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I stumbled upon a regression that I haven't seen before, if anyone can provide some info that'd be great.

Define the dependent variable as: $y_{i,t} = x_{i,t} - x_{i,t-1}$

$y_{i,t} = ax_{i,t} + b$

$\bar{y} = a\bar{x} + b$

$ \bar{y} = -b/a $ assuming $ \bar{x} = 0 $

$ y_{i,t} = a(y_{i,t} - \bar{y}) + \epsilon_{i,t}$

It could be rewritten as an autoregressive process actually:

$ y_{i,t} = \frac{1}{1-a}y_{i,t-1} + \epsilon_{i,t}$

Any source to read about this kind of regressions would be great!

For any QUANT FINANCE person reading this, please read the following as well.

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!

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