R-square, Adjusted R-square and Multicollinearity
Can you explain whether the following statements are correct or not? Thanks!
1)R2 is always equal to 1 when the number of sample observations is equal to the number of coefficients to be estimated.
2)When a new explanatory variable is added to a model, Adjusted R2 will fall if the new variable is insignificant even if R2 rises.
3) When multicollinearity is serious, the likelihood of wrongly rejecting the null of a zero coefficient value increases.
4) When multicollinearity is serious, the likelihood of correctly accepting the null of a zero coefficient value decreases.
5)There are two regressions relating Y to X: one uses male observations and the other uses female observations. Suppose the Adjusted R2 in the male and female regressions are 0.89 and 0.35 respectively. It is legitimate to conclude that the male regression provides better fit to Y than the female regression does in terms of Adjusted R2.