## 29.6 Treatment Intensity

Two-Stage Least Squares (TSLS) can be used to estimate the average causal effect of variable treatment intensity, and it “identifies a weighted average of per-unit treatment effects along the length of a causal response function” . For example

• Drug dosage

• Hours of exam prep on score

• Cigarette smoking on birth weights

• Years of education

• Class size on test score

• Sibship size on earning

The average causal effect here refers to the conditional expectation of the difference in outcomes between the treated and what would have happened in the counterfactual world.

Notes:

• We do not need a linearity assumption of the relationships between the dependent variable, treatment intensities, and instruments.

Example

In their original paper, J. D. Angrist and Imbens (1995) take the example of schooling effect on earnings where they have quarters of birth as the instrumental variable.

For each additional year of schooling, there can be an increase in earnings, and each additional year can be heterogeneous (both in the sense that grade 9th to grade 10th is qualitatively different and one can change to a different school).

$Y = \gamma_0 + \gamma_1 X_1 + \rho S + \epsilon$

where

• $$S$$ is years of schooling (i.e., endogenous regressor)

• $$\rho$$ is the return to a year of schooling

• $$X_1$$ is a matrix of exogenous covariates

Schooling can also be related to the exogenous variable $$X_1$$

$S = \delta_0 + X_1 \delta_1 + X_2 \delta_2 + \eta$

where

• $$X_2$$ is an exogenous instrument

• $$\delta_2$$ is the coefficient of the instrument

by using only the fitted value in the second, the TSLS can give a consistent estimate of the effect of schooling on earning

$Y = \gamma_0 + X_1 \gamma-1 + \rho \hat{S} + \nu$

To give $$\rho$$ a causal interpretation,

1. We first have to have the SUTVA (stable unit treatment value assumption), where the potential outcomes of the same person with different years of schooling are independent.
2. When $$\rho$$ has a probability limit equal to a weighted average of $$E[Y_j - Y_{j-1}] \forall j$$

Even though the first bullet point is not trivial, most of the time we don’t have to defend much about it in a research article, the second bullet point is the harder one to argue and only apply to certain cases.

### References

Angrist, Joshua D, and Guido W Imbens. 1995. “Two-Stage Least Squares Estimation of Average Causal Effects in Models with Variable Treatment Intensity.” Journal of the American Statistical Association 90 (430): 431–42.
Angrist, Joshua D, and Victor Lavy. 1999. “Using Maimonides’ Rule to Estimate the Effect of Class Size on Scholastic Achievement.” The Quarterly Journal of Economics 114 (2): 533–75.
Lavy, Victor, Joshua D Angrist, and Analia Schlosser. 2006. “New Evidence on the Causal Link Between the Quantity and Quality of Children.”
Permutt, Thomas, and J Richard Hebel. 1989. “Simultaneous-Equation Estimation in a Clinical Trial of the Effect of Smoking on Birth Weight.” Biometrics, 619–22.
Powers, Donald E, and Spencer S Swinton. 1984. “Effects of Self-Study for Coachable Test Item Types.” Journal of Educational Psychology 76 (2): 266.