GMVAR: Create a class 'gmvar' object defining a reduced form or structural GMVAR model

Description

GMVAR creates a class 'gmvar' object that defines a reduced form or structural GMVAR model

Usage

GMVAR(
  data,
  p,
  M,
  d,
  params,
  conditional = TRUE,
  parametrization = c("intercept", "mean"),
  constraints = NULL,
  same_means = NULL,
  structural_pars = NULL,
  calc_cond_moments,
  calc_std_errors = FALSE,
  stat_tol = 0.001,
  posdef_tol = 1e-08
)
# S3 method for gmvar
logLik(object, ...)
# S3 method for gmvar
residuals(object, ...)
# S3 method for gmvar
summary(object, ..., digits = 2)
# S3 method for gmvar
plot(x, ...)
# S3 method for gmvar
print(x, ..., digits = 2, summary_print = FALSE)

Arguments

data

a matrix or class 'ts' object with d>1 columns. Each column is taken to represent a single times series. NA values are not supported. Ignore if defining a model without data is desired.

a positive integer specifying the autoregressive order of the model.

a positive integer specifying the number of mixture components.

number of times series in the system, i.e. ncol(data). This can be used to define GMVAR models without data and can be ignored if data is provided.

params

a real valued vector specifying the parameter values.

For unconstrained models:

Should be size $((M(pd^2+d+d(d+1)/2+1)-1)x1)$ and have the form $\theta$$ = $($\upsilon$$_{1}$, ...,$\upsilon$$_{M}$, $\alpha_{1},...,\alpha_{M-1}$), where

$\upsilon$$_{m}$ $ = (\phi_{m,0},$$\phi$$_{m}$$,\sigma_{m})$
$\phi$$_{m}$$ = (vec(A_{m,1}),...,vec(A_{m,p})$
and $\sigma_{m} = vech(\Omega_{m})$, m=1,...,M.

For constrained models:

Should be size $((M(d+d(d+1)/2+1)+q-1)x1)$ and have the form $\theta$$ = (\phi_{1,0},...,\phi_{M,0},$$\psi$, $\sigma_{1},...,\sigma_{M},\alpha_{1},...,\alpha_{M-1})$, where

$\psi$ $(qx1)$ satisfies ($\phi$$_{1}$$,...,$ $\phi$$_{M}) =$ $C \psi$ where $C$ is $(Mpd^2xq)$ constraint matrix.

For same_means models:

Should have the form $\theta$$ = ($$\mu$,$\psi$, $\sigma_{1},...,\sigma_{M},\alpha_{1},...,\alpha_{M-1})$, where

$\mu$$= (\mu_{1},...,\mu_{g})$ where $\mu_{i}$ is the mean parameter for group $i$ and $g$ is the number of groups.
If AR constraints are employed, $\psi$ is as for constrained models, and if AR constraints are not employed, $\psi$$ = $ ($\phi$$_{1}$$,...,$$\phi$$_{M})$.

For structural GMVAR model:

Should have the form $\theta$$ = (\phi_{1,0},...,\phi_{M,0},$$\phi$$_{1},...,$$\phi$$_{M}, vec(W),$$\lambda$$_{2},...,$$\lambda$$_{M},\alpha_{1},...,\alpha_{M-1})$, where

$\lambda$$_{m}=(\lambda_{m1},...,\lambda_{md})$ contains the eigenvalues of the $m$th mixture component.

If AR parameters are constrained:: Replace $\phi$$_{1}$$,...,$ $\phi$$_{M}$ with $\psi$ $(qx1)$ that satisfies ($\phi$$_{1}$$,...,$ $\phi$$_{M}) =$ $C \psi$, as above.
If same_means:: Replace $(\phi_{1,0},...,\phi_{M,0})$ with $(\mu_{1},...,\mu_{g})$, as above.
If $W$ is constrained:: Remove the zeros from $vec(W)$ and make sure the other entries satisfy the sign constraints.
If $\lambda_{mi}$ are constrained:: Replace $\lambda$$_{2},...,$$\lambda$$_{M}$ with $\gamma$ $(rx1)$ that satisfies ($\lambda$$_{2}$$,...,$ $\lambda$$_{M}) =$ $C_{\lambda} \gamma$ where $C_{\lambda}$ is a $(d(M-1) x r)$ constraint matrix.

