Hello Wise mathematicians! I have few quenstions about Multi linear regresstion. I've been asked from my friend, but I have very weak knowledge background from that field. It seems my friend is in hurry, I want to discuss following problems with all yours.
Suppose two predictor linear regression model given as $$ Y = B_{1} + B_{2}x_{2} + B_{3}x_{3} + \epsilon $$
Is is possible to have $\gamma_{yx_{2}} = 0$ but $\hat\beta_{2} \neq 0$? (where $\gamma_{yx_{2}}$ is the correlation coefficient between $y$ and $x_{2}$)
Is it possible to have $\gamma_{yx_{2}} = 0$ and $\gamma_{yx_{3}} = 0$ but model R-square > 0?
Somebody stated "If the variable $Z$ has a positive correlation with both of $X$ and $Y$, then there exists a positive correlation with $Y$ and $Z$" Is it true? or false? Justify it.
My prospective answer only depending on my intuition for 1 and 3 is
- YES, 3. No
For 1, Only considering $Y$ and $x_{2}$, I can make the counterexample by constructing independent event.
For 3, I also can construct the pairwise independent event, but not mutual event.
Could anybody can make the answer more perfect? As I only learned elementary probability theory and Mathematical Statistics, I am uncertain my prospective answer can be extended to the multi linear regresstion.
Thank for your answer in advance :)
So far, What I think is following.
1.Possible. Let $X $~ $N(0,1)$ (normal dist.) and $Y $~ $X^2$. Then $X,Y$ has $0$ correlation but they are not independent. (However, I don't know how to write the answer with respect to $\beta$ coefficient.
3.Let $X = {0,1}$ for the coin A Head or Tail, and $Y = {0,1}$ the coin B Head or Tail. Let $Z = 0,1$ the bell rings or not, and the bell rings only the coin $A$ and $B$ turns out with same sides (Both head, Both tail). Then $Z$ has positive correlation with both $X, Y$ but $X, Y$ are independent.
Am I right? and could anybody add up my answer with $\beta$ coefficient?