A thousand minor improvements
This commit is contained in:
240
index.qmd
240
index.qmd
@@ -36,7 +36,8 @@ revealjs-plugins:
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# - drop
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---
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## Outline {.center}
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# High-Level View {.center visibility="uncounted"}
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<!--
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Render with: quarto preview /home/jonathan/git/PHD-Presentation/25_07_phd_defense/index.qmd --no-browser --port 6074
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@@ -48,7 +49,7 @@ $$
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$$
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:::
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:::: {style="font-size: 150%;"}
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<!-- :::: {style="font-size: 150%;"}
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<i class="fa fa-fw fa-rocket" style="color:var(--col_grey_9);"></i>   [Research Motivation](#motivation)
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@@ -60,7 +61,8 @@ $$
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<i class="fa fa-fw fa-newspaper" style="color:var(--col_grey_9);"></i>   [Contributions](#sec-contributions)
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:::
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:::: -->
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```{r, setup, include=FALSE}
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# Compile with: rmarkdown::render("crps_learning.Rmd")
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@@ -754,7 +756,7 @@ void main(){
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::::
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# CRPS Learning {#sec-crps-learning}
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# CRPS Learning {#sec-crps-learning visibility="uncounted"}
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Berrisch, J., & Ziel, F. [-@BERRISCH2023105221]. *Journal of Econometrics*, 237(2), 105221.
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@@ -843,7 +845,7 @@ plot(w[, 3],
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xlab = "",
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ylab = "", xaxt = "n", yaxt = "n", bty = "n", col = "#FFD44EFF"
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)
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text(6, 0.25, TeX("$w_3(t)$"), cex = 2, col = "#FFD44EFF")
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text(6, 0.25, TeX("$w_2(t)$"), cex = 2, col = "#FFD44EFF")
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arrows(13, 0.75, 15, 1, , lwd = 4, bty = "n", col = "#414141FF")
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```
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@@ -941,7 +943,7 @@ chart = {
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.style('align-self', 'center')
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.style('margin-left', 'auto')
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.on('click', () => {
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selectedMu = 0.5;
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selectedMu = 1;
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muSlider.property('value', selectedMu);
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muDisplay.text(selectedMu.toFixed(1));
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updateChart(filteredData());
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@@ -1053,7 +1055,7 @@ chart = {
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## The Framework of Prediction under Expert Advice
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### The sequential framework
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###
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:::: {.columns}
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@@ -1287,7 +1289,7 @@ $\ell'$ is the subgradient of $\ell$ at forecast combination $\widetilde{X}$.
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\text{CRPS}(F, y) = \int_{\mathbb{R}} {(F(x) - \mathbb{1}\{ x > y \})}^2 dx \label{eq:crps}
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\end{equation*}
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It's strictly proper @gneiting2007strictly.
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It's strictly proper [@gneiting2007strictly].
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Using the CRPS, we can calculate time-adaptive weights $w_{t,k}$. However, what if the experts' performance varies in parts of the distribution?
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@@ -2277,7 +2279,7 @@ weights %>%
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::::
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# Multivariate Probabilistic CRPS Learning with an Application to Day-Ahead Electricity Prices
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# Multivariate Probabilistic CRPS Learning with an Application to Day-Ahead Electricity Prices {visibility="uncounted"}
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Berrisch, J., & Ziel, F. (2024). *International Journal of Forecasting*, 40(4), 1568-1586.
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@@ -3014,14 +3016,12 @@ Pubications:
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::::
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# Modeling Volatility and Dependence of European Carbon and Energy Prices {#sec-voldep}
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# Modeling Volatility and Dependence of European Carbon and Energy Prices {#sec-voldep visibility="uncounted"}
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Berrisch, J., Pappert, S., Ziel, F., & Arsova, A. (2023). *Finance Research Letters*, 52, 103503.
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---
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##
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:::: {.columns}
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::: {.column width="48%"}
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@@ -3037,7 +3037,7 @@ for several fields:
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<i class="fa fa-fw fa-handshake" style="color:var(--col_grey_9);"></i> 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
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How can the dynamics be characterized?
