  Economics and social science research often require analyzing datasets of
sensitive personal information at fine granularity, with models fit to small
subsets of the data. Unfortunately, such fine-grained analysis can easily
reveal sensitive individual information. We study algorithms for simple linear
regression that satisfy differential privacy, a constraint which guarantees
that an algorithm's output reveals little about any individual input data
record, even to an attacker with arbitrary side information about the dataset.
We consider the design of differentially private algorithms for simple linear
regression for small datasets, with tens to hundreds of datapoints, which is a
particularly challenging regime for differential privacy. Focusing on a
particular application to small-area analysis in economics research, we study
the performance of a spectrum of algorithms we adapt to the setting. We
identify key factors that affect their performance, showing through a range of
experiments that algorithms based on robust estimators (in particular, the
Theil-Sen estimator) perform well on the smallest datasets, but that other more
standard algorithms do better as the dataset size increases.
