If we interpret the identification command output in the ‘Collinearity patterns’ figures as correlation coefficients we would expect these figures to be symmetric. In our experience these figures are mostly symmetric, but there are almost always a few values which are not. Can someone explain this to us?

The plot of collinearity patterns does not show graphically a correlation matrix and therefore has not to be symmetric.
Instead, in each row (that correponds to each parameter thetaj ), it shows the group of parameters that fits better (i.e. provides the largest r2 or cosine among all possible combinations of parameters) the derivatives of the moments w.r.t. thetaj. In other words, if J is the Jacobian matrix of the moments w.r.t. all parameters, each column j is regressed with all possible linear combinations (single parameters, groups of two, three parameters and so on) of the other columns and the combination with the best ‘fit’ is reported (such linear regressions do not contain the constant).

In this case, it is always possible that the plot is non symmetric because if column 1 is best regressed by columns 2 and 3 does NOT imply that column 3 is best regressed by columns 1 and 2 but perhaps by column 1 and 4 or even columns 4 and 5 (this is linked to both the missing constant in the regressions and in different norms of the columns that are combined together). See for example Figure 7 at page 45 of publications.jrc.ec.europa.eu/re … iv_312.pdf.