Modeling Volatility and Dependence of European Carbon and Energy Prices

class: center, middle, inverse, title-slide

.title[
# Modeling Volatility and Dependence of European Carbon and Energy Prices
]
.subtitle[
## Jonathan Berrisch, Sven Pappert, Florian Ziel, Antonia Arsova
]
.author[
### University of Duisburg-Essen
]
.date[
### 2023-06-22
]

---

# Motivation

.pull-left[

</br>

- Understanding European Allowances (EUA) dynamics is important
for several fields:
- - Portfolio & Risk Management,
- - Sustainability Planing
- - Political decisions
- - ...

EUA prices are obviously connected to the energy market

How can the dynamics be characterized?

]

.pull-right[

</br>

- Several Questions arise:

- - Data (Pre)processing
- - Modeling Approach
- - Evaluation

]

???

On their own, EUA prices are ’just’ a random walk.
The first differences are stationary and heteroscedastic.
Autoregressive Volatility modeling (ARCH, GARCH, . . .) is sensible!

---
  
# Data

- EUA, natural gas, Brent crude oil, coal 
- March 15, 2010, until October 14, 2022

- Data was normalized w.r.t. `$\text{CO}_2$` emissions
- Emission-adjusted prices reflects one tonne of `$\text{CO}_2$`

- We adjusted for inflation by Eurostat's HICP, excluding energy

- Log transformation of the data to stabilize the variance

- ADF Test: All series are stationary in first differences

- Johansen’s likelihood ratio trace test suggests two cointegrating relationships (levels)
- Johansen’s likelihood ratio trace test suggests no cointegrating relationships (logs)

---

# Data

---

# Modeling Approach: Overview

</br>

#### VECM: Vector Error Correction Model
  - Modeling the expectaion
  - Captures the long-run cointegrating relationship
  - Different cointegrating ranks, including rank zero (no cointegration)

#### GARCH: Generalized Autoregressive Conditional Heteroscedasticity
  - Captures the variance dynamics

#### Copula: Captures the dependence structure
  - Captures: conditional cross-sectional dependence structure
  - Dependence allowed to vary over time

---

# Modeling Approach: Overview

.pull-left[

- Let `$\boldsymbol{X}_t$` be a `$K$`-dimensional vector at time `$t$`
- The forecasting target:
  - Conditional joint distribution 
  - `$F_{\boldsymbol{X}_t|\mathcal{F}_{t-1}}$` 
  - `$\mathcal{F}_{t}$` is the sigma field generated by all information available up to and including time `$t$`

Sklars theorem: decompose target into 
  - marginal distributions: `$F_{X_{k,t}|\mathcal{F}_{t-1}}$` for `$k=1,\ldots, K$`, and
  - copula function: `$C_{\boldsymbol{U}_{t}|\mathcal{F}_{t - 1}}$`

]

.pull-right[

Let  `$\boldsymbol{x}_t= (x_{1,t},\ldots, x_{K,t})^\intercal$` be the realized values

It holds that:

`\begin{align}
    F_{\boldsymbol{X}_t|\mathcal{F}_{t-1}}(\boldsymbol{x}_t) = C_{\boldsymbol{U}_{t}|\mathcal{F}_{t - 1}}(\boldsymbol{u}_t) \nonumber
\end{align}`

with: `$\boldsymbol{u}_t =(u_{1,t},\ldots, u_{K,t})^\intercal$`,  `$u_{k,t} = F_{X_{k,t}|\mathcal{F}_{t-1}}(x_{k,t})$`

For brewity we drop the conditioning on `$\mathcal{F}_{t-1}$`.

The model can be specified as follows

`\begin{align}
    F(\boldsymbol{x}_t) = C \left[\mathbf{F}(\boldsymbol{x}_t; \boldsymbol{\mu}_t, \boldsymbol{ \sigma }_{t}^2, \boldsymbol{\nu}, \boldsymbol{\lambda}); \Xi_t, \Theta\right] \nonumber
\end{align}`

`$\Xi_{t}$` denotes time-varying dependence parameters
`$\Theta$` denotes time-invariant dependence parameters

We take `$C$` as the `$t$`-copula

]

---

# Modeling Approach: Mean and Variance

.pull-left[

#### Individual marginal distributions:

`$$\mathbf{F} = (F_1, \ldots, F_K)^{\intercal}$$`

#### Generalized non-central t-distributions
  - To account for heavy tails
  - Time varying
    - expectation: `$\boldsymbol{\mu}_t = (\mu_{1,t}, \ldots, \mu_{K,t})^{\intercal}$`
    - variance: `$\boldsymbol{\sigma}_{t}^2 = (\sigma_{1,t}^2, \ldots, \sigma_{K,t}^2)^{\intercal}$`
  - Time invariant 
    - degrees of freedom: `$\boldsymbol{\nu} = (\nu_1, \ldots, \nu_K)^{\intercal}$`
    - noncentrality: `$\boldsymbol{\lambda} = (\lambda_1, \ldots, \lambda_K)^{\intercal}$`

]

.pull-right[

#### VECM Model

`\begin{align}
    \Delta \boldsymbol{\mu}_t = \Pi \boldsymbol{x}_{t-1} + \Gamma \Delta \boldsymbol{x}_{t-1} \nonumber
\end{align}`

where `$\Pi = \alpha \beta^{\intercal}$`  is the cointegrating matrix of rank `$r$`, `$0 \leq r\leq K$`.

