other values.

The second list is labeled 'est.results' containing all the results from scest.

The third list is labeled 'inference.results' and contains all the inference-related results.

Author(s)

Matias Cattaneo, Princeton University. cattaneo@princeton.edu.

Yingjie Feng, Tsinghua University. fengyj@sem.tsinghua.edu.cn.

Filippo Palomba, Princeton University (maintainer). fpalomba@princeton.edu.

Rocio Titiunik, Princeton University. titiunik@princeton.edu.

References

Abadie A (2021). “Using Synthetic Controls: Feasibility, Data Requirements, and Methodological Aspects.” Journal of Economic Literature, 59(2), 391–425. ISSN 0022-0515, doi:10.1257/jel.20191450.

Cattaneo MD, Feng Y, Palomba F, Titiunik R (2025). “Uncertainty Quantification in Synthetic Controls with Staggered Treatment Adoption.” Review of Economics and Statistics, 1–46. ISSN 0034-6535, 1530-9142, doi:10.1162/rest_a_01588.

Cattaneo MD, Feng Y, Palomba F, Titiunik R (2025). “scpi: Uncertainty Quantification for Synthetic Control Methods.” Journal of Statistical Software, 113(2), 1–38. doi:10.18637/jss.v113.i01.

Cattaneo MD, Feng Y, Titiunik R (2021). “Prediction Intervals for Synthetic Control Methods.” Journal of the American Statistical Association, 116(536), 1865–1880. ISSN 0162-1459, doi:10.1080/01621459.2021.1979561.

See Also

scdata, scdataMulti, scest, scplot, scplotMulti

Examples

data <- scpi_germany

df <- scdata(df = data, id.var = "country", time.var = "year",
             outcome.var = "gdp", period.pre = (1960:1990),
             period.post = (1991:2003), unit.tr = "West Germany",
             unit.co = setdiff(unique(data$country), "West Germany"),
             constant = TRUE, cointegrated.data = TRUE)

result <- scpi(df, w.constr = list(name = "simplex", Q = 1), cores = 1, sims = 10)
result <- scpi(df, w.constr = list(lb = 0, dir = "==", p = "L1", Q = 1),
               cores = 1, sims = 10)

Replication Dataset for Estimating the Economic Impact of German Reunification

Description

A dataset containing some economic indicators of 17 OECD countries from 1960 to 2003.

Usage

scpi_germany

Format

A data frame with 748 rows and 11 variables:

index

country index.

country

name of the country.

year

time index, in years.

gdp

GDP per Capita (PPP, 2002 USD).

infrate

annual percentage change in consumer prices (base year 1995).

trade

trade openness measured as export plus imports as percentage of GDP.

schooling

percentage of secondary school attained in the total population aged 25 and older.

industry

industry share of value added.

Source

Harvard Dataverse (doi:10.7910/DVN/24714)

Plot Synthetic Control Point Estimates and Prediction Interval

Description

The command plots the actual pre-treatment and post-treatment series of the treated unit and the estimated counterfactual synthetic control unit with corresponding prediction intervals. Prediction intervals can take into account either in-sample uncertainty only or in-sample and out-of-sample uncertainty using the techniques developed in to Cattaneo et al. (2021) and Cattaneo et al. (2025). The input object should come from the command scest or from the command scpi.

Companion Stata and Python packages are described in Cattaneo et al. (2025).

Companion commands are: scdata and scdataMulti for data preparation in the single and multiple treated unit(s) cases, respectively, scest for point estimation, scpi for inference procedures, and scplotMulti for plots with multiple treated units.

Related Stata, R, and Python packages useful for inference in SC designs are described in the following website:

https://nppackages.github.io/scpi/

For an introduction to synthetic control methods, see Abadie (2021) and references therein.

Usage

scplot(
  result,
  fig.path = NULL,
  fig.name = NULL,
  fig.format = "png",
  e.out = TRUE,
  joint = FALSE,
  col.treated = "black",
  col.synth = "mediumblue",
  label.xy = NULL,
  plot.range = NULL,
  x.ticks = NULL,
  event.label = NULL,
  plot.specs = NULL,
  save.data = NULL
)

Arguments

Value

Author(s)

Matias Cattaneo, Princeton University. cattaneo@princeton.edu.

