I have recently discovered Bryan Kelly's paper on Principal Portfolios (https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3623983) and had some doubts about the prediction matrix $\Pi$. He defines $\Pi = \mathbb{E}[R_{i,t+1}S_{j,t}]$ as a matrix in which element $\Pi_{ij}$ shows how the return of asset $i$ is predicted by the signal of asset $j$. Kelly writes $\hat \Pi_t = \frac{1}{120} \sum_{\tau = t-120}^{t-1} R_{\tau+1} S_\tau$ as an estimator of $\Pi$, however he is essentially estimating $N^2$ values with 120 observations, which leads to overfitting if used with large stock universes. Kelly uses Fama French Portfolios, which I'm not fully sure what they are either, but my question is, has anyone successfully tried the theory of this paper on a large number $N$ of assets, or has some insight/advice regarding the estimation of $\Pi$? Thank you in advance !



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