#### GARCH model

`\begin{align}
    \sigma_{i,t}^2 = & \omega_i + \alpha^+_{i} (\epsilon_{i,t-1}^+)^2 + \alpha^-_{i} (\epsilon_{i,t-1}^-)^2 + \beta_i \sigma_{i,t-1}^2 \nonumber
\end{align}`

where `$\epsilon_{i,t-1}^+ = \max\{\epsilon_{i,t-1}, 0\}$` ...

Separate coefficients for positive and negative innovations to capture leverage effects.

]

---

# Modeling Approach: Dependence

.pull-left[

#### Time-varying dependence parameters

`\begin{align*}
    \Xi_{t} = & \Lambda\left(\boldsymbol{\xi}_{t}\right)
    \\
    \xi_{ij,t} = & \eta_{0,ij} + \eta_{1,ij} \xi_{ij,t-1} + \eta_{2,ij} z_{i,t-1} z_{j,t-1},
\end{align*}`

`$\xi_{ij,t}$` is a latent process

`$z_{i,t}$` denotes the `$i$`-th standardized residual from time series `$i$` at time point `$t$`

`$\Lambda(\cdot)$` is a link function
- ensures that `$\Xi_{t}$` is a valid variance covariance matrix
- ensures that `$\Xi_{t}$` does not exceed its support space and remains semi-positive definite
]

.pull-right[

#### Maximum Likelihood Estimation

All parameters can be estimated jointly. Using conditional independence:
`\begin{align*}
    L = f_{X_1} \prod_{i=2}^T f_{X_i|\mathcal{F}_{i-1}},
\end{align*}`
with multivariate conditional density:
`\begin{align*}
    f_{\mathbf{X}_t}(\mathbf{x}_t | \mathcal{F}_{t-1}) = c\left[\mathbf{F}(\mathbf{x}_t;\boldsymbol{\mu}_t, \boldsymbol{\sigma}_{t}^2, \boldsymbol{\nu},
    \boldsymbol{\lambda});\Xi_t, \Theta\right] \cdot \\ \prod_{i=1}^K f_{X_{i,t}}(\mathbf{x}_t;\boldsymbol{\mu}_t, \boldsymbol{\sigma}_{t}^2, \boldsymbol{\nu}, \boldsymbol{\lambda})
\end{align*}`
The copula density `$c$` can be derived analytically.

]

---

# Study Design and Evaluation

.pull-left[

#### Rolling-window forecasting study

- 3257 observations total
- Window size: 1000 days (~ four years)
- Forecasting 30-steps-ahead

=> 2227 potential starting points

We sample 250 to reduce computational cost

We draw `$2^{12}= 2048$` trajectories from the joint predictive distribution

]

.pull-left[

#### Evaluation

Forecasts are evaluated by the energy score (ES)

`\begin{align*}
    \text{ES}_t(F, \mathbf{x}_t) = \mathbb{E}_{F} \left(||\tilde{\mathbf{X}}_t - \mathbf{x}_t||_2\right) - \frac{1}{2} \mathbb{E}_F \left(||\tilde{\mathbf{X}}_t - \tilde{\mathbf{X}}_t'||_2 \right)
\end{align*}`

where `$\mathbf{x}_t$` is the observed `$K$`-dimensional realization and `$\tilde{\mathbf{X}}_t$`, respectively `$\tilde{\mathbf{X}}_t'$` are independent random vectors distributed according to `$F$`

For univariate cases the Energy Score becomes the Continuous Ranked Probability Score (CRPS)

]

---

# Energy Scores

.pull-left[

Relative improvement in ES compared to `$\text{RW}^{\sigma, \rho}$`

Cellcolor: w.r.t test statistic of Diebold-Mariano test (testing wether the model outperformes the benchmark, greener = better).

]

.pull-right[

- Benchmarks:
  - `$\text{RW}^{\sigma, \rho}$`: Random walk with constant volatility and correlation
  - Univariate `$\text{ETS}^{\sigma}$` with constant volatility
  - Vector ETS `$VES^{\sigma}$` with constant volatility

- Heteroscedasticity is a main driver of ES
- The VECM model without cointegration (essentially a VAR) is the best performing model in terms of ES overall
- For EUA, the ETS Benchmark is the best performing model in terms of ES

]

---

# CRPS Scores

.pull-left-3[

- CRPS solely evaluates the marginal distributions
  - The cross-sectional dependence is ignored

- VES models deliver poor performance in short horizons
- For Oil prices the RW Benchmark can't be oupterformed 30 steps ahead
- Both VECM models generally deliver good performance

]

.pull-right-3[

.font80[

Improvement in CRPS of selected models relative to `$\textrm{RW}^{\sigma, \rho}_{}$` in % (higher = better). Colored according to the test statistic of a DM-Test comparing to `$\textrm{RW}^{\sigma, \rho}_{}$`  (greener means lower test statistic i.e., better performance compared to `$\textrm{RW}^{\sigma, \rho}_{}$`).

