1 Introduction

1.1 Introduction

I cover three examples of instrumental variable regressions.
1. Wage regression
2. Demand curve
3. Effects of Voter Turnout (Hansford and Gomez)

2 Wage regression

2.1 Example 1: Wage regression

Use dataset “Mroz”, cross-sectional labor force participation data that accompany “Introductory Econometrics” by Wooldridge.
- Original data from “The Sensitivity of an Empirical Model of Married Women’s Hours of Work to Economic and Statistical Assumptions” by Thomas Mroz published in Econometrica in 1987.
- Detailed description of data: https://www.rdocumentation.org/packages/npsf/versions/0.4.2/topics/mroz

library("foreign")

# You might get a message "cannot read factor labels from Stata 5 files", but you do not have to worry about it. 
data <- read.dta("MROZ.DTA")

## Warning in read.dta("MROZ.DTA"): cannot read factor labels from Stata 5 files

Describe data

library(stargazer)

## 
## Please cite as:

##  Hlavac, Marek (2018). stargazer: Well-Formatted Regression and Summary Statistics Tables.

##  R package version 5.2.2. https://CRAN.R-project.org/package=stargazer

stargazer(data, type = "text")

## 
## ===================================================================
## Statistic  N     Mean     St. Dev.   Min   Pctl(25) Pctl(75)  Max  
## -------------------------------------------------------------------
## inlf      753   0.568      0.496      0       0        1       1   
## hours     753  740.576    871.314     0       0      1,516   4,950 
## kidslt6   753   0.238      0.524      0       0        0       3   
## kidsge6   753   1.353      1.320      0       0        2       8   
## age       753   42.538     8.073      30      36       49      60  
## educ      753   12.287     2.280      5       12       13      17  
## wage      428   4.178      3.310    0.128   2.263    4.971   25.000
## repwage   753   1.850      2.420    0.000   0.000    3.580   9.980 
## hushrs    753 2,267.271   595.567    175    1,928    2,553   5,010 
## husage    753   45.121     8.059      30      38       52      60  
## huseduc   753   12.491     3.021      3       11       15      17  
## huswage   753   7.482      4.231    0.412   4.788    9.167   40.509
## faminc    753 23,080.600 12,190.200 1,500   15,428   28,200  96,000
## mtr       753   0.679      0.083    0.442   0.622    0.721   0.942 
## motheduc  753   9.251      3.367      0       7        12      17  
## fatheduc  753   8.809      3.572      0       7        12      17  
## unem      753   8.624      3.115      3      7.5       11      14  
## city      753   0.643      0.480      0       0        1       1   
## exper     753   10.631     8.069      0       4        15      45  
## nwifeinc  753   20.129     11.635   -0.029  13.025   24.466  96.000
## lwage     428   1.190      0.723    -2.054  0.817    1.604   3.219 
## expersq   753  178.039    249.631     0       16      225    2,025 
## -------------------------------------------------------------------

Consider the wage regression \[ \log(w_i) = \beta_0 + \beta_1 educ_i + \beta_2 exper_i + \beta_3 exper_i^2 + \epsilon_i \]
We assume that \(exper_i\) is exogenous but \(educ_i\) is endogenous.
As an instrument for \(educ_i\), we use the years of schooling for his or her father and mother, which we call \(fathereduc_i\) and \(mothereduc_i\).
Discussion on these IVs will be later.

library("AER")

## Loading required package: car

## Loading required package: carData

## Loading required package: lmtest

## Loading required package: zoo

## 
## Attaching package: 'zoo'

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

## Loading required package: sandwich

## Loading required package: survival

library("dplyr")

## 
## Attaching package: 'dplyr'

## The following object is masked from 'package:car':
## 
##     recode

## The following objects are masked from 'package:stats':
## 
##     filter, lag

## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union

# data cleaning
data %>% 
  select(lwage, educ, exper, expersq, motheduc, fatheduc) %>%
  filter( is.na(lwage) == 0 ) -> data

# OLS regression
result_OLS <- lm( lwage ~ educ + exper + expersq, data = data)

# IV regression using fathereduc and mothereduc
result_IV <- ivreg(lwage ~ educ + exper + expersq | fatheduc + motheduc + exper + expersq, data = data)

# Robust standard errors 
# gather robust standard errors in a list
rob_se <- list(sqrt(diag(vcovHC(result_OLS, type = "HC1"))),
               sqrt(diag(vcovHC(result_IV, type = "HC1"))))

