arguments {BTSR} | R Documentation |
Shared documentation for arguments
Description
This is the common documentation for all parameter in BTSR package.
The package handles function arguments in two compatible formats
-
Legacy structure (pre-1.0.0). Used for models with fixed or no
\nu
parameter. Automatically converted to the new format when processed. -
New structure (1.0.0+). Required for models with time-varying
\nu
parameter.
All functions accept both formats seamlessly, ensuring backward compatibility. The internal processing automatically standardizes to the new structure.
Arguments
model |
character string (case-insensitive) indicating the model to be fitted to the data. Must be one of the options listed in the Section Supported Models. |
n |
the sample size of the output time series |
nnew |
optional; the number of out-of sample predicted values required
(extract and fit only). Default is |
burn |
the length of the ‘burn-in’ period (simulation only).
Default is |
yt |
numeric vector with the observed time series (extract and fit only). Missing values (NA's) are not allowed. |
y.start |
optional; an initial value for |
rho |
the quantile being considered in the conditional distribution of
|
y.lower |
the lower limit for the Kumaraswamy density support. Default
is |
y.upper |
the upper limit for the Kumaraswamy density support. Default
is |
vt.start |
optional; an initial value for |
e2.start |
optional; an initial value for |
xreg |
optional; external regressors. Can be specified as a vector, a
matrix or a list. Default is |
xnew |
optional; |
xreg.start |
optional; initial value for the regressors (to initialize
recursion). Can be specified as a vector or a list. Default is
|
xregar |
a length 1 or 2 logical vector indicating whether |
inf |
a length 1 or 2 integer vector given the truncation point for
infinite sums. Default is |
p |
optional; a length 1 or 2 integer vector given the order of the AR
polynomial (extract and fit only). Default is |
q |
optional; a length 1 or 2 integer vector given the order of the MA
polynomial (extract and fit only). Default is |
d |
a length 1 or 2 logical vector indicating whether the long memory
parameter |
ignore.start |
optional; logical value indicating whether the argument
|
start |
optional; a list with the starting values for the non-fixed
coefficients of the model (fit only). The default is |
coefs |
a list with the coefficients of the model (simulation and
extraction only). The default is |
lags |
optional; a list with the lags (integer values) that the entries
in |
fixed.values |
optional; a list with the values of the coefficients that
are fixed (extract and fit only). The default is |
fixed.lags |
optional; a list with the lags (integer values) that the
fixed values in |
lower |
optional; list with the lower bounds for the parameters (fit
only). Default is |
upper |
optional; list with the upper bounds for the parameters (fit
only). Default is |
map |
a non-negative integer from 1 to 5 corresponding to the map
function. Default is |
error.scale |
either 0 or 1; the scale for the error term. Default is
|
linkg |
link functions. Can be specified as a character, two-character vector or a named list. The corresponding text strings for currently available links are listed in link.btsr. Default values depend on the model. For some models default values override user specifications. See the Section Link defaults for details. |
linkh |
a character indicating which link must be associated to the
chaotic process. See the Section ‘The BTSR structure’ in
btsr-package for details and link.btsr for valid links. Default is
|
configs.linkg |
a list with two elements, |
configs.linkh |
a list with extra configurations for the link |
m |
a non-negative integer indicating the starting time for the sum of
the partial log-likelihood, given by |
llk |
logical; indicates whether the value of the log-likelihood
function should be returned (extract and fit only). Default is |
sco |
logical; indicates whether the score vector should be returned
(extract and fit only). Default is |
info |
logical; indicates whether the information matrix should be
returned (extract and fit only). Default is |
extra |
logical, if |
control |
a list with configurations to be passed to the optimization
subroutines (fit only). Default is |
report |
logical; indicates whether the summary from the fitted model
should be be printed (fit only). Default is |
complete |
logical; if FALSE returns only |
debug |
logical, if |
... |
further arguments passed to the internal functions. See, for instance, summary.btsr for details. |
Supported Models
Internally, all models are handled by the same function and all models can be
obtained from the more general case "*ARFIMAV"
. When a particular model
(e.g. "BREG"
or "BARMA"
) is invoked some default values are assumed.
The following table summarizes the available distributions and the
corresponding string to generate each model type. The character V
at
the end of the string indicates that \nu
is time-varying.