Above, $\phi_{m,0}$ is the intercept parameter, $A_{m,i}$ denotes the $i$th coefficient matrix of the $m$th mixture component, $\Omega_{m}$ denotes the error term covariance matrix of the $m$:th mixture component, and $\alpha_{m}$ is the mixing weight parameter. The $W$ and $\lambda_{mi}$ are structural parameters replacing the error term covariance matrices (see Virolainen, 2020). If $M=1$, $\alpha_{m}$ and $\lambda_{mi}$ are dropped. If parametrization=="mean", just replace each $\phi_{m,0}$ with regimewise mean $\mu_{m}$. $vec()$ is vectorization operator that stacks columns of a given matrix into a vector. $vech()$ stacks columns of a given matrix from the principal diagonal downwards (including elements on the diagonal) into a vector. The notation is in line with the cited article by Kalliovirta, Meitz and Saikkonen (2016) introducing the GMVAR model.

conditional

a logical argument specifying whether the conditional or exact log-likelihood function should be used.

parametrization

"intercept" or "mean" determining whether the model is parametrized with intercept parameters $\phi_{m,0}$ or regime means $\mu_{m}$, m=1,...,M.

constraints

a size $(Mpd^2 x q)$ constraint matrix $C$ specifying general linear constraints to the autoregressive parameters. We consider constraints of form ($\phi$$_{1}$$,...,$$\phi$$_{M}) = $$C \psi$, where $\phi$$_{m}$$ = (vec(A_{m,1}),...,vec(A_{m,p}) (pd^2 x 1), m=1,...,M$, contains the coefficient matrices and $\psi$ $(q x 1)$ contains the related parameters. For example, to restrict the AR-parameters to be the same for all regimes, set $C$= [I:...:I]' $(Mpd^2 x pd^2)$ where I = diag(p*d^2). Ignore (or set to NULL) if linear constraints should not be employed.

same_means

Restrict the mean parameters of some regimes to be the same? Provide a list of numeric vectors such that each numeric vector contains the regimes that should share the common mean parameters. For instance, if M=3, the argument list(1, 2:3) restricts the mean parameters of the second and third regime to be the same but the first regime has freely estimated (unconditional) mean. Ignore or set to NULL if mean parameters should not be restricted to be the same among any regimes. This constraint is available only for mean parametrized models; that is, when parametrization="mean".

structural_pars

If NULL a reduced form model is considered. For structural model, should be a list containing the following elements:

W - a $(dxd)$ matrix with its entries imposing constraints on $W$: NA indicating that the element is unconstrained, a positive value indicating strict positive sign constraint, a negative value indicating strict negative sign constraint, and zero indicating that the element is constrained to zero.
C_lambda - a $(d(M-1) x r)$ constraint matrix that satisfies ($\lambda$$_{2}$$,...,$ $\lambda$$_{M}) =$ $C_{\lambda} \gamma$ where $\gamma$ is the new $(r x 1)$ parameter subject to which the model is estimated (similarly to AR parameter constraints). The entries of C_lambda must be either positive or zero. Ignore (or set to NULL) if the eigenvalues $\lambda_{mi}$ should not be constrained.

See Virolainen (2020) for the conditions required to identify the shocks and for the B-matrix as well (it is $W$ times a time-varying diagonal matrix with positive diagonal entries).

calc_cond_moments

should conditional means and covariance matrices should be calculated? Default is TRUE if the model contains data and FALSE otherwise.

calc_std_errors

should approximate standard errors be calculated?

stat_tol

numerical tolerance for stationarity of the AR parameters: if the "bold A" matrix of any regime has eigenvalues larger that 1 - stat_tol the model is classified as non-stationary. Note that if the tolerance is too small, numerical evaluation of the log-likelihood might fail and cause error.

posdef_tol

numerical tolerance for positive definiteness of the error term covariance matrices: if the error term covariance matrix of any regime has eigenvalues smaller than this, the model is classified as not satisfying positive definiteness assumption. Note that if the tolerance is too small, numerical evaluation of the log-likelihood might fail and cause error.

object

object of class 'gmvar' generated by fitGMVAR or GMVAR.

...

currectly not used.

digits

number of digits to be printed.

object of class 'gmvar' generated by fitGMVAR or GMVAR.

summary_print

if set to TRUE then the print will include log-likelihood and information criteria values.

Value

Returns an object of class 'gmvar' defining the specified reduced form or structural GMVAR model. Can be used to work with other functions provided in gmvarkit.

