Linear regressions with similar coefficients If one has multiple similar dependent variables for which one wants to fit linear regressions
y1 = c11*x1 + c10 + e1
y2 = c21*x2 + c20 + e2
y3 = c31*x3 + c30 + e3

one can estimate the regressions separately for y1, y2, y3 or one can pool the data for y1, y2, y3 and estimate a single regression. This forces c11=c21=c31 and c10=c20=c30. A middle ground is to estimate the regressions separately but with penalties on the variance of [c11 c21 c21] and [c10 c20 c30]. The larger the penalties, the closer one gets to a pooled regression.
Does this model have a name? Should one use cross-validation to determine the sizes of the variance penalties?
 A: Here's an approach I might take, which relies on ridge regression.
First, pool all the data, then create dummy variables $d_1$ to $d_3$ to represent which sample a given row comes from. Then, run a ridge regression with the following coefficients:
$y = c_{\_1}x + c_{11}d_1x + c_{21}d_2x + c_{31}d_3x + c_{\_0} + c_{10}d_1 + c_{20}d_2 + c_{30}d_3 + e$
In this model, the $c_{.1}$ coefficients represent the difference between the overall slope $c_{\_1}$ and the sample-specific slopes. If these are regularized to zero, then the samples will just have the overall slope. The $c_{.0}$ coefficients represent the difference between the overall intercept $c_{\_0}$ and the sample-specific intercepts. A ridge regression puts a penalty on the $l_2$ norm (i.e., the variance) of the coefficients. We put the penalty on all the coefficients except $c_{\_1}$ and $c_{\_0}$ because we're concerned with regularizing the deviations from these overall coefficients rather than the overall coefficients themselves. 
So, our ridge regression loss function looks like this:
$L =\frac{1}{n}\sum_{i}^n{(y - c_{\_1}x - c_{11}d_1x - c_{21}d_2x - c_{31}d_3x - c_{\_0} - c_{10}d_1 - c_{20}d_2 - c_{30}d_3)^2} + \lambda (c_{11}^2+c_{21}^2+c_{31}^2+c_{10}^2+c_{20}^2+c_{30}^2)$
You choose the values of the coefficients that minimize $L$ for a given value of $\lambda$, which you can choose using a variety of methods. Finally, you can get the sample-specific slopes and intercepts by adding the overall slope and intercept to the sample-specific slope and intercept for each sample.
