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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 paperconsiders the case when the regression errors do not have constant variance andheteroskedasticity robust standard errors would normally be needed for teststatistics to provide accurate inference. We show that estimates using datasketched by random projections will behave `as if' the errors werehomoskedastic. Estimation by random sampling would not have this property. Theresult arises because the sketched estimates in the case of random projectionscan be expressed as degenerate $U$-statistics, and under certain conditions,these statistics are asymptotically normal with homoskedastic variance. Weverify that the conditions hold not only in the case of least squaresregression when the covariates are exogenous, but also in instrumentalvariables estimation when the covariates are endogenous. The result impliesthat inference, including first-stage F tests for instrument relevance, can besimpler than the full sample case if the sketching scheme is appropriatelychosen.

Least Squares Estimation Using Sketched Data with Heteroskedastic Errors