# How can one show mathematically that adding many variables will only increase R-squared? [duplicate]

As stated above, How does one show mathematically that adding many variables in a linear regression will only increase R-squared ?

• Add orthogonal variables will not improve R-squared. – SmallChess Mar 26 '17 at 13:16
• You would be correct if you say that it will increase $R^2$ rather than "give a very high value for R-squared. – Michael R. Chernick Mar 26 '17 at 13:53
• @MichaelChernick Not exactly. $R^2$ might not increase. – SmallChess Mar 26 '17 at 14:28
• You are right Student T. I should have said that it will not decrease $R^2$. . – Michael R. Chernick Mar 26 '17 at 16:33

In the specific case of linear regression, the model $\hat{y} = \beta_0 +\beta_1 x_1 + \beta_2 x_2$ must fit the data at least as well as the restricted model $\hat{y} = \beta_0 +\beta_1 x_1 + 0 x_2$ where $\beta_2 = 0$. Or, argue it this way: Let $\beta_0^*$, $\beta_1^*$, and $\beta_2^*$ be the least-squares estimates. Then any other value of $\beta_2$ must make $R^2$ worse; in particular $\beta_2=0$ must make $R^2$ worse (unless $\beta_2^*$ happens already to be exactly $0$).