+--------------+--------+------------+---------+-----------+---------+ | Distribution | i.i.d. | Regression | Short | Long | Chaotic | | | sample | | Memory | Memory | | +--------------+--------+------------+---------+-----------+---------+ | Beta | BETA | BREG | BARMA | BARFIMA | BARC | | | | BREGV | BARMAV | BARFIMAV | | +--------------+--------+------------+---------+-----------+---------+ | Gamma | GAMMA | GREG | GARMA | GARFIMA | | | | | GREGV | GARMAV | GARFIMAV | | +--------------+--------+------------+---------+-----------+---------+ | Kumaraswamy | KUMA | KREG | KARMA | KARFIMA | | | | | KREGV | KARMAV | KARFIMAV | | +--------------+--------+------------+---------+-----------+---------+ | Matsuoka | MATSU | MREG | MARMA | MARFIMA | | +--------------+--------+------------+---------+-----------+---------+ | Unit-Lindley | UL | ULREG | ULARMA | ULARFIMA | | +--------------+--------+------------+---------+-----------+---------+ | Unit-Weibull | UW | UWREG | UWARMA | UWARFIMA | | | | | UWREGV | UWARMAV | UWARFIMAV | | +--------------+--------+------------+---------+-----------+---------+
Default values
All models are special cases of the general "*ARFIMAV"
structure. When a
specific model is selected via model = "NAME"
, the package automatically
applies these default configurations (any parameter that does not appear in
the equations below is ignored)
i.i.d samples (e.g., BETA, GAMMA,...)
\eta_{1t} = \alpha_1 = \mu, \quad \eta_{2t} = \alpha_2 = \nu.
Fixed
p <- q <- d <- 0 xreg <- NULL linkg <- list(g11 = "linear", g2 = "linear", g21 = "linear", g23 = "linear")
Regression models with \nu_t
constant over time (e.g., BREG,
GREG,...)
\eta_{1t} =
g_{11}(\mu_t) = \alpha_1 + \boldsymbol{X}_{1t}'\boldsymbol{\beta}_1,
\quad
\eta_{2t} = \alpha_2 = \nu.
Fixed
p <- q <- d <- 0 xreg <- list(part1 = "user's regressors", part2 = NULL) linkg <- list(g11 = "user's choice", g12 = "linear", g2 = "linear", g21 = "linear", g23 = "linear")
Regression models with \nu_t
varying on time (e.g. BREGV,
GREGV)
\eta_{1t} =
g_{11}(\mu_t) = \alpha_1 + \boldsymbol{X}_{1t}'\boldsymbol{\beta}_1,
\quad
\eta_{2t} =
g_{21}(g_2(\nu_t)) = \alpha_2 + \boldsymbol{X}_{2t}'\boldsymbol{\beta}_2.
Fixed
p <- q <- d <- 0 linkg <- list(g11 = "user's choice", g12 = "linear", g2 = "user's choice", g21 = "user's choice", g22 = "linear", g23 = "linear")
Short-memory models with \nu
constant over time (ARMA-like) (e.g. BARMA,
GARMA,...)
\begin{aligned}
\eta_{1t} & =
g_{11}(\mu_t) = \alpha_1 + \boldsymbol{X}_{1t}'\boldsymbol{\beta}_1 +
\sum_{i=1}^{p_1} \phi_{1i}\bigl(g_{12}(Y_{t-i})-
I_{X_1}\boldsymbol{X}_{1(t-i)}'\boldsymbol{\beta}_1\bigr) +
\sum_{k=1}^{q_1} \theta_{1k} e_{1,t-k}, \\
\eta_{2t} & = \alpha_2 = \nu.
\end{aligned}
Fixed
d <- 0 xreg <- list(part1 = "user's regressors", part2 = NULL) linkg <- list(g11 = "user's choice", g12 = "user's choice", g2 = "linear", g21 = "linear", g23 = "linear")
Short-memory models with \nu_t
varying on time (e.g. BARMAV,
GARMAV,...)
\begin{aligned}
\eta_{1t} & =
g_{11}(\mu_t) =\alpha_1 + \boldsymbol{X}_{1t}'\boldsymbol{\beta}_1 +
\sum_{i=1}^{p_1} \phi_{1i}\big(g_{12}(Y_{t-i})-
I_{X_1}\boldsymbol{X}_{1(t-i)}'\boldsymbol{\beta}_1\big) +
\sum_{k=1}^{q_1} \theta_{1k} r_{t-k},\\
\vartheta_t & = g2(\nu_t)\\
\eta_{2t} &
= g_{21}(\vartheta_t) =\alpha_2 + \boldsymbol{X}_{2t}' \boldsymbol{\beta}_2
+ \sum_{i=1}^{p_2} \phi_{2i}\big(g_{22}(\vartheta_{t-i})-
I_{X_2}\boldsymbol{X}_{2(t-i)}'\boldsymbol{\beta}_2\big) +
\sum_{k=1}^{q_2} \theta_{2k} g_{23}(e_{1,t-k}).
