I have come across a somewhat unusual (I think) estimation problem. I have two "coupled" linear regression models, $$Y = a + b x + \epsilon, \quad Z = c + d x + \nu$$ where $Y,Z,\epsilon,\nu$ are random variables and $a,b,c,d$ are sought parameters. The twist is that the common independent variable $x$ is unobservable; I only have a number of paired observations $$(y_1, z_1), (y_2, z_2), \dots (y_n, z_n) \ .$$ So $y_i, z_i$ originate from the same $x_i$ according to the linear models, but the $x_i$ is not available. If we just consider the regression lines $y = a + b x$ and $z = c + d x$, eliminating $x$ yields $$ \frac{1}{b}(y-a) = \frac{1}{d}(z-c) $$ which I suppose could be solved by regression. It's clear that not all parameters can be estimated from this, but I would guess that the ratio $b/d$ and perhaps the difference $a-c$ could be obtained.

My question is: what can be estimated in this situation, and is there some common model? Is this perhaps a special case of some hidden variable method?

In my scenario, $x_i, y_i$ and $z_i$ are all nonnegative (just to complicate things ...) but I would be interested in any information, on the unconstrained case as well.

I have come across a somewhat unusual (I think) estimation problem. I have two "coupled" linear regression models, $$Y = a + b x + \epsilon, \quad Z = c + d x + \nu$$ where $Y,Z,\epsilon,\nu$ are random variables and $a,b,c,d$ are sought parameters. The twist is that the common independent variable $x$ is unobservable; I only have a number of paired observations $$(y_1, z_1), (y_2, z_2), \dots (y_n, z_n) \ .$$ So $y_i, z_i$ originate from the same $x_i$ according to the linear models, but the $x_i$ is not available. If we just consider the regression lines $y = a + b x$ and $z = c + d x$, eliminating $x$ yields $$ \frac{1}{b}(y-a) = \frac{1}{d}(z-c) $$ which I suppose could be solved by regression. It's clear that not all parameters can be estimated from this, but I would guess that the ratio $b/d$ and perhaps the difference $a-c$ might be suitable parameters.

My question is: what can be estimated in this situation, and is there some well-known model? What form of parameters is most tractable, are small-sample distributions of estimators known (under normality assumptions, for example), etc. Is this perhaps a special case of some hidden variable method?

In my scenario, $x_i, y_i$ and $z_i$ are all nonnegative (just to complicate things ...) but I would be interested in any information on this problem, on the unconstrained case as well.