]

<table class=" lightable-paper table table-condensed" style='font-family: "Arial Narrow", arial, helvetica, sans-serif; width: auto !important; margin-left: auto; margin-right: auto; font-size: 16px; width: auto !important; margin-left: auto; margin-right: auto;'>
 <thead>
<tr>
<th style="empty-cells: hide;" colspan="1"></th>
<th style="padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; " colspan="3"><div style="border-bottom: 1px solid #00000020; padding-bottom: 5px; ">EUA</div></th>
<th style="padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; " colspan="3"><div style="border-bottom: 1px solid #00000020; padding-bottom: 5px; ">Oil</div></th>
<th style="padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; " colspan="3"><div style="border-bottom: 1px solid #00000020; padding-bottom: 5px; ">NGas</div></th>
<th style="padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; " colspan="3"><div style="border-bottom: 1px solid #00000020; padding-bottom: 5px; ">Coal</div></th>
</tr>
  <tr>
   <th style="text-align:left;"> Model </th>
   <th style="text-align:right;"> H1 </th>
   <th style="text-align:right;"> H5 </th>
   <th style="text-align:right;"> H30 </th>
   <th style="text-align:right;"> H1 </th>
   <th style="text-align:right;"> H5 </th>
   <th style="text-align:right;"> H30 </th>
   <th style="text-align:right;"> H1 </th>
   <th style="text-align:right;"> H5 </th>
   <th style="text-align:right;"> H30 </th>
   <th style="text-align:right;"> H1 </th>
   <th style="text-align:right;"> H5 </th>
   <th style="text-align:right;"> H30 </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> $\textrm{RW}^{\sigma, \rho}_{}$ </td>
   <td style="text-align:right;background-color: rgba(189, 189, 189, 1) !important;"> 0.4 </td>
   <td style="text-align:right;background-color: rgba(189, 189, 189, 1) !important;"> 0.9 </td>
   <td style="text-align:right;background-color: rgba(189, 189, 189, 1) !important;"> 2.1 </td>
   <td style="text-align:right;background-color: rgba(189, 189, 189, 1) !important;"> 1.5 </td>
   <td style="text-align:right;background-color: rgba(189, 189, 189, 1) !important;"> 3.4 </td>
   <td style="text-align:right;font-weight: bold;background-color: rgba(189, 189, 189, 1) !important;"> 9.1 </td>
   <td style="text-align:right;background-color: rgba(189, 189, 189, 1) !important;"> 4.7 </td>
   <td style="text-align:right;background-color: rgba(189, 189, 189, 1) !important;"> 11.6 </td>
   <td style="text-align:right;background-color: rgba(189, 189, 189, 1) !important;"> 29.8 </td>
   <td style="text-align:right;background-color: rgba(189, 189, 189, 1) !important;"> 0.3 </td>
   <td style="text-align:right;background-color: rgba(189, 189, 189, 1) !important;"> 0.9 </td>
   <td style="text-align:right;background-color: rgba(189, 189, 189, 1) !important;"> 2.8 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> $\textrm{RW}^{\sigma_t, \rho_t}_{}$ </td>
   <td style="text-align:right;font-weight: bold;background-color: #96CA66 !important;"> 5.6 </td>
   <td style="text-align:right;background-color: #90C866 !important;"> 6.0 </td>
   <td style="text-align:right;background-color: #C8DB60 !important;"> 2.8 </td>
   <td style="text-align:right;background-color: #BED761 !important;"> 2.1 </td>
   <td style="text-align:right;background-color: #98CB65 !important;"> 2.7 </td>
   <td style="text-align:right;background-color: #FFC73E !important;"> -0.8 </td>
   <td style="text-align:right;background-color: #A3CF64 !important;"> 12.6 </td>
   <td style="text-align:right;background-color: #A2CE64 !important;"> 10.5 </td>
   <td style="text-align:right;background-color: #A1CE64 !important;"> 9.6 </td>
   <td style="text-align:right;background-color: #88C567 !important;"> 10.7 </td>
   <td style="text-align:right;background-color: #94CA66 !important;"> 6.5 </td>
   <td style="text-align:right;background-color: #B3D463 !important;"> 2.1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> $\textrm{RW}^{\sigma, \rho_t}_{\textrm{ncp}, \textrm{log}}$ </td>
   <td style="text-align:right;background-color: #9CCC65 !important;"> 5.1 </td>
   <td style="text-align:right;background-color: #7AC068 !important;"> 8.7 </td>
   <td style="text-align:right;background-color: #A9D064 !important;"> 5.0 </td>
