17.1 Model Parameters

• The estimates of the $\beta$ coefficients are the values that minimize the sum of squared errors for the sample.
• Considering $n$ = sample size, $k+1$ = number of $\beta$ coefficients in the model (including the intercept) and $SSE$ = sum of squared errors, $$\textrm{MSE}=\frac{\textrm{SSE}}{n-(k+1)}$$ estimates $\sigma^2,$ the variance of the errors. $S=\sqrt{MSE}$ estimates $\sigma$ and is known as the regression standard error or the residual standard error.
• Each $\beta$ coefficient represents the change in the mean response, $E(y)$, per unit increase in the associated predictor variable when all the other predictors are held constant. For example, $\beta_1$ represents the change in the mean response, $E(y)$, per unit increase in $x_1$ when $x_2, x_3, \ldots, x_k$ are held constant.
• The intercept term, $\beta_0,$ represents the mean response, $E(y)$, when all the predictors $x_1, x_2, \ldots, x_k,$ are all zero (which may or may not have any practical meaning).