a matrix or class 'ts' object with d>1 columns. Each column is taken to represent a univariate time series. NA values are not supported.

a positive integer specifying the autoregressive order of the model.

For GMVAR and StMVAR models:

a positive integer specifying the number of mixture components.

For G-StMVAR models:

a size (2\times 1) 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.

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.

a logical argument specifying whether the conditional or exact log-likelihood function

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

a size (Mpd^2 \times 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 \times pd^2) where I = diag(p*d^2). Ignore (or set to NULL) if linear constraints should not be employed.

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".

a numeric vector of length M-1 specifying fixed parameter values for the mixing weight parameters \alpha_m, \ m=1,...,M-1. Each element should be strictly between zero and one, and the sum of all the elements should be strictly less than one.

If NULL a reduced form model is considered. Reduced models can be used directly as recursively identified structural models. For a structural model identified by conditional heteroskedasticity, should be a list containing at least the first one of 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) \times r) constraint matrix that satisfies (\lambda_{2},..., \lambda_{M}) = C_{\lambda} \gamma where \gamma is the new (r \times 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.

• fixed_lambdas - a length d(M-1) numeric vector (\lambda_{2},..., \lambda_{M}) with elements strictly larger than zero specifying the fixed parameter values for the parameters \lambda_{mi} should be constrained to. This constraint is alternative C_lambda. Ignore (or set to NULL) if the eigenvalues \lambda_{mi} should not be constrained.

See Virolainen (forthcoming) 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).

a positive integer specifying the number of generations to be ran through in the genetic algorithm.

a positive even integer specifying the population size in the genetic algorithm. Default is 10*n_params.

a positive integer specifying the generation after which the random mutations in the genetic algorithm are "smart". This means that mutating individuals will mostly mutate fairly close (or partially close) to the best fitting individual (which has the least regimes with time varying mixing weights practically at zero) so far.

a list of parameter vectors from which the initial population of the genetic algorithm will be generated from. The parameter vectors should be...

For unconstrained models:

Should be size ((M(pd^2+d+d(d+1)/2+2)-M1-1)\times 1) and have the form \theta = (\upsilon_{1}, ...,\upsilon_{M}, \alpha_{1},...,\alpha_{M-1},\nu), 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,

• \nu=(\nu_{M1+1},...,\nu_{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)\times 1) and have the form \theta = (\phi_{1,0},...,\phi_{M,0},\psi, \sigma_{1},...,\sigma_{M},\alpha_{1},...,\alpha_{M-1},\nu), where

• \psi (qx1) satisfies (\phi_{1},..., \phi_{M}) = C \psi where C is a (Mpd^2 \times q) constraint matrix.

For same_means models:

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

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},\nu), where

• \lambda_{m}=(\lambda_{m1},...,\lambda_{md}) contains the eigenvalues of the mth 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 ith coefficient matrix of the mth 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, 2022). 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.

In the GMVAR model, M1=M and \nu 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 \nu should be strictly larger than two.

The notation is similar to the cited literature.

a size (dx1) vector defining means of the normal distributions from which each mean parameter \mu_{m} is drawn from in random mutations. Default is colMeans(data). Note that mean-parametrization is always used for optimization in GAfit - even when parametrization=="intercept". However, input (in initpop) and output (return value) parameter vectors can be intercept-parametrized.

a size (dx1) strictly positive vector defining standard deviations of the normal distributions from which each mean parameter \mu_{m} is drawn from in random mutations. Default is 2*sd(data[,i]), i=1,..,d.

a size (dx1) strictly positive vector specifying the scale and variability of the random covariance matrices in random mutations. The covariance matrices are drawn from (scaled) Wishart distribution. Expected values of the random covariance matrices are diag(omega_scale). Standard deviations of the diagonal elements are sqrt(2/d)*omega_scale[i] and for non-diagonal elements they are sqrt(1/d*omega_scale[i]*omega_scale[j]). Note that for d>4 this scale may need to be chosen carefully. Default in GAfit is var(stats::ar(data[,i], order.max=10)$resid, na.rm=TRUE), i=1,...,d. This argument is ignored if structural model is considered.

a size (dx1) strictly positive vector partly specifying the scale and variability of the random covariance matrices in random mutations. The elements of the matrix W are drawn independently from such normal distributions that the expectation of the main diagonal elements of the first regime's error term covariance matrix \Omega_1 = WW' is W_scale. The distribution of \Omega_1 will be in some sense like a Wishart distribution but with the columns (elements) of W obeying the given constraints. The constraints are accounted for by setting the element to be always zero if it is subject to a zero constraint and for sign constraints the absolute value or negative the absolute value are taken, and then the variances of the elements of W are adjusted accordingly. This argument is ignored if reduced form model is considered.

a length M - 1 vector specifying the standard deviation of the mean zero normal distribution from which the eigenvalue \lambda_{mi} parameters are drawn from in random mutations. As the eigenvalues should always be positive, the absolute value is taken. The elements of lambda_scale should be strictly positive real numbers with the m-1th element giving the degrees of freedom for the mth regime. The expected value of the main diagonal elements ij of the mth (m>1) error term covariance matrix will be W_scale[i]*(d - n_i)^(-1)*sum(lambdas*ind_fun) where the (d x 1) vector lambdas is drawn from the absolute value of the t-distribution, n_i is the number of zero constraints in the ith row of W and ind_fun is an indicator function that takes the value one iff the ijth element of W is not constrained to zero. Basically, larger lambdas (or smaller degrees of freedom) imply larger variance.

If the lambda parameters are constrained with the (d(M - 1) x r) constraint matrix C_lambda, then provide a length r vector specifying the standard deviation of the (absolute value of the) mean zero normal distribution each of the \gamma parameters are drawn from (the \gamma is a (r x 1) vector). The expected value of the main diagonal elements of the covariance matrices then depend on the constraints.

This argument is ignored if M==1 or a reduced form model is considered. Default is rep(3, times=M-1) if lambdas are not constrained and rep(3, times=r) if lambdas are constrained.

