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An Econometric Analysis of Emission Trading Allowances M. Paolella L. Taschini Swiss Banking Institute University of Zurich, Switzerland An Econometric Analysis of Emission Deﬁnition Trading Allowances Motivation Contribution Results SO2 Stylized facts M. Paolella L. Taschini Zero Excess Swiss Banking Institute Econometric Approach University of Zurich, Unconditional Model Switzerland Conditional Model Stable-Mixture GARCH Summary of CO2 November 2006 An Econometric Analysis of Emission Trading Allowances M. Paolella L. Taschini Swiss Banking Institute University of Zurich, Switzerland Deﬁnition Motivation Contribution Results SO2 Stylized facts Zero Excess Econometric Approach Unconditional Model Conditional Model Stable-Mixture GARCH Summary of CO2 An Econometric Analysis of Emission Trading Allowances M. Paolella L. Taschini Swiss Banking Institute University of Zurich, Switzerland Deﬁnition Motivation Contribution Results SO2 Stylized facts Zero Excess Econometric Approach Unconditional Model Conditional Model Stable-Mixture GARCH Summary of CO2 An Econometric Analysis of Emission Trading Allowances M. Paolella L. Taschini Swiss Banking Institute University of Zurich, Switzerland Deﬁnition Motivation Contribution Results SO2 Stylized facts Zero Excess Econometric Approach Unconditional Model Conditional Model Stable-Mixture GARCH Summary of CO2 An Econometric Analysis of Emission Trading Allowances M. Paolella L. Taschini Swiss Banking Institute University of Zurich, Switzerland Deﬁnition Motivation Contribution Results SO2 Stylized facts Zero Excess Econometric Approach Unconditional Model Conditional Model Stable-Mixture GARCH Summary of CO2 An Econometric Analysis of Emission Trading Allowances M. Paolella L. Taschini Swiss Banking Institute University of Zurich, Switzerland Deﬁnition Motivation Contribution Results SO2 Stylized facts Zero Excess Econometric Approach Unconditional Model Conditional Model Stable-Mixture GARCH Summary of CO2 An Econometric Deﬁnition of Tradable Permits Analysis of Emission Trading Allowances M. Paolella L. Taschini Swiss Banking Institute University of Zurich, Tradable permits are a cost-eﬃcient, market-driven approach for Switzerland reducing GHG. Deﬁnition Motivation They are tradable allocation entitled by a government to an Contribution individual ﬁrm to emit a speciﬁc amount of a substance over a Results SO2 Stylized facts speciﬁed interval of time. Zero Excess Econometric Approach They enlists market forces in the quest for cost–eﬀective pollution Unconditional Model control and encouraging technological progress. Conditional Model Stable-Mixture GARCH Tradable emission permits programs are being adopted by Summary of CO2 environmental regulators in applications ranging from local and regional (US-CAAA Title IV) global scale (EU-ETS and from 2008 the Kyoto Protocol) An Econometric Motivation Analysis of Emission Trading Allowances M. Paolella L. Taschini Swiss Banking Institute University of Zurich, Switzerland Deﬁnition Emission Allowances inﬂuence: Motivation Commodity markets and energy market; Contribution Results Business decisions begin to be made with the price of carbon SO2 Stylized facts as a criterion; Zero Excess Firms stock value. Econometric Approach Unconditional Model Conditional Model Emission Allowances market: Stable-Mixture high volatility and market crash; GARCH Summary of CO2 market is working. An Econometric Rhodia daily prices Analysis of Emission Trading Allowances M. Paolella L. Taschini Swiss Banking Institute University of Zurich, Switzerland