\end{aligned}
Fixed
d <- 0
Long-memory models with \nu
constant over time (ARFIMA-like
models) (e.g. BARFIMA, GARFIMA,...)
\begin{aligned}
\eta_{1t} & =
g_{11}(\mu_t) =\alpha_1 + \boldsymbol{X}_{1t}'\boldsymbol{\beta}_1 +
\sum_{i=1}^{p_1} \phi_{1i}\big(g_{12}(Y_{t-i})-
I_{X_1}\boldsymbol{X}_{1(t-i)}'\boldsymbol{\beta}_1\big) +
\sum_{k=1}^\infty c_{1k} r_{t-k},\\
\eta_{2t} & =\alpha_2 = \nu.
\end{aligned}
Fixed
p <- c("user's p", 0) q <- c("user's q", 0) d <- c("user's d", 0) xreg <- list(part1 = "user's regressors", part2 = NULL) linkg <- list(g11 = "user's choice", g12 = "user's choice", g2 = "linear", g21 = "linear", g23 = "linear")
Reproducing Models from the Literature
This section summarizes how to replicate well-known time series models from the literature using the BTSR package. For each model type, we provide the necessary parameter settings and references to the original publications. These configurations act as templates, helping users correctly apply the package to reproduce results or extend established models.
Key arguments (e.g., error.scale
, xregar
, y.lower
,
y.upper
, rho
) should be set to match the specifications in the
referenced articles. While we focus on the btsr.*
functions (see
BTSR.functions), all models can also be implemented using the corresponding
parent model functions (for details, see BTSR.parent.models).
i.i.d. samples: The arguments error.scale
and xregar
are
ignored.
Beta distribution with parameters
shape1
andshape2
compatible with the one from rbeta:model = "BETA" alpha = shape1/(shape1 + shape2) nu = shape1 + shape2
Gamma distribution with parameters
shape
andscale
compatible with the one from rgamma:model = "GAMMA" alpha = shape*scale nu = shape
Kumaraswamy distribution with shape parameters
shape1
andshape2
(respectively denoted bya
andb
in Kumaraswamy 1980):model = "KUMA" alpha = (y.lower - y.upper)*(1 - (1-rho)^1/shape2)*1/shape1 + y.lower nu = shape1
Warning: Choose
\mu
,\nu
and\rho
carefully since|\log(1-\rho)| >> |\log(1 - \mu^\nu)|
may cause numerical instability.Matsuoka distribution with shape parameter
shape
(Matsuoka et al. 2024):model = "MATSU" alpha = (shape/(shape+1))^(3/2)
Unit-Lindley distribution with parameter
theta
(Mazucheli et al. 2018):model = "UL" alpha = 1/(1 + theta)
Unit-Weibull distribution with parameter
mu
,beta
andtau
from (Mazucheli et al. 2019):model = "UW" alpha = mu nu = beta rho = tau
Regression models: the argument error.scale
and all entries but
g11
in linkg
are ignored
Beta regression (Ferrari and Cribari-Neto 2004):
model = "BREG"
Kumaraswamy regression (Mitnik and Baek 2013):
model = "KREG"
.Unit-Lindley regression (Mazucheli et al. 2018):
model = "ULREG"
.Unit-Weibull regression (Mazucheli et al. 2019):
model = "UWREG"
.
