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Matrix formulation for interpolation of data

In the attached article, the author proposes a way to interpolate quarterly values when only annual values are available. The only page of this article which is useful for this problem is page 66 (or the second page in this attachment).

I am trying to understand the very first step, which is how to get from the statement of the problem in equations (2.1) and (2.2), to its matrix formulation in equation (2.4).

If one of you understands how this matrix formulation is derived, I would be grateful if you could explain it step-by-step, in the case where we want to interpolate from quarterly to monthly data.

I would be very grateful to anyone who is able to help.

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Solution Preview

Please see the attached file for the complete answer to your question.


Update: What I presented in my attachment is the case where interpolation was performed from yearly data to quarterly data. ...

Solution Summary

A detailed explanation of how to get from the statement of the problem (of interpolation of quarterly values from annual values) given in equations (2.1) and (2.2) of the student's attached article to the matrix formulation in equation (2.4) is provided (in the OTA's attached file). Then an explanation of how the counterpart of that matrix formulation for the case of interpolating monthly data from quarterly data is given (in the "Problem Solution" box), by indicating the specific differences between the two cases.