Appendix Two: Almon Distributed Lag
This appendix sets out the algebra behind the Almon distributed lag polynomials, both using a first order polynomial (which we use for education lags) and using a second order polynomial (which we use for both domestic and foreign R&D lags).
If we start with equation (9) such that
Then if we assume a second order polynomial, we can replace wi with
Or if we assume a first order polynomial, then
So we now have
or
respectively. That is, we have reduced the number of parameters to be estimated down to 3, in the case of the second order polynomial, or 2 parameters in the case of the first order polynomial.
While we did not constrain the endpoints for the first order polynomial (the lags of the education variable), we did constrain both endpoints of the second order polynomials (domestic and foreign research) such that
So we now have
which further reduced the number of parameters to be estimated to 1 (for each variable).