ARMA-like models
BARMA model (Rocha and Cribari-Neto 2009; Rocha and Cribari-Neto 2017):
model = "BARMA" error.scale = 1 xregar = TRUE
KARMA model (Bayer et al. 2017):
model = "KARMA" error.scale = 1 xregar = TRUE y.lower = 0 y.upper = 1 rho = 0.5
GARMA model (Prass et al. 2025):
model = "GARMA" error.scale = 0
MARMA model (Pumi et al. 2025):
model = "MARMA" error.scale = 1 xregar = TRUE
ULARMA model (Pumi et al. 2025):
model = "ULARMA" error.scale = 1 xregar = TRUE
ARFIMA-like models
BARFIMA model (Pumi et al. 2019):
model = "BARFIMA" error.scale = 1 xregar = TRUE d = TRUE (for fitting)
Chaotic models
BARC model (Pumi et al. 2021): set
model = "BARC"
anderror.scale = 1
.
Regressors format
In-sample (xreg
) and out-of-sample values (xnew
) for regressors can be
provided in two formats
-
Legacy structure: a vector or matrix. Internally
xreg
is converted toxreg = list(part1 = xreg, part2 = NULL)
. The same applies toxnew
-
New structure: a list with elements
part1
(regressors for first model component) andpart2
(regressors for second model component), each being a vector or matrix.
xreg.start
can be provided in two formats
-
Legacy structure: a vector with initial values for each regressor. Internally
xreg.start
is converted toxreg.start = list(part1 = xreg.start, part2 = NULL)
. -
New structure: a list with elements
part1
andpart2
, each a vector of initial values for the respective regressors.
The following rules apply to xreg
, xnew
and xreg.start
if
model
corresponds to a case where\nu
is constant over time (e.g.,model = "BARMA"
),part2
is ignored.For simulation, regressors must include
n + burn
observations.For model fitting, parameter initialization, or component extraction, the number of regressor observations must match the length of the observed time series
yt
.When
xreg = NULL
ornnew = 0
,xnew
is ignored. Ifnnew > 0
and the number of regressors inxnew
does not matchxreg
an error message is issued.If starting values for
xreg
are not provided andp_i > 0
for thei
th part of the model, the default behavior is to assume\displaystyle\boldsymbol{X}_t = \frac{1}{p_i}\sum_{k = 1}^{p_i} \boldsymbol{X}_k, \quad \text{for } t< 1.
Model Order
The coefficients \{c_{ik}\}_{k\geq 0}
are defined through the relation
(see the section ‘The BTSR Structure’ in btsr-package)
c_i(z) := (1-L)^{-d_i}\theta_i(z) = \sum_{k = 0}^\infty c_{ik}z^k, \quad i \in \{1,2\}.
where \theta_i(z) = \sum_{k = 0}^{q_i} \theta_{ik}z^k
is the moving
average characteristic polynomial, with order q_i
. For practical
purposes, the following approximation is used
c_i(z) \approx \sum_{k = 0}^{K_i} c_{ik}z^k,
for some K_i
sufficiently large.
inf
corresponds to the truncation point for all infinite sums using the
coefficients \{c_{ik}\}_{k\geq 0}
, i \in \{1,2\}
, including
samples generation and derivatives calculation. It can be provided as either
a single integer (legacy format) or a length 2 integer vector (new format)
specifying the trunction points for part1
/part2
. If \nu
is
time-varying and a single value is provided the same value is used for both
parts. When d = 0
, Fortran automatically sets inf
to q
(MA
order).
By default p
and q
are set to NULL
, in which case their values are
computed internally, based on the size of the argument phi
and theta
,
respectively, in the lists of coefficients (or staring values), fixed lags,
and fixed values. For fitting purposes, if p
(analogously, q
) and start
are both NULL
, an error message is issued. These parameters can be
provided as either a single integer (legacy format) or a length 2 integer
vector (new format) specifying orders for part1
/part2
. If \nu
is
time-varying and a single value of p
(analogously, q
) is provided it is
assumed that p_1 = p_2 = p
(analogously, q_1 = q_2 = q
).
Model coefficients
start
, coefs
, fixed.values
, lags
and fixed.lags
can be specified in
one of two ways
-
Legacy structure: a list with optional components
alpha
,beta
,phi
,theta
,d
,u0
(BARC only) and required argumentnu
(except for one-parameter models such as ULARMA and MARMA). -
New structure: a list with elements
part1
andpart2
, each being a list with with optional componentsalpha
,beta
,phi
,theta
,d
andu0
(BARC only).
