Question on financial Beta

#1
I've got a question on Beta as it's used in the financial community (covariance of a particular stock with its underlying index, divided by the variance of the index). In this case, I'm not trying to calculate the beta of a particular stock to the market. I'm trying to calculate the Beta of a particular sector of the market, to the overall market. Specifically, the S&P500 Energy Sector's Beta to the overall S&P500. To take things a step further, we'd like to segment the data into days in which the S&P500 is up, and days when it is down. So we're looking for two betas. The S&P500 Energy Sector's beta to the overall market (using the S&P500 as a proxy) on up days, and the Energy Sector's beta to the market on down days.

How I did this was to take the S&P 500 data (I'm using daily performance, year to date) and segment it by up days and down days. For each data set, I take the covariance of the Energy Sector's returns and the market's returns, and divide that by the variance of the market's returns. To make sure the number is correct, I ran a regression using Excel's data analysis tool kit, and made sure that the beta number there is the same that I calculated, and it was.

Here's the problem. At the time in which I ran the numbers (about a week ago), the S&P500 was down about 7% for the year, while the S&P500 Energy Sector was up about 8% for the year. The results I got are as follows: On market up days, the energy sector beta was ~0.75, and on market down days the energy sector beta was ~1.24. My understanding of interpreting beta in this case is that when the market is up 1%, the energy sector will typically be up 0.75%, and when the market is down 1%, the energy sector will typically be down 1.24%. But how can it be that the energy sector will be down more then the market on a down day, and up less then the market on an up day, if its performance overall is so much stronger then the market? I'm pretty sure that the numbers are correct as I've ran them a few times, but using the formula [COV(energy,market)/Var(market)] and the Excel regression tool, coming up with the same beta numbers. I'm wondering if my interpretation of the results is incorrect. If anyone could shed some light on this for me I'd be very appreciative. Thanks.
 
#2
Here's my guess on where I went wrong, please let me know if anyone can confirm or deny the conclusion I come to:

My basic conclusion is that beta isn’t an appropriate analytical tool when splitting data by positive and negative return days. This was prompted by the results that show the S&P Energy Sector having a 1.24 beta on down days and a .74 beta on up days, a result which makes no sense given how strongly the Energy Sector has outperformed the market year to date. My guess is that these results are nonsensical because beta is a linear framework. The regression analysis makes no distinction whether the data is all positive returns, negative, or a mixture. So when, for example, we calculate the beta based on up market days (yielding a beta of .74), the results don’t tell us that for every percentage point that the market is up, the energy sector tends to be up 74 bps. Instead, what it’s telling us is that for every percentage point that the market moves up or down, the energy sector will tend to move 74 bps in the same direction. Again, the regression analysis doesn’t “know” that we’re only using positive market return data points. It’s simply set up to give a sensitivity to the market’s return, whether the market return is positive or negative.
 

mp83

TS Contributor
#3
You can always devide a set using dummy variablew,i.e Z=1 if X>0 else 0, and then run a regression on the dummies.Something like

Y=a+b*Z(x>0)+c*Z(x<=0)