RPIV

The RPIV package implements a residual prediction test for the well specification of linear instrumental variable (IV) models, as presented in Scheidegger, Londschien and Bühlmann (2025). For a response $Y_i\in \mathbb R$, endogenous explanatory variables $X_i\in \mathbb R^p$ and instruments $Z_i\in \mathbb R^d$ ($d\geq p$), it tests the well-specification of the linear IV model (``is the linear IV model appropriate?’’). More formally, it tests the null hypothesis $$H_0: \exists \beta\in \mathbb R^p\text{ s.t. } \mathbb E[Y_i - X_i^T\beta|Z_i] = 0 \text{ a.s., } i =1,\ldots, n,$$ which is implied by the well-specification of the linear IV model (with mean-independence assumption on the errors).

The model allows for additional exogenous explanatory variables (``exogenous controls’’) (denoted by $C$ in the R function) and an intercept, which are added both to $X$ and $Z$.

For a detailed discussion of the method, we refer to Scheidegger, Londschien and Bühlmann (2025). A python implementation is available in the package ivmodels. We now demonstrate, how the RPIV package is used in practice.

Installation

You can install the development version of IVDML from GitHub with

devtools::install_github("cyrillsch/RPIV")

Example

This is a basic example presenting, how the well-specification of linear IV models can be tested with the RPIV package. We simulate a dataset with $n = 200$ observation and three responses.

set.seed(1)
n <- 200
C <- rnorm(n) # exogenous explanatory variable
Z <- cbind(rnorm(n), C + rnorm(n)) # instrumental variable
H <- rnorm(n) # hidden confounding
X <- Z[, 1] - Z[, 2] + rnorm(n) # endogenous explanatory variable
Y1 <- X - C + H + rnorm(n) # linear IV model
Y2 <- X - C + H + Z[, 1]^2 + rnorm(n) # invalid IV -> misspecified
Y3 <- 2 * sign(X - C) + H + rnorm(n) # nonlinear IV model -> misspecified

To apply the well-specification test to the three responses, we use the function , which uses a heteroskedasticity robust variance estimator by default.

library(RPIV)
result1 <- RPIV_test(Y = Y1, X = X, C = C, Z = Z)
result2 <- RPIV_test(Y = Y2, X = X, C = C, Z = Z)
result3 <- RPIV_test(Y = Y3, X = X, C = C, Z = Z)

result1$p_value
#> [1] 0.1575286
result2$p_value
#> [1] 0.0004228503
result3$p_value
#> [1] 0.005525054

We see that, indeed, well-specification is rejected at significance level $\alpha = 0.05$ for the responses $Y_2$ and $Y_3$.

The RPIV package also supports cluster-robust inference. We simulate data with 50 clusters of size 4, but the linear IV model is well-specified otherwise.

set.seed(1)
n <- 200
clustering <- rep(1:50, length.out = n)
Z <- rep(rnorm(1:50), length.out = n) + 0.1 * rnorm(n)
H <- rep(rnorm(1:50), length.out = n) + 0.1 * rnorm(n)
X <- Z + rep(rnorm(1:50), length.out = n) + 0.1 * rnorm(n)
Y <- X + H + rep(rnorm(1:50), length.out = n) + 0.1 * rnorm(n)

We apply the test with three different variance estimators: assuming homoskedasticity, robust to heteroskedasticity, robust to clustering.

result <- RPIV_test(Y = Y, X = X, C = NULL, Z = Z, variance_estimator = 
                      c("homoskedastic", "heteroskedastic", "cluster"), clustering = clustering)
result$homoskedastic$p_value
#> [1] 0.02844595
result$heteroskedastic$p_value
#> [1] 0.01728716
result$cluster$p_value
#> [1] 0.1347029

We see that only using the cluster-robust variance estimator does not reject the null hypothesis at significance level $\alpha = 0.05$.

More Examples

More examples can be found in Scheidegger, Londschien and Bühlmann (2025) and the associated GithHub repository RPIV_Application.

References

Cyrill Scheidegger, Malte Londschien and Peter Bühlmann. A residual prediction test for the well-specification of linear instrumental variable models. Preprint, arXiv:2506.12771, 2025.