IV Estimators
Preparing your workfile
We add the basic libraries needed for this week’s work:
library(tidyverse) # for almost all data handling tasks
library(readxl) # to import Excel data
library(ggplot2) # to produce nice graphiscs
library(stargazer) # to produce nice results tables
library(haven) # to import stata file
library(AER) # access to HS robust standard errors
library(estimatr) # use robust se
source("stargazer_HC.r")Introduction
In this script we will introduce the use of instrumental variables estimation. This is an important and popular technique to potentially reveal causal relationships between variables where simple regression analysis fails as one has to assume that the explanatory variable of interest (here education) is endogenous in a model attempting to explain variation in wages. The data used are a classic dataset used in econometrics which you will find used in multiple econometrics textbooks.
Data Upload - and understanding data structure
Upload the data, which are saved in a STATA datafile (extension
.dta). There is a function which loads STATA file. It is
called read_dta and is supplied by the haven
package.
mroz <- read_dta("../data/mroz.dta")
mroz <- as.data.frame(mroz) # ensure data frame structure
names(mroz)## [1] "inlf" "hours" "kidslt6" "kidsge6" "age" "educ"
## [7] "wage" "repwage" "hushrs" "husage" "huseduc" "huswage"
## [13] "faminc" "mtr" "motheduc" "fatheduc" "unem" "city"
## [19] "exper" "nwifeinc" "lwage" "expersq"The variables have short descriptions: 1. inlf =1 if in labor force,
1975
2. hours hours worked, 1975
3. kidslt6 # kids < 6 years
4. kidsge6 # kids 6-18
5. age woman’s age in yrs
6. educ years of schooling
7. wage estimated wage from earns., hours
8. repwage reported wage at interview in 1976
9. hushrs hours worked by husband, 1975
10. husage husband’s age
11. huseduc husband’s years of schooling
12. huswage husband’s hourly wage, 1975
13. faminc family income, 1975
14. mtr fed. marginal tax rate facing woman
15. motheduc mother’s years of schooling
16. fatheduc father’s years of schooling
17. unem unem. rate in county of resid.
18. city =1 if live in SMSA
19. exper actual labor mkt exper
20. nwifeinc (faminc - wage*hours)/1000
21. lwage log(wage)
22. expersq exper^2
A standard regression
Let’s start by running a standard regression of log wages
(lwage) as dependent variable and a respondents education
(educ) as the explanatory variable.
But before we do this we shall ensure that we remove those observations from the dataset for which we do not have a measure of wage (or log(wage)).
mroz <- mroz %>% filter(!is.na(lwage))ols <- lm(lwage~educ,data = mroz)
stargazer_HC(ols)##
## =========================================================
## Dependent variable:
## -------------------------------------
## lwage
## ---------------------------------------------------------
## educ 0.109***
## (0.013)
##
## Constant -0.185
## (0.171)
##
## ---------------------------------------------------------
## Observations 428
## R2 0.118
## Adjusted R2 0.116
## Residual Std. Error 0.680 (df = 426)
## F Statistic 56.929*** (df = 1; 426)
## =========================================================
## Note: *p<0.1; **p<0.05; ***p<0.01
## Robust standard errors in parenthesisThe IV estimator
Let’s consider a respondent’s father’s education as an instrument for education. We therefore run a first stage regression:
iv_s1 <- lm(educ~fatheduc, data = mroz)
stargazer_HC(iv_s1)##
## =========================================================
## Dependent variable:
## -------------------------------------
## educ
## ---------------------------------------------------------
## fatheduc 0.269***
## (0.029)
##
## Constant 10.237***
## (0.272)
##
## ---------------------------------------------------------
## Observations 428
## R2 0.173
## Adjusted R2 0.171
## Residual Std. Error 2.081 (df = 426)
## F Statistic 88.841*** (df = 1; 426)
## =========================================================
## Note: *p<0.1; **p<0.05; ***p<0.01
## Robust standard errors in parenthesisWhat we learn from this is that the (fatheduc) is indeed
related to the educ variable. Hence we feel justified in
using this in our IV regression. But do remember that you will have to
make an argument why fatheduc is a valid instrument, we
cannot formally show that it is unrelated to the error term.
