BEKKs: Volatility modelling

Description

This package implements estimation, simulation and forecasting techniques for conditional volatility modelling using the BEKK model. We refer the reader to Fülle et al. (2024) for a package overview. The full BEKK(1,1,1) model of Engle and Kroner (1995) \[H_t = CC' + A' r_{t-1} r_{t-1}'A + G' H_{t-1}G ,\] the asymmetric extensions of Kroner and Ng (1998) and Grier et. al. (2004) \[H_t = CC' + A' r_{t-1} r_{t-1}'A +B'\gamma_{t-1} \gamma_{t-1}' B+G'H_{t-1}G\] with \[\gamma_t = r_t I\left(r_t < 0 \right)\] are implemented. Moreover, the diagonal BEKK, where the parameter matrices A, B and G are reduced to diagonal matrices and the scalar BEKK model of Ding and Engle (2001) \[H_t = CC' + a r_{t-1} r_{t-1}' + g H_{t-1},\] where a and g are scalar parameters and are implemented to allow faster but less flexible estimation in higher dimensions.

Details

The main functions are:

• bekk_spec

  Specifies the model type to be estimated.

• bekk_fit

  Estimates a BEKK(1,1,1) model of a given series and specification object bekk_spec.

• simulate

  Simulates a BEKK(1,1,1) process using either a bekk_fit or bekk_spec object.

• predict

  Forecasts conditional volatility using a bekk_fit object.

• VaR	Estimates (portfolio) Value-at-Risk using a fitted BEKK(1,1,1) model.

• backtest

  Backtesting estimated (portfolio) value-at-risks of a fitted BEKK(1,1,1) model.

• virf	Calculates volatility impulse response functions for fitted symmetric BEKK(1,1,1) models.

Author(s)

• Markus J. Fülle fuelle@uni-goettingen.de

• Helmut Herwartz hherwartz@uni-goettingen.de

• Alexander Lange alexander.lange@uni-goettingen.de

• Christian M. Hafner christian.hafner@uclouvain.be

References

Engle, R. F. and K. F. Kroner (1995). Multivariate simultaneous generalized arch. Econometric Theory 11(1),122-150.

Fülle, M. J., A. Lange, C. M. Hafner, and H. Herwartz (2024). BEKKs: An R package for estimation of conditional volatility of multivariate time series. Journal of Statistical Software 111 (4), 1–34. doi:10.18637/jss.v111.i04.

Kroner, K. F. and V. K. Ng (1998). Modeling asymmetric comovements of asset returns. Review of Financial Studies 11(4), 817-44.

Ding, Zhuanxin and Engle, Robert F (2001). Large scale conditional covariance matrix modeling, estimation and testing. NYU working paper No. Fin-01-029.

Grier, K. B., Olan T. Henry, N. Olekalns, and K. Shields (2004). The asymmetric effects of uncertainty on inflation and output growth. Journal of Applied Econometrics 19(5), 551-565.

Hafner CM, Herwartz H (2006). Volatility impulse responses for multivariate GARCH models: An exchange rate illustration. Journal of International Money and Finance,25,719-740.

Gold stock and Bond returns

Description

Trivariate data set consisting of daily gold, S&P 500 and U.S. Treasury Bond Future returns from October 1991 to October 2021.

Usage

data("GoldStocksBonds")

Format

A time series matrix of class mts 7346 observations on the following 3 variables.

Gold

a numeric vector

S&P 500

a numeric vector

US Treasury Bond Future

a numeric vector

Source

Yahoo Finance.

Examples

data(GoldStocksBonds)
## maybe str(GoldStocksBonds) ; plot(GoldStocksBonds) ...

Daily stock and Bond returns

Description

Bivariate data set consisting of daily S&P 500 bond and MSCI World returns from December 1995 to December 2019.

Usage

data("StocksBonds")

Format

A time series matrix of class mts with 6073 observations on the following 2 variables.

S&P 500 Bonds

a numeric vector

MSCI World

a numeric vector

Source

Yahoo Finance.

Examples

data(StocksBonds)
## maybe str(StocksBonds) ; plot(StocksBonds) ...

Calculating Value-at-Risk (VaR)

Description

Method for calculating VaR from estimated covariance processes (bekk_fit) or predicted covariances (predict).

Usage

VaR(x, p = 0.99, portfolio_weights = NULL, distribution = "empirical")

Arguments

Value

Returns a S3 class "var" object containing the VaR forecast and respective confidence bands.