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@@ -3057,23 +3057,20 @@ Several Questions arise:
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### Data
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EUA, natural gas, Brent crude oil, coal
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Daily Observations: 03/15/2010 - 10/14/2022
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March 15, 2010, until October 14, 2022
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EUA, Natural Gas, Brent Crude Oil, Coal
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Data was normalized w.r.t. $\text{CO}_2$ emissions
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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 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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Emission-adjusted prices reflect one tonne of $\text{CO}_2$
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Log transformation of the data to stabilize the variance
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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)
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:::
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@@ -3137,26 +3134,6 @@ readr::read_csv("assets/voldep/2022_10_14_eur_ref_co2_adj_hvpi_ex_nrg.csv") %>%
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scale_y_continuous(trans = "log2")
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```
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## Modeling Approach: Overview
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</br>
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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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<br/>
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@@ -3171,9 +3148,10 @@ readr::read_csv("assets/voldep/2022_10_14_eur_ref_co2_adj_hvpi_ex_nrg.csv") %>%
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- $F_{\boldsymbol{X}_t|\mathcal{F}_{t-1}}$
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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
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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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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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:::
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@@ -3210,7 +3188,7 @@ We take $C$ as the $t$-copula
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::::
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## Modeling Approach: Mean and Variance
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## Modeling Approach: The General Framework
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<br/>
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@@ -3222,22 +3200,13 @@ We take $C$ as the $t$-copula
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$$\mathbf{F} = (F_1, \ldots, F_K)^{\intercal}$$
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### Generalized non-central t-distributions
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- To account for heavy tails
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- Time varying
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- expectation: $\boldsymbol{\mu}_t = (\mu_{1,t}, \ldots, \mu_{K,t})^{\intercal}$
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- variance: $\boldsymbol{\sigma}_{t}^2 = (\sigma_{1,t}^2, \ldots, \sigma_{K,t}^2)^{\intercal}$
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- Time invariant
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- degrees of freedom: $\boldsymbol{\nu} = (\nu_1, \ldots, \nu_K)^{\intercal}$
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- noncentrality: $\boldsymbol{\lambda} = (\lambda_1, \ldots, \lambda_K)^{\intercal}$
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Generalized non-central t-distributions
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:::
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::: {.column width="4%"}
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:::
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::: {.column width="48%"}
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- Time varying: expectation $\boldsymbol{\mu}_t = (\mu_{1,t}, \ldots, \mu_{K,t})^{\intercal}$
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- variance: $\boldsymbol{\sigma}_{t}^2 = (\sigma_{1,t}^2, \ldots, \sigma_{K,t}^2)^{\intercal}$
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- Time invariant
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- degrees of freedom: $\boldsymbol{\nu} = (\nu_1, \ldots, \nu_K)^{\intercal}$
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- noncentrality: $\boldsymbol{\lambda} = (\lambda_1, \ldots, \lambda_K)^{\intercal}$
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### VECM Model
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@@ -3247,6 +3216,14 @@ $$\mathbf{F} = (F_1, \ldots, F_K)^{\intercal}$$
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where $\Pi = \alpha \beta^{\intercal}$ is the cointegrating matrix of rank $r$, $0 \leq r\leq K$.
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:::
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::: {.column width="4%"}
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:::
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::: {.column width="48%"}
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### GARCH model
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\begin{align}
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@@ -3255,19 +3232,7 @@ where $\Pi = \alpha \beta^{\intercal}$ is the cointegrating matrix of rank $r$,
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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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:::
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::::
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## Modeling Approach: Dependence
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<br/>
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:::: {.columns}
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::: {.column width="48%"}
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Positive vs. negative innovations (capture leverage effects).
