Empirical Analysis

Young Geun Kim 15 Jun, 2022

Data
- Split
- Long-range dependency
  - Plot
Model
- VAR
- VHAR
- BVAR
- BVHAR-S
- BVHAR-L
- Hyperparamers
Errors
- Relative Errors
Intervals

etf_data <- "../data/processed/cboe_etf.rds"

# tidyverse----------------------------
library(tidyverse)
# BVHAR custom package-----------------
library(bvhar)
# ggplot grid--------------------------
library(gridExtra)
# Set the number of processor cores----
cl <- parallel::makeCluster(8, type = "FORK")
# set seed for reproducible result-----
set.seed(1)
# width of figure when save------------
fig_width <- 21

# result table-------------------------
source("report-fns.R")
# hyperparameter setting table---------
source("param-fns.R")
# CBOE ETF raw data--------------------
etf_raw <- readRDS(etf_data)

Data

Split

(h <- 30)
#> [1] 30

2000: Dot-com bubble
2001: September 11 terror and Enron scandal
2002: Stock market downturn after 2001
2003: Iraq war (until 2011)
2007 to 2008: Financial crisis
- 2007: Subprime mortgage crisis
- 2008: Bankruptcy of Lehman Brothers
2010 to 2016: European sovereign dept crisis
- 2010: Greek debt crisis
- 2011: Italian default
- 2015: Greek default
- 2016: Brexit
2018: US-China trade war
2019: Brexit
2020: COVID-19

te_date <- tail(etf_raw$data_wide$date, h)[1]
data_plt <- 
  etf_raw$data_long %>% 
  mutate(
    train = date < te_date,
    series_id = str_remove_all(series_id, pattern = "CLS$"),
    xmin = min(date[train == FALSE]),
    xmax = max(date)
  ) %>% 
  ggplot(aes(x = date, y = value)) +
  geom_rect(
    aes(xmin = xmin, xmax = xmax, ymin = -Inf, ymax = Inf),
    alpha = .7,
    fill = "grey" # test set
  ) +
  geom_path() +
  facet_grid(series_id ~ ., scales = "free_y") +
  theme_minimal() +
  theme(
    panel.border = element_rect(fill = NA),
    axis.text.y = element_text(size = 5),
    text = element_text(family = "serif")
  ) +
  labs(
    x = element_blank(),
    y = "Volatility index"
  )
data_plt

Save:

ggsave(
  filename = "../output/figs/analysis-dataplot.pdf", 
  plot = data_plt,
  device = "pdf",
  width = fig_width,
  height = .618 * fig_width,
  units = "cm",
  dpi = 1500,
  limitsize = FALSE
)

etf_split <- 
  divide_ts(
    etf_vix %>% rename_with(~str_remove_all(., pattern = "CLS$")), 
    h
  )
etf_train <- etf_split$train
etf_test <- etf_split$test

Long-range dependency

Plot

ACF:

gvz_acf <- 
  etf_train %>% 
  select(GVZ) %>% 
  forecast::ggAcf(lag.max = 50, plot = FALSE)
#> Registered S3 method overwritten by 'quantmod':
#>   method            from
#>   as.zoo.data.frame zoo

miner_acf <- 
  etf_train %>% 
  select(VXGDX) %>% 
  forecast::ggAcf(lag.max = 50, plot = FALSE)

Prewhitening for CCF:

gvz_ar <- forecast::Arima(etf_train$GVZ, order = c(1L, 0L, 0L), include.mean = FALSE, include.drift = FALSE)
miner_ar <- forecast::Arima(etf_train$VXGDX, order = c(1L, 0L, 0L), include.mean = FALSE, include.drift = FALSE)
# prewhitening---------------
gvz_resid <- gvz_ar$residuals
miner_resid <- miner_ar$residuals

ACF and CCF into one plot:

