Modeling the Effect of Unobservable Variables

• Many social science concepts are difficult or impossible to measure

  • norms, rule of law, state capacity, resolve, costs of voting

• Two distinct approaches:

  • Can use a proxy variable in lieu of the actual variable of interest

  • Can use an instrument variable to help isolate the true effect of our variable (when our variable is endogenous, i. e. correlated with an unobserved variable that also affects y)

Unobserved Explanatory Variables

Consider the following model of $log\left(wage\right)$:

$$log\left(wage\right)={\beta }_0+{\beta }_{1}educ+{\beta }_{2}exper+{\beta }_{3}abil+u$$

• Must account for ability, which is an unobserved variable.

• Since ability is likely correlated with education and experience, excluding it will lead to biased estimates of ${\beta }_1$ and ${\beta }_2$.

• It may be possible to avoid omitted variable bias by using a proxy variable.

• A proxy variable is a variable that is correlated with the unobserved variable (e.g. IQ is a proxy for ability)

Assumptions

• The proxy variable does not affect $y$, it has a correlation with $y$ solely due to its correlation with the unobserved variable for which it is a proxy for.

  • In the above example, IQ only affect wage because IQ is a proxy for ability. IQ does not have its own (separate from its correlation with ability) effect on wage.

• The proxy variable captures all the correlation between the variable it stands for and other variables that affect y.

  • In the above example, ability is only correlated with experience and education due to IQ. Once IQ is accounted for, ability is not correlated with experience or education.

Your Turn

Discuss whether the assumptions from the previous slide hold for each of these scenarios:

1. Suppose we want to model trade between pairs of countries as a function of each country's GDP and population, as well as the distance between them. The problem is that GDP data are not available prior to 1950, so we decided to use country's energy consumption as a proxy variable.

2. Suppose we want to model an individual's decision to vote in an election as a function of their income, education, age, and the opportunity cost of time. Unfortunately, we do not have a good measure of the opportunity cost of time, so we proxy it by the quality of Monday night football game the night before the election.

Assigned Reading

• Matthew Potoski and R Urbatsch. 2017. "Entertainment and the opportunity cost of civic participation: Monday night football game quality suppresses turnout in US elections." The Journal of Politics, 79(2):424--438.

• Emily Hencken Ritter and Courtenay R Conrad. 2016. "Preventing and responding to dissent: The observational challenges of explaining strategic repression." American Political Science Review, 110(1):85--99.

Potoski and Urbatsch (2017)

• Why do people vote? What are the costs and benefits of voting?

• How do Monday Night Football games alter the costs and benefits of voting? What is the causal mechanism?

• How do the authors measure the quality of the game?

• Why does it matter if a local team plays in a pre-election game? How do the authors measure this? Why do the authors say that it is a noisy measure?

Potoski and Urbatsch (2017)

• How does interest in politics interact with the opportunity costs of voting? Why? How do the authors measure interest in politics? Why do they need to use several alternative measures?

• How does the start time of the game affect turnout?

• What are the placebo tests that the authors use? Why do they need these? What do they show?

MNF as a Proxy Variable

$$Turnout={\beta }_0+{\beta }_{1}Benefit+{\beta }_{2}Cost+u,$$ where ${\beta }_0$, ${\beta }_1$, and ${\beta }_2$ are population parameters, and $u$ is the random error.

• MNF game quality is a proxy for cost if two assumptions are met:

  • MNF game quality only affects voting in as much as it increases the opportunity cost of time.

  • Opportunity cost of time is uncorrelated with other variables that are included in the model, once we account for the quality of the pre-election game.

Ritter and Conrad (2016)

• What do the authors mean when they say that the relationship between repression and dissent is endogenous?

• What are the two strategic censoring processes that result in the observed outcomes of dissent and repression?

  • What happens when the government engages in preventive repression? What happens when it does not?

Ritter and Conrad (2016)

• Why is rainfall correlated with dissent?

• Does rainfall affect repression?

• What is the Law of Coercive Responsiveness?

• Why do governments repress?

• What are the alternatives to repression?

Ritter and Conrad (2016)

• How does repression prevent dissent?

• Why do some groups dissent despite the expectation of repression? Why do some groups dissent despite repression?

• If preventive repression deters some dissent, why don't governments always engage in preventive repression?

• Why do the authors use a natural log of daily rainfall rather than rainfall deviations from the norm? Why do they include an additional instrument of percent of annual rainfall the daily amount represents?

