My question in short is as follows: can I take main principal component of historical covariance matrix and use it as historical volatilities when fitting a binomial tree?
Here's more detailed description: let's suppose I want to build Hull-White or BDT tree. They both need current rates curve $r(t)$ and volatility curve $\sigma(t)$.
Let's also suppose I have a history of interest rates in columns
r(1) r(2) r(3) r(4) <---- index means they are of different terms
6% 7% 8% 8.5% <----- example numbers
4% 6.5% 9% 10%
... ... ... ...
7% 11% 13% 14%
What can I do to estimate volatility? Easiest way is to calculate standard deviation of differenced values in each column (hope it is evident what I mean).
Another approach is to obtain whole covariance matrix $\Omega$, which items are $\Omega_{i,j} = \sigma_i \sigma_j \rho_{i,j}$. In this case our standard deviations from previous approach will lie on main diagonal of that matrix.
What we now can do is apply PCA so that to have covariance matrix decomposed into main components. This means I decompose how matrix $\Omega$ multiples by some vector $x$ in the following way: $\Omega x = \left(\lambda_1 u_1 u_1^T + ... + \lambda_N u_N u_N^T\right) x$. And if I take first principal componenent $u_1$ which corresponds to largest $\lambda$, I have some vector with volatilies along the the time, which gives most effect: $\Omega x \approx \lambda_1 u_1 u_1^T x$.
So, I want to understand if I could use that $u_1$ instead of vector of simple standard deviations, in a hope that such choice will help me better capture term structure behavior?