As with omega_scale and W_scale, this argument should be adjusted carefully if specified by hand. NOTE that if lambdas are constrained in some other way than restricting some of them to be identical, this parameter should be adjusted accordingly in order to the estimation succeed!

a positive real number between zero and one, adjusting how large AR parameter values are typically proposed in construction of the initial population: larger value implies larger coefficients (in absolute value). After construction of the initial population, a new scale is drawn from (0, upper_ar_scale) uniform distribution in each iteration. With large p or d, ar_scale is restricted from above, see the details section.

the upper bound for ar_scale parameter (see above) in the random mutations. Setting this too high might lead to failure in proposing new parameters that are well enough inside the parameter space, and especially with large p one might want to try smaller upper bound (e.g., 0.5). With large p or d, upper_ar_scale is restricted from above, see the details section.

a positive real number adjusting how large AR parameter values are typically proposed in some random mutations (if AR constraints are employed, in all random mutations): larger value implies smaller coefficients (in absolute value). Values larger than 1 can be used if the AR coefficients are expected to be very small. If set smaller than 1, be careful as it might lead to failure in the creation of stationary parameter candidates

a non-negative real number specifying how much should natural selection favor individuals with less regimes that have almost all mixing weights (practically) at zero. Set to zero for no favoring or large number for heavy favoring. Without any favoring the genetic algorithm gets more often stuck in an area of the parameter space where some regimes are wasted, but with too much favouring the best genes might never mix into the population and the algorithm might converge poorly. Default is 1 and it gives 2x larger surviving probability weights for individuals with no wasted regimes compared to individuals with one wasted regime. Number 2 would give 3x larger probability weights etc.

a length 2 numeric vector specifying the criteria that is used to determine whether a regime is redundant (or "wasted") or not. Any regime m which satisfies sum(mixingWeights[,m] > red_criteria[1]) < red_criteria[2]*n_obs will be considered "redundant". One should be careful when adjusting this argument (set c(0, 0) to fully disable the 'redundant regime' features from the algorithm).

A number in [0,1] giving a probability of a "smart mutation" occuring randomly in each iteration before the iteration given by the argument smart_mu.

should the genetic algorithm return the best fitting individual which has "positive enough" mixing weights for as many regimes as possible ("alt_ind") or the individual which has the highest log-likelihood in general ("best_ind") but might have more wasted regimes?

a real number defining the minimum value of the log-likelihood function that will be considered. Values smaller than this will be treated as they were minval and the corresponding individuals will never survive. The default is -(10^(ceiling(log10(n_obs)) + d) - 1).

a single value, interpreted as an integer, or NULL, that sets seed for the random number generator in the beginning of the function call. If calling GAfit from fitGSMVAR, use the argument seeds instead of passing the argument seed.

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

What shocks size should be used for all shocks? By the definition of the SGMVAR, SStMVAR, and SG-StMVAR model, the conditional covariance matrix of the structural shock is identity matrix.

a positive integer specifying the horizon how far ahead should the GFEVD be calculated.

What type initial values are used for estimating the GIRFs that the GFEVD is based on?

"data":

Estimate the GIRF for all the possible length p histories in the data.

"random":

Estimate the GIRF for several random initial values generated from the stationary distribution of the process or from the stationary distribution of specific regime(s) chosen with the argument init_regimes. The number of initial values is set with the argument R2.

"fixed":

Estimate the GIRF for a fixed initial value only, which is specified with the argument init_values.

the number of repetitions used to estimate GIRF for each initial value.

the number of initial values to be drawn if initval_type="random".

a numeric vector of length at most M and elements in 1,...,M specifying the regimes from which the initial values should be generated from. The initial values will be generated from a mixture distribution with the mixture components being the stationary distributions of the specific regimes and the (proportional) mixing weights given by the mixing weight parameters of those regimes. Note that if init_regimes=1:M, the initial values are generated from the stationary distribution of the process and if init_regimes=m, the initial value are generated from the stationary distribution of the mth regime. Ignored if the argument init_values is specified.

a size (p\times d) matrix specifying the initial values, where d is the number of time series in the system. The last row will be used as initial values for the first lag, the second last row for second lag etc. If not specified, initial values will be drawn according to mixture distribution specifed by the argument init_regimes.

a numeric vector with values in 1,...,d (d=ncol(data)) specifying which the variables for which the impulse responses should be cumulative. Default is none.

should the GFEVD be estimated for the mixing weights as well? Note that this is ignored if M=1 and if M=2 the GFEVD will be the same for both regime's mixing weights.

the number CPU cores to be used in parallel computing. Only single core computing is supported if an initial value is specified (and the GIRF won't thus be estimated multiple times).

a numeric vector containing the random number generator seed for estimation of each GIRF. Should have the length...

• ...nrow(data) - p + 1 if initval_type="data".

• ...R2 if initval_type="random".

• ...1 if initval_type="fixed.".

Set to NULL for not initializing the seed. Exists for creating reproducible results.

object of class 'gfevd' generated by the function GFEVD.

currently not used.

the number of decimals to print

an integer specifying the horizon how far to print the estimates. The default is that all the values are printed.

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

a numeric vector of length at most d (=ncol(data)) and elements in 1,...,d specifying the structural shocks for which the GIRF should be estimated.

a non-zero scalar value specifying the common size for all scalar components of the structural shock. Note that the conditional covariance matrix of the structural shock is an identity matrix and that the (generalized) impulse responses may not be symmetric to the sign and size of the shock.

a positive integer specifying the horizon how far ahead should the generalized impulse responses be calculated.

the number of repetitions used to estimate GIRF for each initial value.

the number of initial values to be drawn from a stationary distribution of the process or of a specific regime? The confidence bounds will be sample quantiles of the GIRFs based on different initial values. Ignored if the argument init_value is specified.

a numeric vector of length at most M and elements in 1,...,M specifying the regimes from which the initial values should be generated from. The initial values will be generated from a mixture distribution with the mixture components being the stationary distributions of the specific regimes and the (proportional) mixing weights given by the mixing weight parameters of those regimes. Note that if init_regimes=1:M, the initial values are generated from the stationary distribution of the process and if init_regimes=m, the initial value are generated from the stationary distribution of the mth regime. Ignored if the argument init_values is specified.

a size (p\times d) matrix specifying the initial values, where d is the number of time series in the system. The last row will be used as initial values for the first lag, the second last row for second lag etc. If not specified, initial values will be drawn according to mixture distribution specifed by the argument init_regimes.

a numeric vector with values in 1,...,d (d=ncol(data)) specifying which the variables for which the impulse responses should be cumulative. Default is none.

should the GIRFs to some of the shocks be scaled so that they correspond to a specific magnitude of instantaneous or peak response of some specific variable (see the argument scale_type)? Provide a length three vector where the shock of interest is given in the first element (an integer in 1,...,d), the variable of interest is given in the second element (an integer in 1,...,d), and the magnitude of its instantaneous or peak response in the third element (a non-zero real number). If the GIRFs of multiple shocks should be scaled, provide a matrix which has one column for each of the shocks with the columns being the length three vectors described above.

