pwfe: Fitting the Weighted Fixed Effects Model with Propensity Score Weighting

Description

pwfe is used to fit weighted fixed effects model for causal inference after transforming outcome variable based on estimated propensity score. pwfe also derives the regression weights for different causal quantity of interest.

Usage

pwfe(formula, treat = "treat.name", outcome, data, pscore = NULL,
     unit.index, time.index = NULL, method = "unit", within.unit = TRUE,
     qoi = c("ate", "att"), estimator = NULL, C.it = NULL,
     White = TRUE, White.alpha = 0.05,
     hetero.se = TRUE, auto.se = TRUE, unbiased.se = FALSE,
     verbose = TRUE)

Arguments

formula

a symbolic description of the model for estimating propensity score. The formula should not include dummmies for fixed effects. The details of model specifications are given under `Details'.

treat

a character string indicating the name of treatment variable used in the models. The treatment should be binary indicator (integer with 0 for the control group and 1 for the treatment group).

outcome

a character string indicating the name of outcome variable.

data

data frame containing the variables in the model.

pscore

an optional character string indicating the name of estimated propensity score. Note that pre-specified propensity score should be bounded away from zero and one.

unit.index

a character string indicating the name of unit variable used in the models. The index of unit should be factor.

time.index

a character string indicating the name of time variable used in the models. The index of time should be factor.

method

method for weighted fixed effects regression, either unit for unit fixed effects; time for time fixed effects. The default is unit.

within.unit

a logical value indicating whether propensity score is estimated within unit. The default is TRUE.

qoi

one of "ate" or "att". The default is "ate". "fd" and "did" are not compatible with pwfe.

estimator

an optional character string "fd" indicating whether the first-difference estimator will be used.

C.it

an optional non-negative numeric vector specifying relative weights for each unit of analysis.

White

a logical value indicating whether White misspecification statistics should be calculated. The default is TRUE.

White.alpha

level of functional specification test. See White (1980) and Imai . The default is 0.05.

hetero.se

a logical value indicating whether heteroskedasticity across units is allowed in calculating standard errors. The default is TRUE.

auto.se

a logical value indicating whether arbitrary autocorrelation is allowed in calculating standard errors. The default is TRUE.

unbiased.se

logical. If TRUE, bias-asjusted heteroskedasticity-robust standard errors are used. See Stock and Watson (2008). Should be used only for balanced panel. The default is FALSE.

verbose

logical. If TRUE, helpful messages along with a progress report of the weight calculation are printed on the screen. The default is TRUE.

Value

pwfe returns an object of class "pwfe", a list that contains the components listed below.

The function summary (i.e., summary.pwfe) can be used to obtain a table of the results.

coefficients

a named vector of coefficients

residuals

the residuals, that is respons minus fitted values

the degree of freedom

weight matrix calculated from the model. Row and column indices can be found from unit.name, time.name.

call

the matched call

causal

causal quantity of interest

estimator

the estimating method

unit.name

a vector containing unique unit names

unit.index

a vector containing unique unit index number

time.name

a vector containing unique time names

time.index

a vector containing unique time index number

method

call of the method used

vcov

the variance covariance matrix

White.alpha

the alpha level for White specification test

White.pvalue

the p-value for White specification test

White.stat

the White statistics

the design matrix

the response vector

the model frame

Details

To fit the weighted unit (time) fixed effects model with propensity score weighting, use the syntax for the formula, ~ x1 + x2, where x1 and x2 are unit (time) varying covariates.

One can provide his/her own estimated pscore which can be used to transform the outcome varialbe. If so, one does not need to specify formula.

If pscore is not provided, bayesglm will be used to estimate propensity scores. If within.unit = TRUE, propensity score will be separately estimated within time (unit) when method is unit (time). Otherwise, propensity score will be estimated on entire data at once.

The estimated propensity scores will be used to transform the outcome variable as described in Imai and Kim (2018).

pwfe calculates weights based on different underlying causal quantity of interest: Average Treatment Effect (qoi = "ate") or Average Treatment Effect for the Treated (qoi = "att").

One can further set estimating methods: First-Difference (estimator ="fd") or Difference-in-differences (estimator = "did").

To specify different ex-ante weights for each unit of analysis, use non-negative weights C.it. For instance, using the survey weights for C.it enables the estimation fo the average treatement effect for the target population.

References

Imai, Kosuke and In Song Kim. (2018) ``When Should We Use Unit Fixed Effects Regression Models for Causal Inference with Longitudinal Data?" American Journal of Political Science, Forthcoming.

Stock, James and Mark Watson. (2008) ``Heteroskedasticity-Robust Standard Errors for Fixed Effect Panel Data Regression'' Econometrica, 76, 1.

