Modelling conditional volatilities and correlations among multivariate time series in R
The package 'BEKKs' contains functions and methods for a careful analysis, estimation and forecasting of financial asset returns dynamics, and the construction and evaluation of financial portfolios. Modelling correlations and covariances is important to determine portfolios with focus on hedging and asset specialization strategies, as well as to forecast value-at-risk (VaR) thresholds.
Below find a schematic overview of the package functions and methods.
The main function of the package is bekk_spec() which determines the behavior of the subsequent methods with the most important specifications:
bekk_spec(
model = list(
type = "bekk",
asymmetric = FALSE
)
)The model type by default is a symmetric BEKK(1,1) according to Engle and Kroner (1995). Alternative specifications are the diagonal BEKK (type = "dbekk") and scalar BEKK (type = "sbekk"). All types can be specified as either symmetrical or asymmetrical models.
install.packages("BEKKs")Alternatively, install the development version
install.packages("devtools")
devtools::install_github("alexanderlange53/BEKKs")library("BEKKs")To get started, use the example data set which is included in the package. The data set consists of two daily financial time series, i.e. S&P 500 bond returns and MSCI World returns. More details on the data set are provided in the description file ?StocksBonds.
library("ggplot2")
library("ggfortify")
autoplot(StocksBonds , facet = TRUE) + theme_bw()
We estimate the conditional deviations and correlation processes via the symmetric BEKK(1,1) model
s1 <- bekk_spec()
m1 <- bekk_fit(s1, StocksBonds)
summary(m1)
# BEKK estimation results
# -----------------------
# Log-likelihood: -7382.256
# BEKK model stationary: TRUE
# Number of BHHH iterations: 15
# AIC: 14783.51
# BIC: 14794.65
# Estimated paramater matrices:
#
# C
# [,1] [,2]
# [1,] 0.01978368 -0.01496272
# [2,] 0.00000000 0.08526574
#
# A
# [,1] [,2]
# [1,] 0.178859630 -0.03715201
# [2,] -0.007691659 0.29814734
#
# G
# [,1] [,2]
# [1,] 0.981184605 0.01326038
# [2,] 0.001337367 0.95143161
#
# t-values of paramater matrices:
#
# C
# [,1] [,2]
# [1,] 8.979949 1.21244
# [2,] 0.000000 13.41001
#
# A
# [,1] [,2]
# [1,] 29.17318 1.640942
# [2,] 3.55672 42.056807
#
# G
# [,1] [,2]
# [1,] 693.873974 2.099117
# [2,] 2.154099 409.769546The summary includes general information on the estimation (see ?bekk_fit), the estimated parameter matrices C, A and G and the corresponding t-values. The estimated volatility and covariance processes can be shown with plot(m1).
The estimated conditional correlations can be used for risk management purposes, for example, to calculate VaR at the 1% nominal coverage level in accordance with Basel regulations.
v1 <- VaR(m1)
plot(v1)
Alternatively, we can calculate the VaR of a typical portfolio consisting of 30% bonds and 70% stocks:
portfolio_weights <- c(0.3, 0.7)
v2 <- VaR(m1, portfolio_weights = portfolio_weights)
plot(v2)
To predict the future VaR of the 30/70 portfolio, for example for the next two days, we can use the function bekk_forceast and calculate the VaR again.
f1 <- bekk_forecast(m1, n.ahead = 2)
v3 <- VaR(f1, portfolio_weights = portfolio_weights)
plot(v3)
It is possible to perform backtests with the 'BEKKs' package to evaluate the model specification through rolling window fitting and forecasting, e.g., using the time period from January 2018 to January 2019 as testing period.
b1 <- backtest(m1, window_length = 5800, portfolio_weights = portfolio_weights, n.ahead = 2)
plot(b1)
The summary includes information about the hit rate and traditional backtesting risk measures summary(b1).
# BEKK backtesting results
# ------------------------
# Value-at-risk confidence level: 0.99
# Window length: 5800
# Portfolio weights: 0.3 0.7
# ------------------------
# Hit rate: 0.004
#
# Unconditional coverage test of Kupiec:
#
# Test: 1.4624470
# p-value: 0.2265412
#
# conditional coverage test of Christoffesen:
#
# Test: 1.4698271
# p-value: 0.4795469