Examples

data(StocksBonds)
obj_spec <- bekk_spec()
x1 <- bekk_fit(obj_spec, StocksBonds, QML_t_ratios = FALSE, max_iter = 50, crit = 1e-9)

# single VaRs of series
x2 <- VaR(x1, distribution="normal")
plot(x2)

# VaR of equally-weighted portfolio
portfolio_weights <- c(0.5, 0.5)
x3 <- VaR(x1, portfolio_weights = portfolio_weights)
plot(x3)

# VaR of traditional 30/70 weighted bond and stock portfolio
portfolio_weights <- c(0.3, 0.7)
x4 <- VaR(x1, portfolio_weights = portfolio_weights)
plot(x4)

Backtesting via Value-at-Risk (VaR)

Description

Method for backtesting a model obtained from bekk_fit in terms of VaR-forcasting accuracy using a rolling window approach.

Usage

backtest(
  x,
  window_length = 1000,
  p = 0.99,
  portfolio_weights = NULL,
  n.ahead = 1,
  distribution = "empirical",
  nc = 1
)

Arguments

Value

Returns a S3 class "backtest" object containing the VaR forecast, out-of-sample returns and backtest statistics according to the R-package "GAS". conf

Examples

data(StocksBonds)
obj_spec <- bekk_spec()
x1 <- bekk_fit(obj_spec, StocksBonds, QML_t_ratios = FALSE, max_iter = 50, crit = 1e-9)

# backtesting
x2 <- backtest(x1, window_length = 6000, n.ahead = 1, nc = 1)
plot(x2)
# backtesting using 5 day-ahead forecasts
x3 <- backtest(x1, window_length = 6000, n.ahead = 5, nc = 1)
plot(x3)
# backtesting using 20 day-ahead forecasts and portfolio
x4 <- backtest(x1, window_length = 6000, portfolio_weights = c(0.5,0.5), n.ahead = 20, nc = 1)
plot(x4)

Estimating multivariate BEKK-type volatility models

Description

Method for fitting a variety of N-dimensional BEKK models.

Usage

bekk_fit(spec, data, QML_t_ratios = FALSE, max_iter = 50, crit = 1e-09)

Arguments

Details

The BEKK optimization routine is based on the Berndt–Hall–Hall–Hausman (BHHH) algorithm and is inspired by the study of Hafner and Herwartz (2008). The authors provide analytical formulas for the score and Hessian of several MGARCH models in a QML framework and show that analytical derivations significantly outperform numerical methods.

Value

Returns a S3 class "bekkFit" object containing the estimated parameters, t-values, standard errors and volatility process of the model defined by the BEKK_spec object.

References

Hafner and Herwartz (2008). Analytical quasi maximum likelihood inference in multivariate volatility models. Metrika, 67, 219-239.

Fülle, M. J., A. Lange, C. M. Hafner, and H. Herwartz (2024). BEKKs: An R package for estimation of conditional volatility of multivariate time series. Journal of Statistical Software 111 (4), 1–34. <doi:10.18637/jss.v111.i04>.

Examples

data(StocksBonds)

# Fitting a symmetric BEKK model
obj_spec <- bekk_spec()
x1 <- bekk_fit(obj_spec, StocksBonds, QML_t_ratios = FALSE, max_iter = 50, crit = 1e-9)

summary(x1)

plot(x1)

# Fitting an asymmetric BEKK model
obj_spec <- bekk_spec(model = list(type = "bekk", asymmetric = TRUE))
x1 <- bekk_fit(obj_spec, StocksBonds)

summary(x1)

plot(x1)

# Fitting a symmetric diagonal BEKK model
obj_spec <- bekk_spec(model = list(type = "dbekk", asymmetric = FALSE))
x1 <- bekk_fit(obj_spec, StocksBonds, QML_t_ratios = FALSE, max_iter = 50, crit = 1e-9)

summary(x1)

plot(x1)


# Fitting a symmetric scalar BEKK model
obj_spec <- bekk_spec(model = list(type = "sbekk", asymmetric = FALSE))
x1 <- bekk_fit(obj_spec, StocksBonds, QML_t_ratios = FALSE, max_iter = 50, crit = 1e-9)

summary(x1)

plot(x1)

BEKK specification method

Description

Method for creating a N-dimensional BEKK model specification object prior to fitting and/or simulating.

Usage

bekk_spec(
  model = list(type = "bekk", asymmetric = FALSE),
  init_values = NULL,
  signs = NULL,
  N = NULL
)

Arguments

Value

Returns a S3 class "bekkSpec" object containing the specifications of the model to be estimated.

Examples

data(StocksBonds)

# Fitting a symmetric BEKK model using default starting values
# - i.e. fixed values
obj_spec_fixed <- bekk_spec(init_values = NULL)
x1 <- bekk_fit(obj_spec_fixed, StocksBonds, QML_t_ratios = FALSE,
max_iter = 50, crit = 1e-9)
# Fitting a symmetric BEKK model using initial values originating from a
# random grid search algorithm
obj_spec_random <- bekk_spec(init_values = "random")
x2 <- bekk_fit(obj_spec_random, StocksBonds, QML_t_ratios = FALSE,
max_iter = 50, crit = 1e-9)
summary(x1)
summary(x2)
plot(x1)
plot(x2)
# Fitting an asymmetric BEKK model with default starting values
obj_spec_fix <- bekk_spec(model = list(type = "bekk", asymmetric = TRUE),
init_values = NULL)
x1 <- bekk_fit(obj_spec_fix, StocksBonds)
obj_spec_random <- bekk_spec(model = list(type = "bekk", asymmetric = TRUE),
init_values = "random")
x2 <- bekk_fit(obj_spec_random, StocksBonds)
summary(x1)
summary(x2)

Performing a Portmanteau test checking for remaining correlation in the empirical co-variances of the estimated BEKK residuals.

