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analysis_utils.R
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analysis_utils.R
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###############################################################################
# Utility functions for the Backtesting of stock portfolios with the portvine
# package (Mainly visualizations)
###############################################################################
# -----------------------------------------------------------------------------
# NOTE: these functions are not intended for the general usage they do not cover
# input checks, unit tests and are documented quite minimal. If you would like
# to use these functions do this with care. Some of the visualization functions
# can be found in a cleaner setup in the Get Started vignette of the portvine
# package
# -----------------------------------------------------------------------------
# Extract for each asset the residuals of a certain marginal roll and visualize
# Returns named list with visualization for each asset
marg_resid_viz_list <- function(roll, asset_names = NULL, marg_window = 1,
squared = FALSE) {
fitted_marginals <- fitted_marginals(roll)
if (is.null(asset_names)) asset_names <- fitted_vines(roll)[[1]]$names
sapply(
asset_names,
function(asset_name) {
# use again a utility function from the portvine package
model_resid <- roll_residuals(
fitted_marginals[[asset_name]], 1
)
if (squared) model_resid <- model_resid^2
simple_exploratory <- tibble(resid = model_resid) %>%
rowid_to_column(var = "id") %>%
ggplot(aes(x = id, y = resid)) +
geom_line(size = 0.2) +
labs(x = "t", y = ifelse(squared, expression(z[t]^2),expression(z[t])),
title = str_to_title(str_replace_all(asset_name, "_", " "))) +
theme(axis.text.x = element_blank(),
axis.ticks.x = element_blank())
acf_plot <- tibble(
acf = as.numeric(acf(model_resid, type = "cor", lag.max = 20,
plot = FALSE)$acf),
lag = 0:20
) %>%
filter(lag != 0 & lag <= 10) %>%
ggplot() +
geom_hline(yintercept = 0, col = "black", size = 0.3) +
geom_hline(yintercept = qnorm(c(0.025, 0.975)) /
sqrt(length(model_resid)),
linetype = "longdash", col = custom_colors[1], size = 0.5) +
geom_segment(aes(x = lag, xend = lag, y = 0, yend = acf)) +
geom_point(aes(x = lag, y = acf)) +
scale_x_continuous(breaks = seq(1, 10, 1)) +
ylim(-1, 1) +
labs(x = "h", y = "ACF(h)")
ljungplot <- tibble(
pval = sapply(
1:10,
function(i) Box.test(model_resid, lag = i,
type = "Lju")$p.value),
lag = 1:10) %>%
ggplot() +
geom_hline(yintercept = 0, col = "black", size = 0.3) +
geom_hline(yintercept = 0.05,
linetype = "longdash", col = custom_colors[1], size = 0.5) +
geom_line(aes(x = lag, y = pval)) +
geom_point(aes(x = lag, y = pval)) +
scale_x_continuous(breaks = seq(1, 10, 1)) +
labs(x = "h", y = "p-value of Ljung-Box test at lag h")
(simple_exploratory / (ljungplot + acf_plot)) +
plot_layout(nrow = 2)
}, USE.NAMES = TRUE, simplify = FALSE)
}
# creates a heatmap of Ljung Box test pvalues for a portvine_roll object
ljung_heatmap <- function(roll, roll_num = 1) {
