• Imagine \(2 \times T\) design of DID.
• There are \(i = 1, \cdots, N + 1\) units.
• Only the first unit \(i = 1\) is treated since period \(T_0 + 1\):
  • \(G_i = 1\{i = 1\}\).
  • \(D_{it} = G_i \cdot 1\{t > T_0\}\).
• Units of \(i = 2, \cdots, N + 1\) are a set of potential comparisons.
• For each \(i\), we observe the outcome of interest for \(t = 1, \cdots T\):
  • \(Y_{it} = D_{it}Y_{it}(1)+(1 - D_{it})Y_{it}(0)\).
• For each \(i\), we also observe a set of \(K\) predictors of the outcomes:
  • \(X_{1i}, \cdots , X_{Ki}\).
  • This may include the pre-intervention values of \(Y_{it}\) for \(t \le T_0\).

• The effect of intervention of interest for the affected unit in period \(t > T_0\): \[ \tau_{1t} = Y_{1t}(1) - Y_{1t}(0). \]
• How do we estimate \(Y_{1t}(0)\)?
• This is by nature a case study.
• How do we formalize a case study?

• Intervention: The massive arrival of Cuban expatriates to Miam during the 1980 Mariel boatlift.
• \(i = 1\): Miami.
• \(i = 2, \cdots, N + 1\): Atlanta, Los Angeles, Houston, and Tampa-St. Petersburg.
• \(Y_{it}\): Native unemployment.

These four cities were selected both because they had relatively large populations of blacks and Hispanics and because they exhibited a pattern of economic growth similar to that in Miami over the late 1970s and early 1980s. (p.249)

• How to formalize this selection?

• Find a weight \(w_i, i = 2, \cdots, N + 1\).
• Construct a synthetic control estimator of \(Y_{1t}(0)\) for \(t > T_0\) by: \[ \hat{Y}_{1t}(0) = \sum_{i = 2}^{N + 1} w_i Y_{it}. \]
• Construct a synthetic control estimator of \(\tau_{1t}\) for \(t > T_0\) by: \[ \hat{\tau}_{1t} = Y_{1t} - \hat{Y}_{1t}(0). \]

• The weights are restricted to be non-negative and sum up to 1.
• This ensures that \(\hat{Y}_{1t}(0)\) is an interpolation of \(Y_{it}\) for \(i = 2, \cdots, N + 1\).
• A negative value or a value greater than 1 means an extrapolation of the observation.
• We will need to rescale the data to correct the differences in the size between units (e.g., per capita income).

• Simple average with \(w_i = 1/N\): \[ \hat{Y}_{1t}(0) = \frac{1}{N} \sum_{i = 2}^{N + 1}Y_{it}. \]

• Population-weighted average with \(w_i = w_i^{pop}\): \[ \hat{Y}_{1t}(0) = \sum_{i = 2}^{N + 1} w_i^{pop} Y_{it}. \]

• Choose the weight by minimizing a distance between predictors: \[ \min_{w} \sum_{k = 1}^K v_k\Big(X_{k1} - \sum_{i = 2}^{N + 1} w_i X_{ik} \Big)^2 \] subject to: \[ w_i \ge 0, \sum_{i = 2}^{N + 1} w_i = 1, \] with positive constants \(v_k\) for \(k = 1, \cdots, K\) as hyper parameters.

1. Divide the pre-intervention periods into:
   • Initial training period (\(t = 1, \cdots, t_0\));
   • Subsequent validation periods (\(t = t_0 + 1, \cdots, T_0\)).
   • For each \(v\), define \(w_i(v)\) as the minimizer given \(v\).
   • Compute: \[ \sum_{t = t_0 + 1}^{T_0} \Big(Y_{1t} - \sum_{i = 2}^{N + 1} w_i(v) Y_{it} \Big)^2. \]
   • Pick \(v\) that minimize this and denote by \(v^*\).
2. Use \(w^* = w(v^*)\) as the weight.