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### Time-varying dependence parameters
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@@ -3277,39 +3242,15 @@ Separate coefficients for positive and negative innovations to capture leverage
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\xi_{ij,t} = & \eta_{0,ij} + \eta_{1,ij} \xi_{ij,t-1} + \eta_{2,ij} z_{i,t-1} z_{j,t-1},
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\end{align*}
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$\xi_{ij,t}$ is a latent process
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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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$z_{i,t}$ denotes the $i$-th standardized residual from time series $i$ at time point $t$
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$\Lambda(\cdot)$ is a link function
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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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:::
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::: {.column width="4%"}
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:::
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::: {.column width="48%"}
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### Maximum Likelihood Estimation
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All parameters can be estimated jointly. Using conditional independence:
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\begin{align*}
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L = f_{X_1} \prod_{i=2}^T f_{X_i|\mathcal{F}_{i-1}},
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\end{align*}
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with multivariate conditional density:
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\begin{align*}
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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},
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\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})
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\end{align*}
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The copula density $c$ can be derived analytically.
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:::
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::::
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## Study Design and Evaluation
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@@ -3324,13 +3265,19 @@ The copula density $c$ can be derived analytically.
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- 3257 observations total
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- 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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### Estimation
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We sample 250 to reduce computational cost
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Joint maximum lieklihood estimation:
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We draw $2^{12}= 2048$ trajectories from the joint predictive distribution
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\begin{align*}
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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},
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\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})
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\end{align*}
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The copula density $c$ can be derived analytically.
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:::
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@@ -3342,7 +3289,7 @@ We draw $2^{12}= 2048$ trajectories from the joint predictive distribution
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### 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)
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\begin{align*}
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\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)
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@@ -3368,7 +3315,7 @@ For univariate cases the Energy Score becomes the Continuous Ranked Probability
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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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```{r, echo=FALSE, results='asis', width = 'revert-layer', cache = TRUE}
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load("assets/voldep/energy_df.Rdata")
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@@ -3424,6 +3371,23 @@ table_energy %>%
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)
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```
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```{=html}
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<div style="font-size: 0.5em; margin-top: 0.5em;">
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<span style="padding: 2px 6px;">Coloring w.r.t. test statistic: </span>
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<span style="background-color: #66BA6A; padding: 2px 6px;"><-5</span>
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<span style="background-color: #7CC168; padding: 2px 6px;">-4</span>
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<span style="background-color: #91C866; padding: 2px 6px;">-3</span>
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<span style="background-color: #B0D363; padding: 2px 6px;">-2</span>
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<span style="background-color: #D8E05E; padding: 2px 6px;">-1</span>
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<span style="background-color: #FFED58; padding: 2px 6px;">0</span>
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<span style="background-color: #FFD145; padding: 2px 6px;">1</span>
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<span style="background-color: #FFB531; padding: 2px 6px;">2</span>
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<span style="background-color: #FC9733; padding: 2px 6px;">3</span>
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<span style="background-color: #F67744; padding: 2px 6px;">4</span>
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<span style="background-color: #EE5250; padding: 2px 6px;">>5</span>
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</div>
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```
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:::
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::: {.column width="4%"}
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@@ -3438,7 +3402,7 @@ table_energy %>%
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- Vector ETS $VES^{\sigma}$ with constant volatility
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- 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
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- For EUA, the ETS Benchmark is the best performing model in terms of ES
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:::
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@@ -3467,7 +3431,7 @@ table_energy %>%
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::: {.column width="68%"}
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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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```{r, echo=FALSE, results = 'asis', cache = TRUE}
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load("assets/voldep/crps_df.Rdata")
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@@ -3515,6 +3479,23 @@ table_crps %>%
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)
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```
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```{=html}
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<div style="font-size: 0.5em; margin-top: 0.5em;">
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<span style="padding: 2px 6px;">Coloring w.r.t. test statistic: </span>
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<span style="background-color: #66BA6A; padding: 2px 6px;"><-5</span>
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<span style="background-color: #7CC168; padding: 2px 6px;">-4</span>
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<span style="background-color: #91C866; padding: 2px 6px;">-3</span>
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<span style="background-color: #B0D363; padding: 2px 6px;">-2</span>
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<span style="background-color: #D8E05E; padding: 2px 6px;">-1</span>
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<span style="background-color: #FFED58; padding: 2px 6px;">0</span>
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<span style="background-color: #FFD145; padding: 2px 6px;">1</span>
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<span style="background-color: #FFB531; padding: 2px 6px;">2</span>
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<span style="background-color: #FC9733; padding: 2px 6px;">3</span>
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<span style="background-color: #F67744; padding: 2px 6px;">4</span>
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<span style="background-color: #EE5250; padding: 2px 6px;">>5</span>
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</div>
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```
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:::
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::::
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@@ -3527,16 +3508,9 @@ table_crps %>%
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RMSE measures the performance of the forecasts at their mean
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Some models beat the benchmarks at short horizons
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</br>
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- Some models beat the benchmarks at short horizons
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</br>
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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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|
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Conclusion: the Improvements seen before must be attributed to other parts of the multivariate predictive distribution
|
||||
|
||||
:::
|
||||
|
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@@ -3546,7 +3520,7 @@ Conclusion: the Improvements seen before must be attributed to other parts of th
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|
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::: {.column width="68%"}
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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 $\textrm{RW}^{\sigma, \rho}_{}$).
