Package 'BTSR'

Description

The BTSR package provides a unified framework for simulating, fitting, and forecasting bounded time series regression models. It supports a wide range of models, including i.i.d., regression, ARMA-like, and ARFIMA-like models, with a focus on bounded time series data.

Key features of the BTSR package include

• Simulation of bounded time series data using various models.

• Estimation of model parameters using efficient algorithms.

• Forecasting future values based on fitted models.

• Support for both short-memory and long-memory models.

• Flexible link functions and error scales.

Mathematical Notation

The BTSR package is based on the following mathematical framework

• $Y_t$: The bounded random variable at time $t$, with $Y_t \in (a, b)$.

• $\mathcal{F}_t$: The $\sigma$-field generated by information up to time $t$.

• $\mu_t, \nu_t$: Parameters of the conditional distribution of $Y_t$.

• $\vartheta_t$: A transformation of $\nu_t$ (e.g., $\vartheta_t = \nu_t^2$).

• $\eta_{1t}, \eta_{2t}$: Linear predictors for $\mu_t$ and $\vartheta_t$, respectively.

• $\phi, \theta$: Autoregressive (AR) and moving average (MA) coefficients.

• $d$: Fractional differencing parameter, controlling long-memory behavior.

• $e_{1t}, e_{2t}$: Error terms for each part of the model (see the model definition for details).

The BTSR Structure

Let $\{Y_t\}_{t\in \mathbb{Z}}$ be a stochastic process for which $Y_t \in (a, b)$ with probability 1 ($a$ and $b$ not necessarily finite), for all $t \in \mathbb{Z}$, and let $\mathcal{F}_{t}$ denote the $\sigma$-field generated by the information observed up to time $t$. The general structure of a BTSR model is as follows

$\begin{aligned} Y_t | \mathcal{F}_{t-1} & \sim f(\cdot |\mu_t, \nu_t), \quad \vartheta_t = g_2(\nu_t)\\ \eta_{1t} = g_{11}(\mu_t) & =\alpha_1 + \boldsymbol{X}_{1t}'\boldsymbol{\beta}_1 + \sum_{i=1}^{p_1} \phi_{1i}[g_{12}(Y_{t-i})-I_{X_1}\boldsymbol{X}_{1(t-i)}'\boldsymbol{\beta}_1] + \xi_t, \quad \text{(part 1)}\\ \eta_{2t} = g_{21}(\vartheta_t) & =\alpha_2 + \boldsymbol{X}_{2t}' \boldsymbol{\beta}_2 + \sum_{i=1}^{p_2} \phi_{2i}[g_{22}(\vartheta_{t-i})-I_{X_2}\boldsymbol{X}_{2(t-i)}'\boldsymbol{\beta}_2] + \sum_{k=1}^\infty c_{2k} e_{2,t-k}, \quad \text{(part 2)} \end{aligned}$

with $\xi_t$ depending on the model, controlled by the argument model,

$\xi_t = \begin{cases} h(T^{t-1}(U_0)), & \text{if } \code{model = "BARC"},\\ \sum_{k=1}^\infty c_{1k} e_{1,t-k}, & \text{otherwise}, \end{cases}$

$e_{1,t}$ depending on the error.scale adopted

$e_{1,t} = g_{13}(Y_t, \mu_t) = \begin{cases} Y_t - \mu_t, & \text{if } \code{error.scale = 0} \text{ (data scale)},\\ g_{11}(Y_t) - g_{11}(\mu_t), & \text{if } \code{error.scale = 1} \text{ (predictive scale)} \end{cases}$

and $e_{2,t} = g_{23}(e_{1,t})$, where

• $I_{X_1}$, $I_{X_2}$ are indicator functions, which control whether the regressors should be included in the AR part of the equation. These functions are controlled by the argument xregar as follows: xregar = FALSE corresponds to $I_{X} = 0$ and xregar = TRUE corresponds to $I_{X} = 1$. If xregar = FALSE, the corresponding argument xreg is not included in the AR part of the model.

• $g_{13}$ is a function of $Y_t$ and $\mu_t$, defined by the error scale adopted.

