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An EconometricAnalysis of EmissionTrading Allowances
M. Paolella L. TaschiniSwiss Banking
InstituteUniversity of Zurich,
Switzerland
Definition
Motivation
Contribution
Results
SO2 Stylized facts
Zero Excess
Econometric Approach
Unconditional Model
Conditional Model
Stable-MixtureGARCH
Summary of CO2
An Econometric Analysis of EmissionTrading Allowances
M. Paolella L. TaschiniSwiss Banking InstituteUniversity of Zurich,
Switzerland
November 2006
An EconometricAnalysis of EmissionTrading Allowances
M. Paolella L. TaschiniSwiss Banking
InstituteUniversity of Zurich,
Switzerland
Definition
Motivation
Contribution
Results
SO2 Stylized facts
Zero Excess
Econometric Approach
Unconditional Model
Conditional Model
Stable-MixtureGARCH
Summary of CO2
An EconometricAnalysis of EmissionTrading Allowances
M. Paolella L. TaschiniSwiss Banking
InstituteUniversity of Zurich,
Switzerland
Definition
Motivation
Contribution
Results
SO2 Stylized facts
Zero Excess
Econometric Approach
Unconditional Model
Conditional Model
Stable-MixtureGARCH
Summary of CO2
An EconometricAnalysis of EmissionTrading Allowances
M. Paolella L. TaschiniSwiss Banking
InstituteUniversity of Zurich,
Switzerland
Definition
Motivation
Contribution
Results
SO2 Stylized facts
Zero Excess
Econometric Approach
Unconditional Model
Conditional Model
Stable-MixtureGARCH
Summary of CO2
An EconometricAnalysis of EmissionTrading Allowances
M. Paolella L. TaschiniSwiss Banking
InstituteUniversity of Zurich,
Switzerland
Definition
Motivation
Contribution
Results
SO2 Stylized facts
Zero Excess
Econometric Approach
Unconditional Model
Conditional Model
Stable-MixtureGARCH
Summary of CO2
An EconometricAnalysis of EmissionTrading Allowances
M. Paolella L. TaschiniSwiss Banking
InstituteUniversity of Zurich,
Switzerland
Definition
Motivation
Contribution
Results
SO2 Stylized facts
Zero Excess
Econometric Approach
Unconditional Model
Conditional Model
Stable-MixtureGARCH
Summary of CO2
An EconometricAnalysis of EmissionTrading Allowances
M. Paolella L. TaschiniSwiss Banking
InstituteUniversity of Zurich,
Switzerland
Definition
Motivation
Contribution
Results
SO2 Stylized facts
Zero Excess
Econometric Approach
Unconditional Model
Conditional Model
Stable-MixtureGARCH
Summary of CO2
Definition of Tradable Permits
Tradable permits are a cost-efficient, market-driven approach forreducing GHG.
They are tradable allocation entitled by a government to anindividual firm to emit a specific amount of a substance over aspecified interval of time.
They enlists market forces in the quest for cost–effective pollutioncontrol and encouraging technological progress.
Tradable emission permits programs are being adopted byenvironmental regulators in applications ranging from
I local and regional (US-CAAA Title IV)
I global scale (EU-ETS and from 2008 the Kyoto Protocol)
An EconometricAnalysis of EmissionTrading Allowances
M. Paolella L. TaschiniSwiss Banking
InstituteUniversity of Zurich,
Switzerland
Definition
Motivation
Contribution
Results
SO2 Stylized facts
Zero Excess
Econometric Approach
Unconditional Model
Conditional Model
Stable-MixtureGARCH
Summary of CO2
Motivation
Emission Allowances influence:
I Commodity markets and energy market;
I Business decisions begin to be made with the price of carbonas a criterion;
I Firms stock value.
Emission Allowances market:
I high volatility and market crash;
I market is working.
An EconometricAnalysis of EmissionTrading Allowances
M. Paolella L. TaschiniSwiss Banking
InstituteUniversity of Zurich,
Switzerland
Definition
Motivation
Contribution
Results
SO2 Stylized facts
Zero Excess
Econometric Approach
Unconditional Model
Conditional Model
Stable-MixtureGARCH
Summary of CO2
Rhodia daily prices
CDM:a project-based mechanism, according to which the buyerpurchases emission credits from a project that can credibly andverifiable demonstrate that it reduces GHG emissions comparedwith what would have happened otherwise.
