Ordinary least squares
Intervals

Using this asymptotic distribution, approximate two-sided confidence intervals for the j -th component of the vector β ^ {\displaystyle {\hat {\beta }}} can be constructed as

where q denotes the quantile function of standard normal distribution, and is the j -th diagonal element of a matrix.

Similarly, the least squares estimator for σ is also consistent and asymptotically normal (provided that the fourth moment of ε exists) with limiting distribution

These asymptotic distributions can be used for prediction, testing hypotheses, constructing other estimators, etc.. As an example consider the problem of prediction. Suppose x 0 {\displaystyle x_{0}} is some point within the domain of distribution of the regressors, and one wants to know what the response variable would have been at that point. The mean response is the quantity y 0 = x 0 T β {\displaystyle y_{0}=x_{0}^{\mathrm {T} }\beta }, whereas the predicted response is y ^ 0 = x 0 T β ^ {\displaystyle {\hat {y}}_{0}=x_{0}^{\mathrm {T} }{\hat {\beta }}}. Clearly the predicted response is a random variable, its distribution can be derived from that of β ^ {\displaystyle {\hat {\beta }}}:

which allows construct confidence intervals for mean response y 0 {\displaystyle y_{0}} to be constructed:
language:response_language[English]