Resistant Regression

It is a fact that on most days, not much is going on in the stock market. When we estimate the relation of a stock with the market, or the “beta” of a stock, we use all available daily returns. This might not be wise as some days are not really typical and contaminate our estimate. For example, Steve Jobs past away recently, AAPL moved quite a bit as a result. However, this is a distinct event that does not reflect on the relation with the market, but is company specific. Our aim is to exclude such observations, taking into consideration that we don’t want to lose too much information, not all large swings are irrelevant.

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Pairs Trading Issues

A few words for those of you who are not familiar with the “pairs trading” concept. First you should understand that the movement of every stock is dominated not by the companies performance but by the general market movement. This is the origin of many “factor models”, the factor that drives the every stock is the market factor, which is approximated by the S&P index in most cases.

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OLS beta VS. Robust beta

In financial context,  \beta is suppose to reflect the relation between a stock and the general market. A broad based index such as the S&P 500 is often taken as proxy for the general market. The  \beta , without getting into too much detail, is estimated using the regression:

    \[stock_i = \beta_0+\beta_1market_i+e_i\]

A  \widehat{\beta_1} of say, 1.5 means that when the market goes up 1% the specific stock goes up 1.5%. (Ignoring all the biases at the moment!)

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