If we find that one explanatory variable is influenced by possible omitted factors, can we model that with many instrumental variables? In case that it is valid to do that, can anyone provide some examples? Thank you very much.
If I understand your question correctly, the simple answer is "yes."
Expressed in linear-algebra formalism, mathematical models generally take the form:
B = TA
where A is a vector representing your "explanatory" variable(s), T is a tensor representing your model, and B is a vector representing your result(s). The number of elements of A is the number of explanatory variables. The number of elements in A must match the number of columns in T and the number of elements in B match the number of rows of T. There is no requirement that T be a square matrix.
If you find that one element of A results from a set of measurements C, you model that interaction as:
a = MC
where M represents the interaction that combines your measurements (C) to get your explanatory variable a, which is one element of the model input vector A. Since a is a scalar, the number of elements in M must equal the number of elements of C.
I hope this helps.
--- CGM
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