Add voldep slides

.vscode/settings.json

@@ -0,0 +1,3 @@
+{
+  "files.autoSave": "off"
+}

25_07_phd_defense/assets/01_common.R

@@ -53,3 +53,38 @@ cols %>%
   scale_x_discrete(expand = c(0, 0)) +
   scale_y_discrete(expand = c(0, 0)) +
   theme_minimal() -> plot_cols
+
+col_gas <- "blue"
+col_eua <- "green"
+col_oil <- "amber"
+col_coal <- "brown"
+
+col_scale2 <- function(x, rng_t) {
+  ret <- x
+  for (i in seq_along(x)) {
+    if (x[i] < rng_t[1]) {
+      ret[i] <- col_scale(rng_t[1])
+    } else if (x[i] > rng_t[2]) {
+      ret[i] <- col_scale(rng_t[2])
+    } else {
+      ret[i] <- col_scale(x[i])
+    }
+  }
+  return(ret)
+}
+
+rng_t <- c(-5, 5)
+h_sub <- c(1, 5, 30)
+
+col_scale <- scales::gradient_n_pal(
+  c(
+    cols[5, "green"],
+    cols[5, "light-green"],
+    cols[5, "yellow"],
+    # cols[5, "amber"],
+    cols[5, "orange"],
+    # cols[5, "deep-orange"],
+    cols[5, "red"]
+  ),
+  values = seq(rng_t[1], rng_t[2], length.out = 5)
+)

25_07_phd_defense/assets/library.bib

@@ -8,6 +8,15 @@
   year      = {2014},
   publisher = {Taylor \& Francis}
 }
+@article{berrisch2023modeling,
+  title     = {Modeling volatility and dependence of European carbon and energy prices},
+  author    = {Berrisch, Jonathan and Pappert, Sven and Ziel, Florian and Arsova, Antonia},
+  journal   = {Finance Research Letters},
+  volume    = {52},
+  pages     = {103503},
+  year      = {2023},
+  publisher = {Elsevier}
+}
 @incollection{aastveit2019evolution,
   title     = {The Evolution of Forecast Density Combinations in Economics},
   author    = {Aastveit, Knut Are and Mitchell, James and Ravazzolo, Francesco and van Dijk, Herman K},

25_07_phd_defense/index.qmd

@@ -2094,6 +2094,750 @@ Get these slides:
 
 ::::
 
