By Girosi F.,King G.
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Extra info for Demographic Forecasting
To explain what “more important” means, suppose we estimate a specification with two components and ask: If we must drop one of the two principal components, which one should we drop? We clearly should drop the second, since the first principal component is optimal (in the sense of maximum likelihood) for the specifications with only one component. Therefore we should think of the principal components as a nested set of models of the data: We start from a model with one principal component only, which explains a certain percentage of the variance in the data.
Of course, the result is very different since, for example, even identical explanatory variables produce more efficient results and the precise information assumed and modeled is very different. The technology to produce our estimates may seem complicated, but the logic and the end result are quite simple. Indeed, the end result is still a set of regression coefficients, predicted values, and any other quantities of interest that normally come from a linear regression. The only difference is that the new estimates can include considerably more information and will have generally superior statistical properties to the old ones.
However, the usefulness of principal component analysis lies in the fact that in many real data sets, relatively few principal components can provide a good approximation to the original data. Suppose for example that four principal components provide enough flexibility to model the age profiles for a particular combination of country/cause/gender, and that we have 17 age groups (A = 17). That means that instead of having to forecast 17 time series, one for each group, we only have to forecast four time series, those corresponding to γ1 , .