GSMVAR: Create a class 'gsmvar' object defining a reduced form or...

Description Usage Arguments Details Value Methods (by generic) About S3 methods References See Also Examples

View source: R/GSMVARconstruction.R

Description

GSMVAR creates a class 'gsmvar' object that defines a reduced form or structural GMVAR, StMVAR, or G-StMVAR model

Usage

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GSMVAR(
  data,
  p,
  M,
  d,
  params,
  conditional = TRUE,
  model = c("GMVAR", "StMVAR", "G-StMVAR"),
  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,
  df_tol = 1e-08
)

## S3 method for class 'gsmvar'
logLik(object, ...)

## S3 method for class 'gsmvar'
residuals(object, ...)

## S3 method for class 'gsmvar'
summary(object, ..., digits = 2)

## S3 method for class 'gsmvar'
plot(x, ..., type = c("both", "series", "density"))

## S3 method for class 'gsmvar'
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.

p

a positive integer specifying the autoregressive order of the model.

M
For GMVAR and StMVAR models:

a positive integer specifying the number of mixture components.

For G-StMVAR models:

a size (2x1) integer vector specifying the number of GMVAR type components M1 in the first element and StMVAR type components M2 in the second element. The total number of mixture components is M=M1+M2.

d

number of times series in the system, i.e. ncol(data). This can be used to define GSMVAR 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+2)-M1-1)x1) and have the form θ = (υ_{1}, ...,υ_{M}, α_{1},...,α_{M-1},ν), where

  • υ_{m} = (φ_{m,0},φ_{m},σ_{m})

  • φ_{m} = (vec(A_{m,1}),...,vec(A_{m,p})

  • and σ_{m} = vech(Ω_{m}), m=1,...,M,

  • ν=(ν_{M1+1},...,ν_{M})

  • M1 is the number of GMVAR type regimes.

For constrained models:

Should be size ((M(d+d(d+1)/2+2)+q-M1-1)x1) and have the form θ = (φ_{1,0},...,φ_{M,0},ψ, σ_{1},...,σ_{M},α_{1},...,α_{M-1},ν), where

  • ψ (qx1) satisfies (φ_{1},..., φ_{M}) = C ψ where C is a (Mpd^2xq) constraint matrix.

For same_means models:

Should have the form θ = (μ,ψ, σ_{1},...,σ_{M},α_{1},...,α_{M-1},ν), where

  • μ= (μ_{1},...,μ_{g}) where μ_{i} is the mean parameter for group i and g is the number of groups.

  • If AR constraints are employed, ψ is as for constrained models, and if AR constraints are not employed, ψ = (φ_{1},...,φ_{M}).

For structural models:

Should have the form θ = (φ_{1,0},...,φ_{M,0},φ_{1},...,φ_{M}, vec(W),λ_{2},...,λ_{M},α_{1},...,α_{M-1},ν), where

  • λ_{m}=(λ_{m1},...,λ_{md}) contains the eigenvalues of the mth mixture component.

If AR parameters are constrained:

Replace φ_{1},..., φ_{M} with ψ (qx1) that satisfies (φ_{1},..., φ_{M}) = C ψ, as above.

If same_means:

Replace (φ_{1,0},...,φ_{M,0}) with (μ_{1},...,μ_{g}), as above.

If W is constrained:

Remove the zeros from vec(W) and make sure the other entries satisfy the sign constraints.

If λ_{mi} are constrained:

Replace λ_{2},...,λ_{M} with γ (rx1) that satisfies (λ_{2},..., λ_{M}) = C_{λ} γ where C_{λ} is a (d(M-1) x r) constraint matrix.

Above, φ_{m,0} is the intercept parameter, A_{m,i} denotes the ith coefficient matrix of the mth mixture component, Ω_{m} denotes the error term covariance matrix of the m:th mixture component, and α_{m} is the mixing weight parameter. The W and λ_{mi} are structural parameters replacing the error term covariance matrices (see Virolainen, 2020). If M=1, α_{m} and λ_{mi} are dropped. If parametrization=="mean", just replace each φ_{m,0} with regimewise mean μ_{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.