iv <- ivreg(lwage~educ|fatheduc,data=mroz)
stargazer_HC(iv)##
## =========================================================
## Dependent variable:
## -------------------------------------
## lwage
## ---------------------------------------------------------
## educ 0.059
## (0.037)
##
## Constant 0.441
## (0.465)
##
## ---------------------------------------------------------
## Observations 428
## R2 0.093
## Adjusted R2 0.091
## Residual Std. Error 0.689 (df = 426)
## =========================================================
## Note: *p<0.1; **p<0.05; ***p<0.01
## Robust standard errors in parenthesisWe can show all three estimates in the same table (omitting the F statistic as this would make the table very wide).
stargazer_HC(ols,iv,iv_s1, omit.stat = "f")##
## ======================================================================
## Dependent variable:
## ---------------------------------------
## lwage educ
## OLS instrumental OLS
## variable
## (1) (2) (3)
## ----------------------------------------------------------------------
## educ 0.109*** 0.059
## (0.013) (0.037)
##
## fatheduc 0.269***
## (0.029)
##
## Constant -0.185 0.441 10.237***
## (0.171) (0.465) (0.272)
##
## ----------------------------------------------------------------------
## Observations 428 428 428
## R2 0.118 0.093 0.173
## Adjusted R2 0.116 0.091 0.171
## Residual Std. Error (df = 426) 0.680 0.689 2.081
## ======================================================================
## Note: *p<0.1; **p<0.05; ***p<0.01
## Robust standard errors in parenthesisClearly the estimates for the educ variable are
substantially different when comparing ols and
iv. We really only want to revert to the iv
model if there is evidence that the educ variable is indeed
endogenous. The standard test applied in this context is the Wu-Hausmann
test of endogeneity (H0: educ is exogenous). The easiest
way to obtain this is to call
summary(iv,diagnostics = TRUE) where iv is the
name we have given our IV regresison output:
summary(iv, diagnostics = TRUE)##
## Call:
## ivreg(formula = lwage ~ educ | fatheduc, data = mroz)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.0870 -0.3393 0.0525 0.4042 2.0677
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.44110 0.44610 0.989 0.3233
## educ 0.05917 0.03514 1.684 0.0929 .
##
## Diagnostic tests:
## df1 df2 statistic p-value
## Weak instruments 1 426 88.84 <2e-16 ***
## Wu-Hausman 1 425 2.47 0.117
## Sargan 0 NA NA NA
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.6894 on 426 degrees of freedom
## Multiple R-Squared: 0.09344, Adjusted R-squared: 0.09131
## Wald test: 2.835 on 1 and 426 DF, p-value: 0.09294Note that from here you can read that the p-value for the Wu-Hausmann
test is 0.117. So, for instance, at a 5% significance level we would not
reject the null hypothesis that educ is actually
exogenous.
Implications of the different estimators
Recall that the estimated coefficients are merely one draw from an underlying random distribution. The sampling distributions (i.e. our sample estimates of these unknown distributions) are shown in the following graph. The distributions for both are normal distributions where the mean is equal to the respective sample estimate and the sd, is taken from the regression outputs.
#pdf("Lecture8plot_R.pdf",width = 5.5, height = 4) # uncomment to save as pdf
x <- seq(-0.02, 0.16, length=1000)
y_ols <- dnorm(x, mean=0.1086, sd=0.0134)
y_iv <- dnorm(x, mean = 0.0592, sd = 0.0369)
plot(x, y_ols, type="l", col="blue", lwd=2, axes = FALSE,
ylab = "Sampling Distribution", main = "OLS and IV estimator")
lines(x,y_iv,col="green", lwd = 2)
axis(side = 1, at = c(-0.01,0.06,0.08,0.11,0.13,0.14))
#dev.off() # uncomment to save as pdfYou can tell from these that the OLS estimate and its implied distribution suggests that a value of 0 is very unlikely whereas the sampling distribution of the IV estimator does associate significant probability to values of 0 or smaller.