An Endogenous Independent Variable

• Suppose we want to know the effect of X on Y , but we think that Y may also, in part, determine X;

• In this case, we say that X and Y are endogenous;

• Very common in social sciences;

• Examples: Conflict and alliances, armes races, democracy, trade; supply and demand; domestic institutions (political, legal, economic) and dissent.

Why Do Governments Repress?

$$Repression={\beta }_0+{\beta }_{1}Dissent+u,$$

where ${\beta }_0$, and ${\beta }_1$ are population parameters to estimate, and $u$ is the random error.

• Problem: Dissent is correlated with an unobservable variable that affects repression, Protesters' Resolve, which means we cannot obtain an unbiased estimate of ${\beta }_1$.

• Solution: Use an instrumental variable, .

  • Because Rainfall correlates with Dissent, but does not correlate with Resolve, we can estimate the effect of Dissent on Repression in two stages.

The Two-Stage Approach:

[1] Regress Dissent on Rainfall.

$$Dissent={\delta }_0+{\delta }_{1}Rain+\nu ,$$

[2] Regress $Repression$ on the fitted values from the first equation:

$$Repression={\beta }_0+{\beta }_1\widehat{Dissent}+u$$

Assumptions

Suppose our model is:

$$y={\beta }_0+{\beta }_{1}x+u,$$ where we think that x and u are correlated, i. e. $Cov(x,u)\ne 0$.

A valid instrument for x is a variable $z$, such that:

1. $z$ is uncorrelated with $u$, or $Cov(z,u)=0$;

   • This condition is known as instrument exogeneity

   • I. e., $z$ is only correlated with $y$ because it is correlated with $x$; $z$ does not have its own partial effect on $y$ after we account for $x$ and other controls.

2. $z$ is correlated with $x$, or $Cov(x,z)\ne 0$.

   • This condition is called instrument relevance.

   • We can, and should, test this assumption by regressing $x$ on $z$.

Example

• Suppose we are interested in the effect of education on income. Why can't we simply estimate this effect using OLS?

• Load the mroz data from the wooldridge package.

library(tidyverse)
library(wooldridge)
data(mroz) 
mydata<-mroz %>% filter(!is.na(lwage))
summary(m1<-lm(lwage~educ, data=mydata))

## 
## Call:
## lm(formula = lwage ~ educ, data = mydata)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -3.10256 -0.31473  0.06434  0.40081  2.10029 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  -0.1852     0.1852  -1.000    0.318    
## educ          0.1086     0.0144   7.545 2.76e-13 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.68 on 426 degrees of freedom
## Multiple R-squared:  0.1179,    Adjusted R-squared:  0.1158 
## F-statistic: 56.93 on 1 and 426 DF,  p-value: 2.761e-13

Using Parents' Education as an Instrument

Regress educ on fatheduc to check the instrument relevance.

summary(m2<-lm(educ~fatheduc, data=mydata))

## 
## Call:
## lm(formula = educ ~ fatheduc, data = mydata)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -8.4704 -1.1231 -0.1231  0.9546  5.9546 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 10.23705    0.27594  37.099   <2e-16 ***
## fatheduc     0.26944    0.02859   9.426   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.081 on 426 degrees of freedom
## Multiple R-squared:  0.1726,    Adjusted R-squared:  0.1706 
## F-statistic: 88.84 on 1 and 426 DF,  p-value: < 2.2e-16

Implement the Two-Stage Approach

• What do you conclude?

summary(m3<-lm(lwage~m2$fitted.values, data=mydata))

## 
## Call:
## lm(formula = lwage ~ m2$fitted.values, data = mydata)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.2126 -0.3763  0.0563  0.4173  2.0604 
## 
## Coefficients:
##                  Estimate Std. Error t value Pr(>|t|)
## (Intercept)       0.44110    0.46711   0.944    0.346
## m2$fitted.values  0.05917    0.03680   1.608    0.109
## 
## Residual standard error: 0.7219 on 426 degrees of freedom
## Multiple R-squared:  0.006034,    Adjusted R-squared:  0.003701 
## F-statistic: 2.586 on 1 and 426 DF,  p-value: 0.1086

Your Turn

• Evaluate the validity of each of the following instruments:

  • Rain as an instrument of election turnout if the DV is Democrats' vote share in a US election;

  • Rain as an instrument of protests if the DV is government repression;

  • Mountains as an instrument for governmemt's ability to detect and track rebel groups if the DV is civil war.

  • IQ as an instrument for education if the DV is wage.

  • Distance from a four-year college as an instrument for education if the DV is wage.

  • Sibling's education as an instrument for education if the DV is wage.

  • Cloud coverage as an instrument for drone strikes if the DV is civilian casualties.