• $g_2$ and $g_{ij}$, $i,j \in \{1,2\}$, and $g_{23}$ are link functions defined in the argument linkg. Notice that the links $g_{ij}$ are only used in the AR part of the model and, typically, $g_{i1} = g_{i2}$ for each $i \in \{1,2\}$. The link $g_2$ might depend on the distribution adopted. Finally, $g_{23}$ is a link function that transforms the error term $e_{1t}$.

• $h$ is the link function for BARC models, defined in the argument linkh.

• $\{c_{ik}\}_{k \geq 1}$ are the coefficients obtained through the relation

  $\theta_i(z) = \sum_{k = 0}^{q_i} \theta_{ik}z^k \quad \text{and} \quad (1-L)^{-d_i}\theta_i(z) = \sum_{k = 0}^\infty c_{ik}z^k, \quad i \in \{1,2\}.$

  In particular, if $d_i = 0$, then $c_{ik} = \theta_{ik}$, for $k = 1, \dots, q_i$.

Author(s)

Taiane Schaedler Prass [email protected], Guilherme Pumi [email protected]

References

Bayer FM, Bayer DM, Pumi G (2017). “Kumaraswamy autoregressive moving average models for double bounded environmental data.” Journal of Hydrology, 555, 385–396. doi:10.1016/j.jhydrol.2017.10.006.

Pumi G, Valk M, Bisognin C, Bayer FM, Prass TS (2019). “Beta autoregressive fractionally integrated moving average models.” Journal of Statistical Planning and Inference, 200, 196–212. doi:10.1016/j.jspi.2018.10.001.

Pumi G, Prass TS, Souza RR (2021). “A dynamic model for double bounded time series with chaotic driven conditional averages.” Scandinavian Journal of Statistics, 48(1), 68–86. doi:10.1111/sjos.12439.

Pumi G, Prass TS, Taufemback CG (2024). “Unit-Weibull autoregressive moving average models.” TEST, 33, 204–229. doi:10.1007/s11749-023-00893-8.

Pumi G, Prass TS, Taufemback CG (2024). “Publisher Correction: Unit-Weibull autoregressive moving average models.” TEST, 33, 358–359. doi:10.1007/s11749-023-00905-7.

Pumi G, Matsuoka DH, Prass TS (2025). “A GARMA Framework for Unit-Bounded Time Series Based on the Unit-Lindley Distribution with Application to Renewable Energy Data.” doi:10.48550/arXiv.2504.07351.

Pumi G, Matsuoka DH, Prass TS, Palm BG (2025). “A Matsuoka-Based GARMA Model for Hydrological Forecasting: Theory, Estimation, and Applications.” doi:10.48550/arXiv.2502.18645.

Prass TS, Pumi G, Taufemback CG, Carlos JH (2025). “Positive time series regression models: theoretical and computational aspects.” Computational Statistics, 40, 1185–1215. doi:10.1007/s00180-024-01531-z.

See Also

For detailed examples and usage instructions, see the documentation for individual functions

• btsr.sim: Simulate bounded time series data.

• btsr.extract: Extract components of a BTSR model, for a given set of parameters

• btsr.fit: Fit a BTSR model to data.

• predict: Forecast future values using a fitted model.

• arguments: Shared documentation for arguments

Examples

#----------------------------
# Quickstart examples.
#----------------------------

# Example 1: Simulate i.i.d. samples
set.seed(1234)
y1 <- btsr.sim(model = "BETA", n = 1000, coefs = list(alpha = 0.2, nu = 20))
hist(y1)

# Example 2: Simulate ARMA-like model with fixed nu
y2 <- btsr.sim(
  model = "BARMA", n = 100, link = "logit",
  coefs = list(alpha = 0.2, phi = 0.5, theta = 0.3, nu = 20)
)
plot(y2, type = "l")

Description

This is the common documentation for arguments related to the coefficients in BTSR models.

Arguments

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 argument nu (except for one-parameter models such as ULARMA and MARMA).

• New structure: a list with elements part1 and part2, each being a list with with optional components alpha, beta, phi, theta, d and u0 (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 in xreg.

• 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 variable $U_0$. 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 and fixed.values (consequently, fixed.lags) are not used.

