When and to what extent should the money supply of a pegged currency be included in the "parent" currency?

Question

For example, the Manx pound is in a de facto currency union with GBP. The Isle of Man Bank chooses to back IMP one-to-one with Bank of England notes, but is not legally required to do so. At one-to-one this would automatically be part of the GBP money supply. In principle it could expand the IMP money supply, and since IMP is tied to GBP, one could argue this increase should be included in the GBP money supply.

The Singapore dollar and Brunei Dollar are pegged to each other and the total money supply managed by the MAS.

On the other hand, the HKD is pegged to the USD without one-to-one backing and the federal reserve has no control over it (HKD being managed by Hong Kong Monetary Authority).

During the Irish Pound-Sterling pegging era, we again had separate monetary authorities, but currency union. First through Bretton-Woods and later "managed". In this case GBP was mostly accepted for trade in Ireland, while IEP was rarely accepted in the UK. Would GBP be in the IEP money supply, but the IEP not in the GBP?

When performing an analysis on currencies of this nature, how should we determine what to include and not in the money supply calculation?

Answer 1

During the Irish Pound-Sterling pegging era, we again had separate monetary authorities, but currency union. (...) In this case GBP was mostly accepted for trade in Ireland, while IEP was rarely accepted in the UK. Would GBP be in the IEP money supply, but the IEP not in the GBP?

Let the states and the institutions declare what they want. To study economic phenomena we must look at what economies actually do. Think of something totally unrelated: each state has a specific tax structure, a range of various taxes, progressive, proportional, whatever. The official structure matters, because it anchors the effective taxation, but effective taxation itself matters more.
In your case, the Irish Economy incorporated in its workings the official monetary union, while the UK Economy didn't.

So one has to go per case, which creates a testable hypothesis:

a) postulate an hypothesis based on "casual" (i.e. true but unquantified) observations like the 2nd sentence in the above quote
b) derive the theoretical implications of the hypothesis
c) send your theory and your theoretical results out to meet the data.

For your example, assume that UK increased steadily its money supply to accommodate a growing economy, while the Irish economy was stable (or stagnant, depending on your philosophical point of view), or grew at a lower rate. If we add the GBP quantity in the money supply variable of the Irish Economy as equivalent, $1:1$ to the IEP, shouldn't we observe excessive inflation? We should. Did we? Let's say "partly". Then we can estimate the effective ratio between the two currencies as regards their effect on the Irish Economy.