Least Squares Estimation Using Sketched Data with Heteroskedastic Errors

Researchers may use a sketch of data of size $m$ instead of the full sample of size $n$ sometimes to relieve computation burden, and other times to maintain data privacy. This paper considers the case when full sample estimation would have required the Eicker-Huber-White robust standard errors to account for heteroskedasticity. We show that random projections have a smoothing effect on the sketched data, with the consequence that the least squares estimates using such sketched data behave 'as if' the errors were homoskedastic. This result is obtained by expressing the difference between the moments computed from the full sample and the sketched data as a degenerate $U$-statistic which is asymptotically normal with a homoskedastic variance when the conditions in Hall (1984) are satisfied. This result also holds for two-stage least squares for which algorithmic and statistical properties are analyzed. Sketches produced by random sampling will not, however, have the effect of homogenizing the error variances.