If argument scale is specified, should the GIRFs be scaled to match an instantaneous response ("instant") or peak response ("peak"). If "peak", the scale is based on the largest magnitude of peak response in absolute value. Ignored if scale is not specified.

If scale_type == "peak" what the maximum horizon up to which peak response is expected? Scaling won't based on values after this.

a numeric vector with elements in (0, 1) specifying the confidence levels of the confidence intervals.

should the generalized impulse response be calculated for the mixing weights as well? TRUE or FALSE.

the number CPU cores to be used in parallel computing. Only single core computing is supported if an initial value is specified (and the GIRF won't thus be estimated multiple times).

TRUE if the results should be plotted, FALSE if not.

a length R2 vector containing the random number generator seed for estimation of each GIRF. A single number of an initial value is specified. or NULL for not initializing the seed. Exists for creating reproducible results.

arguments passed to grid which plots grid to the figure.

object of class 'girf' generated by the function GIRF.

should grid be added to the plots?

numeric vector of length four that adjusts the [bottom_marginal, left_marginal, top_marginal, right_marginal] as the relative sizes of the marginals to the figures of the responses.

the number of decimals to print

an integer specifying the horizon how far to print the estimates and confidence intervals. The default is that all the values are printed.

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.

For GMVAR and StMVAR models:

a positive integer specifying the number of mixture components.

For G-StMVAR models:

a size (2\times 1) 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.

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.

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 \theta = (\upsilon_{1}, ...,\upsilon_{M}, \alpha_{1},...,\alpha_{M-1},\nu), 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,

• \nu=(\nu_{M1+1},...,\nu_{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 \theta = (\phi_{1,0},...,\phi_{M,0},\psi, \sigma_{1},...,\sigma_{M},\alpha_{1},...,\alpha_{M-1},\nu), where

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

For same_means models:

Should have the form \theta = (\mu,\psi, \sigma_{1},...,\sigma_{M},\alpha_{1},...,\alpha_{M-1},\nu), 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 models with weight_constraints:

Drop \alpha_1,...,\alpha_{M-1} from the parameter vector.

For structural models:

Reduced form models can be directly used as recursively identified structural models. If the structural model is identified by conditional heteroskedasticity, the parameter vector 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},\nu), where

• \lambda_{m}=(\lambda_{m1},...,\lambda_{md}) contains the eigenvalues of the mth 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 via C_lambda:

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.

If \lambda_{mi} are constrained via fixed_lambdas:

Drop \lambda_{2},..., \lambda_{M} from the parameter vector.

Above, \phi_{m,0} is the intercept parameter, A_{m,i} denotes the ith coefficient matrix of the mth 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, 2022). 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.

In the GMVAR model, M1=M and \nu 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 \nu should be strictly larger than two.

The notation is similar to the cited literature.

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

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.

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

a size (Mpd^2 \times 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 \times pd^2) where I = diag(p*d^2). Ignore (or set to NULL) if linear constraints should not be employed.

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".

a numeric vector of length M-1 specifying fixed parameter values for the mixing weight parameters \alpha_m, \ m=1,...,M-1. Each element should be strictly between zero and one, and the sum of all the elements should be strictly less than one.

If NULL a reduced form model is considered. Reduced models can be used directly as recursively identified structural models. For a structural model identified by conditional heteroskedasticity, should be a list containing at least the first one of 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) \times r) constraint matrix that satisfies (\lambda_{2},..., \lambda_{M}) = C_{\lambda} \gamma where \gamma is the new (r \times 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.

• fixed_lambdas - a length d(M-1) numeric vector (\lambda_{2},..., \lambda_{M}) with elements strictly larger than zero specifying the fixed parameter values for the parameters \lambda_{mi} should be constrained to. This constraint is alternative C_lambda. Ignore (or set to NULL) if the eigenvalues \lambda_{mi} should not be constrained.

See Virolainen (forthcoming) 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).

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

should approximate standard errors be calculated?

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.

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.

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 of class 'gsmvar' generated by fitGSMVAR or GSMVAR.

currently not used.

number of digits to be printed.

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

which type figure should be produced? Or both?

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

an object of class 'gsmvar' generated by fitGSMVAR or GSMVAR, containing the freely estimated model.

an object of class 'gsmvar' generated by fitGSMVAR or GSMVAR, containing the constrained model.

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

Should the residuals be standardized? Use FALSE to obtain raw residuals.

an object of class 'gsmvar' generated by fitGSMVAR or GSMVAR, containing the model specified by the null hypothesis (i.e., the constrained model).

a positive integer specifying the autoregressive order of the model.

the number of time series in the system.

[d, d, p] array containing the AR coefficient matrices

the dxd error term covariance matrix

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

a size (k \times n_params) matrix with full row rank specifying part of the null hypothesis where n_params is the number of parameters in the (unconstrained) model. See details for more information.

a length k vector specifying part of the null hypothesis. See details for more information.

a numeric vector with the same length as x specifying the difference h for each dimension separately. If NULL (default), then the difference 1e-6 used for all but overly large degrees of freedom parameters. For them, the difference is adjusted to avoid numerical problems.

a size (dxd) square matrix to be vectorized.

a matrix or class 'ts' object with d>1 columns. Each column is taken to represent a univariate time series. NA values are not supported.

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

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

should approximate standard errors be calculated?

a vector containing the elements to be tested.

