# Why are cumulative residuals from regression on stock and index returns mean reverting

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, $X_{60}=0$?

• @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. – mpiktas Apr 11 '11 at 8:23

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.