  Researchers may perform regressions using a sketch of data of size $m$
instead of the full sample of size $n$ for a variety of reasons. This paper
considers the case when the regression errors do not have constant variance and
heteroskedasticity robust standard errors would normally be needed for test
statistics to provide accurate inference. We show that estimates using data
sketched by random projections will behave `as if' the errors were
homoskedastic. Estimation by random sampling would not have this property. The
result arises because the sketched estimates in the case of random projections
can be expressed as degenerate $U$-statistics, and under certain conditions,
these statistics are asymptotically normal with homoskedastic variance. We
verify that the conditions hold not only in the case of least squares
regression when the covariates are exogenous, but also in instrumental
variables estimation when the covariates are endogenous. The result implies
that inference, including first-stage F tests for instrument relevance, can be
simpler than the full sample case if the sketching scheme is appropriately
chosen.