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

based on which estimation round should the model be constructed? An integer value in 1,...,ncalls.

based on estimation round with which largest log-likelihood should the model be constructed? An integer value in 1,...,ncalls. For example, which_largest=2 would take the second largest log-likelihood and construct the model based on the corresponding estimates. If used, then which_round is ignored.

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

should approximate standard errors be calculated?

a numeric vector specifying the point where the gradient or Hessian should be calculated.

a function that takes in argument x as the first argument.

difference used to approximate the derivatives.

a numeric vector with the same length as x specifying the difference h for each dimension separately. If NULL (default), then the difference given as parameter h will be used for all dimensions.

other arguments passed to fn

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

same as varying_h except thaqt if NULL (default), then the difference used for differentiating overly large degrees of freedom parameters is adjusted to avoid numerical problems, and the difference 6e-6 is used for the other parameters.

a positive integer specifying the autoregressive order of the model.

For GMVAR and StMVAR models:

a positive integer specifying the number of mixture components.

For G-StMVAR models:

a size (2\times 1) 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.

the number of time series in the system.

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 \theta = (\upsilon_{1}, ...,\upsilon_{M}, \alpha_{1},...,\alpha_{M-1},\nu), 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,

• \nu=(\nu_{M1+1},...,\nu_{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 \theta = (\phi_{1,0},...,\phi_{M,0},\psi, \sigma_{1},...,\sigma_{M},\alpha_{1},...,\alpha_{M-1},\nu), where

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

For same_means models:

Should have the form \theta = (\mu,\psi, \sigma_{1},...,\sigma_{M},\alpha_{1},...,\alpha_{M-1},\nu), 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 models with weight_constraints:

Drop \alpha_1,...,\alpha_{M-1} from the parameter vector.

For structural models:

Reduced form models can be directly used as recursively identified structural models. If the structural model is identified by conditional heteroskedasticity, the parameter vector 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},\nu), where

• \lambda_{m}=(\lambda_{m1},...,\lambda_{md}) contains the eigenvalues of the mth 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 via C_lambda:

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.

If \lambda_{mi} are constrained via fixed_lambdas:

Drop \lambda_{2},..., \lambda_{M} from the parameter vector.

Above, \phi_{m,0} is the intercept parameter, A_{m,i} denotes the ith coefficient matrix of the mth 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, 2022). 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.

In the GMVAR model, M1=M and \nu 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 \nu should be strictly larger than two.

The notation is similar to the cited literature.

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.

a size (Mpd^2 \times 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 \times pd^2) where I = diag(p*d^2). Ignore (or set to NULL) if linear constraints should not be employed.

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".

a numeric vector of length M-1 specifying fixed parameter values for the mixing weight parameters \alpha_m, \ m=1,...,M-1. Each element should be strictly between zero and one, and the sum of all the elements should be strictly less than one.

If NULL a reduced form model is considered. Reduced models can be used directly as recursively identified structural models. For a structural model identified by conditional heteroskedasticity, should be a list containing at least the first one of 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) \times r) constraint matrix that satisfies (\lambda_{2},..., \lambda_{M}) = C_{\lambda} \gamma where \gamma is the new (r \times 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.

• fixed_lambdas - a length d(M-1) numeric vector (\lambda_{2},..., \lambda_{M}) with elements strictly larger than zero specifying the fixed parameter values for the parameters \lambda_{mi} should be constrained to. This constraint is alternative C_lambda. Ignore (or set to NULL) if the eigenvalues \lambda_{mi} should not be constrained.

See Virolainen (forthcoming) 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).

either "intercept" or "mean" specifying to which parametrization it should be switched to. If set to "intercept", it's assumed that params is mean-parametrized, and if set to "mean" it's assumed that params is intercept-parametrized.

a positive integer specifying the autoregressive order of the model.

For GMVAR and StMVAR models:

a positive integer specifying the number of mixture components.

For G-StMVAR models:

a size (2\times 1) 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.

number of time series in the system, i.e. the dimension.

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 \theta = (\upsilon_{1}, ...,\upsilon_{M}, \alpha_{1},...,\alpha_{M-1},\nu), 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,

• \nu=(\nu_{M1+1},...,\nu_{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 \theta = (\phi_{1,0},...,\phi_{M,0},\psi, \sigma_{1},...,\sigma_{M},\alpha_{1},...,\alpha_{M-1},\nu), where

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

For same_means models:

Should have the form \theta = (\mu,\psi, \sigma_{1},...,\sigma_{M},\alpha_{1},...,\alpha_{M-1},\nu), 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 models with weight_constraints:

Drop \alpha_1,...,\alpha_{M-1} from the parameter vector.

For structural models:

Reduced form models can be directly used as recursively identified structural models. If the structural model is identified by conditional heteroskedasticity, the parameter vector 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},\nu), where

• \lambda_{m}=(\lambda_{m1},...,\lambda_{md}) contains the eigenvalues of the mth 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 via C_lambda:

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.

If \lambda_{mi} are constrained via fixed_lambdas:

Drop \lambda_{2},..., \lambda_{M} from the parameter vector.

Above, \phi_{m,0} is the intercept parameter, A_{m,i} denotes the ith coefficient matrix of the mth 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, 2022). 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.

In the GMVAR model, M1=M and \nu 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 \nu should be strictly larger than two.

The notation is similar to the cited literature.

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.

which component?

For reduced form models:

a size ((pd^2+d+d(d+1)/2)x1) vector (\upsilon_{m},\nu_m) = (\phi_{m,0},\phi_{m},\sigma_{m},\nu_m).

For structural models:

a length pd^2 + d vector (\phi_{m,0},\phi_{m},\nu_m).

In the case of a GMVAR type regime, \nu_m is omitted.

If NULL a reduced form model is considered. Reduced models can be used directly as recursively identified structural models. For a structural model identified by conditional heteroskedasticity, should be a list containing at least the first one of 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) \times r) constraint matrix that satisfies (\lambda_{2},..., \lambda_{M}) = C_{\lambda} \gamma where \gamma is the new (r \times 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.

• fixed_lambdas - a length d(M-1) numeric vector (\lambda_{2},..., \lambda_{M}) with elements strictly larger than zero specifying the fixed parameter values for the parameters \lambda_{mi} should be constrained to. This constraint is alternative C_lambda. Ignore (or set to NULL) if the eigenvalues \lambda_{mi} should not be constrained.

