The goal of gmvarkit is to provide tools to analyse structural and
reduced form Gaussian mixture vector autoregressive (GMVAR) model,
Student’s t mixtue vector autoregressive (StMVAR) model, and Gaussian
and Student’s t mixture vector autoregressive (G-StMVAR) model.
`gmvarkit`

provides functions for unconstrained and
constrained maximum likelihood estimation of the model parameters,
quantile residual based model diagnostics, simulation from the
processes, forecasting, estimation of generalized impulse response
function, and more.

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

`install.packages("gmvarkit")`

And the development version from GitHub with:

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

This is a basic example on how to use `gmvarkit`

in time
series analysis. The estimation process is computationally demanding and
takes advantage of parallel computing. After estimating the model, it is
shown by simple examples how to conduct some further analysis. StMVAR
and G-StMVAR model are briefly covered after giving examples related to
the reduced form and structural GMVAR model.

```
# These examples use the data 'gdpdef' which comes with the package, and contains the quarterly percentage growth rate
# of real U.S. GDP and quarterly percentage growth rate of U.S. GDP implicit price deflator, covering the period
# from 1959Q1 to 2019Q4.
data(gdpdef, package="gmvarkit")
## Reduced form GMVAR model ##
# Estimate a GMVAR(2, 2) model: 20 estimation rounds and seeds for reproducible results
# (note: many empirical applications require more estimation rounds, e.g., hundreds
# or thousands).
<- fitGMVAR(gdpdef, p=2, M=2, ncalls=20, seeds=1:20, ncores=2)
fit
fit
# Estimate a GMVAR(2, 2) model with autoregressive parameters restricted to be the same for all regimes
<- rbind(diag(2*2^2), diag(2*2^2))
C_mat <- fitGMVAR(gdpdef, p=2, M=2, constraints=C_mat, ncalls=20, seeds=1:20, ncores=2)
fitc
fitc
# Estimate a GMVAR(2, 2) model with autoregressive parameters and the unconditional means
# restricted to be the same in both regimes (only the covariance matrix varies)
<- fitGMVAR(gdpdef, p=2, M=2, parametrization="mean", constraints=C_mat, same_means=list(1:2),
fitcm ncalls=20, seeds=1:20, ncores=2)
fitcm
# Test the above constraints on the AR parameters with likelihood ratio test:
LR_test(fit, fitc)
# Further information on the estimated model:
plot(fit)
summary(fit)
print_std_errors(fit)
get_foc(fit) # The first order condition (gradient of the log-likelihood function)
get_soc(fit) # The second order condition (eigenvalues of approximated Hessian)
profile_logliks(fit) # Profile log-likelihood functions
# Note: models can be built based the results from any estimation round
# conveniently with the function 'alt_gmvar'.
# Quantile residual diagnostics
diagnostic_plot(fit) # type=c("all", "series", "ac", "ch", "norm")
<- quantile_residual_tests(fit, nsimu=10000)
qrt
# Simulate a sample path from the estimated process
<- simulateGMVAR(fit, nsimu=100)
sim plot.ts(sim$sample)
# Forecast future values of the process
predict(fit, n_ahead=12)
## Structural GMVAR model ##
# Estimate structural GMVAR(2,2) model identified with sign constraints:
<- matrix(c(1, 1, -1, 1), nrow=2, byrow=FALSE)
W22 <- fitGMVAR(gdpdef, p=2, M=2, structural_pars=list(W=W22),
fit22s ncalls=20, seeds=1:20, ncores=2)
fit22s
# Alternatively, if there are two regimes (M=2), a stuctural model can
# be build based on the reduced form model:
<- gmvar_to_sgmvar(fit)
fit22s_2
fit22s_2