   <td style="text-align:right;background-color: #E3E45D !important;"> 0.7 </td>
   <td style="text-align:right;background-color: #E3E45C !important;"> 0.8 </td>
   <td style="text-align:right;background-color: #FFE04F !important;"> -0.4 </td>
   <td style="text-align:right;background-color: #8EC766 !important;"> 11.4 </td>
   <td style="text-align:right;background-color: #96CA66 !important;"> 11.5 </td>
   <td style="text-align:right;background-color: #8FC866 !important;"> 12.4 </td>
   <td style="text-align:right;background-color: #77BF69 !important;"> 8.0 </td>
   <td style="text-align:right;background-color: #7FC268 !important;"> 7.3 </td>
   <td style="text-align:right;background-color: #7CC168 !important;"> 6.7 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> $\textrm{RW}^{\sigma, \rho}_{\textrm{log}}$ </td>
   <td style="text-align:right;background-color: #A3CF64 !important;"> 4.7 </td>
   <td style="text-align:right;background-color: #76BF69 !important;"> 8.9 </td>
   <td style="text-align:right;background-color: #A2CE64 !important;"> 5.2 </td>
   <td style="text-align:right;background-color: #FFEC57 !important;"> 0.0 </td>
   <td style="text-align:right;background-color: #F3E95A !important;"> 0.3 </td>
   <td style="text-align:right;background-color: #FFD84A !important;"> -0.6 </td>
   <td style="text-align:right;background-color: #8EC766 !important;"> 11.2 </td>
   <td style="text-align:right;background-color: #95CA66 !important;"> 11.4 </td>
   <td style="text-align:right;background-color: #8FC866 !important;"> 12.4 </td>
   <td style="text-align:right;background-color: #77C069 !important;"> 7.7 </td>
   <td style="text-align:right;background-color: #7FC268 !important;"> 7.5 </td>
   <td style="text-align:right;background-color: #7BC168 !important;"> 6.6 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> $\textrm{VECM}^{\textrm{r0}, \sigma_t, \rho_t}_{\textrm{lev}, \textrm{ncp}}$ </td>
   <td style="text-align:right;background-color: #9BCC65 !important;"> 3.6 </td>
   <td style="text-align:right;background-color: #E7E55C !important;"> 0.6 </td>
   <td style="text-align:right;background-color: #FFA928 !important;"> -1.6 </td>
   <td style="text-align:right;font-weight: bold;background-color: #B1D363 !important;"> 2.7 </td>
   <td style="text-align:right;font-weight: bold;background-color: #9DCD65 !important;"> 3.0 </td>
   <td style="text-align:right;background-color: #FFED58 !important;"> 0.0 </td>
   <td style="text-align:right;font-weight: bold;background-color: #9CCC65 !important;"> 13.1 </td>
   <td style="text-align:right;font-weight: bold;background-color: #A1CE64 !important;"> 12.2 </td>
   <td style="text-align:right;background-color: #98CB65 !important;"> 10.4 </td>
   <td style="text-align:right;font-weight: bold;background-color: #8EC766 !important;"> 11.8 </td>
   <td style="text-align:right;background-color: #92C966 !important;"> 7.2 </td>
   <td style="text-align:right;background-color: #C5DA60 !important;"> 1.5 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> $\textrm{VECM}^{\textrm{r0}, \sigma, \rho_t}_{\textrm{log}}$ </td>
   <td style="text-align:right;background-color: #B2D363 !important;"> 4.2 </td>
   <td style="text-align:right;font-weight: bold;background-color: #72BE69 !important;"> 8.9 </td>
   <td style="text-align:right;background-color: #99CB65 !important;"> 5.1 </td>
   <td style="text-align:right;background-color: #FAEB59 !important;"> 0.2 </td>
   <td style="text-align:right;background-color: #EFE85B !important;"> 0.4 </td>
   <td style="text-align:right;background-color: #FFD044 !important;"> -0.8 </td>
   <td style="text-align:right;background-color: #94C966 !important;"> 9.9 </td>
   <td style="text-align:right;background-color: #9ACC65 !important;"> 11.8 </td>
   <td style="text-align:right;font-weight: bold;background-color: #90C866 !important;"> 12.7 </td>
   <td style="text-align:right;background-color: #7BC168 !important;"> 7.8 </td>
   <td style="text-align:right;font-weight: bold;background-color: #7DC168 !important;"> 7.9 </td>
   <td style="text-align:right;font-weight: bold;background-color: #79C068 !important;"> 7.3 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> $\textrm{ETS}^{\sigma}$ </td>