An EconometricAnalysis of EmissionTrading Allowances
M. Paolella L. TaschiniSwiss Banking
InstituteUniversity of Zurich,
Switzerland
Definition
Motivation
Contribution
Results
SO2 Stylized facts
Zero Excess
Econometric Approach
Unconditional Model
Conditional Model
Stable-MixtureGARCH
Summary of CO2
CO2 daily prices
CO2 daily price
Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov2005 2006
1214
1618
2022
2426
28
Figure: Daily CO2 allowance prices over the period June 25, 2005 toNovember 3, 2006.
An EconometricAnalysis of EmissionTrading Allowances
M. Paolella L. TaschiniSwiss Banking
InstituteUniversity of Zurich,
Switzerland
Definition
Motivation
Contribution
Results
SO2 Stylized facts
Zero Excess
Econometric Approach
Unconditional Model
Conditional Model
Stable-MixtureGARCH
Summary of CO2
SO2 daily prices
SO2 daily price
Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2
1999 2000 2001 2002 2003 2004 2005 2006
200
400
600
800
1000
1200
1400
1600
Figure: Daily SO2 allowance prices over the period January 1999 - May2006.
An EconometricAnalysis of EmissionTrading Allowances
M. Paolella L. TaschiniSwiss Banking
InstituteUniversity of Zurich,
Switzerland
Definition
Motivation
Contribution
Results
SO2 Stylized facts
Zero Excess
Econometric Approach
Unconditional Model
Conditional Model
Stable-MixtureGARCH
Summary of CO2
Contribution
Econometric investigation of emission allowances price:
I Measure the tail index (tail thickness) of the unconditionaldistribution:
Useful for long-term risk assessment and probability ofextreme movements
I Fit nonstandard GARCH-type models for the conditionaldistribution:
Short term risk and volatility prediction
An EconometricAnalysis of EmissionTrading Allowances
M. Paolella L. TaschiniSwiss Banking
InstituteUniversity of Zurich,
Switzerland
Definition
Motivation
Contribution
Results
SO2 Stylized facts
Zero Excess
Econometric Approach
Unconditional Model
Conditional Model
Stable-MixtureGARCH
Summary of CO2
Results
I Actual forecast methods based on supply and demandfundamental analysis:
does not suffice due to the complexity of the market......we believe that fundamentals drive value and a future stepis to implement a fundamental-like analysis into the mean
equation of the return process...
I A pure statistical model designed to capture the uniquestylized facts of the data (abundance of zero-returns andcomplicated conditional heteroskedasticity)
An EconometricAnalysis of EmissionTrading Allowances
M. Paolella L. TaschiniSwiss Banking
InstituteUniversity of Zurich,
Switzerland
Definition
Motivation
Contribution
Results
SO2 Stylized facts
Zero Excess
Econometric Approach
Unconditional Model
Conditional Model
Stable-MixtureGARCH
Summary of CO2
SO2 Stylized facts
0 200 400 600 800 1000 1200 1400 1600 1800
−15
−10
−5
0
5
10
15
0 10 20 30 40 50−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0 10 20 30 40 50−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
Figure: Daily SO2 returns (top), the SACF of the absolute returns(middle) and the SACF of the zeros-removed absolute returns (bottom).
An EconometricAnalysis of EmissionTrading Allowances
M. Paolella L. TaschiniSwiss Banking
InstituteUniversity of Zurich,
Switzerland
Definition
Motivation
Contribution
Results
SO2 Stylized facts
Zero Excess
Econometric Approach
Unconditional Model
Conditional Model
Stable-MixtureGARCH
Summary of CO2
Zero Excess
700 800 900 1000 1100 1200
−5
−4
−3
−2
−1
0
1
2
3
4
Time in days
Per
cent
age
SO2 return
Figure: Magnifying Zero excess around 500 days.
An EconometricAnalysis of EmissionTrading Allowances
M. Paolella L. TaschiniSwiss Banking
InstituteUniversity of Zurich,
Switzerland
Definition
Motivation
Contribution
Results
SO2 Stylized facts
Zero Excess
Econometric Approach
Unconditional Model
Conditional Model
Stable-MixtureGARCH
Summary of CO2
Unconditional Distributional Fit: SαSI Large number of zeros precludes use of typical fat-tailed
distributions (t, hyperbolic, stable, etc) because the centerwill be too peaked, forcing the tails to be unnecessarily thick.
