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.