27.1 Applications
27.1.1 Example 1
simulate data for 10 states and 30 years. State A receives the treatment T = 20 after year 15.
set.seed(1)
year <- rep(1:30, 10)
state <- rep(LETTERS[1:10], each = 30)
X1 <- round(rnorm(300, mean = 2, sd = 1), 2)
X2 <- round(rbinom(300, 1, 0.5) + rnorm(300), 2)
Y <- round(1 + 2 * X1 + rnorm(300), 2)
df <- as.data.frame(cbind(Y, X1, X2, state, year))
df$Y <- as.numeric(as.character(df$Y))
df$X1 <- as.numeric(as.character(df$X1))
df$X2 <- as.numeric(as.character(df$X2))
df$year <- as.numeric(as.character(df$year))
df$state.num <- rep(1:10, each = 30)
df$state <- as.character(df$state)
df$`T` <- ifelse(df$state == "A" & df$year >= 15, 1, 0)
df$Y <- ifelse(df$state == "A" & df$year >= 15,
df$Y + 20, df$Y)str(df)
#> 'data.frame': 300 obs. of 7 variables:
#> $ Y : num 2.29 4.51 2.07 8.87 4.37 1.32 8 7.49 6.98 3.72 ...
#> $ X1 : num 1.37 2.18 1.16 3.6 2.33 1.18 2.49 2.74 2.58 1.69 ...
#> $ X2 : num 1.96 0.4 -0.75 -0.56 -0.45 1.06 0.51 -2.1 0 0.54 ...
#> $ state : chr "A" "A" "A" "A" ...
#> $ year : num 1 2 3 4 5 6 7 8 9 10 ...
#> $ state.num: int 1 1 1 1 1 1 1 1 1 1 ...
#> $ T : num 0 0 0 0 0 0 0 0 0 0 ...dataprep.out <-
dataprep(
df,
predictors = c("X1", "X2"),
dependent = "Y",
unit.variable = "state.num",
time.variable = "year",
unit.names.variable = "state",
treatment.identifier = 1,
controls.identifier = c(2:10),
time.predictors.prior = c(1:14),
time.optimize.ssr = c(1:14),
time.plot = c(1:30)
)
synth.out <- synth(dataprep.out)
#>
#> X1, X0, Z1, Z0 all come directly from dataprep object.
#>
#>
#> ****************
#> searching for synthetic control unit
#>
#>
#> ****************
#> ****************
#> ****************
#>
#> MSPE (LOSS V): 9.831789
#>
#> solution.v:
#> 0.3888387 0.6111613
#>
#> solution.w:
#> 0.1115941 0.1832781 0.1027237 0.312091 0.06096758 0.03509706 0.05893735 0.05746256 0.07784853print(synth.tables <- synth.tab(
dataprep.res = dataprep.out,
synth.res = synth.out)
)
#> $tab.pred
#> Treated Synthetic Sample Mean
#> X1 2.028 2.028 2.017
#> X2 0.513 0.513 0.394
#>
#> $tab.v
#> v.weights
#> X1 0.389
#> X2 0.611
#>
#> $tab.w
#> w.weights unit.names unit.numbers
#> 2 0.112 B 2
#> 3 0.183 C 3
#> 4 0.103 D 4
#> 5 0.312 E 5
#> 6 0.061 F 6
#> 7 0.035 G 7
#> 8 0.059 H 8
#> 9 0.057 I 9
#> 10 0.078 J 10
#>
#> $tab.loss
#> Loss W Loss V
#> [1,] 9.761708e-12 9.831789path.plot(synth.res = synth.out,
dataprep.res = dataprep.out,
Ylab = c("Y"),
Xlab = c("Year"),
Legend = c("State A","Synthetic State A"),
Legend.position = c("topleft")
)
abline(v = 15,
lty = 2)
Gaps plot:
gaps.plot(synth.res = synth.out,
dataprep.res = dataprep.out,
Ylab = c("Gap"),
Xlab = c("Year"),
Ylim = c(-30, 30),
Main = ""
)
abline(v = 15,
lty = 2)
Alternatively, gsynth provides options to estimate iterative fixed effects, and handle multiple treated units at tat time.
