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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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