data |
a matrix or class 'ts' object with d>1 columns. Each column is taken to represent
a univariate time series. Missing values are not supported.
|
p |
a positive integer specifying the autoregressive order
|
M |
a positive integer specifying the number of regimes
|
weight_function |
What type of transition weights \alpha_{m,t} should be used?
"relative_dens":\alpha_{m,t}=
\frac{\alpha_mf_{m,dp}(y_{t-1},...,y_{t-p+1})}{\sum_{n=1}^M\alpha_nf_{n,dp}(y_{t-1},...,y_{t-p+1})}, where
\alpha_m\in (0,1) are weight parameters that satisfy \sum_{m=1}^M\alpha_m=1 and
f_{m,dp}(\cdot) is the dp-dimensional stationary density of the mth regime corresponding to p
consecutive observations. Available for Gaussian conditional distribution only.
"logistic":M=2, \alpha_{1,t}=1-\alpha_{2,t},
and \alpha_{2,t}=[1+\exp\lbrace -\gamma(y_{it-j}-c) \rbrace]^{-1}, where y_{it-j} is the lag j
observation of the ith variable, c is a location parameter, and \gamma > 0 is a scale parameter.
"mlogit":\alpha_{m,t}=\frac{\exp\lbrace \gamma_m'z_{t-1} \rbrace}
{\sum_{n=1}^M\exp\lbrace \gamma_n'z_{t-1} \rbrace}, where \gamma_m are coefficient vectors, \gamma_M=0,
and z_{t-1} (k\times 1) is the vector containing a constant and the (lagged) switching variables.
"exponential":M=2, \alpha_{1,t}=1-\alpha_{2,t},
and \alpha_{2,t}=1-\exp\lbrace -\gamma(y_{it-j}-c) \rbrace, where y_{it-j} is the lag j
observation of the ith variable, c is a location parameter, and \gamma > 0 is a scale parameter.
"threshold":\alpha_{m,t} = 1 if r_{m-1}<y_{it-j}\leq r_{m} and 0 otherwise, where
-\infty\equiv r_0<r_1<\cdots <r_{M-1}<r_M\equiv\infty are thresholds y_{it-j} is the lag j
observation of the ith variable.
"exogenous":Exogenous nonrandom transition weights, specify the weight series in weightfun_pars.
See the vignette for more details about the weight functions.
|
weightfun_pars |
- If
weight_function == "relative_dens": Not used.
- If
weight_function %in% c("logistic", "exponential", "threshold"): a numeric vector with the switching variable
i\in\lbrace 1,...,d \rbrace in the first and the lag j\in\lbrace 1,...,p \rbrace in the second element.
- If
weight_function == "mlogit": a list of two elements:
- The first element
$vars: a numeric vector containing the variables that should used as switching variables
in the weight function in an increasing order, i.e., a vector with unique elements in \lbrace 1,...,d \rbrace.
- The second element
$lags: an integer in \lbrace 1,...,p \rbrace specifying the number of lags to be
used in the weight function.
- If
weight_function == "exogenous": a size (nrow(data) - p x M) matrix containing the exogenous
transition weights as [t, m] for time t and regime m. Each row needs to sum to one and only weakly positive
values are allowed.
|
cond_dist |
specifies the conditional distribution of the model as "Gaussian", "Student", or "ind_Student",
where the latest is the Student's t distribution with independent components.
|
parametrization |
"intercept" or "mean" determining whether the model is parametrized with intercept
parameters \phi_{m,0} or regime means \mu_{m}, m=1,...,M.
|
AR_constraints |
a size (Mpd^2 x q) constraint matrix C specifying linear constraints
to the autoregressive parameters. The constraints are of the form
(\varphi_{1},...,\varphi_{M}) = C\psi, where \varphi_{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 identical across the regimes, set C =
[I:...:I]' (Mpd^2 x pd^2) where I = diag(p*d^2).
|
mean_constraints |
Restrict the mean parameters of some regimes to be identical? 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
identical 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".
|
weight_constraints |
a list of two elements, R in the first element and r in the second element,
specifying linear constraints on the transition weight parameters \alpha.
The constraints are of the form \alpha = R\xi + r, where R is a known (a\times l)
constraint matrix of full column rank (a is the dimension of \alpha), r is a known (a\times 1) constant,
and \xi is an unknown (l\times 1) parameter. Alternatively, set R=0 in order to constrain the
the weight parameter to the constant r (in this case, \alpha is dropped from the constrained parameter vector).
|
ngen |
a positive integer specifying the number of generations to be ran through in
the genetic algorithm.
|
popsize |
a positive even integer specifying the population size in the genetic algorithm.
Default is 10*n_params.
|
smart_mu |
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.
|
initpop |
a list of parameter vectors from which the initial population of the genetic algorithm
will be generated from. The parameter vectors should have the form
\theta = (\phi_{1,0},...,\phi_{M,0},\varphi_1,...,\varphi_M,\sigma,\alpha,\nu),
where
\phi_{m,0} = the (d \times 1) intercept (or mean) vector of the mth regime.
\varphi_m = (vec(A_{m,1}),...,vec(A_{m,p})) (pd^2 \times 1).
-
- if
cond_dist="Gaussian" or "Student": \sigma = (vech(\Omega_1),...,vech(\Omega_M))
(Md(d + 1)/2 \times 1).
- if
cond_dist="ind_Student": \sigma = (vec(B_1),...,vec(B_M) (Md^2 \times 1).
\alpha contains the transition weights parameters (see below)
-
- if
cond_dist = "Gaussian"): Omit \nu from the parameter vector.
