# Constructing a Replicating Portfolio : Regression on Individual Constituents or their Average?

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.