Range of $R^2$ of the model with two predictors given the $R^2$s of univariate models of each predictor We have a dependent variable $Y$ and two predictors $X_1,X_2$.
Given a fixed dataset, $R^2$ of the model $Y$~$X_1=t_1$ and
$R^2$ of the model $Y$~$X_2=t_2$.
What is the range of $R^2$ of the model $Y$~$X_1+X_2$?
My attempt
I tried to use the fact that $R^2$ of the model with a single predictor is $(correlation)^2$ but that didn't help me.
Also, I feel that if the predictors are fully correlated, then the $R^2$ model with two predictors shouldn't change. However, we don't have $t_1=t_2$, so we can't have $1$ or $-1$ correlation.
If the predictors are orthogonal, then I can see that the coefficients of the predictors in the joint model is the same as their coeff in their univariate model(by basic matrix calculations).
Note that Feel free to answer different versions of the problem like models with the intercepts.
 A: The range is $[max(t_1, t_2),\ min(t_1+t_2, 1)]$.
Of course you can't obtain a $R^2$ lesser than $t_1$ or $t_2$, because $R^2$ always increases when you add variables. Also, you can't have any $R^2 > 1$, but there's more to it, let's see...
In order to compute the value of $R^2$, you must decompose the total sum of squares of $Y$: $SS_Y$. This amount behaves like a pie chart, or maybe more like a Venn diagram: part of $SS_Y$ is explained by $X_1$, part of it is explained by $X_2$, and part of it by neither of them. Also, the parts explained by $X_1$ and $X_2$ likely overlap (if they don't, it means that $X_1 \perp X_2$). If you don't know how much is this overlap, you can tell that the final $R^2$ won't be greater than $t_1+t_2$ anyway.
It is possible that the part of $SS_Y$ explained by one variable is totally included in the part explained by the other, so the resulting $R^2$ will be no higher than $max(t_1, t_2)$. For this to happen the predictor that's less correlated to $Y$ must also be orthogonal to the subspace generated by the other predictor and $Y$.
Why is this
If you are not good with thinking of $SS_Y$ as a pie being divided among explanatory variables, you must know that this happens because least squares regression maps a target variable $Y$ to its projection on the subspace generated by the matrix $X$. You can get some context in this lesson I've found with some quick googling. The explained sum of squares $SSM$ is the square euclidean norm of this projected vector, and as you add another vector to it, the resulting squared norm will be the sum of the squared norm of the addends, as long as they are orthogonal. Add this to what I've said before and you will figure out the passages.
