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QRM 9 - Laptop session

创建日期：2024-04-27 16:20:21

所属分类：计算机computer science

标签： QRM 9

Laptop session

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QRM 9 - Laptop session

Introduction
The objective of the session is
(a). To understand how to use the ghyp package to undertake parametric estimation,
and to obtain risk measures, in both the case of a single asset portfolio, and a
multiple asset portfolio, for non-Normal risk factors.
The ghyp package
As before, to load the package, so that we can use the functions and data it supplies,
we type
library(ghyp)
In session 8 we saw how to use the package to facilitate working with random vari-
ables, in particular Normally distributed ones, and we also saw how to estimate risk
measures for the case of Normal return. Today we look at relaxing the restriction
of Normality. The focus is on estimating risk measures for a more general class of
distributions.
Estimation
In addition to the Normal, we can fit six possible parametric models to the data. They
are called:
(a). The Generalised Hyperbolic (GH)
(b). The Hyperbolic (H)
(c). The Normal Inverse Gaussian (NIG)
(d). The Variance Gamma (VG)
(e). The t distribution (t)
Each one has a symmetric and an asymmetric (skewed) variant. We can fit univariate
or multivariate data in each case. The models are also hierarchically connected, in
that they share the same set of parameters, but for some of the models certain of
the parameters are fixed, and for others they can vary. The parameters are:
Parameter Role
lambda Shape parameter. Free in GH. Constrained to be
positive for VG. Constrained to be less than -1
for t. Fixed for H. Fixed at -0.5 for NIG.
alpha.bar Shape parameter. In all cases, constrained to
be non-negative. Otherwise free for GH, H and
NIG. Fixed at zero for VG and t.
mu Location parameter. Free in all cases.
sigma Dispersion parameter. Free in all cases.
gamma Skew parameter. Zero for symmetric, free for
asymmetric distributions.
The point about all this is not that one needs to learn the details, but to point out the
connections among these different members of the same family. For some parameter
values, the GH and the t are the same. For other parameter values, the GH and the
VG are the same. This is important because other things equal we would prefer a
simple model (one with fewer free parameters) over a more complex model, if the
fit of the former was not much worse than the fit of the latter. In econometrics this
is known as efficiency. This gives a reason for wanting to fit several alternatives and
select the best according to competing criteria of goodness of fit and parsimony.
We now explore how to fit the parameters of a model using returns data. ghyp
comes with some data on Swiss stock returns. They are stored in the data set called
smi.stocks, which we first have to load into the workspace with the data command,
see below. Then I pick out columns 4 and 5 (Nestle and Swisscom) and store in d.
Then I pick out the first column of the result and store in d1 for Univariate estimation.
are already
data(smi.stocks)
d <- smi.stocks[,c(4,5)]
d1 <- d[,1]
Univariate estimation
Possibly the most useful approach is to estimate all possible models (at least, within
a specific class of models), and then select the one which ”fits best”. This judgement
can be made using a statistic called Akaike’s Information Criterion (AIC). Lower is
better. AIC falls, when the goodness of fit, as measured by the Likelihood, rises1. It
rises, when the model has more free parameters, so in this way models with fewer
free parameters are favoured.
The function which does this estimation is stepAIC.ghyp.
stepAIC.ghyp(d1)
After running this you will get something like below, though I have rounded and cutoff
the last two columns
1AIC is calculated as AIC = 2(k − LL), where k is the number of parameters, and LL is the logarithm
of the likelihood.
model symmetric lambda alpha.bar mu sigma gamma aic llh
8 NIG TRUE -0.50 0.65 0.00032 0.013 0.00000 -10658 5332
6 ghyp TRUE -0.84 0.60 0.00032 0.013 0.00000 -10656 5332
3 NIG FALSE -0.50 0.65 0.00048 0.013 -0.00024 -10656 5332
1 ghyp FALSE -0.84 0.60 0.00045 0.013 -0.00021 -10655 5332
10 t TRUE -1.67 0.00 0.00030 0.014 0.00000 -10654 5330
5 t FALSE -1.67 0.00 0.00039 0.014 -0.00016 -10652 5330
7 hyp TRUE 1.00 0.39 0.00035 0.013 0.00000 -10645 5325
2 hyp FALSE 1.00 0.39 0.00059 0.013 -0.00035 -10643 5326
9 VG TRUE 1.23 0.00 0.00029 0.013 0.00000 -10639 5322
4 VG FALSE 1.23 0.00 0.00064 0.013 -0.00040 -10637 5323
11 gauss TRUE NA Inf 0.00024 0.013 0.00000 -10362 5183

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