I Otherwise, the stable Paretian would be a great candidatedistribution (GCLT, closed under summation, good fit tofinancial returns data, easy VaR approximations)
I The downside is its computation: For the symmetric stable,the characteristic function is
ϕX (t;α) = E[exp{itX}
]= exp {− |t|α} , 0 < α ≤ 2.
and the usual inversion formula reduces to:
fX (x) =1
π
∫ ∞
0
cos (tx) e−tα
dt.
I We use the FFT and linear interpolation to speed up.
I Except for the normal (α = 2) case, the α-stable distributionhas infinite variance. For α ≤ 1, its tails are so heavy thateven the mean does not exist.
An EconometricAnalysis of EmissionTrading Allowances
M. Paolella L. TaschiniSwiss Banking
InstituteUniversity of Zurich,
Switzerland
Definition
Motivation
Contribution
Results
SO2 Stylized facts
Zero Excess
Econometric Approach
Unconditional Model
Conditional Model
Stable-MixtureGARCH
Summary of CO2
Asymmetric Stable
I The general stable Paretian distribution, with skewnessparameter β, location µ and scale σ, is denoted Sα,β (µ, σ)and its characteristic function is E
(e iεtθ
)via
log E(e iεtθ
)=
{iµθ − |σθ|α
[1− iβsgn (θ) tan πα
2
], α 6= 1,
iµθ − |σθ|[1 + iβ 2
π sgn (θ) log |θ|], α = 1,
for α ∈ (0, 2], β ∈ [−1, 1], σ > 0, and µ ∈ R.
I For the SO2 data, the estimate of β was practically andstatistically zero.
I MLE of α for SO2 returns with the zeros removed is 1.45.(With zeros, it is near Cauchy).
An EconometricAnalysis of EmissionTrading Allowances
M. Paolella L. TaschiniSwiss Banking
InstituteUniversity of Zurich,
Switzerland
Definition
Motivation
Contribution
Results
SO2 Stylized facts
Zero Excess
Econometric Approach
Unconditional Model
Conditional Model
Stable-MixtureGARCH
Summary of CO2
Kernel
−10 −5 0 5 100
0.05
0.1
0.15
0.2
0.25
kernelnormalstable
6 7 8 9 10 11 12 130
1
2
3
4
5
6x 10
−3
kernelnormalstable
Figure: Kernel density (solid) of the SO2 return series, with thebest-fitting normal density (dashed) and best-fitting symmetric stabledensity (dash-dot). Right panel is just the magnified view of the righttail.
An EconometricAnalysis of EmissionTrading Allowances
M. Paolella L. TaschiniSwiss Banking
InstituteUniversity of Zurich,
Switzerland
Definition
Motivation
Contribution
Results
SO2 Stylized facts
Zero Excess
Econometric Approach
Unconditional Model
Conditional Model
Stable-MixtureGARCH
Summary of CO2
Zeros Problem: StructureI If the occurrence of the zeros throughout the data have
(Markov) structure, then this needs to be modeled.I Use standard combinatoric runs test, plot p-values
200 400 600 800 1000 1200 1400 16000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure: The p-values from the runs–test performed on segments of theSO2 return series. The first segment is the returns in the whole series,the second is from the second return to the end, etc., up to the(T − 50)th observation to the end.
An EconometricAnalysis of EmissionTrading Allowances
M. Paolella L. TaschiniSwiss Banking
InstituteUniversity of Zurich,
Switzerland
Definition
Motivation
Contribution
Results
SO2 Stylized facts
Zero Excess
Econometric Approach
Unconditional Model
Conditional Model
Stable-MixtureGARCH
Summary of CO2
Zeros Problem: Effect on α
The tail index α is biased because of the overabundance of zeros.
−15 −10 −5 0 5 10 150
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5 Symmetric Stable Paretian Density
x
Figure: Symmetric Stable densities: α=1.027 for blue line and α=1.64for green line.