# Compute ccf-----------------------------------
ccf_compute <- 
  data.frame(
    gvz = gvz_resid,
    miner = miner_resid
  ) %>% 
  forecast::ggAcf(lag.max = 50, plot = FALSE)
# testing--------------------------------------
bartlett_line <- qnorm(.05 / 2, lower.tail = FALSE) / sqrt(ccf_compute$n.used)
# Change acf values to not-prewhitened one-----
ccf_compute$acf[, 1, 1] <- gvz_acf$acf[,, 1]
ccf_compute$acf[, 2, 2] <- miner_acf$acf[,, 1]
# Data frame-----------------------------------
ccf_gvz <- ccf_compute$acf[-1,, 1]
colnames(ccf_gvz) <- c("GVZ", "VXGDX")
ccf_gvz <- cbind(ccf_gvz, data.frame(idx = "GVZ", lag = 1:50))
ccf_miner <- ccf_compute$acf[-1,, 2]
colnames(ccf_miner) <- c("GVZ", "VXGDX")
ccf_miner <- cbind(ccf_miner, data.frame(idx = "VXGDX", lag = 1:50))
ccf_df <- 
  rbind(ccf_gvz, ccf_miner) %>% 
  pivot_longer(c(GVZ, VXGDX), names_to = "series_id", values_to = "acf") %>% 
  unite(col = "index", c(idx, series_id)) %>% 
  mutate(index = factor(index, levels = c("GVZ_GVZ", "VXGDX_GVZ", "GVZ_VXGDX", "VXGDX_VXGDX")))
# Title of each plot
give_title <- function(string) {
  str_split(string, pattern = "\\_", simplify = TRUE) %>% 
    apply(
      1,
      function(x) {
        if (x[1] == x[2]) {
          return(paste("ACF for", x[1]))
        } else {
          y1 <- paste("Prewhitened", x[1])
          y2 <- paste("Prewhitened", x[2])
          return(paste("CCF for", y1, "and", y2))
        }
      }
    )
}
# Draw CCF-------------------------------------
resid_ccf <- 
  ccf_df %>%
  ggplot(aes(x = lag)) +
  geom_linerange(aes(ymin = 0, ymax = acf)) +
  facet_wrap(
    index ~ .,
    scales = "free_y",
    labeller = labeller(index = give_title)
  ) +
  geom_hline(yintercept = bartlett_line, col = "blue", linetype = "dashed") +
  geom_hline(yintercept = -bartlett_line, col = "blue", linetype = "dashed") +
  theme_minimal() +
  theme(
    panel.border = element_rect(fill = NA),
    axis.text.y = element_text(size = 5),
    axis.text.x = element_text(size = 5),
    text = element_text(family = "serif")
  ) +
  labs(
    title = element_blank(),
    x = "Lag",
    y = "SACF"
  )
resid_ccf

Save:

ggsave(
  filename = "../output/figs/analysis-lrdccf.pdf", 
  plot = resid_ccf,
  device = "pdf",
  width = fig_width, 
  height = .618 * fig_width, 
  units = "cm",
  dpi = 1500,
  limitsize = FALSE
)

Model

VAR

choose_var(etf_train, lag_max = 10)
#> $ic
#>      AIC     BIC      HQ     FPE
#> 1  -1.39 -0.9015 -1.2050 -1.2050
#> 2  -1.40 -0.4632 -1.0403 -1.0403
#> 3  -1.43 -0.0467 -0.8979 -0.8979
#> 4  -1.37  0.4529 -0.6729 -0.6729
#> 5  -1.35  0.9208 -0.4801 -0.4801
#> 6  -1.32  1.3935 -0.2829 -0.2829
#> 7  -1.30  1.8662 -0.0864 -0.0864
#> 8  -1.28  2.3290  0.0999  0.0999
#> 9  -1.20  2.8602  0.3542  0.3542
#> 10 -1.17  3.3427  0.5592  0.5592
#> 
#> $min_lag
#> AIC BIC  HQ FPE 
#>   3   1   1   1

(var_lag <- 3)
#> [1] 3

fit_var <- var_lm(etf_train, var_lag)

VHAR

fit_vhar <- vhar_lm(etf_train)

BVAR

(bvar_lag <- var_lag)
#> [1] 3

n_asset <- ncol(etf_train)
bvar_init <- set_bvar(
  sigma = apply(etf_train, 2, sd),
  lambda = .2,
  delta = rep(.1, n_asset)
)

