When setting up and establishing regression-based hedges, there are two schools of thought. Some regress changes in yields on changes in yields, as demonstrated previously, but an alternative approach is to regress yields on yields.
Using a single-variable approach, the formula for a change-on-change regression with dependent variable y and independent variable x is as follows:
Ayt = ol + (3Axt + Aet
where: Ayt = yt – yt-i Axt = xt – xt l
Alternatively, the formula for a level-on-level regression is as follows:
yt = a + (3xt + t
With both approaches, the estimated regression coefficients are unbiased and consistent; however, the error terms are unlikely to be independent of each other. Thus, since the error terms are correlated over time (i.e., serially correlated), the estimated regression coefficients are not efficient. As a result, there is a third way to model the relationship between two bond yields (for some constant correlation < 1):
t = Pt-l + vt
This formula assumes that todays error term consists of some part of yesterdays error term, plus a new random fluctuation.