Two sets of input variables for the same unknown dependent variable I currently have two sets of input variables say, $X$ and $Y$ with one output variable $Z$. That is:
$$Z = a_0 + a_1X_1 + a_2X_2... + a_{11}X_{11} = b_0 + b_1Y_1 + b_2Y_2 + b_3Y_3 + b_4Y_4$$
I have the independent $X$ and $Y$ values but don't have the dependent variable $Z$ values.
Is there anyway that I can estimate coefficients $a$ and $b$ and also the value of R squared?
 A: This sounds to me like problem where a canonical correlation study might help. In canonical correlation, we are given a random vector W that is partitioned into two sub-random vectors X and Y; and the issue is to find linear combinations of the two subvectors that have maximal correlation and are orthogonal to one another.
The end result is the discovery of correlates that will point to relationships that have some physical meaning.
If you are comfortable with matrices, then Google Wolfgang Hardle and Leopold Simar's online book and read Chapter 14 on Canonical Correlation. If you use R, then read Chapter 10 of Paul Hewson's online book "Multivariate Statistics with R". 
A: Are you sure there's a Z variable?    Since you have an equation of X's versus Y's, isn't one of the X's or Y's the dependent variable?  For example:
$$X_1 = - newa_2X_2... - newa_{11}X_{11} + newb_0 + newb_1Y_1 + newb_2Y_2 + new b_3Y_3 + newb_4Y_4$$
A: Canonical Correlation Analysis was one way to go and it works! Credits to @schenectady for this. Thanks a lot for your help.
I want to write this for future reference and for others who might have a similar query, if you want to perform a regression analysis in such a situation, you should attempt to minimize the squared errors from each of your equations, i.e.
$$\left\{\begin{array}{l} a_0 + a_1X_1 + a_2X_2 + \dots + a_{11}X_{11}\\ b_0 + b_1Y_1 + b_2Y_2 + b_3Y_3 + b_4Y_4\end{array}\right.$$
The simplest way is to use the excel solver to arrive at the coefficients with the aim of minimizing the mean of the squared errors from the above two equations.
