help for aaniv

Unbiased IV regression

Syntax

aaniv [varlist] [if] [in]
[, deltase ivreg2 ivoptions(options for IV regression)
sureg]

Table of Contents

General description of estimator

Examples

Description of options

Remarks and saved results

References

Acknowledgements

Citation of aaniv

Author information

Description

The conventional instrumental variable (IV) or two-stage least squares (2SLS) estimator may be
badly biased in overidentified models with weak instruments. While the 2SLS estimator performs better in the
exactly identified case, in the sense that its median rapidly approaches the true value as instruments become strong, it has
no first moment. That is, the estimator has no mean, and no finite higher moments, either.
For papers on the finite-sample properties of IV estimators, see e.g. Phillips (1980), Phillips (1983), and Hillier (2006), and references therein.

The estimator implemented in aaniv is an unbiased IV estimator for a special case of an exactly identified model
with one endogenous variable and one instrument, from Andrews and Armstrong (2017), which relies on a
sign restriction in the first stage. That is, if we know the sign of the effect of of the instrument on the endogenous treatment variable,
we can construct an unbiased estimate of the effect of treatment on the treated.
Per Andrews and Armstrong (2017), "the only other paper to treat finite-sample mean-unbiased estimation
in IV models is Hirano and Porter (2015), who find that unbiased estimators do
not exist when the parameter space is unrestricted." In that setting, "the sign restriction
on the first-stage coefficient leads to a parameter space that violates the assumptions of
Hirano and Porter (2015), so that the negative results in that paper do not apply.
In particular, the sign restriction violates Assumption 2.4 of Hirano and Porter (2015), and so renders
the negative result in Theorem 2.5 of that paper inapplicable.
The
nonexistence of unbiased estimators has been noted in other nonstandard econometric
contexts by Hirano and Porter (2012)."

The unbiased estimator in Andrews and Armstrong (2017), "differs from the conventional IV estimator only in that it replaces the plug-in
estimate [the reciprocal of the first stage coefficient] for 1/pi [the true value] by the unbiased estimate [tau-hat]."
The unbiased estimator of the reciprocal of the first stage coefficient (tau-hat) is less than the reciprocal of the first stage coefficient
when the first stage coefficient exceeds zero, so in this case,
the unbiased estimator shrinks the conventional IV estimator toward the covariance of the two errors scaled by the variance of the first-stage error.

Examples

The following example offers a comparison of standard IV/2SLS (using ivregress) and the unbiased IV implemented in aaniv.

An example from Card (1995):

use http://fmwww.bc.edu/ec-p/data/wooldridge/card, clear

loc x "exper* smsa* south mar black reg662-reg669"

ivregress 2sls lw `x' (educ=nearc4), noid

est sto iv2sls

aaniv lwage `x' (educ=nearc4)

est sto aaniv

est tab iv2sls aaniv, t keep(educ)

Options summary

deltase; by default, the standard IV/2SLS standard error is used as asymptotically correct but the option deltase requests a first-order delta-method approximation
to the Andrews & Armstrong (2017) estimator's (nonfinite) variance. Neither of these estimates of variance are correct, as the true variance is nonfinite, so approximations are
by necessity incorrect and dramatically understate the true variance. Nevertheless, simulations show that inference is quite good in practice, using either approximation.

ivreg2; by default, the standard IV/2SLS standard error is computed using ivregress.
To use the user-written ivreg2 instead, specify the option ivreg2.

ivoptions; any options to pass to ivregress, or it the option ivreg2 is also specified, to ivreg2.

level(#); see
[R] estimation options.

Remarks and saved results

The command saves the following results in e():

Scalars
e(N) Number of observations used in estimation

Macros
e(cmd) aaniv
e(version) Version number
e(depvar) Name of dependent variable

Matrices

e(b) Coefficient vector

e(V) VCE estimate

Functions
e(sample) Marks estimation sample

References

Andrews, Isaiah, and Timothy B. Armstrong. 2017. "Unbiased instrumental variables estimation
under known first-stage sign." Quantitative Economics, 8(2): 479-503.

Hillier, G. 2006. "Yet more on the exact properties of IV estimators." Econometric Theory,
22: 913-931.

Hirano, K. and J. R. Porter. 2012 "Impossibility results for nondifferentiable functionals."
Econometrica, 80(4): 1769-1790.

Hirano, K. and J. Porter. 2015. "Location properties of point estimators in linear instrumental
variables and related models." Econometric Reviews, 34(6-10): 719-733.

Phillips, Peter C. B. 1980. "The Exact Distribution of Instrumental Variable Estimators in an Equation Containing n+1 Endogenous Variables." Econometrica, 48: 861–878.

Phillips, Peter C. B. 1983. "Exact Small Sample Theory in the Simultaneous Equations Model." Chapter in Zvi Griliches and Michael D. Intriligator, ed.
Handbook of Econometrics, Volume 1, 449–516, North Holland, Amsterdam.
See also http://cowles.yale.edu/sites/default/files/files/pub/d06/d0617.pdf

The first example is based on:

Card, David E. 1995. "Using Geographic Variation in College Proximity to Estimate the
Return to Schooling" in Aspects of Labour Economics: Essays in Honour of John Vanderkamp,
edited by Louis Christofides, E. Kenneth Grant and Robert Swindinsky.
University of Toronto Press. See also NBER WP 4483.

Citation of aaniv

aaniv is not an official Stata command. It is a free contribution
to the research community, like a paper. Please cite it as such:

Andrews, Isaiah, Timothy B. Armstrong, and Austin Nichols. 2019.
aaniv: Stata module for Unbiased IV Regression.
http://ideas.repec.org/c/boc/bocode/s456890.html

Contact for support

Austin Nichols
Washington, DC, USA
austinnichols@gmail.com

Also see

Manual: [U] 23 Estimation and post-estimation commands

[R] bootstrap

[R] regress

[R] ivregress

On-line: help for (if installed) ivreg2,
overid, ivendog, ivhettest, ivreset,
xtivreg2, xtoverid, ranktest, weakivtest,
condivreg.