Endogeneity: Is all Hope Lost?
We have discussed why confounders are a source of endogeneity.
Imagine the confounder is unobserved. How can we fend off this threat to identification?
The basic idea is: If we can find a so-called instrumental variable that explains the endogenous regressor, we can use this variable to circumvent the issue. We can also use this technique to deal with other sources of endogeneity.
How Does This Work?
Intuitively, we can think of the depicted situation like this:
- We cannot identify the causal effect from \(\text{studying econometrics}\) on \(\text{being really smart}\) since they are confounded.
- But if we find an instrumental variable that explains only the endogenous regressor, we can isolate the part of the co-variation that is our causal effect.
When Does This Work?
There are two conditions our instrumental variable must fulfill.
- Relevance Condition: The instrument must be correlated with the endogenous regressor, i.e., it must actually explain this endogenous regressor.
- Exclusion Restriction: The instrument must affect the outcome only through the endogenous regressor.
What does “Using an Instrument” Mean?
We can think of the process as containing two steps:
- In the First Stage, we regress the endogenous regressor \(\boldsymbol{X}\) on the instrument \(\boldsymbol{Z}\).
- In the Second Stage, we take the predictions \(\hat{\boldsymbol{X}}\) from the first stage and regress the outcome \(\boldsymbol{y}\) on the predictions \(\hat{\boldsymbol{X}}\).
This is the intuition behind what we will call the Two Stage Least Squares (2SLS) estimator.
How Do We Use This?
We now know what instrumental variables are, and how we can estimate \(\boldsymbol{\beta}\) in an instrumental variables setting. We know that instruments must be exogenous and relevant, and that we need at least one instrument per endogenous variable. Now consider the following example:
Say we want to find out the effect of education \(\text{X}\) on income \(\text{Y}\).
But we know that both parental education \(\text{PE}\) and parental income \(\text{PI}\) influence the level of education. We could control for these since we can observe them.
However, there are likely other background factors \(\text{BG}\) that influence parental education, education and income. These background factors are unobserved, meaning we cannot identify a causal effect.
Only if we find an instrument \(\text{IV}\), we can bypass this restriction.
An IV for Education
This example is from Angrist & Krueger (1991)1. They came up with a novel instrument for education: the quarter of birth of a given individual.
How does this work?
In the United States, students must attend school from the calendar year in which they turn six until their 16th birthday. School entry is once per year, so the length of schooling at age 16 differs, and students who drop out at 16 create variation in education.
Quarter of Birth as an Instrument
Is this instrument both exogenous and relevant?
- Exogeneity implies that the quarter of birth does not directly affect income. There is no statistical test for this, and so whether you believe this depends on how much you trust the authors’ argument.
- Relevance is easier to investigate quantitatively. Angrist & Krueger (1991) show that men born earlier in the year tend to have lower education on average.
There is a rule of thumb that a good instrument must seem ridiculous, since it is then likely fulfilling the exclusion restriction.
Figures from Angrist & Krueger (1991)
The length of completed education shows a clear cyclical pattern when plotted against the quarter and year of birth.
Log weekly earnings also show a cyclical pattern.
Overidentification
If we have more instruments than endogenous regressors, we have overidentification.
In a setting of overidentification, we can use Sargan’s \(J\)-test to assess exogeneity of our instruments. The idea is to compare estimates using different instruments:
- If instruments are exogenous, estimates should be the same.
- The test’s null hypothesis is that all instruments are exogenous.
Unfortunately, we do not learn which instrument is not valid, and estimates could always be similar or different by chance.
Family Size and Female Labor
Say we want to learn about the way family size affects the labour supply of women — e.g. to better understand discrimination, or to design policies for more equality.
- Women with more children tend to work less (outside the home).
- This is unlikely to be exogenous since children are not randomly assigned.
Now consider the fact that mothers whose first two children are of the same gender work fewer hours than others. How is this related to labour supply?
It probably is not related to labor supply. But it may be related to family size since parents may have a preference for mixed genders and choose to have a third kid.
Fish Market
Suppose we want to understand how the price of fish affects the quantity sold at a fish market.
- This is a simultaneity issue: Price and quantity are determined simultaneously by supply and demand.
However, on days after a period with especially high waves, prices on the fish market are usually higher. How and why?
When waves are high, it is more difficult to fish, which means that the quantity sold at the fish market will be lower. Note, however, that we need to rely on the assumption that the kind of fish caught is not affected by waves.