Compare $SSreg (X_1,X_2)$ and $SSreg (X_1)+SSreg (X_2)$ with intuitive explanation for this relationship

Question

Please I need help from the experts regaring this question: Regression with uncorrelated predictors: Suppose that we fit the following model:

Compare . Give an intuitive explanation for this relationship.

Answer 1

Here is the sketch. try to work out the technical details by yourself. Note/Show that for the simple (univariate) regression you have $SSreg(X_1) =\hat{\beta_1}^2\sum (X_{1i} - \bar{X}_{1n})^2 $, and that for the bivariate model you have $$ \bar{Y}_n = {\hat{\beta}_0} + {\hat{\beta}_1}\bar{X}_{1n} + {\hat{\beta}_0}\bar{X}_{2n}, $$ next plug it in the SSreg of the bivariate model \begin{align} SSreg(X_1, X_2) &= \sum( \hat{Y} - \bar{Y}_n) ^2\\ & = \sum ( {\hat{\beta}_0} + {\hat{\beta}_1}X_{1i} + {\hat{\beta}_0}X_{2i} - {\hat{\beta}_0} - {\hat{\beta}_1}\bar{X}_{1n} - {\hat{\beta}_0}\bar{X}_{2n})^2\\ & = \sum (\hat{\beta}_1(X_{1i} - \bar{X}_{1n}) + \hat{\beta}_2(X_{1i} - \bar{X}_{2n}))\\ &= \hat{\beta_1}^2\sum (X_{1i} - \bar{X}_{1n})^2 + \hat{\beta_2}^2\sum (X_{2i} - \bar{X}_{2n})^2\\ & = SSreg(X_1) + SSreg(X_2), \end{align} note that the middle term $$ \hat{\beta}_1\hat{\beta}_2 \sum (X_{1i} - \bar{X}_{1n})(X_{2i} - \bar{X}_{2n}) = 0, $$ because $X_1$ and $X_2$ are uncorrelated.