In the paper

M. Avellaneda and J. H. Lee, Statistical arbitrage in the U.S. equities market, July 2008,

in the Appendix on page 44, I have some questions.

First he runs the regression of stock-return ($R_n^S$) with index/ETF-return ($R_n^I$).

$R_n^S = \beta_0 + \beta R_n^I + \epsilon_n, ~~~~~n=1,2,...,60$

Then he defines an auxiliary process $X_n$ as sum of the residuals from regression of $R_n$, and estimates it to be a mean reverting process.

$X_n = \sum_{j=1}^{n} \epsilon_j ~~~~~n=1,2,...,60$

I have two related questions?

What is the significance of taking cumulative sum of residuals as opposed to taking just residuals as a mean reverting process? I have a basic intuition but lack a good understanding.

Second, he says that the regression on stock-returns "forces" the residuals to have mean zero. Why is this? How does this imply that the sum of all residuals, $X_{60}=0$?

  • 2
    $\begingroup$ @user862, The sum of residuals equal to zero is a standard feature of regression with intercept. Differentiate sum of squares with respect to intercept to see why. $\endgroup$
    – mpiktas
    Apr 11, 2011 at 8:23

1 Answer 1


The significance of modeling the cumulative sum of residuals is to better approximate the Ornstein-Uhlembeck process of equation $(12)$ with discrete real-life data.

This process $X_i(t)$ represents the idiosyncratic above- or below- market fluctuations of the particular stock. More specifically, it is the difference between the stock's return and that of its industry sector (ETF). The expected value of the infinitesimal increment $dX_i(t)$ of the $X_i(t)$ process is based on the previous value of the process:

$$ E[dX_i(t)|X_i(s),s{\le}t] = {\kappa}_i(m_i-X_i(t))dt $$

Note the $X_i(t)$ on the right-hand side, suggesting a cumulative process.

The authors approximate a stock's $X_i(t)$ process with actual market data by first regressing the stock on its industry ETF (top of p.45), and then summing the residuals up to a certain point in time. This represents the cumulative above- or below- market return of the stock before the end of the regression time window.


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