|
||||
Relative improvement in RMSE compared to $\text{RW}^{\sigma, \rho}$
|
||||
|
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```{r, echo=FALSE, results = 'asis', cache = TRUE}
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load("assets/voldep/rmsq_df.Rdata")
|
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@@ -3593,6 +3567,23 @@ table_rmsq %>%
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)
|
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```
|
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||||
```{=html}
|
||||
<div style="font-size: 0.5em; margin-top: 0.5em;">
|
||||
<span style="padding: 2px 6px;">Coloring w.r.t. test statistic: </span>
|
||||
<span style="background-color: #66BA6A; padding: 2px 6px;"><-5</span>
|
||||
<span style="background-color: #7CC168; padding: 2px 6px;">-4</span>
|
||||
<span style="background-color: #91C866; padding: 2px 6px;">-3</span>
|
||||
<span style="background-color: #B0D363; padding: 2px 6px;">-2</span>
|
||||
<span style="background-color: #D8E05E; padding: 2px 6px;">-1</span>
|
||||
<span style="background-color: #FFED58; padding: 2px 6px;">0</span>
|
||||
<span style="background-color: #FFD145; padding: 2px 6px;">1</span>
|
||||
<span style="background-color: #FFB531; padding: 2px 6px;">2</span>
|
||||
<span style="background-color: #FC9733; padding: 2px 6px;">3</span>
|
||||
<span style="background-color: #F67744; padding: 2px 6px;">4</span>
|
||||
<span style="background-color: #EE5250; padding: 2px 6px;">>5</span>
|
||||
</div>
|
||||
```
|
||||
|
||||
:::
|
||||
|
||||
::::
|
||||
@@ -3757,8 +3748,8 @@ Accounting for heteroscedasticity or stabilizing the variance via log transforma
|
||||
- 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
|
||||
- the tails
|
||||
- the dependence structure between the marginals
|
||||
|
||||
:::
|
||||
|
||||
@@ -3778,7 +3769,7 @@ Accounting for heteroscedasticity or stabilizing the variance via log transforma
|
||||
|
||||
::::
|
||||
|
||||
---
|
||||
# Final Remarks {visibility="uncounted"}
|
||||
|
||||
## Contributions {#sec-contributions}
|
||||
|
||||
@@ -3786,8 +3777,6 @@ Accounting for heteroscedasticity or stabilizing the variance via log transforma
|
||||
|
||||
::: {.column width="48%"}
|
||||
|
||||
<p style="margin:1.5em;"></p>
|
||||
|
||||
**Theoretical**
|
||||
|
||||
Probabilistic Online Learning:
|
||||
@@ -3821,8 +3810,6 @@ Applications
|
||||
|
||||
::: {.column width="48%"}
|
||||
|
||||
<p style="margin:1.5em;"></p>
|
||||
|
||||
**Software**
|
||||
|
||||
R Packages:
|
||||
@@ -3852,5 +3839,8 @@ Berrisch, J., Narajewski, M., & Ziel, F. [-@BERRISCH2023100236]:
|
||||
|
||||
::::
|
||||
|
||||
## Questions! {visibility="uncounted"}
|
||||
|
||||
](assets/allisonhorst/hiding.png)
|
||||
|
||||
## References {visibility="uncounted"}
|
||||
Reference in New Issue
Block a user