a numeric vector or class 'ts' object containing the data. NA values are not supported.

a positive integer specifying the autoregressive order of the model.

For GMAR and StMAR models:

a positive integer specifying the number of mixture components.

For G-StMAR models:

a size (2\times 1) integer vector specifying the number of GMAR type components M1 in the first element and StMAR type components M2 in the second element. The total number of mixture components is M=M1+M2.

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

a logical argument stating whether the AR coefficients \phi_{m,1},...,\phi_{m,p} are restricted to be the same for all regimes.

specifies linear constraints imposed to each regime's autoregressive parameters separately.

For non-restricted models:

a list of size (p \times q_{m}) constraint matrices C_{m} of full column rank satisfying \phi_{m}=C_{m}\psi_{m} for all m=1,...,M, where \phi_{m}=(\phi_{m,1},...,\phi_{m,p}) and \psi_{m}=(\psi_{m,1},...,\psi_{m,q_{m}}).

For restricted models:

a size (p\times q) constraint matrix C of full column rank satisfying \phi=C\psi, where \phi=(\phi_{1},...,\phi_{p}) and \psi=\psi_{1},...,\psi_{q}.

The symbol \phi denotes an AR coefficient. Note that regardless of any constraints, the autoregressive order is always p for all regimes. Ignore or set to NULL if applying linear constraints is not desired.

is the model parametrized with the "intercepts" \phi_{m,0} or "means" \mu_{m} = \phi_{m,0}/(1-\sum\phi_{i,m})?

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

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*d where d is the number of parameters.

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 so far.

a real valued vector of length two specifying the mean (the first element) and standard deviation (the second element) of the normal distribution from which the \mu_{m} mean-parameters are generated in random mutations in the genetic algorithm. Default is c(mean(data), sd(data)). Note that the genetic algorithm optimizes with mean-parametrization even when parametrization=="intercept", but input (in initpop) and output (return value) parameter vectors may be intercept-parametrized.

a positive real number specifying the standard deviation of the (zero mean, positive only by taking absolute value) normal distribution from which the component variance parameters are generated in the random mutations in the genetic algorithm. Default is var(stats::ar(data, order.max=10)$resid, na.rm=TRUE).

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

For non-restricted models:

Size (M(p+3)+M-M1-1\times 1) vector \theta=(\upsilon_{1},...,\upsilon_{M}, \alpha_{1},...,\alpha_{M-1},\nu) where

• \upsilon_{m}=(\phi_{m,0},\phi_{m},\sigma_{m}^2)

• \phi_{m}=(\phi_{m,1},...,\phi_{m,p}), m=1,...,M

• \nu=(\nu_{M1+1},...,\nu_{M})

• M1 is the number of GMAR type regimes.

In the GMAR model, M1=M and the parameter \nu dropped. In the StMAR model, M1=0.

If the model imposes linear constraints on the autoregressive parameters: Replace the vectors \phi_{m} with the vectors \psi_{m} that satisfy \phi_{m}=C_{m}\psi_{m} (see the argument constraints).

For restricted models:

Size (3M+M-M1+p-1\times 1) vector \theta=(\phi_{1,0},...,\phi_{M,0},\phi, \sigma_{1}^2,...,\sigma_{M}^2,\alpha_{1},...,\alpha_{M-1},\nu), where \phi=(\phi_{1},...,\phi_{p}) contains the AR coefficients, which are common for all regimes.

If the model imposes linear constraints on the autoregressive parameters: Replace the vector \phi with the vector \psi that satisfies \phi=C\psi (see the argument constraints).

Symbol \phi denotes an AR coefficient, \sigma^2 a variance, \alpha a mixing weight, and \nu a degrees of freedom parameter. If parametrization=="mean", just replace each intercept term \phi_{m,0} with the regimewise mean \mu_m = \phi_{m,0}/(1-\sum\phi_{i,m}). In the G-StMAR model, the first M1 components are GMAR type and the rest M2 components are StMAR type. Note that in the case M=1, the mixing weight parameters \alpha are dropped, and in the case of StMAR or G-StMAR model, the degrees of freedom parameters \nu have to be larger than 2. If not specified (or NULL as is default), the initial population will be drawn randomly.

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 (see red_criteria), i.e., with less "redundant regimes". 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 favoring 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 probabilities etc.

a length 2 numeric vector specifying the criteria that is used to determine whether a regime is redundant or not. Any regime m which satisfies sum(mixing_weights[,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).

should the genetic algorithm return the best fitting individual which has the least "redundant" regimes ("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(length(data))) + 1) - 1), and one should be very careful if adjusting this.

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 fitGSMAR, use the argument seeds instead of passing the argument seed.

We currently use this to catch deprecated arguments.

a numeric vector or class 'ts' object containing the data. NA values are not supported.

a positive integer specifying the autoregressive order of the model.

For GMAR and StMAR models:

a positive integer specifying the number of mixture components.

For G-StMAR models:

a size (2\times 1) integer vector specifying the number of GMAR type components M1 in the first element and StMAR type components M2 in the second element. The total number of mixture components is M=M1+M2.

a real valued parameter vector specifying the model.

For non-restricted models:

Size (M(p+3)+M-M1-1\times 1) vector \theta=(\upsilon_{1},...,\upsilon_{M}, \alpha_{1},...,\alpha_{M-1},\nu) where

• \upsilon_{m}=(\phi_{m,0},\phi_{m},\sigma_{m}^2)

• \phi_{m}=(\phi_{m,1},...,\phi_{m,p}), m=1,...,M

• \nu=(\nu_{M1+1},...,\nu_{M})

• M1 is the number of GMAR type regimes.

In the GMAR model, M1=M and the parameter \nu dropped. In the StMAR model, M1=0.

If the model imposes linear constraints on the autoregressive parameters: Replace the vectors \phi_{m} with the vectors \psi_{m} that satisfy \phi_{m}=C_{m}\psi_{m} (see the argument constraints).

For restricted models:

Size (3M+M-M1+p-1\times 1) vector \theta=(\phi_{1,0},...,\phi_{M,0},\phi, \sigma_{1}^2,...,\sigma_{M}^2,\alpha_{1},...,\alpha_{M-1},\nu), where \phi=(\phi_{1},...,\phi_{p}) contains the AR coefficients, which are common for all regimes.

If the model imposes linear constraints on the autoregressive parameters: Replace the vector \phi with the vector \psi that satisfies \phi=C\psi (see the argument constraints).

Symbol \phi denotes an AR coefficient, \sigma^2 a variance, \alpha a mixing weight, and \nu a degrees of freedom parameter. If parametrization=="mean", just replace each intercept term \phi_{m,0} with the regimewise mean \mu_m = \phi_{m,0}/(1-\sum\phi_{i,m}). In the G-StMAR model, the first M1 components are GMAR type and the rest M2 components are StMAR type. Note that in the case M=1, the mixing weight parameters \alpha are dropped, and in the case of StMAR or G-StMAR model, the degrees of freedom parameters \nu have to be larger than 2.

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

a logical argument stating whether the AR coefficients \phi_{m,1},...,\phi_{m,p} are restricted to be the same for all regimes.

specifies linear constraints imposed to each regime's autoregressive parameters separately.

For non-restricted models:

a list of size (p \times q_{m}) constraint matrices C_{m} of full column rank satisfying \phi_{m}=C_{m}\psi_{m} for all m=1,...,M, where \phi_{m}=(\phi_{m,1},...,\phi_{m,p}) and \psi_{m}=(\psi_{m,1},...,\psi_{m,q_{m}}).

For restricted models:

a size (p\times q) constraint matrix C of full column rank satisfying \phi=C\psi, where \phi=(\phi_{1},...,\phi_{p}) and \psi=\psi_{1},...,\psi_{q}.

The symbol \phi denotes an AR coefficient. Note that regardless of any constraints, the autoregressive order is always p for all regimes. Ignore or set to NULL if applying linear constraints is not desired.

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

is the model parametrized with the "intercepts" \phi_{m,0} or "means" \mu_{m} = \phi_{m,0}/(1-\sum\phi_{i,m})?

should quantile residuals be calculated? Default is TRUE iff the model contains data.

should conditional means and variances be calculated? Default is TRUE iff the model contains data.

should approximate standard errors be calculated?

A numeric vector with same the length as the parameter vector: i:th element of custom_h is the difference used in central difference approximation for partial differentials of the log-likelihood function for the i:th parameter. If NULL (default), then the difference used for differentiating overly large degrees of freedom parameters is adjusted to avoid numerical problems, and the difference is 6e-6 for the other parameters.

object of class 'gsmar' created with fitGSMAR or GSMAR.

in the plot method: arguments passed to the function density which calculates the kernel density estimate of the data.

number of digits to be printed (max 20)

object of class 'gsmar' created with fitGSMAR or GSMAR.

Plot also kernel density estimate of the data and model implied stationary density with regimewise densities? See the details.

if set to TRUE then the print will include approximate standard errors for the estimates, log-likelihood, information criteria values, modulus of the roots of the characteristic AR polynomials for each regime, and several unconditional moments.

an object of class 'gsmar' generated by fitGSMAR or GSMAR, containing the unconstrained model.

an object of class 'gsmar' generated by fitGSMAR or GSMAR, containing the constrained model.

the value of the likelihood ratio statistics.

the degrees of freedom of the likelihood ratio statistic.

the p-value of the test.

a character string describing the alternative hypothesis.

a character string indicating the type of the test (likelihood ratio test).

a character string giving the names of the supplied models, gsmar1 and gsmar2.

the supplied argument gsmar1

the supplied argument gsmar2

a class 'gsmar' object, typically generated by fitGSMAR or GSMAR.

a size (k \times n_params) matrix with full row rank specifying a 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 a part of the null hypothesis. See details for more information.

the difference used to approximate the derivatives.

the value of the Wald statistics.

the degrees of freedom of the Wald statistic.

the p-value of the test.

a character string describing the alternative hypothesis.

a character string indicating the type of the test (Wald test).

a character string giving the names of the supplied model, constraint matrix A, and vector c.

the supplied argument gsmar.

the supplied argument A.

the supplied argument c.

the supplied argument h.

a numeric vector or class 'ts' object containing the data. NA values are not supported.

a class 'gsmar' object, typically generated by fitGSMAR or GSMAR.

should quantile residuals be calculated? Default is TRUE iff the model contains data.

should conditional means and variances be calculated? Default is TRUE iff the model contains data.

should approximate standard errors be calculated?

A numeric vector with same the length as the parameter vector: i:th element of custom_h is the difference used in central difference approximation for partial differentials of the log-likelihood function for the i:th parameter. If NULL (default), then the difference used for differentiating overly large degrees of freedom parameters is adjusted to avoid numerical problems, and the difference is 6e-6 for the other parameters.

a vector to add the dfs to

how many dfs?

a vector containing the elements to be tested.

a class 'gsmar' object, typically generated by fitGSMAR or GSMAR.

