How do you choose the best instruments for two stage least squares?

1 Validity criteria

The first thing to check is whether your instruments satisfy the two validity criteria: relevance and exogeneity. Relevance means that the instruments are strongly correlated with the endogenous variable, so that they can explain its variation. Exogeneity means that the instruments are uncorrelated with the error term, so that they do not introduce bias or inconsistency. You can test the relevance of your instruments by using the first stage F-statistic, which measures how well the instruments predict the endogenous variable. You can test the exogeneity of your instruments by using the overidentification test, which compares the 2SLS estimates with those obtained by using different subsets of instruments.

2 Efficiency trade-off

The second thing to consider is the efficiency trade-off between using more or fewer instruments. In general, using more instruments can increase the precision of your 2SLS estimates, but it can also increase the potential for bias or inconsistency if some of the instruments are weak or invalid. Using fewer instruments can reduce the risk of bias or inconsistency, but it can also reduce the precision of your 2SLS estimates, especially if the instruments are weak. Therefore, you should balance the benefits and costs of adding or dropping instruments, and use diagnostic tests such as the weak instrument test, the Hausman test, and the Sargan test to assess the quality of your instruments.

3 Example: Angrist and Krueger (1991)

One of the most famous examples of 2SLS is the study by Angrist and Krueger (1991), who estimated the effect of education on earnings using quarter of birth as an instrument. The idea is that quarter of birth affects the age at which people leave school, due to compulsory schooling laws, but it does not affect their earnings directly. Therefore, quarter of birth satisfies the validity criteria of relevance and exogeneity. However, quarter of birth is also a weak instrument, as it explains only a small fraction of the variation in education. To improve the efficiency of their 2SLS estimates, Angrist and Krueger used two other instruments: state of birth and year of birth. These instruments capture the variation in schooling laws and economic conditions across states and years, and they are also plausibly exogenous. By using these three instruments, Angrist and Krueger obtained more precise and consistent estimates of the effect of education on earnings than by using OLS or single instruments.

4 Example: Card (1995)

Another famous example of 2SLS is the study by Card (1995), who estimated the effect of college proximity on college attendance and earnings using distance to the nearest college as an instrument. The idea is that distance to the nearest college affects the cost and convenience of attending college, but it does not affect earnings directly. Therefore, distance to the nearest college satisfies the validity criteria of relevance and exogeneity. However, distance to the nearest college is also a weak instrument, as it explains only a small fraction of the variation in college attendance. To improve the efficiency of his 2SLS estimates, Card used two other instruments: the number of colleges in the county and the average tuition of public colleges in the state. These instruments capture the variation in college supply and price across counties and states, and they are also plausibly exogenous. By using these three instruments, Card obtained more precise and consistent estimates of the effect of college proximity on college attendance and earnings than by using OLS or single instruments.

5 Example: Acemoglu et al. (2001)

A third famous example of 2SLS is the study by Acemoglu et al. (2001), who estimated the effect of institutions on economic growth using settler mortality as an instrument. The idea is that settler mortality affected the type of institutions that colonial powers established in different countries, but it does not affect economic growth directly. Therefore, settler mortality satisfies the validity criteria of relevance and exogeneity. However, settler mortality is also a weak instrument, as it explains only a small fraction of the variation in institutions. To improve the efficiency of their 2SLS estimates, Acemoglu et al. used two other instruments: latitude and the fraction of the population speaking English or a major European language. These instruments capture the variation in geographic and cultural factors that influenced colonial policies and institutions, and they are also plausibly exogenous. By using these three instruments, Acemoglu et al. obtained more precise and consistent estimates of the effect of institutions on economic growth than by using OLS or single instruments.