Linear Regression: reversing the roles of X and Y

#1
Simple linear regression:
Y = β0 + β1 *X + ε , where ε is random error


Fitted (predicted) value of Y for each X is:
^
Y = b0 + b1 *X (e.g. Y hat = 7.2 + 2.6 X)


Consider
^
X = b0' + b1' *Y

[the b0,b1,b0', and b1' are least-square estimates of the β's]

Prove whether or not we can get the values of bo,b1 from bo',b1'. If not, why not?


Completely clueless...Any help is greatly appreciated!
 

vinux

Dark Knight
#2
Simple linear regression:
Y = β0 + β1 *X + ε , where ε is random error


Fitted (predicted) value of Y for each X is:
^
Y = b0 + b1 *X (e.g. Y hat = 7.2 + 2.6 X)


Consider
^
X = b0' + b1' *Y

[the b0,b1,b0', and b1' are least-square estimates of the β's]

Prove whether or not we can get the values of bo,b1 from bo',b1'. If not, why not?


Completely clueless...Any help is greatly appreciated!
We will not be get the values (bo,b1) from (bo',b1') until if we know some information or assumption.

In the first line
^
Y = b0 + b1 *X
where

ie we need, Sxy, Sx2,xbar , ybar to build the Y on X line.

But out of this four we will have only two parameters.. ie if we know b0 and b1.. we will not be able to say what is xbar, ybar etc...

And for X on Y line we need Sxy, Sy2, xbar,ybar. now we can mathematically prove that we can't estimate X on Y line from Y on X (vice versa).
 
#3
We shall assume that pairs of data for variables x & y are available.
y = a1 + b1*x and x= a2 + b2 * y
Then b2 =b1 * (Sxx/Syy) and a2= a1 +b1*x-b2*x
 
#4
We will not be get the values (bo,b1) from (bo',b1') until if we know some information or assumption.

In the first line
^
Y = b0 + b1 *X
where

ie we need, Sxy, Sx2,xbar , ybar to build the Y on X line.

But out of this four we will have only two parameters.. ie if we know b0 and b1.. we will not be able to say what is xbar, ybar etc...

And for X on Y line we need Sxy, Sy2, xbar,ybar. now we can mathematically prove that we can't estimate X on Y line from Y on X (vice versa).
But if we don't have the values of X bar, Y bar, how can we find bo and b1 in the first place?