Linear regression through the origin

In the model $Y_i=\beta x_i+{\epsilon }_i$ without an intercept, with ${\sum }_i{x}_i^2>0$, the ordinary least-squares estimator isFor independent errors of common variance ${\sigma }^2$, it is unbiased and has variance ${\sigma }^2/{\sum }_i{x}_i^2$.

For independent errors of common unknown variance in $Y_i=\beta x_i+{\epsilon }_i$, the ordinary least squares estimate is $\hat{\beta }={\sum }_i{x}_i{Y}_i/{\sum }_i{x}_i^2$. Fitting one parameter leaves $n-1$ residual degrees of freedom. Therefore $s^2={\sum }_i(Y_i-\hat{\beta }{x}_i{)}^2/(n-1)$ estimates the common error variance, and $s^2/{\sum }_i{x}_i^2$ estimates the slope variance. For errors with a Gaussian distribution, the slope's standardized deviation follows a Student's t-distribution with $n-1$ degrees of freedom. A common calibration bias does not appear as independent residual scatter.

For independent $Y_i\sim N(\beta x_i,{\sigma }^2)$ with $n\ge 2$ and $\sum x_i^2>0$, the least-squares estimator is $\hat{\beta }=\sum x_i{Y}_i/\sum x_i^2$. The projection of the isotropic Gaussian vector onto the design direction is independent of its orthogonal residual; hence $\hat{\beta }\sqrt{\sum x_i^2}/\sigma$ is standard normal under $\beta =0$ and $\mathrm{SSE}/{\sigma }^2$ is independent chi-squared with $n-1$ degrees of freedom. The displayed statistic therefore has Student's t-distribution with $n-1$ degrees of freedom. A two-sided test rejects at the corresponding upper/lower quantiles. The zero design has no information about the slope; with one observation there is no residual variance degree of freedom.