From 9bc402b926556555f4f1a0bf4e7010a19620376f Mon Sep 17 00:00:00 2001 From: Jonathan Berrisch Date: Fri, 20 Jun 2025 11:30:44 +0200 Subject: [PATCH] A thousand minor improvements --- assets/hecf_logo.svg | 567 +++++++------- assets/hecf_logo_reduced.svg | 1382 ++++++++++++++++++++++++++++++++++ assets/wiwi_logo.svg | 13 + assets/wiwi_signet.svg | 14 + index.html | 973 ++++++++++++------------ index.qmd | 240 +++--- 6 files changed, 2309 insertions(+), 880 deletions(-) create mode 100644 assets/hecf_logo_reduced.svg create mode 100644 assets/wiwi_logo.svg create mode 100644 assets/wiwi_signet.svg diff --git a/assets/hecf_logo.svg b/assets/hecf_logo.svg index f8864b7..f3a1c14 100644 --- a/assets/hecf_logo.svg +++ b/assets/hecf_logo.svg @@ -1,8 +1,8 @@ - + + diff --git a/assets/hecf_logo_reduced.svg b/assets/hecf_logo_reduced.svg new file mode 100644 index 0000000..ac4b792 --- /dev/null +++ b/assets/hecf_logo_reduced.svg @@ -0,0 +1,1382 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/assets/wiwi_logo.svg b/assets/wiwi_logo.svg new file mode 100644 index 0000000..64253ca --- /dev/null +++ b/assets/wiwi_logo.svg @@ -0,0 +1,13 @@ + + + + + + + + + + + + + \ No newline at end of file diff --git a/assets/wiwi_signet.svg b/assets/wiwi_signet.svg new file mode 100644 index 0000000..53b278a --- /dev/null +++ b/assets/wiwi_signet.svg @@ -0,0 +1,14 @@ + + + + + + + + + + + + + + \ No newline at end of file diff --git a/index.html b/index.html index 61a0edd..240691f 100644 --- a/index.html +++ b/index.html @@ -25531,8 +25531,9 @@ Jonathan Berrisch

2025-06-30

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Outline

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High-Level View

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The beginning: June 2020

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CRPS Learning

Berrisch, J., & Ziel, F. (2023). Journal of Econometrics, 237(2), 105221.

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The Framework of Prediction under Expert Advice

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The sequential framework

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Each day, \(t = 1, 2, ... T\)

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@@ -26302,7 +26309,7 @@ w_{t,k}^{\text{Naive}} = \frac{1}{K}\label{eq:naive_combination}

\[\begin{equation*} \text{CRPS}(F, y) = \int_{\mathbb{R}} {(F(x) - \mathbb{1}\{ x > y \})}^2 dx \label{eq:crps} \end{equation*}\]

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It’s strictly proper Gneiting & Raftery (2007).

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It’s strictly proper (Gneiting & Raftery, 2007).

Using the CRPS, we can calculate time-adaptive weights \(w_{t,k}\). However, what if the experts’ performance varies in parts of the distribution?

Utilize this relation:

\[\begin{equation*} @@ -26614,7 +26621,7 @@ w_{t,k}^{\text{smooth}} = \sum_{l=1}^L \beta_l \varphi_l = \beta'\varphi

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Multivariate Probabilistic CRPS Learning with an Application to Day-Ahead Electricity Prices

Berrisch, J., & Ziel, F. (2024). International Journal of Forecasting, 40(4), 1568-1586.

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Modeling Volatility and Dependence of European Carbon and Energy Prices

Berrisch, J., Pappert, S., Ziel, F., & Arsova, A. (2023). Finance Research Letters, 52, 103503.

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Motivation

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Portfolio & Risk Management,

Sustainability Planing

Political decisions

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EUA prices are obviously connected to the energy market

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EUA prices are connected to energy markets

How can the dynamics be characterized?

Several Questions arise:

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Data

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EUA, natural gas, Brent crude oil, coal

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March 15, 2010, until October 14, 2022

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Data was normalized w.r.t. \(\text{CO}_2\) emissions

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Emission-adjusted prices reflects one tonne of \(\text{CO}_2\)

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We adjusted for inflation by Eurostat’s HICP, excluding energy

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Daily Observations: 03/15/2010 - 10/14/2022

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EUA, Natural Gas, Brent Crude Oil, Coal

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  • normalized w.r.t. \(\text{CO}_2\) emissions
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  • Adjusted for inflation by Eurostat’s HICP, excluding energy
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Emission-adjusted prices reflect one tonne of \(\text{CO}_2\)

Log transformation of the data to stabilize the variance

ADF Test: All series are stationary in first differences

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Johansen’s likelihood ratio trace test suggests two cointegrating relationships (levels)

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

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Johansen’s likelihood ratio trace test suggests two cointegrating relationships (only in levels)

Data

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Modeling Approach: Overview

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VECM: Vector Error Correction Model

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  • Modeling the expectaion
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  • Captures the long-run cointegrating relationship
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  • Different cointegrating ranks, including rank zero (no cointegration)
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GARCH: Generalized Autoregressive Conditional Heteroscedasticity

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  • Captures dynamics in conditional variance
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Copula: Captures the dependence structure

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  • Captures: conditional cross-sectional dependencies
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  • Dependence allowed to vary over time
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Modeling Approach: Notation


