Data observed with multiple but unpaired/unmatching observations I have a dataset with repeated measures, in which for each unit $i$, we observe multiple observations (for the same variable) $y_{ij}$ and multiple $x_{ik}$, but we don't know which $y_{ij}$ observations correspond to which $x_{ik}$ observations, (i.e. we don't necessarily have $j=k$). In fact, I usually observe 3 $y_{ij}$ and 10 $x_{ik}$ for each $i=1 \ldots N, N \approx 1000$. Written differently, my data is like $\{ \{y_{1a}, y_{1b}, y_{1c}; x_{1,1}, \ldots x_{1,10}\}, \ldots, \{y_{Na}, y_{Nb}, y_{Nc}; x_{N,1}, \ldots x_{N,10}\}\}$
The goal of the analysis is to run a regression of y on x. My assumption is that the true model is $y_{ij}=\alpha + \beta x_{ij}+\epsilon_{ij}$. I can't run this regression since I don't know the pairing of each j and k.  For now I average observations for each unit, and regress  $y_{i\cdot}\sim x_{i\cdot}$. I believe this leads to an unbiased/consistent estimator only if $K\to \infty$ and $J\to \infty$, which is not my case (for me $J=3$).
My questions:


*

*Is there a more specific name for this kind of unpaired/unmatched data ? 

*More importantly, is there a literature associated to this? Can I do better than averaging my $x_i$ and $y_i$? One idea could be to match quantiles (over i) of $y_{ij}$ to those of $x_{ik}$, or run the regression for a lot of combinations of $y_{ij}$ and $x_{ik}$... Or do some clustering?
 A: For each Y_i randomly select a j and for the corresponding X_i randomly select a k. Now fit the model Y ~ mX + c. Now repeat the above for 1000 iteration by randomly selecting values of j and k for each i. At the end, average the models and provide confidence interval.
A: For every specimen $i$ you can compute means and error bars for your observations: 
$\bar y \pm \sigma_y$ and $\bar x \pm \sigma_x$
It is then usual (as implemented in the python statsmodels package, weighted least squares class) to perform a regression on the $\bar y, \bar x$ where the contribution of each specimen to the error function is weighted inversely with the standard deviation of the specimen's observations.
This way, the uncertainty in the $x_i, y_i$ is taken into account when regressing, the result is reproducable and limites for $K \to \infty, J \to \infty$ are sensible. Moreover, this method obviousely scales to more specimen and more observations.
A: Using averages is just a between estimator in the literature on panel data analysis. 
Using averages is also the main justification for variance weights, e.g. the explanation of aweights in Stata. (as in Neuneck's answer.)
Another related issue is estimation with aggregates, e.g. if you only observe group totals, where groups could be states or countries or industries
Consistency is obtained as N goes to infinity, K, and J can be fixed and finite.
My guess is that you cannot do better under your assumption that the linear model is correctly specified if you have enough variation of x across units.
Without within variation it would be a problem with unobserved heterogeneity, e.g. if there is a unit specific effect or there is not enough between information, e.g. every one has the same number of repeat observations for each levels of a categorical variable.