• Suppose that the control outcome follows a linear factor model: \[ Y_{it}(0) = \delta_t + \theta_t Z_i + \lambda_t \mu_i + \epsilon_{it}, \] where:
• \(\delta_t\): time trend.
• \(Z_i\) and \(\mu_i\): observed and unobserved factor loading.
• \(\theta_t\) and \(\lambda_t\): unobserved common factors.
• \(\epsilon_{it}\): mean 0 i.i.d. shocks (can be generalized).
• Intuitively, SCM tries to replicate \(\lambda_t \mu_i\) by weighting.

• Suppose that there exist a weight \(w^*\) such that: \[ X_{1k} = \sum_{i = 2}^{N + 1}w_i^* X_{ik} \] for \(k = 1, \cdots, K\).
• Moreover, suppose that \(\sum_{t = 1}^{T_0} \lambda_t' \lambda_t\) is non-singular. Then, \[ \begin{split} Y_{1t}(0) - \sum_{i = 2}^{N + 1} w_i^* Y_{it} & = \sum_{i = 2}^{N + 1} w_i^* \sum_{s = 1}^{T_0} \lambda_t (\sum_{n = 1}^{T_0} \lambda_n' \lambda_n)^{-1} \lambda_s' (\epsilon_{is} - \epsilon_{1s})\\ &- \sum_{i = 2}^{N + 1}w_i^*(\epsilon_{it} - \epsilon_{1t}). \end{split} \]

• The bias is the sum of: \[ R_{1t} = \sum_{i = 2}^{N + 1} w_i^* \sum_{s = 1}^{T_0} \lambda_t (\sum_{n = 1}^{T_0} \lambda_n' \lambda_n)^{-1} \lambda_s' \epsilon_{is}, \] \[ R_{2t} = - \sum_{s = 1}^{T_0} \lambda_t (\sum_{n = 1}^{T_0} \lambda_n' \lambda_n)^{-1} \lambda_s' \epsilon_{1s}, \] \[ R_{3t} = - \sum_{i = 2}^{N + 1}w_i^*(\epsilon_{it} - \epsilon_{1t}). \]
• For \(t > T_0\), \(R_{2t}\) and \(R_{3t}\) are mean zero but \(R_{1t}\) is not because \(w^*\) and \(\epsilon_s\) are correlated.

• Suppose that the \(p\)-th moment of \(|\epsilon_{it}|\) exists.
• Let \(m_{p,it} = \mathbb{E}|\epsilon_{it}|^p\), \(m_{p,i} = 1/T_0 \sum_{t = 1}^{T_0} m_{p, it}\), \(\overline{m}_p = \max_{i = 2, \cdots, N + 1} m_{p, i}\).
• Then, we can bound the bias by: \[ \mathbb{E}|R_{1t}| \lesssim \max\{\frac{\overline{m}_p^{1/p}}{T_0^{1 - 1/p}}, \frac{\overline{m}_2^{1/2}}{T_0^{1/2}}\}. \]
• The bias is smaller if:
  • Pre-treatment period \(T_0\) is longer.
  • The standard deviations of the idiosyncratic shocks are smaller.

• Then, should we just take a long enough pre-treatment period?
  • Not necessarily true, because the model is more likely to be misspecified if the data goes back to the past.
  • Then, the assumption of the presence of perfect-match \(w_i^*\) will be violated.

• Use a randomization test.
• Abadie et al. (2010) suggest a test statistics measuring the ratio of post-treatment fit relative to the pre-intervention fit.
• Consider a placebo treatment to unit \(i\) and \(\hat{Y}_{it}(0)\) is the SC estimator of \(Y_{it}(0)\) for \(t > T_0\).
• Consider: \[ R_i(t_1, t_2) = \Bigg(\frac{1}{t_2 - t_1 + 1} \sum_{t = t_1}^{t_2} [Y_{it} - \hat{Y}_{it}(0)]^2 \Bigg)^{1/2}, \] \[ r_i = \frac{R_i(T_0 + 1, T)}{R_i(1, T_0)}. \]

• Then, calculate the p-value of the sharp null hypothesis \(\tau_{it} = 0\) for \(i = 1, \cdots, N + 1\) and \(t = 1, \cdots, T\) is: \[ p = \frac{1}{N + 1} \sum_{i = 1}^{N + 1}1\{r_i \ge r_1\}. \]

• Remember that SCM was a formalization of a case study.