See Virolainen (forthcoming) 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).

a positive integer specifying the autoregressive order of the model.

For GMVAR and StMVAR models:

a positive integer specifying the number of mixture components.

For G-StMVAR models:

a size (2\times 1) 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.

the number of time series in the system.

a size (Mpd^2 \times 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 \times pd^2) where I = diag(p*d^2). Ignore (or set to NULL) if linear constraints should not be employed.

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".

a numeric vector of length M-1 specifying fixed parameter values for the mixing weight parameters \alpha_m, \ m=1,...,M-1. Each element should be strictly between zero and one, and the sum of all the elements should be strictly less than one.

If NULL a reduced form model is considered. Reduced models can be used directly as recursively identified structural models. For a structural model identified by conditional heteroskedasticity, should be a list containing at least the first one of 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) \times r) constraint matrix that satisfies (\lambda_{2},..., \lambda_{M}) = C_{\lambda} \gamma where \gamma is the new (r \times 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.

• fixed_lambdas - a length d(M-1) numeric vector (\lambda_{2},..., \lambda_{M}) with elements strictly larger than zero specifying the fixed parameter values for the parameters \lambda_{mi} should be constrained to. This constraint is alternative C_lambda. Ignore (or set to NULL) if the eigenvalues \lambda_{mi} should not be constrained.

See Virolainen (forthcoming) 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).

a matrix or class 'ts' object with d>1 columns. Each column is taken to represent a univariate time series. NA values are not supported.

a positive integer specifying the autoregressive order of the model.

S3 object to be tested

what is the name of the object that should of class 'gsmvar'?

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

a positive integer specifying the autoregressive order of the model.

For GMVAR and StMVAR models:

a positive integer specifying the number of mixture components.

For G-StMVAR models:

a size (2\times 1) 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.

the number of time series in the system.

a positive integer specifying the autoregressive order of the model.

For GMVAR and StMVAR models:

a positive integer specifying the number of mixture components.

For G-StMVAR models:

a size (2\times 1) 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.

the number of time series in the system.

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 \theta = (\upsilon_{1}, ...,\upsilon_{M}, \alpha_{1},...,\alpha_{M-1},\nu), 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,

• \nu=(\nu_{M1+1},...,\nu_{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 \theta = (\phi_{1,0},...,\phi_{M,0},\psi, \sigma_{1},...,\sigma_{M},\alpha_{1},...,\alpha_{M-1},\nu), where

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

For same_means models:

Should have the form \theta = (\mu,\psi, \sigma_{1},...,\sigma_{M},\alpha_{1},...,\alpha_{M-1},\nu), 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 models with weight_constraints:

Drop \alpha_1,...,\alpha_{M-1} from the parameter vector.

For structural models:

Reduced form models can be directly used as recursively identified structural models. If the structural model is identified by conditional heteroskedasticity, the parameter vector 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},\nu), where

• \lambda_{m}=(\lambda_{m1},...,\lambda_{md}) contains the eigenvalues of the mth 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 via C_lambda:

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.

If \lambda_{mi} are constrained via fixed_lambdas:

Drop \lambda_{2},..., \lambda_{M} from the parameter vector.

Above, \phi_{m,0} is the intercept parameter, A_{m,i} denotes the ith coefficient matrix of the mth 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, 2022). 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.

In the GMVAR model, M1=M and \nu 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 \nu should be strictly larger than two.

The notation is similar to the cited literature.

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.

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

a size (Mpd^2 \times 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 \times pd^2) where I = diag(p*d^2). Ignore (or set to NULL) if linear constraints should not be employed.

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".

a numeric vector of length M-1 specifying fixed parameter values for the mixing weight parameters \alpha_m, \ m=1,...,M-1. Each element should be strictly between zero and one, and the sum of all the elements should be strictly less than one.

If NULL a reduced form model is considered. Reduced models can be used directly as recursively identified structural models. For a structural model identified by conditional heteroskedasticity, should be a list containing at least the first one of 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) \times r) constraint matrix that satisfies (\lambda_{2},..., \lambda_{M}) = C_{\lambda} \gamma where \gamma is the new (r \times 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.

• fixed_lambdas - a length d(M-1) numeric vector (\lambda_{2},..., \lambda_{M}) with elements strictly larger than zero specifying the fixed parameter values for the parameters \lambda_{mi} should be constrained to. This constraint is alternative C_lambda. Ignore (or set to NULL) if the eigenvalues \lambda_{mi} should not be constrained.

See Virolainen (forthcoming) 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).

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.

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.

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

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

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".

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

should conditional means or variances be plotted?

add grid to the plots?

additional paramters passed to grid(...) plotting the grid if grid == TRUE.

a matrix or class 'ts' object with d>1 columns. Each column is taken to represent a univariate time series. NA values are not supported.

a positive integer specifying the autoregressive order of the model.

For GMVAR and StMVAR models:

a positive integer specifying the number of mixture components.

For G-StMVAR models:

a size (2\times 1) 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.

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 \theta = (\upsilon_{1}, ...,\upsilon_{M}, \alpha_{1},...,\alpha_{M-1},\nu), 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,

• \nu=(\nu_{M1+1},...,\nu_{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 \theta = (\phi_{1,0},...,\phi_{M,0},\psi, \sigma_{1},...,\sigma_{M},\alpha_{1},...,\alpha_{M-1},\nu), where

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

For same_means models:

Should have the form \theta = (\mu,\psi, \sigma_{1},...,\sigma_{M},\alpha_{1},...,\alpha_{M-1},\nu), 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 models with weight_constraints:

Drop \alpha_1,...,\alpha_{M-1} from the parameter vector.

For structural models:

Reduced form models can be directly used as recursively identified structural models. If the structural model is identified by conditional heteroskedasticity, the parameter vector 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},\nu), where

• \lambda_{m}=(\lambda_{m1},...,\lambda_{md}) contains the eigenvalues of the mth 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 via C_lambda:

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.

If \lambda_{mi} are constrained via fixed_lambdas:

Drop \lambda_{2},..., \lambda_{M} from the parameter vector.