Deﬁnition Motivation Contribution Results SO2 Stylized facts Zero Excess Econometric Approach Unconditional Model Conditional Model Stable-Mixture GARCH Summary of CO2 CDM:a project-based mechanism, according to which the buyer purchases emission credits from a project that can credibly and veriﬁable demonstrate that it reduces GHG emissions compared with what would have happened otherwise. An Econometric CO2 daily prices Analysis of Emission Trading Allowances M. Paolella L. Taschini Swiss Banking CO2 daily price Institute University of Zurich, Switzerland 28 Deﬁnition 26 Motivation Contribution 24 Results 22 SO2 Stylized facts 20 Zero Excess 18 Econometric Approach Unconditional Model 16 Conditional Model 14 Stable-Mixture GARCH 12 Summary of CO2 Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov 2005 2006 Figure: Daily CO2 allowance prices over the period June 25, 2005 to November 3, 2006. An Econometric SO2 daily prices Analysis of Emission Trading Allowances M. Paolella L. Taschini Swiss Banking Institute SO2 daily price University of Zurich, Switzerland 1600 Deﬁnition 1400 Motivation Contribution 1200 Results 1000 SO2 Stylized facts Zero Excess 800 Econometric Approach 600 Unconditional Model Conditional Model 400 Stable-Mixture GARCH 200 Summary of CO2 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 Figure: Daily SO2 allowance prices over the period January 1999 - May 2006. An Econometric Contribution Analysis of Emission Trading Allowances M. Paolella L. Taschini Swiss Banking Institute University of Zurich, Switzerland Deﬁnition Motivation Econometric investigation of emission allowances price: Contribution Measure the tail index (tail thickness) of the unconditional Results distribution: SO2 Stylized facts Useful for long-term risk assessment and probability of Zero Excess extreme movements Econometric Approach Unconditional Model Fit nonstandard GARCH-type models for the conditional Conditional Model distribution: Stable-Mixture GARCH Short term risk and volatility prediction Summary of CO2 An Econometric Results Analysis of Emission Trading Allowances M. Paolella L. Taschini Swiss Banking Institute University of Zurich, Switzerland Deﬁnition Actual forecast methods based on supply and demand Motivation fundamental analysis: Contribution Results SO2 Stylized facts does not suﬃce due to the complexity of the market... Zero Excess ...we believe that fundamentals drive value and a future step Econometric Approach is to implement a fundamental-like analysis into the mean Unconditional Model equation of the return process... Conditional Model Stable-Mixture A pure statistical model designed to capture the unique GARCH stylized facts of the data (abundance of zero-returns and Summary of CO2 complicated conditional heteroskedasticity) An Econometric SO2 Stylized facts Analysis of Emission Trading Allowances M. Paolella L. Taschini Swiss Banking 15 Institute 10 University of Zurich, 5 Switzerland 0 −5 Deﬁnition −10 Motivation −15 0 200 400 600 800 1000 1200 1400 1600 1800 Contribution 0.4 0.3 Results 0.2 0.1 SO2 Stylized facts 0 Zero Excess −0.1 −0.2 Econometric Approach −0.3 −0.4 Unconditional Model 0 10 20 30 40 50 0.4 Conditional Model 0.3 0.2 Stable-Mixture 0.1 GARCH 0 −0.1 Summary of CO2 −0.2 −0.3 −0.4 0 10 20 30 40 50 Figure: Daily SO2 returns (top), the SACF of the absolute returns (middle) and the SACF of the zeros-removed absolute returns (bottom). An Econometric Zero Excess Analysis of Emission Trading Allowances M. Paolella L. Taschini SO return Swiss Banking 2 Institute 4 University of Zurich, Switzerland 3 Deﬁnition Motivation 2 Contribution 