can we correlate two regression equations whose input datasets have different lengths?

If you mean correlated errors like in a seemingly unrelated regressions (SUR) model, then that only makes sense if the observations for the two regressions are paired. If some of the inputs are paired and some are missing one of the components, that’s also easy to fit if you’re using multivariate normal errors because the marginals for any subset of the data is easy to calculate by just taking the appropriate rows/columns of mean and covariance.

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