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 specified, then which_round is ignored.

should quantile residuals be calculated? Default is TRUE iff the model contains data.

should conditional means and variances be calculated? Default is TRUE iff the model contains data.

should approximate standard errors be calculated?

A numeric vector with same the length as the parameter vector: i:th element of custom_h is the difference used in central difference approximation for partial differentials of the log-likelihood function for the i:th parameter. If NULL (default), then the difference used for differentiating overly large degrees of freedom parameters is adjusted to avoid numerical problems, and the difference is 6e-6 for the other parameters.

a numeric vector specifying the point at which the gradient or Hessian should be evaluated.

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

the 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.

a class 'gsmar' object, typically generated by fitGSMAR or GSMAR.

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

a positive integer specifying the autoregressive order of the model.

For GMAR and StMAR models:

a positive integer specifying the number of mixture components.

For G-StMAR models:

a size (2\times 1) integer vector specifying the number of GMAR type components M1 in the first element and StMAR type components M2 in the second element. The total number of mixture components is M=M1+M2.

a real valued parameter vector specifying the model.

For non-restricted models:

Size (M(p+3)+M-M1-1\times 1) vector \theta=(\upsilon_{1},...,\upsilon_{M}, \alpha_{1},...,\alpha_{M-1},\nu) where

• \upsilon_{m}=(\phi_{m,0},\phi_{m},\sigma_{m}^2)

• \phi_{m}=(\phi_{m,1},...,\phi_{m,p}), m=1,...,M

• \nu=(\nu_{M1+1},...,\nu_{M})

• M1 is the number of GMAR type regimes.

In the GMAR model, M1=M and the parameter \nu dropped. In the StMAR model, M1=0.

If the model imposes linear constraints on the autoregressive parameters: Replace the vectors \phi_{m} with the vectors \psi_{m} that satisfy \phi_{m}=C_{m}\psi_{m} (see the argument constraints).

For restricted models:

Size (3M+M-M1+p-1\times 1) vector \theta=(\phi_{1,0},...,\phi_{M,0},\phi, \sigma_{1}^2,...,\sigma_{M}^2,\alpha_{1},...,\alpha_{M-1},\nu), where \phi=(\phi_{1},...,\phi_{p}) contains the AR coefficients, which are common for all regimes.

If the model imposes linear constraints on the autoregressive parameters: Replace the vector \phi with the vector \psi that satisfies \phi=C\psi (see the argument constraints).

Symbol \phi denotes an AR coefficient, \sigma^2 a variance, \alpha a mixing weight, and \nu a degrees of freedom parameter. If parametrization=="mean", just replace each intercept term \phi_{m,0} with the regimewise mean \mu_m = \phi_{m,0}/(1-\sum\phi_{i,m}). In the G-StMAR model, the first M1 components are GMAR type and the rest M2 components are StMAR type. Note that in the case M=1, the mixing weight parameters \alpha are dropped, and in the case of StMAR or G-StMAR model, the degrees of freedom parameters \nu have to be larger than 2.

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

a logical argument stating whether the AR coefficients \phi_{m,1},...,\phi_{m,p} are restricted to be the same for all regimes.

specifies linear constraints imposed to each regime's autoregressive parameters separately.

For non-restricted models:

a list of size (p \times q_{m}) constraint matrices C_{m} of full column rank satisfying \phi_{m}=C_{m}\psi_{m} for all m=1,...,M, where \phi_{m}=(\phi_{m,1},...,\phi_{m,p}) and \psi_{m}=(\psi_{m,1},...,\psi_{m,q_{m}}).

For restricted models:

a size (p\times q) constraint matrix C of full column rank satisfying \phi=C\psi, where \phi=(\phi_{1},...,\phi_{p}) and \psi=\psi_{1},...,\psi_{q}.

The symbol \phi denotes an AR coefficient. Note that regardless of any constraints, the autoregressive order is always p for all regimes. Ignore or set to NULL if applying linear constraints is not desired.

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 GMAR and StMAR models:

a positive integer specifying the number of mixture components.

For G-StMAR models:

a size (2\times 1) integer vector specifying the number of GMAR type components M1 in the first element and StMAR type components M2 in the second element. The total number of mixture components is M=M1+M2.

a real valued parameter vector specifying the model.

For non-restricted models:

Size (M(p+3)+M-M1-1\times 1) vector \theta=(\upsilon_{1},...,\upsilon_{M}, \alpha_{1},...,\alpha_{M-1},\nu) where

• \upsilon_{m}=(\phi_{m,0},\phi_{m},\sigma_{m}^2)

• \phi_{m}=(\phi_{m,1},...,\phi_{m,p}), m=1,...,M

• \nu=(\nu_{M1+1},...,\nu_{M})

• M1 is the number of GMAR type regimes.

In the GMAR model, M1=M and the parameter \nu dropped. In the StMAR model, M1=0.

If the model imposes linear constraints on the autoregressive parameters: Replace the vectors \phi_{m} with the vectors \psi_{m} that satisfy \phi_{m}=C_{m}\psi_{m} (see the argument constraints).

For restricted models:

Size (3M+M-M1+p-1\times 1) vector \theta=(\phi_{1,0},...,\phi_{M,0},\phi, \sigma_{1}^2,...,\sigma_{M}^2,\alpha_{1},...,\alpha_{M-1},\nu), where \phi=(\phi_{1},...,\phi_{p}) contains the AR coefficients, which are common for all regimes.

If the model imposes linear constraints on the autoregressive parameters: Replace the vector \phi with the vector \psi that satisfies \phi=C\psi (see the argument constraints).

Symbol \phi denotes an AR coefficient, \sigma^2 a variance, \alpha a mixing weight, and \nu a degrees of freedom parameter. If parametrization=="mean", just replace each intercept term \phi_{m,0} with the regimewise mean \mu_m = \phi_{m,0}/(1-\sum\phi_{i,m}). In the G-StMAR model, the first M1 components are GMAR type and the rest M2 components are StMAR type. Note that in the case M=1, the mixing weight parameters \alpha are dropped, and in the case of StMAR or G-StMAR model, the degrees of freedom parameters \nu have to be larger than 2.

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

a logical argument stating whether the AR coefficients \phi_{m,1},...,\phi_{m,p} are restricted to be the same for all regimes.

specifies linear constraints imposed to each regime's autoregressive parameters separately.

For non-restricted models:

a list of size (p \times q_{m}) constraint matrices C_{m} of full column rank satisfying \phi_{m}=C_{m}\psi_{m} for all m=1,...,M, where \phi_{m}=(\phi_{m,1},...,\phi_{m,p}) and \psi_{m}=(\psi_{m,1},...,\psi_{m,q_{m}}).

For restricted models:

a size (p\times q) constraint matrix C of full column rank satisfying \phi=C\psi, where \phi=(\phi_{1},...,\phi_{p}) and \psi=\psi_{1},...,\psi_{q}.

The symbol \phi denotes an AR coefficient. Note that regardless of any constraints, the autoregressive order is always p for all regimes. Ignore or set to NULL if applying linear constraints is not desired.

a numeric vector specifying the parameter values that should be inserted to the specified regime.

For non-restricted models:

For GMAR model:

Size (p+2\times 1) vector (\phi_{m,0},\phi_{m,1},...,\phi_{m,p}, \sigma_{m}^2).

For StMAR model:

Size (p+3\times 1) vector (\phi_{m,0},\phi_{m,1},...,\phi_{m,p}, \sigma_{m}^2, \nu_{m}).

For G-StMAR model:

Same as GMAR for GMAR type regimes and same as StMAR for StMAR type regimes.

With linear constraints:

Parameter vector as described above, but vector \phi_{m} replaced with vector \psi_{m} that satisfies \phi_{m}=R_{m}\psi_{m}.

For restricted models:

For GMAR model:

Size (2\times 1) vector (\phi_{m,0}, \sigma_{m}^2).

For StMAR model:

Size (3\times 1) vector (\phi_{m,0}, \sigma_{m}^2, \nu_{m}).

For G-StMAR model:

Same as GMAR for GMAR type regimes and same as StMAR for StMAR type regimes.

With linear constraints:

Parameter vector as described above.

a positive integer in the interval [1, M] defining which regime should be changed.

a numeric vector or class 'ts' object containing the data. NA values are not supported.

a positive integer specifying the autoregressive order of the model.

a positive integer specifying the autoregressive order of the model.

For GMAR and StMAR models:

a positive integer specifying the number of mixture components.

For G-StMAR models:

a size (2\times 1) integer vector specifying the number of GMAR type components M1 in the first element and StMAR type components M2 in the second element. The total number of mixture components is M=M1+M2.

a logical argument stating whether the AR coefficients \phi_{m,1},...,\phi_{m,p} are restricted to be the same for all regimes.

specifies linear constraints imposed to each regime's autoregressive parameters separately.

For non-restricted models:

a list of size (p \times q_{m}) constraint matrices C_{m} of full column rank satisfying \phi_{m}=C_{m}\psi_{m} for all m=1,...,M, where \phi_{m}=(\phi_{m,1},...,\phi_{m,p}) and \psi_{m}=(\psi_{m,1},...,\psi_{m,q_{m}}).

For restricted models:

a size (p\times q) constraint matrix C of full column rank satisfying \phi=C\psi, where \phi=(\phi_{1},...,\phi_{p}) and \psi=\psi_{1},...,\psi_{q}.

The symbol \phi denotes an AR coefficient. Note that regardless of any constraints, the autoregressive order is always p for all regimes. Ignore or set to NULL if applying linear constraints is not desired.

an object to be tested

an object to be tested

the name of the tested object

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

a positive integer specifying the autoregressive order of the model.

For GMAR and StMAR models:

a positive integer specifying the number of mixture components.

For G-StMAR models:

a size (2\times 1) integer vector specifying the number of GMAR type components M1 in the first element and StMAR type components M2 in the second element. The total number of mixture components is M=M1+M2.

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

a positive integer specifying the autoregressive order of the model.

For GMAR and StMAR models:

a positive integer specifying the number of mixture components.

For G-StMAR models:

a size (2\times 1) integer vector specifying the number of GMAR type components M1 in the first element and StMAR type components M2 in the second element. The total number of mixture components is M=M1+M2.

a real valued parameter vector specifying the model.

For non-restricted models:

Size (M(p+3)+M-M1-1\times 1) vector \theta=(\upsilon_{1},...,\upsilon_{M}, \alpha_{1},...,\alpha_{M-1},\nu) where

• \upsilon_{m}=(\phi_{m,0},\phi_{m},\sigma_{m}^2)

• \phi_{m}=(\phi_{m,1},...,\phi_{m,p}), m=1,...,M

• \nu=(\nu_{M1+1},...,\nu_{M})

• M1 is the number of GMAR type regimes.