+# Modeling Volatility and Dependence of European Carbon and Energy Prices
+
+---
+
+## Motivation
+
+:::: {.columns}
+
+::: {.column width="48%"}
+
+- 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?
+
+:::
+
+::: {.column width="2%"}
+
+:::
+
+::: {.column width="48%"}
+
+</br>
+
+- Several Questions arise:
+
+- - Data (Pre)processing
+- - Modeling Approach
+- - Evaluation
+
+:::
+
+::::
+
+## 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
+
+```{r, echo=FALSE, fig.width = 12, fig.height = 6, fig.align="center"}
+readr::read_csv("assets/voldep/2022_10_14_eur_ref_co2_adj_hvpi_ex_nrg.csv") %>%
+  select(-EUR_USD, -hvpi_x_nrg) %>%
+  pivot_longer(-Date) %>%
+  mutate(name = factor(name,
+    levels = c("EUA", "Oil", "NGas", "Coal")
+  )) %>%
+  ggplot(aes(x = Date, y = value, color = name)) +
+  geom_line(
+    linewidth = linesize
+  ) + # The ggplot2 call got a little simpler now :)
+  geom_vline(
+    # Beginning of the war
+    xintercept = as.Date("2022-02-24"),
+    col = as.character(cols[8, "grey"]),
+    linetype = "dashed"
+  ) +
+  ylab("Prices EUR") +
+  xlab("Date") +
+  scale_color_manual(
+    values = as.character(c(
+      cols[8, col_eua],
+      cols[8, col_oil],
+      cols[8, col_gas],
+      cols[8, col_coal]
+    )),
+    labels = c(
+      "EUA",
+      "Oil",
+      "NGas",
+      "Coal"
+    )
+  ) +
+  guides(color = guide_legend(override.aes = list(size = 2))) +
+  theme_minimal() +
+  theme(
+    legend.position = "bottom",
+    # legend.key.size = unit(1.5, "cm"),
+    legend.title = element_blank(),
+    text = element_text(
+      size = text_size
+    )
+  ) +
+  scale_x_date(
+    breaks = as.Date(
+      paste0(
+        as.character(seq(2010, 2022, 2)),
+        "-01-01"
+      )
+    ),
+    date_labels = "%Y"
+  ) +
+  xlab(NULL) +
+  scale_y_continuous(trans = "log2")
+```
+
+## 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
+
+:::: {.columns}
+
+::: {.column width="48%"}
+
+- 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: $C_{\boldsymbol{U}_{t}|\mathcal{F}_{t - 1}}$
+
+:::
+
+::: {.column width="2%"}
+
+:::
+
+::: {.column width="48%"}
+
+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
+
+:::: {.columns}
+
+::: {.column width="48%"}
+
+#### 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: $\boldsymbol{\lambda} = (\lambda_1, \ldots, \lambda_K)^{\intercal}$
+
+:::
+
+::: {.column width="2%"}
+
+:::
+
+::: {.column width="48%"}
+
+#### 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
+
+:::: {.columns}
+
+::: {.column width="48%"}
+
+#### 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
+
+:::
+
+::: {.column width="2%"}
+
+:::
+
+::: {.column width="48%"}
+
+#### 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
+
+:::: {.columns}
+
+::: {.column width="48%"}
+
+#### 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 
+
+:::
+
+::: {.column width="2%"}
+
+:::
+
+::: {.column width="48%"}
+
+#### 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
+
+:::: {.columns}
+
+::: {.column width="48%"}
+
+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).
+
+```{r, echo=FALSE, results='asis'}
+load("assets/voldep/energy_df.Rdata")
+table_energy <- energy %>%
+  kbl(
+    digits = 2,
+    col.names = c(
+      "Model",
+      "\\(\\text{ES}^{\\text{All}}_{1-30}\\)",
+      "\\(\\text{ES}^{\\text{EUA}}_{1-30}\\)",
+      "\\(\\text{ES}^{\\text{Oil}}_{1-30}\\)",
+      "\\(\\text{ES}^{\\text{NGas}}_{1-30}\\)",
+      "\\(\\text{ES}^{\\text{Coal}}_{1-30}\\)",
+      "\\(\\text{ES}^{\\text{All}}_{1}\\)",
+      "\\(\\text{ES}^{\\text{All}}_{5}\\)",
+      "\\(\\text{ES}^{\\text{All}}_{30}\\)"
+    ),
+    # linesep = "",
+    bootstrap_options = "condensed",
+    # Dont replace any string, dataframe has to be valid latex code ...
+    escape = TRUE,
+    format = "html",
+    align = c("l", rep("r", ncol(energy) - 1))
+  ) %>%
