Interpreting Market Beta Across Return Quantiles and Market Regimes

# Market beta across quantiles: interpretation of increasing beta in the tails


# Market beta across quantiles: interpretation of increasing beta in the tails












Let's say that you measure the market beta across quantiles :

$Q_{r_i|r_M}=a_i(\tau) + \beta_i(\tau)r_M + \varepsilon_{i}(\tau)$

where $r_i$ is a specific asset return and $r_M$ is the market return. You would obtain a $\beta$ across asset returns regimes (during normal periods, i.e. for $\tau \in \{0.1,0.9 \}$, in extreme negative returns $\tau < 0.1$ and in extreme positive return $\tau > 0.9$). You would have many possible scenarios but just to simplify, the $\beta$ can be stable across regimes, increasing in the tails or decreasing in the tails.

I'm just trying to clearly clarify the exact terms that can be used for these scenarios. If the market beta increases during extreme periods, it would represent increasing comovements with the market during times of extreme (positive or negative) returns periods. Following Forbes & Rigobon (2002), increasing comovements during extreme negative periods represents Contagion. However, I find it harder to define the other scenarios : what would represent increasing comovements with the market during extreme positive periods ? My idea is that it is some sort of speculation but I'm not sure. (Note that I'm putting the hypothesis of positive $\beta \ \forall \ \tau$)

Overall, how would you define the increasing beta in the right-tail, or decreasing beta in the right/left tail ?

## Answer by Richard Hardy (score 2)

https://quant.stackexchange.com/a/82459

The answer applies to the original question before the edit of 9 May 2025

Your model specifies how $Q_{r_i|r_M}$ (the conditional quantiles of $r_i$) vary with the values of $r_M$. There is nothing in the model that would reflect the dependence of $Q_{r_i|r_M}$ on market regimes. Let us define market regimes continuously as $Q_{r_M}$ (quantiles of $r_M$).$^*$ If you want a model where the $Q_{r_i|r_M}$ vary with the market regimes, you would need to replace $r_M$ with $Q_{r_M}$: $$ Q_{r_i|r_M}=a_i(\tau) + \beta_i(\tau)Q_{r_M} + \varepsilon_{i}(\tau). $$ $^*$Alternatively, you could define some discrete regimes separated by quantile levels of $r_M$ such as $\tau=0.1$ and $\tau=0.9$ that would produce three regimes: low, middle, high.