16 - Instrumental Variable Analysis

This notebook introduces instrumental variable analysis. We look the conditions that must be satisfied to perform an IV analysis, how the two-stage-least-squares approach works, and how to interpret the results.

Prerequisites

1. Run OLS regressions.

Learning Outcomes

1. Understand what an instrumental variable is and the conditions that must be satisfied to address the endogeneity problem.
2. Implement a Two Stage Least Squares (2SLS) regression-based approach using an instrument.
3. Describe the weak instrument problem.
4. Interpret the first stage test of whether or not the instrument is weak.

16.1 The Linear Instrumental Variable Model

Consider a case where we want to know the effect of education on earnings. We may want to estimate a model like the following:

\[ Y_{i} = \alpha + \beta X_i + \epsilon_i, \]

where \(Y_i\) is earnings of individual \(i\) and \(X_i\) is years of education of individual \(i\).

A possible issue with this model comes from omitted variable bias: it is possible that the decision to attend school is influenced by other individual characteristics that are also correlated with earnings. For example, think of individuals with high innate ability. They may want to enroll in school for longer and obtain higher-level degrees. Moreover, their employers may compensate them for their high ability, regardless of their years of schooling.

Instrumental variables (IVs) can help us when there are hidden factors affecting both the treatment (in our case, years of education) and the outcome (in our case, earnings). The instrumental variable approach relies on finding something that affects the treatment and affects the outcome, but that affects the outcome solely through the treatment. In short, the instrument must satisfy two assumptions:

1. Relevance: the instrument should be correlated with the explanatory variable; in our case, it should be correlated with the years of education \(X_i\);
2. Exclusion restriction: the instrument should be correlated with the dependent variable only through the explanatory variable; in our case, it should be correlated with \(Y_i\) only through its correlation with \(X_i\).

Let’s say we have found an instrumental variable \(Z_i\) for the variable \(X_i\). Then, using an IV analyis implies estimating the following model: \[ \begin{align} Y_i &= \alpha_1 + \beta \hat{X_i} + u_i \quad \text{(Structural Equation)}\\ X_i &= \alpha_2 + \gamma Z_i + e_i \quad \text{(First Stage Equation)} \end{align} \]

where the two conditions we have seen above imply that:

1. \(\gamma \neq 0\);
2. \(Z_i\) is uncorrelated with \(u_i\).

In practice, using an IV analysis often implies using a Two-Stages Least Square (2SLS) estimator. The two steps of 2SLS are:

1. Estimate the first stage equation by OLS and obtain the predicted value of \(X_i\). In this way, we have effectively split \(X_i\) into \[ X_i = \underbrace{\hat{X}_i}_\text{exogenous part} + \underbrace{\hat{e}_i}_\text{endogenous part} \]

where \(\hat{X_i} \equiv \hat{\alpha_2} + \hat{\gamma} {Z}_{i}\).

2. Plug \(\hat{X_i}\) instead of \(X_i\) into the structural equation and estimate via OLS. We are then using the “exogenous” part of \(X_i\) to capture \(\beta\).

Warning: We can run 2SLS following the steps above, but when we want to do inference we need to be sure we’re using the true residuals in the structural equation \(\hat{u}_i\). The built-in R command ivreg automatically give us the right residuals.

Let’s see how to estimate this in R. Once again, we can use our fictional data set simulating wages of workers in the years 1982-2012 in a fictional country.

# Load in our packages
library(tidyverse)
library(haven)

# Import dataset
fake_data <- read_dta("../econ490-stata/fake_data.dta")

In R, we can perform IV analysis with a 2SLS estimator by using the command ivreg. This command is a part of the package AER which we will have to install.
The syntax is

ivreg(<Y> ~ <X> | <Z>, data=<data>)

where instead of <Y>, <X>, and <Z>, we write the names of the corresponding dependent, independent, and instrument variables of our model, and instead of <data> we write the name of our data frame.

We now have to choose an IV that can work in our setting. A well-known example for an instrument for years of schooling is studied by Angrist and Krueger (1991): they propose using \(Z\), the quarter of birth. The premise behind their IV is that students are required to enter school in the year they turn 6 but not necessarily when they are already 6 years old, creating a relationship between quarter of birth and schooling. At the same time, the time of the year one is born shouldn’t affect one’s earnings aside from its effect on schooling.

Let’s see how to estimate a simple IV in R using our data and ivregr.

# Install AER package
#uncomment to install the package! install.packages("AER")
library(AER)

modelIV <- ivreg(earnings ~ schooling | quarter_birth, data = fake_data)
summary(modelIV)

We obtain a standard output: the values of the coefficients, standard errors, p-values, and 95% confidence intervals. From the regression output, years of schooling does not seem to have any effect on earnings. However, before trusting these results we should check that the two IV assumptions are met in this case.

16.2 Weak Instrument Test

While we cannot really test for the exclusion restriction, we can check whether our instrument is relevant. We do that by looking directly at the coefficients in the first stage.

In R, we need to estimate the first stage equation and then perform a simple F-test.

Recall that our first stage is \[ X_i = \alpha_2 + \gamma Z_i + e_i, \]

where \(X_i\) is years of education and \(Z_i\) is quarter of birth.

We simply have to estimate this regression with OLS. We have seen how to do it using the function lm.

first_stage <- lm(schooling ~ quarter_birth, data = fake_data)
summary(first_stage)

Among the outputs given to us by summary, we can see the F-statistic and its p-value. we can see that the IV we have chosen is not relevant for our explanatory variable \(X\). quarter_birth is not correlated with schooling. Another indicator of the lack of relevance is given by the F-statistic reported by R: as a rule of thumb, every time its value is less than 10, the instrument is not relevant.

Whenever the correlation between \(X\) and \(Z\) is very close to zero (as in our case), we say we have a weak instrument problem. In practice, this problem will result in severe finite-sample bias and large variance in our estimates. Since our instrument is not valid, we cannot trust the results we have obtained.

16.3 Wrap Up

In this module, we studied the linear IV model and how to estimate it using the 2SLS method using ivreg. We learned that we can overcome the endogeneity problem when we have access to a different type of variable: an instrumental variable. A good instrument must satisfy two important conditions:

1. It must be uncorrelated with the error term (also referred to as the exclusion restriction).
2. It must be correlated, after controlling for observables, with the variable of interest (there must be a first stage).

While the second condition can be checked using the regression results of the first stage, the first condition is inherently not testable. Therefore, any project that uses IVs must include a discussion, using contextual knowledge, of why the first condition may hold.

Finally, do not forget that for every endogenous variable in our regression, we require at least one instrument. For example, if we have a regression with two endogenous variables, we require at least two IVs!

16.4 Wrap-up Table

Command	Function
ivreg(<Y> ~ <X> | <Z>, data=<data>)	It performs Instrumental Variable analysis using a Two-Stage Least Squares estimator by default.