Estimate Univariate Gaussian and Student's t Mixture Autoregressive Models.

uGMAR

uGMAR provides tools for estimating and analyzing Gaussian mixture autoregressive (GMAR), Student’s t mixture Autoregressive (StMAR) and Gaussian and Student’s t mixture autoregressive (G-StMAR) models, including functions for unconstrained and constrained maximum likelihood estimation of the model parameters, quantile residual based model diagnostics, simulation from the processes, and forecasting.

Installation

You can install the released version of uGMAR from CRAN with:

install.packages("uGMAR")

And the development version from GitHub with:

# install.packages("devtools")
devtools::install_github("saviviro/uGMAR")

Example

This is a basic example how to estimate GSMAR model and further analyze it.. The example data is simulated from a GMAR p=1, M=2 process. The estimation process is computationally demanding and takes advantage of parallel computing.

## Estimate a GMAR(1, 2) model and examine the estimates
data(simudata, package="uGMAR")
fit <- fitGSMAR(data=simudata, p=1, M=2, model="GMAR", ncalls=10, seeds=1:10)
fit
summary(fit) # Approximate standard errors in brackets
plot(fit)

get_gradient(fit) # The first order condition
get_soc(fit) # The second order condition (eigenvalues of approximated Hessian)
profile_logliks(fit) # Plot the profile log-likelihood functions

## Quantile residual diagnostics
quantile_residual_plot(fit)
diagnostic_plot(fit)
qrt <- quantile_residual_tests(fit)

## Simulate a sample path from the estimated process
sim <- simulate(fit, nsim=100)
plot.ts(sim$sample)

## Forecast future values of the process
predict(fit, n_ahead=10, pi=c(0.95, 0.8))

# Estimate a GMAR(1, 2) model with the autoregressive coefficients restricted
# to be the same in both regimes:
fitr <- fitGSMAR(data=simudata, p=1, M=2, model="GMAR", restricted=TRUE,
                 ncalls=10, seeds=1:10)

# Test with likelihood ratio tests whether the AR parameters are the same in
# both regimes (see also the function 'Wald_test'):
LR_test(fit, fitr)

# Conditional mean and variance plots:
cond_moment_plot(fit, which_moment="mean")
cond_moment_plot(fit, which_moment="variance")

References

• Kalliovirta L., Meitz M. and Saikkonen P. 2015. Gaussian Mixture Autoregressive model for univariate time series. Journal of Time Series Analysis, 36(2), 247-266.

• Meitz M., Preve D., Saikkonen P. 2023. A mixture autoregressive model based on Student’s t-distribution. Communications in Statistics - Theory and Methods52(2), 499-515.

• Virolainen S. 2022. A mixture autoregressive model based on Gaussian and Student’s t-distributions. Studies in Nonlinear Dynamics & Econometrics, 26(4), 559-580.