asset_names <- fitted_vines(roll)[[1]]$names
roll_marginals <- fitted_marginals(roll)
ljung_data <- sapply(asset_names, function(asset_name) {
# use again a utility function from the portvine package
model_resid <- roll_residuals(
roll_marginals[[asset_name]], roll_num = roll_num
)
sapply(
1:10,
function(i) Box.test(model_resid, lag = i, type = "Lju")$p.value
)
}, USE.NAMES = TRUE, simplify = TRUE)
ljung_data %>%
as_tibble() %>%
rowid_to_column("lag") %>%
pivot_longer(-lag, names_to = "asset", values_to = "pval") %>%
ggplot(aes(x = lag, y = asset, fill = pval)) +
geom_tile() +
scale_y_discrete(labels = ~ str_to_title(str_replace_all(.x, "_", " "))) +
scale_x_continuous(breaks = 1:10) +
labs(y = "", x = "h", fill = "p-value",
title = "Results of the Ljung-Box tests",
caption = paste("Rolling window:", roll_num)) +
scale_fill_gradientn(colours = c(custom_colors[2],"#C37285" ,
custom_colors[1], "#2a82db"),
values = scales::rescale(c(0, 0.05 - 0.01,
0.05, 1)),
breaks = c(0.05),
labels = c(0.05),
guide = guide_colourbar(nbin = 1000)) +
theme(legend.position = "right",
panel.grid.minor.x = element_blank())
}
# animated version of ljung_heatmap that animates over all marginal windows
library(gganimate)
ljung_heatmap_animation <- function(roll) {
asset_names <- fitted_vines(roll)[[1]]$names
roll_marginals <- fitted_marginals(roll)
n_marginal_rolls <- roll_marginals[[1]]@model$n.refits
anim <- map_df(seq(n_marginal_rolls), function(roll_num) {
ljung_data <- sapply(asset_names, function(asset_name) {
# use again a utility function from the portvine package
model_resid <- roll_residuals(
roll_marginals[[asset_name]], roll_num = roll_num
)
sapply(
1:10,
function(i) Box.test(model_resid, lag = i, type = "Lju")$p.value
)
}, USE.NAMES = TRUE, simplify = TRUE)
ljung_data %>%
as_tibble() %>%
rowid_to_column("lag") %>%
pivot_longer(-lag, names_to = "asset", values_to = "pval") %>%
mutate(roll = roll_num)
}) %>%
ggplot(aes(x = lag, y = asset, fill = pval, group = roll)) +
geom_tile() +
transition_states(roll,
transition_length = 1,
state_length = 4) +
scale_y_discrete(labels = ~ str_to_title(str_replace_all(.x, "_", " "))) +
scale_x_continuous(breaks = 1:10) +
labs(y = "", x = "h", fill = "p-value",
title = "Results of the Ljung-Box tests",
caption = "Rolling window: {closest_state}") +
scale_fill_gradientn(colours = c(custom_colors[2],"#C37285" ,
custom_colors[1], "#2a82db"),
values = scales::rescale(c(0, 0.05 - 0.01,
0.05, 1)),
breaks = c(0.05),
labels = c(0.05),
guide = guide_colourbar(nbin = 1000)) +
theme(legend.position = "right",
panel.grid.minor.x = element_blank())
animate(
anim +
enter_fade() +
exit_shrink(),
width = 600,
height = 300,
nframes = 200
)
}
# utility function for labeled vinecop tree plots
labeled_vinecop_plot <- function(vine, tree = 1) {
rvinecopulib:::plot.vinecop(vine, tree = tree,
var_names = "use",
edge_labels = "family_tau")
}
# utility for table of used bivariate copula families
bicops_used <- function(vine) {
table(unlist(vine$pair_copulas)[names(unlist(vine$pair_copulas)) == "family"])
