Yes, I believe that’s the case. Thank you for tagging @verdi_green, I would really appreciate his help 
Just two quick follow up questions:
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If a time lag was the issue, could that be sorted by changing the lag structure in the recursive matrix multiplication used to simulate the data, or would that be a deeper rooted problem?
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When you wrote this:
did you mean to imply that in case of limited information techniques we would not expect to retrieve the observed data exactly?