RobustIV

This package provides functions for Inference for the treatment effect with possibly invalid instrumental variables, including the Two-Stage Hard Thresholding(TSHT) method, the Endogeneity Testing method, and Searching-Sampling method.

Installation

The package can be installed from Github using the following code: R devtools::install_github("https://github.com/zijguo/RobustIV") Before using the package, we can use the following code: R library(RobustIV)

Low-dimensional Examples

We use pseudodata provided by Youjin Lee, which is generated mimicing the structure of Framingham Heart Study data. We assume Y is a linear model of D,Z, and X, and D is a linear model of Z and X.

TSHT

```R

data("lineardata") Y <- lineardata[,"Y"] D <- lineardata[,"D"] Z <- as.matrix(lineardata[,c("Z.1","Z.2","Z.3","Z.4","Z.5","Z.6","Z.7","Z.8")]) X <- as.matrix(lineardata[,c("age","sex")]) TSHT.model <- TSHT(Y=Y,D=D,Z=Z,X=X) summary(TSHT.model)

Relevant IVs: Z.3 Z.4 Z.5

betaHat Std.Error CI(2.5%) CI(97.5%) Valid IVs
0.05166598 0.02015546 0.012162 0.09116995 Z.3 Z.4 Z.5

```

Reference

Guo, Z., Kang, H., Tony Cai, T. and Small, D.S. (2018), Confidence intervals for causal effects with invalid instruments by using two-stage hard thresholding with voting, J. R. Stat. Soc. B, 80: 793-815.

Searching-Sampling

```R

Y <- lineardata[,"Y"] D <- lineardata[,"D"] Z <- as.matrix(lineardata[,c("Z.1","Z.2","Z.3","Z.4","Z.5","Z.6","Z.7","Z.8")]) X <- as.matrix(lineardata[,c("age","sex")]) Searching.model <- SearchingSampling(Y,D,Z,X, Sampling = FALSE) summary(Searching.model)

Initial set of Valid Instruments: Z.3 Z.4 Z.5

Plurality rule holds.

Confidence Interval for Beta: [-0.0356797,0.1422332]

Sampling.model <- SearchingSampling(Y,D,Z,X) summary(Sampling.model)

Initial set of Valid Instruments: Z.3 Z.4 Z.5

Plurality rule holds.

Confidence Interval for Beta: [-0.02297164,0.1295251] ```

Reference

Guo, Z. (2021), Causal Inference with Invalid Instruments: Post-selection Problems and A Solution Using Searching and Sampling, Preprint arXiv:2104.06911.

High-dimensional examples (Simulated data)

In this section, we consider the following linear models.

Y = D\beta + Z\alpha + X\phi +u

D = Z\gamma + X\psi + v

TSHT

```R

set.seed(1) n = 500; L = 600; s = 3; k = 10; px = 10; alpha = c(rep(3,s),rep(0,L-s)); beta = 1; gamma = c(rep(1,k),rep(0,L-k)) phi<-(1/px)seq(1,px)+0.5; psi<-(1/px)seq(1,px)+1 epsilonSigma = matrix(c(1,0.8,0.8,1),2,2) Z = matrix(rnorm(nL),n,L) X = matrix(rnorm(npx),n,px) epsilon = MASS::mvrnorm(n,rep(0,2),epsilonSigma) D = 0.5 + Z %% gamma + X %% psi + epsilon[,1] Y = -0.5 + Z %% alpha + D * beta + X %% phi + epsilon[,2] TSHT.model <- TSHT(Y,D,Z,X,method = "Fast.DeLasso") summary(TSHT.model)

Relevant IVs: 1 2 3 4 5 6 7 8 9 10

betaHat Std.Error CI(2.5%) CI(97.5%) Valid IVs
0.9880546 0.0157771 0.9571321 1.018977 4 5 6 7 8 9 10

```

Endogeneity test in high dimension

It uses same reduced form estimator as TSHT in each setting.

```R

set.seed(1) n = 500; L = 600; s = 3; k = 10; px = 10; alpha = c(rep(3,s),rep(0,L-s)); beta = 1; gamma = c(rep(1,k),rep(0,L-k)) phi<-(1/px)seq(1,px)+0.5; psi<-(1/px)seq(1,px)+1 epsilonSigma = matrix(c(1,0.8,0.8,1),2,2) Z = matrix(rnorm(nL),n,L) X = matrix(rnorm(npx),n,px) epsilon = MASS::mvrnorm(n,rep(0,2),epsilonSigma) D = 0.5 + Z %% gamma + X %% psi + epsilon[,1] Y = -0.5 + Z %% alpha + D * beta + X %% phi + epsilon[,2] endo.test.model <- endo.test(Y,D,Z,X, invalid = TRUE) summary(endo.test.model)

Valid Instruments: 4 5 6 7 8 9 10

Estimated covariance: 0.7690532 Test statistics Q = 13.47976 P-value = 0 'H0 : Sigma12 = 0' is rejected at the significance level 0.05 . ```

Reference

Guo, Z., Kang, H., Tony Cai, T. and Small, D.S. (2018), Testing endogeneity with high dimensional covariates, Journal of Econometrics, Elsevier, vol. 207(1), pages 175-187.