An EconometricAnalysis of EmissionTrading Allowances
M. Paolella L. TaschiniSwiss Banking
InstituteUniversity of Zurich,
Switzerland
Definition
Motivation
Contribution
Results
SO2 Stylized facts
Zero Excess
Econometric Approach
Unconditional Model
Conditional Model
Stable-MixtureGARCH
Summary of CO2
Magnified view of tail
−14 −12 −10 −8 −6 −4 −2 0
−0.04
−0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
Symmetric Stable Paretian Density
x
Figure: Magnifying tail decay.
An EconometricAnalysis of EmissionTrading Allowances
M. Paolella L. TaschiniSwiss Banking
InstituteUniversity of Zurich,
Switzerland
Definition
Motivation
Contribution
Results
SO2 Stylized facts
Zero Excess
Econometric Approach
Unconditional Model
Conditional Model
Stable-MixtureGARCH
Summary of CO2
Hill Estimator
To avoid the zeros-problem and still estimate α, use a tailestimator.The Hill estimator is by far the most commonly used tail estimatorfor the tail index of a distribution:
αHill(k) =1
(1/k)∑k
j=1 ln (Xn+1−j :n)− lnXn−k:n
with standard error
SE (αHill;k) =kαHill(k)
(k − 1) (k − 2)1/2, k > 2,
where Xj :n denotes the jth order statistic of sample X1, . . . ,Xn.
If the right tail of the distribution is asymptotically Pareto, i.e., forlarge x , 1− F (x) ≈ cx−α, then, given an appropriate choice of k,αHill provides an estimate of Pareto tail index α.
An EconometricAnalysis of EmissionTrading Allowances
M. Paolella L. TaschiniSwiss Banking
InstituteUniversity of Zurich,
Switzerland
Definition
Motivation
Contribution
Results
SO2 Stylized facts
Zero Excess
Econometric Approach
Unconditional Model
Conditional Model
Stable-MixtureGARCH
Summary of CO2
Hill Estimator
We plot the Hill estimates of the the SO2 spot price as a functionof k, based on the 1,780 sorted absolute returns.
0 200 400 600 800 10000
1
2
3
4
5
6
The graph is typical of Hill estimator plots applied to financialreturns data, and a sizeable region for which αHill is “flat”, orroughly constant in k, cannot be found.
An EconometricAnalysis of EmissionTrading Allowances
M. Paolella L. TaschiniSwiss Banking
InstituteUniversity of Zurich,
Switzerland
Definition
Motivation
Contribution
Results
SO2 Stylized facts
Zero Excess
Econometric Approach
Unconditional Model
Conditional Model
Stable-MixtureGARCH
Summary of CO2
Hill Intercept Estimator
A tail estimator designed explicitly for stable Paretian data andwhich exhibits excellent small sample properties was developed inMittnik and Paolella (1999).
It is based on a set of Hill estimators for a range of k–values, andcomputed as
αHint = −0.8110− 0.3079 b + 2.0278 b0.5
where b is the intercept in the simple linear regression of αHill (k)on k/1000.
The main feature is that αHint is unbiased for α ∈ [1, 2] andvirtually exactly normally distributed.
For the SO2 returns, we obtain αHint = 1.46 with standard error0.043.
An EconometricAnalysis of EmissionTrading Allowances
M. Paolella L. TaschiniSwiss Banking
InstituteUniversity of Zurich,
Switzerland
Definition
Motivation
Contribution
Results
SO2 Stylized facts
Zero Excess
Econometric Approach
Unconditional Model
Conditional Model
Stable-MixtureGARCH
Summary of CO2
GARCH framework for SO2
I Because of the massive volatility clustering, a GARCH model(with a fat–tailed distribution) suggests itself:
rt = µt + εt = µt + σtzt , σ2t = θ0 +
r∑i=1
θiε2t−i +
s∑j=1
φjσ2t−j ,
where ztiid∼ fZ (·) and where fZ is a zero-location, unit-scale
probability density (in Bollerslev 1986, Gaussian. In Bollerslev1987, Student’s t).
I The problem is still the zeros! The GARCH model does notaccount for this. We get the same problems as in theunconditional case.
I A mixture distribution suggests itself, with one componentcapturing the zeros.