   <td style="text-align:right;background-color: #FCEC58 !important;"> 0.2 </td>
   <td style="text-align:right;background-color: #8DC766 !important;"> 6.8 </td>
   <td style="text-align:right;background-color: #A4CF64 !important;"> 5.7 </td>
   <td style="text-align:right;background-color: #C4D961 !important;"> 1.1 </td>
   <td style="text-align:right;background-color: #B1D363 !important;"> 0.9 </td>
   <td style="text-align:right;background-color: #FFDF4F !important;"> -0.2 </td>
   <td style="text-align:right;background-color: #99CB65 !important;"> 10.9 </td>
   <td style="text-align:right;background-color: #98CB65 !important;"> 11.3 </td>
   <td style="text-align:right;background-color: #94C966 !important;"> 10.9 </td>
   <td style="text-align:right;background-color: #B7D562 !important;"> 7.5 </td>
   <td style="text-align:right;background-color: #BCD762 !important;"> 6.7 </td>
   <td style="text-align:right;background-color: #C7DA60 !important;"> 5.6 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> $\textrm{ETS}^{\sigma}_{\textrm{log}}$ </td>
   <td style="text-align:right;background-color: #EFE85B !important;"> 1.0 </td>
   <td style="text-align:right;background-color: #80C268 !important;"> 8.6 </td>
   <td style="text-align:right;font-weight: bold;background-color: #A0CE64 !important;"> 8.0 </td>
   <td style="text-align:right;background-color: #FCEC58 !important;"> 0.1 </td>
   <td style="text-align:right;background-color: #E9E65C !important;"> 0.7 </td>
   <td style="text-align:right;background-color: #FFDA4B !important;"> -0.6 </td>
   <td style="text-align:right;background-color: #B8D562 !important;"> 8.9 </td>
   <td style="text-align:right;background-color: #BBD762 !important;"> 9.4 </td>
   <td style="text-align:right;background-color: #C9DB60 !important;"> 7.1 </td>
   <td style="text-align:right;background-color: #C1D861 !important;"> 7.3 </td>
   <td style="text-align:right;background-color: #B7D562 !important;"> 7.8 </td>
   <td style="text-align:right;background-color: #C8DB60 !important;"> 6.7 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> $\textrm{VES}^{\sigma}$ </td>
   <td style="text-align:right;background-color: #F1634B !important;"> -38.5 </td>
   <td style="text-align:right;background-color: #FFBE38 !important;"> -6.4 </td>
   <td style="text-align:right;background-color: #FFBC36 !important;"> -5.4 </td>
   <td style="text-align:right;background-color: #F05D4D !important;"> -33.3 </td>
   <td style="text-align:right;background-color: #FFB934 !important;"> -6.1 </td>
   <td style="text-align:right;background-color: #FFC43B !important;"> -2.4 </td>
   <td style="text-align:right;background-color: #FE9E2D !important;"> -26.6 </td>
   <td style="text-align:right;background-color: #FFDE4E !important;"> -2.6 </td>
   <td style="text-align:right;background-color: #D7E05E !important;"> 3.6 </td>
   <td style="text-align:right;background-color: #F57146 !important;"> -37.5 </td>
   <td style="text-align:right;background-color: #FFCB40 !important;"> -5.5 </td>
   <td style="text-align:right;background-color: #CBDC60 !important;"> 4.7 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> $\textrm{VES}^{\sigma}_{\textrm{log}}$ </td>
   <td style="text-align:right;background-color: #F88140 !important;"> -32.4 </td>
   <td style="text-align:right;background-color: #E3E45C !important;"> 2.8 </td>
   <td style="text-align:right;background-color: #E8E55C !important;"> 1.8 </td>
   <td style="text-align:right;background-color: #F2654B !important;"> -30.4 </td>
   <td style="text-align:right;background-color: #FFB632 !important;"> -6.2 </td>
   <td style="text-align:right;background-color: #FFB934 !important;"> -3.2 </td>
   <td style="text-align:right;background-color: #FFA626 !important;"> -22.0 </td>
   <td style="text-align:right;background-color: #F2E95A !important;"> 1.8 </td>
   <td style="text-align:right;background-color: #CFDD5F !important;"> 5.4 </td>
   <td style="text-align:right;background-color: #FC9733 !important;"> -27.0 </td>
   <td style="text-align:right;background-color: #ECE75B !important;"> 2.3 </td>
   <td style="text-align:right;background-color: #AAD164 !important;"> 6.4 </td>
  </tr>
</tbody>
</table>

