# Fitting the Weighted Fixed Effects Model with Propensity Score Weighting `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. ```r 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`. ## Returns `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 - **df**: the degree of freedom - **W**: 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 - **x**: the design matrix - **y**: the response vector - **mf**: 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. ## See Also `wfe` for fitting weighted fixed effect models. ## Examples ```r ### 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) ## End(Not run) ``` ## Author(s) In Song Kim, Massachusetts Institute of Technology, insong@mit.edu and Kosuke Imai, Princeton University, imai@harvard.edu

pwfe function