The optional arguments in this lists are
-
alpha
: a numeric value corresponding to the intercept. For i.i.d. corresponds to the mean of the distribution. -
beta
: a vector of coefficients corresponding to the regressors inxreg
. -
phi
: a vector of autoregressive coefficients. -
theta
: for BARC models, this is the parameter for the map function (see BARC.functions for details). For any other model, this is a vector of moving average coefficients corresponding to the MA order. d
: a numeric value corresponding to the long memory parameter.-
u0
: a numeric value in the interval(0,1)
, corresponding to the value of the random variableU_0
. See BARC.functions for details. nu
: distribution related parameter, usually the dispersion.
The following rules apply for these lists and their arguments.
Simulation:
Passing
coefs
as an empty list will result in an error message.-
start
andfixed.values
(consequently,fixed.lags
) are not used. If
xreg
is provided butcoefs
does not include abeta
argument, an error message is issued.-
phi
must be a vector of lengthp
(the AR order), meaning all coefficients must be provided, including zeros. -
theta
(non-BARC models) must be a vector of lengthq
(the MA order), meaning all coefficients must be provided, including zeros.
Extraction:
One dimensional parameters (e.g.
alpha
) that do not appear incoefs
are assume to be fixed.An error message will be issued if both
coefs
andfixed.values
are both empty.If
\nu
is not constant over time andnu
is missing in bothcoefs
andfixed.values
, an error message is issued (except for one-parameter models such as ULARMA and MARMA). Ignored if the new format is used.
Fitting:
One dimensional parameters (e.g.
alpha
) cannot appear in bothstart
andfixed.values
.-
coefs
is not used.
Extraction and fitting:
Coefficients may include both fixed lags (with values in
fixed.values
) and non-fixed lags (with values incoefs
orstart
).-
lags
andfixed.lags
are complementary. Either suffices, or mix them (e.g.,lags
for some parameters,fixed.lags
for others). For one dimensional parameters, the lag is obviously always 1 and can be suppressed when the parameter added to the
fixed.values
list.For extraction, if
coefs = NULL
, one dimensional parameters that do not appear infixed.values
are assumed to be non-fixed. The same goes for fitting whenstart = NULL
orignore.start = TRUE
.If
coefs
/start
is provided, one dimensional parameters that do not appear in this list are assumed to be fixed.By default, if a given vector has fixed lags and the corresponding entry in
fixed.values
is empty, the fixed values are set as zero.If parameter values are provided in
coefs
,start
orfixed.values
and the size of the vector is not the same as the dimension of the parameters, eitherlags
orfixed.lags
must also be provided.
The map function
The map function T:[0,1] \to [0,1]
in BARC models is a dynamical
system, i.e., a function, potentially depending on a r
-dimensional
vector of parameters \theta
. As for today, for all implemented maps,
r = 1
.
Available choices are
-
map = 1
,\theta = k
, fork
integer greater or equal to 2.T(u) = (ku)(\text{mod } 1)
-
map = 2
,0 \le \theta \le 1
T(u) = \dfrac{u}{\theta}I( u < \theta) + \theta\dfrac{(u - \theta)}{(1 - \theta)}I(u \ge \theta)
-
map = 3
(logistic map),0 \le \theta \le 4
,T(u) = \theta(1-\theta)
-
map = 4
(Manneville-Pomeau map),0 < \theta < 1
T(u) = (u + u^{1+\theta})(\text{mod } 1)
-
map = 5
(Lasota-Mackey's map),T(u) = \dfrac{u}{(1 - u)}I(u \le 0.5) + (2u - 1)I(u > 0.5)
Link defaults
linkh
and configs.linkh
only apply to BARC models.
linkg
can be specified in one of two ways
-
Legacy structure: a character or two-character vector. If only one string is provided, the same link name is used for
g11
andg12
. Internally, this structure is automatically converted to the new format withg2 = g21 = g22 = g23 = "linear"
. -
New structure: a named list with optional elements (order is irrelevant)
g11
,g12
,g2
,g21
,g22
andg23
. These links apply, respectively, to\mu_t
,Y_t
(in the AR recursion or part 1),\nu_t
,\vartheta_t = g_2(\nu_t)
,\vartheta_t
(in the AR recursion of part 2) ande_{1t}
(to build the error term in part 2).
For models that do not have the \nu
parameter, the links g2
, g21
,
g22
and g23
are set to "linear"
for compatibility with Fortran
subroutines.