• If xreg is provided but coefs does not include a beta argument, an error message is issued.

• phi must be a vector of length $p$ (the AR order), meaning all coefficients must be provided, including zeros.

• theta (non-BARC models) must be a vector of length $q$ (the MA order), meaning all coefficients must be provided, including zeros.

Extraction:

• One dimensional parameters (e.g. alpha) that do not appear in coefs are assume to be fixed.

• An error message will be issued if both coefs and fixed.values are both empty.

• If $\nu$ is not constant over time and nu is missing in both coefs and fixed.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 both start and fixed.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 in coefs or start).

• lags and fixed.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 in fixed.values are assumed to be non-fixed. The same goes for fitting when start = NULL or ignore.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 or fixed.values and the size of the vector is not the same as the dimension of the parameters, either lags or fixed.lags must also be provided.

Description

This is the common documentation for arguments related the configurations for fitting models and printing reports.

Arguments

Description

This is the common documentation for arguments related link functions in BTSR models.

Arguments

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 and g12. Internally, this structure is automatically converted to the new format with g2 = g21 = g22 = g23 = "linear".

• New structure: a named list with optional elements (order is irrelevant) g11, g12, g2, g21, g22 and g23. 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) and $e_{1t}$ (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 or g12 is missing (but not both), internally it is set g12 = g11.

• If both g11 and g12 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 or g22 is missing (but not both), internally it is set g22 = g21.

• If both g21 and g22 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, and g22 are forced to "linear", with $a = 1$.
  Overrides any user specifications.

• iid samples:
  Overrides any user specifications.
  linkg = "linear" (with $a = 1$). Internally converted to

  linkg = list(g11 = "linear", g12 = "linear", g2 = "linear"
               g21 = "linear", g22 = "linear", g23 = "linear")
  

• BARFIMA, KARFIMA, ULARFIMA, UWARFIMA:
  linkg = "logit". Internally converted to

  linkg = list(g11 = "logit", g12 = "logit", g2 = "linear"
               g21 = "linear", g22 = "linear", g23 = "linear")
  

• GARFIMA:
  linkg = "log". Internally converted to

  linkg = list(g11 = "log", g12 = "log", g2 = "linear"
               g21 = "linear", g22 = "linear", g23 = "linear")
  

• MARFIMA:
  linkg = "cloglog". Internally converted to

  linkg = list(g11 = "cloglog", g12 = "cloglog", g2 = "linear"
               g21 = "linear", g22 = "linear", g23 = "linear")
  

• BARFIMAV, GARFIMAV, KARFIMAV, UWARFIMAV:

  g11 and g12 have the same default values as the particular model where $\nu$ is constant over time.

  g2 = "default", meaning that g2 is set as the the default link for the model.

  • For BARFIMAV "default" = SIP with $a = b = 1$.

  • For GARFIMAV "default" = SIP with $a = 0$ and $b = 1$.

  • For remaining models "default" = "linear" with $a = 1$.

  g21 depends on the model.

  • For BARFIMAV g21 = "logit"

  • For any other model g21 = "log".

  For g22, the default is to assume g22 = g21.

  Finally, ⁠g23 = "polynomial⁠, with $a = 1$ and $b = 2$ (set in configs.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 and g12.

• New structure: a named list with optional elements (order is irrelevant) g11, g12, g2, g21, g22 and g23.

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.

See Also

BTSR.model.defaults: function to print default settings for a specified model

Description

This is the common documentation for arguments related the log-likelihood functions, score vector and information matrix for BTSR models.

Arguments

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

• $D_{\boldsymbol\rho}$, $D_{\boldsymbol\lambda}$ and $M_{\boldsymbol\rho}$ are the matrices for which the $(i,j)$th elements are given, respectively, by

  $[D_{\boldsymbol\rho}]_{i,j} = \dfrac{\partial \eta_{1i}}{\partial \rho_j}, \quad [D_{\boldsymbol\lambda}]_{i,j} =\dfrac{\partial \eta_{2i}}{\partial \lambda_j} \quad \mbox{and} \quad [M_{\boldsymbol\rho}]_{i,j} = \dfrac{\partial \eta_{2i}}{\partial \rho_j},$