Above, \phi_{m,0} is the intercept parameter, A_{m,i} denotes the ith coefficient matrix of the mth 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, 2022). 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.

In the GMVAR model, M1=M and \nu 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 \nu should be strictly larger than two.

The notation is similar to the cited literature.

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.

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

a size (Mpd^2 \times 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 \times pd^2) where I = diag(p*d^2). Ignore (or set to NULL) if linear constraints should not be employed.

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".

a numeric vector of length M-1 specifying fixed parameter values for the mixing weight parameters \alpha_m, \ m=1,...,M-1. Each element should be strictly between zero and one, and the sum of all the elements should be strictly less than one.

If NULL a reduced form model is considered. Reduced models can be used directly as recursively identified structural models. For a structural model identified by conditional heteroskedasticity, should be a list containing at least the first one of 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) \times r) constraint matrix that satisfies (\lambda_{2},..., \lambda_{M}) = C_{\lambda} \gamma where \gamma is the new (r \times 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.

• fixed_lambdas - a length d(M-1) numeric vector (\lambda_{2},..., \lambda_{M}) with elements strictly larger than zero specifying the fixed parameter values for the parameters \lambda_{mi} should be constrained to. This constraint is alternative C_lambda. Ignore (or set to NULL) if the eigenvalues \lambda_{mi} should not be constrained.

See Virolainen (forthcoming) 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).

should the regimewise conditional means, total conditional means, or total conditional covariance matrices be returned?

the value that will be returned if the parameter vector does not lie in the parameter space (excluding the identification condition).

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.

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.

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

An integer representing the dimension of the identity matrix.

An integer representing the factor by which to extend the matrix with zeros.

a positive definite (dxd) covariance matrix (d>1)

another positive definite (dxd) covariance matrix

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

which type of diagnostic plot should be plotted?

• "all" all below sequentially.

• "series" the quantile residual time series.

• "ac" the quantile residual autocorrelation and cross-correlation functions.

• "ch" the squared quantile residual autocorrelation and cross-correlation functions.

• "norm" the quantile residual histogram with theoretical standard normal density (dashed line) and standard normal QQ-plots.

the maximum lag considered in types "ac" and "ch".

if type == all how many seconds to wait before showing next figure?

dimension (T x k) matrix where each row is a k-dimensional vector

dimension (T x k) matrix where each row is the mean of the k-dimensional vector in corresponding row of y.

inverse of the (k x k) covariance matrix Omega.

logarithm of the determinant of the covariance matrix Omega.

dimension (T x k) matrix where each row is a k-dimensional vector

dimension (T x k) matrix where each row is the mean of the k-dimensional vector in corresponding row of y.

inverse of the (k x k) covariance matrix Omega.

logarithm of the determinant of the covariance matrix Omega.

length T numeric vector containing the coefficients that multiply the covariance matrix Omega.

the degrees of freedom parameter that is common for all t=1,...,T.

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

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.

the number of estimation rounds that should be performed.

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

the maximum number of iterations in the variable metric algorithm.

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.

a matrix or class 'ts' object with d>1 columns. Each column is taken to represent a univariate time series. NA values are not supported.

a positive integer specifying the autoregressive order of the model.

For GMVAR and StMVAR models:

a positive integer specifying the number of mixture components.

For G-StMVAR models:

a size (2\times 1) 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.

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.

a logical argument specifying whether the conditional or exact log-likelihood function

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

a size (Mpd^2 \times 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 \times pd^2) where I = diag(p*d^2). Ignore (or set to NULL) if linear constraints should not be employed.

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".

a numeric vector of length M-1 specifying fixed parameter values for the mixing weight parameters \alpha_m, \ m=1,...,M-1. Each element should be strictly between zero and one, and the sum of all the elements should be strictly less than one.

If NULL a reduced form model is considered. Reduced models can be used directly as recursively identified structural models. For a structural model identified by conditional heteroskedasticity, should be a list containing at least the first one of 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) \times r) constraint matrix that satisfies (\lambda_{2},..., \lambda_{M}) = C_{\lambda} \gamma where \gamma is the new (r \times 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.

• fixed_lambdas - a length d(M-1) numeric vector (\lambda_{2},..., \lambda_{M}) with elements strictly larger than zero specifying the fixed parameter values for the parameters \lambda_{mi} should be constrained to. This constraint is alternative C_lambda. Ignore (or set to NULL) if the eigenvalues \lambda_{mi} should not be constrained.

See Virolainen (forthcoming) 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).

the number of estimation rounds that should be performed.

the number CPU cores to be used in parallel computing.

the maximum number of iterations in the variable metric algorithm.

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.

should summaries of estimation results be printed?

employ parallel computing? If FALSE, no progression bar etc will be generated.

should the likely inappropriate estimates be filtered? See details.

calculate approximate standard errors for the estimates?

additional settings passed to the function GAfit employing the genetic algorithm.

a positive integer specifying the autoregressive order of the model.

For GMVAR and StMVAR models:

a positive integer specifying the number of mixture components.

For G-StMVAR models:

a size (2\times 1) 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.

the number of time series in the system.

4D array containing all coefficient matrices A_{m,i}, obtained from pick_allA.

the number of decimals to print

log-likelihood value

number of (freely estimated) parameters in the model

numbers of observations with starting values excluded for conditional models.

a positive integer specifying the autoregressive order of the model.

For GMVAR and StMVAR models:

a positive integer specifying the number of mixture components.

For G-StMVAR models:

a size (2\times 1) 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.

the number of time series in the system.

4D array containing all coefficient matrices A_{m,i}, obtained from pick_allA.

3D array containing the ((dp)\times (dp)) "bold A" matrices related to each mixture component VAR-process, obtained from form_boldA. Will be computed if not given.

a [d, d, M] array containing the covariance matrix Omegas

For GMVAR and StMVAR models:

a positive integer specifying the number of mixture components.

For G-StMVAR models:

a size (2\times 1) 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.

T x M matrix containing the log multivariate normal densities.

M x 1 vector containing the mixing weight pa

the smallest number such that its exponent is wont classified as numerically zero (around -698 is used).

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

return also l_0 (the first term in the exact log-likelihood function)?