• Without the formalization, we could not apply the “same procedure” to estimate the placebo statistics.

• In this sense, this inference became possible only by the formalization of SCM.

• Exclude states that had a pre-treatment mean-squared prediction error (MSPE) is more than 20 times the treated state’s MSPE.Figure 5 of Abadie et al. (2010)

• No extrapolation.
• Transparency of fit.
  • If comparison group are not appropriate, the pre-treatment fit shows this.
• Safeguard against specification searches.
  • It does not use the post-treatment data for constructing weights.
• Transparency of the counterfactual.
  • Weights can be interpreted by domain knowledge.

• The treatment effect should be large enough.
• Similar enough comparison group should exist.
• No anticipation should hold.
• No spillover effects.

• Backdating: Set placebo \(T_0\) and repeat the same analysis.Figure 3 of Abadie (2021)

• Change donor pool.
• Change predictors.

• The minimization problem for finding weights often has multiple solutions.
• This is especially true when we extend the analysis to multiple treated units.
• Abadie and L’Hour (2021) proposes the following penalized estimator: \[ \min_{w} \sum_{k = 1}^K v_k\Big(X_{k1} - \sum_{i = 2}^{N + 1} w_i X_{ik} \Big)^2 + \lambda \sum_{i = 2}^{N + 1} w_i \sum_{k = 1}^K v_k (X_{1k} - X_{ik})^2 \] subject to: \[ w_i \ge 0, \sum_{i = 2}^{N + 1} w_i = 1, \]
• The penalty \(\lambda\) can be chosen by validation data as well as \(v\).

• Consider \(2 \times T\) DID design.
• For simplicity, consider no covariate.
• There are \(N\) units and \(T\) periods.
• There can be multiple treated units.
• \(i = 1, \cdots, N_0\) are control and \(i = N_0 + 1, \cdots N\) is treated, with \(N_1 = N - N_0\).
• Treatment is assigned on \(T_0 + 1\) with \(T_1 = T - T_0\).
• We estimate the average treatment effect on treated by a class of weighted double-differencing estimators: \[ \hat{\tau} = \frac{1}{N_1} \sum_{i = N_0 + 1}^N \hat{\delta}_i - \sum_{i = 1}^{N_0} \hat{w}_i \hat{\delta}_i. \]

• \(\hat{\delta}_i\) is calculated as: \[ \hat{\delta}_i^{sc} = \frac{1}{T_1} \sum_{t = T_0 + 1} Y_{it}. \]
• \(\hat{w}_i\) is constructed according to the SCM as \(\hat{w}^{sc}\).

• \(\hat{\delta}_i\) is calculated as: \[ \hat{\delta}_i^{did} = \frac{1}{T_1} \sum_{t = T_0 + 1} Y_{it} - \frac{1}{T_0} \sum_{t = 1}^{T_0} Y_{it}. \]
• \(\hat{w}_i\) is constructed as \(\hat{w}^{did} = 1/N_0\).

• Arkhangelsky et al. (2021) propose a synthetic DID estimator that combines the features of SCM and DID and adding extra features as follows.
• \(\hat{\delta}_i\) is calculated as: \[ \hat{\delta}_i^{sdid} = \frac{1}{T_1} \sum_{t = T_0 + 1} Y_{it} - \sum_{t = 1}^{T_0} \hat{\lambda}_t^{sdid} Y_{it}, \] which borrows the first-differencing from DID and augments by allowing for a flexible time weight \(\hat{\lambda}_t^{sdid}\) instead of \(1/T_0\).
• \(\hat{w}_i\) is constructed according to a procedure similar to the SCM as \(\hat{w}^{sdid}\).

