Z2t = J2 i2 Xt-i = (Xt-1 + 4 Xt-2 + 9 Xt-3 + 16 Xt-4 + 25 Xt-5) i=0
Notice that the Z's are linear combinations of the original X's. Also notice why the Z's are likely to exhibit multicollinearity.
Before proceeding to a numerical example, note the advantages of the Almon method. First, it provides a flexible method of incorporating a variety of lag structures (see exercise 17.17). The Koyck technique, on the other hand, is quite rigid in that it assumes that the 0's decline geometrically. Second, unlike the Koyck technique, in the Almon method we do not have to worry about the presence of the lagged dependent variable as an explanatory variable in the model and the problems it creates for estimation. Finally, if a sufficiently low-degree polynomial can be fitted, the number of coefficients to be estimated (the a's) is considerably smaller than the original number of coefficients (the 0's).
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But let us re-emphasize the problems with the Almon technique. First, the degree of the polynomial as well as the maximum value of the lag is largely a subjective decision. Second, for reasons noted previously, the Z variables are likely to exhibit multicollinearity. Therefore, in models like (17.13.9) the estimated as are likely to show large standard errors (relative to the values of these coefficients), thereby rendering one or more such coefficients statistically insignificant on the basis of the conventional t test. But this does not necessarily mean that one or more of the original ¡3 coefficients will also be statistically insignificant. (The proof of this statement is slightly involved but is suggested in exercise 17.18.) As a result, the multi-collinearity problem may not be as serious as one might think. Besides, as we know, in cases of multicollinearity even if we cannot estimate an individual coefficient precisely, a linear combination of such coefficients (the estimable function) can be estimated more precisely.
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