}
# utility to get the copula famlies with their kendalls tau for the copulas
# that are directly associated with the conditional asset i.e. the rightmost
# bicop of of every tree level.
index_copulas <- function(vine) {
n_trees <- length(vine$pair_copulas)
res <- data.frame("Tree" = seq(n_trees),
"Copula_familiy" = "",
"Kendalls_tau" = 0)
for (i in seq(n_trees)) {
res[i, 2] <- vine$pair_copulas[[i]][[1]]$family
res[i, 3] <- rvinecopulib::par_to_ktau(vine$pair_copulas[[i]][[1]])
}
res
}
# -----------------------------------------------------------------------------
# backtesting utilities
# -----------------------------------------------------------------------------
# traditional backtests for a portvine_roll object
get_traditional_backtests_uncond <- function(roll, alphas) {
res <- tribble(
~"Risk measure", ~"Backtest",
"VaR", "unconditional coverage",
"VaR", "conditional coverage",
"ES", "exceedance residuals (two sided)",
"ES", "exceedance residuals (one sided)",
"ES", "simple conditional calibration (two sided)",
"ES", "simple conditional super-calibration (one sided)",
"ES", "strict ESR test (two sided)",
"ES", "one sided intercept ESR test",
)
for (alpha in alphas) {
res_vec <- numeric(8)
### VaR
temp <- rugarch::VaRTest(
alpha = alpha,
actual = risk_estimates(roll, "VaR", alpha) %>%
pull(realized),
VaR = risk_estimates(roll, "VaR", alpha) %>%
pull(risk_est)
)[c("uc.LRp", "cc.LRp")]
res_vec[1] <- temp[[1]]
res_vec[2] <- temp[[2]]
### ES
# exceedance residuals
temp <- esback::er_backtest(
r = risk_estimates(roll, "VaR", alpha) %>%
pull(realized),
q = risk_estimates(roll, "VaR", alpha) %>%
pull(risk_est),
e = risk_estimates(roll, "ES_mean", alpha) %>%
pull(risk_est)
)
res_vec[3] <- temp$pvalue_twosided_simple
res_vec[4] <- temp$pvalue_onesided_simple
# conditional calibration
temp <- esback::cc_backtest(
r = risk_estimates(roll, "VaR", alpha) %>%
pull(realized),
q = risk_estimates(roll, "VaR", alpha) %>%
pull(risk_est),
e = risk_estimates(roll, "ES_mean", alpha) %>%
pull(risk_est),
alpha = alpha
)
res_vec[5] <- temp$pvalue_twosided_simple
res_vec[6] <- temp$pvalue_onesided_simple
# ESR tests
temp <- try(
esback::esr_backtest(
r = risk_estimates(roll, "VaR", alpha) %>%
pull(realized),
q = risk_estimates(roll, "VaR", alpha) %>%
pull(risk_est),
e = risk_estimates(roll, "ES_mean", alpha) %>%
pull(risk_est),
alpha = alpha,
version = 1
), silent = TRUE)
if ("try-error" %in% class(temp)) {
res_vec[7] <- -1
} else {
res_vec[7] <- temp$pvalue_twosided_asymptotic
}
temp <- try(
esback::esr_backtest(
r = risk_estimates(roll, "VaR", alpha) %>%
pull(realized),
q = risk_estimates(roll, "VaR", alpha) %>%
pull(risk_est),
e = risk_estimates(roll, "ES_mean", alpha) %>%
pull(risk_est),
alpha = alpha,
version = 3
), silent = TRUE)
if ("try-error" %in% class(temp)) {
res_vec[8] <- -1
} else {
res_vec[8] <- temp$pvalue_onesided_asymptotic
}
res <- res %>%
bind_cols("temp" = res_vec) %>%
rename(!!paste("alpha:", alpha) := temp)
}
res
}
# the corresponding version dealing with cond_portvine_roll objects
get_traditional_backtests_cond <- function(roll, alphas, cond_u) {
res <- tribble(
~"Risk measure", ~"Backtest",
"VaR", "unconditional coverage",
"VaR", "conditional coverage",
"ES", "exceedance residuals (two sided)",
"ES", "exceedance residuals (one sided)",
"ES", "simple conditional calibration (two sided)",
"ES", "simple conditional super-calibration (one sided)",
"ES", "strict ESR test (two sided)",
"ES", "one sided intercept ESR test",
)
for (alpha in alphas) {
res_vec <- numeric(8)
### VaR
temp <- rugarch::VaRTest(
alpha = alpha,
actual = risk_estimates(roll, "VaR", alpha, cond_u = cond_u) %>%
pull(realized),
