I am currently looking at a regression which tries to model EWMA volatility in the presence of negative interest rates. The regression is as follows and uses absolute return instead of relative in order to avoid the issue with the negative rates:

  1. Last 50 days are taken: $y_{i,50}, ..., y_{i,0}$

  2. 50 absolute returns are calculated: $r_{i,t} = y_{i,t} - y_{i,t-1}$

  3. $r_{i,t}$ is regressed on $y_{i,t}$

  4. Residuals ($\epsilon_{i,t}$) are computed as: $\epsilon_{i,t} = r_{i,t} + a(\bar{y} - y_{i,t})$, where $\bar{y} = -b/a$. It assumes here that $\bar{r}$ is zero which I personally think it can be included as well.

  5. EWMA volatility is then calculated in the usual way using $\epsilon_{i,t}$ instead of $r_{i,t}$. So it decays the residuals from this regressions instead of the returns.

Has anyone seen anything similar or can explain to me why would this be a suitable model for negative interest rates? Is anyone aware of any other alternatives. Thanks!


1 Answer 1


By regressing $r_{i,t}$ on $y_{i,t}$ you are implying that:

$$ r_{i,t} \equiv y_{i,t} - y_{i,t-1} = c_1 y_{i,t} + c_2 + \epsilon_{i,t}$$

This seems quite odd to me initially.

If you assume that daily yield changes are independent with mean zero, then;

$$ y_{i,t} = y_{i,t-1} + \xi_{i,t} \; \quad E[\xi_{i,t}]=0$$

Which can be replicated in the linear regression by asserting that $c_1 = 0$ and $c_2=0$ and $\epsilon_{i,t} \equiv \xi_{i,t}$.

(If you were to incorporate a constant drift $\mu$ then $c_2 \equiv \mu$)

Perhaps I misunderstand the formulation of your question..


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