Why does adding more terms into a linear model always increase the r-squared value? Many statistics textbooks state that adding more terms into a linear model always reduces the sum of squares and in turn increases the r-squared value. This has led to the use of the adjusted r-squared. But is it possible that adding a term into a linear model reduces the sum of squares by zero and in turns keep the r squared value exactly the same? 
 A: Certainly this can happen: if the new predictor is contained in the linear span of the predictors already in the model.
Think about it geometrically: your new "fitting subspace" (the possible linear combinations of your predictors) is exactly the same as the old one, so the optimal fit and the sum of squares is unchanged.
However, this is only a sufficient condition for $R^2$ to be unchanged, not a necessary one. Consider three points like this:
xx <- c(-1,0,1)
yy <- c(1,-2,1)
plot(xx,yy,pch=19)
abline(h=0)
abline(v=0)

model.1 <- lm(yy~1)
abline(model.1,col="red",lty=2)
summary(model.1)

model.2 <- lm(yy~xx)
abline(model.2,col="green",lty=3)
summary(model.2)


If we add xx as a predictor to the simple mean model, we get the same fit and the same $R^2$. Such a construction should be possible with larger models, as well.
A: Adding more terms into a linear model may keep the r squared value exactly the same or increase the r squared value. It is called non-decreasing property of R square.
To demonstrate this property, first recall that the objective of least squares linear regression is
$$
min{SSE}=min\displaystyle\sum\limits_{i=1}^n \left(e_i \right)^2= min_{\beta}\sum_{i=1}^n\left(y_i -\beta_0 - \beta_1x_{i,1} - \beta_2x_{i,2} -…- \beta_px_{i,p}\right)^2
$$
R square is
$$
R^2=1-\frac{SSE}{SST}
$$
When the extra variable is included, the objective of least squares linear regression becomes
$$
min{SSE}=min_{\beta}\sum_{i=1}^n\left(y_i -\beta_0 - \beta_1x_{i,1} - \beta_2x_{i,2} -…- \beta_px_{i,p}-\beta_{p+1}x_{i,p+1}\right)^2
$$
If extra estimated coefficient($\beta_{p+1}$) is zero, the SSE and the R square will stay unchanged. Or if extra estimated coefficient($\beta_{p+1}$) takes a nonzero value , the SSE will reduce. In this case, the R square will increase, because it improves the quality of the fit.