Yingjie Feng, Tsinghua University. fengyj@sem.tsinghua.edu.cn.

Filippo Palomba, Princeton University (maintainer). fpalomba@princeton.edu.

Rocio Titiunik, Princeton University. titiunik@princeton.edu.

References

Abadie A (2021). “Using Synthetic Controls: Feasibility, Data Requirements, and Methodological Aspects.” Journal of Economic Literature, 59(2), 391–425. ISSN 0022-0515, doi:10.1257/jel.20191450.

Cattaneo MD, Feng Y, Palomba F, Titiunik R (2025). “Uncertainty Quantification in Synthetic Controls with Staggered Treatment Adoption.” Review of Economics and Statistics, 1–46. ISSN 0034-6535, 1530-9142, doi:10.1162/rest_a_01588.

Cattaneo MD, Feng Y, Palomba F, Titiunik R (2025). “scpi: Uncertainty Quantification for Synthetic Control Methods.” Journal of Statistical Software, 113(2), 1–38. doi:10.18637/jss.v113.i01.

Cattaneo MD, Feng Y, Titiunik R (2021). “Prediction Intervals for Synthetic Control Methods.” Journal of the American Statistical Association, 116(536), 1865–1880. ISSN 0162-1459, doi:10.1080/01621459.2021.1979561.

See Also

scdata, scdataMulti, scest, scpi, scplotMulti

Examples

data <- scpi_germany

df <- scdata(df = data, id.var = "country", time.var = "year",
             outcome.var = "gdp", period.pre = (1960:1990),
             period.post = (1991:2003), unit.tr = "West Germany",
             unit.co = setdiff(unique(data$country), "West Germany"),
             constant = TRUE, cointegrated.data = TRUE)

result <- scest(df, w.constr = list(name = "simplex", Q = 1))

scplot(result)

Plot Synthetic Control Point Estimates and Prediction Interval With Multiple Treated units and Staggered Adoption

Description

The command produces a wide range of plots of Synthetic Control estimates and corresponding prediction intervals. The command allows form multiple treated units and staggered adoption. Prediction intervals can take into account either in-sample uncertainty only or in-sample and out-of-sample uncertainty using the techniques developed in to Cattaneo et al. (2021) and Cattaneo et al. (2025). The input object should come from the command scest or from the command scpi.

Companion Stata and Python packages are described in Cattaneo et al. (2025).

Companion commands are: scdata and scdataMulti for data preparation in the single and multiple treated unit(s) cases, respectively, scest for point estimation, scpi for inference procedures, and scplotMulti for plots with multiple treated units.

Related Stata, R, and Python packages useful for inference in SC designs are described in the following website:

https://nppackages.github.io/scpi/

For an introduction to synthetic control methods, see Abadie (2021) and references therein.

Usage

scplotMulti(
  result,
  type = "series",
  e.out = TRUE,
  joint = FALSE,
  col.treated = "black",
  col.synth = "mediumblue",
  scales = "fixed",
  point.size = 1.5,
  ncols = 3,
  save.data = NULL,
  verbose = TRUE
)

Arguments

Value

Author(s)

Matias Cattaneo, Princeton University. cattaneo@princeton.edu.

Yingjie Feng, Tsinghua University. fengyj@sem.tsinghua.edu.cn.

Filippo Palomba, Princeton University (maintainer). fpalomba@princeton.edu.

Rocio Titiunik, Princeton University. titiunik@princeton.edu.

References

Abadie A (2021). “Using Synthetic Controls: Feasibility, Data Requirements, and Methodological Aspects.” Journal of Economic Literature, 59(2), 391–425. ISSN 0022-0515, doi:10.1257/jel.20191450.

Cattaneo MD, Feng Y, Palomba F, Titiunik R (2025). “Uncertainty Quantification in Synthetic Controls with Staggered Treatment Adoption.” Review of Economics and Statistics, 1–46. ISSN 0034-6535, 1530-9142, doi:10.1162/rest_a_01588.