In the GMAR model, M1=M and the parameter \nu dropped. In the StMAR model, M1=0.

If the model imposes linear constraints on the autoregressive parameters: Replace the vectors \phi_{m} with the vectors \psi_{m} that satisfy \phi_{m}=C_{m}\psi_{m} (see the argument constraints).

For restricted models:

Size (3M+M-M1+p-1\times 1) vector \theta=(\phi_{1,0},...,\phi_{M,0},\phi, \sigma_{1}^2,...,\sigma_{M}^2,\alpha_{1},...,\alpha_{M-1},\nu), where \phi=(\phi_{1},...,\phi_{p}) contains the AR coefficients, which are common for all regimes.

If the model imposes linear constraints on the autoregressive parameters: Replace the vector \phi with the vector \psi that satisfies \phi=C\psi (see the argument constraints).

Symbol \phi denotes an AR coefficient, \sigma^2 a variance, \alpha a mixing weight, and \nu a degrees of freedom parameter. If parametrization=="mean", just replace each intercept term \phi_{m,0} with the regimewise mean \mu_m = \phi_{m,0}/(1-\sum\phi_{i,m}). In the G-StMAR model, the first M1 components are GMAR type and the rest M2 components are StMAR type. Note that in the case M=1, the mixing weight parameters \alpha are dropped, and in the case of StMAR or G-StMAR model, the degrees of freedom parameters \nu have to be larger than 2.

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

a logical argument stating whether the AR coefficients \phi_{m,1},...,\phi_{m,p} are restricted to be the same for all regimes.

specifies linear constraints imposed to each regime's autoregressive parameters separately.

For non-restricted models:

a list of size (p \times q_{m}) constraint matrices C_{m} of full column rank satisfying \phi_{m}=C_{m}\psi_{m} for all m=1,...,M, where \phi_{m}=(\phi_{m,1},...,\phi_{m,p}) and \psi_{m}=(\psi_{m,1},...,\psi_{m,q_{m}}).

For restricted models:

a size (p\times q) constraint matrix C of full column rank satisfying \phi=C\psi, where \phi=(\phi_{1},...,\phi_{p}) and \psi=\psi_{1},...,\psi_{q}.

The symbol \phi denotes an AR coefficient. Note that regardless of any constraints, the autoregressive order is always p for all regimes. Ignore or set to NULL if applying linear constraints is not desired.

a class 'gsmar' object, typically generated by fitGSMAR or GSMAR.

should conditional means or variances be plotted?

a numeric vector or class 'ts' object containing the data. NA values are not supported.

a positive integer specifying the autoregressive order of the model.

For GMAR and StMAR models:

a positive integer specifying the number of mixture components.

For G-StMAR models:

a size (2\times 1) integer vector specifying the number of GMAR type components M1 in the first element and StMAR type components M2 in the second element. The total number of mixture components is M=M1+M2.

a real valued parameter vector specifying the model.

For non-restricted models:

Size (M(p+3)+M-M1-1\times 1) vector \theta=(\upsilon_{1},...,\upsilon_{M}, \alpha_{1},...,\alpha_{M-1},\nu) where

• \upsilon_{m}=(\phi_{m,0},\phi_{m},\sigma_{m}^2)

• \phi_{m}=(\phi_{m,1},...,\phi_{m,p}), m=1,...,M

• \nu=(\nu_{M1+1},...,\nu_{M})

• M1 is the number of GMAR type regimes.

In the GMAR model, M1=M and the parameter \nu dropped. In the StMAR model, M1=0.

If the model imposes linear constraints on the autoregressive parameters: Replace the vectors \phi_{m} with the vectors \psi_{m} that satisfy \phi_{m}=C_{m}\psi_{m} (see the argument constraints).

For restricted models:

Size (3M+M-M1+p-1\times 1) vector \theta=(\phi_{1,0},...,\phi_{M,0},\phi, \sigma_{1}^2,...,\sigma_{M}^2,\alpha_{1},...,\alpha_{M-1},\nu), where \phi=(\phi_{1},...,\phi_{p}) contains the AR coefficients, which are common for all regimes.

If the model imposes linear constraints on the autoregressive parameters: Replace the vector \phi with the vector \psi that satisfies \phi=C\psi (see the argument constraints).

Symbol \phi denotes an AR coefficient, \sigma^2 a variance, \alpha a mixing weight, and \nu a degrees of freedom parameter. If parametrization=="mean", just replace each intercept term \phi_{m,0} with the regimewise mean \mu_m = \phi_{m,0}/(1-\sum\phi_{i,m}). In the G-StMAR model, the first M1 components are GMAR type and the rest M2 components are StMAR type. Note that in the case M=1, the mixing weight parameters \alpha are dropped, and in the case of StMAR or G-StMAR model, the degrees of freedom parameters \nu have to be larger than 2.

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

a logical argument stating whether the AR coefficients \phi_{m,1},...,\phi_{m,p} are restricted to be the same for all regimes.

specifies linear constraints imposed to each regime's autoregressive parameters separately.

For non-restricted models:

a list of size (p \times q_{m}) constraint matrices C_{m} of full column rank satisfying \phi_{m}=C_{m}\psi_{m} for all m=1,...,M, where \phi_{m}=(\phi_{m,1},...,\phi_{m,p}) and \psi_{m}=(\psi_{m,1},...,\psi_{m,q_{m}}).

For restricted models:

a size (p\times q) constraint matrix C of full column rank satisfying \phi=C\psi, where \phi=(\phi_{1},...,\phi_{p}) and \psi=\psi_{1},...,\psi_{q}.

The symbol \phi denotes an AR coefficient. Note that regardless of any constraints, the autoregressive order is always p for all regimes. Ignore or set to NULL if applying linear constraints is not desired.

is the model parametrized with the "intercepts" \phi_{m,0} or "means" \mu_{m} = \phi_{m,0}/(1-\sum\phi_{i,m})?

calculate regimewise conditional means (regime_cmeans), regimewise conditional variances (regime_cvars), total conditional means (total_cmeans), or total conditional variances (total_cvars)?

a class 'gsmar' object, typically generated by fitGSMAR or GSMAR.

a positive integer specifying how many lags should be calculated for the autocorrelation and conditional heteroscedasticity statistics.

a positive integer specifying to how many simulated values from the process the covariance matrix "Omega" (used to compute the tests) should be based on. Larger number of simulations may result more reliable tests but takes longer to compute. If smaller than data size, then "Omega" will be based on the given data. Ignored if plot_indstats==FALSE.

set TRUE if the individual statistics discussed in Kalliovirta (2012) should be plotted with their approximate 95% critical bounds (this may take some time).

a positive integer specifying the autoregressive order of the model.

For GMAR and StMAR models:

a positive integer specifying the number of mixture components.

For G-StMAR models:

a size (2\times 1) integer vector specifying the number of GMAR type components M1 in the first element and StMAR type components M2 in the second element. The total number of mixture components is M=M1+M2.

a real valued parameter vector specifying the model.

For non-restricted models:

Size (M(p+3)+M-M1-1\times 1) vector \theta=(\upsilon_{1},...,\upsilon_{M}, \alpha_{1},...,\alpha_{M-1},\nu) where

• \upsilon_{m}=(\phi_{m,0},\phi_{m},\sigma_{m}^2)

• \phi_{m}=(\phi_{m,1},...,\phi_{m,p}), m=1,...,M

• \nu=(\nu_{M1+1},...,\nu_{M})

• M1 is the number of GMAR type regimes.

In the GMAR model, M1=M and the parameter \nu dropped. In the StMAR model, M1=0.

If the model imposes linear constraints on the autoregressive parameters: Replace the vectors \phi_{m} with the vectors \psi_{m} that satisfy \phi_{m}=C_{m}\psi_{m} (see the argument constraints).

For restricted models:

Size (3M+M-M1+p-1\times 1) vector \theta=(\phi_{1,0},...,\phi_{M,0},\phi, \sigma_{1}^2,...,\sigma_{M}^2,\alpha_{1},...,\alpha_{M-1},\nu), where \phi=(\phi_{1},...,\phi_{p}) contains the AR coefficients, which are common for all regimes.

If the model imposes linear constraints on the autoregressive parameters: Replace the vector \phi with the vector \psi that satisfies \phi=C\psi (see the argument constraints).

Symbol \phi denotes an AR coefficient, \sigma^2 a variance, \alpha a mixing weight, and \nu a degrees of freedom parameter. If parametrization=="mean", just replace each intercept term \phi_{m,0} with the regimewise mean \mu_m = \phi_{m,0}/(1-\sum\phi_{i,m}). In the G-StMAR model, the first M1 components are GMAR type and the rest M2 components are StMAR type. Note that in the case M=1, the mixing weight parameters \alpha are dropped, and in the case of StMAR or G-StMAR model, the degrees of freedom parameters \nu have to be larger than 2.

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

a logical argument stating whether the AR coefficients \phi_{m,1},...,\phi_{m,p} are restricted to be the same for all regimes.

specifies linear constraints imposed to each regime's autoregressive parameters separately.

For non-restricted models:

a list of size (p \times q_{m}) constraint matrices C_{m} of full column rank satisfying \phi_{m}=C_{m}\psi_{m} for all m=1,...,M, where \phi_{m}=(\phi_{m,1},...,\phi_{m,p}) and \psi_{m}=(\psi_{m,1},...,\psi_{m,q_{m}}).

For restricted models:

a size (p\times q) constraint matrix C of full column rank satisfying \phi=C\psi, where \phi=(\phi_{1},...,\phi_{p}) and \psi=\psi_{1},...,\psi_{q}.

The symbol \phi denotes an AR coefficient. Note that regardless of any constraints, the autoregressive order is always p for all regimes. Ignore or set to NULL if applying linear constraints is not desired.

a positive integer in the interval [1, M] defining which regime should be extracted.

Should the degrees of freedom parameter (if any) be included?

a numeric vector or class 'ts' object containing the data. NA values are not supported.

a positive integer specifying the autoregressive order of the model.

For GMAR and StMAR models:

a positive integer specifying the number of mixture components.

For G-StMAR models:

a size (2\times 1) integer vector specifying the number of GMAR type components M1 in the first element and StMAR type components M2 in the second element. The total number of mixture components is M=M1+M2.

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

a logical argument stating whether the AR coefficients \phi_{m,1},...,\phi_{m,p} are restricted to be the same for all regimes.

specifies linear constraints imposed to each regime's autoregressive parameters separately.

For non-restricted models:

a list of size (p \times q_{m}) constraint matrices C_{m} of full column rank satisfying \phi_{m}=C_{m}\psi_{m} for all m=1,...,M, where \phi_{m}=(\phi_{m,1},...,\phi_{m,p}) and \psi_{m}=(\psi_{m,1},...,\psi_{m,q_{m}}).