An EconometricAnalysis of EmissionTrading Allowances
M. Paolella L. TaschiniSwiss Banking
InstituteUniversity of Zurich,
Switzerland
Definition
Motivation
Contribution
Results
SO2 Stylized facts
Zero Excess
Econometric Approach
Unconditional Model
Conditional Model
Stable-MixtureGARCH
Summary of CO2
Mixture Models
{εt} is generated by an n–component Mixed Normal GARCH(r , s)process, if the conditional distribution of εt is an n–componentmixed normal with zero mean, i.e.,
εt |Ft−1 ∼MN(ω,µ,σ2
t
), (1)
and the mixed normal density is given by
fMN(y ;ω,µ,σ2
)=
n∑j=1
ωjφ(y ;µj , σ
2jt
), (2)
φ is the normal pdf, ωj ∈ (0, 1) with∑n
j=1 ωj = 1 and, to ensure
zero mean, µn = −∑n−1
j=1 (ωj/ωn) µj .The n x 1 component variances evolves according to aGARCH–like structure
σ(2)t = γ0 +
r∑i=1
γ iε2t−i +
s∑j=1
Ψjσ(2)t−j , (3)
An EconometricAnalysis of EmissionTrading Allowances
M. Paolella L. TaschiniSwiss Banking
InstituteUniversity of Zurich,
Switzerland
Definition
Motivation
Contribution
Results
SO2 Stylized facts
Zero Excess
Econometric Approach
Unconditional Model
Conditional Model
Stable-MixtureGARCH
Summary of CO2
The Component Variance
I We denote by MixN(n, g) the model with n componentdensities, but such that only g , g ≤ n, follow a GARCH(r , s)process.
I To avoid a degenerate component, we replace the zeros withsmall iid normal noise, with zero mean and constant smallvariance. This is NOT as ad hoc as it seems!
I In our notation the component variance for the MN(3,2)takes the form:
σ21t
σ22t
σ23t
=
γ01
γ02
γ03
+
γ11
γ12
0
ε2t−1+
Ψ11 0 00 Ψ22 00 0 0
σ21,t−1
σ22,t−1
σ23,t−1
.
An EconometricAnalysis of EmissionTrading Allowances
M. Paolella L. TaschiniSwiss Banking
InstituteUniversity of Zurich,
Switzerland
Definition
Motivation
Contribution
Results
SO2 Stylized facts
Zero Excess
Econometric Approach
Unconditional Model
Conditional Model
Stable-MixtureGARCH
Summary of CO2
Likelihood-based goodness-of-fit
Model K L AIC BICMN(1,1) 5 −4072.0 8134.0 8181.42MN(2,1) 8 −2919.6 5855.2 5899.07MN(2,2) 10 −2919.3 5858.6 5913.44MN(3,1) 11 −2873.8 5769.6 5829.93MN(3,2) 13 −2835.7 5697.4 5768.70MN(4,2) 16 −2834.5 5701.0 5781.75MN(4,3) 18 −2831.6 5699.2 5797.92
Table: Likelihood-based goodness-of-fit for SO2. The best values foreach criteria are marked in boldface.
An EconometricAnalysis of EmissionTrading Allowances
M. Paolella L. TaschiniSwiss Banking
InstituteUniversity of Zurich,
Switzerland
Definition
Motivation
Contribution
Results
SO2 Stylized facts
Zero Excess
Econometric Approach
Unconditional Model
Conditional Model
Stable-MixtureGARCH
Summary of CO2
Empirical Results
Param MN(2,1) MN(3,1) MN(3,2)a0 0.046 0.040 0.041a1 0.000 0.000 0.000
γ01 0.137 0.157 0.621γ11 0.243 0.235 0.427Ψ11 0.797 0.867 0.846ω1 0.709 0.440 0.165µ1 0.019 0.012 −0.001
γ02 0.001 0.505 0.121γ12 - - 0.212Ψ22 - - 0.649ω2 0.290 0.334 0.595µ2 −0.047 0.019 0.001
γ03 - 0.012 0.013γ13 - - -Ψ33 - - -ω3 - 0.226 0.239µ3 - −0.046 −0.045
Table: Maximum likelihood parameter estimates of the mixed normalGARCH models for the SO2 allowances price return 1999-2006.