VaR = risk_estimates(roll, "VaR", alpha, cond_u = cond_u) %>%
pull(risk_est)
)[c("uc.LRp", "cc.LRp")]
res_vec[1] <- temp[[1]]
res_vec[2] <- temp[[2]]
### ES
# exceedance residuals
temp <- esback::er_backtest(
r = risk_estimates(roll, "VaR", alpha, cond_u = cond_u) %>%
pull(realized),
q = risk_estimates(roll, "VaR", alpha, cond_u = cond_u) %>%
pull(risk_est),
e = risk_estimates(roll, "ES_mean", alpha, cond_u = cond_u) %>%
pull(risk_est)
)
res_vec[3] <- temp$pvalue_twosided_simple
res_vec[4] <- temp$pvalue_onesided_simple
# conditional calibration
temp <- esback::cc_backtest(
r = risk_estimates(roll, "VaR", alpha, cond_u = cond_u) %>%
pull(realized),
q = risk_estimates(roll, "VaR", alpha, cond_u = cond_u) %>%
pull(risk_est),
e = risk_estimates(roll, "ES_mean", alpha, cond_u = cond_u) %>%
pull(risk_est),
alpha = alpha
)
res_vec[5] <- temp$pvalue_twosided_simple
res_vec[6] <- temp$pvalue_onesided_simple
# ESR tests
temp <- try(
esback::esr_backtest(
r = risk_estimates(roll, "VaR", alpha, cond_u = cond_u) %>%
pull(realized),
q = risk_estimates(roll, "VaR", alpha, cond_u = cond_u) %>%
pull(risk_est),
e = risk_estimates(roll, "ES_mean", alpha, cond_u = cond_u) %>%
pull(risk_est),
alpha = alpha,
version = 1
), silent = TRUE)
if ("try-error" %in% class(temp)) {
res_vec[7] <- -1
} else {
res_vec[7] <- temp$pvalue_twosided_asymptotic
}
temp <- try(
esback::esr_backtest(
r = risk_estimates(roll, "VaR", alpha, cond_u = cond_u) %>%
pull(realized),
q = risk_estimates(roll, "VaR", alpha, cond_u = cond_u) %>%
pull(risk_est),
e = risk_estimates(roll, "ES_mean", alpha, cond_u = cond_u) %>%
pull(risk_est),
alpha = alpha,
version = 3
), silent = TRUE)
if ("try-error" %in% class(temp)) {
res_vec[8] <- -1
} else {
res_vec[8] <- temp$pvalue_onesided_asymptotic
}
res <- res %>%
bind_cols("temp" = res_vec) %>%
rename(!!paste("alpha:", alpha) := temp)
}
res
}
### comparative backtesting as in Nolde and Ziegel 2017
# first up the scoring function for the ES
es_scoring <- function(realized, es, var, alpha) {
if (any(es >= 0)) stop("ES must be always negative for this scoring function")
((realized < var) * (realized - var) / es) +
alpha * ((var / es) - 1 + log(-es))
}
# interpretation if confidence_result is smaller than the specified nu then the
# model corresponding to roll1 is considered at least as good as the roll2 model
es_comparative_backtest <- function(roll1, roll2, alpha) {
realized1 <- risk_estimates(roll1, "ES_mean", alpha) %>%
pull(realized)
var1 <- risk_estimates(roll1, "VaR", alpha) %>%
pull(risk_est)
es1 <- risk_estimates(roll1, "ES_mean", alpha) %>%
pull(risk_est)
scoring1 <- es_scoring(
realized = realized1,
es = es1,
var = var1,
alpha = alpha
)
realized2 <- risk_estimates(roll2, "ES_mean", alpha) %>%
pull(realized)
var2 <- risk_estimates(roll2, "VaR", alpha) %>%
pull(risk_est)
es2 <- risk_estimates(roll2, "ES_mean", alpha) %>%
pull(risk_est)
scoring2 <- es_scoring(
realized = realized2,
es = es2,
var = var2,
alpha = alpha
)
n <- length(realized1)
sigma <- sd(scoring1 - scoring2)
test_statistic <- mean(scoring1 - scoring2) * sqrt(n) / sigma
confidence_result <- pnorm(test_statistic)
c(test_statistic = test_statistic, confidence_result = confidence_result)
}
# the same but roll1 is a cond_portvine_roll object
es_cond_comparative_backtest <- function(cond_roll1, roll2, alpha, cond_u) {
realized1 <- risk_estimates(cond_roll1, "ES_mean", alpha, cond_u = cond_u) %>%
pull(realized)
var1 <- risk_estimates(cond_roll1, "VaR", alpha, cond_u = cond_u) %>%
pull(risk_est)
es1 <- risk_estimates(cond_roll1, "ES_mean", alpha, cond_u = cond_u) %>%
pull(risk_est)
scoring1 <- es_scoring(
realized = realized1,
es = es1,
var = var1,
alpha = alpha
)
realized2 <- risk_estimates(roll2, "ES_mean", alpha) %>%
