In danmrc/weak-instruments-r: Suite Test for Weak Instruments in R

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)

library(weakinstruments)
library(AER)
library(stargazer)

Introduction

Instrumental Variables must satisfies two hypotesis: the exclusion restriction (exogenity) and the inclusion restriction (relevance). This package focus on the second: are the instruments correlated enough the endogenous variables? When using one endogenous variable, one can easily check it by using a F test on the first stage: a F-statistics above 10 means a strong instrument. I will generate two examples, one with strong instrument and another with a weak instrument. I set the sample size at 2000, to show that it is not a small sample issue!

u <- rnorm(2000) #this will generate the endogenity

z_1 <- rnorm(2000) #this will be the strong instrument
z_2 <- rnorm(2000) #this will be the weak instrument

x <- u + z_1 + 0.01*z_2 + rnorm(2000)

y <- u + 2*x + rnorm(2000)

first_stage_strong <- lm(x ~ z_1)
first_stage_weak <- lm(x ~ z_2)

tab <- c(summary(first_stage_strong)$fstatistic[1],summary(first_stage_weak)$fstatistic[1])
names(tab) <- c("Strong Instrument", "Weak Instrument")
tab <- round(tab,digits = 4)

knitr::kable(tab,caption = "F-Statistics for Instruments")

The results of a weak instrument are coefficients that can be absolutely anywhere. I will use ivreg from AER to show that:

iv_reg <- ivreg(y ~ x|z_2)

stargazer(iv_reg, type = "html")

More than one endogenous variable

While the F-test is absolutely adequate for a single endogenous variable, there is a test for when you have two or three endogenous variables. This is the test by Stock & Yogo (2005). We will create two endogenous variables and 4 instruments. The test is done using the command stock_yogo_test:

u <- rnorm(200)

z.1 <- rnorm(200)
z.2 <- rnorm(200)
z.3 <- rnorm(200)
z.4 <- rnorm(200)

x.1 <- u + 0.5*z.1 + z.2 + z.4 + rnorm(200)
x.2 <- u + 0.4*z.1 - 0.9*z.2 + 2*z.3 + rnorm(200)

y <- 1 + u + 2*x.1 - 2*x.2 + rnorm(200)

lm(y ~ x.1 + x.2)

X <- cbind(x.1,x.2)
Z <- cbind(z.1,z.2,z.3,z.4)

print(stock_yogo_test(X,Z))

Now lets generate a case in which the instruments are weak:

x.1 <- 2*u + 0.1*z.1 - 0.02*z.2 + rnorm(200)
x.2 <- 2*u + 0.0074*z.4 -0.09*z.2 - 0.012*z.3 + rnorm(200)

y <- 1 + u + 2*x.1 - 2*x.2 + rnorm(200)

X <- cbind(x.1,x.2)
Z <- cbind(z.1,z.2,z.3,z.4)

stock_yogo_test(X,Z)