Rolling Beta and VAR for Comparing Directional Sensitivity
Summary
The question asks how to measure asymmetric responses between two related series, using equity returns and inversely related Treasury yields as an example. The responses suggest rolling historical beta, computed from recent log returns, as a way to track how the magnitude of one series’ moves relative to a reference series changes over time. A new observation that departs from the prior relationship will shift the estimated beta. One response offers a 21-day window as a personal choice, not a generally established optimum.
A vector autoregression (VAR) is also suggested for examining lagged interactions and possible directional relationships. Neither rolling beta nor VAR by itself proves causation, and ordinary beta summarizes a linear average relationship rather than explicitly modeling different responses to up and down moves. The document provides conceptual suggestions only: it gives no data, model specification, diagnostic checks, or performance evidence. Window choice and the treatment of yield changes versus returns would need to match the research question.
Key ideas
- Rolling beta on recent log returns can track changing relative sensitivity.
- A new observation changes the rolling estimate according to how it compares with prior observations.
- A VAR can examine lagged relationships between series, but does not establish causality by itself.
- A standard beta captures an average linear relationship and may miss asymmetry between rises and falls.
- The suggested window length is an individual preference rather than a validated rule.
Tags
Full text
# How can i compare two time series in order to detect directionality preference? # How can i compare two time series in order to detect directionality preference? Let’s say I am looking at two assets which are correlated, for example a=S&P500 and b=10-year US Treasury Yields (inversed). I want to detect when the strength of the change is different. For example, When b move up by 1x, a move up by 2x. When b move down by 1x, a move down by 0.5x or even flat or up. What’s the best way to capture this? ## Answer by Drew (score 2) https://quant.stackexchange.com/a/80921 I think constructing and studying the results of a vector autoregression might help you understand the direction of "causality" (if such exists). Incorporating the terms suggested by @nbbo2 in comment above into that VAR is a possibility. ## Answer by KaiSqDist (score 1) https://quant.stackexchange.com/a/80906 I would probably use a rolling window for historical beta (probably with the less volatile asset as the reference; or the market, as it is usually known) that updates itself with the most recent observations (in this case, log returns). Personally, I prefer to use the past 21 days of log returns to compute historical beta as it is not too long (short) to be barely (overly) affected by the incorporation of the most recent observation. If we go by your example, the beta ($a$, S&P 500 to $b$, US T yields, where US T yields is the "market") would originally be $+2$ (or somewhere around that if we assume the history perfectly gives us this value). However, incorporating the most recent observation, say when $b$ moves down by 1x, and instead of $a$ moving down by 2; it goes down by 0.5x, stays flat or moves up - this would mean that the most recent observation offsets/lowers the original historical (rolling window) beta to a lower value. That would mean that the newest observation would lower the beta to below $+2$. ## Answer by Content_Quantinsti (score 0) https://quant.stackexchange.com/a/80958 Rolling Beta (Dynamic Correlation) Beta measures the sensitivity of an asset's returns relative to a benchmark. In your case, the S&P 500 (a) would be the asset, and the inverse 10-year US Treasury yield (b) could act as the benchmark. This will give you insight into how the relationship between the two assets changes over time, capturing periods where the S&P 500 moves with a greater or lesser magnitude compared to the yield.
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