FACTOR MODEL ESTIMATION
In this section, we provide first a step-by-step procedure for estimating the factor model based on the popular and implementable approach, the principal components analysis (PCA), to which a detailed and intuitive introduction is provided in the appendix. PCA is a statistical tool that is used by statisticians to determine factors with statistical learning techniques when factors are not observable. That is, given a variance–covariance matrix, a statisician can determine factors using the technique of PCA. Then, after introducing the computational procedure, we provide an application to identify three factors for bond returns. Finally, we outline some alternative procedures for estimating the factor models and their extensions.
Computational Procedure
Based on our latent models, we need to consider only how to estimate the latent factors 
from the K-factor model,
where
from the K-factor model,
This version of the factor model is obtained in two steps. We de-mean first the factor ft so that the alphas are the expected returns of the assets. Second, we de-mean again the asset returns. In other words, we let .
In practice, suppose that we have return data on N risky assets ...
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