# 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?

• Why not regress 0 on $1, X_1, \ldots, X_{11}, Y_1, \ldots, Y_4$? The difference $a_0 - b_0$ will not be identifiable, but in retrospect that's obvious. And of course you cannot obtain an $R^2$ for either of your models, because you have absolutely no information about the variation of $Z$; however, the $R^2$ for the regression involving all $X$ and $Y$ together is informative: it tells you how closely the two models can typically be made to agree (although the MSE is much more useful in that regard).
– whuber
Mar 23 '11 at 16:42
• There is one problem with the method you have suggested tho, i.e. all my co-eff & intercept comes out as zero... Mar 24 '11 at 6:57
• That's the point, isn't it? Your problem is not solvable because you have no observations of $Z$! It's possible you could make progress by constraining the coefficients to be nonzero, perhaps through a Bayesian hierarchical model. That at least would indicate whether some linear combinations of the $X$ tend to agree with some linear combination of the $Y$. But that's as far as you can get.
– whuber
Mar 24 '11 at 14:13

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".

• Canonical correlation was perfect! I have completed my analysis and got appropriate results as well. Thanks a lot! Apr 9 '11 at 7:41

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:

$$X_1 = - newa_2X_2... - newa_{11}X_{11} + newb_0 + newb_1Y_1 + newb_2Y_2 + new b_3Y_3 + newb_4Y_4$$

• This is theoretically incorrect simply because none of the X & Y variables are dependent variables. Hence, I doubt if doing a regression would be the right thing to do. Mar 24 '11 at 12:55
• @Santosh: If your above equation is correct (not overspecified), then not all of those variables are independent. One is a linear combination of the others. Mar 24 '11 at 15:27
• @Santosh: Is this a homework problem? Mar 24 '11 at 16:49
• No this isn't a homework problem :) but on a serious note let me explain it with my actual problem, it might clear things up. I am looking at different aspects of a product with consumer satisfaction as the indicator. Sadly, I don't have the overall consumer satisfaction but I have the satisfaction ratings for individual characteristics of the product - intrinsic factors like features & I also have the extrinsic satisfaction factors like satisfying needs, expectations, etc. Each of the extrinsic factors are independent to each other but dependent on the intrinsic factors. Apr 9 '11 at 7:39

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.