]

---

# RMSE

.pull-right-3[

Improvement in RMSE score of selected models relative to `$\textrm{RW}^{\sigma, \rho}_{}$` in % (higher = better). Colored according to the test statistic of a DM-Test comparing to `$\textrm{RW}^{\sigma, \rho}_{}$`  (greener means lower test statistic i.e., better performance compared to `$\textrm{RW}^{\sigma, \rho}_{}$`).

<table class=" lightable-paper table table-condensed" style='font-family: "Arial Narrow", arial, helvetica, sans-serif; width: auto !important; margin-left: auto; margin-right: auto; font-size: 14px; width: auto !important; margin-left: auto; margin-right: auto;'>
 <thead>
  <tr>
   <th style="text-align:left;"> Model </th>
   <th style="text-align:right;"> H1 </th>
   <th style="text-align:right;"> H5 </th>
   <th style="text-align:right;"> H30 </th>
   <th style="text-align:right;"> H1 </th>
   <th style="text-align:right;"> H5 </th>
   <th style="text-align:right;"> H30 </th>
   <th style="text-align:right;"> H1 </th>
   <th style="text-align:right;"> H5 </th>
   <th style="text-align:right;"> H30 </th>
   <th style="text-align:right;"> H1 </th>
   <th style="text-align:right;"> H5 </th>
   <th style="text-align:right;"> H30 </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> $\textrm{RW}^{\sigma, \rho}_{}$ </td>
   <td style="text-align:right;font-weight: bold;background-color: rgba(189, 189, 189, 1) !important;"> 0.9 </td>
   <td style="text-align:right;background-color: rgba(189, 189, 189, 1) !important;"> 2.0 </td>
   <td style="text-align:right;background-color: rgba(189, 189, 189, 1) !important;"> 5.0 </td>
   <td style="text-align:right;background-color: rgba(189, 189, 189, 1) !important;"> 2.9 </td>
   <td style="text-align:right;background-color: rgba(189, 189, 189, 1) !important;"> 6.4 </td>
   <td style="text-align:right;font-weight: bold;background-color: rgba(189, 189, 189, 1) !important;"> 16.7 </td>
   <td style="text-align:right;background-color: rgba(189, 189, 189, 1) !important;"> 17.8 </td>
   <td style="text-align:right;background-color: rgba(189, 189, 189, 1) !important;"> 42.8 </td>
   <td style="text-align:right;background-color: rgba(189, 189, 189, 1) !important;"> 85.4 </td>
   <td style="text-align:right;background-color: rgba(189, 189, 189, 1) !important;"> 0.9 </td>
   <td style="text-align:right;background-color: rgba(189, 189, 189, 1) !important;"> 2.9 </td>
   <td style="text-align:right;background-color: rgba(189, 189, 189, 1) !important;"> 7.0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> $\textrm{RW}^{\sigma_t, \rho_t}_{}$ </td>
   <td style="text-align:right;background-color: #FFE653 !important;"> -0.1 </td>
   <td style="text-align:right;background-color: #FFE150 !important;"> -0.1 </td>
   <td style="text-align:right;background-color: #B4D462 !important;"> 0.7 </td>
   <td style="text-align:right;background-color: #FBEC59 !important;"> 0.0 </td>
   <td style="text-align:right;background-color: #FFC93F !important;"> -0.3 </td>
   <td style="text-align:right;background-color: #FFDF4E !important;"> -0.1 </td>
   <td style="text-align:right;background-color: #FFE14F !important;"> -0.2 </td>
   <td style="text-align:right;background-color: #CADB60 !important;"> 0.3 </td>
   <td style="text-align:right;font-weight: bold;background-color: #CFDD5F !important;"> 1.3 </td>
   <td style="text-align:right;background-color: #FFE754 !important;"> -0.2 </td>
   <td style="text-align:right;background-color: #FFED58 !important;"> 0.0 </td>
   <td style="text-align:right;background-color: #FFCE42 !important;"> -1.8 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> $\textrm{RW}^{\sigma, \rho_t}_{\textrm{ncp}, \textrm{log}}$ </td>
   <td style="text-align:right;background-color: #FFD145 !important;"> -270.5 </td>
   <td style="text-align:right;background-color: #FFB833 !important;"> -154.1 </td>
   <td style="text-align:right;background-color: #FD9C2F !important;"> -139.9 </td>
   <td style="text-align:right;background-color: #C0D861 !important;"> 0.5 </td>
   <td style="text-align:right;background-color: #FFDD4D !important;"> -0.5 </td>
   <td style="text-align:right;background-color: #FFD547 !important;"> -2.9 </td>
   <td style="text-align:right;background-color: #FFD044 !important;"> -0.8 </td>
   <td style="text-align:right;background-color: #E5E45C !important;"> 0.7 </td>
   <td style="text-align:right;background-color: #FFE250 !important;"> -1.6 </td>
   <td style="text-align:right;background-color: #E2E35D !important;"> 0.3 </td>
   <td style="text-align:right;background-color: #FFD044 !important;"> -31.2 </td>
   <td style="text-align:right;background-color: #FFBB35 !important;"> -24.5 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> $\textrm{RW}^{\sigma, \rho}_{\textrm{log}}$ </td>
   <td style="text-align:right;background-color: #FFCB40 !important;"> -705.0 </td>
   <td style="text-align:right;background-color: #FFC93F !important;"> -265.4 </td>
   <td style="text-align:right;background-color: #FFAD2B !important;"> -125.2 </td>
   <td style="text-align:right;background-color: #B0D363 !important;"> 0.6 </td>
   <td style="text-align:right;font-weight: bold;background-color: #F5EA5A !important;"> 0.2 </td>
   <td style="text-align:right;background-color: #FFE754 !important;"> -0.2 </td>
   <td style="text-align:right;background-color: #FFDE4D !important;"> -0.4 </td>
   <td style="text-align:right;background-color: #FBEC59 !important;"> 0.1 </td>
   <td style="text-align:right;background-color: #FFE251 !important;"> -1.6 </td>
   <td style="text-align:right;background-color: #FFC13A !important;"> -0.9 </td>
   <td style="text-align:right;background-color: #FFE250 !important;"> -0.3 </td>
   <td style="text-align:right;background-color: #FFC73D !important;"> -8.3 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> $\textrm{VECM}^{\textrm{r0}, \sigma_t, \rho_t}_{\textrm{lev}, \textrm{ncp}}$ </td>
   <td style="text-align:right;background-color: #FFDA4B !important;"> -0.9 </td>
   <td style="text-align:right;background-color: #F1E85A !important;"> 0.2 </td>
   <td style="text-align:right;background-color: #C6DA60 !important;"> 0.5 </td>
   <td style="text-align:right;background-color: #E4E45C !important;"> 0.5 </td>