Missing entries in the linkg
list follow these rules
If either
g11
org12
is missing (but not both), internally it is setg12 = g11
.If both
g11
andg12
are missing, use the default values for the particular model (see below).If
phi = NULL
for part 1,g12
is not required, hence set to"linear"
and ignored in Fortran.If
phi = NULL
for part 2,g22
is not required, hence set to"linear"
and ignored in Fortran.If either
g21
org22
is missing (but not both), internally it is setg22 = g21
.If both
g21
andg22
are missing, use the default values for the particular model (see below).
Default linkg
values are model-dependent (based on the string provided with
model
):
For all models where
\nu
is constant over time:
internally,g2
,g21
, andg22
are forced to"linear"
, witha = 1
.
Overrides any user specifications.iid samples:
Overrides any user specifications.
linkg = "linear"
(witha = 1
). Internally converted tolinkg = list(g11 = "linear", g12 = "linear", g2 = "linear" g21 = "linear", g22 = "linear", g23 = "linear")
BARFIMA, KARFIMA, ULARFIMA, UWARFIMA:
linkg = "logit"
. Internally converted tolinkg = list(g11 = "logit", g12 = "logit", g2 = "linear" g21 = "linear", g22 = "linear", g23 = "linear")
GARFIMA:
linkg = "log"
. Internally converted tolinkg = list(g11 = "log", g12 = "log", g2 = "linear" g21 = "linear", g22 = "linear", g23 = "linear")
MARFIMA:
linkg = "cloglog"
. Internally converted tolinkg = list(g11 = "cloglog", g12 = "cloglog", g2 = "linear" g21 = "linear", g22 = "linear", g23 = "linear")
BARFIMAV, GARFIMAV, KARFIMAV, UWARFIMAV:
g11
andg12
have the same default values as the particular model where\nu
is constant over time.g2 = "default"
, meaning thatg2
is set as the the default link for the model.For BARFIMAV
"default" = SIP
witha = b = 1
.For GARFIMAV
"default" = SIP
witha = 0
andb = 1
.For remaining models
"default" = "linear"
witha = 1
.
g21
depends on the model.For BARFIMAV
g21 = "logit"
For any other model
g21 = "log"
.
For
g22
, the default is to assumeg22 = g21
.Finally,
g23 = "polynomial
, witha = 1
andb = 2
(set inconfigs.link
)Particular cases (e.g., BREG, BREGV) inherit defaults from parent models (except iid samples).
configs.linkg
if provided, it must be provided as a list with optional
elements, ctt
and power
, which define the constant a
and the
exponent b
in the link function g(x) = a x^b
. Each element in
this list can be specified in one of two ways
-
Legacy structure: a numeric value (applied uniformly across all linear links) or a numeric vector of length 2, which will be associated to
g11
andg12
. -
New structure: a named list with optional elements (order is irrelevant)
g11
,g12
,g2
,g21
,g22
andg23
.
For now, the arguments ctt
and power
are only used when the link function
is "linear"
or "polynomial"
. If NULL
, default is to assume that ctt
and power
are both equal to 1 for all links.
The log-likelihood
Let \boldsymbol\gamma = (\boldsymbol \rho', \boldsymbol \lambda')'
be
the vector of unknown parameters in the model where
-
\boldsymbol\rho
is the vector of unknown parameters in part 1 -
\boldsymbol\lambda
is the vector of unknown parameters in part 2.
The log-likelihood function, conditioned on a set of initial conditions
\mathcal{F}_m
is given by
\ell(\boldsymbol\gamma) = \sum_{t = m+1}^n \ell_t =
\displaystyle\sum_{t=m+1}^n\log\!\big(f(Y_t \mid \mathcal{F}_{t-1},
\boldsymbol{\gamma})\big).