For restricted models:

a size (p\times q) constraint matrix C of full column rank satisfying \phi=C\psi, where \phi=(\phi_{1},...,\phi_{p}) and \psi=\psi_{1},...,\psi_{q}.

The symbol \phi denotes an AR coefficient. Note that regardless of any constraints, the autoregressive order is always p for all regimes. Ignore or set to NULL if applying linear constraints is not desired.

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

is the model parametrized with the "intercepts" \phi_{m,0} or "means" \mu_{m} = \phi_{m,0}/(1-\sum\phi_{i,m})?

a positive integer specifying how many rounds of estimation should be conducted. The estimation results may vary from round to round because of multimodality of the log-likelihood function and the randomness associated with the genetic algorithm.

the number of CPU cores to be used in the estimation process.

the maximum number of iterations for 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 the purpose of creating reproducible results.

should the estimation results be printed?

should likely inappropriate estimates be filtered? See details.

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

number of digits to use

log-likelihood value

the number of (freely estimated) parameters in the model.

the number of observations with initial values excluded for conditional models.

For GMAR and StMAR models:

a positive integer specifying the number of mixture components.

For G-StMAR models:

a size (2\times 1) integer vector specifying the number of GMAR type components M1 in the first element and StMAR 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.

should the returned object be the log-likelihood value, mixing weights, mixing weights including value for alpha_{m,T+1}, a list containing log-likelihood value and mixing weights, the terms l_{t}: t=1,..,T in the log-likelihood function (see KMS 2015, eq.(13)), the densities in the terms, regimewise conditional means, regimewise conditional variances, total conditional means, total conditional variances, or quantile residuals?

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

a class 'gsmar' object, typically generated by fitGSMAR or GSMAR.

a numeric vector or class 'ts' object containing the data. NA values are not supported.

a class 'gsmar' object, typically generated by fitGSMAR or GSMAR.

a class 'gsmar' object, typically generated by fitGSMAR or GSMAR.

a class 'gsmar' object, typically generated by fitGSMAR or GSMAR.

a numeric vector or class 'ts' object containing the data. NA values are not supported.

a positive integer specifying the autoregressive order of the model.

For GMAR and StMAR models:

a positive integer specifying the number of mixture components.

For G-StMAR models:

a size (2\times 1) integer vector specifying the number of GMAR type components M1 in the first element and StMAR type components M2 in the second element. The total number of mixture components is M=M1+M2.

a real valued parameter vector specifying the model.

For non-restricted models:

Size (M(p+3)+M-M1-1\times 1) vector \theta=(\upsilon_{1},...,\upsilon_{M}, \alpha_{1},...,\alpha_{M-1},\nu) where

• \upsilon_{m}=(\phi_{m,0},\phi_{m},\sigma_{m}^2)

• \phi_{m}=(\phi_{m,1},...,\phi_{m,p}), m=1,...,M

• \nu=(\nu_{M1+1},...,\nu_{M})

• M1 is the number of GMAR type regimes.

In the GMAR model, M1=M and the parameter \nu dropped. In the StMAR model, M1=0.

If the model imposes linear constraints on the autoregressive parameters: Replace the vectors \phi_{m} with the vectors \psi_{m} that satisfy \phi_{m}=C_{m}\psi_{m} (see the argument constraints).

For restricted models:

Size (3M+M-M1+p-1\times 1) vector \theta=(\phi_{1,0},...,\phi_{M,0},\phi, \sigma_{1}^2,...,\sigma_{M}^2,\alpha_{1},...,\alpha_{M-1},\nu), where \phi=(\phi_{1},...,\phi_{p}) contains the AR coefficients, which are common for all regimes.

If the model imposes linear constraints on the autoregressive parameters: Replace the vector \phi with the vector \psi that satisfies \phi=C\psi (see the argument constraints).

Symbol \phi denotes an AR coefficient, \sigma^2 a variance, \alpha a mixing weight, and \nu a degrees of freedom parameter. If parametrization=="mean", just replace each intercept term \phi_{m,0} with the regimewise mean \mu_m = \phi_{m,0}/(1-\sum\phi_{i,m}). In the G-StMAR model, the first M1 components are GMAR type and the rest M2 components are StMAR type. Note that in the case M=1, the mixing weight parameters \alpha are dropped, and in the case of StMAR or G-StMAR model, the degrees of freedom parameters \nu have to be larger than 2.

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

a logical argument stating whether the AR coefficients \phi_{m,1},...,\phi_{m,p} are restricted to be the same for all regimes.

specifies linear constraints imposed to each regime's autoregressive parameters separately.

For non-restricted models:

a list of size (p \times q_{m}) constraint matrices C_{m} of full column rank satisfying \phi_{m}=C_{m}\psi_{m} for all m=1,...,M, where \phi_{m}=(\phi_{m,1},...,\phi_{m,p}) and \psi_{m}=(\psi_{m,1},...,\psi_{m,q_{m}}).

For restricted models:

a size (p\times q) constraint matrix C of full column rank satisfying \phi=C\psi, where \phi=(\phi_{1},...,\phi_{p}) and \psi=\psi_{1},...,\psi_{q}.

The symbol \phi denotes an AR coefficient. Note that regardless of any constraints, the autoregressive order is always p for all regimes. Ignore or set to NULL if applying linear constraints is not desired.

is the model parametrized with the "intercepts" \phi_{m,0} or "means" \mu_{m} = \phi_{m,0}/(1-\sum\phi_{i,m})?

a function specifying the transformation.

output dimension of the transformation g.

a positive integer specifying the autoregressive order of the model.

For GMAR and StMAR models:

a positive integer specifying the number of mixture components.

For G-StMAR models:

a size (2\times 1) integer vector specifying the number of GMAR type components M1 in the first element and StMAR type components M2 in the second element. The total number of mixture components is M=M1+M2.

a real valued parameter vector specifying the model.

For non-restricted models:

Size (M(p+3)+M-M1-1\times 1) vector \theta=(\upsilon_{1},...,\upsilon_{M}, \alpha_{1},...,\alpha_{M-1},\nu) where

• \upsilon_{m}=(\phi_{m,0},\phi_{m},\sigma_{m}^2)

• \phi_{m}=(\phi_{m,1},...,\phi_{m,p}), m=1,...,M

• \nu=(\nu_{M1+1},...,\nu_{M})

• M1 is the number of GMAR type regimes.

In the GMAR model, M1=M and the parameter \nu dropped. In the StMAR model, M1=0.

If the model imposes linear constraints on the autoregressive parameters: Replace the vectors \phi_{m} with the vectors \psi_{m} that satisfy \phi_{m}=C_{m}\psi_{m} (see the argument constraints).

For restricted models:

Size (3M+M-M1+p-1\times 1) vector \theta=(\phi_{1,0},...,\phi_{M,0},\phi, \sigma_{1}^2,...,\sigma_{M}^2,\alpha_{1},...,\alpha_{M-1},\nu), where \phi=(\phi_{1},...,\phi_{p}) contains the AR coefficients, which are common for all regimes.

If the model imposes linear constraints on the autoregressive parameters: Replace the vector \phi with the vector \psi that satisfies \phi=C\psi (see the argument constraints).

Symbol \phi denotes an AR coefficient, \sigma^2 a variance, \alpha a mixing weight, and \nu a degrees of freedom parameter. If parametrization=="mean", just replace each intercept term \phi_{m,0} with the regimewise mean \mu_m = \phi_{m,0}/(1-\sum\phi_{i,m}). In the G-StMAR model, the first M1 components are GMAR type and the rest M2 components are StMAR type. Note that in the case M=1, the mixing weight parameters \alpha are dropped, and in the case of StMAR or G-StMAR model, the degrees of freedom parameters \nu have to be larger than 2.

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

a positive integer specifying the autoregressive order of the model.

For GMAR and StMAR models:

a positive integer specifying the number of mixture components.

For G-StMAR models:

a size (2\times 1) integer vector specifying the number of GMAR type components M1 in the first element and StMAR type components M2 in the second element. The total number of mixture components is M=M1+M2.

a real valued parameter vector specifying the model.

For non-restricted models:

Size (M(p+3)+M-M1-1\times 1) vector \theta=(\upsilon_{1},...,\upsilon_{M}, \alpha_{1},...,\alpha_{M-1},\nu) where

• \upsilon_{m}=(\phi_{m,0},\phi_{m},\sigma_{m}^2)

• \phi_{m}=(\phi_{m,1},...,\phi_{m,p}), m=1,...,M

• \nu=(\nu_{M1+1},...,\nu_{M})

• M1 is the number of GMAR type regimes.

In the GMAR model, M1=M and the parameter \nu dropped. In the StMAR model, M1=0.

If the model imposes linear constraints on the autoregressive parameters: Replace the vectors \phi_{m} with the vectors \psi_{m} that satisfy \phi_{m}=C_{m}\psi_{m} (see the argument constraints).

For restricted models:

Size (3M+M-M1+p-1\times 1) vector \theta=(\phi_{1,0},...,\phi_{M,0},\phi, \sigma_{1}^2,...,\sigma_{M}^2,\alpha_{1},...,\alpha_{M-1},\nu), where \phi=(\phi_{1},...,\phi_{p}) contains the AR coefficients, which are common for all regimes.

If the model imposes linear constraints on the autoregressive parameters: Replace the vector \phi with the vector \psi that satisfies \phi=C\psi (see the argument constraints).

Symbol \phi denotes an AR coefficient, \sigma^2 a variance, \alpha a mixing weight, and \nu a degrees of freedom parameter. If parametrization=="mean", just replace each intercept term \phi_{m,0} with the regimewise mean \mu_m = \phi_{m,0}/(1-\sum\phi_{i,m}). In the G-StMAR model, the first M1 components are GMAR type and the rest M2 components are StMAR type. Note that in the case M=1, the mixing weight parameters \alpha are dropped, and in the case of StMAR or G-StMAR model, the degrees of freedom parameters \nu have to be larger than 2.

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

a logical argument stating whether the AR coefficients \phi_{m,1},...,\phi_{m,p} are restricted to be the same for all regimes.

specifies linear constraints imposed to each regime's autoregressive parameters separately.

For non-restricted models:

a list of size (p \times q_{m}) constraint matrices C_{m} of full column rank satisfying \phi_{m}=C_{m}\psi_{m} for all m=1,...,M, where \phi_{m}=(\phi_{m,1},...,\phi_{m,p}) and \psi_{m}=(\psi_{m,1},...,\psi_{m,q_{m}}).

For restricted models:

a size (p\times q) constraint matrix C of full column rank satisfying \phi=C\psi, where \phi=(\phi_{1},...,\phi_{p}) and \psi=\psi_{1},...,\psi_{q}.

The symbol \phi denotes an AR coefficient. Note that regardless of any constraints, the autoregressive order is always p for all regimes. Ignore or set to NULL if applying linear constraints is not desired.

a positive integer specifying the autoregressive order of the model.

