= Ordinary least squares estimators
{title2=$\widehat\beta=(X^TX)^{-1}X^TY$}
= Least-squares estimator
{synonym}
For a <linear regression> design matrix $X$ with <full column rank>, the <ordinary least squares> coefficient vector is $\widehat\beta=(X^TX)^{-1}X^TY$. In a <normal linear model> with error <covariance matrix> $\sigma^2I$, it is an <unbiased estimator> with <covariance matrix> $\sigma^2(X^TX)^{-1}$. Replacing $\sigma^2$ by the residual variance estimate gives the reported coefficient <standard errors>.
In simple <linear regression> with a nonconstant predictor,
$$
\widehat\beta=\frac{\sum_i(x_i-\bar x)(Y_i-\bar Y)}
{\sum_i(x_i-\bar x)^2},
\qquad
\widehat\alpha=\bar Y-\widehat\beta\bar x.
$$
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