# Columns of the matrix W can be permutated and all signs in any column
# can be swapped without affecting the implied reduced form model, as
# long as the lambda parameters are also rearranged accordingly:
<- reorder_W_columns(fit22s_2, perm=c(2, 1))
fit22s_3
fit22s_3
<- swap_W_signs(fit22s_3, which_to_swap=2)
fit22s_4 # The same model as fit22s
fit22s_4
all.equal(fit22s$loglik, fit$loglik)
all.equal(fit22s$loglik, fit22s_2$loglik)
all.equal(fit22s$loglik, fit22s_3$loglik)
all.equal(fit22s$loglik, fit22s_4$loglik)
# Check out also the function: estimate_sgsmvar
# for estimating overidentified structural models.
# Estimate generalized impulse response function (GIRF) with starting values
# generated from the stationary distribution of the process:
<- GIRF(fit22s, N=20, ci=c(0.95, 0.8), R1=200, R2=200, ncores=2)
girf1
# Estimate GIRF with starting values generated from the stationary distribution
# of the first regime:
<- GIRF(fit22s, N=20, ci=c(0.95, 0.8), init_regimes=1, R1=200, R2=200, ncores=2)
girf2
# Estimate GIRF with starting values given by the last p observations of the
# data:
<- GIRF(fit22s, N=20, init_values=fit22s$data, R1=1000, ncores=2)
girf3
# Estimate generalized forecast error variance decmposition (GFEVD) with the
# initial values being all possible lenght p the histories in the data:
<- GFEVD(fit22s, N=20, R1=100, initval_type="data", ncores=2)
gfevd1 plot(gfevd1)
# Estimate GFEVD with the initial values generated from the stationary
# distribution of the second regime:
<- GFEVD(fit22s, N=20, R1=100, R2=100, initval_type="random",
gfevd2 init_regimes=2, ncores=2)
plot(gfevd2)
# Estimate GFEVD with fixed starting values that are the unconditional mean
# of the process:
<- rbind(fit22s$uncond_moments$uncond_mean,
myvals $uncond_moments$uncond_mean)
fit22s<- GFEVD(fit22s, N=48, R1=250, initval_type="fixed",
gfevd3 init_values=myvals, ncores=2)
plot(gfevd3)
# Test with Wald test whether the diagonal elements of the first AR coefficient
# matrix of the second regime are identical:
# fit22s has parameter vector of length 27 with the diagonal elements of the
# first A-matrix of the second regime are in elements 13 and 16.
<- matrix(c(rep(0, times=12), 1, 0, 0, -1, rep(0, times=27-16)), nrow=1, ncol=27)
A <- 0
c Wald_test(fit22s, A, c)
# The same functions used in the demonstration of the reduced form model also
# work with structural models.
## StMVAR and G-StMVAR models ##
# Fit a StMVAR(2, 2) model
<- fitGSMVAR(gdpdef, p=2, M=2, model="StMVAR", ncalls=1, seeds=1)
fit22t
# Printout shows that there is an overly large degrees of freedom estimate
# in the 2nd regime!
fit22t
# So we change it to GMVAR type by switching to the appropariate G-StMVAR model
<- stmvar_to_gstmvar(fit22t)
fit22gs
# Printout of the appropriate G-StMVAR model that is based on the above StMVAR model.
fit22gs
# Switch to the stastitically identified structural model
<- gsmvar_to_sgsmvar(fit22gs)
fit22gss
# Printout of the structural model that is implied by the reduced form model
fit22gss
```

- Kalliovirta L., Meitz M. and Saikkonen P. (2016) Gaussian mixture
vector autoregression.
*Journal of Econometrics*,**192**, 485-498. - Kalliovirta L. and Saikkonen P. (2010) Reliable Residuals for
Multivariate Nonlinear Time Series Models.
*Unpublished Revision of HECER Discussion Paper No. 247*. - Virolainen S. 2022. Structural Gaussian mixture vector autoregressive model with application to the asymmetric effects of monetary policy shocks. Unpublished working paper, available as arXiv:2007.04713.
- Virolainen S. 2022. Gaussian and Student’s t mixture vector autoregressive model with application to the assymetric effects of monetary policy shocks in the Euro area. Unpublished working paper, available as arXiv:2109.13648.