Two-Stage Least Squares (2SLS)

Do people with more years of college education also earn more? Can you tell whether higher wages are due to the years of education or to underlying ability or family background? If having a college near home affects how long people stay in school, could you use that in the analysis?

When an explanatory variable is tangled up with other, unobserved factors, two-stage least squares (2SLS) is the method that uses an instrumental variable to estimate the change in the explanatory variable and then uses that change to find the effect on the outcome.

Ordinary regression estimates the conditional relationship between years of education and wages, but if there are omitted variables such as ability or family background, or possible reverse causation in the choice of education, it's hard to read that coefficient directly as a causal effect. 2SLS goes through two rounds of estimation to use the part of the endogenous years of education that is linked to the instrument. The instrument must, however, satisfy relevance and the exclusion restriction.

2SLS is used in labor economics, the economics of education, health and policy evaluation, and other fields. In this article we'll take the distance from home to a college as a candidate instrument to look at the relationship between years of education and wages, and cover what the two stages mean, the key assumptions, how to read the results table, and how it differs from OLS and instrumental variable analysis.

2SLS is the standard estimation method for linear instrumental variable models. In stage 1, years of education, the endogenous variable, is explained by the instrument and appropriate control variables; in stage 2, the variation in years of education explained by the instrument is used to estimate the effect on wages.

The intuition is often introduced by plugging the predicted years of education into a second regression, but the actual standard errors and confidence intervals must be computed with a dedicated estimator that reflects the whole 2SLS estimation process. You should not use the second-stage standard errors from simply running ordinary regression twice.

Years of education can be associated with a person's ability, academic inclination, family finances, regional characteristics, and so on. If these factors also affect wages, the OLS coefficient on years of education can include influences other than the effect of education.

For example, living near a college may make people more likely to stay in school longer, but if areas close to colleges are themselves places with higher incomes or better job opportunities, distance may not be a valid instrument. So the reason for using 2SLS and the reason the instrument holds up need to be explained separately.

In practice, 2SLS uses the variation obtained by projecting years of education onto the instrument and the exogenous control variables. The same set of control variables must be used in both stages, and the predicted values in stage 2 are not simply treated as if they were new observations.

Assume a study that observed 600 hypothetical adult workers. Years of education is defined as the total years of formal schooling, and the wage as the natural log of the hourly wage. The candidate instrument is the distance from the place a person lived as a teenager to the nearest four-year college. All numbers are examples for learning.

Access to college may be related to the opportunity to go on to school, but a region's industrial structure or educational environment may also affect wages directly. So here we only explain the procedure for using a candidate instrument, and we don't assume the exclusion restriction has been verified.

① Define the variables and the causal assumptions

Set years of education (X), log wages (Y), distance to college (Z), and background controls, and examine other paths through which Z might affect Y.

② Estimate the first-stage regression

Explain years of schooling with distance to college and the controls. Check information on the relevance and strength of the instrument, such as the partial F statistic.

③ Estimate the effect in the second stage

Use a dedicated IV/2SLS estimator to produce the effect of years of education on wages along with correct standard errors and confidence intervals.

④ Compare with ordinary regression and report the limitations

Compare the OLS and IV estimates, but don't conclude that 2SLS must be right just because the estimates differ. Address weak instruments and possible violations of the exclusion restriction separately.

F = 16.4 is a hypothetical value for illustration. An F above 10 does not always mean you're safe; consider weak-instrument diagnostics and robust inference suited to the conditions of the analysis, such as heteroskedasticity or clustered data.

The hypothetical 2SLS coefficient of 0.091 can be read as hourly wages being about 9.5% higher for each additional year of education (exp(0.091) − 1 ≈ 0.095). As a simple approximation, it's about 9.1%. The numbers here are illustrative values, not results from an actual fit.

To call this a causal effect, the instrumental variable assumptions must hold. And if the effect differs from person to person, the IV result may be a local effect for the group that responds to the instrument, so don't generalize it as an average effect for everyone.

The 2SLS confidence interval can be wider than the OLS one. That's because using only the limited variation that comes through the instrument can lower the precision of the estimate. A bigger coefficient doesn't simply mean a better analysis.

Could distance to college affect wages directly?

If areas near colleges have more jobs or higher wages, distance can be linked to wages through more than years of education. Alternative paths like this are a major threat to the exclusion restriction.

Are you using a weak instrument?

If first-stage relevance is weak, the 2SLS coefficient and the usual confidence intervals can become very unstable. Consider diagnostics suited to your data structure and tests that are robust to weak instruments.

Did you calculate the standard errors correctly?

Don't run the second stage as a separate OLS and report its standard errors as they are. Use a 2SLS estimator with appropriate heteroskedasticity-robust or cluster-robust standard errors.

Are the estimated population and the scope of generalization clear?

It may be the effect for the group whose education choices change with access to college. Whether it generalizes to the whole population has to be judged separately, based on the study design and identifying assumptions.

You don't have to think of instrumental variable analysis and 2SLS as entirely separate, competing methods. 2SLS is one of several estimators you can use once a suitable instrumental variable design is in place.

In a paper or report, lay out the variable definitions, the rationale for the instrument, the sample and controls, first-stage relevance and strength diagnostics, the 2SLS coefficient and confidence interval, how standard errors were handled, the comparison with OLS, and the weak points of the exclusion restriction.

Don't write that a causal effect has been proven just because relevance was confirmed. In particular, discuss other paths through which regional differences in access to college could affect wages.

The instruments in the table are only candidates a study could examine. Whether the exclusion restriction holds, and for whom the effect is identified, has to be checked anew in each study.

Just because people with more years of education earn more, you can't immediately conclude that adding years of education will raise wages by that much. Two-stage least squares is an estimation method that uses variation linked to an instrument to deal with the endogeneity of years of education. The key is that stage 1 checks how much distance to college explains years of education, and stage 2 estimates the relationship with wages based on that variation.

But finishing the two stages of calculation doesn't automatically confirm a causal effect, because distance to college may also be related to the local labor market, family background, and where people choose to live. So researchers need to examine not only the instrument's relevance and strength but also independence and the exclusion restriction, and to be clear about the confidence interval and the scope of what is being estimated. Without a valid instrument, it is better not to claim a causal relationship from 2SLS numbers alone.