For simplicity of notation assume m = 0
. The score vector
U(\boldsymbol\gamma) = \big(U_{\boldsymbol\rho}(\boldsymbol\gamma)',
U_{\boldsymbol\lambda}(\boldsymbol\gamma)'\big)'
can be written as
U_{\boldsymbol\rho}(\boldsymbol\gamma) = D_{\boldsymbol\rho}' T_1\boldsymbol h_1 +
M_{\boldsymbol\rho}' T_2\boldsymbol h_2 \qquad \mbox{and} \qquad
U_{\boldsymbol\lambda}(\boldsymbol\gamma) = D_{\boldsymbol\lambda}' T_2\boldsymbol h_2,
where
-
D_{\boldsymbol\rho}
,D_{\boldsymbol\lambda}
andM_{\boldsymbol\rho}
are the matrices for which the(i,j)
th elements are given, respectively, by[D_{\boldsymbol\rho}]_{i,j} = \dfrac{\partial \eta_{1i}}{\partial \rho_j}, \quad [D_{\boldsymbol\lambda}]_{i,j} =\dfrac{\partial \eta_{2i}}{\partial \lambda_j} \quad \mbox{and} \quad [M_{\boldsymbol\rho}]_{i,j} = \dfrac{\partial \eta_{2i}}{\partial \rho_j},
-
T_1
andT_2
are diagonal matrices given byT_1 = \mathrm{diag}\bigg\{\dfrac{\partial \mu_1}{\partial \eta_{1t}},\dots, \dfrac{\partial \mu_n}{\partial \eta_{1n}}\bigg\}, \quad T_2 = \mathrm{diag}\bigg\{\dfrac{\partial \nu_1}{\partial \eta_{2t}},\dots, \dfrac{\partial \nu_n}{\partial \eta_{2n}}\bigg\},
-
\boldsymbol{h}_1
and\boldsymbol{h}_2
are the vectors defined by\boldsymbol{h}_1 = \bigg(\dfrac{\partial \ell_1}{\partial \mu_1}, \cdots, \dfrac{\partial \ell_n}{\partial \mu_n}\bigg)' \quad \mbox{and} \quad \boldsymbol{h}_2 = \bigg(\dfrac{\partial \ell_1}{\partial \nu_1}, \cdots, \dfrac{\partial \ell_n}{\partial \nu_n}\bigg)'.
For the models implemented so far, \partial\eta_{1t}/\partial\lambda_j =
0
so that we don't need a matrix for these derivatives.
The conditional Fisher information matrix for
\boldsymbol\gamma
is given by
K_n(\boldsymbol\gamma) =
\begin{pmatrix}
K_{\boldsymbol\rho,\boldsymbol\rho} & K_{\boldsymbol\rho,\boldsymbol\lambda}\\
K_{\boldsymbol\lambda,\boldsymbol\rho}& K_{\boldsymbol\lambda,\boldsymbol\lambda}
\end{pmatrix}
with
\begin{aligned}
K_{\boldsymbol\rho,\boldsymbol\rho} &=
D'_{\boldsymbol \rho}T_1E_\mu T_1 D_{\boldsymbol \rho}
+ M'_{\boldsymbol \rho}T_2E_{\mu\nu}T_1 D_{\boldsymbol \rho}
+ D'_{\boldsymbol \rho}T_1E_{\mu\nu} T_2 M_{\boldsymbol \rho}
+ M'_{\boldsymbol \rho}T_2 E_\nu T_2 M_{\boldsymbol \rho}\\
K_{\boldsymbol\rho,\boldsymbol\lambda} &= K_{\boldsymbol\lambda,\boldsymbol\rho}'=
D_{\boldsymbol \rho}' T_1E_{\mu\nu}T_2D_{\boldsymbol \lambda}
+ M_{\boldsymbol \rho}' T_2 E_\nu T_2 D_{\boldsymbol \lambda},\\
K_{\boldsymbol\lambda,\boldsymbol\lambda} &=
D_{\boldsymbol \lambda}' T_2E_\nu T_2D_{\boldsymbol \lambda}
\end{aligned}
where E_\mu
, E_{\mu\nu}
and E_\nu
are diagonal matrices for which the (t,t)
th element is given by
[E_\mu ]_{t,t} =
-\mathbb{E}\bigg(\dfrac{\partial^2 \ell_t}{\partial \mu_t^2} \bigg|
\mathcal{F} _{t-1} \bigg), \quad
[E_{\mu\nu}]_{t,t} =
-\mathbb{E}\bigg(\dfrac{\partial^2 \ell_t}{\partial\mu_t\partial \nu_t}
\bigg| \mathcal{F} _{t-1} \bigg) \quad \mbox{and} \quad
[E_\nu]_{t,t} =
- \mathbb{E}\bigg(\dfrac{\partial^2 \ell_t}{ \partial \nu_t^2} \bigg|
\mathcal{F} _{t-1} \bigg).
References
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See Also
BTSR.model.defaults: function to print default settings for a specified model