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I would like to replicate a portfolio of stocks $S_1, \cdots, S_n$ using other instruments, $X_1, \cdots, X_m$. Using the letters above with a subscript $t$ to denote the forward returns over some horizon of the portfolio at time $t$, the return of the portfolio (and the variable I am targeting) is:

$$ P_t = \frac{1}{n} \sum_{i=1}^n \epsilon_{it}S_{it}, \quad \epsilon_{it} \in \{-1,1\} $$ e.g. we hold an equal weighted portfolio of each stock, long and short.

To construct the hedge ratios, I considered estimating $n$ regressions (on a rolling basis, backwards in time):

$$ S_{it} = \sum_{j=1}^m \beta_{ij}X_{jt}, \quad 1 \leq i \leq n $$ and getting coefficients $\beta_{ij}$, representing the number of shares of the hedging instrument $X_j$ that need to be bought/sold-short to replicate stock $i$. The final allocations of the portfolio to each $X_j$ would then be: $$ \beta_j : = \frac{1}{n}\sum_{i=1}^n \beta_{ij}, \quad 1 \leq j \leq m $$ This means - replicate each stock $i$ individually, and average the coefficients across stocks to get the final amount to buy/sell-short for the portfolio.

However, I also considered just reconstructing the $P_t$ backwards in time (averaging the signed past returns of the selected stocks), and then directly estimating: $$ P_t = \sum_{j=1}^m\beta_j X_{jt} $$ This is a (marginally) computationally less intensive procedure, that uses a less noisy left hand side variable.

My question is - from a theoretical/methodological point of view, is there anything incorrect with this second method? Or is it just a question of getting my hands dirty with the data and seeing what works best? Of course I'd be interesting in minimizing my tracking error here, and computational considerations aren't really a problem as these are all OLS regressions.

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