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While reading the paper Statistical Arbitrage in the U.S. Equities Market by Marco Avellaneda and Jeong-Hyun Lee on statistical arbitrage using PCA

I realized that the author sums the residuals of regression against PCA factors and says that is mean reverting. By standard regression principles aren't residuals IID normal and hence their sum should be a random walk? Then how can the sum of residuals be mean reverting?

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    $\begingroup$ Error terms might be independent (but probably not in finance and economics), but residuals are not. In OLS, if you know all but one residual, you can determine that last residual. $\endgroup$
    – Dave
    Dec 31, 2021 at 14:20
  • $\begingroup$ Ideally for each residual you would take its cumulative sum, and then check if that is stationary through some kind of test (e.g. ADF) $\endgroup$ Feb 28, 2023 at 20:17

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The estimation of the parameters by regression ensures that the mean of the residuals is 0. So, technically the residuals are not IID as if the number of observations is $n$, any $n-1$ residuals completely determine the last one.

In practice, the assumption of iid-ness is not realistic anyway.

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