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I was given the returns of a cross-asset class portfolio of ETFs and I conducted PCA to obtain factors on dates from T-n, T-3, T-2,..., T. What I would like to do is decompose the market moves from T+1, T+2, ... onwards into combinations of the PCs.

My questions is, what sort of algorithm or optimization method can I use to obtain a set of factor weightings that explain the maximal amount of variance in the market moves going forward. Furthermore, this optimization method should be capable of outputting a list of maxima , not just one, so that the stability of the optimal factor weightings can be assessed over time, since the optimal factor weightings for one day/time period may not be the same as the optimal factor weightings over a larger period.

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If you have a time-varying covariance matrix (you could construct with Garch volatility and either a constant or time-varying dependence structure), then you can perform PCA on each period or project out to the future. Not sure if that's what you're looking to do. –  John Jan 6 at 21:42
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Your approach is a good one. But before you venture too far, you should be aware of issues related to zero eigenvalues (positive semi-definiteness) of your correlation matrix $\mathbf{R}$ or covariance matrix $\mathbf{C}$. Let $p$ be the number of assets, and $t$ the number of, for example, day or bars. You probably have many more times in the time series than you do assets, and thus, $t\gg p$. Since you are turning the dataset on its side, and treating days like variables, and assets like observations, there will be $t-p$ zero eigenvalues in your covariance matrix $\mathbf{C}$.

Given, the above, don't ever lose sight of the basic premise of, for example, Markowitzian portfolio optimization, where the covariance between assets (not days) is determined, and the number of days is exceedingly large (250 trading days per year) when compared with the number of assets. By default, Markowitz did not need to worry too much about zero eigenvalues of $\mathbf{C}$ when introducing his theory since $t\gg p$.

As a solution, you could therefore use singular value decomposition (SVD) on your $\mathbf{C}$, which is ideal for large dimension and low example datasets. When done, what you will observe is that certain days (bars) will "load" on certain principal components -- so you essentially be identifying groups of trading days which are similar. Since principal components are by definition orthogonal (zero correlation between them), you will need to think of a way to project your results into the future. Overall, however, there will be high merit in what you are doing for the ETFs involved, since you will be able to understand the structure of trading days for the basket of ETFs and days used.

It's a complex undertaking, and since so many issues are involved surrounding the problem of $t \gg p$, you will undoubtedly run into the Marcenko-Pastur law involving $\gamma=p/t$ and random data matrices. Look at, for example, Johnstone's talk

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Have you considered using 'incremental' singular value decomposition to calculate your component scores? Each future market move (or increment) forces a recalculation of component scores given the new data.

This paper outlines an algorithm to do this Fast Low-Rank Modifications of the Think Singular Value Decomposition

This paper develops an identity for additive modifications of a singular value decomposition (SVD) to reflect updates, downdates, shifts, and edits of the data matrix. This sets the stage for fast and memory-efficient sequential algorithms for tracking singular values and subspaces.

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