1.1 The statistical model

We can express this in formula, where we first predict X1 using Z:

\[\begin{equation} \text{X}_1 = \text{b}_0 + \text{b}_1 \text{Z}_1 + \text{e}_x \tag{1.3} \end{equation}\]

And then use the predicted value of X1 (the A section in figure 1.5) to predict Y. In this way we remove from the model the correlation between X1 and X2 (which is part of error term \(\text{e}_x\))

\[\begin{equation} \text{Y}_1 = \text{b}_0 + \text{b}_1 \text{X}_1 + \text{e}_y \tag{1.4} \end{equation}\]

2.1 Data

Data contain the following variables:

Our dependent variable represents an individual’s hourly wage in a given year in the United States. Our policy variable is the number of years that individuals have spent in school. We can look at some key summary statistics.

stargazer( dat,  
           type = "html", 
           omit = c("ability", "experience", "fathereduc"),
           omit.summary.stat = c("p25", "p75"), 
           covariate.labels = c("Wage", "Education"),
           digits = 2 )

When estimating the relationship between wage and education, scholars have encountered issues in dealing with the so-called “ability bias”: individuals who have higher ability are more likely both to stay longer in school and get a more highly paid job. We can represent these relationships in this diagram:

Figure 2.1: Ability bias

Ability, however, is difficult to measure and cannot be included in the model. It is an omitted variable, which is correlated with both our outcome variable and our policy variable. Excluding “ability” from the analysis creates a bias in the estimation of \(\beta_1\).

We can observe this bias with our simulated data. Let’s assume we can create a variable measuring ability and include it in our model (full model) based on equation (1.1) such that:

Let’s then run a more realistic model (naive model) based on equation (1.2) where the ability variable is excluded:

We can now compare the different estimates:

full.model  <- lm( wage ~ educ + ability + experience ) # Full Model if we were able to measure ability
naive.model <- lm( wage ~ educ + experience )      # Naive Model where ability is excluded (omitted variable bias)

stargazer( full.model, naive.model, 
           type = "html", 
           dep.var.labels = ("Wage"),
           column.labels = c("Full Model", "Naive Model"),
           covariate.labels = c("Education", "Ability", "Experience"),
           omit.stat = "all", 
           digits = 2 )

We note that the coefficient of the education variable is significantly different in the naive and the full model, where the naive model overestimate the effect of education on wage.

2.2 Analysis

In order to avoid the omitted variable bias we can introduce an instrumental variable. As mentioned above, the instrumental variable has these key characterististics:

1. It is correlated with the policy variable.
2. It is correlated to the dependent variable only through the policy variable.
3. It is not correlated with the omitted variable.

While point #1 and #2 can be empirically tested, point #3 needs to be argued by theory.

We based our example on previous studies which utilize an individual’s father’s education as an instrumental variable ( Blackburn & Newmark, 1993; Hoogerheide et al. 2012 ). Individuals who come from more educated families are more likely to stay in school longer, especially when fathers are more educatedbut one’s father’s education has little to no direct effect on hourly wage.

We can empirically test these hypotheses by looking at correlations across the variables in our dataset. In the first graph, we can see that father’s education is positively correlated with education. In the second graph, there is a slight correlation between a father’s education and hourly wage. We would expect this, because education and hourly wage are correlated. However, we theoretically argue that there is no direct correlation: a father’s education increases the hourly wage only because it increases one’s education, which in turn leads to a higher salary. Finally, in the third graph, we can see that there is no correlation between father’s education and ability.

par( mfrow=c(1,3) )

plot(fathereduc, educ, xlab = "Father's education", ylab = "Education" )   # Plot

abline( lm( educ ~ fathereduc ), col = "red")                              # regression line 

plot( fathereduc, wage, xlab = "Father's education", ylab = "Hourly wage" )

abline(lm( wage ~ fathereduc ), col = "green") 

plot( fathereduc, ability, xlab = "Father's education", ylab = "Ability")

abline(lm( ability ~ fathereduc ), col = "blue")

We now run the model using our instrumental variable. The model is defined in two stages: first, we estimate our policy variable based on our instrumental variable (equation (1.3)):

First stage:

first.stage  <- lm( educ ~ fathereduc + experience)        #Run the first stage predict the education variable

stargazer( first.stage,
           type = "html", 
           dep.var.labels = ("Education"),
           column.labels = c("First stage"),
           omit.stat = c("adj.rsq", "ser"), 
           covariate.labels = c("Father's education", "Experience"),
           digits = 2 )

Then, we estimate our dependent variable (Wage) using the predicted values of education estimated in the first model (equation (1.4)):

Second stage:

x1_hat       <- fitted( first.stage )   
#Saved the predicted values of education from the first stage

second.stage <- lm( wage ~ x1_hat + experience )   
#Run the 2nd stage using the predicted values of education to look at the effect on income.


stargazer( full.model, naive.model, second.stage,
           type = "html", 
           dep.var.labels = ("Wage"),
           column.labels = c("Full Model", "Naive Model", "IV model"),
           omit.stat = c("adj.rsq", "ser"), 
           covariate.labels = c("Education", "Ability", "Predicted Education", "Experience"),
           digits = 2 )

Note that the table report the “true”" model (full model) where we include all variables, the naive model where we omit the ability variable, and the instrumental variable model where we use the predicted values of the first stage. The IV model has almost completely recovered the true slope.