# Show result
stargazer(result_OLS, result_IV, type ="text", se = rob_se)

## 
## ===================================================================
##                                        Dependent variable:         
##                                ------------------------------------
##                                               lwage                
##                                          OLS           instrumental
##                                                          variable  
##                                          (1)               (2)     
## -------------------------------------------------------------------
## educ                                  0.107***            0.061*   
##                                        (0.013)           (0.033)   
##                                                                    
## exper                                 0.042***           0.044***  
##                                        (0.015)           (0.016)   
##                                                                    
## expersq                                -0.001*           -0.001**  
##                                       (0.0004)           (0.0004)  
##                                                                    
## Constant                              -0.522***           0.048    
##                                        (0.202)           (0.430)   
##                                                                    
## -------------------------------------------------------------------
## Observations                             428               428     
## R2                                      0.157             0.136    
## Adjusted R2                             0.151             0.130    
## Residual Std. Error (df = 424)          0.666             0.675    
## F Statistic                    26.286*** (df = 3; 424)             
## ===================================================================
## Note:                                   *p<0.1; **p<0.05; ***p<0.01

How about the first stage? You should always check this!!

# First stage regression 

result_1st <- lm(educ ~ motheduc + fatheduc + exper + expersq, data = data)

# F test
linearHypothesis(result_1st, 
                 c("fatheduc = 0", "motheduc = 0" ), 
                 vcov = vcovHC, type = "HC1")

2.2 Discussion on IV

Labor economists have used family background variables as IVs for education.
Relevance: OK from the first stage regression.
Independence: A bit suspicious. Parents’ education would be correlated with child’s ability through quality of nurturing at an early age.
Still, we can see that these IVs can mitigate (though may not eliminate completely) the omitted variable bias.
Discussion on the validity of instruments is crucial in empirical research.

3 Demand curve

3.1 Example 2: Estimation of the Demand for Cigaretts

Demand model is a building block in many branches of Economics.
For example, health economics is concerned with the study of how health-affecting behavior of individuals is influenced by the health-care system and regulation policy.
Smoking is a prominent example as it is related to many illnesses and negative externalities.
It is plausible that cigarette consumption can be reduced by taxing cigarettes more heavily.
Question: how much taxes must be increased to reach a certain reduction in cigarette consumption? -> Need to know price elasticity of demand for cigaretts.

Use CigarrettesSW in the package AER.
a panel data set that contains observations on cigarette consumption and several economic indicators for all 48 continental federal states of the U.S. from 1985 to 1995.
What is panel data? The data involves both time series and cross-sectional information.
- The variable is denoted as \(y_{it}\), which indexed by individual \(i\) and time \(t\).
- Cross section data \(y_i\): information for a particular individual \(i\) (e.g., income for person \(i\)).
- Time series data \(y_t\): information for a particular time period (e.g., GDP in year \(y\))
- Panel data \(y_{it}\): income of person \(i\) in year \(t\).
We will see more on panel data later in this course. For now, we use the panel data as just cross-sectional data (pooled cross-sections)

# load the data set and get an overview
library(AER)
data("CigarettesSW")
summary(CigarettesSW)

##      state      year         cpi          population           packs       
##  AL     : 2   1985:48   Min.   :1.076   Min.   :  478447   Min.   : 49.27  
##  AR     : 2   1995:48   1st Qu.:1.076   1st Qu.: 1622606   1st Qu.: 92.45  
##  AZ     : 2             Median :1.300   Median : 3697472   Median :110.16  
##  CA     : 2             Mean   :1.300   Mean   : 5168866   Mean   :109.18  
##  CO     : 2             3rd Qu.:1.524   3rd Qu.: 5901500   3rd Qu.:123.52  
##  CT     : 2             Max.   :1.524   Max.   :31493524   Max.   :197.99  
##  (Other):84                                                                
##      income               tax            price             taxs       
##  Min.   :  6887097   Min.   :18.00   Min.   : 84.97   Min.   : 21.27  
##  1st Qu.: 25520384   1st Qu.:31.00   1st Qu.:102.71   1st Qu.: 34.77  
##  Median : 61661644   Median :37.00   Median :137.72   Median : 41.05  
##  Mean   : 99878736   Mean   :42.68   Mean   :143.45   Mean   : 48.33  
##  3rd Qu.:127313964   3rd Qu.:50.88   3rd Qu.:176.15   3rd Qu.: 59.48  
##  Max.   :771470144   Max.   :99.00   Max.   :240.85   Max.   :112.63  
##

