diff --git a/.vscode/settings.json b/.vscode/settings.json new file mode 100644 index 0000000..c3d3372 --- /dev/null +++ b/.vscode/settings.json @@ -0,0 +1,3 @@ +{ + "files.autoSave": "off" +} \ No newline at end of file diff --git a/25_07_phd_defense/assets/01_common.R b/25_07_phd_defense/assets/01_common.R index 1f13e12..94e4993 100644 --- a/25_07_phd_defense/assets/01_common.R +++ b/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) +) diff --git a/25_07_phd_defense/assets/library.bib b/25_07_phd_defense/assets/library.bib index d8a31ee..64db732 100644 --- a/25_07_phd_defense/assets/library.bib +++ b/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}, diff --git a/25_07_phd_defense/assets/voldep/crps_classic.Rdata b/25_07_phd_defense/assets/voldep/crps_classic.Rdata new file mode 100644 index 0000000..12f8728 Binary files /dev/null and b/25_07_phd_defense/assets/voldep/crps_classic.Rdata differ diff --git a/25_07_phd_defense/assets/voldep/crps_df.Rdata b/25_07_phd_defense/assets/voldep/crps_df.Rdata new file mode 100644 index 0000000..7d0f609 Binary files /dev/null and b/25_07_phd_defense/assets/voldep/crps_df.Rdata differ diff --git a/25_07_phd_defense/assets/voldep/energy_classic.Rdata b/25_07_phd_defense/assets/voldep/energy_classic.Rdata new file mode 100644 index 0000000..19038fe Binary files /dev/null and b/25_07_phd_defense/assets/voldep/energy_classic.Rdata differ diff --git a/25_07_phd_defense/assets/voldep/energy_df.Rdata b/25_07_phd_defense/assets/voldep/energy_df.Rdata new file mode 100644 index 0000000..26a31e7 Binary files /dev/null and b/25_07_phd_defense/assets/voldep/energy_df.Rdata differ diff --git a/25_07_phd_defense/assets/voldep/plot_quant_df.Rdata b/25_07_phd_defense/assets/voldep/plot_quant_df.Rdata new file mode 100644 index 0000000..ffd3f9b Binary files /dev/null and b/25_07_phd_defense/assets/voldep/plot_quant_df.Rdata differ diff --git a/25_07_phd_defense/assets/voldep/plot_rho_df.Rdata b/25_07_phd_defense/assets/voldep/plot_rho_df.Rdata new file mode 100644 index 0000000..ae35a77 Binary files /dev/null and b/25_07_phd_defense/assets/voldep/plot_rho_df.Rdata differ diff --git a/25_07_phd_defense/assets/voldep/rmsq_df.Rdata b/25_07_phd_defense/assets/voldep/rmsq_df.Rdata new file mode 100644 index 0000000..74dd41b Binary files /dev/null and b/25_07_phd_defense/assets/voldep/rmsq_df.Rdata differ diff --git a/25_07_phd_defense/index.qmd b/25_07_phd_defense/index.qmd index 1aa6ad1..590b12c 100644 --- a/25_07_phd_defense/index.qmd +++ b/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%"} + +
+ +- 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 + +
+ +### 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 + + +
+ + +- Some models beat the benchmarks at short horizons + +
+ +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%"} + +
+ +
+ +
+ +`r fontawesome::fa("newspaper")` `r Citet(my_bib, "berrisch2023modeling")` + +::: + +:::: ## Columns Template