   <td style="text-align:right;background-color: #F0E85A !important;"> 0.2 </td>
   <td style="text-align:right;background-color: #FFEB56 !important;"> 0.0 </td>
   <td style="text-align:right;background-color: #FFE754 !important;"> -0.4 </td>
   <td style="text-align:right;background-color: #CBDC60 !important;"> 0.7 </td>
   <td style="text-align:right;background-color: #EFE85B !important;"> 0.2 </td>
   <td style="text-align:right;font-weight: bold;background-color: #E0E35D !important;"> 1.4 </td>
   <td style="text-align:right;font-weight: bold;background-color: #F1E85A !important;"> 0.1 </td>
   <td style="text-align:right;background-color: #EAE65B !important;"> 0.2 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> $\textrm{VECM}^{\textrm{r0}, \sigma, \rho_t}_{\textrm{log}}$ </td>
   <td style="text-align:right;background-color: #FFD145 !important;"> -271.5 </td>
   <td style="text-align:right;background-color: #FFB934 !important;"> -191.3 </td>
   <td style="text-align:right;background-color: #FE9F2D !important;"> -114.3 </td>
   <td style="text-align:right;font-weight: bold;background-color: #CBDC60 !important;"> 1.7 </td>
   <td style="text-align:right;background-color: #FFD145 !important;"> -12.3 </td>
   <td style="text-align:right;background-color: #FFC43C !important;"> -3.6 </td>
   <td style="text-align:right;background-color: #FFDF4E !important;"> -0.6 </td>
   <td style="text-align:right;font-weight: bold;background-color: #E1E35D !important;"> 1.6 </td>
   <td style="text-align:right;background-color: #FFDA4B !important;"> -4.1 </td>
   <td style="text-align:right;background-color: #FFEB57 !important;"> 0.0 </td>
   <td style="text-align:right;background-color: #FFD447 !important;"> -0.8 </td>
   <td style="text-align:right;background-color: #FFDD4D !important;"> -6.7 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> $\textrm{ETS}^{\sigma}$ </td>
   <td style="text-align:right;background-color: #FFE452 !important;"> -0.3 </td>
   <td style="text-align:right;background-color: #F0E85B !important;"> 0.3 </td>
   <td style="text-align:right;background-color: #D8E05E !important;"> 1.6 </td>
   <td style="text-align:right;background-color: #D0DD5F !important;"> 0.7 </td>
   <td style="text-align:right;background-color: #F4E95A !important;"> 0.1 </td>
   <td style="text-align:right;background-color: #FFDD4D !important;"> -0.1 </td>
   <td style="text-align:right;font-weight: bold;background-color: #F0E85A !important;"> 0.1 </td>
   <td style="text-align:right;background-color: #FFD648 !important;"> -0.1 </td>
   <td style="text-align:right;background-color: #E5E45C !important;"> 0.2 </td>
   <td style="text-align:right;background-color: #FFD849 !important;"> -2.4 </td>
   <td style="text-align:right;background-color: #FFC63D !important;"> -3.9 </td>
   <td style="text-align:right;font-weight: bold;background-color: #E6E55C !important;"> 2.5 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> $\textrm{ETS}^{\sigma}_{\textrm{log}}$ </td>
   <td style="text-align:right;background-color: #FFCE43 !important;"> -1.0 </td>
   <td style="text-align:right;font-weight: bold;background-color: #F5EA5A !important;"> 0.4 </td>
   <td style="text-align:right;font-weight: bold;background-color: #ECE75B !important;"> 1.6 </td>
   <td style="text-align:right;background-color: #C7DA60 !important;"> 0.9 </td>
   <td style="text-align:right;background-color: #FFED58 !important;"> 0.0 </td>
   <td style="text-align:right;background-color: #FFEA56 !important;"> -0.1 </td>
   <td style="text-align:right;background-color: #FFD447 !important;"> -1.9 </td>
   <td style="text-align:right;background-color: #FFE150 !important;"> -1.9 </td>
   <td style="text-align:right;background-color: #FFCE42 !important;"> -13.9 </td>
   <td style="text-align:right;background-color: #FFE250 !important;"> -0.3 </td>
   <td style="text-align:right;background-color: #FFCD42 !important;"> -3.6 </td>
   <td style="text-align:right;background-color: #FFE452 !important;"> -1.8 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> $\textrm{VES}^{\sigma}$ </td>
   <td style="text-align:right;background-color: #FFA828 !important;"> -37.4 </td>
   <td style="text-align:right;background-color: #FFBC36 !important;"> -8.9 </td>
   <td style="text-align:right;background-color: #FFBC36 !important;"> -6.0 </td>
   <td style="text-align:right;background-color: #FD9931 !important;"> -27.9 </td>
   <td style="text-align:right;background-color: #FFC53C !important;"> -7.4 </td>
   <td style="text-align:right;background-color: #FFB430 !important;"> -2.8 </td>
   <td style="text-align:right;background-color: #FFAB2A !important;"> -27.2 </td>
   <td style="text-align:right;background-color: #FFB732 !important;"> -9.5 </td>
   <td style="text-align:right;background-color: #FFDD4D !important;"> -2.4 </td>
   <td style="text-align:right;background-color: #FFC038 !important;"> -41.7 </td>
   <td style="text-align:right;background-color: #FFDA4B !important;"> -1.2 </td>
   <td style="text-align:right;background-color: #D4DF5F !important;"> 1.6 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> $\textrm{VES}^{\sigma}_{\textrm{log}}$ </td>
   <td style="text-align:right;background-color: #FFAA29 !important;"> -37.6 </td>
   <td style="text-align:right;background-color: #FFBD37 !important;"> -9.2 </td>
   <td style="text-align:right;background-color: #FFBA35 !important;"> -7.8 </td>
   <td style="text-align:right;background-color: #FD9A30 !important;"> -26.8 </td>
   <td style="text-align:right;background-color: #FFC63D !important;"> -7.3 </td>
   <td style="text-align:right;background-color: #FFBE37 !important;"> -3.0 </td>
   <td style="text-align:right;background-color: #FFAC2A !important;"> -27.0 </td>
   <td style="text-align:right;background-color: #FFC33B !important;"> -6.8 </td>
   <td style="text-align:right;background-color: #FFD144 !important;"> -3.5 </td>
   <td style="text-align:right;background-color: #FFC139 !important;"> -41.2 </td>
   <td style="text-align:right;background-color: #FFC43C !important;"> -2.2 </td>
   <td style="text-align:right;background-color: #FFE855 !important;"> -0.3 </td>
  </tr>
</tbody>
</table>