Here, we use two=way fixed effects and bootstrapped standard errors
gsynth.out <- gsynth(
Y ~ `T` + X1 + X2,
data = df,
index = c("state", "year"),
force = "two-way",
CV = TRUE,
r = c(0, 5),
se = TRUE,
inference = "parametric",
nboots = 1000,
parallel = F # TRUE
)
#> Cross-validating ...
#> r = 0; sigma2 = 1.13533; IC = 0.95632; PC = 0.96713; MSPE = 1.65502
#> r = 1; sigma2 = 0.96885; IC = 1.54420; PC = 4.30644; MSPE = 1.33375
#> r = 2; sigma2 = 0.81855; IC = 2.08062; PC = 6.58556; MSPE = 1.27341*
#> r = 3; sigma2 = 0.71670; IC = 2.61125; PC = 8.35187; MSPE = 1.79319
#> r = 4; sigma2 = 0.62823; IC = 3.10156; PC = 9.59221; MSPE = 2.02301
#> r = 5; sigma2 = 0.55497; IC = 3.55814; PC = 10.48406; MSPE = 2.79596
#>
#> r* = 2
#>
#>
Simulating errors .............
Bootstrapping ...
#> ..........
27.1.2 Example 2
by Leihua Ye
library(Synth)
data("basque")
dim(basque) #774*17
#> [1] 774 17
head(basque)
#> regionno regionname year gdpcap sec.agriculture sec.energy sec.industry
#> 1 1 Spain (Espana) 1955 2.354542 NA NA NA
#> 2 1 Spain (Espana) 1956 2.480149 NA NA NA
#> 3 1 Spain (Espana) 1957 2.603613 NA NA NA
#> 4 1 Spain (Espana) 1958 2.637104 NA NA NA
#> 5 1 Spain (Espana) 1959 2.669880 NA NA NA
#> 6 1 Spain (Espana) 1960 2.869966 NA NA NA
#> sec.construction sec.services.venta sec.services.nonventa school.illit
#> 1 NA NA NA NA
#> 2 NA NA NA NA
#> 3 NA NA NA NA
#> 4 NA NA NA NA
#> 5 NA NA NA NA
#> 6 NA NA NA NA
#> school.prim school.med school.high school.post.high popdens invest
#> 1 NA NA NA NA NA NA
#> 2 NA NA NA NA NA NA
#> 3 NA NA NA NA NA NA
#> 4 NA NA NA NA NA NA
#> 5 NA NA NA NA NA NA
#> 6 NA NA NA NA NA NAtransform data to be used in synth()
dataprep.out <- dataprep(
foo = basque,
predictors = c(
"school.illit",
"school.prim",
"school.med",
"school.high",
"school.post.high",
"invest"
),
predictors.op = "mean",
# the operator
time.predictors.prior = 1964:1969,
#the entire time frame from the #beginning to the end
special.predictors = list(
list("gdpcap", 1960:1969, "mean"),
list("sec.agriculture", seq(1961, 1969, 2), "mean"),
list("sec.energy", seq(1961, 1969, 2), "mean"),
list("sec.industry", seq(1961, 1969, 2), "mean"),
list("sec.construction", seq(1961, 1969, 2), "mean"),
list("sec.services.venta", seq(1961, 1969, 2), "mean"),
list("sec.services.nonventa", seq(1961, 1969, 2), "mean"),
list("popdens", 1969, "mean")
),
dependent = "gdpcap",
# dv
unit.variable = "regionno",
#identifying unit numbers
unit.names.variable = "regionname",
#identifying unit names
time.variable = "year",
#time-periods
treatment.identifier = 17,
#the treated case
controls.identifier = c(2:16, 18),
#the control cases; all others #except number 17
time.optimize.ssr = 1960:1969,
#the time-period over which to optimize
time.plot = 1955:1997
) #the entire time period before/after the treatmentwhere
\(X_1\) = the control case before the treatment
\(X_0\) = the control cases after the treatment
\(Z_1\): the treatment case before the treatment
\(Z_0\): the treatment case after the treatment