White, Halbert. (1980) `Using Least Squares to Approximate Unknown Regression Functions.'' International Economic Review, 21, 1, 149--170.

Examples

Run this code

# NOT RUN {
### NOTE: this example illustrates the use of wfe function with randomly
### generated panel data with arbitrary number of units and time.

## generate panel data with number of units = N, number of time = Time
# }
# NOT RUN {
N <- 10 # number of distinct units
Time <- 15 # number of distinct time

## generate treatment variable
treat <- matrix(rbinom(N*Time, size = 1, 0.25), ncol = N)
## make sure at least one observation is treated for each unit
while ((sum(apply(treat, 2, mean) == 0) > 0) | (sum(apply(treat, 2, mean) == 1) > 0) |
       (sum(apply(treat, 1, mean) == 0) > 0) | (sum(apply(treat, 1, mean) == 1) > 0)) {
  treat <- matrix(rbinom(N*Time, size = 1, 0.25), ncol = N)
}
treat.vec <- c(treat)

## unit fixed effects
alphai <- rnorm(N, mean = apply(treat, 2, mean))

## geneate two random covariates
x1 <- matrix(rnorm(N*Time, 0.5,1), ncol=N)
x2 <- matrix(rbeta(N*Time, 5,1), ncol=N)
pscore <- matrix(runif(N*Time, 0,1), ncol=N)
x1.vec <- c(x1)
x2.vec <- c(x2)
pscore <- c(pscore)

## generate outcome variable
y <- matrix(NA, ncol = N, nrow = Time)
for (i in 1:N) {
    y[, i] <- alphai[i] + treat[, i] + x1[,i] + x2[,i] + rnorm(Time)
}
y.vec <- c(y)

## generate unit and time index
unit.index <- rep(1:N, each = Time)
time.index <- rep(1:Time, N)

Data.str <- as.data.frame(cbind(y.vec, treat.vec, unit.index, x1.vec, x2.vec))
colnames(Data.str) <- c("y", "tr", "strata.id", "x1", "x2")

Data.obs <- as.data.frame(cbind(y.vec, treat.vec, unit.index, time.index, x1.vec, x2.vec, pscore))
colnames(Data.obs) <- c("y", "tr", "unit", "time", "x1", "x2", "pscore")


############################################################
# Example 1: Stratified Randomized Experiments
############################################################

## run the weighted fixed effect regression with strata fixed effect.
## Note: the quantity of interest is Average Treatment Effect ("ate")
## and the standard errors allow heteroskedasticity and arbitrary
## autocorrelation.


### Average Treatment Effect
ps.ate <- pwfe(~ x1+x2, treat = "tr", outcome = "y", data = Data.str, 
               unit.index = "strata.id", method = "unit", within.unit = TRUE,
               qoi = "ate", hetero.se=TRUE, auto.se=TRUE)
## summarize the results
summary(ps.ate)

### Average Treatment Effect for the Treated
ps.att <- pwfe(~ x1+x2, treat = "tr", outcome = "y", data = Data.str, 
               unit.index = "strata.id", method = "unit", within.unit = TRUE,
               qoi = "att", hetero.se=TRUE, auto.se=TRUE)
## summarize the results
summary(ps.att)


############################################################
# Example 2: Observational Studies with Unit Fixed-effects
############################################################

## run the weighted fixed effect regression with unit fixed effect.
## Note: the quantity of interest is Average Treatment Effect ("ate")
## and the standard errors allow heteroskedasticity and arbitrary
## autocorrelation.

### Average Treatment Effect
ps.obs <- pwfe(~ x1+x2, treat = "tr", outcome = "y", data = Data.obs, 
               unit.index = "unit", time.index = "time",
               method = "unit", within.unit = TRUE,
               qoi = "ate", hetero.se=TRUE, auto.se=TRUE)

## summarize the results
summary(ps.obs)

## extracting weigths
summary(ps.obs)$Weights

### Average Treatment Effect with First-difference

ps.fd <- pwfe(~ x1+x2, treat = "tr", outcome = "y", data = Data.obs,
              unit.index = "unit", time.index = "time",
              method = "unit", within.unit = TRUE,
              qoi = "ate", estimator = "fd", hetero.se=TRUE, auto.se=TRUE)

## summarize the results
summary(ps.fd)


############################################################
# Example 3: Estimation with pre-specified propensity score
############################################################

### Average Treatment Effect with Pre-specified Propensity Scores

mod.ps <- pwfe(treat = "tr", outcome = "y", data = Data.obs, pscore = "pscore",
               unit.index = "unit", time.index = "time",
               method = "unit", within.unit = TRUE,
               qoi = "ate", hetero.se=TRUE, auto.se=TRUE)

## summarize the results
summary(mod.ps)
# }
# NOT RUN {
# }

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