(bvar_optim <- choose_bvar(
  bvar_init, 
  lower = c(
    rep(1, n_asset), # sigma
    1e-4, # lambda
    rep(1e-2, n_asset) # delta
  ), 
  upper = c(
    rep(15, n_asset), # sigma
    Inf, # lambda
    rep(1, n_asset) # delta
  ), 
  y = etf_train, 
  p = bvar_lag, 
  include_mean = TRUE,
  parallel = list(cl = cl, forward = FALSE, loginfo = FALSE)
))
#> Model Specification for BVAR
#> 
#> Parameters: Coefficent matrice and Covariance matrix
#> Prior: Minnesota
#> # Type '?bvar_minnesota' in the console for some help.
#> ========================================================
#> 
#> Setting for 'sigma':
#>   GVZ    OVX  VXFXI  VXEEM  VXSLV    EVZ  VXXLE  VXGDX  VXEWZ  
#>  1.89   6.11   2.49   1.80   4.23   1.30   1.94   5.47   5.74  
#> 
#> Setting for 'lambda':
#>         
#> 0.0279  
#> 
#> Setting for 'delta':
#>                                                                
#> 0.895  0.976  0.931  0.947  0.881  0.966  0.965  0.943  0.970  
#> 
#> Setting for 'eps':
#> [1]  1e-04

fit_bvar <- bvar_optim$fit

BVHAR-S

bvhar_init <- set_bvhar(
  sigma = apply(etf_train, 2, sd),
  lambda = .2,
  delta = rep(.1, n_asset)
)

(bvhar_var_optim <- choose_bvhar(
  bvhar_init, 
  lower = c(
    rep(1, n_asset), # sigma
    1e-4, # lambda
    rep(1e-2, n_asset) # delta
  ), 
  upper = c(
    rep(15, n_asset), # sigma
    Inf, # lambda
    rep(1, n_asset) # delta
  ), 
  y = etf_train, 
  har = c(5, 22),
  include_mean = TRUE,
  parallel = list(cl = cl, forward = FALSE, loginfo = FALSE)
))
#> Model Specification for BVHAR
#> 
#> Parameters: Coefficent matrice and Covariance matrix
#> Prior: MN_VAR
#> # Type '?bvhar_minnesota' in the console for some help.
#> ========================================================
#> 
#> Setting for 'sigma':
#>   GVZ    OVX  VXFXI  VXEEM  VXSLV    EVZ  VXXLE  VXGDX  VXEWZ  
#>  2.36   4.34   2.64   2.38   3.65   1.22   2.47   4.71   5.30  
#> 
#> Setting for 'lambda':
#>         
#> 0.0332  
#> 
#> Setting for 'delta':
#>                                                                
#> 0.878  0.973  0.936  0.945  0.861  0.956  0.956  0.938  0.973  
#> 
#> Setting for 'eps':
#> [1]  1e-04

fit_bvhar <- bvhar_var_optim$fit

BVHAR-L

bvhar_vhar_init <- set_weight_bvhar(
  sigma = apply(etf_train, 2, sd),
  lambda = .2,
  daily = rep(.1, n_asset),
  weekly = rep(.1, n_asset),
  monthly = rep(.1, n_asset)
)

(bvhar_vhar_optim <- choose_bvhar(
  bvhar_vhar_init, 
  lower = c(
    rep(1, n_asset), # sigma
    1e-4, # lambda
    rep(1e-2, n_asset), # daily
    rep(1e-2, n_asset), # weekly
    rep(1e-2, n_asset) # monthly
  ), 
  upper = c(
    rep(15, n_asset), # sigma
    Inf, # lambda
    rep(1, n_asset), # daily
    rep(1, n_asset), # weekly
    rep(1, n_asset) # monthly
  ), 
  y = etf_train, 
  har = c(5, 22),
  include_mean = TRUE,
  parallel = list(cl = cl, forward = FALSE, loginfo = FALSE)
))
#> Model Specification for BVHAR
#> 
#> Parameters: Coefficent matrice and Covariance matrix
#> Prior: MN_VHAR
#> # Type '?bvhar_minnesota' in the console for some help.
#> ========================================================
#> 
#> Setting for 'sigma':
#>   GVZ    OVX  VXFXI  VXEEM  VXSLV    EVZ  VXXLE  VXGDX  VXEWZ  
#>  3.07   9.51   3.25   3.48   5.54   1.00   3.82   7.01   6.70  
#> 
#> Setting for 'lambda':
#>        
#> 1e-04  
#> 
#> Setting for 'eps':
#> [1]  1e-04
#> 
#> Setting for 'daily':
#>                                                                
#> 0.811  0.913  0.936  0.864  0.766  0.930  0.908  0.846  0.959  
#> 
#> Setting for 'weekly':
#>                                                                         
#> 0.1634  0.1569  0.0171  0.0987  0.2383  0.0100  0.0100  0.2000  0.0100  
#> 
#> Setting for 'monthly':
#>                                                                         
#> 0.2121  0.0100  0.0737  0.2057  0.2335  0.1662  0.1886  0.1019  0.0376

fit_bvhar_vhar <- bvhar_vhar_optim$fit

parallel::stopCluster(cl)

