34.8 Treatment Intensity

Two-Stage Least Squares is a powerful method for estimating the average causal effect when treatment intensity varies across units. Rather than simple binary treatment (treated vs. untreated), many empirical applications involve treatments that can take on a range of values. TSLS can identify causal effects in these settings, capturing “a weighted average of per-unit treatment effects along the length of a causal response function” (J. D. Angrist and Imbens 1995, 431).

Common examples of treatment intensity include:

• Drug dosage (e.g., milligrams administered)
• Hours of exam preparation on test performance (Powers and Swinton 1984)
• Cigarette consumption (e.g., packs per day) on infant birth weights (Permutt and Hebel 1989)
• Years of education and their effect on earnings
• Class size and its impact on student test scores (J. D. Angrist and Lavy 1999)
• Sibship size and later-life earnings (Lavy, Angrist, and Schlosser 2006)
• Social media adoption intensity (e.g., time spent, number of platforms)

The average causal effect here refers to the conditional expectation of the difference in outcomes between the treated unit (at a given treatment intensity) and what would have happened in the counterfactual scenario (at a different treatment intensity). Importantly:

• Linearity is not required in the relationships between the dependent variable, treatment intensities, and instruments. TSLS can accommodate nonlinear causal response functions, provided the assumptions of the method hold.

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34.8.1 Example: Schooling and Earnings

In their seminal paper, J. D. Angrist and Imbens (1995) estimate the causal effect of years of schooling on earnings, using quarter of birth as an instrumental variable. The intuition is that individuals born in different quarters are subject to different compulsory schooling laws, which affect educational attainment but are plausibly unrelated to unobserved ability or motivation (the typical omitted variables in this context).

The structural outcome equation is:

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

where:

• $Y$ is the log of earnings (the dependent variable)
• $S$ is years of schooling (the endogenous regressor)
• $X_1$ is a vector (or matrix) of exogenous covariates (e.g., demographic characteristics)
• $\rho$ is the causal return to schooling we wish to estimate
• $\varepsilon$ is the error term, capturing unobserved factors

Because schooling $S$ may be endogenous (e.g., correlated with $\varepsilon$), we model its first-stage relationship with the exogenous variables and instruments:

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

where:

• $X_2$ represents the instrumental variables (e.g., quarter of birth)
• $\delta_2$ is the coefficient on the instrument
• $\eta$ is the first-stage error term

The Two-Stage Procedure

1. First Stage Regression
   Regress $S$ on $X_1$ and $X_2$ to obtain the predicted (fitted) values $\hat{S}$.

$$\hat{S} = \widehat{\delta_0} + X_1 \widehat{\delta_1} + X_2 \widehat{\delta_2}$$

2. Second Stage Regression
   Replace $S$ with $\hat{S}$ in the structural equation and estimate:

$$Y = \gamma_0 + \gamma_1 X_1 + \rho \hat{S} + \nu$$

where $\nu$ is the new error term (different from $\varepsilon$ because $\hat{S}$ is constructed to be exogenous).

Under the standard IV assumptions, $\rho$ is a consistent estimator of the causal effect of schooling on earnings.

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34.8.2 Causal Interpretation of $\rho$

For $\rho$ to have a valid causal interpretation, two key assumptions are essential:

1. SUTVA (Stable Unit Treatment Value Assumption)
   • The potential outcomes of each individual are not affected by the treatment assignments of other units.
   • There are no hidden variations of the treatment; “one year of schooling” means the same treatment type across individuals.
   • While important, SUTVA is often assumed without extensive defense in empirical work, though violations (e.g., spillovers in education settings) should be acknowledged when plausible.
2. [Local Average Treatment Effect] (LATE)
   • TSLS identifies a weighted average of marginal effects at the points where the instrument induces variation in treatment intensity.
   • Formally, $\rho$ converges in probability to a weighted average of causal increments:

$$\text{plim } \hat{\rho} = \sum_j w_j \cdot E[Y_j - Y_{j-1} \mid \text{Compliers at level } j]$$

where $w_j$ are weights determined by the distribution of the instrument and treatment intensity.

• This LATE interpretation means that TSLS estimates apply to compliers whose treatment intensity changes in response to the instrument. If there is heterogeneity in treatment effects across units, the interpretation of $\rho$ becomes instrument-dependent and may not generalize to the entire population.

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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.