Exercise 5.2 Suppose that the means and standard deviations of Y and X are the s
ID: 3354565 • Letter: E
Question
Exercise 5.2 Suppose that the means and standard deviations of Y and X are the same: Y=Xand Sy = S (a) Show that, under these circumstances, where Byx is the least-squares slope for the simple regression of Y on X, Bxir is the least-squares slope for the simple regression of X on Y, and rxr is the correlation between the two variables. Show that the intercepts are also the same, Arx = Ax Why, if Arx = Ary and Brix-Bry, is the least-squares line for the regression of Y on X different from the line for the regression ofX on Y (as long as (b)Explanation / Answer
Y =A(Y/X) +B(Y/X) X
X=A(X/Y) +B(X/Y) Y
A(Y/X)=A(X/Y) and B(Y/X)=B(X/Y)
The intercepts and the slopes are equal to each other and for both regressions
Also if you may visualize the scatter plot , it will be such that the trend line is Y=X and equal amount of data points will be on either side of the trend line. Also each of the data point will have mirror image point(another data point on the other side of the trend line at the same perpendicular distance from it) with respect to the trend line.
The trend line will be the mean line.
Clearly from above we can conclude that the data points in both the sets (Y and X) are the same but they do not correspond to each other.
This can happen when few of the son's have height equal to few of the fathers so much so that each of the son must have a father equal to his height and vice versa.
Now think about a father whose height is below mean.
Clearly his son must be someone whose height is also less than the mean .this is so because on the father -son height point lies below the trend line (mean) .Hence if Y is negetive with respect to mean (trend line) the its corresponding X will also be negetive.
Simliarly , if father's height is above trend line then ,the point on sacatter plot corresponding to father son height will lie on the upper side (+ve) of the trend line. Hence if father's height is higher than mean, then son's height will also have to be hiegher than mean.
e). This is a weak research design because:-
It can be improved by:-
1. Taking into account all variations.
2. smothening of the equation.
3. Dummy variables.
4. Expanding the data set to more years of observation.