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I'm sorry is this is too obvious, but I'm having a hard time trying to find a distribution for my data. It is clearly not a normal distribution. It does not seem to be skewed, but seems to have fat tails. Is that right? I thought a Student distribution would be the closest but I'm not sure. I need the distribution to be able to fit a GARCH model. Thanks in advance! QQ plot Density Histogram

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    $\begingroup$ Density plot looks a smidge similar to a Laplace (i.e. double exponential) distribution. $\endgroup$ – Alexis Oct 1 '18 at 16:00
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    $\begingroup$ You can't readily use the raw (marginal) response to choose a conditional distribution for a GARCH model - if the model is approximately correct, then the marginal distribution of the data will be a scale mixture that will look heavier tailed than the conditional distribution you need to choose. [This is a similar problem to trying to choose an error distribution with a regression model from the raw response, except in that case you have a location-mixture.] $\endgroup$ – Glen_b -Reinstate Monica Oct 1 '18 at 16:44
  • $\begingroup$ @Glen_b I'm not sure I understand. Do you know what can I use for the GARCH conditional distribution? Is a normal distribution okay? $\endgroup$ – Narjems Oct 2 '18 at 10:43
  • $\begingroup$ I don't know what you should use -- I don't have your data. Generally speaking, normals are not heavy tailed enough (and log returns tend to be slightly left skew). Sometimes people try t-errors, but you can't estimate the d.f. from the above information (indeed d.f. are very hard to estimate; it may be better to choose a d.f.); you could assume something and then look at whether it reasonably approximates the estimated errors in the model. $\endgroup$ – Glen_b -Reinstate Monica Oct 2 '18 at 11:12
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    $\begingroup$ It's probably time I turn the above comments into some form of answer and then if you have additional things to ask perhaps you can post a new question $\endgroup$ – Glen_b -Reinstate Monica Oct 2 '18 at 12:33
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You can't readily use the raw (marginal) response to choose a conditional distribution for a GARCH model - if the model is approximately correct, then the marginal distribution of the data will be a scale mixture that will look heavier tailed than the conditional distribution you need to choose. [This is a similar problem to trying to choose an error distribution with a regression model from the raw response, except in that case you have a location-mixture.]

Since I don't have your data, I don't know what model you should choose. Generally speaking, normals are not heavy tailed enough (and log returns tend to be slightly left skew).

Sometimes people try t-errors, but you can't estimate the d.f. from the above marginal information (indeed d.f. in the t are hard to estimate in any case; it may be better to choose a d.f. somewhat arbitrarily). You could for example assume something (low d.f like 5 or 7 seem to be fairly common choices, but it's not really my area) and then look at whether it reasonably approximates the estimated errors in the model (eg. via a QQ plot).

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