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If you look at changes of the points on the yield curve, then you probably find something stationary - right? Applying PCA on the covariance of these changes makes sense. E.g. you will find out that on PC describes a parallel shift (a change in the yield curve). Look at this question too: What do eigenvalues/eigenvectors of the yield/forward rates ...

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There is a vast literature on modelling time-series with periodcities. Rob Hyndman is one of the leading reseaerchers in this area. He has published the R package forecast and a free online text book on this subject (with another package and R code in the book). Your task is covered starting here.

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You have the risk factor $F$ and the asset that it is correlated to $r_m$. You can calculate the variances of each of these, say $\sigma^2_F$ and $\sigma^2_m$. If you do not care about the distribution but just work with variances and correlations then can look at an OLS setting: $$F = \beta r_m + \epsilon$$ with $\beta = \rho \frac{\sigma_F}{\sigma_m}$ ...

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There is a wide knowledge on correlation estimation, see other questions and answers: principal component analysis (PCA) - Equity Risk Model Using PCA random matrix theory (RMT) - Cleansing covariance matrices via Random matrix theory or Random matrix theory (RMT) in finance shrinkage - Portfolio Optimization : Shrinkage of Covariance Matrix when data is ...

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