smart_ind: Create random parameter vector that is fairly close to a given parameter vector

Description

smart_ind creates random mean parametrized parameter vector that is model fairly close to a given parameter vector. The result may not be satisfy the stability condition.

Usage

smart_ind(
  p,
  M,
  d,
  params,
  weight_function = c("relative_dens", "logistic", "mlogit", "exponential", "threshold",
    "exogenous"),
  weightfun_pars = NULL,
  cond_dist = c("Gaussian", "Student", "ind_Student", "ind_skewed_t"),
  AR_constraints = NULL,
  mean_constraints = NULL,
  weight_constraints = NULL,
  accuracy = 1,
  which_random = numeric(0),
  mu_scale,
  mu_scale2,
  omega_scale,
  B_scale,
  ar_scale = 1,
  ar_scale2 = 1,
  fixed_params = NULL
)

Value

Returns random mean parametrized parameter vector that has the same form as the argument params

in the other functions, for instance, in the function loglikelihood.

Arguments

p

a positive integer specifying the autoregressive order

M

a positive integer specifying the number of regimes

params

a real valued vector specifying the parameter values. Should have the form $\theta = (\phi_{1},...,\phi_{M},\varphi_1,...,\varphi_M,\sigma,\alpha,\nu)$, where (see exceptions below):

$\phi_{m} = $ the $(d \times 1)$ intercept (or mean) vector of the $m$th 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" or "ind_skewed_t":

$\sigma = (vec(B_1),...,vec(B_M)$ $(Md^2 \times 1)$.
$\alpha = $ the $(a\times 1)$ vector containing the transition weight 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":

$\nu = (\nu_1,...,\nu_d)$ $(d \times 1)$, $\nu_i > 2$.

if cond_dist="ind_skewed_t":

$\nu = (\nu_1,...,\nu_d,\lambda_1,...,\lambda_d)$ $(2d \times 1)$, $\nu_i > 2$ and $\lambda_i \in (0, 1)$.

For models with...

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 $m$th 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 thresholds.

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 $\phi_{1},...,\phi_{M}$ 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.

identification="heteroskedasticity":

$\sigma = (vec(W),\lambda_2,...,\lambda_M)$, where $W$ $(d\times d)$ and $\lambda_m$ $(d\times 1)$, $m=2,...,M$, satisfy $\Omega_1=WW'$ and $\Omega_m=W\Lambda_mW'$, $\Lambda_m=diag(\lambda_{m1},...,\lambda_{md})$, $\lambda_{mi}>0$, $m=2,...,M$, $i=1,...,d$.

B_constraints:

For models identified by heteroskedasticity, replace $vec(W)$ with $\tilde{vec}(W)$ that stacks the columns of the matrix $W$ in to vector so that the elements that are constrained to zero are not included. For models identified by non-Gaussianity, replace $vec(B_1),...,vec(B_M)$ with similarly with vectorized versions $B_m$ so that the elements that are constrained to zero are not included.

Above, $\phi_{m}$ is the intercept parameter, $A_{m,i}$ denotes the $i$th coefficient matrix of the $m$th regime, $\Omega_{m}$ denotes the positive definite error term covariance matrix of the $m$th regime, and $B_m$ is the invertible $(d\times d)$ impact matrix of the $m$th regime. $\nu_m$ is the degrees of freedom parameter of the $m$th regime. If parametrization=="mean", just replace each $\phi_{m}$ 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.

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 $m$th 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 $i$th 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 $i$th 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 $i$th 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", "ind_Student", or "ind_skewed_t", where "ind_Student" the Student's $t$ distribution with independent components, and "ind_skewed_t" is the skewed $t$ distribution with independent components (see Hansen, 1994).

AR_constraints

a size $(Mpd^2 \times q)$ constraint matrix $C$ specifying linear constraints to the autoregressive parameters. The constraints are of the form $(\varphi_{1},...,\varphi_{M}) = C\psi$, where $\varphi_{m} = (vec(A_{m,1}),...,vec(A_{m,p})) \ (pd^2 \times 1),\ m=1,...,M$, contains the coefficient matrices and $\psi$ $(q \times 1)$ contains the related parameters. For example, to restrict the AR-parameters to be the identical across the regimes, set $C =$ [I:...:I]' $(Mpd^2 \times pd^2)$ 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 $R=0$ to constrain the weight parameters to the constant $r$ (in this case, $\alpha$ is dropped from the constrained parameter vector).

accuracy

a positive real number adjusting how close to the given parameter vector the returned individual should be. Larger number means larger accuracy. Read the source code for details.

which_random

a vector with length between 1 and M specifying the mixture components that should be random instead of close to the given parameter vector. This does not consider constrained AR or lambda parameters.

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".

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.

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.

fixed_params

a vector containing fixed parameter values for intercept, autoregressive, and weight parameters that should be fixed in the initial population. Should have the form: $(\phi_{1},...,\phi_{M},\varphi_1,...,\varphi_M,\alpha$, where

$(\phi_{m} = $ the $(d \times 1)$ intercept vector of the $m$th regime.
$\varphi_m = (vec(A_{m,1}),...,vec(A_{m,p}))$ $(pd^2 \times 1)$.
$\alpha$ vector of the weight parameters.

For models with...

AR_constraints:: Replace $\varphi_1,...,\varphi_M$ with $\psi$ as described in the argument AR_constraints.

weight_constraints:

Note that fixed_params should always be in the intercept parametrization (and parametrization="intercept" should always be used). Passing this argument from fitSTVAR in does not do anything, as it is designed to be used with the three-phase estimation procedure only. Also, this argument does not do anything if the initial population is specified in the argument initpop.

Details

Structural models are not supported!

References

Ansley C.F., Kohn R. 1986. A note on reparameterizing a vector autoregressive moving average model to enforce stationarity. Journal of statistical computation and simulation, 24:2, 99-106.