An EconometricAnalysis of EmissionTrading Allowances
M. Paolella L. TaschiniSwiss Banking
InstituteUniversity of Zurich,
Switzerland
Definition
Motivation
Contribution
Results
SO2 Stylized facts
Zero Excess
Econometric Approach
Unconditional Model
Conditional Model
Stable-MixtureGARCH
Summary of CO2
Time Varying Moments
0 200 400 600 800 1000 1200 1400 1600 1800−1.5
−1
−0.5
0x 10
−3
Time in days
Implied Skewness
0 200 400 600 800 1000 1200 1400 16004.5
5
5.5
6
6.5
7
7.5
8
Time in days
Implied Kurtosis
An EconometricAnalysis of EmissionTrading Allowances
M. Paolella L. TaschiniSwiss Banking
InstituteUniversity of Zurich,
Switzerland
Definition
Motivation
Contribution
Results
SO2 Stylized facts
Zero Excess
Econometric Approach
Unconditional Model
Conditional Model
Stable-MixtureGARCH
Summary of CO2
QQ–plot simulated and actual series
−15 −10 −5 0 5 10 15−15
−10
−5
0
5
10
15
X Quantiles
Y Q
uant
iles
−15 −10 −5 0 5 10 15−15
−10
−5
0
5
10
15
X Quantiles
Y Q
uant
iles
−15 −10 −5 0 5 10 15−15
−10
−5
0
5
10
15
X Quantiles
Y Q
uant
iles
−15 −10 −5 0 5 10 15−15
−10
−5
0
5
10
15
20
X Quantiles
Y Q
uant
iles
An EconometricAnalysis of EmissionTrading Allowances
M. Paolella L. TaschiniSwiss Banking
InstituteUniversity of Zurich,
Switzerland
Definition
Motivation
Contribution
Results
SO2 Stylized facts
Zero Excess
Econometric Approach
Unconditional Model
Conditional Model
Stable-MixtureGARCH
Summary of CO2
Stable-Mixture GARCH
I stable-GARCH and mix-norm-GARCH are completelydifferent classes of models. Both have theoretically niceproperties and admirable in- and out-of-sample fit.
I Only the mixture model can support the zeros problem.
I The models can be COMBINED:
fεt |Ft−1(x ;α,ω,µ,σ
(δ)t ) =
n∑j=1
ωj fS(x ;αj , µj , σδjt),
where α = (α1, . . . , αn)′ and the component scale terms are
σ(δ)t = γ0 +
r∑i=1
γ i
∣∣εt−i
∣∣δ +s∑
j=1
Ψjσ(δ)t−j .
An EconometricAnalysis of EmissionTrading Allowances
M. Paolella L. TaschiniSwiss Banking
InstituteUniversity of Zurich,
Switzerland
Definition
Motivation
Contribution
Results
SO2 Stylized facts
Zero Excess
Econometric Approach
Unconditional Model
Conditional Model
Stable-MixtureGARCH
Summary of CO2
Summary of CO2 Returns Analysis
I Only 337 returns.
I The Hint estimate of the unconditional tail index is1.25(0.091), and removing the single massive negative returngives 1.304(0.092).
I For the normal-mixture models, MN(3,3) and MN(2,2) arenearly as good, according to AIC.
I The best model, by far, is the stable-mixture-GARCH,according to all criteria, despite the large number ofparameters and the small sample size.
An EconometricAnalysis of EmissionTrading Allowances
M. Paolella L. TaschiniSwiss Banking
InstituteUniversity of Zurich,
Switzerland
Definition
Motivation
Contribution
Results
SO2 Stylized facts
Zero Excess
Econometric Approach
Unconditional Model
Conditional Model
Stable-MixtureGARCH
Summary of CO2
Summary of CO2 Returns Analysis
Model K L AIC BICSα,0-GARCH 6 −799.25 1610.49 1633.42Sα,β-GARCH 7 −793.18 1600.36 1627.10MN(1,1) 5 −983.26 1976.53 1995.60MN(2,1) 8 −805.30 1626.60 1657.16MN(2,2) 10 −788.40 1596.80 1635.00MN(3,1) 11 −800.60 1623.20 1665.22MN(3,2) 13 −785.57 1597.14 1646.80MN(3,3) 15 −784.30 1598.60 1655.90MSα(2, 2) 12 −785.83 1595.75 1647.36MSα(3, 2) 16 −749.29 1530.60 1591.72MSα(3, 3) 18 −748.17 1532.38 1601.11
Table: Likelihood-based goodness-of-fit for CO2. Sα,β-GARCH refers tothe AR(1)-stable-GARCH(1,1) model with Z ∼ Sα,β (0, 1).