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


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


And the development version from GitHub with:

# install.packages("devtools")


This is a basic example how to estimate a GMAR model to data. 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. After estimating the model, it’s shown by simple examples how to conduct some further analysis.

## Estimate a GMAR(1, 2) model and examine the estimates
data(simudata, package="uGMAR")
fit <- fitGSMAR(data=simudata, p=1, M=2, model="GMAR", ncores=4)
summary(fit) # Approximate standard errors in brackets

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
qrt <- quantileResidualTests(fit)

## Simulate a sample path from the estimated process
sim <- simulateGSMAR(fit, nsimu=100)

## 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,

# 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:
condmomentPlot(fit, which_moment="mean")
condmomentPlot(fit, which_moment="variance")