Cattaneo MD, Feng Y, Palomba F, Titiunik R (2025). “scpi: Uncertainty Quantification for Synthetic Control Methods.” Journal of Statistical Software, 113(2), 1–38. doi:10.18637/jss.v113.i01.

Cattaneo MD, Feng Y, Titiunik R (2021). “Prediction Intervals for Synthetic Control Methods.” Journal of the American Statistical Association, 116(536), 1865–1880. ISSN 0162-1459, doi:10.1080/01621459.2021.1979561.

See Also

scdata, scdataMulti, scest, scpi, scplotMulti

Examples

datager <- scpi_germany

datager$tr_id <- 0
datager$tr_id[(datager$country == "West Germany" & datager$year > 1990)] <- 1
datager$tr_id[(datager$country == "Italy" & datager$year > 1992)] <- 0

outcome.var <- "gdp"
id.var <- "country"
treatment.var <- "tr_id"
time.var <- "year"
df.unit <- scdataMulti(datager, id.var = id.var, outcome.var = outcome.var,
                       treatment.var = treatment.var,
                       time.var = time.var, features = list(c("gdp", "trade")),
               		    cointegrated.data = TRUE, constant = TRUE)

res.unit <- scpi(df.unit, sims = 10, cores = 1)
scplotMulti(res.unit, joint = TRUE)

Summary Method for Synthetic Control Prediction

Description

The summary method for synthetic control prediction objects.

Usage

## S3 method for class 'scdata'
summary(object, ...)

Arguments

Value

No return value, called to summarize scdata results.

Author(s)

Matias Cattaneo, Princeton University. cattaneo@princeton.edu.

Yingjie Feng, Tsinghua University. fengyj@sem.tsinghua.edu.cn.

Filippo Palomba, Princeton University (maintainer). fpalomba@princeton.edu.

Rocio Titiunik, Princeton University. titiunik@princeton.edu.

See Also

scdata

Supported methods: print.scdata, summary.scdata.

Summary Method for Synthetic Control Prediction

Description

The summary method for synthetic control prediction objects.

Usage

## S3 method for class 'scdataMulti'
summary(object, ...)

Arguments

Value

No return value, called to summarize scdataMulti results.

Author(s)

Matias Cattaneo, Princeton University. cattaneo@princeton.edu.

Yingjie Feng, Tsinghua University. fengyj@sem.tsinghua.edu.cn.

Filippo Palomba, Princeton University (maintainer). fpalomba@princeton.edu.

Rocio Titiunik, Princeton University. titiunik@princeton.edu.

See Also

scdataMulti

Supported methods: print.scdataMulti, summary.scdataMulti.

Summary Method for Synthetic Control Prediction

Description

The summary method for synthetic control prediction fitted objects.

Usage

## S3 method for class 'scest'
summary(object, ...)

Arguments

Value

No return value, called to summarize scest results.

Author(s)

Matias Cattaneo, Princeton University. cattaneo@princeton.edu.

Yingjie Feng, Tsinghua University. fengyj@sem.tsinghua.edu.cn.

Filippo Palomba, Princeton University (maintainer). fpalomba@princeton.edu.

Rocio Titiunik, Princeton University. titiunik@princeton.edu.

See Also

scest

Supported methods: print.scest, summary.scest, coef.scest.

Summary Method for Synthetic Control Inference

Description

The summary method for synthetic control inference objects.

Usage

## S3 method for class 'scpi'
summary(object, ...)

Arguments

Value

No return value, called to summarize scpi results.

Author(s)

Matias Cattaneo, Princeton University. cattaneo@princeton.edu.

Yingjie Feng, Tsinghua University. fengyj@sem.tsinghua.edu.cn.

Filippo Palomba, Princeton University (maintainer). fpalomba@princeton.edu.

Rocio Titiunik, Princeton University. titiunik@princeton.edu.

See Also

scpi

Supported methods: print.scpi, summary.scpi.