2.3 Additional explanation: plotting the results

Using our simulated data, we can have a look at what is happening when we estimate the model with an instrument variable.

We know that omitted variable bias results from the correlation of our policy variable (Education) and the omitted variable (Ability).

plot(ability, educ, xlab = "Ability", ylab = "Education")

abline(lm( educ ~ ability ), col = "green")

Figure 2.2: Relationship between Y and the omitted variable

The instrumental variable, however, is correlated only with the policy variable, not the omitted variable. Therefore, we can use the instrumental variable to partition the variance of education into a component that is highly correlated with ability (the residual), and a component that is uncorrelated with ability (X1-hat). Remember the figure 1.5 we discussed at the beginning of the lecture:

Figure 2.3: Instrumental variable model

To see this in practice, we can plot the predicted values (x1_hat) and the residual (e_x1) of the first stage in two graphs and look at the correlation with the omitted variable ability. You can see that there is almost no correlation between h1_hat and ability, but ability is highly correlated with the residual (e_x1).

first.stage <- lm( educ ~ fathereduc + experience )          #Run again the first stage to predict education (x1)

x1_hat      <- fitted( first.stage )      #Save the predicted values of education (x1_hat)

e_x1        <- residuals( first.stage )   #Save residuals of education (e_x1)

par( mfrow=c(1,2) )

#Plot the predicted values against ability
plot( ability, x1_hat, xlab = "Ability (omitted variable)", ylab = "Predicted values of Education (x1_hat)")

abline( lm( x1_hat ~ ability), col = "blue")

#Plot the residual against ability
plot( ability, e_x1, xlab = "Ability (omitted variable)", ylab = "Education - Residual (e_X1)" )

abline( lm( e_x1 ~ ability), col = "green")

Figure 2.4: Partitioning the variance of Education

One interesting thing about the instrumental variable model is that Education will be more correlated with Wage than X1-hat, but X1-hat does a better job of recovering the true slope B1 from the full model.

par( mfrow=c(1,2) )

plot( educ, wage, xlab = "Education", ylab = "Wage" )

abline( lm( wage ~ educ), col = "red")

plot( x1_hat, wage, xlab = "Predicted values of Education (x1_hat)", ylab = "Wage" )

abline( lm( wage ~ x1_hat), col = "blue")

Figure 2.5: Relationship between the ‘parts’ of education and Y

2.4 Summary: How to select a valid instrumental variable

Some things you should keep in mind about instrumental variables and how to select a valid one.

1. An instrumental variable should be strongly correlated with your policy variable; the higher the correlation, the better. You can empirically test how ‘good’ the correlation is, when running the first stage. The rule of thumb is that the first stage should have an F Statistic of at least 10 in the first stage model to avoid the weak instrument problem.`

first.stage <- lm( educ ~ fathereduc + ability )

stargazer( first.stage,
           column.labels = c("First Stage"),
           type="html",
           covariate.labels = c("Father's education", "Experience"),
           omit.stat = c("rsq","ser"), 
           digits = 2 )

If your instrumental variable is weak, it means that it is little correlated with the policy variable, so it does a “bad” job in approximating its effect on the policy outcome. You might find no policy effect even if there is one! You need to have strong evidence that the instrumental variable does a good job in approximating the policy variable (e.g., the F test) before concluding that the policy has no effect.

2. Because the instrumental variable is correlated with the policy variable and the policy variable is correlated with the outcome variable, the instrumental variable is also correlated with the outcome. See for instance the correlation between father’s education and wage.

plot( fathereduc, wage, xlab = "Father's education", ylab = "Hourly wage")
abline(lm( wage ~ fathereduc ), col = "green") 

Figure 2.6: Correlation between Y and the omitted variable

However, it is important to be able to argue that such correlation occurs only through the policy variable and there is no direct correlation between the instrumental and the outcome variable.

Figure 2.7: Instrumental variable model, path diagram

In our previous example, a father’s education increases the hourly wage only if it makes an individual stays longer in school. If not, it has no effect on the wage.

The lack of direct correlation between the instrumental and the oucome variable is argued through theory and previous studies.

