Filtering time series from common variance

Dear All,

perhaps a more methodological question than a coding one. I want to filter time series from their common covariance, in order to uncover their respective idiosyncratic variability. Speaking of stock returuns, that is to filter the series from the unobservable aggregate market componet and be left with a series of which variability depends on stock-intrinsic factors.

My go-to method would be principal component analysis, so that I can calculate the share of the leading principal component on the series.

But now I'm not sure on how to complete the analysis, i.e. how to use the leading principal component to filter the original series.

Thank you

Answer 1

[Shivam Lahoti]

Hi Marco,

I can understand that you want to filter time series from common variance to identify the intrinsic factors driving the stock movement leaving the market component.

Principal Component Analysis (PCA) is indeed a common technique used to isolate common factors in a set of time series, such as stock returns. The first principal component often represents the market component or systematic risk that affects all stocks, while subsequent components may capture other common factors or sector-specific risks.

• The methodological way to analyze the data would be:
• first standardize the data before performing PCA.
• Then apply PCA to decompose your data into orthogonal components where the first principal component explains the most variance, which is often interpreted as the market component.
• Analyze the eigenvalues and the explained variance ratios to identify the leading principal component.
• Compute the scores for the leading principal component for each time point. These scores represent the contribution of the market component to the variance of the returns at each time point.
• Multiply the standardized data by the loadings of the leading principal component to reconstruct the market component time series. Subtract the reconstructed market component from the original standardized returns to obtain the residual returns.
• These residuals represent the idiosyncratic or stock-specific returns, filtered from the common market component.

Please refer to the following documentation to understand the implementation of PCA in MATLAB.

https://www.mathworks.com/help/stats/pca.html

I hope this was helpful.

Regards,

Shivam Lahoti.