wfe {wfe}

Fitting the Weighted Fixed Effects Model for Causal Inference

Description

wfe is used to fit weighted fixed effects model for causal inference. wfe also derives the regression weights for different causal quantity of interest.

Usage

wfe(formula, data, treat = "treat.name",
    unit.index, time.index = NULL, method = "unit",
    dyad1.index = NULL, dyad2.index = NULL,
    qoi = "ate", estimator = NULL, C.it = NULL,
    hetero.se = TRUE, auto.se = TRUE, 
    dyad.se = FALSE,
    White = TRUE, White.alpha = 0.05,
    verbose = TRUE, unbiased.se = FALSE, unweighted = FALSE,
    store.wdm = FALSE, maxdev.did = NULL,
    tol = sqrt(.Machine$double.eps))

Arguments

formula

a symbolic description of the model to be fitted. The formula should not include dummmies for fixed effects. The details of model specifications are given under ‘Details’.

data

data frame containing the variables in the model.

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).

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. For two-way weighted fixed effects regression models, set method to the default value unit.

dyad1.index

a character string indicating the variable name of first unit of a given dyad. The default is NULL. This is required to calculate robust standard errors with dyadic data.

dyad2.index

a character string indicating the variable name of second unit of a given dyad. The default is NULL. This is required to calculate robust standard errors with dyadic data.

qoi

one of "ate" or "att". The default is "ate". If set to "att" in implementing "fd" and "did" estimators, the comparison of the treated observation is restricted to the control observation from the previous time period but not with the control observation from the next time period.

estimator

an optional character string indicating the estimating method. One of "fd", "did", or "Mdid". "fd" is for First-Difference Design. "did" is for multi-period Difference-in-Differences design. The default is NULL. Setting estimator to be "Mdid" implements the Difference-in-Differences design with Matching on the pretreatment outcome variables.

C.it

an optional non-negative numeric vector specifying relative weights for each unit of analysis. If not specified, the weights will be calculated based on the estimator and quantity of interest.

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.

dyad.se

a logical value indicating whether correlations across dyads exist. The default is FALSE.

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 and Kim (2018). The default is 0.05.

verbose

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

unweighted

logical. If TRUE, standard unweighted fixed effects model is estimated. The default is FALSE. Note: users do not need to specify qoi when unweighted=TRUE. For standard two-way fixed effects model (unit and time), set estimator="did" and unweighted="TRUE".

store.wdm

logical. If TRUE, weighted demeaned dataframe will be stored. The default is FALSE.

maxdev.did

an optional positive numeric value specifying the maximum deviation in pre-treatment outcome when "Mdid" is implemented. The default is NULL, which implements nearest-neighbor matching.

tol

a relative tolerance to detect zero singular values for generalized inverse. The default is sqrt(.Machine$double.eps)

Details

To fit the weighted unit (time) fixed effects model, use the syntax for the formula, y ~ x1 + x2, where y is a dependent variable and x1 and x2 are unit (time) varying covariates.

wfe 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"). For the two-way fixed effects model, set 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.

An object of class "wfe" contains vectors of unique unit(time) names and unique unit(time) indices.

Value

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

The function summary (i.e., summary.wfe) 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

df

the degree of freedom

W

a dataframe containing unit and time indices along with the weights used for the observation. If method=unit, integer numbers corresponding to the order of input data will be used for generating time index.

Num.nonzero

Number of observations with non-zero weights

call

the matched call

causal

causal quantity of interest

estimator

the estimating method

units

a dataframe containing unit names used for W

times

a dataframe containing time names used for W

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

X

the design matrix

Y

the response vector

X.wdm

the demeaned design matrix

Y.wdm

the demeaned response vector

mf

the model frame where the last column is the weights used for the analysis

Author(s)

In Song Kim, Massachusetts Institute of Technology, [e-mail] and Kosuke Imai, Princeton University, [e-mail]

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.

Aronow, Peter M., Cyrus Samii, and Valentina A. Assenova (2015) “Cluster–robust Variance Estimation for Dyadic Data." Political Analysis 23, no. 4, 564–577.

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.

See Also

pwfe for fitting weighted fixed effects models with propensity score weighting

Examples

### 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
N <- 10 # number of distinct units
Time <- 15 # number of distinct time

## treatment effect
beta <- 1

## 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)
x1.vec <- c(x1)
x2.vec <- c(x2)
## 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))
colnames(Data.obs) <- c("y", "tr", "unit", "time", "x1", "x2")

############################################################
# 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
mod.ate <- wfe(y~ tr+x1+x2, data = Data.str, treat = "tr",
               unit.index = "strata.id", method = "unit",
               qoi = "ate", hetero.se=TRUE, auto.se=TRUE)
## summarize the results
summary(mod.ate)

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

mod.obs <- wfe(y~ tr+x1+x2, data = Data.obs, treat = "tr",
               unit.index = "unit", time.index = "time", method = "unit",
               qoi = "ate", hetero.se=TRUE, auto.se=TRUE,
               White = TRUE, White.alpha = 0.05)

## summarize the results
summary(mod.obs)

## extracting weigths
summary(mod.obs)$W


## Not run: 
###################################################################
# Example 3: Observational Studies with differences-in-differences
###################################################################

## run difference-in-differences estimator.
## Note: the quantity of interest is Average Treatment Effect ("ate")
## and the standard errors allow heteroskedasticity and arbitrary
## autocorrelation.

mod.did <- wfe(y~ tr+x1+x2, data = Data.obs, treat = "tr",
               unit.index = "unit", time.index = "time", method = "unit",
               qoi = "ate", estimator ="did", hetero.se=TRUE, auto.se=TRUE,
               White = TRUE, White.alpha = 0.05, verbose = TRUE)

## summarize the results
summary(mod.did)

## extracting weigths
summary(mod.did)$W

#########################################################################
# Example 4: DID with Matching on Pre-treatment Outcomes
#########################################################################

## implements matching on pre-treatment outcomes where the maximum
## deviation is specified as 0.5 

mod.Mdid <- wfe(y~ tr+x1+x2, data = Data.obs, treat = "tr",
               unit.index = "unit", time.index = "time", method = "unit",
               qoi = "ate", estimator ="Mdid", hetero.se=TRUE, auto.se=TRUE,
               White = TRUE, White.alpha = 0.05, maxdev.did = 0.5, verbose = TRUE)

## summarize the results
summary(mod.Mdid)

## Note: setting the maximum deviation to infinity (or any value
## bigger than the maximum pair-wise difference in the outcome) will
## return the same result as Example 3.

dev <- 1000+max(Data.obs$y)-min(Data.obs$y)
mod.did2 <- wfe(y~ tr+x1+x2, data = Data.obs, treat = "tr",
               unit.index = "unit", time.index = "time", method = "unit",
               qoi = "ate", estimator ="Mdid", hetero.se=TRUE, auto.se=TRUE,
               White = TRUE, White.alpha = 0.05, maxdev.did = dev, verbose = TRUE)

## summarize the results
summary(mod.did2)
mod.did2$coef[1] ==  mod.did$coef[1]

## End(Not run)