1 Results SO2 Stylized facts 0 Percentage Zero Excess Econometric Approach −1 Unconditional Model −2 Conditional Model Stable-Mixture −3 GARCH Summary of CO2 −4 −5 700 800 900 1000 1100 1200 Time in days Figure: Magnifying Zero excess around 500 days. An Econometric Unconditional Distributional Fit: SαS Analysis of Emission Trading Allowances M. Paolella L. Taschini Large number of zeros precludes use of typical fat-tailed Swiss Banking Institute distributions (t, hyperbolic, stable, etc) because the center University of Zurich, Switzerland will be too peaked, forcing the tails to be unnecessarily thick. Deﬁnition Otherwise, the stable Paretian would be a great candidate Motivation distribution (GCLT, closed under summation, good ﬁt to Contribution ﬁnancial returns data, easy VaR approximations) Results The downside is its computation: For the symmetric stable, SO2 Stylized facts the characteristic function is Zero Excess Econometric Approach α ϕX (t; α) = E exp{itX } = exp {− |t| } , 0 < α ≤ 2. Unconditional Model Conditional Model and the usual inversion formula reduces to: Stable-Mixture GARCH ∞ 1 α Summary of CO2 fX (x) = cos (tx) e −t dt. π 0 We use the FFT and linear interpolation to speed up. Except for the normal (α = 2) case, the α-stable distribution has inﬁnite variance. For α ≤ 1, its tails are so heavy that even the mean does not exist. An Econometric Asymmetric Stable Analysis of Emission Trading Allowances M. Paolella L. Taschini Swiss Banking Institute University of Zurich, Switzerland The general stable Paretian distribution, with skewness Deﬁnition parameter β, location µ and scale σ, is denoted Sα,β (µ, σ) Motivation and its characteristic function is E e i t θ via Contribution Results α iµθ − |σθ| 1 − iβsgn (θ) tan πα , α = 1, SO2 Stylized facts log E e i tθ = 2 2 Zero Excess iµθ − |σθ| 1 + iβ π sgn (θ) log |θ| , α = 1, Econometric Approach Unconditional Model for α ∈ (0, 2], β ∈ [−1, 1], σ > 0, and µ ∈ R. Conditional Model For the SO2 data, the estimate of β was practically and Stable-Mixture GARCH statistically zero. Summary of CO2 MLE of α for SO2 returns with the zeros removed is 1.45. (With zeros, it is near Cauchy). An Econometric Kernel Analysis of Emission Trading Allowances M. Paolella L. Taschini Swiss Banking Institute −3 University of Zurich, x 10 kernel 6 Switzerland normal kernel stable normal 0.25 Deﬁnition 5 stable Motivation 0.2 Contribution 4 Results 0.15 3 SO2 Stylized facts Zero Excess 0.1 2 Econometric Approach Unconditional Model 0.05 1 Conditional Model Stable-Mixture 0 0 GARCH −10 −5 0 5 10 6 7 8 9 10 11 12 13 Summary of CO2 Figure: Kernel density (solid) of the SO2 return series, with the best-ﬁtting normal density (dashed) and best-ﬁtting symmetric stable density (dash-dot). Right panel is just the magniﬁed view of the right tail. An Econometric Zeros Problem: Structure Analysis of Emission Trading Allowances If the occurrence of the zeros throughout the data have M. Paolella L. Taschini Swiss Banking (Markov) structure, then this needs to be modeled. Institute University of Zurich, Use standard combinatoric runs test, plot p-values Switzerland Deﬁnition 1 Motivation 0.9 Contribution 0.8 Results 0.7 SO2 Stylized facts 0.6 Zero Excess 0.5 Econometric Approach 0.4 Unconditional Model Conditional Model 0.3 Stable-Mixture 0.2 GARCH 0.1 Summary of CO2 0 200 400 600 800 1000 1200 1400 1600 Figure: The p-values from the runs–test performed on segments of the SO2 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 Econometric Zeros Problem: Eﬀect on α Analysis of Emission Trading Allowances M. Paolella L. Taschini The tail