  In the presence of confounders, the ordinary least squares (OLS) estimator is
known to be biased. This problem can be remedied by using the two-stage least
squares (TSLS) estimator, based on the availability of valid instrumental
variables (IVs). This reduction in bias, however, is offset by an increase in
variance. Under standard assumptions, the OLS has indeed a larger bias than the
TSLS estimator; and moreover, one can prove that the sample variance of the OLS
estimator is no greater than the one of the TSLS. Therefore, it is natural to
ask whether one could combine the desirable properties of the OLS and TSLS
estimators. Such a trade-off can be achieved through a convex combination of
these two estimators, thereby producing our proposed convex least squares (CLS)
estimator. The relative contribution of the OLS and TSLS estimators is here
chosen to minimize a sample estimate of the mean squared error (MSE) of their
convex combination. This proportion parameter is proved to be unique, whenever
the OLS and TSLS differ in MSEs. Remarkably, we show that this proportion
parameter can be estimated from the data, and that the resulting CLS estimator
is consistent. We also show how the CLS framework can incorporate other
asymptotically unbiased estimators, such as the jackknife IV estimator (JIVE).
The finite-sample properties of the CLS estimator are investigated using Monte
Carlo simulations, in which we independently vary the amount of confounding and
the strength of the instrument. Overall, the CLS estimator is found to
outperform the TSLS estimator in terms of MSE. The method is also applied to a
classic data set from econometrics, which models the financial return to
education.