Remark that the first autocovariance/correlation matrix in $uncond_moments is for the lag zero, the second one for the lag one, etc.

Methods (by generic)

logLik: Log-likelihood method
residuals: residuals method to extract multivariate quantile residuals
summary: summary method
plot: plot method for class 'gmvar'
print: print method

About S3 methods

Only the print method is available if data is not provided. If data is provided, then in addition to the ones listed above, the predict method is also available.

Details

If data is provided, then also multivariate quantile residuals (Kalliovirta and Saikkonen 2010) are computed and included in the returned object.

If the function fails to calculate approximative standard errors and the parameter values are near the border of the parameter space, it might help to use smaller numerical tolerance for the stationarity and positive definiteness conditions.

The first plot displays the time series together with estimated mixing weights. The second plot displays (Gaussian) kernel density estimates of the individual series together with the marginal stationary density implied by the model. The colored regimewise stationary densities are multiplied with the mixing weight parameter estimates.

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. 2020. Structural Gaussian mixture vector autoregressive model. Unpublished working paper, available as arXiv:2007.04713.

Examples

Run this code

# NOT RUN {
# These examples use the data 'eurusd' which comes with the
# package, but in a scaled form.
data <- cbind(10*eurusd[,1], 100*eurusd[,2])
colnames(data) <- colnames(eurusd)

# GMVAR(1,2), d=2 model:
params122 <- c(0.623, -0.129, 0.959, 0.089, -0.006, 1.006, 1.746,
 0.804, 5.804, 3.245, 7.913, 0.952, -0.037, -0.019, 0.943, 6.926,
 3.982, 12.135, 0.789)
mod122 <- GMVAR(data, p=1, M=2, params=params122)
mod122

# GMVAR(1,2), d=2 model without data
mod122_2 <- GMVAR(p=1, M=2, d=2, params=params122)
mod122_2

# GMVAR(2,2), d=2 model with mean-parametrization:
params222 <- c(-11.904, 154.684, 1.314, 0.145, 0.094, 1.292, -0.389,
 -0.070, -0.109, -0.281, 0.920, -0.025, 4.839, 11.633, 124.983, 1.248,
  0.077, -0.040, 1.266, -0.272, -0.074, 0.034, -0.313, 5.855, 3.570,
  9.838, 0.740)
mod222 <- GMVAR(data, p=2, M=2, params=params222, parametrization="mean")
mod222

# Structural GMVAR(2, 2), d=2 model identified with sign-constraints:
params222s <- c(-11.964, 155.024, 11.636, 124.988, 1.314, 0.145, 0.094, 1.292,
 -0.389, -0.07, -0.109, -0.281, 1.248, 0.077, -0.04, 1.266, -0.272, -0.074,
  0.034, -0.313, 0.903, 0.718, -0.324, 2.079, 7.00, 1.44, 0.742)
W_222 <- matrix(c(1, 1, -1, 1), nrow=2, byrow=FALSE)
mod222s <- GMVAR(data, p=2, M=2, params=params222s, parametrization="mean",
 structural_pars=list(W=W_222))
mod222s

# GMVAR(2,2), d=2 model with AR-parameters restricted to be
# the same for both regimes:
C_mat <- rbind(diag(2*2^2), diag(2*2^2))
params222c <- c(1.031, 2.356, 1.786, 3.000, 1.250, 0.060, 0.036,
 1.335, -0.290, -0.083, -0.047, -0.356, 0.934, -0.152, 5.201, 5.883,
 3.560, 9.799, 0.368)
mod222c <- GMVAR(data, p=2, M=2, params=params222c, constraints=C_mat)
mod222c

# GMVAR(2,2), d=2 model with AR-parameters restricted to be
# the same for both regimes and the non-diagonal elements of
# the coefficient matrices constrained to zero.
tmp <- matrix(c(1, rep(0, 10), 1, rep(0, 8), 1, rep(0, 10), 1),
 nrow=2*2^2, byrow=FALSE)
C_mat2 <- rbind(tmp, tmp)
params222c2 <- c(0.355, 3.193, -0.114, 2.829, 1.263, 1.338, -0.292,
 -0.362, 5.597, 3.456, 9.622, 0.982, -0.327, 5.236, 0.650)
mod222c2 <- GMVAR(data, p=2, M=2, params=params222c2,
  constraints=C_mat2)
mod222c2
# }

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