Are high R square and low t-stats a signal for multicollinearity? What is the nature of this problem and correction? Also, how do you generally decide if the problem is high enough to be corrected?
It’s an indicator but I personally would never decide based on them.
$t$- statistics tests the null that individual coefficient $\beta_i$ is 0 against alternative that it is statistically different from 0.
$F$-test which is increasing function of $R^2$:
Which tests the hypothesis that all $\beta$ coefficients are 0 against alternative that some of them aren’t.
$F$-test is not affected by multicollinearity and it will be high if $R^2$ is high. Hence if all your $t$-tests say all variables are insignificant but $F$-test (based on high $R^2$) says some of them should be significant it’s an indicator for multicollinearity. However, it could be also due to other problems so it’s always best to do some direct test - calculate correlation between independent regressors or even better something like calculating variance inflation factor (VIF) or some more fancy model.
Generally the rule of thumb is to remove variable if it’s VIF is above 5 some sources say 10.