In the GMVAR model, M1=M and ν is dropped from the parameter vector. In the StMVAR model, M1=0. In the G-StMVAR model, the first M1 regimes are GMVAR type and the rest M2 regimes are StMVAR type. In StMVAR and G-StMVAR models, the degrees of freedom parameters in ν should be strictly larger than two.

The notation is similar to the cited literature.

conditional

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

model

is "GMVAR", "StMVAR", or "G-StMVAR" model considered? In the G-StMVAR model, the first M1 components are GMVAR type and the rest M2 components are StMVAR type.

parametrization

"intercept" or "mean" determining whether the model is parametrized with intercept parameters φ_{m,0} or regime means μ_{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 (φ_{1},...,φ_{M}) = C ψ, where φ_{m} = (vec(A_{m,1}),...,vec(A_{m,p}) (pd^2 x 1), m=1,...,M, contains the coefficient matrices and ψ (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 (λ_{2},..., λ_{M}) = C_{λ} γ where γ 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 λ_{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.

df_tol

the parameter vector is considered to be outside the parameter space if all degrees of freedom parameters are not larger than 2 + df_tol.

object

object of class 'gsmvar' generated by fitGSMVAR or GSMVAR.

...

currently not used.

digits

number of digits to be printed.

x

object of class 'gsmvar' generated by fitGSMVAR or GSMVAR.

type

which type figure should be produced? Or both?

summary_print

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

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 a (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.

Value

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

Note 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)

About S3 methods

If data is not provided, only the print and simulate methods are available. If data is provided, then in addition to the ones listed above, predict method is also available. See ?simulate.gsmvar and ?predict.gsmvar for details about the usage.

References

See Also

fitGSMVAR, add_data, swap_parametrization, GIRF, gsmvar_to_sgsmvar, stmvar_to_gstmvar, reorder_W_columns, swap_W_signs, update_numtols

Examples

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# GMVAR(1, 2), d=2 model:
params12 <- c(0.55, 0.112, 0.344, 0.055, -0.009, 0.718, 0.319, 0.005,
  0.03, 0.619, 0.173, 0.255, 0.017, -0.136, 0.858, 1.185, -0.012,
  0.136, 0.674)
mod12 <- GSMVAR(gdpdef, p=1, M=2, params=params12)
mod12

# GMVAR(1, 2), d=2 model without data
mod12_2 <- GSMVAR(p=1, M=2, d=2, params=params12)
mod12_2

# StMVAR(1, 2), d=2 model:
mod12t <- GSMVAR(gdpdef, p=1, M=2, params=c(params12, 10, 20),
                 model="StMVAR")
mod12t

# G-StMVAR(1, 1, 1), d=2 model:
mod12gs <- GSMVAR(gdpdef, p=1, M=c(1, 1), params=c(params12, 20),
                  model="G-StMVAR")
mod12gs

# GMVAR(2, 2), d=2 model with mean-parametrization:
params22 <- c(0.869, 0.549, 0.223, 0.059, -0.151, 0.395, 0.406,
 -0.005, 0.083, 0.299, 0.215, 0.002, 0.03, 0.576, 1.168, 0.218,
 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 1.101, -0.004,
 0.105, 0.58)
mod22 <- GSMVAR(gdpdef, p=2, M=2, params=params22, parametrization="mean")
mod22

# Structural GMVAR(2, 2), d=2 model identified with sign-constraints:
params22s <- c(0.36, 0.121, 0.484, 0.072, 0.223, 0.059, -0.151, 0.395,
  0.406, -0.005, 0.083, 0.299, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032,
  0.044, 0.191, 0.057, 0.172, -0.46, 0.016, 3.518, 5.154, 0.58)
W_22 <- matrix(c(1, 1, -1, 1), nrow=2, byrow=FALSE)
mod22s <- GSMVAR(gdpdef, p=2, M=2, params=params22s,
 structural_pars=list(W=W_22))
mod22s

saviviro/gmvarkit documentation built on Oct. 25, 2021, 2:14 a.m.