For GMAR and StMAR models:

a positive integer specifying the number of mixture components.

For G-StMAR models:

a size (2\times 1) integer vector specifying the number of GMAR type components M1 in the first element and StMAR type components M2 in the second element. The total number of mixture components is M=M1+M2.

a real valued parameter vector specifying the model.

For non-restricted models:

Size (M(p+3)+M-M1-1\times 1) vector \theta=(\upsilon_{1},...,\upsilon_{M}, \alpha_{1},...,\alpha_{M-1},\nu) where

• \upsilon_{m}=(\phi_{m,0},\phi_{m},\sigma_{m}^2)

• \phi_{m}=(\phi_{m,1},...,\phi_{m,p}), m=1,...,M

• \nu=(\nu_{M1+1},...,\nu_{M})

• M1 is the number of GMAR type regimes.

In the GMAR model, M1=M and the parameter \nu dropped. In the StMAR model, M1=0.

If the model imposes linear constraints on the autoregressive parameters: Replace the vectors \phi_{m} with the vectors \psi_{m} that satisfy \phi_{m}=C_{m}\psi_{m} (see the argument constraints).

For restricted models:

Size (3M+M-M1+p-1\times 1) vector \theta=(\phi_{1,0},...,\phi_{M,0},\phi, \sigma_{1}^2,...,\sigma_{M}^2,\alpha_{1},...,\alpha_{M-1},\nu), where \phi=(\phi_{1},...,\phi_{p}) contains the AR coefficients, which are common for all regimes.

If the model imposes linear constraints on the autoregressive parameters: Replace the vector \phi with the vector \psi that satisfies \phi=C\psi (see the argument constraints).

Symbol \phi denotes an AR coefficient, \sigma^2 a variance, \alpha a mixing weight, and \nu a degrees of freedom parameter. If parametrization=="mean", just replace each intercept term \phi_{m,0} with the regimewise mean \mu_m = \phi_{m,0}/(1-\sum\phi_{i,m}). In the G-StMAR model, the first M1 components are GMAR type and the rest M2 components are StMAR type. Note that in the case M=1, the mixing weight parameters \alpha are dropped, and in the case of StMAR or G-StMAR model, the degrees of freedom parameters \nu have to be larger than 2.

a logical argument stating whether the AR coefficients \phi_{m,1},...,\phi_{m,p} are restricted to be the same for all regimes.

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

specifies linear constraints imposed to each regime's autoregressive parameters separately.

For non-restricted models:

a list of size (p \times q_{m}) constraint matrices C_{m} of full column rank satisfying \phi_{m}=C_{m}\psi_{m} for all m=1,...,M, where \phi_{m}=(\phi_{m,1},...,\phi_{m,p}) and \psi_{m}=(\psi_{m,1},...,\psi_{m,q_{m}}).

For restricted models:

a size (p\times q) constraint matrix C of full column rank satisfying \phi=C\psi, where \phi=(\phi_{1},...,\phi_{p}) and \psi=\psi_{1},...,\psi_{q}.

The symbol \phi denotes an AR coefficient. Note that regardless of any constraints, the autoregressive order is always p for all regimes. Ignore or set to NULL if applying linear constraints is not desired.

a class 'gsmar' object, typically generated by fitGSMAR or GSMAR.

the maximum number of iterations for the variable metric algorithm.

A numeric vector with same the length as the parameter vector: i:th element of custom_h is the difference used in central difference approximation for partial differentials of the log-likelihood function for the i:th parameter. If NULL (default), then the difference used for differentiating overly large degrees of freedom parameters is adjusted to avoid numerical problems, and the difference is 6e-6 for the other parameters.

should approximate standard errors be calculated?

a numeric vector or class 'ts' object containing the data. NA values are not supported.

a positive integer specifying the autoregressive order of the model.

For GMAR and StMAR models:

a positive integer specifying the number of mixture components.

For G-StMAR models:

a size (2\times 1) integer vector specifying the number of GMAR type components M1 in the first element and StMAR type components M2 in the second element. The total number of mixture components is M=M1+M2.

a real valued parameter vector specifying the model.

For non-restricted models:

Size (M(p+3)+M-M1-1\times 1) vector \theta=(\upsilon_{1},...,\upsilon_{M}, \alpha_{1},...,\alpha_{M-1},\nu) where

• \upsilon_{m}=(\phi_{m,0},\phi_{m},\sigma_{m}^2)

• \phi_{m}=(\phi_{m,1},...,\phi_{m,p}), m=1,...,M

• \nu=(\nu_{M1+1},...,\nu_{M})

• M1 is the number of GMAR type regimes.

In the GMAR model, M1=M and the parameter \nu dropped. In the StMAR model, M1=0.

If the model imposes linear constraints on the autoregressive parameters: Replace the vectors \phi_{m} with the vectors \psi_{m} that satisfy \phi_{m}=C_{m}\psi_{m} (see the argument constraints).

For restricted models:

Size (3M+M-M1+p-1\times 1) vector \theta=(\phi_{1,0},...,\phi_{M,0},\phi, \sigma_{1}^2,...,\sigma_{M}^2,\alpha_{1},...,\alpha_{M-1},\nu), where \phi=(\phi_{1},...,\phi_{p}) contains the AR coefficients, which are common for all regimes.

If the model imposes linear constraints on the autoregressive parameters: Replace the vector \phi with the vector \psi that satisfies \phi=C\psi (see the argument constraints).

Symbol \phi denotes an AR coefficient, \sigma^2 a variance, \alpha a mixing weight, and \nu a degrees of freedom parameter. If parametrization=="mean", just replace each intercept term \phi_{m,0} with the regimewise mean \mu_m = \phi_{m,0}/(1-\sum\phi_{i,m}). In the G-StMAR model, the first M1 components are GMAR type and the rest M2 components are StMAR type. Note that in the case M=1, the mixing weight parameters \alpha are dropped, and in the case of StMAR or G-StMAR model, the degrees of freedom parameters \nu have to be larger than 2.

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

a logical argument stating whether the AR coefficients \phi_{m,1},...,\phi_{m,p} are restricted to be the same for all regimes.

specifies linear constraints imposed to each regime's autoregressive parameters separately.

For non-restricted models:

a list of size (p \times q_{m}) constraint matrices C_{m} of full column rank satisfying \phi_{m}=C_{m}\psi_{m} for all m=1,...,M, where \phi_{m}=(\phi_{m,1},...,\phi_{m,p}) and \psi_{m}=(\psi_{m,1},...,\psi_{m,q_{m}}).

For restricted models:

a size (p\times q) constraint matrix C of full column rank satisfying \phi=C\psi, where \phi=(\phi_{1},...,\phi_{p}) and \psi=\psi_{1},...,\psi_{q}.

The symbol \phi denotes an AR coefficient. Note that regardless of any constraints, the autoregressive order is always p for all regimes. Ignore or set to NULL if applying linear constraints is not desired.

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

is the model parametrized with the "intercepts" \phi_{m,0} or "means" \mu_{m} = \phi_{m,0}/(1-\sum\phi_{i,m})?

should the terms l_{t}: t=1,..,T in the log-likelihood function (see KMS 2015, eq.(13) or MPS 2018, eq.(15)) be returned instead of the log-likelihood value?

this will be returned when the parameter vector is outside the parameter space and boundaries==TRUE.

a numeric vector or class 'ts' object containing the data. NA values are not supported.

a positive integer specifying the autoregressive order of the model.

For GMAR and StMAR models:

a positive integer specifying the number of mixture components.

For G-StMAR models:

a size (2\times 1) integer vector specifying the number of GMAR type components M1 in the first element and StMAR type components M2 in the second element. The total number of mixture components is M=M1+M2.

a real valued parameter vector specifying the model.

For non-restricted models:

Size (M(p+3)+M-M1-1\times 1) vector \theta=(\upsilon_{1},...,\upsilon_{M}, \alpha_{1},...,\alpha_{M-1},\nu) where

• \upsilon_{m}=(\phi_{m,0},\phi_{m},\sigma_{m}^2)

• \phi_{m}=(\phi_{m,1},...,\phi_{m,p}), m=1,...,M

• \nu=(\nu_{M1+1},...,\nu_{M})

• M1 is the number of GMAR type regimes.

In the GMAR model, M1=M and the parameter \nu dropped. In the StMAR model, M1=0.

If the model imposes linear constraints on the autoregressive parameters: Replace the vectors \phi_{m} with the vectors \psi_{m} that satisfy \phi_{m}=C_{m}\psi_{m} (see the argument constraints).

For restricted models:

Size (3M+M-M1+p-1\times 1) vector \theta=(\phi_{1,0},...,\phi_{M,0},\phi, \sigma_{1}^2,...,\sigma_{M}^2,\alpha_{1},...,\alpha_{M-1},\nu), where \phi=(\phi_{1},...,\phi_{p}) contains the AR coefficients, which are common for all regimes.

If the model imposes linear constraints on the autoregressive parameters: Replace the vector \phi with the vector \psi that satisfies \phi=C\psi (see the argument constraints).

Symbol \phi denotes an AR coefficient, \sigma^2 a variance, \alpha a mixing weight, and \nu a degrees of freedom parameter. If parametrization=="mean", just replace each intercept term \phi_{m,0} with the regimewise mean \mu_m = \phi_{m,0}/(1-\sum\phi_{i,m}). In the G-StMAR model, the first M1 components are GMAR type and the rest M2 components are StMAR type. Note that in the case M=1, the mixing weight parameters \alpha are dropped, and in the case of StMAR or G-StMAR model, the degrees of freedom parameters \nu have to be larger than 2.

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

a logical argument stating whether the AR coefficients \phi_{m,1},...,\phi_{m,p} are restricted to be the same for all regimes.

specifies linear constraints imposed to each regime's autoregressive parameters separately.

For non-restricted models:

a list of size (p \times q_{m}) constraint matrices C_{m} of full column rank satisfying \phi_{m}=C_{m}\psi_{m} for all m=1,...,M, where \phi_{m}=(\phi_{m,1},...,\phi_{m,p}) and \psi_{m}=(\psi_{m,1},...,\psi_{m,q_{m}}).

For restricted models:

a size (p\times q) constraint matrix C of full column rank satisfying \phi=C\psi, where \phi=(\phi_{1},...,\phi_{p}) and \psi=\psi_{1},...,\psi_{q}.

The symbol \phi denotes an AR coefficient. Note that regardless of any constraints, the autoregressive order is always p for all regimes. Ignore or set to NULL if applying linear constraints is not desired.

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

is the model parametrized with the "intercepts" \phi_{m,0} or "means" \mu_{m} = \phi_{m,0}/(1-\sum\phi_{i,m})?

a logical argument. If TRUE, then loglikelihood returns minval if...