+  kable_paper(full_width = FALSE)
+
+for (i in 2:ncol(energy)) {
+  bold_cells <- rep(FALSE, times = nrow(energy))
+
+  if (all(energy[-1, i, drop = TRUE] < 0)) {
+    bold_cells[1] <- TRUE
+  } else {
+    bold_cells[which.max(energy[-1, i, drop = TRUE]) + 1] <- TRUE
+  }
+
+  table_energy <- table_energy %>%
+    column_spec(i,
+      background = c(
+        cols[5, "grey"],
+        col_scale2(
+          energy_dm[, i, drop = TRUE][-1],
+          rng_t
+        )
+      ),
+      bold = bold_cells
+    )
+}
+table_energy %>%
+  kable_styling(
+    bootstrap_options = c("condensed"),
+    full_width = FALSE,
+    font_size = 16
+  )
+```
+
+:::
+
+::: {.column width="2%"}
+
+:::
+
+::: {.column width="48%"}
+
+- 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
+
+:::: {.columns}
+
+::: {.column width="28%"}
+
+- 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 
+
+:::
+
+::: {.column width="2%"}
+
+:::
+
+::: {.column width="68%"}
+
+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 $\textrm{RW}^{\sigma, \rho}_{}$).
+
+```{r, echo=FALSE, results = 'asis'}
+load("assets/voldep/crps_df.Rdata")
+
+table_crps <- crps %>%
+  kbl(
+    col.names = c("Model", rep(paste0("H", h_sub), 4)),
+    digits = 1,
+    linesep = "",
+    # Dont replace any string, dataframe has to be valid latex code ...
+    escape = FALSE,
+    format = "html",
+    booktabs = TRUE,
+    align = c("l", rep("r", ncol(crps) - 1))
+  ) %>%
+  kable_paper(full_width = FALSE)
+
+for (i in 2:ncol(crps)) {
+  bold_cells <- rep(FALSE, times = nrow(crps))
+
+  if (all(crps[-1, i, drop = TRUE] < 0)) {
+    bold_cells[1] <- TRUE
+  } else {
+    bold_cells[which.max(crps[-1, i, drop = TRUE]) + 1] <- TRUE
+  }
+
+  table_crps <- table_crps %>%
+    column_spec(i,
+      background = c(cols[5, "grey"], col_scale2(
+        crps_dm[, i, drop = TRUE][-1], rng_t
+      )),
+      bold = bold_cells
+    )
+}
+
+table_crps <- table_crps %>%
+  add_header_above(c(" ", "EUA" = 3, "Oil" = 3, "NGas" = 3, "Coal" = 3))
+
+table_crps %>%
+  kable_styling(
+    bootstrap_options = c("condensed"),
+    full_width = FALSE,
+    font_size = 16
+  )
+```
+
+:::
+
+::::
+
+## RMSE
+
+:::: {.columns}
+
+::: {.column width="28%"}
+
+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
+
+
+:::
+
+::: {.column width="2%"}
+
+:::
+
+::: {.column width="68%"}
+
+
+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 $\textrm{RW}^{\sigma, \rho}_{}$).
+
+```{r, echo=FALSE, results = 'asis'}
+load("assets/voldep/rmsq_df.Rdata")
+
+table_rmsq <- rmsq %>%
+  kbl(
+    col.names = c("Model", rep(paste0("H", h_sub), 4)),
+    digits = 1,
+    # Dont replace any string, dataframe has to be valid latex code ...
+    escape = FALSE,
+    linesep = "",
+    format = "html",
+    align = c("l", rep("r", ncol(rmsq) - 1))
+  ) %>%
+  kable_paper(full_width = FALSE)
+
+for (i in 2:ncol(rmsq)) {
+  bold_cells <- rep(FALSE, times = nrow(rmsq))
+
+  if (all(rmsq[-1, i, drop = TRUE] < 0)) {
+    bold_cells[1] <- TRUE
+  } else {
+    bold_cells[which.max(rmsq[-1, i, drop = TRUE]) + 1] <- TRUE
+  }
+
+  table_rmsq <- table_rmsq %>%
+    column_spec(i,
+      background = c(cols[5, "grey"], col_scale2(
+        rmsq_dm[, i, drop = TRUE][-1], rng_t
+      )),
+      bold = bold_cells
+    )
+}
+
+table_rmsq <- table_rmsq %>%
+  add_header_above(c(" ", "EUA" = 3, "Oil" = 3, "NGas" = 3, "Coal" = 3))
+
+table_rmsq %>%
+  kable_styling(
+    bootstrap_options = c("condensed"),
+    full_width = FALSE,
+    font_size = 14
+  )
+```
+
+:::
+
+::::
+
+## Evolution of Linear Dependence $\Xi$
+
+```{r, echo=FALSE, fig.width = 12, fig.height = 6, fig.align="center"}
+load("assets/voldep/plot_rho_df.Rdata")
+ggplot() +
+  geom_line(
+    size = linesize,
+    data = plot_data[plot_data$name == "5Oil-Coal", ],
+    aes(x = idx, y = value, col = name)
+  ) +
+  geom_line(
+    size = linesize,
