∂b ∂θ′(θ0) = ∂2ψ∞ ∂β∂β′[θ0, b(θ0)] −1 ∂2ψ∞ ∂β∂θ′[θ0, b(θ0)] =J0−1∂ 2ψ ∞ ∂β∂θ′[θ0, b(θ0)] (3.12)
We can thus derive an alternative expression of the asymptotic variance- covariance matrix of the optimal I.I. estimator ˆθ∗ which may be directly com- puted from the criterion function:
WS∗=1 + 1 S ∂2ψ∞ ∂θ∂β′(I0−K0) −1∂2ψ∞ ∂β∂θ′ −1 (3.13) As far as an estimation of W∗
S is concerned, a consistent estimator of ψ∞
is needed. Such an estimator can be obtained by a numerical derivation of
∂ψT
∂β′[yst(θ); ˆβ] with respect to θ, evaluated at ˆθ∗. For the derivation of a con- sistent estimator of (I0−K0), see Gouri´eroux et al. (1993), Appendix 2.
3.4
A simple example
To fix ideas, it might be useful to consider a simple example7 involving a nonlinear data generating process (dgp) like
yt= exp{zt′θ}+ǫt, ǫt∼N(0, σ2) (3.14)
Let the auxiliary model be
yt=zt′β+ηt, ηt∼N(0, σ2η) (3.15)
Note that:
∂E[yt|zt]
∂zt
=β (under the auxiliary model)
∂lnE[yt|zt] ∂zt = ∂E[yt|zt] ∂zt · 1 E[yt|zt]
=θ (under the model of interest) One can thus deduce that the binding function is
β =θE[yt|zt], or
θ= E[yt|zt]−1β (3.16)
Note that dim(θ) = dim(β).
A naive estimation of the auxiliary parameter, drawn on the observed data
yt,zt, t= 1, ..., T, can be easily obtained (e.g. by least squares). Let us denote
such an (inconsistent) estimate by ˆβ. Now, given a T-dimensional pseudo- random draw, denoted byǫ(0), and chosen ˜θ(0)= ˆβ, say, it is easy to generate
yt(1), t= 1, ..., T using
yt(1)= exp{zt′θ˜(0)}+ǫ(0)t
and obtain a revised estimator ˆβ(1)= (P
tztzt′)−1Ptztyt(1), which in turn is
used to derive θ(p) from (3.16) and to generate a new set of pseudo-random observation from (3.14). The entire simulation cycle is repeated, holding ǫ(0)
fixed, until [ ˆβ−βˆ(˜θ(p))]′[ ˆβ−βˆ(˜θ(p))] is minimized, i.e. until the calibration procedure has corrected for the bias of the naive estimator. The resulting estimate ofθ is the indirect inference estimate.
Chapter 4
Indirect Estimation of Stable
GARCH Processes
Several studies have highlighted the fact that heavy-tailedness of asset re- turns can be the consequence of conditional heteroskedasticity. ARCH and GARCH models have thus become very popular, given their ability to account for volatility clustering and, implicitly, heavy tails. However, as outlined in chapter 1, these models encounter some difficulties in handling financial time series, as they respond equally to positive and negative shocks; in addition, some empirical studies (for instance, Yang & Brorsen, 1993 [44]) indicate that the tail behavior of GARCH models remains too short even with Student-t
error terms1. To overcome these weaknesses we apply GARCH models with
α-stable innovations2. Since simulated values fromα-stable distributions can be straightforwardly obtained (see section 2.7), the indirect inference approach (described in chapter 3) is particularly suited to the situation at hand. Here we provide a description of how to implement such a method by using a GARCH with skewed Student’stinnovations as auxiliary model. This distribution has four parameters which have a clear and interpretable matching with those of
1Furthermore, the Student-tdistribution lacks the stability-under-addition property. Sta-
bility is desirable because stable distributions, having domains of attraction, provide a very good approximation for large classes of distributions.
2GARCH models with symmetric stable innovations have been first proposed by McCul-
theα-stable distribution. Among the many proposals of skew-tdensity func- tions appeared in the literature, we have adopted the one recently introduced by Azzalini & Capitanio (2003)[2], which is briefly reviewed in the following section3. In section 4.2 the models implemented are presented and the simu- lations results are shown in section 4.3. Finally, the proposed models are used to estimate the IBM weekly returns series, to see how they perform on real data.
