In ph-rast/bmgarch: Bayesian Multivariate GARCH Models

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bmgarch

bmgarch estimates Bayesian multivariate generalized autoregressive conditional heteroskedasticity (MGARCH) models. Currently, bmgarch supports a variety of MGARCH(P,Q) parameterizations and simultaneous estimation of ARMA(1,1), VAR(1) and intercept-only (Constant) mean structures. In increasing order of complexity:

• CCC(P, Q): Constant Conditional Correlation
• DCC(P, Q): Dynamic Conditional Correlation
• BEKK(P, Q): Baba, Engle, Kraft, and Kroner
• pdBEKK(P, Q): BEKK(P, Q) with positive diagonal constraints

Installation

bmgarch is available on CRAN and can be installed with:

install.packages('bmgarch')

Linux

Linux users may need to install libv8 prior to installing bmgarch. For example, in Ubuntu, run sudo apt install libv8-dev before installing the package from CRAN or github. For those who's distro installs libnode-dev instead of libv8-dev, run install.packages("V8") in R prior to installing bmgarch (during installationrstan looks explicitly for V8).

Development Version

The development version can be installed from GitHub with:

devtools::install_github("ph-rast/bmgarch")

How to cite this package

Please add at least one of the following citations when referring to to this package:

Rast, P., & Martin, S. R. (2021). bmgarch: An R-Package for Bayesian Multivariate GARCH models. Journal of Open Source Software, 6, 3452 - 4354. doi: https://joss.theoj.org/papers/10.21105/joss.03452

Rast, P., Martin, S. R., Liu, S., & Williams, D. R. (2022). A New Frontier for Studying Within-Person Variability: Bayesian Multivariate Generalized Autoregressive Conditional Heteroskedasticity Models. Psychological Methods, 27, 856--873. https://doi.apa.org/10.1037/met0000357; Preprint-doi: https://psyarxiv.com/j57pk

Examples:

We present two examples, one with behavioral data and one with stocks from three major Japanese automakers.

Example 1: Behavioral Data

In this example, we use the pdBEKK(1,1) model for the variances, and an intercept-only model for the means.

library(bmgarch)

data(panas)
head(panas)

## Fit pdBEKK(1, 1) with ARMA(1,1) on the mean structure.
fit <- bmgarch(panas,
               parameterization = "pdBEKK",
               iterations = 1000,
               P = 1, Q = 1,
               distribution = "Student_t",
               meanstructure = "arma")

Parameter estimates

summary(fit)

Forecasted values

fit.fc <- forecast(fit, ahead = 5)

fit.fc

plot(fit.fc, askNewPage = FALSE, type = "var")

plot(fit.fc, askNewPage = FALSE, type = "cor")

Example 2: Stocks

Here we use the first 100 days (we only base our analyses on 100 days to reduce wait time -- this is not meant to be a serious analysis) of Stata's stocks data on daily returns of three Japanese automakers, Toyota, Nissan, and Honda.

library(bmgarch)

data(stocks)
head(stocks)

Ease computation by first standardizing the time series

stocks.z <- scale(stocks[,c("toyota", "nissan", "honda")])
head(stocks.z )

# Fit CCC(1, 1) with constant on the mean structure.
fit1 <- bmgarch(stocks.z[1:100, c("toyota", "nissan", "honda")],
                parameterization = "CCC",
                iterations = 1000,
                P = 1, Q = 1,
                distribution = "Student_t",
                meanstructure = "constant")

Parameter Estimates

summary( fit1 )

Forecasted Values

Forecast volatility 10 days ahead

fc <- forecast(fit1, ahead = 10 )
fc

plot(fc,askNewPage = FALSE, type = 'var' )

Ensemble Methods

Here we illustrate how to obtain model weights across three models. These weights will be used to compute weighted forecasts, thus, taking into account that we do not have a single best model.

