The famous fat tails characteristic of stock returns. https://www.nasdaq.com/articles/fat-tail-risk-what-it-means-and-why-you-should-be-aware-it-2015-11-02

fattails.R

library(tidyverse)
library(quantmod)
library(gamlss)
select <- dplyr::select
posix <- function(x) { as.POSIXct(x, origin = "1970-01-01") }

## "Fat tails"
## Here we compare the residuals from the normal and t distributional models.
## Notice standardized error is worse in the normal model. This happens
## because returns are leptokurtic (large surprises should be expected, there's risk in stock returns),
#  https://www.nasdaq.com/articles/fat-tail-risk-what-it-means-and-why-you-should-be-aware-it-2015-11-02

getSymbols("SHOP", src = "yahoo", from = "2017-01-01", to = "2020-12-01", auto.assign = FALSE) -> SHOP

SHOP %>%
  as_tibble() %>%
  mutate(date = posix(attributes(SHOP)$index)) %>%
  arrange(date) %>%
  mutate(first_difference = log(SHOP.Close) - log(lag(SHOP.Close))) %>%
  filter(!is.na(first_difference)) -> SHOP

gamlss(first_difference ~ 1, data = SHOP) -> nfit
gamlss(first_difference ~ 1, data = SHOP, family = "TF", method = mixed(20, 20)) -> tfit

tibble(normal = residuals(nfit), t = residuals(tfit)) %>%
  pivot_longer(cols = c(normal, t)) %>%
  ggplot(aes(x = value, color = factor(name))) +
  stat_ecdf() +
  geom_point(stat = "ecdf") +
  geom_vline(xintercept = 0) +
  coord_cartesian(ylim = c(0, 0.05), xlim = c(-8, 0)) +
  ggtitle("Lower tail of residuals") +
  xlab("Scaled residuals") +
  ylab("Quantiles") +
  ggthemes::scale_color_colorblind(name = "Model") +
  theme_bw() +
  theme(axis.text = element_text(color = "black", size = 20))


plot(pacf(SHOP$first_difference))
Box.test (SHOP$first_difference, lag = 1, type = "Ljung")
plot(acf(SHOP$first_difference))

library(rugarch)
spec <- ugarchspec(variance.model = list(garchOrder = c(1, 1)))
fit <- ugarchfit(spec = spec, data = SHOP$first_difference, out.sample = 1)
show(fit)
plot(fit)