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

a matrix or class 'ts' object with d>1 columns. Each column is taken to represent a univariate time series. NA values are not supported.

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

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

a positive integer specifying the autoregressive order of the model.

For GMVAR and StMVAR models:

a positive integer specifying the number of mixture components.

For G-StMVAR models:

a size (2\times 1) 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.

the number of time series in the system.

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 \theta = (\upsilon_{1}, ...,\upsilon_{M}, \alpha_{1},...,\alpha_{M-1},\nu), 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,

• \nu=(\nu_{M1+1},...,\nu_{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 \theta = (\phi_{1,0},...,\phi_{M,0},\psi, \sigma_{1},...,\sigma_{M},\alpha_{1},...,\alpha_{M-1},\nu), where

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

For same_means models:

Should have the form \theta = (\mu,\psi, \sigma_{1},...,\sigma_{M},\alpha_{1},...,\alpha_{M-1},\nu), 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 models with weight_constraints:

Drop \alpha_1,...,\alpha_{M-1} from the parameter vector.

For structural models:

Reduced form models can be directly used as recursively identified structural models. If the structural model is identified by conditional heteroskedasticity, the parameter vector 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},\nu), where

• \lambda_{m}=(\lambda_{m1},...,\lambda_{md}) contains the eigenvalues of the mth 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 via C_lambda:

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.

If \lambda_{mi} are constrained via fixed_lambdas:

Drop \lambda_{2},..., \lambda_{M} from the parameter vector.

Above, \phi_{m,0} is the intercept parameter, A_{m,i} denotes the ith coefficient matrix of the mth 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, 2022). 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.

In the GMVAR model, M1=M and \nu 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 \nu should be strictly larger than two.

The notation is similar to the cited literature.

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.

a size (Mpd^2 \times 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 \times pd^2) where I = diag(p*d^2). Ignore (or set to NULL) if linear constraints should not be employed.

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".

a numeric vector of length M-1 specifying fixed parameter values for the mixing weight parameters \alpha_m, \ m=1,...,M-1. Each element should be strictly between zero and one, and the sum of all the elements should be strictly less than one.

If NULL a reduced form model is considered. Reduced models can be used directly as recursively identified structural models. For a structural model identified by conditional heteroskedasticity, should be a list containing at least the first one of 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) \times r) constraint matrix that satisfies (\lambda_{2},..., \lambda_{M}) = C_{\lambda} \gamma where \gamma is the new (r \times 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.

• fixed_lambdas - a length d(M-1) numeric vector (\lambda_{2},..., \lambda_{M}) with elements strictly larger than zero specifying the fixed parameter values for the parameters \lambda_{mi} should be constrained to. This constraint is alternative C_lambda. Ignore (or set to NULL) if the eigenvalues \lambda_{mi} should not be constrained.

See Virolainen (forthcoming) 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).

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

a positive integer specifying the autoregressive order of the model.

For GMVAR and StMVAR models:

a positive integer specifying the number of mixture components.

For G-StMVAR models:

a size (2\times 1) 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.

the number of time series in the system.

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 \theta = (\upsilon_{1}, ...,\upsilon_{M}, \alpha_{1},...,\alpha_{M-1},\nu), 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,

• \nu=(\nu_{M1+1},...,\nu_{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 \theta = (\phi_{1,0},...,\phi_{M,0},\psi, \sigma_{1},...,\sigma_{M},\alpha_{1},...,\alpha_{M-1},\nu), where

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

For same_means models:

Should have the form \theta = (\mu,\psi, \sigma_{1},...,\sigma_{M},\alpha_{1},...,\alpha_{M-1},\nu), 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 models with weight_constraints:

Drop \alpha_1,...,\alpha_{M-1} from the parameter vector.

For structural models:

Reduced form models can be directly used as recursively identified structural models. If the structural model is identified by conditional heteroskedasticity, the parameter vector 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},\nu), where

• \lambda_{m}=(\lambda_{m1},...,\lambda_{md}) contains the eigenvalues of the mth 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 via C_lambda:

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.

If \lambda_{mi} are constrained via fixed_lambdas:

Drop \lambda_{2},..., \lambda_{M} from the parameter vector.

Above, \phi_{m,0} is the intercept parameter, A_{m,i} denotes the ith coefficient matrix of the mth 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, 2022). 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.

In the GMVAR model, M1=M and \nu 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 \nu should be strictly larger than two.

The notation is similar to the cited literature.

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.

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

a size (Mpd^2 \times 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 \times pd^2) where I = diag(p*d^2). Ignore (or set to NULL) if linear constraints should not be employed.

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".

a numeric vector of length M-1 specifying fixed parameter values for the mixing weight parameters \alpha_m, \ m=1,...,M-1. Each element should be strictly between zero and one, and the sum of all the elements should be strictly less than one.

If NULL a reduced form model is considered. Reduced models can be used directly as recursively identified structural models. For a structural model identified by conditional heteroskedasticity, should be a list containing at least the first one of 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) \times r) constraint matrix that satisfies (\lambda_{2},..., \lambda_{M}) = C_{\lambda} \gamma where \gamma is the new (r \times 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.

• fixed_lambdas - a length d(M-1) numeric vector (\lambda_{2},..., \lambda_{M}) with elements strictly larger than zero specifying the fixed parameter values for the parameters \lambda_{mi} should be constrained to. This constraint is alternative C_lambda. Ignore (or set to NULL) if the eigenvalues \lambda_{mi} should not be constrained.

See Virolainen (forthcoming) 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).

a positive definite covariance matrix

a matrix or class 'ts' object with d>1 columns. Each column is taken to represent a univariate time series. NA values are not supported.

a positive integer specifying the autoregressive order of the model.

For GMVAR and StMVAR models:

a positive integer specifying the number of mixture components.

For G-StMVAR models:

a size (2\times 1) 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.

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 \theta = (\upsilon_{1}, ...,\upsilon_{M}, \alpha_{1},...,\alpha_{M-1},\nu), 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,

• \nu=(\nu_{M1+1},...,\nu_{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 \theta = (\phi_{1,0},...,\phi_{M,0},\psi, \sigma_{1},...,\sigma_{M},\alpha_{1},...,\alpha_{M-1},\nu), where

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

For same_means models:

Should have the form \theta = (\mu,\psi, \sigma_{1},...,\sigma_{M},\alpha_{1},...,\alpha_{M-1},\nu), 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 models with weight_constraints:

Drop \alpha_1,...,\alpha_{M-1} from the parameter vector.