• Obtain unit weight \(\hat{w}_i^{sdid}\) and level adjustment \(\hat{w}_0\) by solving the constrained minimization problem: \[ \min_{w_0, w} \sum_{t = 1}^{T_0}(w_0 + \sum_{i = 1}^{N_0} w_i Y_{it} - \frac{1}{N_1} \sum_{i = N_0 + 1}^N Y_{it})^2 + \zeta^2 T_0 \sum_{i = 1}^N w_i^2, \] subject to: \[ w_i \ge 0, \sum_{i = 1}^{N_0} w_i = 1. \]
• Allowing for a level difference \(w_0\) is different from SCM.
• Ridge penalty instead of lasso penalty (minor).

• Obtain time weight \(\hat{\lambda}_t^{sdid}\) and level adjustment \(\hat{\lambda}_0\) by solving the constrained minimization problem: \[ \min_{\lambda_0, \lambda} \sum_{i = 1}^{N_0} (\lambda_0 + \sum_{t = 1}^{T_0} \lambda_t Y_{it} - \frac{1}{T_1} \sum_{t = T_0 + 1}^T Y_{it})^2, \] subject to: \[ \lambda_t \ge 0, \sum_{t = 1}^{T_0} \lambda_t = 1. \]

• They suggest to set the ridge penalty \(\zeta\) at: \[ \zeta = (N_1 T_1)^{1/4} \hat{\sigma}, \] \[ \hat{\sigma}^2 = \frac{1}{N_0(T_0 - 1)} \sum_{i = 1}^{N_0} \sum_{t = 1}^{T_0 - 1} (\Delta_{it} - \overline{\Delta})^2, \] \[ \Delta_{it} = Y_{i, t + 1} - Y_{it}, \] \[ \overline{\Delta} = \frac{1}{N_0(T_0 - 1)} \sum_{i = 1}^{N_0} \sum_{t = 1}^{T_0 - 1} \Delta_{it}. \]

• DID puts an equal weight across units and time.
• Because the pre-trend is different, DID is dubious.

• SCM puts weights on a few states.
• Synthetic control targets both level and trend.

• Re-weight units and time.
• Then, apply DID to the re-weighted unit.
• Synthetic control only targets the trend.

• Suppose that the control outcome follows a linear factor model: \[ Y_{it}(0) = \delta_t + \theta_t Z_i + \lambda_t \mu_i + \epsilon_{it}. \]
• Synthetic DID works if the combination of i) double differencing, ii) the unit re-weighting, and iii) the time re-weighting enables us to trace out \(\lambda_t \mu_i\).
• Formal asymptotic result is found in Arkhangelsky et al. (2021).

• If multiple units are treated, we can use a block bootstrap for inference.
• If only one unit is treated, we can use a randomization test.

install.packages("Synth")

data("basque", package = "Synth")
basque %>% head() %>% kbl() %>% kable_styling()

period <- seq(1961, 1969, 2)
df <-
  Synth::dataprep(
    foo = basque,
    predictors = c("school.illit", "school.prim", "school.med", "school.high", "school.post.high", "invest"),
    predictors.op = "mean",
    time.predictors.prior = 1964:1969,
    special.predictors = list(
      list("gdpcap", 1960:1969, "mean"), list("sec.agriculture", period, "mean"), list("sec.energy", period, "mean"),
      list("sec.industry", period, "mean"), list("sec.construction", period, "mean"), list("sec.services.venta", period, "mean"),
      list("sec.services.nonventa", period, "mean"), list("popdens", 1969, "mean")
    ),
    dependent = "gdpcap", unit.variable = "regionno", unit.names.variable = "regionname", time.variable = "year",
    treatment.identifier = 17, controls.identifier = c(2:16, 18), time.optimize.ssr = 1960:1969, time.plot = 1955:1997
  )

names(df)