• $T_1$ and $T_2$ are diagonal matrices given by

  $T_1 = \mathrm{diag}\bigg\{\dfrac{\partial \mu_1}{\partial \eta_{1t}},\dots, \dfrac{\partial \mu_n}{\partial \eta_{1n}}\bigg\}, \quad T_2 = \mathrm{diag}\bigg\{\dfrac{\partial \nu_1}{\partial \eta_{2t}},\dots, \dfrac{\partial \nu_n}{\partial \eta_{2n}}\bigg\},$

• $\boldsymbol{h}_1$ 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

$[E_\mu ]_{t,t} = -\mathbb{E}\bigg(\dfrac{\partial^2 \ell_t}{\partial \mu_t^2} \bigg| \mathcal{F} _{t-1} \bigg), \quad [E_{\mu\nu}]_{t,t} = -\mathbb{E}\bigg(\dfrac{\partial^2 \ell_t}{\partial\mu_t\partial \nu_t} \bigg| \mathcal{F} _{t-1} \bigg) \quad \mbox{and} \quad [E_\nu]_{t,t} = - \mathbb{E}\bigg(\dfrac{\partial^2 \ell_t}{ \partial \nu_t^2} \bigg| \mathcal{F} _{t-1} \bigg).$

Description

This documentation describes the map argument in BARC models and the map functions implemented in the BTSR package.

Arguments

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, $\theta = k$, for $k$ 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),

  $T(u) = \dfrac{u}{(1 - u)}I(u \le 0.5) + (2u - 1)I(u > 0.5)$

Description

The BTSR package supports a variety of models, including

• i.i.d structure,

• regression models,

• short- and long-memory time series models

• chaotic processes.

This documentation describes

• the model argument and available model strings,

• default configurations for specific models,

• how to reproduce models from literature.

Arguments

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 and shape2 compatible with the one from rbeta:

  model = "BETA"
  alpha = shape1/(shape1 + shape2)
  nu = shape1 + shape2
  

• Gamma distribution with parameters shape and scale compatible with the one from rgamma:

  model = "GAMMA"
  alpha = shape*scale
  nu = shape
  

• Kumaraswamy distribution with shape parameters shape1 and shape2 (respectively denoted by $a$ and $b$ 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 and tau 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" and error.scale = 1.

References

Bayer FM, Bayer DM, Pumi G (2017). “Kumaraswamy autoregressive moving average models for double bounded environmental data.” Journal of Hydrology, 555, 385–396. doi:10.1016/j.jhydrol.2017.10.006.

Ferrari SLP, Cribari-Neto F (2004). “Beta Regression for Modelling Rates and Proportions.” Journal of Applied Statistics, 31(7), 799–815. doi:10.1080/0266476042000214501.

Kumaraswamy P (1980). “A generalized probability density function for double-bounded random processes.” Journal of Hydrology, 46(1-2), 79–88. doi:10.1016/0022-1694(80)90036-0.

Matsuoka DH, Pumi G, Torrent HS, Valk M (2024). “A three-step approach to production frontier estimation and the Matsuoka's distribution.” doi:10.48550/arXiv.2311.06086.

Mazucheli J, Menezes AFB, Fernandes LB, de Oliveira RP, Ghitany ME (2019). “The unit-Weibull distribution as an alternative to the Kumaraswamy distribution for the modeling of quantiles conditional on covariates.” Journal of Applied Statistics. doi:10.1080/02664763.2019.1657813.

Mazucheli J, Menezes AJB, Chakraborty S (2018). “On the one parameter unit-Lindley distribution and its associated regression model for proportion data.” Journal of Applied Statistics. doi:10.1080/02664763.2018.1511774.

Mitnik PA, Baek S (2013). “The Kumaraswamy distribution: median-dispersion re-parameterizations for regression modeling and simulation-based estimation.” Statistical Papers, 54, 177–192. doi:10.1007/s00362-011-0417-y.

Prass TS, Pumi G, Taufemback CG, Carlos JH (2025). “Positive time series regression models: theoretical and computational aspects.” Computational Statistics, 40, 1185–1215. doi:10.1007/s00180-024-01531-z.

