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An instrument is valid when it moves the treatment but does not affect the outcome in any other way. When the effect differs from person to person, the IV estimate still does not identify the average effect for everyone. It identifies the effect for the compliers, those the instrument moves, and only when no one responds in the opposite direction. The Handbook chapter by Mogstad and Torgovitsky (2024) adds what is needed to go further: which target parameter the question concerns, marginal treatment effects that extrapolate from the compliers, and bounds for what the data cannot pin down.
Here you run the estimators on simulated data where ATE and LATE are known. Judge designs, where the instrument is the judge the case is assigned to, are on Judge and examiner designs.
The papers · four sources
What each of them contributes
- Imbens and Angrist (1994)
- Econometrica 62(2). Under monotonicity the Wald ratio identifies LATE, the average effect for the compliers. Linear IV with a multivalued instrument is a weighted average of pairwise LATEs.
- Imbens and Rubin (1997)
- Review of Economic Studies 64(4). The whole outcome distribution for the compliers can be estimated, with and without treatment. Negative density in the estimate reveals violations of the assumptions.
- Mogstad, Torgovitsky and Walters (2021)
- NBER w25691. With multiple instruments, 2SLS is a weighted average over choice groups. The weights can turn negative when the instruments are negatively correlated, even if each instrument on its own satisfies monotonicity.
- Mogstad and Torgovitsky (2024)
- NBER w32927, Handbook of Labor Economics. Target parameters such as ATE, ATT, ATU and PRTE. Marginal treatment effects that build a bridge from the compliers to them. Bounds when the bridge does not hold, and a checklist for practice.
Simulate · choose the design
The estimates against the true effect
Each person has a resistance to treatment, uniformly distributed from 0 to 1. The effect is 1 at the middle of the scale and rises with resistance as much as the spread says. The compliers lie between the always-takers and the never-takers. These make up one third and two thirds of the rest, so the compliers have lower resistance than average. Defiers respond to the instrument in the opposite direction and violate monotonicity. Everything is computed on the server with the same tools as the MCP server.
| Quantity | True value | Estimate | Standard error | What it says |
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Checklist · from the tool iv_checklist
What each estimator identifies, and what can go wrong
The list comes from the tool iv_checklist. Page numbers refer to the NBER versions of Mogstad and Torgovitsky (MT) and Mogstad, Torgovitsky and Walters (MTW). For Imbens and Rubin (IR) and Imbens and Angrist (IA) they refer to the journal pages.
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The assumptions
The recommendations for practice
Method and sources
How the numbers are computed
The papers are Imbens, G. W. and Angrist, J. D. (1994). Identification and Estimation of Local Average Treatment Effects. Econometrica 62(2): 467-475. Imbens, G. W. and Rubin, D. B. (1997). Estimating Outcome Distributions for Compliers in Instrumental Variables Models. Review of Economic Studies 64(4): 555-574. Mogstad, M., Torgovitsky, A. and Walters, C. R. (2021). The Causal Interpretation of Two-Stage Least Squares with Multiple Instrumental Variables. NBER Working Paper 25691. Mogstad, M. and Torgovitsky, A. (2024). Instrumental Variables with Unobserved Heterogeneity in Treatment Effects. NBER Working Paper 32927, chapter in Handbook of Labor Economics.
The computations are done by ivadvisor 0.1.0, a Python package written for this project. None of the papers supplies data, so the package is checked against the printed numbers. The intersection bounds in MT Table 7 are matched within 0.0015 for all 28 numbers. The linear program with constant splines gives the logical IV bounds −0.227 and 0.252 from footnote 69. The shares in Imbens and Rubin, 0.762, 0.218 and 0.020, and the coefficients 39.10 and −38.10 are exact. LATE from the printed cells is off by 0.02 to 0.03. The cell means are rounded to three decimals and the complier share is 0.02. MT eq. 18 gives −0.16 with the numbers in the paper.
The bounds from the linear program are compared with the R package ivmte 1.4.0 on the same data. With constant splines at the propensity knots the two programs are identical: the ATE bounds −0.86309 and 0.49776 differ by 4 billionths. With fifth-degree polynomials they differ by 0.0044, because the packages enforce the bounds on the MTR on different grids. With decreasing MTRs in both arms the ATT bounds differ by 0.024. There the moments are not matched exactly, and the two criteria weigh the deviations differently. The point estimates differ by at most 0.0015. The server does not include ivmte.
Limits on what is included. The two maximum likelihood estimators in Table 2 of Imbens and Rubin, multinomial and normal, are not implemented. The tool iv_mte takes at most 10,000 rows here, and at that limit the target parameters have sampling noise. Take the default design with multiple instrument values and seed 2024. There a quadratic MTR gives an ATE of −0.367 against a true −0.267, ATT −0.362 against −0.368 and ATU −0.371 against −0.194. The bootstrap standard errors are 0.049, 0.055 and 0.070. The design with two instruments has no covariates. No real data are stored on the server.
The tools are also available as an MCP server at https://andreaskostol.no/mcp. Their names begin with iv_. Result files come back as download links that are deleted after one day. Judge designs, with the FLL and GHK tests, CJIVE and UJIVE, are on Judge and examiner designs. How to connect your assistant.