- if
cond_dist="Student": \nu > 2 is the single degrees of freedom parameter.
- if
cond_dist="ind_Student": \nu = (\nu_1,...,\nu_M) (M \times 1), nu_m > 2.
weight_function="relative_dens":\alpha = (\alpha_1,...,\alpha_{M-1})
(M - 1 \times 1), where \alpha_m (1\times 1), m=1,...,M-1 are the transition weight parameters.
weight_function="logistic":\alpha = (c,\gamma)
(2 \times 1), where c\in\mathbb{R} is the location parameter and \gamma >0 is the scale parameter.
weight_function="mlogit":\alpha = (\gamma_1,...,\gamma_M) ((M-1)k\times 1),
where \gamma_m (k\times 1), m=1,...,M-1 contains the multinomial logit-regression coefficients
of the mth regime. Specifically, for switching variables with indices in I\subset\lbrace 1,...,d\rbrace, and with
\tilde{p}\in\lbrace 1,...,p\rbrace lags included, \gamma_m contains the coefficients for the vector
z_{t-1} = (1,\tilde{z}_{\min\lbrace I\rbrace},...,\tilde{z}_{\max\lbrace I\rbrace}), where
\tilde{z}_{i} =(y_{it-1},...,y_{it-\tilde{p}}), i\in I. So k=1+|I|\tilde{p}
where |I| denotes the number of elements in I.
weight_function="exponential":\alpha = (c,\gamma)
(2 \times 1), where c\in\mathbb{R} is the location parameter and \gamma >0 is the scale parameter.
weight_function="threshold":\alpha = (r_1,...,r_{M-1})
(M-1 \times 1), where r_1,...,r_{M-1} are the threshold values.
weight_function="exogenous":Omit \alpha from the parameter vector.
- AR_constraints:
Replace \varphi_1,...,\varphi_M with \psi as described in the argument AR_constraints.
- mean_constraints:
Replace \phi_{1,0},...,\phi_{M,0} with (\mu_{1},...,\mu_{g}) where
\mu_i, \ (d\times 1) is the mean parameter for group i and g is the number of groups.
- weight_constraints:
If linear constraints are imposed, replace \alpha with \xi as described in the
argument weigh_constraints. If weight functions parameters are imposed to be fixed values, simply drop \alpha
from the parameter vector.
Above, \phi_{m,0} is the intercept parameter, A_{m,i} denotes the ith coefficient matrix of the mth
regime, \Omega_{m} denotes the positive definite error term covariance matrix of the mth regime, and B_m
is the invertible (d\times d) impact matrix of the mth regime. \nu_m is the degrees of freedom parameter
of the mth regime.
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. Bvec()
is a vectorization operator that stacks the columns of a given impact matrix B_m into a vector so that the elements
that are constrained to zero by the argument B_constraints are excluded.
|
mu_scale |
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.
|
mu_scale2 |
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 vapply(1:d, function(i1) sd(data[,i1]), numeric(1)).
|
omega_scale |
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
cond_dist == "ind_Student".
|
B_scale |
a size (d \times 1) strictly positive vector specifying the mean and variability of the
random impact matrices in random mutations. In Regime 1, the mean of the error term covariance matrix
implied by the random impact matrix will be 0.95*diag(B_scale) and in the rest of the regimes diag(B_scale),
whereas the variability increases with B_scale.
Default in GAfit is var(stats::ar(data[,i], order.max=10)$resid, na.rm=TRUE), i=1,...,d.
This argument is ignored if cond_dist != "ind_Student".
|
weight_scale |
For...
weight_function %in% c("relative_dens", "exogenous"):not used.
weight_function %in% c("logistic", "exponential"):length three vector with the mean (in the first element)
and standard deviation (in the second element) of the normal distribution the location parameter is drawn from
in random mutations. The third element is the standard deviation of the normal distribution from whose absolute value
the location parameter is drawn from.
weight_function == "mlogit":length two vector with the mean (in the first element)
and standard deviation (in the second element) of the normal distribution the coefficients of the logit sub model's
constant terms are drawn from in random mutations. The third element is the standard deviation of the normal distribution
from which the non-constant regressors' coefficients are drawn from.
weight_function == "threshold":a lenght two vector with the lower bound, in the first element
and the upper bound, in the second element, of the uniform distribution threshold parameters are drawn from
in random mutations.
|
ar_scale |
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.
|
upper_ar_scale |
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.
|
ar_scale2 |
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 parameter candidates
that satisfy the stability condition.
|
regime_force_scale |
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.
|
red_criteria |
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(transitionWeights[,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).
|
pre_smart_mu_prob |
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.
|
to_return |
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?
|
minval |
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).
|
seed |
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 fitSTVAR, use the argument seeds
instead of passing the argument seed.
|
By "redundant" or "wasted" regimes we mean regimes that have the time varying mixing weights practically at
zero for almost all t. A model including redundant regimes would have about the same log-likelihood value without
the redundant regimes and there is no purpose to have redundant regimes in a model.
Some of the AR coefficients are drawn with the algorithm by Ansley and Kohn (1986). However,
when using large ar_scale with large p or d, numerical inaccuracies caused
by the imprecision of the float-point presentation may result in errors or nonstationary AR-matrices.
Using smaller ar_scale facilitates the usage of larger p or d. Therefore, we bound
upper_ar_scale from above by 1-pd/150 when p*d>40 and by 1 otherwise.
Structural models are not supported here, as they are best estimated based on reduced form parameter estimates
using the function fitSSTVAR.