PCA factors not uncorrelated

Question

I ran into an interesting case recently. I am trying to construct a set of uncorrelated factors for a statistical factor model. I have started with picking a certain amount of assets (indices) which I believe would fully capture the different dynamics of the US market. Naturally the set of indices are not uncorrelated so after making the data stationary (i.e taking the time series of daily returns) I perform PCA to extract the common independent factors. After choosing an appropriate number of components I can then use the matrix with factor loadings of the different PCs and the matrix of the time series of the different assets to obtain historical factor returns for my new PC factors.

Here is where it gets interesting and unexpected for me. I tried performing the procedure first by standard scaling the returns of each asset (mean 0 and unit variance). Interestingly enough doing it this way yielded PC factors whose returns are quite correlated. I then tried directly performing the PCA on the returns without scaling them and then obtained factor returns which as expected are uncorrelated. Furthermore I was particularly surprised that the factors obtained via the non-scaled returns had on average better explanatory power when running regressions for different assets I am trying to explain with the factor model. What is the deal here? I have always thought that it is better to scale data when performing PCA while in this case my empirical results show the opposite.

The only explanation I can come up with so far is that by taking the returns of the price series in the first place I have already 'scaled' and 'centered' the data in some sense and that the subsequent scaling simply washes away important information from my data. Any further ideas?

Answer 1

If $X1$ and $X2$ are almost perfectly correlated (but variance of x1 is a million times variance of x2) so that the first (normalised) pca factor is (1/sqrt(2),1/sqrt(2)) and second is (1/sqrt(2),-1/sqrt(2)), they are orthogonal.

However if you replace the normalised assets in eigenvectors by NON normalised assets, they are now correlated because x2 is insignificant in comparison to x1. Infact they will now be perfectly correlated.

There is absolutely nothing funny going on, it's just how the data is.

If you create factors out of non scaled returns, you are (mostly) ranking assets in terms of their variances.

Thus your factors are (mostly) assets themselves. If you take k of these, k of your regressions will have perfect R squares. Again this is not surprising.

In the other case, you changed your PCA factors to un-normalized, and they no longer reflect the "PCA" decomposition of your real returns. So poorer performance is expected, as these are not the factors that govern real data. In other words, by rescaling the eigen vectors, you drowned out the assets with small variances, and no wonder you will not be able to explain the real returns of these assets.