Description

Method for a Portmanteau test of the null hypothesis of no remaining correlation in the co-variances of the estimated BEKK residuals.

Usage

portmanteau.test(x, lags = 5)

Arguments

Details

Here, the multivariate Portmanteau test of Hosking (1980) is implemented.

Value

Returns an Object of class "htest" containing the p-value and test statistic.

References

J. R. M. Hosking (1980). The Multivariate Portmanteau Statistic, Journal of the American Statistical Association, 75:371, 602-608.

Forecasting conditional volatilities with BEKK models

Description

Method for predicting a N-dimensional BEKK covariances.

Usage

## S3 method for class 'bekk'
predict(object, n.ahead = 1, ci = 0.95, ...)

## S3 method for class 'bekka'
predict(object, n.ahead = 1, ci = 0.95, ...)

## S3 method for class 'dbekk'
predict(object, n.ahead = 1, ci = 0.95, ...)

## S3 method for class 'dbekka'
predict(object, n.ahead = 1, ci = 0.95, ...)

## S3 method for class 'sbekk'
predict(object, n.ahead = 1, ci = 0.95, ...)

## S3 method for class 'sbekka'
predict(object, n.ahead = 1, ci = 0.95, ...)

Arguments

Value

Returns a S3 class "bekkForecast" object containing the conditional volatility forecasts and respective confindence bands.

Examples

#'
data(StocksBonds)
obj_spec <- bekk_spec()
x1 <- bekk_fit(obj_spec, StocksBonds, QML_t_ratios = FALSE, max_iter = 50, crit = 1e-9)

x2 <- predict(x1, n.ahead = 1)

bekkFit method

Description

Generic 'bekkFit' methods. More details on 'bekkFit' are described in bekk_fit

Usage

## S3 method for class 'bekkFit'
print(x, ...)

## S3 method for class 'bekkFit'
residuals(object, ...)

## S3 method for class 'bekkFit'
logLik(object, ..., k = 2)

Arguments

Examples

data(StocksBonds)

# Fitting a symmetric BEKK model
obj_spec <- bekk_spec()
x1 <- bekk_fit(obj_spec, StocksBonds, QML_t_ratios = FALSE, max_iter = 50, crit = 1e-9)

logLik(x1)

Simulating BEKK models

Description

Method for simulating a N-dimensional BEKK model.

Usage

## S3 method for class 'bekk'
simulate(object, nsim, ...)

## S3 method for class 'bekka'
simulate(object, ..., nsim)

## S3 method for class 'dbekk'
simulate(object, ..., nsim)

## S3 method for class 'dbekka'
simulate(object, ..., nsim)

## S3 method for class 'sbekk'
simulate(object, ..., nsim)

## S3 method for class 'sbekka'
simulate(object, ..., nsim)

Arguments

Value

Returns a simulated time series S3 class object using the parameters of passed "bekkSpec" or "bekkFit".

Examples

# simulate a BEKK with estimated parameter
obj_spec <- bekk_spec()
x1 <- bekk_fit(obj_spec, StocksBonds)

x2 <- simulate(x1, nsim = 3000)

plot(x2)

Estimating multivariate volatility impulse response functions (VIRF) for BEKK models

Description

Method for estimating VIRFs of N-dimensional BEKK models. Currently, only VIRFs for symmetric BEKK models are implemented.

Usage

virf(
  x,
  time = 1,
  q = 0.05,
  index_series = 1,
  n.ahead = 10,
  ci = 0.9,
  time_shock = FALSE
)

Arguments

Value

Returns an object of class "virf".

References

Hafner CM, Herwartz H (2006). Volatility impulse responses for multivariate GARCH models: An exchange rate illustration. Journal of International Money and Finance,25,719–740.

Examples

data(StocksBonds)
obj_spec <- bekk_spec()
x1 <- bekk_fit(obj_spec, StocksBonds, QML_t_ratios = FALSE, max_iter = 50, crit = 1e-9)

# 250 day ahead VIRFs and 90% CI for a Shock in the 1% quantile  of Bonds (i.e. series=2)
# shock is supposed to occur at day 500
x2 <- virf(x1, time = 500, q = 0.01, index_series=2, n.ahead = 500, ci = 0.90)
plot(x2)