Consider the following model \[ \log (Q_{it}) = \beta_0 + \beta_1 \log (P_{it}) + \beta_2 \log(income_{it}) + u_{it} \] where
- \(Q_{it}\) is the number of packs per capita in state \(i\) in year \(t\),
- \(P_{it}\) is the after-tax average real price per pack of cigarettes, and
- \(income_{it}\) is the real income per capita. This is demand shifter.
As an IV for the price, we use the followings:
- \(SalesTax_{it}\): the proportion of taxes on cigarettes arising from the general sales tax.
  - Relevant as it is included in the after-tax price
  - Exogenous(indepndent) since the sales tax does not influence demand directly, but indirectly through the price.
- \(CigTax_{it}\): the cigarett-specific taxes

library(dplyr)
CigarettesSW %>% 
  mutate( rincome = (income / population) / cpi) %>% 
  mutate( rprice  = price / cpi ) %>% 
  mutate( salestax = (taxs - tax) / cpi ) %>% 
  mutate( cigtax = tax/cpi ) -> Cigdata

Let’s run the regressions

cig_ols <- lm(log(packs) ~ log(rprice) + log(rincome) ,  data = Cigdata)
#coeftest(cig_ols, vcov = vcovHC, type = "HC1")


cig_ivreg <- ivreg(log(packs) ~ log(rprice) + log(rincome)  | 
                    log(rincome) +  salestax +  cigtax, data = Cigdata)
#coeftest(cig_ivreg, vcov = vcovHC, type = "HC1")

# Robust standard errors 
rob_se <- list(sqrt(diag(vcovHC(cig_ols, type = "HC1"))),
               sqrt(diag(vcovHC(cig_ivreg, type = "HC1"))))

# Show result
stargazer(cig_ols, cig_ivreg, type ="text", se = rob_se)

## 
## =================================================================
##                                       Dependent variable:        
##                               -----------------------------------
##                                           log(packs)             
##                                        OLS           instrumental
##                                                        variable  
##                                        (1)               (2)     
## -----------------------------------------------------------------
## log(rprice)                         -1.334***         -1.229***  
##                                      (0.154)           (0.155)   
##                                                                  
## log(rincome)                         0.318**            0.257*   
##                                      (0.154)           (0.153)   
##                                                                  
## Constant                            10.067***          9.736***  
##                                      (0.502)           (0.514)   
##                                                                  
## -----------------------------------------------------------------
## Observations                            96                96     
## R2                                    0.552             0.549    
## Adjusted R2                           0.542             0.539    
## Residual Std. Error (df = 93)         0.165             0.165    
## F Statistic                   57.185*** (df = 2; 93)             
## =================================================================
## Note:                                 *p<0.1; **p<0.05; ***p<0.01

The first stage regression

# First stage regression 

result_1st <- lm(log(rprice) ~ log(rincome) + log(rincome) + salestax + cigtax , data= Cigdata)

# F test
linearHypothesis(result_1st, 
                 c("salestax = 0", "cigtax = 0" ), 
                 vcov = vcovHC, type = "HC1")

4 Voting

4.1 Example 3: Effects of Turnout on Partisan Voting

THOMAS G. HANSFORD and BRAD T. GOMEZ “Estimating the Electoral Effects of Voter Turnout” The American Political Science Review Vol. 104, No. 2 (May 2010), pp. 268-288
- Link: https://www.cambridge.org/core/journals/american-political-science-review/article/estimating-the-electoral-effects-of-voter-turnout/8A880C28E79BE770A5CA1A9BB6CF933C
Here, we will see a simplified version of their analysis.
The dataset is here

library(readr)

HGdata <- read_csv("HansfordGomez_Data.csv")

## Parsed with column specification:
## cols(
##   .default = col_double()
## )

## See spec(...) for full column specifications.

stargazer::stargazer(as.data.frame(HGdata), type="text")

