Instrumental Variables in Simple Regression

Instrumental Variables in Simple Regression

$$ y = \beta_0 + \beta_1 x + \varepsilon \tag{15.1} $$

The OLS estimator is:

$$ \beta^{OLS}_1 = \frac{cov(x, y)}{var(x)} $$

If the regressor $x$ is correlated with the error term $\varepsilon$, then the OLS estimator is biased.

Given a valid instrumental variable $z$, the IV estimator becomes:

$$ \beta^{VI}_1 = \frac{cov(z, y)}{cov(z,x)} $$

Implementing IV in R

Example 15.1: Returns to Education for Married Women (Wooldridge, 2019)

  • We will use the mroz dataset from the wooldridge package to estimate the following model:
$$ \log(\text{wage}) = \beta_0 + \beta_1 \text{educ} + \varepsilon $$
  • For comparison, we first estimate the model by OLS:
data(mroz, package="wooldridge") # loading the dataset
mroz = mroz[!is.na(mroz$wage),] # dropping missing wage observations

reg.ols = lm(lwage ~ educ, mroz) # OLS regression
round( summary(reg.ols)$coef, 5 )

##             Estimate Std. Error  t value Pr(>|t|)
## (Intercept) -0.18520    0.18523 -0.99984  0.31795
## educ         0.10865    0.01440  7.54513  0.00000

Using ivreg()

  • To estimate an instrumental-variables regression, we can use the ivreg() function from the AER package.
  • After specifying the endogenous regressor educ, we add the instrument on the right-hand side of |. In this example, the father’s education (fatheduc) is used as the instrument:
library(AER) # loading the package that includes ivreg

## Carregando pacotes exigidos: car

## Warning: package 'car' was built under R version 4.2.3

## Carregando pacotes exigidos: carData

## Carregando pacotes exigidos: lmtest

## Carregando pacotes exigidos: zoo

## Warning: package 'zoo' was built under R version 4.2.3

## 
## Attaching package: 'zoo'

## The following objects are masked from 'package:base':
## 
##     as.Date, as.Date.numeric

## Carregando pacotes exigidos: sandwich

## Carregando pacotes exigidos: survival

## Warning: package 'survival' was built under R version 4.2.3

reg.iv = ivreg(lwage ~ educ | fatheduc, data=mroz) # IV regression
round( summary(reg.iv)$coef, 5 )

##             Estimate Std. Error t value Pr(>|t|)
## (Intercept)  0.44110    0.44610 0.98880  0.32332
## educ         0.05917    0.03514 1.68385  0.09294
## attr(,"df")
## [1] 426
## attr(,"nobs")
## [1] 428

By-hand derivation

(1) IV estimate $$ \beta^{VI} $$
  • The simple-regression formula above is useful for intuition.
  • In practice, the multivariate IV setup is more relevant in econometrics, so the full matrix derivation is developed in the next section.

Instrumental Variables in Multiple Regression

  • Section 15.2 of Heiss (2020)
  • The next section extends the IV estimator to the multivariate case, where we combine endogenous and exogenous regressors in matrix form.

Testing Regressor Exogeneity

  • Section 15.4 of Heiss (2020)
  • Once we allow for endogeneity, an important empirical question is whether OLS and IV differ enough to justify treating a regressor as endogenous.

Testing Overidentifying Restrictions

  • Section 15.5 of Heiss (2020)
  • When the model has more instruments than endogenous regressors, we can test whether the extra instruments are jointly consistent with the exogeneity assumptions.

Two-Stage Least Squares

  • Section 15.3 of Heiss (2020)
  • Two-stage least squares (2SLS) is the standard operational form of IV estimation in multivariate models and will be the main focus of the following section.

Simultaneous Equations Models

  • Section 15.3 of Heiss (2020)
  • Simultaneous-equations settings provide a classic motivation for IV methods, since endogenous variables are determined jointly within the system.

Last updated on Sep 9, 2018