• some component variance is not larger than zero,

• some parametrized mixing weight \alpha_{1},...,\alpha_{M-1} is not larger than zero,

• sum of the parametrized mixing weights is not smaller than one,

• if the model is not stationary,

• or if model=="StMAR" or model=="G-StMAR" and some degrees of freedom parameter \nu_{m} is not larger than two.

Argument minval will be used only if boundaries==TRUE.

TRUE or FALSE specifying whether argument checks, such as stationarity checks, should be done.

should the returned object be the log-likelihood value, mixing weights, mixing weights including value for alpha_{m,T+1}, a list containing log-likelihood value and mixing weights, the terms l_{t}: t=1,..,T in the log-likelihood function (see KMS 2015, eq.(13)), the densities in the terms, regimewise conditional means, regimewise conditional variances, total conditional means, total conditional variances, or quantile residuals?

this will be returned when the parameter vector is outside the parameter space and boundaries==TRUE.

a numeric vector or class 'ts' object containing the data. NA values are not supported.

a positive integer specifying the autoregressive order of the model.

For GMAR and StMAR models:

a positive integer specifying the number of mixture components.

For G-StMAR models:

a size (2\times 1) integer vector specifying the number of GMAR type components M1 in the first element and StMAR type components M2 in the second element. The total number of mixture components is M=M1+M2.

a real valued parameter vector specifying the model.

For non-restricted models:

Size (M(p+3)+M-M1-1\times 1) vector \theta=(\upsilon_{1},...,\upsilon_{M}, \alpha_{1},...,\alpha_{M-1},\nu) where

• \upsilon_{m}=(\phi_{m,0},\phi_{m},\sigma_{m}^2)

• \phi_{m}=(\phi_{m,1},...,\phi_{m,p}), m=1,...,M

• \nu=(\nu_{M1+1},...,\nu_{M})

• M1 is the number of GMAR type regimes.

In the GMAR model, M1=M and the parameter \nu dropped. In the StMAR model, M1=0.

If the model imposes linear constraints on the autoregressive parameters: Replace the vectors \phi_{m} with the vectors \psi_{m} that satisfy \phi_{m}=C_{m}\psi_{m} (see the argument constraints).

For restricted models:

Size (3M+M-M1+p-1\times 1) vector \theta=(\phi_{1,0},...,\phi_{M,0},\phi, \sigma_{1}^2,...,\sigma_{M}^2,\alpha_{1},...,\alpha_{M-1},\nu), where \phi=(\phi_{1},...,\phi_{p}) contains the AR coefficients, which are common for all regimes.

If the model imposes linear constraints on the autoregressive parameters: Replace the vector \phi with the vector \psi that satisfies \phi=C\psi (see the argument constraints).

Symbol \phi denotes an AR coefficient, \sigma^2 a variance, \alpha a mixing weight, and \nu a degrees of freedom parameter. If parametrization=="mean", just replace each intercept term \phi_{m,0} with the regimewise mean \mu_m = \phi_{m,0}/(1-\sum\phi_{i,m}). In the G-StMAR model, the first M1 components are GMAR type and the rest M2 components are StMAR type. Note that in the case M=1, the mixing weight parameters \alpha are dropped, and in the case of StMAR or G-StMAR model, the degrees of freedom parameters \nu have to be larger than 2.

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

a logical argument stating whether the AR coefficients \phi_{m,1},...,\phi_{m,p} are restricted to be the same for all regimes.

specifies linear constraints imposed to each regime's autoregressive parameters separately.

For non-restricted models:

a list of size (p \times q_{m}) constraint matrices C_{m} of full column rank satisfying \phi_{m}=C_{m}\psi_{m} for all m=1,...,M, where \phi_{m}=(\phi_{m,1},...,\phi_{m,p}) and \psi_{m}=(\psi_{m,1},...,\psi_{m,q_{m}}).

For restricted models:

a size (p\times q) constraint matrix C of full column rank satisfying \phi=C\psi, where \phi=(\phi_{1},...,\phi_{p}) and \psi=\psi_{1},...,\psi_{q}.

The symbol \phi denotes an AR coefficient. Note that regardless of any constraints, the autoregressive order is always p for all regimes. Ignore or set to NULL if applying linear constraints is not desired.

is the model parametrized with the "intercepts" \phi_{m,0} or "means" \mu_{m} = \phi_{m,0}/(1-\sum\phi_{i,m})?

a numeric vector or class 'ts' object containing the data. NA values are not supported.

a positive integer specifying the autoregressive order of the model.

For GMAR and StMAR models:

a positive integer specifying the number of mixture components.

For G-StMAR models:

a size (2\times 1) integer vector specifying the number of GMAR type components M1 in the first element and StMAR type components M2 in the second element. The total number of mixture components is M=M1+M2.

a real valued parameter vector specifying the model.

For non-restricted models:

Size (M(p+3)+M-M1-1\times 1) vector \theta=(\upsilon_{1},...,\upsilon_{M}, \alpha_{1},...,\alpha_{M-1},\nu) where

• \upsilon_{m}=(\phi_{m,0},\phi_{m},\sigma_{m}^2)

• \phi_{m}=(\phi_{m,1},...,\phi_{m,p}), m=1,...,M

• \nu=(\nu_{M1+1},...,\nu_{M})

• M1 is the number of GMAR type regimes.

In the GMAR model, M1=M and the parameter \nu dropped. In the StMAR model, M1=0.

If the model imposes linear constraints on the autoregressive parameters: Replace the vectors \phi_{m} with the vectors \psi_{m} that satisfy \phi_{m}=C_{m}\psi_{m} (see the argument constraints).

For restricted models:

Size (3M+M-M1+p-1\times 1) vector \theta=(\phi_{1,0},...,\phi_{M,0},\phi, \sigma_{1}^2,...,\sigma_{M}^2,\alpha_{1},...,\alpha_{M-1},\nu), where \phi=(\phi_{1},...,\phi_{p}) contains the AR coefficients, which are common for all regimes.

If the model imposes linear constraints on the autoregressive parameters: Replace the vector \phi with the vector \psi that satisfies \phi=C\psi (see the argument constraints).

Symbol \phi denotes an AR coefficient, \sigma^2 a variance, \alpha a mixing weight, and \nu a degrees of freedom parameter. If parametrization=="mean", just replace each intercept term \phi_{m,0} with the regimewise mean \mu_m = \phi_{m,0}/(1-\sum\phi_{i,m}). In the G-StMAR model, the first M1 components are GMAR type and the rest M2 components are StMAR type. Note that in the case M=1, the mixing weight parameters \alpha are dropped, and in the case of StMAR or G-StMAR model, the degrees of freedom parameters \nu have to be larger than 2.

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

a logical argument stating whether the AR coefficients \phi_{m,1},...,\phi_{m,p} are restricted to be the same for all regimes.

specifies linear constraints imposed to each regime's autoregressive parameters separately.

For non-restricted models:

a list of size (p \times q_{m}) constraint matrices C_{m} of full column rank satisfying \phi_{m}=C_{m}\psi_{m} for all m=1,...,M, where \phi_{m}=(\phi_{m,1},...,\phi_{m,p}) and \psi_{m}=(\psi_{m,1},...,\psi_{m,q_{m}}).

For restricted models:

a size (p\times q) constraint matrix C of full column rank satisfying \phi=C\psi, where \phi=(\phi_{1},...,\phi_{p}) and \psi=\psi_{1},...,\psi_{q}.

The symbol \phi denotes an AR coefficient. Note that regardless of any constraints, the autoregressive order is always p for all regimes. Ignore or set to NULL if applying linear constraints is not desired.

is the model parametrized with the "intercepts" \phi_{m,0} or "means" \mu_{m} = \phi_{m,0}/(1-\sum\phi_{i,m})?

TRUE or FALSE specifying whether argument checks, such as stationarity checks, should be done.

should the returned object contain mixing weights for t=1,..,T ("mw") or for t=1,..,T+1 ("mw_tplus1")?

a positive integer specifying the autoregressive order of the model.

For GMAR and StMAR models:

a positive integer specifying the number of mixture components.

For G-StMAR models:

a size (2\times 1) integer vector specifying the number of GMAR type components M1 in the first element and StMAR type components M2 in the second element. The total number of mixture components is M=M1+M2.

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

a logical argument stating whether the AR coefficients \phi_{m,1},...,\phi_{m,p} are restricted to be the same for all regimes.

specifies linear constraints imposed to each regime's autoregressive parameters separately.

For non-restricted models:

a list of size (p \times q_{m}) constraint matrices C_{m} of full column rank satisfying \phi_{m}=C_{m}\psi_{m} for all m=1,...,M, where \phi_{m}=(\phi_{m,1},...,\phi_{m,p}) and \psi_{m}=(\psi_{m,1},...,\psi_{m,q_{m}}).

For restricted models:

a size (p\times q) constraint matrix C of full column rank satisfying \phi=C\psi, where \phi=(\phi_{1},...,\phi_{p}) and \psi=\psi_{1},...,\psi_{q}.

The symbol \phi denotes an AR coefficient. Note that regardless of any constraints, the autoregressive order is always p for all regimes. Ignore or set to NULL if applying linear constraints is not desired.

a positive integer specifying the autoregressive order of the model.

For GMAR and StMAR models:

a positive integer specifying the number of mixture components.

For G-StMAR models:

a size (2\times 1) integer vector specifying the number of GMAR type components M1 in the first element and StMAR type components M2 in the second element. The total number of mixture components is M=M1+M2.

a real valued parameter vector specifying the model.

For non-restricted models:

Size (M(p+3)+M-M1-1\times 1) vector \theta=(\upsilon_{1},...,\upsilon_{M}, \alpha_{1},...,\alpha_{M-1},\nu) where

• \upsilon_{m}=(\phi_{m,0},\phi_{m},\sigma_{m}^2)

• \phi_{m}=(\phi_{m,1},...,\phi_{m,p}), m=1,...,M

• \nu=(\nu_{M1+1},...,\nu_{M})

• M1 is the number of GMAR type regimes.

In the GMAR model, M1=M and the parameter \nu dropped. In the StMAR model, M1=0.

If the model imposes linear constraints on the autoregressive parameters: Replace the vectors \phi_{m} with the vectors \psi_{m} that satisfy \phi_{m}=C_{m}\psi_{m} (see the argument constraints).

For restricted models:

Size (3M+M-M1+p-1\times 1) vector \theta=(\phi_{1,0},...,\phi_{M,0},\phi, \sigma_{1}^2,...,\sigma_{M}^2,\alpha_{1},...,\alpha_{M-1},\nu), where \phi=(\phi_{1},...,\phi_{p}) contains the AR coefficients, which are common for all regimes.

If the model imposes linear constraints on the autoregressive parameters: Replace the vector \phi with the vector \psi that satisfies \phi=C\psi (see the argument constraints).

