1 What Problem Does It Solve?

Two-stage least squares (2SLS) with a binary instrument identifies the Local Average Treatment Effect (LATE): the average causal effect for compliers—units induced to take treatment by the instrument. But policymakers rarely want to know the effect on compliers specifically. They want to know what would happen under a new policy that changes the fraction of the population treated by some amount, which may differ from the compliance margin induced by any particular instrument.  

The ivmte package (Mogstad et al., 2018) implements the Mogstad et al. (2018) approach to computing sharp identified sets for any Policy-Relevant Treatment Effect (PRTE) using the Marginal Treatment Effect (MTE) function as a building block . Rather than extrapolating a parametric MTE beyond the data, it uses linear programming to find the bounds on the PRTE consistent with all data-generating processes that satisfy the IV assumptions and any additional shape restrictions.  

2 The MTE Framework

The marginal treatment effect framework originates with Heckman and Vytlacil (2005). Assume binary treatment D ∈ {0, 1} is determined by a threshold-crossing model:  

where Zᵢ is an instrument, p(Zᵢ) = Pr(Dᵢ = 1|Zᵢ) is the propensity score, and Vᵢ ∼ Uniform (0, 1) is an unobserved resistance to treatment (standardised to be uniform by the probability integral transform). The MTE at unobserved resistance u ∈ (0, 1) is:  

This is the average treatment effect for the "marginal person" with resistance u: individuals with Vᵢ just below a propensity score p(z) will take treatment when offered instrument z, and their average effect is MTE(p(z)).

Standard treatment effect parameters are weighted integrals of the MTE:  

The LATE is the average MTE over the complier range [p(z), p(z′)]. The ATE, ATT, and ATU integrate the MTE over regions that may extend beyond the support of the propensity score—regions where the MTE is not identified by the instrument variation.  

3 The Mogstad-Santos-Torgovitsky (2018) Approach

The key insight of Mogstad et al. (2018) is that without imposing a parametric form for MTE(u) over unidentified regions, the best we can do is bound it. They compute sharp bounds—the tightest possible interval for any target PRTE—consistent with:  

1. The IV assumptions (independence, exclusion, monotonicity).  
2. The observed data moments (regression of Y on D and Z, conditional on X).  
3. Optional shape restrictions on MTE(u): monotone treatment response (MTR), monotone treatment selection (MTS) .  

Bounds are computed via linear programming over the space of all MTE functions consistent with the constraints. The Local Instrumental Variable (LIV) estimator identifies MTE(u) at interior points of the propensity score support:  

The LIV is estimated by a derivative of a nonparametric regression of Y on the propensity score.  

4 Installation and Setup

5 A Minimal Working Example

The following example uses the Card (1995) college proximity data, estimating bounds on the Average Treatment Effect (ATE) for returns to schooling .  

The function returns a list with:  

• bounds: the lower and upper bounds on the target estimand.  

• point: a point estimate under the parametric MTE assumption (if the MTE is identified at the target by the instrument support) .  

• sens: sensitivity analysis over shape restrictions.  

6 Key Arguments

7 Tightening Bounds with Shape Restrictions

Without any shape restrictions, the bounds on the ATE can be very wide—often wider than [Γ_OLS, Γ_OLS + c] for some large c. Two common restrictions substantially tighten bounds:  

• Monotone treatment response (MTR). If the treatment can only help (or can only harm), MTE(u) ≥ 0 for all u.  

• Monotone treatment selection (MTS). If individuals with lower resistance to treatment (lower u) have higher potential outcomes, MTE(u) is non-increasing. This is plausible when individuals self-select into treatment based on comparative advantage.  

When both MTR and MTS are imposed, Manski and Pepper (1997) bounds on the ATE can be substantially tighter . ivmte reports bounds under each combination of restrictions.  

8 Comparison to Alternative Approaches

9 Pitfalls and Practical Advice

1. Wide bounds are the honest answer. Do not be disappointed if the identified set is large—it reflects what the data can and cannot say. Wide bounds indicate extrapolation beyond the instrument support, not a failure of the method.  ‍
2. Instrument support matters. If the instrument moves the propensity score only over a narrow range, large regions of the MTE are unidentified. The bounds for the ATE will be correspondingly wide.  ‍
3. Shape restrictions are substantive. MTR and MTS are economic assumptions, not statistical regularisation. Verify them with domain knowledge.  ‍
4. First-stage strength. Weak instruments make the LIV derivative imprecise, broadening the identified set. Report the first-stage F-statistic as always.  

10 Conclusion

The ivmte package implements the Mogstad et al. (2018) sharp IV bounds framework, enabling researchers to characterise what IV assumptions and shape restrictions imply about any target treatment effect parameter—not just the LATE. It is especially valuable when the policy question asks about a different margin than the one identified by the available instrument: instead of asserting a parametric extrapolation, ivmte reports the honest identified set. For researchers working in the MTE tradition of Heckman and Vytlacil (2005) and Carneiro et al. (2011), this package provides a rigorous, transparent approach to estimation and inference .  

References

1. Carneiro, P., Heckman, J. J., and Vytlacil, E. J. (2011). Estimating marginal returns to education. American Economic Review, 101(6):2754-2781.  
2. Heckman, J. J. and Vytlacil, E. (2005). Structural equations, treatment effects, and econometric policy evaluation. Econometrica, 73(3):669-738.  
3. Imbens, G. W. and Angrist, J. D. (1994). Identification and estimation of local average treatment effects. Econometrica, 62(2):467-475.  
4. Manski, C. F. and Pepper, J. V. (1997). Monotone instrumental variables with an application to the returns to schooling. Econometrica, 68(4):997-1010 .  
5. Mogstad, M., Santos, A., and Torgovitsky, A. (2018). Using instrumental variables for inference about policy relevant treatment parameters. Econometrica, 86(5):1589-1619 .  
6. Walters, C. R. (2018). The demand for effective charter schools. Journal of Political Economy, 126(1):103-154.