estimate_sgsmvar {gmvarkit}

Maximum likelihood estimation of a structural GMVAR, StMVAR, or G-StMVAR model with preliminary estimates

Description

estimate_gsmvar uses a genetic algorithm and variable metric algorithm to estimate the parameters of a structural GMVAR, StMVAR, or G-StMVAR model with the method of maximum likelihood and preliminary estimates from (typically identified) reduced form or structural GSMVAR model.

Usage

estimate_sgsmvar(
  gsmvar,
  new_W,
  ncalls = 16,
  ncores = 2,
  maxit = 1000,
  seeds = NULL
)

Arguments

gsmvar

an object of class 'gsmvar', typically created with fitGSMVAR or GSMVAR.

new_W

What should be the constraints on the W-matrix (or equally B-matrix)? Provide a (d x d) matrix (where d is the number of time series in the system) expressing the constraints such that NA signifies that the element is not constrained, strictly positive real number signifies strict positive sign constraint, strictly negative real number signified strict negative sign constraints, and zero signifies a zero constraints.

ncalls

the number of estimation rounds that should be performed.

ncores

the number of CPU cores to be used in numerical differentiation. Multiple cores are not supported on Windows, though.

maxit

the maximum number of iterations in the variable metric algorithm.

seeds

a length ncalls vector containing the random number generator seed for each call to the genetic algorithm, or NULL for not initializing the seed. Exists for creating reproducible results.

Details

The purpose of estimate_gsmvar is to provide a convenient tool to estimate (typically over)identified structural GSMVAR models when preliminary estimates are available from a fitted reduced form or structural GMVAR model. Often one estimates a two-regime reduced form model and then uses the function gsmvar_to_sgsmvar to obtain the corresponding, statistically identified structural model. After obtaining the statistically identified structural model, overidentifying constraints may be placed the W-matrix (or equally B-matrix). This function makes imposing the overidentifying constraints and estimating the overidentified structural model convenient. Reduced form models can be directly used as lower-triangular Cholesky identified SVARs without having to estimate a structural model separately.

Note that the surface of the log-likelihood function is extremely multimodal, and this function is designed to only explore the neighbourhood of the preliminary estimates, so it finds its way reliably to the correct MLE only the preliminary estimates are close it in the first place. Use the function directly fitGSMVAR for a more thorough search of the parameter space, if necessary. This function calls fitGSMVAR by construction an initial population to the genetic algorithm from a preliminary guess of the new estimates. The smart mutations are set to begin from the first generation.

In order to the impose the constraints you wish, it might be useful to first run the model through the functioons reorder_W_columns and swap_W_signs. Particularly check that the sign constraints are readily satisfied. If not, the estimated solution might not be correct MLE.

estimate_sgsmvar can also be used to estimate models that are not identified, i.e., one regime models. If it supplied with a reduced form model, it will first apply the function gsmvar_to_sgsmvar, then impose the constraints and finally estimate the model.

Value

Returns an object of class 'gsmvar' defining the estimated GMVAR, StMVAR, or G-StMVAR model.

References

• 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. (forthcoming). A statistically identified structural vector autoregression with endogenously switching volatility regime. Journal of Business & Economic Statistics.

• Virolainen S. 2022. Gaussian and Student's t mixture vector autoregressive model with application to the asymmetric effects of monetary policy shocks in the Euro area. Unpublished working paper, available as arXiv:2109.13648.

See Also

fitGSMVAR, GSMVAR, optim, profile_logliks, iterate_more gsmvar_to_sgsmvar

Examples

## These are long running examples that use parallel computing!
## Running the below examples takes 30 seconds

# GMVAR(1,2) model
fit12 <- fitGSMVAR(gdpdef, p=1, M=2, ncalls=2, seeds=1:2) # Reduced form
fit12s <- gsmvar_to_sgsmvar(fit12) # Structural
fit12s
# Constrain the lower right element of W (or B-matrix) to zero, and for
# global identification the first elements of each column to strictly positive.
(new_W <- matrix(c(1, NA, 1, 0), nrow=2))
new_fit12s <- estimate_sgsmvar(fit12s, new_W, ncalls=2, ncores=2, seeds=1:2)
new_fit12s # Overidentified model

# Cholesky VAR(1)
fit11 <- fitGSMVAR(gdpdef, p=1, M=1, ncalls=2, seeds=1:2) # Reduced form
(new_W <- matrix(c(1, NA, 0, 1), nrow=2))
new_fit11s <- estimate_sgsmvar(fit11, new_W, ncalls=2, ncores=2, seeds=1:2)
print(new_fit11s, digits=4)
# Also: gsmvar_to_sgsmvar(fit11, cholesky=TRUE) gives Cholesky VAR