Symbol \phi denotes an AR coefficient, \sigma^2 a variance, \alpha a mixing weight, and \nu a degrees of freedom parameter. If parametrization=="mean", just replace each intercept term \phi_{m,0} with the regimewise mean \mu_m = \phi_{m,0}/(1-\sum\phi_{i,m}). In the G-StMAR model, the first M1 components are GMAR type and the rest M2 components are StMAR type. Note that in the case M=1, the mixing weight parameters \alpha are dropped, and in the case of StMAR or G-StMAR model, the degrees of freedom parameters \nu have to be larger than 2.

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

a logical argument stating whether the AR coefficients \phi_{m,1},...,\phi_{m,p} are restricted to be the same for all regimes.

specifies linear constraints imposed to each regime's autoregressive parameters separately.

For non-restricted models:

a list of size (p \times q_{m}) constraint matrices C_{m} of full column rank satisfying \phi_{m}=C_{m}\psi_{m} for all m=1,...,M, where \phi_{m}=(\phi_{m,1},...,\phi_{m,p}) and \psi_{m}=(\psi_{m,1},...,\psi_{m,q_{m}}).

For restricted models:

a size (p\times q) constraint matrix C of full column rank satisfying \phi=C\psi, where \phi=(\phi_{1},...,\phi_{p}) and \psi=\psi_{1},...,\psi_{q}.

The symbol \phi denotes an AR coefficient. Note that regardless of any constraints, the autoregressive order is always p for all regimes. Ignore or set to NULL if applying linear constraints is not desired.

a positive integer specifying the autoregressive order of the model.

For GMAR and StMAR models:

a positive integer specifying the number of mixture components.

For G-StMAR models:

a size (2\times 1) integer vector specifying the number of GMAR type components M1 in the first element and StMAR type components M2 in the second element. The total number of mixture components is M=M1+M2.

a real valued parameter vector specifying the model.

For non-restricted models:

Size (M(p+3)+M-M1-1\times 1) vector \theta=(\upsilon_{1},...,\upsilon_{M}, \alpha_{1},...,\alpha_{M-1},\nu) where

• \upsilon_{m}=(\phi_{m,0},\phi_{m},\sigma_{m}^2)

• \phi_{m}=(\phi_{m,1},...,\phi_{m,p}), m=1,...,M

• \nu=(\nu_{M1+1},...,\nu_{M})

• M1 is the number of GMAR type regimes.

In the GMAR model, M1=M and the parameter \nu dropped. In the StMAR model, M1=0.

If the model imposes linear constraints on the autoregressive parameters: Replace the vectors \phi_{m} with the vectors \psi_{m} that satisfy \phi_{m}=C_{m}\psi_{m} (see the argument constraints).

For restricted models:

Size (3M+M-M1+p-1\times 1) vector \theta=(\phi_{1,0},...,\phi_{M,0},\phi, \sigma_{1}^2,...,\sigma_{M}^2,\alpha_{1},...,\alpha_{M-1},\nu), where \phi=(\phi_{1},...,\phi_{p}) contains the AR coefficients, which are common for all regimes.

If the model imposes linear constraints on the autoregressive parameters: Replace the vector \phi with the vector \psi that satisfies \phi=C\psi (see the argument constraints).

Symbol \phi denotes an AR coefficient, \sigma^2 a variance, \alpha a mixing weight, and \nu a degrees of freedom parameter. If parametrization=="mean", just replace each intercept term \phi_{m,0} with the regimewise mean \mu_m = \phi_{m,0}/(1-\sum\phi_{i,m}). In the G-StMAR model, the first M1 components are GMAR type and the rest M2 components are StMAR type. Note that in the case M=1, the mixing weight parameters \alpha are dropped, and in the case of StMAR or G-StMAR model, the degrees of freedom parameters \nu have to be larger than 2.

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

a logical argument stating whether the AR coefficients \phi_{m,1},...,\phi_{m,p} are restricted to be the same for all regimes.

specifies linear constraints imposed to each regime's autoregressive parameters separately.

For non-restricted models:

a list of size (p \times q_{m}) constraint matrices C_{m} of full column rank satisfying \phi_{m}=C_{m}\psi_{m} for all m=1,...,M, where \phi_{m}=(\phi_{m,1},...,\phi_{m,p}) and \psi_{m}=(\psi_{m,1},...,\psi_{m,q_{m}}).

For restricted models:

a size (p\times q) constraint matrix C of full column rank satisfying \phi=C\psi, where \phi=(\phi_{1},...,\phi_{p}) and \psi=\psi_{1},...,\psi_{q}.

The symbol \phi denotes an AR coefficient. Note that regardless of any constraints, the autoregressive order is always p for all regimes. Ignore or set to NULL if applying linear constraints is not desired.

a positive integer specifying the autoregressive order of the model.

For GMAR and StMAR models:

a positive integer specifying the number of mixture components.

For G-StMAR models:

a size (2\times 1) integer vector specifying the number of GMAR type components M1 in the first element and StMAR type components M2 in the second element. The total number of mixture components is M=M1+M2.

a real valued parameter vector specifying the model.

For non-restricted models:

Size (M(p+3)+M-M1-1\times 1) vector \theta=(\upsilon_{1},...,\upsilon_{M}, \alpha_{1},...,\alpha_{M-1},\nu) where

• \upsilon_{m}=(\phi_{m,0},\phi_{m},\sigma_{m}^2)

• \phi_{m}=(\phi_{m,1},...,\phi_{m,p}), m=1,...,M

• \nu=(\nu_{M1+1},...,\nu_{M})

• M1 is the number of GMAR type regimes.

In the GMAR model, M1=M and the parameter \nu dropped. In the StMAR model, M1=0.

If the model imposes linear constraints on the autoregressive parameters: Replace the vectors \phi_{m} with the vectors \psi_{m} that satisfy \phi_{m}=C_{m}\psi_{m} (see the argument constraints).

For restricted models:

Size (3M+M-M1+p-1\times 1) vector \theta=(\phi_{1,0},...,\phi_{M,0},\phi, \sigma_{1}^2,...,\sigma_{M}^2,\alpha_{1},...,\alpha_{M-1},\nu), where \phi=(\phi_{1},...,\phi_{p}) contains the AR coefficients, which are common for all regimes.

If the model imposes linear constraints on the autoregressive parameters: Replace the vector \phi with the vector \psi that satisfies \phi=C\psi (see the argument constraints).

Symbol \phi denotes an AR coefficient, \sigma^2 a variance, \alpha a mixing weight, and \nu a degrees of freedom parameter. If parametrization=="mean", just replace each intercept term \phi_{m,0} with the regimewise mean \mu_m = \phi_{m,0}/(1-\sum\phi_{i,m}). In the G-StMAR model, the first M1 components are GMAR type and the rest M2 components are StMAR type. Note that in the case M=1, the mixing weight parameters \alpha are dropped, and in the case of StMAR or G-StMAR model, the degrees of freedom parameters \nu have to be larger than 2.

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

a positive integer specifying the autoregressive order of the model.

For GMAR and StMAR models:

a positive integer specifying the number of mixture components.

For G-StMAR models:

a size (2\times 1) integer vector specifying the number of GMAR type components M1 in the first element and StMAR type components M2 in the second element. The total number of mixture components is M=M1+M2.

a real valued parameter vector specifying the model.

For non-restricted models:

Size (M(p+3)+M-M1-1\times 1) vector \theta=(\upsilon_{1},...,\upsilon_{M}, \alpha_{1},...,\alpha_{M-1},\nu) where

• \upsilon_{m}=(\phi_{m,0},\phi_{m},\sigma_{m}^2)

• \phi_{m}=(\phi_{m,1},...,\phi_{m,p}), m=1,...,M

• \nu=(\nu_{M1+1},...,\nu_{M})

• M1 is the number of GMAR type regimes.

In the GMAR model, M1=M and the parameter \nu dropped. In the StMAR model, M1=0.

If the model imposes linear constraints on the autoregressive parameters: Replace the vectors \phi_{m} with the vectors \psi_{m} that satisfy \phi_{m}=C_{m}\psi_{m} (see the argument constraints).

For restricted models:

Size (3M+M-M1+p-1\times 1) vector \theta=(\phi_{1,0},...,\phi_{M,0},\phi, \sigma_{1}^2,...,\sigma_{M}^2,\alpha_{1},...,\alpha_{M-1},\nu), where \phi=(\phi_{1},...,\phi_{p}) contains the AR coefficients, which are common for all regimes.

If the model imposes linear constraints on the autoregressive parameters: Replace the vector \phi with the vector \psi that satisfies \phi=C\psi (see the argument constraints).

Symbol \phi denotes an AR coefficient, \sigma^2 a variance, \alpha a mixing weight, and \nu a degrees of freedom parameter. If parametrization=="mean", just replace each intercept term \phi_{m,0} with the regimewise mean \mu_m = \phi_{m,0}/(1-\sum\phi_{i,m}). In the G-StMAR model, the first M1 components are GMAR type and the rest M2 components are StMAR type. Note that in the case M=1, the mixing weight parameters \alpha are dropped, and in the case of StMAR or G-StMAR model, the degrees of freedom parameters \nu have to be larger than 2.

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

a logical argument stating whether the AR coefficients \phi_{m,1},...,\phi_{m,p} are restricted to be the same for all regimes.

specifies linear constraints imposed to each regime's autoregressive parameters separately.

For non-restricted models:

a list of size (p \times q_{m}) constraint matrices C_{m} of full column rank satisfying \phi_{m}=C_{m}\psi_{m} for all m=1,...,M, where \phi_{m}=(\phi_{m,1},...,\phi_{m,p}) and \psi_{m}=(\psi_{m,1},...,\psi_{m,q_{m}}).

For restricted models:

a size (p\times q) constraint matrix C of full column rank satisfying \phi=C\psi, where \phi=(\phi_{1},...,\phi_{p}) and \psi=\psi_{1},...,\psi_{q}.

The symbol \phi denotes an AR coefficient. Note that regardless of any constraints, the autoregressive order is always p for all regimes. Ignore or set to NULL if applying linear constraints is not desired.

a positive integer specifying the autoregressive order of the model.

For GMAR and StMAR models:

a positive integer specifying the number of mixture components.

For G-StMAR models:

a size (2\times 1) integer vector specifying the number of GMAR type components M1 in the first element and StMAR type components M2 in the second element. The total number of mixture components is M=M1+M2.

a real valued parameter vector specifying the model.

For non-restricted models:

Size (M(p+3)+M-M1-1\times 1) vector \theta=(\upsilon_{1},...,\upsilon_{M}, \alpha_{1},...,\alpha_{M-1},\nu) where

• \upsilon_{m}=(\phi_{m,0},\phi_{m},\sigma_{m}^2)

• \phi_{m}=(\phi_{m,1},...,\phi_{m,p}), m=1,...,M

• \nu=(\nu_{M1+1},...,\nu_{M})

• M1 is the number of GMAR type regimes.