3. To recap, the covariance structure among the variables in an omitted variable model can be summarized in the following table, where:

• Wage is the outcome variable
• Education is the policy variable
• Ability is the omitted variable
• Experience is a control variable
• Father’s education is the instrumental variable

Figure 2.8: Correlation matrix structure

4. Finding a valid instrumental variable is challenging, requires creativity and some review of previous studies to get ideas and preliminary assess the correlation between the model variables.

3.1 Data

The dataset contains 3,010 observations collected in 1966/1976.

data( "card" ) ## we attach the data to our R working space
nrow( card ) ## To check if data were properly load, we can look if the number of obs is equal to 3010

## [1] 3010

For this exercise we are going to use the following variables. You can find a description of all variables here.

3.2 Analysis

As we did before, Card wants to estimate the impact of education on wage. But to solve the ability bias, he utilizes a different instrumental variable: proximity to college. He provides arguments to support each of three main characteristics of a good instrumental variable:

1. It is correlated with the policy variable: Individuals who live close to a 4-year college have easier access to education at a lower costs (no communiting costs and time nor accomodation costs), so they have greater incentives to pursue education.
2. It is not correlated with the omitted variable: Individual ability does not depend on proximity to a college.
3. It is correlated to the dependent variable only through the policy variable: Proximity to a college has no effect on your annual income, unless you decide to pursue further education because of the nearby college.

Therefore, he estimates a model where:

First stage (equation (1.3))

Second stage (equation (1.4))

We replicate the study starting by having a look at the summary statistics and distribution of the 3 key variables:

stargazer( card,  
           type = "html", 
           keep = c("lwage", "^educ$", "nearc4"), 
           omit.summary.stat = c("p25", "p75"),
           digits = 2 )

#Note: the symbols ^ & $ around educ prevents the function from grabbing other variables containing educ, such as fatheduc or mothedut

par( mfrow=c(1,2) )

plot( density(card$lwage), xlab = "Wage", main = "Dependent variable", ylab = "Frequency")

plot( density(card$educ), xlab = "Education", main = "Policy variable", ylab = "Frequency")

Figure 3.1: Distribution of Y, X, and Z

barplot( table(card$nearc4), xlab = "Living near a four-year college", ylab = "Frequency", main = "Instrumental variable")

Figure 3.1: Distribution of Y, X, and Z

We then estimate the model using two stages. As before, in the first stage we estimate education using our instrumental variable; you can see that the F statistic is greater than 10, suggesting that \(\text{nearc4}\) is a valid instrumental variable. Then, we estimate the impact of education on wage using the predicted values of education.

first.stage = lm(educ ~ 
nearc4 +  
smsa66 +          
exper +          
expersq +        
black +         
south66, data = card)

x1_hat <- fitted( first.stage ) #Save predicted values for education

second.stage <- lm(lwage ~
x1_hat +
smsa66 +          
exper +          
expersq +        
black +         
south66, data = card)


stargazer( first.stage, second.stage,
           column.labels = c("First Stage", "Second Stage"),
           type = "html",
           omit.stat = c("rsq","ser"), 
           digits = 2 )

As you can see, our policy variable has a different estimate in the two models.

naive.model <- lm(lwage ~
educ +
smsa66 +          
exper +          
expersq +        
black +         
south66, data = card)

stargazer( naive.model, second.stage,
           model.numbers = F,
           type = "html", 
           omit.stat = "all", 
           digits = 2,  keep = c("educ", "x1_hat") )

4.1 ivreg function

If you like to explore new R functions, you can use the ivreg function from the package AER to calculate your instrumental variable model. See the example below using the same data as in the Card’s example.

Note the syntax for using the ivreg function.

simple.model <- lm(
lwage ~              
educ +
smsa66 +
exper +    
expersq + 
black +
south66, data = card)

ivreg.model <- 
ivreg(            ## Call the ivreg function
lwage ~           ## Dependent variable of the second stage 
educ +            ## Education, as estimated in the first stage below 
smsa66 +          ## Control variable - Living in a metropolitan area
exper +           ## Control variable - Expertise
expersq +         ## Control variable - Squared term of the expertise variable
black +           ## Control variable - If race is black = 1, otherwise = 0
south66           ## Control variable - If living in the south = 1, otherwise = 0 
|                 ## Separate first and second stage   
.-educ +          ## we include all variable of the first stage, minus the education variable that is estimated     
nearc4,           ## The instrumental variable is included only here, in the first stage
data = card )     ## Set the dataset


stargazer( simple.model, ivreg.model,
           column.labels = c("No IV model", "IV model"),
           type="html", 
           omit.stat = c("rsq","ser"), 
           digits=2 )

4.2 Additional examples

Several studies have used instrumental variables to solve omitted variable issues. Here are some examples: