Why use R^2 adjusted over R^2 in multiple regression? I read that when adding variables or predictors to the multiple regression model, R^2 tends to go up even if the variable or predictor doesn't add any predictive power. Why is this?
In linear regression, I understand $R^2$ to equal:
$$R^2 = 1 - SSE/SST$$  
Why does this go up when we add a new variable or predictor to the model regardless of whether the new variable adds predictive value? Why do we need adjusted $R^2$?
I understand $R^2$ to equal the % of variability explained by the model.
 A: The reason is because adding variables to a model can just randomly align to some kind of random variance (i.e. noise). Pure chance could make it seems like its a meaningful predictor rather than its real explanatory power. If you throw mud on the wall, some of it will stick. 
So what an adjusted r squared does is subtract the expected random luck of prediction from your r squared value. 
Its like the old saying, a broken clock is right twice a day. R squared counts those two times and adjusted r square kicks them out. 
A: In your formula $SST$ is a constant and is the same regardless of how you want to model your multiple linear regression model.
$SSE$ is an non-decreasing function over the number of predictors. It goes up (bounded by $SST$) unless the new predictor is perfectly correlated with existing predictors. You can think like this: in the worse case the new predictor can get a zero coefficient, so adding a new predictor should never worsen the performance.
Thus, as you add more predictors into the model the $R^2$ measure never decrease. This is bad for forming a predictive model because $R^2$ encourages overfitting. Adjusted $R^2$ takes that into account and put a penalty for the number of predictors you have in the model. 
However, if your goal is not to predict but for exploratory analysis. It's reasonable to use the $R^2$ measure.