pull(realized)
var2 <- risk_estimates(roll2, "VaR", alpha) %>%
pull(risk_est)
es2 <- risk_estimates(roll2, "ES_mean", alpha) %>%
pull(risk_est)
scoring2 <- es_scoring(
realized = realized2,
es = es2,
var = var2,
alpha = alpha
)
n <- length(realized1)
sigma <- sd(scoring1 - scoring2)
test_statistic <- mean(scoring1 - scoring2) * sqrt(n) / sigma
confidence_result <- pnorm(test_statistic)
c(test_statistic = test_statistic, confidence_result = confidence_result)
}
# heatmap that displays at which quantile level the conditional risk measures
# would pass the traditional backtests. This can be used to infer knowledge
# about the influence of the certain conditional asset on the portfolio level
# risk measures
cond_backtest_heatmap <- function(roll, alpha, cond_u) {
plot_df <- map_df(cond_u, function(u) {
get_traditional_backtests_cond(
roll,
alphas = alpha,
cond_u = u
) %>%
mutate(cond_u = u)
}) %>%
mutate(test = paste0(`Risk measure`, ": ", Backtest)) %>%
select(starts_with("alpha"), cond_u, test)
colnames(plot_df)[1] <- "pval"
plot_df <- plot_df %>%
mutate(pval = if_else(is.nan(pval) | pval == -1, NA_real_, pval),
test = fct_rev(fct_inorder(factor(test))))
if (all(plot_df$pval >= 0.05 | is.na(plot_df$pval))) {
legend_scale <- scale_fill_gradient(
low = "#92B8DE", high = "#2a82db"
)
} else {
legend_scale <- scale_fill_gradientn(
colours = c("#db4f59","#C37285" ,
"#92B8DE", "#2a82db"),
values = scales::rescale(c(0, 0.05 - 0.01, 0.05, 1)),
breaks = c(0.05),
labels = c(0.05),
guide = guide_colourbar(nbin = 1000))
}
plot_df %>%
ggplot(aes(x = factor(cond_u), y = test, fill = pval)) +
geom_tile() +
legend_scale +
labs(y = "", x = "Quantile level", fill = "P-value",
title = "Traditional backtests on the conditional risk measures",
subtitle = paste("Alpha level:", alpha))
}
#' Extract a filtered ugarch model from a uGARCHroll object
#'
#' The [`rugarch::ugarchroll`] class object encompasses fitting information
#' about a number of
#' models fitted in a rolling window fashion. This utility function gives an
#' easy interface to extract the filtered ugarch model for a specified roll.
#'
#' @param ugarchroll Object of class [`rugarch::ugarchroll`].
#' @param roll_num Count that specifies the fitted model to extract the
#' residuals from.
#'
#' @return filtered ugarch model
#' @export
roll_filtered_model <- function(ugarchroll, roll_num = 1) {
checkmate::assert_class(ugarchroll, classes = "uGARCHroll")
total_roll_num <- ugarchroll@model$n.refits
checkmate::assert_integerish(roll_num,
lower = 0, upper = total_roll_num,
len = 1
)
train_end_index <- ugarchroll@model$n.start
refit_size <- ugarchroll@model$refit.every
distribution <- ugarchroll@model$spec@model$modeldesc$distribution
coefs <- ugarchroll@model$coef[[roll_num]]$coef[, 1]
spec <- rugarch::ugarchspec(
distribution.model = distribution,
fixed.pars = coefs,
variance.model = list(
model = ugarchroll@model$spec@model$modeldesc$vmodel,
garchOrder = c(
marginals$iberdrola@model$spec@model$modelinc[["alpha"]],
marginals$iberdrola@model$spec@model$modelinc[["beta"]]
),
submodel = NULL,
external.regressors = NULL,
variance.targeting = FALSE
),
mean.model = list(
armaOrder = c(
marginals$iberdrola@model$spec@model$modelinc[["ar"]],
marginals$iberdrola@model$spec@model$modelinc[["ma"]]
),
include.mean = marginals$iberdrola@model$spec@model$modelinc[["mu"]] == 1,
archm = FALSE,
archpow = 1,
arfima = FALSE,
external.regressors = NULL,
archex = FALSE
)
)
filtered_model <- rugarch::ugarchfilter(
spec = spec,
data = ugarchroll@model$data[seq(
1 + refit_size * (roll_num - 1),
min(
refit_size * (roll_num - 1) +
train_end_index,
length(ugarchroll@model$data)
)
)]
)
filtered_model
}