## [1] "X0"                "X1"                "Z0"                "Z1"                "Y0plot"            "Y1plot"           
## [7] "names.and.numbers" "tag"

df$X1 %>% kbl() %>% kable_styling()

result <- 
  Synth::synth(
  data.prep.obj = df,
  method = "BFGS"
)

## 
## X1, X0, Z1, Z0 all come directly from dataprep object.
## 
## 
## **************** 
##  searching for synthetic control unit  
##  
## 
## **************** 
## **************** 
## **************** 
## 
## MSPE (LOSS V): 0.008864606 
## 
## solution.v:
##  0.02773094 1.194e-07 1.60609e-05 0.0007163836 1.486e-07 0.002423908 0.0587055 0.2651997 0.02851006 0.291276 0.007994382 0.004053188 0.009398579 0.303975 
## 
## solution.w:
##  2.53e-08 4.63e-08 6.44e-08 2.81e-08 3.37e-08 4.844e-07 4.2e-08 4.69e-08 0.8508145 9.75e-08 3.2e-08 5.54e-08 0.1491843 4.86e-08 9.89e-08 1.162e-07

table <- Synth::synth.tab(dataprep.res = df, synth.res = result)
table$tab.pred %>% kbl() %>% kable_styling()

Synth::path.plot(synth.res = result, dataprep.res = df, Ylab = "real per-capita GDP (1986 USD, thousand",
                 Xlab = "year", Ylim = c(0, 12), Legend = c("Basque country", "Synthetic Basque country"),
                 Legend.position = "bottomright")

Synth::gaps.plot(synth.res = result, dataprep.res = df, Ylab = "real per-capita GDP (1986 USD, thousand",
                 Xlab = "year", Ylim = c(-1.5, 1.5), Main = NA)

• Install the package from GitHub repository.

devtools::install_github("synth-inference/synthdid")

data("california_prop99", package = "synthdid")
california_prop99 %>% head() %>% kbl() %>% kable_styling()

df <-
  california_prop99 %>%
  synthdid::panel.matrices(unit = "State", time = "Year", outcome = "PacksPerCapita", treatment = "treated")

names(df)

## [1] "Y"  "N0" "T0" "W"

result_did <- synthdid::did_estimate(Y = df$Y, N0 = df$N0, T0 = df$T0)
plot(result_did)

synthdid::synthdid_units_plot(result_did)

result_sc <- synthdid::sc_estimate(Y = df$Y, N0 = df$N0, T0 = df$T0)
plot(result_sc)

synthdid::synthdid_units_plot(result_sc)

result_synthdid <- synthdid::synthdid_estimate(Y = df$Y, N0 = df$N0, T0 = df$T0)
plot(result_synthdid)

synthdid::synthdid_units_plot(result_synthdid)

• Abadie, Alberto. 2021. “Using Synthetic Controls: Feasibility, Data Requirements, and Methodological Aspects.” Journal of Economic Literature 59 (2): 391–425.
• 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. https://doi.org/10.1198/jasa.2009.ap08746.
• Abadie, Alberto, Alexis Diamond, and Jens Hainmueller. 2011. “Synth: An R Package for Synthetic Control Methods in Comparative Case Studies,” June. https://dspace.mit.edu/handle/1721.1/71234?show=full.
• Abadie, Alberto, and Javier Gardeazabal. 2003. “The Economic Costs of Conflict: A Case Study of the Basque Country.” The American Economic Review 93 (1): 113–32.

• Abadie, Alberto, and Jérémy L’Hour. 2021. “A Penalized Synthetic Control Estimator for Disaggregated Data.” Journal of the American Statistical Association, August, 1–18.
• Arkhangelsky, Dmitry, Susan Athey, David A. Hirshberg, Guido W. Imbens, and Stefan Wager. n.d. “Synthetic Difference in Differences.” The American Economic Review. Accessed September 2, 2021. https://doi.org/10.1257/aer.20190159.
• “Synthdid Introduction.” n.d. Accessed November 13, 2021. https://synth-inference.github.io/synthdid/articles/synthdid.html.