## 
## =========================================================================================
## Statistic           N       Mean       St. Dev.      Min   Pctl(25)  Pctl(75)     Max    
## -----------------------------------------------------------------------------------------
## Year              27,401  1,973.972     16.111      1,948    1,960     1,988     2,000   
## FIPS_County       27,401 29,985.500   13,081.250    4,001   20,013    39,157     56,045  
## Turnout           27,401   65.562       10.514     20.366   58.477    72.613    100.000  
## Closing2          27,401   23.053       13.042        0       11        30        125    
## Literacy          27,401    0.058        0.234        0        0         0         1     
## PollTax           27,401    0.001        0.023        0        0         0         1     
## Motor             27,401    0.211        0.408        0        0         0         1     
## GubElection       27,401    0.434        0.496        0        0         1         1     
## SenElection       27,401    0.680        0.467        0        0         1         1     
## GOP_Inc           27,401    0.501        0.500        0        0         1         1     
## Yr52              27,401    0.071        0.258        0        0         0         1     
## Yr56              27,401    0.071        0.258        0        0         0         1     
## Yr60              27,401    0.071        0.258        0        0         0         1     
## Yr64              27,401    0.071        0.258        0        0         0         1     
## Yr68              27,401    0.071        0.258        0        0         0         1     
## Yr72              27,401    0.071        0.258        0        0         0         1     
## Yr76              27,401    0.071        0.258        0        0         0         1     
## Yr80              27,401    0.071        0.258        0        0         0         1     
## Yr84              27,401    0.072        0.258        0        0         0         1     
## Yr88              27,401    0.072        0.258        0        0         0         1     
## Yr92              27,401    0.072        0.258        0        0         0         1     
## Yr96              27,401    0.072        0.258        0        0         0         1     
## Yr2000            27,401    0.070        0.256        0        0         0         1     
## DNormPrcp_KRIG    27,401    0.005        0.208     -0.419   -0.093     0.001     2.627   
## GOPIT             27,401   33.282       34.066        0        0       66.3       100    
## DemVoteShare2_3MA 27,401   44.250       10.606     10.145   37.006    50.996     88.982  
## DemVoteShare2     27,401   43.622       12.415      6.420   34.954    51.858     97.669  
## RainGOPI          27,401    0.007        0.142       -0      -0.03       0         2     
## TO_DVS23MA        27,401  2,886.877     792.530    473.161 2,321.025 3,384.772 8,526.616 
## Rain_DVS23MA      27,401    0.355       10.188     -25.054  -4.019     0.028    144.257  
## dph               27,401    0.021        0.145        0        0         0         1     
## dvph              27,401    0.018        0.133        0        0         0         1     
## rph               27,401    0.025        0.155        0        0         0         1     
## rvph              27,401    0.025        0.155        0        0         0         1     
## state_del         27,401    0.037        0.187     -0.821   -0.090     0.172     0.619   
## dph_StateVAP      27,401 77,525.150   597,474.000     0        0         0     6,150,988 
## dvph_StateVAP     27,401 63,138.400   663,707.600     0        0         0     12,700,000
## rph_StateVAP      27,401 243,707.900 1,720,659.000    0        0         0     18,300,000
## rvph_StateVAP     27,401 142,166.500 1,071,445.000    0        0         0     12,800,000
## State_DVS_lag     27,401   46.896        8.317     22.035   40.767    52.197     80.872  
## State_DVS_lag2    27,401  2,268.381     786.199    485.533 1,661.934 2,724.515 6,540.244 
## -----------------------------------------------------------------------------------------

Data description:

Name	Description
Year	Election Year
FIPS_County	FIPS County Code
Turnout	Turnout as Pcnt VAP
Closing2	Days between registration closing date and election
Literacy	Literacy Test
PollTax	Poll Tax
Motor	Motor Voter
GubElection	Gubernatorial Election in State
SenElection	U.S. Senate Election in State
GOP_Inc	Republican Incumbent
Yr52	1952 Dummy
Yr56	1956 Dummy
Yr60	1960 Dummy
Yr64	1964 Dummy
Yr68	1968 Dummy
Yr72	1972 Dummy
Yr76	1976 Dummy
Yr80	1980 Dummy
Yr84	1984 Dummy
Yr88	1988 Dummy
Yr92	1992 Dummy
Yr96	1996 Dummy
Yr2000	2000 Dummy
DNormPrcp_KRIG	Election day rainfall - differenced from normal rain for the day
GOPIT	Turnout x Republican Incumbent
DemVoteShare2_3MA	Partisan composition measure = 3 election moving avg. of Dem Vote Share
DemVoteShare2	Democratic Pres Candidate’s Vote Share
RainGOPI	Rainfall measure x Republican Incumbent
TO_DVS23MA	Turnout x Partisan Composition measure
Rain_DVS23MA	Rainfall measure x Partisan composition measure
dph	=1 if home state of Dem pres candidate
dvph	=1 if home state of Dem vice pres candidate
rph	=1 if home state of Rep pres candidate
rvph	=1 if home state of Rep vice pres candidate
state_del	avg common space score for the House delegation
dph_StateVAP	= dph*State voting age population
dvph_StateVAP	= dvph*State voting age population
rph_StateVAP	= rph*State voting age population
rvph_StateVAP	= rvph*State voting age population
State_DVS_lag	State-wide Dem vote share, lagged one election
State_DVS_lag2	State_DVS_lag squared

Consider the following regression \[ DemoShare_{it} = \beta_0 + \beta_1 Turnout_{it} + u_t + u_{it} \] where
- \(Demoshare_{it}\): Two-party vote share for Democrat candidate in county \(i\) in the presidential election in year \(t\)
- \(Turnout_{it}\): Turnout rate in county \(i\) in the presidential election in year \(t\)
- \(u_t\): Year fixed effects. Time dummies for each presidential election year
As an IV, we use the rainfall measure denoted by DNormPrcp_KRIG