Pumi G, Matsuoka DH, Prass TS (2025). “A GARMA Framework for Unit-Bounded Time Series Based on the Unit-Lindley Distribution with Application to Renewable Energy Data.” doi:10.48550/arXiv.2504.07351.

Pumi G, Matsuoka DH, Prass TS, Palm BG (2025). “A Matsuoka-Based GARMA Model for Hydrological Forecasting: Theory, Estimation, and Applications.” doi:10.48550/arXiv.2502.18645.

Pumi G, Prass TS, Souza RR (2021). “A dynamic model for double bounded time series with chaotic driven conditional averages.” Scandinavian Journal of Statistics, 48(1), 68–86. doi:10.1111/sjos.12439.

Pumi G, Valk M, Bisognin C, Bayer FM, Prass TS (2019). “Beta autoregressive fractionally integrated moving average models.” Journal of Statistical Planning and Inference, 200, 196–212. doi:10.1016/j.jspi.2018.10.001.

Rocha AV, Cribari-Neto F (2009). “Beta autoregressive moving average models.” Test, 18, 529–545. doi:10.1007/s11749-008-0112-z.

Rocha AV, Cribari-Neto F (2017). “Erratum to: Beta autoregressive moving average models.” Test, 26, 451–459. doi:10.1007/s11749-017-0528-4.

See Also

BTSR.models, BTSR.model.defaults, get.defaults

Description

This is the common documentation for arguments related to order of polynomials and truncation points for infinite sums, presented in BTSR models.

Arguments

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, $q_1 = q_2 = q$).

Description

This is the common documentation for arguments related to the regressors.

Arguments

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 to xreg = list(part1 = xreg, part2 = NULL). The same applies to xnew

• New structure: a list with elements part1 (regressors for first model component) and part2 (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 to xreg.start = list(part1 = xreg.start, part2 = NULL).

• New structure: a list with elements part1 and part2, 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 or nnew = 0, xnew is ignored. If nnew > 0 and the number of regressors in xnew does not match xreg an error message is issued.

• If starting values for xreg are not provided and $p_i > 0$ for the $i$th part of the model, the default behavior is to assume

  $\displaystyle\boldsymbol{X}_t = \frac{1}{p_i}\sum_{k = 1}^{p_i} \boldsymbol{X}_k, \quad \text{for } t< 1.$

Description

This is the common documentation for arguments related to the observed/simulated time series and its conditional distribution.

Arguments

Description

These functions can be used to simulate, extract components and fit any model of the class barc. A model with class barc is a special case of a model with class btsr. See the Section ‘The BTSR structure’ in btsr-package for more details on the general structure. See also ‘Details’ below.

Usage

BARC.sim(n = 1, burn = 0, y.start = NULL, xreg = NULL,
  xreg.start = NULL, xregar = TRUE, coefs = NULL, map = 4,
  error.scale = 0, linkg = "linear", configs.linkg = NULL,
  linkh = "linear", configs.linkh = list(ctt = 1, power = 1),
  complete = FALSE, debug = FALSE)

BARC.extract(yt, y.start = NULL, xreg = NULL, xreg.start = NULL,
  xnew = NULL, xregar = TRUE, nnew = 0, p = NULL, coefs = NULL,
  lags = NULL, fixed.values = NULL, fixed.lags = NULL, error.scale = 0,
  map = 4, linkg = "linear", configs.linkg = NULL, linkh = "linear",
  configs.linkh = list(ctt = 1, power = 1), llk = TRUE, sco = FALSE,
  info = FALSE, debug = FALSE)

BARC.fit(yt, y.start = NULL, xreg = NULL, xreg.start = NULL,
  xregar = TRUE, xnew = NULL, nnew = 0, p = NULL,
  ignore.start = FALSE, start = NULL, lags = NULL, fixed.values = NULL,
  fixed.lags = NULL, lower = NULL, upper = NULL, map = 4,
  linkg = "linear", configs.linkg = NULL, linkh = "linear",
  configs.linkh = list(ctt = 1, power = 1), sco = FALSE, info = FALSE,
  error.scale = 0, control = NULL, report = TRUE, debug = FALSE, ...)

