Prove That the Regression R² Is Identical to the Square

Question

Examlex · Accepted Answer

The Answer is The regression R²=

\frac { E S S } { T S S }

, where ESS is given by

\sum _ { i = 1 } ^ { n } ( \hat { Y } - \bar { Y } ) ^ { 2 }

But

\hat { Y }

_i =

\hat { \beta }

₀ +

\hat { \beta }

₁X_iand

\bar { Y }

=

\hat { \beta }

₀ +

\hat { \beta }

₁

\bar { X }

Hence (

\hat { Y }

_i -

\bar { Y }

)² =

\hat { \beta } _ { 1 } ^ { 2 }

(X_i -

\bar { X }

)² and therefore ESS =

\hat { \beta } _ { 1 } ^ { 2 }

\sum _ { i = 1 } ^ { n } \left( X _ { i } - \bar { X } \right) ^ { 2 }

Using small letters to indicate deviations from mean, i.e., z_i = Z_i -

\bar { Z }

, we get that the regression R²=

\frac { \hat { \beta } _ { 1 } ^ { 2 } \sum _ { i = 1 } ^ { n } x _ { i } ^ { 2 } } { \sum _ { i = 1 } ^ { n } y _ { i } ^ { 2 } }

The square of the correlation coefficient is

r ^ { 2 } = \frac { \sum _ { i = 1 } ^ { n } \left( y _ { i } x _ { i } \right) ^ { 2 } } { \sum _ { i = 1 } ^ { n } x _ { i } ^ { 2 } \sum _ { i = 1 } ^ { n } y _ { i } ^ { 2 } } = \frac { \sum _ { i = 1 } ^ { n } \left( y _ { i } x _ { i } \right) ^ { 2 } \sum _ { i = 1 } ^ { n } x _ { i } ^ { 2 } } { \left( \sum _ { i = 1 } ^ { n } x _ { i } ^ { 2 } \right) ^ { 2 } \sum _ { i = 1 } ^ { n } y _ { i } ^ { 2 } } = \frac { \hat { \beta } _ { i } ^ { 2 } \sum _ { i = 1 } ^ { n } x _ { i } ^ { 2 } } { \sum _ { i = 1 } ^ { n } y _ { i } ^ { 2 } }

. Hence the two are the same. Correlation does not imply causation. Income is a regressor in the consumption function, yet consumption enters on the right-hand side of the GDP identity. Regressing the weight of individuals on the height is a situation where causality is without doubt, since the author of this test bank should be seven feet tall otherwise. The authors of the textbook use weather data to forecast orange juice prices later in the text.

Prove That the Regression R2 Is Identical to the Square

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Prove That the Regression R² Is Identical to the Square