Hyperparamers

spec_list <- list(
  bvar = bvar_optim$spec,
  bvhar_s = bvhar_var_optim$spec,
  bvhar_l = bvhar_vhar_optim$spec
)

report_hyperparam(spec_list = spec_list, report_true = TRUE, caption = "Hyperparameter Selection", label = "analysis-emp") %>% 
  writeLines()
\begin{table}

\caption{\label{tab:analysis-emp}Hyperparameter Selection}
\centering
\resizebox{\linewidth}{!}{
\begin{tabular}[t]{llrrrrrrrrr}
\toprule
 &  & GVZ & OVX & VXFXI & VXEEM & VXSLV & EVZ & VXXLE & VXGDX & VXEWZ\\
\midrule
\addlinespace[0.3em]
\multicolumn{11}{l}{\textbf{BVAR}}\\
\hspace{1em} & $\sigma_j$ & 1.8924 & 6.107 & 2.4877 & 1.7998 & 4.226 & 1.301 & 1.936 & 5.474 & 5.7440\\

\hspace{1em} & $\lambda$ & 0.0279 &  &  &  &  &  &  &  & \\

\hspace{1em} & $\delta_j$ & 0.8953 & 0.976 & 0.9306 & 0.9475 & 0.881 & 0.966 & 0.965 & 0.944 & 0.9703\\
\cmidrule{1-11}
\addlinespace[0.3em]
\multicolumn{11}{l}{\textbf{BVHAR-S}}\\
\hspace{1em} & $\sigma_j$ & 2.3644 & 4.341 & 2.6413 & 2.3780 & 3.646 & 1.223 & 2.469 & 4.712 & 5.2994\\

\hspace{1em} & $\lambda$ & 0.0332 &  &  &  &  &  &  &  & \\

\hspace{1em} & $\delta_j$ & 0.8784 & 0.973 & 0.9357 & 0.9454 & 0.861 & 0.956 & 0.956 & 0.938 & 0.9729\\
\cmidrule{1-11}
\addlinespace[0.3em]
\multicolumn{11}{l}{\textbf{BVHAR-L}}\\
\hspace{1em} & $\sigma_j$ & 3.0711 & 9.511 & 3.2453 & 3.4786 & 5.544 & 1.000 & 3.818 & 7.012 & 6.6986\\

\hspace{1em} & $\lambda$ & 0.0001 &  &  &  &  &  &  &  & \\

\hspace{1em} & $d_j$ & 0.8108 & 0.913 & 0.9362 & 0.8638 & 0.766 & 0.930 & 0.908 & 0.846 & 0.9587\\

\hspace{1em} & $w_j$ & 0.1634 & 0.157 & 0.0171 & 0.0987 & 0.238 & 0.010 & 0.010 & 0.200 & 0.0100\\

\hspace{1em} & $m_j$ & 0.2121 & 0.010 & 0.0737 & 0.2057 & 0.234 & 0.166 & 0.189 & 0.102 & 0.0376\\
\bottomrule
\end{tabular}}
\end{table}

Errors

mod_list <- list(
  fit_var,
  fit_vhar,
  fit_bvar,
  fit_bvhar,
  fit_bvhar_vhar
)
# 1-step-----------
roll1 <- 
  mod_list %>% 
  parallel::mclapply(
    function(mod) {
      forecast_roll(mod, 1, etf_test)
    },
    mc.cores = 4
  )
# 5-step-----------
roll2 <- 
  mod_list %>% 
  parallel::mclapply(
    function(mod) {
      forecast_roll(mod, 5, etf_test)
    },
    mc.cores = 4
  )
# 20-step----------
roll3 <- 
  mod_list %>% 
  parallel::mclapply(
    function(mod) {
      forecast_roll(mod, 20, etf_test)
    },
    mc.cores = 4
  )

roll_list <- 
  parallel::mclapply(
    c(1, 5, 20),
    function(h) {
      mod_list %>% 
        lapply(
          function(mod) {
            forecast_roll(mod, h, etf_test)
          }
        )
    },
    mc.cores = 8
  )

