Regression coefficient ($\beta$)

A regression coefficient is a component of $\beta$ measuring the change in the linear predictor associated with its design-matrix column.

These are the parameters of a linear regression predictor, rather than the regression procedure itself.

In an ordinary linear regression with nonzero response and predictor standard deviations, the coefficient after rescaling predictor $j$ and the response to standard-deviation units is the displayed quantity. An unstandardized coefficient table does not automatically report it. Standardization and reference coding must be explicit.

In a simple linear regression with conditional mean $\mathbb{E}[Y\mid X=x]=\alpha +\beta x$, the slope $\beta$ is the change in that mean per unit increase in the predictor. The ordinary least squares estimate is $\hat{\beta }=S_{xy}/S_{xx}$ when $S_{xx}>0$. It describes a statistical association; a causal interpretation requires further assumptions.