9.4.4 Principal-Component Remappings
A principal-component remapping is a variables remapping in which 1R is approximated with a linear polynomial of some of its principal components. Such remappings can replace a high-dimensional, multicollinear key vector with a lower-dimensional, non-multicollinear key vector .
Consider a portfolio mapping 1P = θ(1R), where 1R is n-dimensional with conditional mean vector 1|0μ and multicollinear conditional covariance matrix 1|0Σ. We can represent 1R in terms of its principal components:
where ν is the n × n matrix whose columns are orthonormal eigenvectors of 1|0Σ, and 1D is an n-dimensional column vector of the principal components of 1R. We convert [9.71] to an approximate relationship by discarding principal components 1Di that have variances close to 0.9 Suppose we retain m principal components and discard n – m. Let be the m-dimensional vector of retained principal components. Let be the n × n matrix whose columns are the eigenvectors corresponding to those principal components. We obtain
Our portfolio remapping has form
When implementing a principal-component remapping, it is important to be aware of the scaling issue discussed in Section 3.7.3. This can be addressed by changing the units of measure for key factors, as appropriate.