For structural models:

Reduced form models can be directly used as recursively identified structural models. If the structural model is identified by conditional heteroskedasticity, the parameter vector 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},\nu), where

• \lambda_{m}=(\lambda_{m1},...,\lambda_{md}) contains the eigenvalues of the mth 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 via C_lambda:

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.

If \lambda_{mi} are constrained via fixed_lambdas:

Drop \lambda_{2},..., \lambda_{M} from the parameter vector.

Above, \phi_{m,0} is the intercept parameter, A_{m,i} denotes the ith coefficient matrix of the mth 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, 2022). 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.

In the GMVAR model, M1=M and \nu 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 \nu should be strictly larger than two.

The notation is similar to the cited literature.

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.

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

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

a size (Mpd^2 \times 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 \times pd^2) where I = diag(p*d^2). Ignore (or set to NULL) if linear constraints should not be employed.

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".

a numeric vector of length M-1 specifying fixed parameter values for the mixing weight parameters \alpha_m, \ m=1,...,M-1. Each element should be strictly between zero and one, and the sum of all the elements should be strictly less than one.

If NULL a reduced form model is considered. Reduced models can be used directly as recursively identified structural models. For a structural model identified by conditional heteroskedasticity, should be a list containing at least the first one of 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) \times r) constraint matrix that satisfies (\lambda_{2},..., \lambda_{M}) = C_{\lambda} \gamma where \gamma is the new (r \times 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.

• fixed_lambdas - a length d(M-1) numeric vector (\lambda_{2},..., \lambda_{M}) with elements strictly larger than zero specifying the fixed parameter values for the parameters \lambda_{mi} should be constrained to. This constraint is alternative C_lambda. Ignore (or set to NULL) if the eigenvalues \lambda_{mi} should not be constrained.

See Virolainen (forthcoming) 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).

function g specifying the transformation.

output dimension of the transformation g.

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

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.

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.

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

If NULL a reduced form model is considered. Reduced models can be used directly as recursively identified structural models. For a structural model identified by conditional heteroskedasticity, should be a list containing at least the first one of 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) \times r) constraint matrix that satisfies (\lambda_{2},..., \lambda_{M}) = C_{\lambda} \gamma where \gamma is the new (r \times 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.

• fixed_lambdas - a length d(M-1) numeric vector (\lambda_{2},..., \lambda_{M}) with elements strictly larger than zero specifying the fixed parameter values for the parameters \lambda_{mi} should be constrained to. This constraint is alternative C_lambda. Ignore (or set to NULL) if the eigenvalues \lambda_{mi} should not be constrained.

See Virolainen (forthcoming) 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).

For GMVAR and StMVAR models:

a positive integer specifying the number of mixture components.

For G-StMVAR models:

a size (2\times 1) 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.

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 \theta = (\upsilon_{1}, ...,\upsilon_{M}, \alpha_{1},...,\alpha_{M-1},\nu), 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,

• \nu=(\nu_{M1+1},...,\nu_{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 \theta = (\phi_{1,0},...,\phi_{M,0},\psi, \sigma_{1},...,\sigma_{M},\alpha_{1},...,\alpha_{M-1},\nu), where

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

For same_means models:

Should have the form \theta = (\mu,\psi, \sigma_{1},...,\sigma_{M},\alpha_{1},...,\alpha_{M-1},\nu), 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 models with weight_constraints:

Drop \alpha_1,...,\alpha_{M-1} from the parameter vector.

For structural models:

Reduced form models can be directly used as recursively identified structural models. If the structural model is identified by conditional heteroskedasticity, the parameter vector 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},\nu), where

• \lambda_{m}=(\lambda_{m1},...,\lambda_{md}) contains the eigenvalues of the mth 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 via C_lambda:

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.

If \lambda_{mi} are constrained via fixed_lambdas:

Drop \lambda_{2},..., \lambda_{M} from the parameter vector.

Above, \phi_{m,0} is the intercept parameter, A_{m,i} denotes the ith coefficient matrix of the mth 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, 2022). 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.

In the GMVAR model, M1=M and \nu 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 \nu should be strictly larger than two.

The notation is similar to the cited literature.

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.

a class 'gmvar' or 'gsmvar' object.

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

should approximate standard errors be calculated?

if M == 1, should the lower triangular Cholesky identification be employed? See details for using Cholesky identification with M > 1.

a positive integer specifying the autoregressive order of the model.

For GMVAR and StMVAR models:

a positive integer specifying the number of mixture components.

For G-StMVAR models:

a size (2\times 1) 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.

the number of time series in the system.

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 \theta = (\upsilon_{1}, ...,\upsilon_{M}, \alpha_{1},...,\alpha_{M-1},\nu), 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,

• \nu=(\nu_{M1+1},...,\nu_{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 \theta = (\phi_{1,0},...,\phi_{M,0},\psi, \sigma_{1},...,\sigma_{M},\alpha_{1},...,\alpha_{M-1},\nu), where

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

For same_means models:

Should have the form \theta = (\mu,\psi, \sigma_{1},...,\sigma_{M},\alpha_{1},...,\alpha_{M-1},\nu), 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 models with weight_constraints:

Drop \alpha_1,...,\alpha_{M-1} from the parameter vector.

For structural models:

Reduced form models can be directly used as recursively identified structural models. If the structural model is identified by conditional heteroskedasticity, the parameter vector 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},\nu), where

• \lambda_{m}=(\lambda_{m1},...,\lambda_{md}) contains the eigenvalues of the mth 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 via C_lambda:

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.

If \lambda_{mi} are constrained via fixed_lambdas:

Drop \lambda_{2},..., \lambda_{M} from the parameter vector.

Above, \phi_{m,0} is the intercept parameter, A_{m,i} denotes the ith coefficient matrix of the mth 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, 2022). 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.

In the GMVAR model, M1=M and \nu 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 \nu should be strictly larger than two.