ivmte: An R Package for Marginal Treatment Effect Methods

Joshua Shea and Alexander Torgovitsky

Introduction
Scope of this Vignette
Installation and Requirements
Usage Demonstration
Help, Feature Requests and Bug Reports
References

Introduction

Heckman and Vytlacil (2005) introduced the marginal treatment effect (MTE) to provide a choice-theoretic interpretation for the widely used instrumental variables model of Imbens and Angrist (1994). The MTE can be used to formally extrapolate from the compliers to estimate treatment effects for other subpopulations.

The ivmte package provides a flexible set of methods for conducting this extrapolation. The package uses the moment-based framework developed by Mogstad, Santos, and Torgovitsky (2018), which accommodates both point identification and partial identification (bounds), both parametric and nonparametric models, and allows the user to maintain additional shape constraints.

Scope of this Vignette

This vignette is intended as a guide to using ivmte for users already familiar with MTE methods. The key definitions and concepts from that literature are used without further explanation. We have written a paper (Shea and Torgovitsky 2021) that discusses both the MTE methodology and the usage of ivmte, which should be helpful for users unfamiliar with MTE methods. The survey article by Mogstad and Torgovitsky (2018) provides additional theoretical background on MTE methods, including the moment-based implementation used in this module.

Installation and Requirements

ivmte can be installed from CRAN via

install.packages("ivmte")

If you have the devtools package, you can install the latest version of the module directly from our GitHub repository via

devtools::install_github("jkcshea/ivmte")

Up to four additional packages are also required for ivmte:

splines2 for specifying models with polynomial splines. This package is available on CRAN.
A package for solving linear programs. There are four options here:
1. Gurobi and the Gurobi R package gurobi, which can be obtained from Gurobi Optimization. This option requires a Gurobi software license, which Gurobi Optimization offers at no cost to academic researchers.
2. MOSEK and the package Rmosek, which can be obtained from MOSEK ApS. This option requires a MOSEK software license, which MOSEK ApS offers at no cost to academic researchers.
3. CPLEX and the package cplexAPI, which is available on CRAN. CPLEX can be obtained from IBM. This option requires a CPLEX software license, which IBM offers at no cost to academic researchers.
4. The lpSolveAPI package, which is free and open-source, and available from CRAN. Note that lpSolveAPI is a wrapper for lp_solve, which is no longer actively developed.
A package for solving quadratically constrained quadratic programs. gurobi and Rmosek are able to solve such programs. ivmte can still be used if the user only has access to cplexAPI or lpSolveAPI, but features of ivmte involving quadratically constrained quadratic programs will not be available.
lsei for performing constrained least squares. This package is available on CRAN.

ivmte tries to automatically choose a solver from those available, with preference being given to Gurobi, MOSEK, and CPLEX. We have provided the option to use lpSolveAPI because it appears to be the only interface for solving linear programs that can be installed solely through install.packages. However, we strongly recommend using Gurobi, MOSEK, or CPLEX, since these are actively developed, much more stable, and typically an order of magnitude faster than lpSolveAPI. A very clear installation guide for Gurobi can be found here

Usage Demonstration

Data and Background

We will use a subsample of 209133 women from the data used in Angrist and Evans (1998). The data is included with ivmte and has the following structure.

library(ivmte)
knitr::kable(head(AE, n = 10))

| worked | hours | morekids | samesex | yob | black | hisp | other | |-------:|------:|---------:|--------:|----:|------:|-----:|------:| | 0 | 0 | 0 | 0 | 52 | 0 | 0 | 0 | | 0 | 0 | 0 | 0 | 45 | 1 | 0 | 0 | | 1 | 35 | 0 | 1 | 49 | 0 | 0 | 0 | | 0 | 0 | 0 | 0 | 50 | 0 | 0 | 0 | | 0 | 0 | 0 | 0 | 50 | 0 | 0 | 0 | | 0 | 0 | 1 | 1 | 51 | 0 | 0 | 0 | | 1 | 40 | 0 | 0 | 51 | 0 | 0 | 0 | | 1 | 40 | 0 | 1 | 46 | 0 | 0 | 0 | | 0 | 0 | 0 | 1 | 45 | 0 | 0 | 0 | | 1 | 30 | 1 | 0 | 44 | 0 | 0 | 0 |

We will use these variables as follows:

worked is a binary outcome variable that indicates whether the woman was working.
hours is a multivalued outcome variable that measures the number of hours the woman worked per week.
morekids is a binary treatment variable that indicates whether a woman had more than two children or exactly two children.
- Note that Angrist and Evans (1998) removed women with fewer than two children when constructing their sample.
- Also note that the treatment variable must be binary.
samesex is an indicator for whether the woman’s first two children had the same sex (male-male or female-female). This is used as an instrument for morekids.
yob is the woman’s year of birth (i.e. her age), which will be used as a conditioning variable.

A goal of Angrist and Evans (1998) was to estimate the causal effect of morekids on either worked or hours. A linear regression of worked on morekids yields:

lm(data = AE, worked ~ morekids)
#> 
#> Call:
#> lm(formula = worked ~ morekids, data = AE)
#> 
#> Coefficients:
#> (Intercept)     morekids  
#>      0.5822      -0.1423

The regression shows that women with three or more children are about 14–15% less likely to be working than women with exactly two children. Is this because children have a negative impact on a woman’s labor supply? Or is it because women with a weaker attachment to the labor market choose to have more children?

Angrist and Evans (1998) address this question by using samesex as an instrument for morekids. We expect that samesex is randomly assigned, so that among women with two or more children, those with weak labor market attachment are just as likely as those with strong labor market attachment to have to have two boys or two girls for their first two children. A regression shows that women whose first two children had the same sex are also more likely to go on to have a third child:

lm(data = AE, morekids ~ samesex)
#> 
#> Call:
#> lm(formula = morekids ~ samesex, data = AE)
#> 
#> Coefficients:
#> (Intercept)      samesex  
#>     0.30214      0.05887

Thus, samesex constitutes a potential instrumental variable for morekids. We can run a simple instrumental variable regression using the ivreg command from the AER package.

library("AER")
ivreg(data = AE, worked ~ morekids | samesex )
#> 
#> Call:
#> ivreg(formula = worked ~ morekids | samesex, data = AE)
#> 
#> Coefficients:
#> (Intercept)     morekids  
#>     0.56315     -0.08484

The coefficient on morekids is smaller in magnitude than it was in the linear regression of worked on morekids. This suggests that the linear regression overstates the negative impact that children have on a woman’s labor supply. The likely explanation is that women who have more children were less likely to work anyway.

An important caveat to this reasoning, first discussed by Imbens and Angrist (1994), is that it applies only to the group of compliers who would have had a third child if and only if their first two children were same sex. (This interpretation requires the so-called monotonicity condition.) The first stage regression of morekids on samesex shows that this group comprises less than 6% of the entire population. Thus, the complier subpopulation is a small and potentially unrepresentative group of individuals.

Is the relationship between fertility and labor supply for the compliers the same as for other groups? The answer is important if we want to use an instrumental variable estimator to inform policy questions. The purpose of ivmte is to provide a formal framework for

Ivmte

Install / Use

README