synth.out = synth(data.prep.obj = dataprep.out, 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-07Calculate the difference between the real basque region and the synthetic control
gaps = dataprep.out$Y1plot - (dataprep.out$Y0plot
%*% synth.out$solution.w)
gaps[1:3,1]
#> 1955 1956 1957
#> 0.15023473 0.09168035 0.03716475synth.tables = synth.tab(dataprep.res = dataprep.out,
synth.res = synth.out)
names(synth.tables)
#> [1] "tab.pred" "tab.v" "tab.w" "tab.loss"
synth.tables$tab.pred[1:13,]
#> Treated Synthetic Sample Mean
#> school.illit 39.888 256.337 170.786
#> school.prim 1031.742 2730.104 1127.186
#> school.med 90.359 223.340 76.260
#> school.high 25.728 63.437 24.235
#> school.post.high 13.480 36.153 13.478
#> invest 24.647 21.583 21.424
#> special.gdpcap.1960.1969 5.285 5.271 3.581
#> special.sec.agriculture.1961.1969 6.844 6.179 21.353
#> special.sec.energy.1961.1969 4.106 2.760 5.310
#> special.sec.industry.1961.1969 45.082 37.636 22.425
#> special.sec.construction.1961.1969 6.150 6.952 7.276
#> special.sec.services.venta.1961.1969 33.754 41.104 36.528
#> special.sec.services.nonventa.1961.1969 4.072 5.371 7.111Relative importance of each unit
synth.tables$tab.w[8:14, ]
#> w.weights unit.names unit.numbers
#> 9 0.000 Castilla-La Mancha 9
#> 10 0.851 Cataluna 10
#> 11 0.000 Comunidad Valenciana 11
#> 12 0.000 Extremadura 12
#> 13 0.000 Galicia 13
#> 14 0.149 Madrid (Comunidad De) 14
#> 15 0.000 Murcia (Region de) 15# plot the changes before and after the treatment
path.plot(
synth.res = synth.out,
dataprep.res = dataprep.out,
Ylab = "real per-capita gdp (1986 USD, thousand)",
Xlab = "year",
Ylim = c(0, 12),
Legend = c("Basque country",
"synthetic Basque country"),
Legend.position = "bottomright"
)
gaps.plot(
synth.res = synth.out,
dataprep.res = dataprep.out,
Ylab = "gap in real per - capita GDP (1986 USD, thousand)",
Xlab = "year",
Ylim = c(-1.5, 1.5),
Main = NA
)
Doubly Robust Difference-in-Differences
Example from DRDID package
library(DRDID)
data(nsw_long)
# Form the Lalonde sample with CPS comparison group
eval_lalonde_cps <- subset(nsw_long, nsw_long$treated == 0 |
nsw_long$sample == 2)Estimate Average Treatment Effect on Treated using Improved Locally Efficient Doubly Robust DID estimator
out <-
drdid(
yname = "re",
tname = "year",
idname = "id",
dname = "experimental",
xformla = ~ age + educ + black + married + nodegree + hisp + re74,
data = eval_lalonde_cps,
panel = TRUE
)
summary(out)
#> Call:
#> drdid(yname = "re", tname = "year", idname = "id", dname = "experimental",
#> xformla = ~age + educ + black + married + nodegree + hisp +
#> re74, data = eval_lalonde_cps, panel = TRUE)
#> ------------------------------------------------------------------
#> Further improved locally efficient DR DID estimator for the ATT:
#>
#> ATT Std. Error t value Pr(>|t|) [95% Conf. Interval]
#> -901.2703 393.6247 -2.2897 0.022 -1672.7747 -129.766
#> ------------------------------------------------------------------
#> Estimator based on panel data.
