Why Regression Residuals Are Not Independent in PCA Arbitrage

Summary

The document asks whether residuals from regressing equity returns on principal component factors can support a mean-reversion strategy, given the common regression assumptions of normally distributed, independent errors. The response clarifies that fitting regression parameters imposes a zero-sum constraint on the residuals in the fitted sample. Consequently, when the sample size is fixed, all but one residual determine the last, so the residuals cannot be independent in the strict sense described in the question.

This constraint alone does not establish that residuals form a useful mean-reverting trading signal. The response also cautions that independent, identically distributed errors are rarely a realistic description of financial data. The exchange offers a narrow statistical clarification rather than an analysis of the cited PCA strategy, its time-series behavior, or its trading performance. It provides no empirical evidence that the residual portfolio reverts or that a strategy based on it is profitable.

Key ideas

• Regression fitting constrains the residuals to sum to zero over the fitted sample.
• That constraint makes the sample residuals dependent rather than independent.
• A zero residual sum does not by itself demonstrate time-series mean reversion.
• Independent, identically distributed errors are often an unrealistic model for financial data.

Tags

# statistical arbitrage using PCA


# statistical arbitrage using PCA












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?

## Answer by Bob Jansen (score 2)

https://quant.stackexchange.com/a/43732

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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.