index α is biased because of the overabundance of zeros. Swiss Banking Institute University of Zurich, Symmetric Stable Paretian Density Switzerland 0.5 Deﬁnition 0.45 Motivation 0.4 Contribution 0.35 Results 0.3 SO2 Stylized facts Zero Excess pdf 0.25 Econometric Approach 0.2 Unconditional Model 0.15 Conditional Model Stable-Mixture 0.1 GARCH 0.05 Summary of CO2 0 −15 −10 −5 0 5 10 15 x Figure: Symmetric Stable densities: α=1.027 for blue line and α=1.64 for green line. An Econometric Magniﬁed view of tail Analysis of Emission Trading Allowances M. Paolella L. Taschini Symmetric Stable Paretian Density Swiss Banking 0.12 Institute University of Zurich, Switzerland 0.1 Deﬁnition Motivation 0.08 Contribution Results 0.06 SO2 Stylized facts Zero Excess 0.04 pdf Econometric Approach Unconditional Model 0.02 Conditional Model 0 Stable-Mixture GARCH Summary of CO2 −0.02 −0.04 −14 −12 −10 −8 −6 −4 −2 0 x Figure: Magnifying tail decay. An Econometric Hill Estimator Analysis of Emission Trading Allowances M. Paolella L. Taschini Swiss Banking Institute To avoid the zeros-problem and still estimate α, use a tail University of Zurich, Switzerland estimator. The Hill estimator is by far the most commonly used tail estimator Deﬁnition for the tail index of a distribution: Motivation Contribution 1 Results ˆ αHill (k) = k SO2 Stylized facts (1/k) j=1 ln (Xn+1−j:n ) − ln Xn−k:n Zero Excess with standard error Econometric Approach Unconditional Model ˆ k αHill (k) Conditional Model ˆ SE (αHill;k ) = 1/2 , k > 2, (k − 1) (k − 2) Stable-Mixture GARCH Summary of CO2 where Xj:n denotes the jth order statistic of sample X1 , . . . , Xn . If the right tail of the distribution is asymptotically Pareto, i.e., for large x, 1 − F (x) ≈ cx −α , then, given an appropriate choice of k, ˆ αHill provides an estimate of Pareto tail index α. An Econometric Hill Estimator Analysis of Emission Trading Allowances M. Paolella L. Taschini Swiss Banking We plot the Hill estimates of the the SO2 spot price as a function Institute University of Zurich, of k, based on the 1,780 sorted absolute returns. Switzerland Deﬁnition 6 Motivation 5 Contribution Results 4 SO2 Stylized facts Zero Excess 3 Econometric Approach Unconditional Model 2 Conditional Model Stable-Mixture 1 GARCH Summary of CO2 0 0 200 400 600 800 1000 The graph is typical of Hill estimator plots applied to ﬁnancial ˆ returns data, and a sizeable region for which αHill is “ﬂat”, or roughly constant in k, cannot be found. An Econometric Hill Intercept Estimator Analysis of Emission Trading Allowances M. Paolella L. Taschini Swiss Banking Institute A tail estimator designed explicitly for stable Paretian data and University of Zurich, Switzerland which exhibits excellent small sample properties was developed in Deﬁnition Mittnik and Paolella (1999). Motivation Contribution It is based on a set of Hill estimators for a range of k–values, and Results computed as SO2 Stylized facts Zero Excess ˆ ˆ αHint = −0.8110 − 0.3079 b + 2.0278 b ˆ0.5 Econometric Approach Unconditional Model ˆ ˆ where b is the intercept in the simple linear regression of αHill (k) Conditional Model on k/1000. Stable-Mixture GARCH Summary of CO2 The main feature is that αHint is unbiased for α ∈ [1, 2] and ˆ virtually exactly normally distributed. ˆ For the SO2 returns, we obtain αHint = 1.46 with standard error 0.043. An Econometric GARCH framework for SO2 Analysis of Emission Trading Allowances M. Paolella L. Taschini Swiss Banking Institute University of Zurich, Because of the massive volatility clustering, a GARCH model Switzerland (with a fat–tailed distribution) suggests