Factor Estimation: Filtering and Black-Litterman

We mentioned in Chapter 2 that the factor process X(t) in our models has two possible interpretations. Its components Xi(t) may represent observable data series, either financial data such as stock indices, bond yield spreads etc., or macroeconomic data such as GDP growth, employment data or confidence indices. Alternatively, Xi(t) may be an unobserved ‘latent variable’ factor whose role is to represent the inherent uncertainty of asset growth rates. In previous chapters we have always assumed that the X(t) process is observable, so implicitly we are adopting the first interpretation. In this chapter we outline what to do if the factors are not observable, in the context of the diffusion factor model $$\hat X(t)$$ used in Chapters 2–7. In a word, we can use the Kalman filter to estimate X(t) and then replace (15.1) by the Kalman filter equation and the asset price model by the so-called innovations representation. We are then back in a ‘completely observable’ world where X(t) has been replaced by its Kalman filter estimate $$\hat X(t)$$, and can apply all the results for the ‘observable’ models of the preceding chapters. This idea was used by Nagai and Peng (2002) and Nagai (2004), but in concept goes back at least to work by Wonham (1968) on the ‘separation theorem’ of stochastic control…