#> Outcome regression est. method: weighted least squares.
#> Propensity score est. method: inverse prob. tilting.
#> Analytical standard error.
#> ------------------------------------------------------------------
#> See Sant'Anna and Zhao (2020) for details.27.1.3 Example 3
by Synth package’s authors
synth() requires
\(X_1\) vector of treatment predictors
\(X_0\) matrix of same variables for control group
\(Z_1\) vector of outcome variable for treatment group
\(Z_0\) matrix of outcome variable for control group
use dataprep() to prepare data in the format that can be used throughout the Synth package
dataprep.out <- 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", seq(1961, 1969, 2), "mean"),
list("sec.energy", seq(1961, 1969, 2), "mean"),
list("sec.industry", seq(1961, 1969, 2), "mean"),
list("sec.construction", seq(1961, 1969, 2), "mean"),
list("sec.services.venta", seq(1961, 1969, 2), "mean"),
list("sec.services.nonventa", seq(1961, 1969, 2), "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
)find optimal weights that identifies the synthetic control for the treatment group
synth.out <- synth(data.prep.obj = dataprep.out, 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-07gaps <- dataprep.out$Y1plot - (dataprep.out$Y0plot %*% synth.out$solution.w)
gaps[1:3, 1]
#> 1955 1956 1957
#> 0.15023473 0.09168035 0.03716475synth.tables <-
synth.tab(dataprep.res = dataprep.out, synth.res = synth.out)
names(synth.tables) # you can pick tables to see
#> [1] "tab.pred" "tab.v" "tab.w" "tab.loss"path.plot(
synth.res = synth.out,
dataprep.res = dataprep.out,
Ylab = "real per-capita GDP (1986 USD, thousand)",
Xlab = "year",
Ylim = c(0, 12),
Legend = c("Basque country",
"synthetic Basque country"),
Legend.position = "bottomright"
)
gaps.plot(
synth.res = synth.out,
dataprep.res = dataprep.out,
Ylab = "gap in real per-capita GDP (1986 USD, thousand)",
Xlab = "year",
Ylim = c(-1.5, 1.5),
Main = NA
)
You could also run placebo tests
27.1.4 Example 4
by Michael Robbins and Steven Davenport who are authors of MicroSynth with the following improvements:
Standardization
use.survey = TRUEand permutation (perm = 250andjack = TRUE) for placebo testsOmnibus statistic (set to
omnibus.var) for multiple outcome variablesincorporate multiple follow-up periods
end.post
Notes:
Both predictors and outcome will be used to match units before intervention
Outcome variable has to be time-variant
Predictors are time-invariant
# right now the package is not availabe for R version 4.2
library(microsynth)
data("seattledmi")
cov.var <-
c(
"TotalPop",
"BLACK",
"HISPANIC",
"Males_1521",
"HOUSEHOLDS",
"FAMILYHOUS",
"FEMALE_HOU",
"RENTER_HOU",
"VACANT_HOU"
)
match.out <- c("i_felony", "i_misdemea", "i_drugs", "any_crime")sea1 <- microsynth(
seattledmi,
idvar = "ID",
timevar = "time",
intvar = "Intervention",
start.pre = 1,
end.pre = 12,
end.post = 16,
match.out = match.out, # outcome variable will be matched on exactly
match.covar = cov.var, # specify covariates will be matched on exactly
result.var = match.out, # used to report results
omnibus.var = match.out, # feature in the omnibus p-value
test = "lower",
n.cores = min(parallel::detectCores(), 2)
)
sea1
summary(sea1)sea2 <- microsynth(
seattledmi,
idvar = "ID",
timevar = "time",
intvar = "Intervention",
start.pre = 1,
end.pre = 12,
end.post = c(14, 16),
match.out = match.out,
match.covar = cov.var,
result.var = match.out,
omnibus.var = match.out,
test = "lower",
perm = 250,
jack = TRUE,
n.cores = min(parallel::detectCores(), 2)
)