# Asymptotic Variance and Confidence Interval Estimation of the ATE `AsyVar` estimates the asymptotic variance of the ATE obtained with the CBPS or oCBPS method. It also returns the finite variance estimate, the finite standard error, and a CI for the ATE. ```r AsyVar( Y, Y_1_hat = NULL, Y_0_hat = NULL, CBPS_obj, method = "CBPS", X = NULL, TL = NULL, pi = NULL, mu = NULL, CI = 0.95 ) ``` ## Arguments - `Y`: The vector of actual outcome values (observations). - `Y_1_hat`: The vector of estimated outcomes according to the treatment model. (AsyVar automatically sets the treatment model as a linear regression model and it is fitted within the function.) If `CBPS_obj` is specified, or if `X` $and$ `TL` are specified, this is unnecessary. - `Y_0_hat`: The vector of estimated outcomes according to the control model. (AsyVar automatically sets the control model as a linear regression model and it is fitted within the function.) If `CBPS_obj` is specified, or if `X` $and$ `TL` are specified, this is unnecessary. - `CBPS_obj`: An object obtained with the CBPS function. If this object is not sepecified, then `X`, `TL`, `pi`, and `mu` must $all$ be specified instead. - `method`: The specific method to be considered. Either `"CBPS"` or `"oCBPS"` must be selected. - `X`: The matrix of covariates with the rows corresponding to the observations and the columns corresponding to the variables. The left most column must be a column of 1's for the intercept. (`X` is not necessary if `CBPS_obj` is specified.) - `TL`: The vector of treatment labels. More specifically, the label is 1 if it is in the treatment group and 0 if it is in the control group. (`TL` is not necessary if `CBPS_obj` is specified.) - `pi`: The vector of estimated propensity scores. (`pi` is not necessary if `CBPS_obj` is specified.) - `mu`: The estimated average treatment effect obtained with either the CBPS or oCBPS method. (`mu` is not necessary if `CBPS_obj` is specified.) - `CI`: The specified confidence level (between 0 and 1) for calculating the confidence interval for the average treatment effect. Default value is 0.95. ## Returns - **mu.hat**: The estimated average treatment effect, $hat{\mu}$. - **asy.var**: The estimated asymptotic variance of $\sqrt{n}*hat{\mu}$ obtained with the CBPS or oCBPS method. - **var**: The estimated variance of $hat{\mu}$ obtained with the CBPS or oCBPS method. - **std.err**: The standard error of $hat{\mu}$ obtained with the CBPS or oCBPS method. - **CI.mu.hat**: The confidence interval of $hat{\mu}$ obtained with the CBPS or oCBPS method with the confidence level specified in the input argument. ## Examples ```r #GENERATING THE DATA n=300 #Initialize the X matrix. X_v1 <- rnorm(n,3,sqrt(2)) X_v2 <- rnorm(n,0,1) X_v3 <- rnorm(n,0,1) X_v4 <- rnorm(n,0,1) X_mat <- as.matrix(cbind(rep(1,n), X_v1, X_v2, X_v3, X_v4)) #Initialize the Y_1 and Y_0 vector using the treatment model and the control model. Y_1 <- X_mat %*% matrix(c(200, 27.4, 13.7, 13.7, 13.7), 5, 1) + rnorm(n) Y_0 <- X_mat %*% matrix(c(200, 0 , 13.7, 13.7, 13.7), 5, 1) + rnorm(n) #True Propensity Score calculation. pre_prop <- X_mat[,2:5] %*% matrix(c(0, 0.5, -0.25, -0.1), 4, 1) propensity_true <- (exp(pre_prop))/(1+(exp(pre_prop))) #Generate T_vec, the treatment vector, with the true propensity scores. T_vec <- rbinom(n, size=1, prob=propensity_true) #Now generate the actual outcome Y_outcome (accounting for treatment/control groups). Y_outcome <- Y_1*T_vec + Y_0*(1-T_vec) #Use oCBPS. ocbps.fit <- CBPS(T_vec ~ X_mat, ATT=0, baseline.formula = ~X_mat[,c(1,3:5)], diff.formula = ~X_mat[,2]) #Use the AsyVar function to get the asymptotic variance of the #estimated average treatment effect and its confidence interval when using oCBPS. AsyVar(Y=Y_outcome, CBPS_obj=ocbps.fit, method="oCBPS", CI=0.95) #Use CBPS. cbps.fit <- CBPS(T_vec ~ X_mat, ATT=0) #Use the AsyVar function to get the asymptotic variance of the #estimated average treatment effect and its confidence interval when using CBPS. AsyVar(Y=Y_outcome, CBPS_obj=cbps.fit, method="CBPS", CI=0.95) ``` ## References Fan, Jianqing and Imai, Kosuke and Lee, Inbeom and Liu, Han and Ning, Yang and Yang, Xiaolin. 2021. ``Optimal Covariate Balancing Conditions in Propensity Score Estimation.'' Journal of Business & Economic Statistics. [https://imai.fas.harvard.edu/research/CBPStheory.html](https://imai.fas.harvard.edu/research/CBPStheory.html) ## Author(s) Inbeom Lee