+    data = plot_data[plot_data$name == "2EUA-NGas", ],
+    aes(x = idx, y = value, col = name)
+  ) +
+  geom_line(
+    size = linesize,
+    data = plot_data[plot_data$name == "1EUA-Oil", ],
+    aes(x = idx, y = value, col = name)
+  ) +
+  geom_line(
+    size = linesize,
+    data = plot_data[plot_data$name == "3EUA-Coal", ],
+    aes(x = idx, y = value, col = name)
+  ) +
+  geom_line(
+    size = linesize,
+    data = plot_data[plot_data$name == "6NGas-Coal", ],
+    aes(x = idx, y = value, col = name)
+  ) +
+  geom_line(
+    size = linesize,
+    data = plot_data[plot_data$name == "4Oil-NGas", ],
+    aes(x = idx, y = value, col = name)
+  ) +
+  geom_vline(
+    xintercept = as.Date("2022-02-24"), # Beginning of the war
+    col = as.character(cols[8, "grey"]),
+    linetype = "dashed"
+  ) +
+  geom_vline(
+    xintercept = as.Date("2022-02-28"), # Central bank of russia was blocked from access to foreign reserves
+    col = as.character(cols[8, "grey"]),
+    linetype = "dotted"
+  ) +
+  scale_colour_manual(
+    values = as.character(
+      cols[
+        6,
+        c("green", "purple", "blue", "red", "orange", "brown")
+      ]
+    ),
+    labels = c(
+      paste0(varnames[comb[, 1] + 1], "-", varnames[comb[, 2] + 1])
+    )
+  ) +
+  theme_minimal() +
+  theme(
+    zoom.x = element_rect(fill = cols[4, "grey"], colour = NA),
+    legend.position = "bottom",
+    text = element_text(size = text_size)
+  ) +
+  ylab("Correlation") +
+  scale_x_date(
+    breaks = date_breaks_fct,
+    labels = date_labels_fct
+  ) +
+  xlab(NULL) +
+  guides(col = guide_legend(
+    title = NULL,
+    nrow = 1,
+    byrow = TRUE,
+    override.aes = list(size = 2)
+  )) +
+  # scale_y_continuous(breaks = seq(-1.5, 1.0, 0.5)) +
+  ggforce::facet_zoom(
+    xlim = c(
+      as.Date("2022-02-02"),
+      as.Date("2022-04-30")
+    ),
+    zoom.size = 1.5,
+    show.area = TRUE
+  )
+```
+
+## Predictive Quantiles (Russian Invasion)
+
+```{r, echo=FALSE, fig.width = 12, fig.height = 6, fig.align="center"}
+load("assets/voldep/plot_quant_df.Rdata")
+
+plot_quant_data %>% ggplot(aes(x = date, y = value)) +
+  geom_line(size = 1, aes(col = quant)) +
+  geom_vline(
+    xintercept = as.Date("2022-02-24"), # Beginning of the war
+    col = as.character(cols[8, "grey"]),
+    linetype = "dashed"
+  ) +
+  geom_vline(
+    xintercept = as.Date("2022-02-28"), # Central bank of russia was blocked from access to foreign reserves
+    col = as.character(cols[8, "grey"]),
+    linetype = "dotted"
+  ) +
+  scale_colour_manual(
+    values = as.character(
+      c(
+        cols[
+          5,
+          c(1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15)
+        ],
+        cols[9, 18]
+      )
+    )
+  ) +
+  facet_grid(var ~ ., scales = "free_y") +
+  theme_minimal() +
+  theme(
+    legend.position = "bottom",
+    text = element_text(size = text_size),
+    strip.background = element_rect(
+      fill = cols[1, 18],
+      colour = cols[1, 18]
+    )
+  ) +
+  ylab("EUR") +
+  scale_x_date(
+    breaks = (seq(
+      as.Date("2022-02-15"),
+      as.Date("2022-03-31"),
+      by = "4 day"
+    ) + 1)[-12],
+    date_labels = "%b %d",
+    limits = c(as.Date("2022-02-15"), as.Date("2022-03-31"))
+  ) +
+  xlab(NULL) +
+  guides(col = guide_legend(
+    title = "Quantiles [\\%]",
+    nrow = 2,
+    byrow = TRUE,
+    override.aes = list(size = 2)
+  )) +
+  scale_y_continuous(
+    trans = "log2",
+    breaks = y_breaks
+  )
+
+```
+
+## Conclusion
+
+:::: {.columns}
+
+::: {.column width="48%"}
+
+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
+
+:::
+
+::: {.column width="2%"}
+
+:::
+
+::: {.column width="48%"}
+
+</br>
+
+<center>
+<img src="assets/voldep/frame.png">
+</center>
+
+`r fontawesome::fa("newspaper")` `r Citet(my_bib, "berrisch2023modeling")`
+
+:::
+
+::::
 
 ## Columns Template