  Truncated linear regression is a classical challenge in Statistics, wherein a
label, $y = w^T x + \varepsilon$, and its corresponding feature vector, $x \in
\mathbb{R}^k$, are only observed if the label falls in some subset $S \subseteq
\mathbb{R}$; otherwise the existence of the pair $(x, y)$ is hidden from
observation. Linear regression with truncated observations has remained a
challenge, in its general form, since the early works
of~\citet{tobin1958estimation,amemiya1973regression}. When the distribution of
the error is normal with known variance, recent work
of~\citet{daskalakis2019truncatedregression} provides computationally and
statistically efficient estimators of the linear model, $w$.
  In this paper, we provide the first computationally and statistically
efficient estimators for truncated linear regression when the noise variance is
unknown, estimating both the linear model and the variance of the noise. Our
estimator is based on an efficient implementation of Projected Stochastic
Gradient Descent on the negative log-likelihood of the truncated sample.
Importantly, we show that the error of our estimates is asymptotically normal,
and we use this to provide explicit confidence regions for our estimates.