Relative Errors

get_rmafetex_tr_2(
  roll_list, 
  etf_test, 
  ahead_list = c("$h = 1$", "$h = 5$", "$h = 20$"), 
  benchmark_id = 1,
  caption = "Out-of-sample forecasting performance measures with VAR(3) as benchmark", 
  label = "losscboe"
) %>% 
  writeLines()
#> \begin{table}[H]
#> 
#> \caption{\label{tab:losscboe}Out-of-sample forecasting performance measures with VAR(3) as benchmark}
#> \centering
#> \resizebox{\linewidth}{!}{
#> \begin{tabular}[t]{c|ccc|ccc|ccc|ccc|}
#> \toprule
#> \multicolumn{1}{c}{ } & \multicolumn{3}{c}{RMAFE} & \multicolumn{3}{c}{RMSFE} & \multicolumn{3}{c}{RMAPE} & \multicolumn{3}{c}{RMASE} \\
#> \cmidrule(l{3pt}r{3pt}){2-4} \cmidrule(l{3pt}r{3pt}){5-7} \cmidrule(l{3pt}r{3pt}){8-10} \cmidrule(l{3pt}r{3pt}){11-13}
#>  & $h = 1$ & $h = 5$ & $h = 20$ & $h = 1$ & $h = 5$ & $h = 20$ & $h = 1$ & $h = 5$ & $h = 20$ & $h = 1$ & $h = 5$ & $h = 20$\\
#> \midrule
#> VHAR & \textcolor{black}{\num{.964}} & \textcolor{black}{\num{.895}} & \textcolor{black}{\num{.734}} & \textcolor{black}{\num{.943}} & \textcolor{black}{\num{.799}} & \textcolor{black}{\num{.552}} & \textcolor{black}{\num{.970}} & \textcolor{black}{\num{.891}} & \textcolor{black}{\num{.744}} & \textcolor{black}{\num{.958}} & \textcolor{black}{\num{.875}} & \textcolor{black}{\num{.737}}\\
#> \cmidrule{1-13}
#> BVAR & \textcolor{black}{\num{.943}} & \textcolor{black}{\num{.830}} & \textcolor{black}{\num{.703}} & \textcolor{black}{\num{.916}} & \textcolor{black}{\num{.737}} & \textcolor{black}{\num{.494}} & \textcolor{black}{\num{.945}} & \textcolor{black}{\num{.811}} & \textcolor{black}{\num{.718}} & \textcolor{black}{\num{.932}} & \textcolor{black}{\num{.806}} & \textcolor{black}{\num{.710}}\\
#> \cmidrule{1-13}
#> BVHAR-S & \textcolor{black}{\num{.945}} & \textcolor{black}{\num{.828}} & \textcolor{black}{\num{.681}} & \textcolor{black}{\num{.915}} & \textcolor{black}{\num{.731}} & \textcolor{black}{\num{.457}} & \textcolor{black}{\num{.947}} & \textcolor{black}{\num{.812}} & \textcolor{black}{\num{.701}} & \textcolor{black}{\num{.934}} & \textcolor{black}{\num{.806}} & \textcolor{black}{\num{.688}}\\
#> \cmidrule{1-13}
#> BVHAR-L & \textcolor{red}{\num{.937}} & \textcolor{red}{\num{.798}} & \textcolor{red}{\num{.538}} & \textcolor{red}{\num{.880}} & \textcolor{red}{\num{.679}} & \textcolor{red}{\num{.300}} & \textcolor{red}{\num{.935}} & \textcolor{red}{\num{.773}} & \textcolor{red}{\num{.531}} & \textcolor{red}{\num{.918}} & \textcolor{red}{\num{.787}} & \textcolor{red}{\num{.540}}\\
#> \bottomrule
#> \end{tabular}}
#> \end{table}

get_rmfe_tr(
  roll_list,
  etf_test,
  ahead_list = c("$h = 1$", "$h = 5$", "$h = 20$"),
  benchmark_id = 1
)
#>           VHAR  BVAR BVHAR-S BVHAR-L
#> $h = 1$  0.964 0.943   0.945   0.937
#> $h = 5$  0.895 0.830   0.828   0.798
#> $h = 20$ 0.734 0.703   0.681   0.538

get_rmfe_tr(
  roll_list,
  etf_test,
  ahead_list = c("$h = 1$", "$h = 5$", "$h = 20$"),
  benchmark_id = 1,
  error = "rmsfe"
)
#>           VHAR  BVAR BVHAR-S BVHAR-L
#> $h = 1$  0.943 0.916   0.915   0.880
#> $h = 5$  0.799 0.737   0.731   0.679
#> $h = 20$ 0.552 0.494   0.457   0.300

