Spurious Regression problem dates back to Yule (1926): “Why Do We Sometimes Get Nonsense Correlations between Time-series?”. Lets see what is the problem, and how can we fix it. I am using Morgan Stanley (MS) symbol for illustration, pre-crisis time span. Take a look at the following figure, generated from the regression of MS on the S&P, actual prices of the stock, actual prices of the S&P, when we use actual prices we term it regression in levels, as in price levels, as oppose to log transformed or returns.
Category: Statistics and Econometrics
Piecewise Regression
A beta of a stock generally means its relation with the market, how many percent move we should expect from the stock when the market moves one percent.
Market, being a somewhat vague notion is approximated here, as usual, using the S&P 500. This aforementioned relation (henceforth, beta) is detrimental to many aspects of trading and risk management. It is already well established that volatility has different dynamics for rising markets and for declining market. Recently, I read few papers that suggest the same holds true for beta, specifically that the beta is not the same for rising markets and for declining markets. We anyway use regression for estimation of beta, so piecewise linear regression can fit right in for an investor/speculator who wishes to accommodate himself with this asymmetry.
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.
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.
What is important for a loan?
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!)