itself: Deﬁnition Motivation r s Contribution 2 rt = µt + εt = µt + σt zt , σt = θ0 + θ i ε2 t−i + 2 φj σt−j , Results SO2 Stylized facts i=1 j=1 Zero Excess iid Econometric Approach where zt ∼ fZ (·) and where fZ is a zero-location, unit-scale Unconditional Model probability density (in Bollerslev 1986, Gaussian. In Bollerslev Conditional Model 1987, Student’s t). Stable-Mixture GARCH The problem is still the zeros! The GARCH model does not Summary of CO2 account for this. We get the same problems as in the unconditional case. A mixture distribution suggests itself, with one component capturing the zeros. An Econometric Mixture Models Analysis of Emission Trading Allowances M. Paolella L. Taschini { t } is generated by an n–component Mixed Normal GARCH(r , s) Swiss Banking Institute process, if the conditional distribution of t is an n–component University of Zurich, Switzerland mixed normal with zero mean, i.e., Deﬁnition Motivation t |Ft−1 ∼ MN ω, µ, σ 2 t , (1) Contribution and the mixed normal density is given by Results SO2 Stylized facts n Zero Excess fMN y ; ω, µ, σ 2 = 2 ωj φ y ; µj , σjt , (2) Econometric Approach j=1 Unconditional Model n Conditional Model φ is the normal pdf, ωj ∈ (0, 1) with j=1 ωj = 1 and, to ensure Stable-Mixture n−1 GARCH zero mean, µn = − j=1 (ωj /ωn ) µj . Summary of CO2 The n x 1 component variances evolves according to a GARCH–like structure r s (2) 2 (2) σt = γ0 + γi t−i + Ψj σ t−j , (3) i=1 j=1 An Econometric The Component Variance Analysis of Emission Trading Allowances M. Paolella L. Taschini Swiss Banking Institute University of Zurich, We denote by MixN(n, g ) the model with n component Switzerland densities, but such that only g , g ≤ n, follow a GARCH(r , s) Deﬁnition process. Motivation Contribution Results To avoid a degenerate component, we replace the zeros with SO2 Stylized facts small iid normal noise, with zero mean and constant small Zero Excess variance. This is NOT as ad hoc as it seems! Econometric Approach In our notation the component variance for the MN(3,2) Unconditional Model takes the form: Conditional Model Stable-Mixture GARCH Summary of CO2 2 2 σ1t γ01 γ11 Ψ11 0 0 σ1,t−1 2 2 2 σ2t = γ02 + γ12 t−1 + 0 Ψ22 0 σ2,t−1 . 2 2 σ3t γ03 0 0 0 0 σ3,t−1 An Econometric Likelihood-based goodness-of-ﬁt Analysis of Emission Trading Allowances M. Paolella L. Taschini Swiss Banking Institute University of Zurich, Switzerland Deﬁnition Model K L AIC BIC Motivation MN(1,1) 5 −4072.0 8134.0 8181.42 Contribution MN(2,1) 8 −2919.6 5855.2 5899.07 Results SO2 Stylized facts MN(2,2) 10 −2919.3 5858.6 5913.44 Zero Excess MN(3,1) 11 −2873.8 5769.6 5829.93 Econometric Approach MN(3,2) 13 −2835.7 5697.4 5768.70 Unconditional Model MN(4,2) 16 −2834.5 5701.0 5781.75 Conditional Model MN(4,3) 18 −2831.6 5699.2 5797.92 Stable-Mixture GARCH Table: Likelihood-based goodness-of-ﬁt for SO2 . The best values for Summary of CO2 each criteria are marked in boldface. An Econometric Empirical Results Analysis of Emission Trading Allowances M. Paolella L. Taschini Swiss Banking Param MN(2,1) MN(3,1) MN(3,2) Institute a0 0.046 0.040 0.041 University of Zurich, Switzerland a1 0.000 0.000 0.000 Deﬁnition γ01 0.137 0.157 0.621 Motivation γ11 0.243 0.235 0.427 Contribution Ψ11 0.797 0.867 0.846 ω1 0.709 0.440 0.165 Results µ1 0.019 0.012 −0.001 SO2 Stylized facts Zero Excess γ02 0.001 0.505 0.121 Econometric Approach γ12 - - 0.212 Unconditional Model Ψ22 - - 0.649 ω2 0.290 0.334 0.595 Conditional Model µ2 −0.047 0.019 0.001 Stable-Mixture GARCH γ03 - 0.012 0.013 Summary of CO2 γ13 - - - Ψ33 - - - ω3 - 0.226 0.239 µ3 - −0.046 −0.045 Table: Maximum likelihood