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I fit a GARCH(1,1) model on the spread of 2 correlated assets :

enter image description here

the GARCH model shows this summary:

==========================================================================
                 coef    std err          t      P>|t|    95.0% Conf. Int.
--------------------------------------------------------------------------
omega          0.2066  5.839e-02      3.537  4.042e-04 [9.211e-02,  0.321]
alpha[1]       0.6416  5.479e-02     11.712  1.107e-31   [  0.534,  0.749]
beta[1]        0.3584  6.020e-02      5.953  2.640e-09   [  0.240,  0.476]
==========================================================================

From this point, nothing weird, but then when I plot the standardized residual by their conditional volatility :

Standardized residual

In order to retrieve entry/exit signals for my strategy, I'm doing a 2-tails test on the distribution of these standardized residual. However, as you can see, the distribution is very weird :

enter image description here

Is it normal to have such bimodal distribution for a standardized residual by a GARCH model ? I'm asking because this is definitely not something I was expecting (standard normal distribution, or at least t-student with fatter tails), neither something I found on the Internet as what we can expect for a GARCH std residual.. What did I miss here ?

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1 Answer 1

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This is not normal. I bet that you have not specified the distribution of the standardized innovations to be a bimodal one when specifying the GARCH model. If so, your model is misspecified. And even if there was a possibility to specify a bimodal distribution for standardized innovations, you would probably rather prefer to account for the bimodality in the conditional mean (and perhaps the conditional variance) equation instead.

To give an unrelated, hypothetical example, consider the wage distribution of a population. If the distribution for males has a different peak than the one for females, you might end up with a bimodal distribution for the total population. Why not use a sex dummy for the conditional mean (and probably for some higher-order moments) to account for that instead of trying to find a suitable bimodal distribution? The sex dummy approach makes matters more transparent.

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  • $\begingroup$ @RichThank you for your answer If I understand correctly, one should capture as much information as possible in the mean process, and then apply a GARCH only on a residual with constant mean over time, but only with changing variance. Indeed, the spread of the 2 correlated assets that I showed has changing mean over time as you can see. If I apply a VECM on my spread as a conditional mean process, and take its residual, it gives a real constant mean over time (0) and then the std residual looks "unimodal". $\endgroup$ Apr 17, 2023 at 10:07
  • $\begingroup$ But the problem is that we need an estimate of the variance of the overall spread next period to be able to twinkle entry/exit signals. With a variance of the VECM, I'm not able to have any idea about this variance (because I compute the variance of something else basically). How I could get back to the variance of the spread, in your opinion ? $\endgroup$ Apr 17, 2023 at 10:08
  • $\begingroup$ @JeremLachkar, not sure if I understand your question. Suppose the cond. mean model is $x_t=\mu_t+\varepsilon_t$ where $\mu_t$ is a function of the past or of an exogenous variable that is available at time $t$. Then the cond. variance of $x_t$ is the (cond.) variance of $\varepsilon_t$. The unconditional variance of $x_t$ is something else, but you probably (?) do not need that as it probably does not make sense to ignore conditioning information that is readily available. $\endgroup$ Apr 17, 2023 at 10:38
  • $\begingroup$ Sorry, actually I wanted to estimate the conditional variance of the spread, at all time. This enables to twinkle entry/exit signals (if spread cond. variance is high, then increase the level at which we take a position). But I have no idea of this cond. variance if I have only a VECM residual conditional variance estimate, since they have no link (or a link that I can't see ?) with the cond. variance of the spread itself. That's why I wanted to estimate the cond. variance of the spread in the first place, or get back from the VECM cond. var to the spread cond. var with some formula ? $\endgroup$ Apr 17, 2023 at 10:49
  • $\begingroup$ @JeremLachkar, does my previous comment not address your question? If you have an estimate of the cond. variance of the residual, that also serves as an estimate of the cond. variance of the dependent variable. The regressors in the cond. mean equation are given, so they do not affect the cond. variance of the dependent variable. $\endgroup$ Apr 17, 2023 at 10:55

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