# You can do this, but it is tedious.
hg_ols <- lm( DemVoteShare2 ~ Turnout + Yr52 + Yr56 + Yr60 + Yr64 + Yr68 + 
                Yr72 + Yr76 + Yr80 + Yr84 + Yr88 + Yr92 + Yr96 + Yr2000,  data = HGdata)
#coeftest(hg_ols, vcov = vcovHC, type = "HC1")

# By using "factor(Year)" as an explanatory variable, the regression automatically incorporates the dummies for each value.
hg_ols <- lm( DemVoteShare2 ~ Turnout + factor(Year)   ,  data = HGdata)
#coeftest(hg_ols, vcov = vcovHC, type = "HC1")

# Iv regression
hg_ivreg <- ivreg( DemVoteShare2 ~ Turnout + factor(Year) | 
                    factor(Year) + DNormPrcp_KRIG, data = HGdata)
#coeftest(hg_ivreg, vcov = vcovHC, type = "HC1")

# Robust standard errors 
rob_se <- list(sqrt(diag(vcovHC(hg_ols, type = "HC1"))),
               sqrt(diag(vcovHC(hg_ivreg, type = "HC1"))))

# Show result
stargazer(hg_ols, hg_ivreg, type ="text", se = rob_se)

## 
## =========================================================================
##                                            Dependent variable:           
##                                  ----------------------------------------
##                                               DemVoteShare2              
##                                              OLS             instrumental
##                                                                variable  
##                                              (1)                 (2)     
## -------------------------------------------------------------------------
## Turnout                                   -0.157***            0.363**   
##                                            (0.008)             (0.178)   
##                                                                          
## factor(Year)1952                         -10.215***           -15.832*** 
##                                            (0.374)             (1.987)   
##                                                                          
## factor(Year)1956                          -8.756***           -13.656*** 
##                                            (0.357)             (1.746)   
##                                                                          
## factor(Year)1960                          -3.862***           -11.094*** 
##                                            (0.351)             (2.531)   
##                                                                          
## factor(Year)1964                          10.851***            6.837***  
##                                            (0.337)             (1.441)   
##                                                                          
## factor(Year)1968                          -6.477***           -8.514***  
##                                            (0.356)             (0.811)   
##                                                                          
## factor(Year)1972                         -13.749***           -16.473*** 
##                                            (0.335)             (1.022)   
##                                                                          
## factor(Year)1976                           -0.367             -2.111***  
##                                            (0.337)             (0.715)   
##                                                                          
## factor(Year)1980                         -10.346***           -11.696*** 
##                                            (0.356)             (0.620)   
##                                                                          
## factor(Year)1984                         -13.134***           -13.515*** 
##                                            (0.352)             (0.404)   
##                                                                          
## factor(Year)1988                          -5.712***           -4.951***  
##                                            (0.349)             (0.447)   
##                                                                          
## factor(Year)1992                           -0.327              -1.008**  
##                                            (0.362)             (0.469)   
##                                                                          
## factor(Year)1996                          -1.193***             0.811    
##                                            (0.377)             (0.782)   
##                                                                          
## factor(Year)2000                          -9.013***           -8.130***  
##                                            (0.386)             (0.465)   
##                                                                          
## Constant                                  59.085***            26.910**  
##                                            (0.560)             (11.024)  
##                                                                          
## -------------------------------------------------------------------------
## Observations                               27,401               27,401   
## R2                                          0.281               0.130    
## Adjusted R2                                 0.280               0.130    
## Residual Std. Error (df = 27386)           10.533               11.582   
## F Statistic                      763.153*** (df = 14; 27386)             
## =========================================================================
## Note:                                         *p<0.1; **p<0.05; ***p<0.01

# First stage regression 
hg_1st <- lm(Turnout ~  factor(Year) + DNormPrcp_KRIG, data= HGdata)

# F test
linearHypothesis(hg_1st, 
                 c("DNormPrcp_KRIG = 0" ), 
                 vcov = vcovHC, type = "HC1")

Instrumental Variable Estimation 2: Implementation in R

Instructor: Yuta Toyama

Last updated: 2020-04-14

1 Introduction

1.1 Introduction

2 Wage regression

2.1 Example 1: Wage regression

2.2 Discussion on IV

3 Demand curve

3.1 Example 2: Estimation of the Demand for Cigaretts

4 Voting

4.1 Example 3: Effects of Turnout on Partisan Voting