parameter estimates of the mixed normal GARCH models for the SO2 allowances price return 1999-2006. An Econometric Time Varying Moments Analysis of Emission Trading Allowances M. Paolella L. Taschini Swiss Banking −3 Implied Skewness Institute x 10 0 University of Zurich, Switzerland Deﬁnition −0.5 Motivation Contribution −1 Results SO2 Stylized facts Zero Excess −1.5 0 200 400 600 800 1000 1200 1400 1600 1800 Time in days Econometric Approach Unconditional Model Implied Kurtosis 8 Conditional Model 7.5 Stable-Mixture GARCH 7 Summary of CO2 6.5 6 5.5 5 4.5 0 200 400 600 800 1000 1200 1400 1600 Time in days An Econometric QQ–plot simulated and actual series Analysis of Emission Trading Allowances M. Paolella L. Taschini Swiss Banking Institute University of Zurich, 15 15 Switzerland 10 10 Deﬁnition 5 5 Motivation Y Quantiles Y Quantiles 0 0 Contribution Results −5 −5 SO2 Stylized facts −10 −10 Zero Excess −15 −15 −15 −10 −5 0 5 10 15 −15 −10 −5 0 5 10 15 Econometric Approach X Quantiles X Quantiles Unconditional Model 15 20 Conditional Model 10 15 Stable-Mixture GARCH 10 5 Summary of CO2 Y Quantiles Y Quantiles 5 0 0 −5 −5 −10 −10 −15 −15 −10 −5 0 5 10 15 −15 −15 −10 −5 0 5 10 15 X Quantiles X Quantiles An Econometric Stable-Mixture GARCH Analysis of Emission Trading Allowances M. Paolella L. Taschini Swiss Banking Institute University of Zurich, Switzerland stable-GARCH and mix-norm-GARCH are completely diﬀerent classes of models. Both have theoretically nice Deﬁnition Motivation properties and admirable in- and out-of-sample ﬁt. Contribution Only the mixture model can support the zeros problem. Results The models can be COMBINED: SO2 Stylized facts Zero Excess n (δ) Econometric Approach δ f t |Ft−1 (x; α, ω, µ, σ t ) = ωj fS (x; αj , µj , σjt ), Unconditional Model j=1 Conditional Model Stable-Mixture where α = (α1 , . . . , αn ) and the component scale terms are GARCH Summary of CO2 r s (δ) δ (δ) σt = γ0 + γi t−i + Ψj σ t−j . i=1 j=1 An Econometric Summary of CO2 Returns Analysis Analysis of Emission Trading Allowances M. Paolella L. Taschini Swiss Banking Institute University of Zurich, Switzerland Deﬁnition Only 337 returns. Motivation Contribution The Hint estimate of the unconditional tail index is Results 1.25(0.091), and removing the single massive negative return SO2 Stylized facts gives 1.304(0.092). Zero Excess For the normal-mixture models, MN(3,3) and MN(2,2) are Econometric Approach nearly as good, according to AIC. Unconditional Model Conditional Model The best model, by far, is the stable-mixture-GARCH, Stable-Mixture according to all criteria, despite the large number of GARCH parameters and the small sample size. Summary of CO2 An Econometric Summary of CO2 Returns Analysis Analysis of Emission Trading Allowances M. Paolella L. Taschini Swiss Banking Institute University of Zurich, Switzerland Model K L AIC BIC Sα,0 -GARCH 6 −799.25 1610.49 1633.42 Deﬁnition Motivation Sα,β -GARCH 7 −793.18 1600.36 1627.10 Contribution MN(1,1) 5 −983.26 1976.53 1995.60 Results MN(2,1) 8 −805.30 1626.60 1657.16 SO2 Stylized facts MN(2,2) 10 −788.40 1596.80 1635.00 Zero Excess MN(3,1) 11 −800.60 1623.20 1665.22 Econometric Approach MN(3,2) 13 −785.57 1597.14 1646.80 Unconditional Model MN(3,3) 15 −784.30 1598.60 1655.90 Conditional Model MSα (2, 2) 12 −785.83 1595.75 1647.36 Stable-Mixture GARCH MSα (3, 2) 16 −749.29 1530.60 1591.72 Summary of CO2 MSα (3, 3) 18 −748.17 1532.38 1601.11 Table: Likelihood-based goodness-of-ﬁt for CO2 . Sα,β -GARCH refers to the AR(1)-stable-GARCH(1,1) model with Z ∼ Sα,β (0, 1).

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