In the GMAR model, M1=M and the parameter \nu dropped. In the StMAR model, M1=0.

If the model imposes linear constraints on the autoregressive parameters: Replace the vectors \phi_{m} with the vectors \psi_{m} that satisfy \phi_{m}=C_{m}\psi_{m} (see the argument constraints).

For restricted models:

Size (3M+M-M1+p-1\times 1) vector \theta=(\phi_{1,0},...,\phi_{M,0},\phi, \sigma_{1}^2,...,\sigma_{M}^2,\alpha_{1},...,\alpha_{M-1},\nu), where \phi=(\phi_{1},...,\phi_{p}) contains the AR coefficients, which are common for all regimes.

If the model imposes linear constraints on the autoregressive parameters: Replace the vector \phi with the vector \psi that satisfies \phi=C\psi (see the argument constraints).

Symbol \phi denotes an AR coefficient, \sigma^2 a variance, \alpha a mixing weight, and \nu a degrees of freedom parameter. If parametrization=="mean", just replace each intercept term \phi_{m,0} with the regimewise mean \mu_m = \phi_{m,0}/(1-\sum\phi_{i,m}). In the G-StMAR model, the first M1 components are GMAR type and the rest M2 components are StMAR type. Note that in the case M=1, the mixing weight parameters \alpha are dropped, and in the case of StMAR or G-StMAR model, the degrees of freedom parameters \nu have to be larger than 2.

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

a logical argument stating whether the AR coefficients \phi_{m,1},...,\phi_{m,p} are restricted to be the same for all regimes.

specifies linear constraints imposed to each regime's autoregressive parameters separately.

For non-restricted models:

a list of size (p \times q_{m}) constraint matrices C_{m} of full column rank satisfying \phi_{m}=C_{m}\psi_{m} for all m=1,...,M, where \phi_{m}=(\phi_{m,1},...,\phi_{m,p}) and \psi_{m}=(\psi_{m,1},...,\psi_{m,q_{m}}).

For restricted models:

a size (p\times q) constraint matrix C of full column rank satisfying \phi=C\psi, where \phi=(\phi_{1},...,\phi_{p}) and \psi=\psi_{1},...,\psi_{q}.

The symbol \phi denotes an AR coefficient. Note that regardless of any constraints, the autoregressive order is always p for all regimes. Ignore or set to NULL if applying linear constraints is not desired.

object of class 'gsmarpred' created with predict.gsmar.

arguments passed to function grid.

a positive integer specifying the number of observations to be plotted along with the prediction. Default is round(length(data)*0.15).

TRUE if forecasts for mixing weights should be plotted, FALSE in not.

should grid a be added to the plots?

object of class 'qrtest' created with the function quantile_residual_tests.

graphical parameters passed to segments in plot.qrtest. Currently not used in print.qrtest

the number of digits to be print

a class 'gsmar' object, typically generated by fitGSMAR or GSMAR.

a numeric vector of positive integers specifying the lags for which autocorrelation is tested.

a numeric vector of positive integers specifying the lags for which conditional heteroscedasticity is tested.

a positive integer specifying to how many simulated observations the covariance matrix Omega (see Kalliovirta (2012)) should be based on. If smaller than data size, then omega will be based on the given data and not on simulated data. Having the covariance matrix omega based on a large simulated sample might improve the tests size properties.

a logical argument defining whether the results should be printed or not.

object of class 'gsmar' created with function fitGSMAR or GSMAR.

additional arguments passed to grid (ignored if plot_res==FALSE).

a positive integer specifying how many steps in the future should be forecasted.

a positive integer specifying to how many simulations the forecast should be based on.

a numeric vector specifying confidence levels for the prediction intervals.

should the prediction be based on sample "median" or "mean"? Or should it be one-step-ahead forecast based on the exact conditional mean ("cond_mean")? prediction intervals won't be calculated if the exact conditional mean is used.

should the prediction intervals be "two-sided", "upper", or "lower"?

a logical argument defining whether the forecast should be plotted or not.

TRUE if forecasts for mixing weights should be plotted, FALSE in not.

a positive integer specifying the number of observations to be plotted along with the prediction. Default is round(length(data)*0.15).

Point forecasts

Prediction intervals

Point forecasts for mixing weights

Individual prediction intervals for mixing weights, as [, , m], m=1,..,M.

object of class 'gsmarpred' generated by predict.gsmar.

currently not in use.

the number of digits to be printed

object of class 'gsmarsum' generated by summary.gsmar.

currently not in use.

the number of digits to be printed

a class 'gsmar' object, typically generated by fitGSMAR or GSMAR.

a numeric scalar specifying the interval plotted for each estimate: the estimate plus-minus abs(scale*estimate).

how many rows should be in the plot-matrix? The default is max(ceiling(log2(nparams) - 1), 1).

how many columns should be in the plot-matrix? The default is ceiling(nparams/nrows). Note that nrows*ncols should not be smaller than the number of parameters.

at how many points should each profile log-likelihood be evaluated at?

a class 'gsmar' object, typically generated by fitGSMAR or GSMAR.

a numeric vector or class 'ts' object containing the data. NA values are not supported.

a positive integer specifying the autoregressive order of the model.

For GMAR and StMAR models:

a positive integer specifying the number of mixture components.

For G-StMAR models:

a size (2\times 1) integer vector specifying the number of GMAR type components M1 in the first element and StMAR type components M2 in the second element. The total number of mixture components is M=M1+M2.

a real valued parameter vector specifying the model.

For non-restricted models:

Size (M(p+3)+M-M1-1\times 1) vector \theta=(\upsilon_{1},...,\upsilon_{M}, \alpha_{1},...,\alpha_{M-1},\nu) where

• \upsilon_{m}=(\phi_{m,0},\phi_{m},\sigma_{m}^2)

• \phi_{m}=(\phi_{m,1},...,\phi_{m,p}), m=1,...,M

• \nu=(\nu_{M1+1},...,\nu_{M})

• M1 is the number of GMAR type regimes.

In the GMAR model, M1=M and the parameter \nu dropped. In the StMAR model, M1=0.

If the model imposes linear constraints on the autoregressive parameters: Replace the vectors \phi_{m} with the vectors \psi_{m} that satisfy \phi_{m}=C_{m}\psi_{m} (see the argument constraints).

For restricted models:

Size (3M+M-M1+p-1\times 1) vector \theta=(\phi_{1,0},...,\phi_{M,0},\phi, \sigma_{1}^2,...,\sigma_{M}^2,\alpha_{1},...,\alpha_{M-1},\nu), where \phi=(\phi_{1},...,\phi_{p}) contains the AR coefficients, which are common for all regimes.

If the model imposes linear constraints on the autoregressive parameters: Replace the vector \phi with the vector \psi that satisfies \phi=C\psi (see the argument constraints).

Symbol \phi denotes an AR coefficient, \sigma^2 a variance, \alpha a mixing weight, and \nu a degrees of freedom parameter. If parametrization=="mean", just replace each intercept term \phi_{m,0} with the regimewise mean \mu_m = \phi_{m,0}/(1-\sum\phi_{i,m}). In the G-StMAR model, the first M1 components are GMAR type and the rest M2 components are StMAR type. Note that in the case M=1, the mixing weight parameters \alpha are dropped, and in the case of StMAR or G-StMAR model, the degrees of freedom parameters \nu have to be larger than 2.

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

a logical argument stating whether the AR coefficients \phi_{m,1},...,\phi_{m,p} are restricted to be the same for all regimes.

specifies linear constraints imposed to each regime's autoregressive parameters separately.

For non-restricted models:

a list of size (p \times q_{m}) constraint matrices C_{m} of full column rank satisfying \phi_{m}=C_{m}\psi_{m} for all m=1,...,M, where \phi_{m}=(\phi_{m,1},...,\phi_{m,p}) and \psi_{m}=(\psi_{m,1},...,\psi_{m,q_{m}}).

For restricted models:

a size (p\times q) constraint matrix C of full column rank satisfying \phi=C\psi, where \phi=(\phi_{1},...,\phi_{p}) and \psi=\psi_{1},...,\psi_{q}.

The symbol \phi denotes an AR coefficient. Note that regardless of any constraints, the autoregressive order is always p for all regimes. Ignore or set to NULL if applying linear constraints is not desired.

is the model parametrized with the "intercepts" \phi_{m,0} or "means" \mu_{m} = \phi_{m,0}/(1-\sum\phi_{i,m})?

a numeric vector or class 'ts' object containing the data. NA values are not supported.

a positive integer specifying the autoregressive order of the model.

For GMAR and StMAR models:

a positive integer specifying the number of mixture components.

For G-StMAR models:

a size (2\times 1) integer vector specifying the number of GMAR type components M1 in the first element and StMAR type components M2 in the second element. The total number of mixture components is M=M1+M2.

a real valued parameter vector specifying the model.

For non-restricted models:

Size (M(p+3)+M-M1-1\times 1) vector \theta=(\upsilon_{1},...,\upsilon_{M}, \alpha_{1},...,\alpha_{M-1},\nu) where

• \upsilon_{m}=(\phi_{m,0},\phi_{m},\sigma_{m}^2)

• \phi_{m}=(\phi_{m,1},...,\phi_{m,p}), m=1,...,M

• \nu=(\nu_{M1+1},...,\nu_{M})

• M1 is the number of GMAR type regimes.

In the GMAR model, M1=M and the parameter \nu dropped. In the StMAR model, M1=0.

If the model imposes linear constraints on the autoregressive parameters: Replace the vectors \phi_{m} with the vectors \psi_{m} that satisfy \phi_{m}=C_{m}\psi_{m} (see the argument constraints).

For restricted models:

Size (3M+M-M1+p-1\times 1) vector \theta=(\phi_{1,0},...,\phi_{M,0},\phi, \sigma_{1}^2,...,\sigma_{M}^2,\alpha_{1},...,\alpha_{M-1},\nu), where \phi=(\phi_{1},...,\phi_{p}) contains the AR coefficients, which are common for all regimes.

If the model imposes linear constraints on the autoregressive parameters: Replace the vector \phi with the vector \psi that satisfies \phi=C\psi (see the argument constraints).

Symbol \phi denotes an AR coefficient, \sigma^2 a variance, \alpha a mixing weight, and \nu a degrees of freedom parameter. If parametrization=="mean", just replace each intercept term \phi_{m,0} with the regimewise mean \mu_m = \phi_{m,0}/(1-\sum\phi_{i,m}). In the G-StMAR model, the first M1 components are GMAR type and the rest M2 components are StMAR type. Note that in the case M=1, the mixing weight parameters \alpha are dropped, and in the case of StMAR or G-StMAR model, the degrees of freedom parameters \nu have to be larger than 2.