Add two additional models, one with CCC(2,2) and a DCC(1,1)

# Fit CCC(1, 1) with constant on the mean structure.
fit2 <- bmgarch(stocks.z[1:100, c("toyota", "nissan", "honda")],
                parameterization = "CCC",
                iterations = 1000,
                P = 2, Q = 2,
                distribution = "Student_t",
                meanstructure = "constant")

fit3 <- bmgarch(stocks.z[1:100, c("toyota", "nissan", "honda")],
                parameterization = "DCC",
                iterations = 1000,
                P = 1, Q = 1,
                distribution = "Student_t",
                meanstructure = "arma")

The DCC(1,1) model also incorporates an ARMA(1,1) meanstructure. The output will have the according information:

summary( fit3 )
fc <- forecast(fit3, ahead =  10)

plot( fc,askNewPage = FALSE, type =  'mean' ) 

Compute Model Weights

Obtain model weights with either the stacking or the pseudo BMA method. These methods are inherited from the loo package.

First, gather models to a bmgarch_list.

## use bmgarch_list function to collect bmgarch objects
modfits <- bmgarch_list(fit1, fit2, fit3)

Compute model weights with the stacking method (default) and the approximate (default) leave-future-out cross validation (LFO CV). L defines the minimal length of the time series before we start engaging in cross-validation. Eg., for a time series with length 100, L = 50 reserves values 51--100 as the cross-validation sample. Note that the standard is to use the approximate backward method to CV as it results in fewest refits. Exact CV is also available with exact but not encouraged as it results in refitting all CV models.

mw <- model_weights(modfits, L = 50, method = 'stacking')

## Return model weights:
mw

Weighted Forecasting

Use model weights to obtain weighted forecasts. Here we will forecast 5 days ahead.

w_fc <- forecast(modfits, ahead = 5, weights = mw )
w_fc

Plot the weighted forecast. Save plots into a ggplot object and post-process

plt <- plot(w_fc, askNewPage = FALSE, type =  'var' )

library( patchwork )
( plt$honda  + ggplot2::coord_cartesian(ylim = c(0, 2.5 ) ) ) /
( plt$toyota + ggplot2::coord_cartesian(ylim = c(0, 2.5 ) ) ) /
( plt$nissan + ggplot2::coord_cartesian(ylim = c(0, 2.5 ) ) ) 

Predictors for Constant Variance (C)

We can add predictors for the constant variance term, c or C, in the MGARCH model with the option xC = The predictors need to be of the same dimension as the time-series object. For example, with three time-series of length 100, the predictor needs to be entered as a 100 by 3 matrix as well.

To illustrate, we will add nissan as the predictor for C in a bivariate MGARCH:

# Fit CCC(1, 1) with constant on the mean structure.
fitx <- bmgarch(stocks.z[1:100, c("toyota", "honda")],
                xC = stocks.z[1:100, c("nissan", "nissan")],
                parameterization = "CCC",
                iterations = 1000,
                P = 2, Q = 2,
                distribution = "Student_t",
                meanstructure = "constant")

The estimates for the predictors for C are on a log scale in section Exogenous predictor:

summary(fitx)

The predictor results in a linear model (on the log scale) with an intercept $\beta_0$ and the effect of the predictor in the slope $\beta_1$.

We can generate forecasts given the known values of the predictor. Note that the dimension of the predictor needs to match the number of timepoints that we predict ahead and the number of variables, 5 by 2, in this example:

fc2x <- forecast(fitx, ahead = 5, xC = stocks.z[101:105, c("nissan", "nissan")])
fc2x

Variational Approximation

The package features the option to use Stan's variational Bayes (sampling_algorithm = "VB") algorithm. Currently, this feature is lagging behind CmdStan's version and is considered to be experimental and mostly a placeholder for future improvements.

Community Guidelines

1. Contributions and suggestions to the software are always welcome. Please consult our contribution guidelines prior to submitting a pull request.
2. Report issues or problems with the software using github's issue tracker.
3. Contributors must adhere to the Code of Conduct.

Acknowledgment

This work was supported by the National Institute On Aging of the National Institutes of Health under Award Number R01AG050720 to PR. The content is solely the responsibility of the authors and does not necessarily represent the official views of the funding agency.

ph-rast/bmgarch documentation built on Nov. 4, 2024, 6:50 a.m.