Question

You see scatter plots of three pairs: Y versus X1, Y versus X2, and Y versus X3. If each of the first two scatters is fairly close to a straight line, but the third is more a shapeless swarm of points, which of the following can you conclude?

a. It is possible that a regression with all three X's could have an appreciably higher R-square value than a regression with only X1 and X2, due to the correlation structure among the X variables.

b. The R-square value for a regression with all three X's could actually be slightly smaller than the R-square value for a regression with only X1 and X2.

c. The R-square value for a regression with all three X's will be at best marginally larger than the R-square value for a regression with only X1 and X2.

d. None of these choices are true.

Answer 1

If Y shows linear correlation with X1 and X2 but not X3 , it means , Y depends on X1 and X2 and is mostly independent of X3 .

But R2 is so designed that , if you keep adding independent variables , it will keep increasing even if the new variable has no correlation with dependent variable . But the amount of increase depends on whether the new variable is correlated or not .

So, option c is correct .

NOTE : But , if we calculate adjusted R2 , we would get a decreased value after adding X3 in these scenario.