]

.pull-left-3[

RMSE measures the performance of the forecasts at their mean

</br>

- Some models beat the benchmarks at short horizons

</br>

Conclusion: the Improvements seen before must be attributed to other parts of the multivariate probabilistic predictive distribution

]

---

# Evolution of Linear Dependence `$\Xi$`

---

# Predictive Quantiles (Russian Invasion)

---

# Conclusion

.pull-left[

Accounting for heteroscedasticity or stabilizing the variance via log transformation is crucial for good performance in terms of ES

- Price dynamics emerged way before the russian invaion into ukraine
- Linear dependence between the series reacted only right after the invasion 
- Improvements in forecasting performance is mainly attributed to:
  -  the tails multivariate probabilistic predictive distribution
  -  the dependence structure between the marginals

]

.pull-right[

</br>

]

<a href="https://github.com/BerriJ" class="github-corner" aria-label="View source on Github"><svg width="80" height="80" viewBox="0 0 250 250" style="fill:#f2f2f2; color:#212121; position: absolute; top: 0; border: 0; right: 0;" aria-hidden="true"><path d="M0,0 L115,115 L130,115 L142,142 L250,250 L250,0 Z"></path><path d="M128.3,109.0 C113.8,99.7 119.0,89.6 119.0,89.6 C122.0,82.7 120.5,78.6 120.5,78.6 C119.2,72.0 123.4,76.3 123.4,76.3 C127.3,80.9 125.5,87.3 125.5,87.3 C122.9,97.6 130.6,101.9 134.4,103.2" fill="currentColor" style="transform-origin: 130px 106px;" class="octo-arm"></path><path d="M115.0,115.0 C114.9,115.1 118.7,116.5 119.8,115.4 L133.7,101.6 C136.9,99.2 139.9,98.4 142.2,98.6 C133.8,88.0 127.5,74.4 143.8,58.0 C148.5,53.4 154.0,51.2 159.7,51.0 C160.3,49.4 163.2,43.6 171.4,40.1 C171.4,40.1 176.1,42.5 178.8,56.2 C183.1,58.6 187.2,61.8 190.9,65.4 C194.5,69.0 197.7,73.2 200.1,77.6 C213.8,80.2 216.3,84.9 216.3,84.9 C212.7,93.1 206.9,96.0 205.4,96.6 C205.1,102.4 203.0,107.8 198.3,112.5 C181.9,128.9 168.3,122.5 157.7,114.1 C157.9,116.9 156.7,120.9 152.7,124.9 L141.0,136.5 C139.8,137.7 141.6,141.9 141.8,141.8 Z" fill="currentColor" class="octo-body"></path></svg></a><style>.github-corner:hover .octo-arm{animation:octocat-wave 560ms ease-in-out}@keyframes octocat-wave{0%,100%{transform:rotate(0)}20%,60%{transform:rotate(-25deg)}40%,80%{transform:rotate(10deg)}}@media (max-width:500px){.github-corner:hover .octo-arm{animation:none}.github-corner .octo-arm{animation:octocat-wave 560ms ease-in-out}}</style>

---
name:references

# References 1

<p><cite><a id='bib-berrisch2023modeling'></a><a href="#cite-berrisch2023modeling">Berrisch, J., S. Pappert, F. Ziel, et al.</a>
(2023).
&ldquo;Modeling volatility and dependence of European carbon and energy prices&rdquo;.
In: <em>Finance Research Letters</em> 52, p. 103503.</cite></p>