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  • \(\mathcal{F}_{t}\) is the sigma field generated by all information available up to and including time \(t\)
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    Sklars theorem: decompose target into - marginal distributions: \(F_{X_{k,t}|\mathcal{F}_{t-1}}\) for \(k=1,\ldots, K\), and - copula function: \(C_{\boldsymbol{U}_{t}|\mathcal{F}_{t - 1}}\)

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    Sklars theorem: decompose target into

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    • marginal distributions: \(F_{X_{k,t}|\mathcal{F}_{t-1}}\) for \(k=1,\ldots, K\), and
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    • copula function: \(C_{\boldsymbol{U}_{t}|\mathcal{F}_{t - 1}}\)
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    We take \(C\) as the \(t\)-copula

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    Modeling Approach: Mean and Variance

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    Modeling Approach: The General Framework


    Individual marginal distributions:

    \[\mathbf{F} = (F_1, \ldots, F_K)^{\intercal}\]

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    Generalized non-central t-distributions

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    Generalized non-central t-distributions

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    • To account for heavy tails
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    • Time varying +
    • Time varying: expectation \(\boldsymbol{\mu}_t = (\mu_{1,t}, \ldots, \mu_{K,t})^{\intercal}\)
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      • 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 @@ -27641,46 +27632,32 @@ Y_{t} = \mu + Y_{t-1} + \varepsilon_t \quad \text{with} \quad \varepsilon_t = \
    • noncentrality: \(\boldsymbol{\lambda} = (\lambda_1, \ldots, \lambda_K)^{\intercal}\)
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    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 \(r\), \(0 \leq r\leq K\).

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    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 \(\epsilon_{i,t-1}^+ = \max\{\epsilon_{i,t-1}, 0\}\) …

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

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    Modeling Approach: Dependence

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    Positive vs. negative innovations (capture leverage effects).

    Time-varying dependence parameters

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

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    \(\xi_{ij,t}\) is a latent process

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    \(z_{i,t}\) denotes the \(i\)-th standardized residual from time series \(i\) at time point \(t\)

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    \(\Lambda(\cdot)\) 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

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    Maximum Likelihood Estimation

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    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.

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    \(z_{i,t}\) is the \(i\)-th standardized residual from time series \(i\)

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    \(\Lambda(\cdot)\) is a link function:

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    • ensures that \(\Xi_{t}\) is a valid variance covariance matrix
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    • ensures that \(\Xi_{t}\) does not exceed its support space and remains semi-positive definite
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    • 3257 observations total
    • Window size: 1000 days (~ four years)
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    • Forecasting 30-steps-ahead
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    • We sample 250 of 2227 starting points
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    • We draw \(2^{12}= 2048\) trajectories 30 steps ahead
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    => 2227 potential starting points

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    We sample 250 to reduce computational cost

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    We draw \(2^{12}= 2048\) trajectories from the joint predictive distribution

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    Estimation

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    Joint maximum lieklihood estimation:

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    \[\begin{align*} + f_{\mathbf{X}_t}(\mathbf{x}_t | \mathcal{F}_{t-1}) = c\left[\mathbf{F}(\mathbf{x}_t;\boldsymbol{\mu}_t, \boldsymbol{\sigma}_{t}^2, \boldsymbol{\nu}, + \boldsymbol{\lambda});\Xi_t, \Theta\right] \cdot \\ \prod_{i=1}^K f_{X_{i,t}}(\mathbf{x}_t;\boldsymbol{\mu}_t, \boldsymbol{\sigma}_{t}^2, \boldsymbol{\nu}, \boldsymbol{\lambda}) +\end{align*}\]

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    The copula density \(c\) can be derived analytically.

    Evaluation

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    Forecasts are evaluated by the energy score (ES)

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    Our main objective is the Energy Score (ES)

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

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    Relative improvement in ES compared to \(\text{RW}^{\sigma, \rho}\)

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

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

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  • Vector ETS \(VES^{\sigma}\) with constant volatility
  • Heteroscedasticity is a main driver of ES
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  • The VECM model without cointegration (essentially a VAR) is the best performing model in terms of ES overall
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  • The VECM model without cointegration (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
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    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 \(\textrm{RW}^{\sigma, \rho}_{}\)).

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    Relative improvement in CRPS compared to \(\text{RW}^{\sigma, \rho}\)

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    RMSE measures the performance of the forecasts at their mean

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

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    Some models beat the benchmarks at short horizons

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

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    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 \(\textrm{RW}^{\sigma, \rho}_{}\)).

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    Relative improvement in RMSE compared to \(\text{RW}^{\sigma, \rho}\)

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    + Coloring w.r.t. test statistic: + <-5 + -4 + -3 + -2 + -1 + 0 + 1 + 2 + 3 + 4 + >5 +
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  • Linear dependence between the series reacted only right after the invasion
  • Improvements in forecasting performance is mainly attributed to:
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    • the tails
    • the dependence structure between the marginals
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    Berrisch, J., Pappert, S., Ziel, F., & Arsova, A. (2023). Modeling volatility and dependence of European carbon and energy prices. Finance Research Letters, 52, 103503.

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    Final Remarks

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    Contributions

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    Theoretical

    Probabilistic Online Learning:

    Aggregation
    Regression

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    Software

    R Packages:

    profoc, rcpptimer, dccpp

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    Won Western Power Distribution Competition
    Won Best-Student-Presentation Award

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    Questions!

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    Artwork by @allison_horst

    References

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