get_rmfe_tr(
  roll_list,
  etf_test,
  ahead_list = c("$h = 1$", "$h = 5$", "$h = 20$"),
  benchmark_id = 1,
  error = "mape"
)
#>           VHAR  BVAR BVHAR-S BVHAR-L
#> $h = 1$  0.970 0.945   0.947   0.935
#> $h = 5$  0.891 0.811   0.812   0.773
#> $h = 20$ 0.744 0.718   0.701   0.531

get_rmfe_tr(
  roll_list,
  etf_test,
  ahead_list = c("$h = 1$", "$h = 5$", "$h = 20$"),
  benchmark_id = 1,
  error = "mase"
)
#>           VHAR  BVAR BVHAR-S BVHAR-L
#> $h = 1$  0.958 0.932   0.934   0.918
#> $h = 5$  0.875 0.806   0.806   0.787
#> $h = 20$ 0.737 0.710   0.688   0.540

Intervals

# predict-----------------------
pred_var <- predict(fit_var, h)
pred_vhar <- predict(fit_vhar, h)
pred_bvar <- predict(fit_bvar, h)
pred_bvhar <- predict(fit_bvhar, h)
pred_bvhar_vhar <- predict(fit_bvhar_vhar, h)
# model name---------------------
bayesmod_name <- c("BVAR_Minnesota" = "BVAR", "BVHAR_MN_VAR" = "BVHAR-S", "BVHAR_MN_VHAR" = "BVHAR-L", "VAR" = "VAR", "VHAR" = "VHAR")

interval_plt <- 
  autoplot(pred_var, x_cut = 860, ci_alpha = .8, type = "wrap") +
  autolayer(pred_vhar, ci_alpha = .7) +
  autolayer(pred_bvar, ci_alpha = .6) +
  autolayer(pred_bvhar, ci_alpha = .5) +
  autolayer(pred_bvhar_vhar, ci_alpha = .4) +
  geom_eval(etf_test, num_train = nrow(etf_train), colour = "#000000", alpha = .5) +
  theme_minimal() +
  theme(
    panel.border = element_rect(fill = NA),
    axis.text.x = element_blank(),
    legend.position = "top",
    text = element_text(family = "serif")
  ) +
  scale_color_discrete(labels = bayesmod_name) +
  scale_fill_discrete(labels = bayesmod_name)
interval_plt

bayesmod_plt <- 
  autoplot(pred_bvar, x_cut = 860, ci_alpha = .8, type = "wrap") +
  autolayer(pred_bvhar, ci_alpha = .7) +
  autolayer(pred_bvhar_vhar, ci_alpha = .6) +
  geom_eval(etf_test, num_train = nrow(etf_train), colour = "#000000", alpha = .5) +
  theme_minimal() +
  theme(
    panel.border = element_rect(fill = NA),
    axis.text.x = element_blank(),
    legend.position = "top",
    text = element_text(family = "serif")
  ) +
  scale_color_discrete(labels = bayesmod_name) +
  scale_fill_discrete(labels = bayesmod_name)
bayesmod_plt

# Every result---------------------------
ggsave(
  filename = "../output/figs/analysis-credplot.pdf", 
  plot = interval_plt,
  device = "pdf",
  width = fig_width, 
  height = .618 * fig_width, 
  units = "cm",
  dpi = 1500,
  limitsize = FALSE
)
# Bayesian result-----------------------
ggsave(
  filename = "../output/figs/analysis-bayespred.pdf", 
  plot = bayesmod_plt,
  device = "pdf",
  width = fig_width, 
  height = .618 * fig_width, 
  units = "cm",
  dpi = 1500,
  limitsize = FALSE
)

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analysis.md

analysis.md

Empirical Analysis

Data

Split

Long-range dependency

Plot

Model

VAR

VHAR

BVAR

BVHAR-S

BVHAR-L

Hyperparamers

Errors

Relative Errors

Intervals

Files

analysis.md

Latest commit

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analysis.md

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Empirical Analysis

Data

Split

Long-range dependency

Plot

Model

VAR

VHAR

BVAR

BVHAR-S

BVHAR-L

Hyperparamers

Errors

Relative Errors

Intervals