I am working on alternative ways for the estimation of variance-covariance matrices. For this I have already estimated the sample variance-covariance matrix, single index covariance matrix. I also want to estimate the covariance matrix by principal component analysis (PCA). As I have 5 different types of asset returns and factors which are assumed to drive these returns are 6 in numbers like (Inflation, interest rate etc).
Kindly guide me what is the procedure to estimate this covariance matrix by PCA.
Best Answer
Quoting from the link in the above question, the methodology is as follow
This technique is often used when the number of assets N is close to the number samples T, leading to spurious correlations in the sample covariance and when N > T, a sample covariance matrix which is singular.
Example
As a concrete example, here is an implementation in
R
for returns generated from the 1 factor model$$ R_{t} = m_{t}\beta + \epsilon_{t} $$
where $R_{t}$ is an Nx1 vector of returns at time t, $m_t$ is the market return at time t, $\beta$ are the Nx1 betas of the assets to the market return and $\epsilon_{t}$ is Nx1 gaussian noise at time t
Keeping all the factors, we can reconstruct the sample variance exactly (modulo machine precision)
Or we can drop PCs with less variance. A detailed answer discussing this is Relationship between SVD and PCA. Here we choose only the first PC, with the omniscience that this is a 1 factor model.
Comparing the PCA covariance and sample covariance to the model covariance, $Var(m_{t}\beta)$, we can see improvements across a variety of distance metrics.