4.1
The skew-t
distribution
To be better informed about the four stable parameters (α, β, γ, δ), it is in- tuitively to go through a quasi-likelihood function which entails similar pa- rameters with similar interpretations. Therefore, the family of skew-Student’s
t distributions introduced by Azzalini & Capitanio (2003)[2] seems to be a natural choice.
The idea follows from an extension of the skew-normal distribution (Azza- lini, 1985 [1]), in which the symmetry of the density is perturbated by means of the distribution function evaluated at a certain point. More formally, the univariate skew-normal density function is defined as:
f(x; ˜β, µ, σ) = 2φ(z) Φ( ˜βz) (4.1) where φ(·) and Φ(·) denote, respectively, the density and the distribution function of the standard normal distribution and z = x−σµ. The parameter4
˜
β ∈Rplays the role of shape parameter dealing with the degree of skewness; when ˜β= 0 we recover the regular normal density and we write SN(µ, σ,0) = N(µ, σ). Among the many formal properties shared with the normal class, a noteworthy fact is that ifX ∼SN(µ, σ,β˜), then X−σµ2
∼χ2 1.
The usual construction of thetdistribution is by means of the ratio of a normal variate and an appropriate transformation of a chi-square. Hence, replacing
3A widely used alternative, adopted for instance in Garcia et al. (2011)[18] is the version
introduced by Fern`andez & Steel in 1998.
4In the original paper, ˜βis denoted byα; this different notation is adopted here to avoid
4.1 The skew-t distribution
the normal variate above by a skew one, leads to an asymmetric variant of the
tdistribution, whose density is given by
f(x;ν,β, σ, µ˜ ) = 2 σft(z;ν)Ft ˜ βz r ν+ 1 z2+ν;ν+ 1 = 2 σ Γ(ν+12 ) Γ(ν2)√πν h 1 +z 2 ν i−ν+12 Ft ˜ βz r ν+ 1 z2+ν;ν+ 1 (4.2)
where z is defined as before, ft(·) and Ft(·) denote density and distribution
function of a Student-tvariable with ν degrees of freedom. Distribution (4.2) is called skew-t and we writeX ∼St(ν,β, σ, µ˜ ). Figure 4.1 shows the pdf of a SN(0,1,8) (left panel) and of a St(2,3.5,1,0) (right panel).
Figure 4.1: Probability density function of a skew-normal withµ= 0, σ= 1,β˜= 8 (left) and a skew-t withν= 2,β˜= 3.5, σ= 1, µ= 0 (right).
The four parameters of the skew-tdistribution all have a clear interpretation:
µ∈Rand σ∈R+ model location and dispersion, respectively; the additional parameter ˜β ∈ R influences the asymmetry; ν ∈ R+ captures the thickness of the tails5. In an indirect inference framework, one can thus expect the skew-t auxiliary parameters to be very informative about the stable ones. In fact, Garcia et al. (2011)[18] prove four analytical results that show the correspondence between these two set of auxiliary and structural parameters, as summarized by Table 4.1.
5The first four moments of a skew-tdistribution with ν degrees of freedom are defined
only forνlarger than the corresponding order of the moment. Note the similarity with the moments ofα-stable distributions (Property 2.4).
Characteristic Structural Auxiliary
Tail thickness α ν
Asymmetry β β˜
Scale γ σ
Location δ µ
Table 4.1: Relation between structural and auxiliary parameters.
For skew-t-based models, maximum likelihood (ML) is a feasible estimator. The log-likelihood function for a skew-tsample ofnobservation is:
lnL(ν,β, σ, µ˜ |x) = n ln2σ + ln Γ(ν+12 )−ln Γ(ν2)−12ln(πν) + n X i=1 lnFt ˜ βzi s ν+ 1 zi2+ν;ν+ 1 −ν+ 12 n X i=1 ln 1 + z 2 i ν (4.3) The log-likelihood of the auxiliary models presented in the following section have been computed exploiting the skew GAUSS library6 implemented by Roncalli & Lagache (2004)[37].