Addition of skewed distributions for the innovation terms

[chm-von-tla]

Hello,

I did my thesis on a comparative analysis of Value-at-Risk methodologies and I used this package quite a lot (thank you!)

One addition I had to make was to include skewed distributions for the innovation terms since this usually leads to superior VaR forecasting performance (see: https://doi.org/10.1002/for.2303). The distribution I implemented was Hansen's Standardized Skewed T Distribution (see: https://doi.org/10.2307/2527081) which is defined as:

Its analytical quantile function is provided in https://doi.org/10.1016/s0165-1889(02)00079-9 and is defined as:

with A_η^-1 being the inverse cdf of a t-distribution with η dof

Another approach that could be taken is to adopt the framework found in (https://dx.doi.org/10.2139/ssrn.821) under which any unimodal continuous symmetric distribution can become asymmetric by changing the scale at each side of the mode. In the same paper they demonstrate their framework by creating a skewed version of the Standardized GED.

If you think that any of the above would be a worthwhile addition to this package, I provide as a starting point my implementation of Hansen's SKT-Distribution. (#86) There are at least two problems with my code:

• startingvals is essentially duplicated from the corresponding function for StdT. I tried to do
  startingvals(::Type{<:StdSkewT}, data::Array{T}) where {T<:AbstractFloat} = [startingvals(StdT,data), zero(T)] but it didn't work
• performance is bad. On the BG96 dataset fitting a GARCH(1,1) with StdSkewT errors was almost 10 times slower than the same model with StdT errors.

[s-broda]

Thanks for your contribution! I'll be happy to incorporate this.

To get this merged, could I ask you to do the following:

• Add some tests. Ideally coverage won't drop below the current almost 100%
• CI is failing at the moment. This is because of a problem in the doctests, see here: https://travis-ci.org/github/s-broda/ARCHModels.jl/jobs/761289701. The problem is in line 332 of usage.md. You could either exclude the StdSkewT from consideration there, or make sure that the test doesn't fail.
• To avoid the problem with the duplicated code for the starting values, you can use startingvals(::Type{<:StdSkewT}, data::Array{T}) where {T<:AbstractFloat} = [startingvals(StdT, data)..., zero(T)] (note the splatting operator ...).
• I've left some more comments in the PR. Most importantly, rand and quantile didn't seem to work before,

Do you have an idea why this is so much slower? I didn't see any type instabilities, so maybe it's just a result of there being another parameter to estimate.

[chm-von-tla]

Thanks for the excellent feedback. I will be making the changes soon. I should probably also make changes to the documentation to reflect that another error distribution is available.

Do you have an idea why this is so much slower? I didn't see any type instabilities, so maybe it's just a result of there being another parameter to estimate.

I haven't done any profiling but I highly suspect that it is because of all those expensive a, b, c, S functions in logkernel [1]

Consider

@inline logkernel(::Type{<:StdT}, x, coefs, iv) = (-(coefs[1] + 1) / 2) * log1p(abs2(x) *iv)

versus

@inline logkernel(d::Type{<:StdSkewT}, x, coefs, iv) = (-(coefs[1] + 1) / 2) * log1p(1/abs2(1+coefs[2]*S(d,x,coefs)) * abs2(b(d,coefs)*x+a(d,coefs)) *iv)

I just spotted an inefficiency in my logkernel function. I call the function S which itself calls a and b and later I call again a and b. I am not sure if the compiler can optimize away this problem. I will change it and see if there is any considerable gain. I suppose this code could be optimized even further, but I suspect that it will always be noticeable slower than the StdT case because of the added complexity.

[1] (the a,b,c functions correspond to the equations (11,12,13) and the S function determines on which side of the mode we are [the mode being (-a/b)] so that we choose the correct sign (hence the name S), for the parameter λ in equation (10)

[chm-von-tla]

The inefficiency on logkernel was huge. With the change introduced by my new commits a 9x slowdown on fitting a GARCH(1,1) model on BG96 with StdSkewT errors compared to StdT has now been reduced to about only 3x slower, which, taking the added complexity into account, I would deem as acceptable performance

[chm-von-tla]

Since my pull request for the addition of Hansen's Skewed T Distribution has been merged to master it is time to close this issue.

As closing thoughts I will point to some directions for further improvement on this subject. If anyone else (or my future self :) ) has the time and motivation, a skewed standardized GED, as found in https://dx.doi.org/10.2139/ssrn.821 could be also added.