Arguments

Details

Sim, Extract and Fit functions

The function BARC.sim generates a random sample from a BARC($p$) model.

The function BARC.extract allows the user to extract the components $Y_t$, $\mu_t$, $\eta_t = g(\mu_t)$, $e_t$, $T^t(U_0)$, the log-likelihood, the score vector and the information matrix associated to a given set of parameters. This function can be used by any user to create an objective function that can be passed to optimization algorithms not available in the BTSR Package.

The function BARC.fit fits a BARC model to a given univariate time series. For now, available optimization algorithms are "L-BFGS-B" and "Nelder-Mead". Both methods accept bounds for the parameters. For "Nelder-Mead", bounds are set via parameter transformation.

Particular cases

Neither the beta regression or an i.i.d. sample from a beta distribution can be obtained as special cases of the BARC model since the term $h(T(U_0))$ is always present.

The model from Pumi et al. (2021) is obtained by setting xregar = TRUE (so that the regressors are included in the AR part of the model) and using the same link for $Y_t$ and $\mu_t$.

Value

By default, the function BARC.sim returns the simulated time series yt. If complete = TRUE, it returns a list with the following components

• model: string with the text "BARC"

• yt: the simulated time series $Y_t$

• mut: the conditional mean $\mu_t$

• etat: the linear predictor $\eta_t = g_{11}(\mu_t)$

• u0: the starting values of $U_0$

• Ts: the chaotic process $T^t(U_0)$

• error: the error term $e_{1t}$

• out.Fortran: the output from FORTRAN (if requested).

The function BARC.extract returns a list with the following components.

• model: string with the text "BARC"

• yt: the observed time series $Y_t$

• TS: the chaotic process $T^t(U_0)$.

• mut: the conditional mean $\mu_t$

• etat: the linear predictor $\eta_t = g_{11}(\mu_t)$

• error: the error term $e_{1t}$

• forecast: the out-of-sample forecast (if requested)

• xnew: the out-of-sample values of xreg provided by the user (only present if the model includes regressors and forecast is requested)

• sll: the sum of the conditional log-likelihood (if requested)

• score: the score vector (if requested)

• info.Matrix.: the score vector (if requested)

• out.Fortran: FORTRAN output (if requested)

The function BARC.fit returns a list with the following components.

• model: string with the text "BARC"

• call: string with a complete description of the model, including the AR and MA order.

• n: the sample size used for estimation.

• series: the observed time series $Y_t$

• gyt: a vector or a matrix with the transformed time series $g_{11}(Y_t)$ and $g_{12}(Y_t)$. Only returns a matrix if the links $g_{11}$ and $g_{12}$ are not the same.

• xreg: a vector or matrix of regressors $\boldsymbol{X}_t$ (if included in the model).

• control: a list of control parameters.

• convergence: An integer code. 0 indicates successful completion. The error codes depend on the algorithm used.

• message: A character string giving any additional information returned by the optimizer (if any), or NULL.

• counts: an integer giving the number of function evaluations.

• start: the starting values used by the algorithm.

• coefficients: The best set of parameters found.

• fitted.values: the conditional time series $\mu_t$ and the chaotic process $T^t(U_0)$, which corresponds to the in-sample forecast, also denoted fitted values.

• etat: the linear predictor $\eta_{1t} = g_{11}(\mu_t)$

• error: the error term $e_{1t}$

• residual: the observed values $Y_t$ minus the fitted values $\mu_t$. The same as the error term if error.scale = 0.

• forecast: a matrix with the out-of-sample forecast (if requested) for $\mu_t$ and $\eta_{1t}$

• xnew: the observations of the regressors observed/predicted values corresponding to the period of out-of-sample forecast. Only included if xreg is not NULL and nnew > 0.

• sll: the sum of the conditional log-likelihood (if requested)

• score: the score vector (if requested)

• info.Matrix: the information matrix (if requested)

• link: the codes for the link functions (for summary purposes)

• configs: a list with the configurations passed to FORTRAN to fit the model. This information is used by the prediction function.

• out.Fortran: FORTRAN output (if requested).

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, $\theta = k$, for $k$ 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),

  $T(u) = \dfrac{u}{(1 - u)}I(u \le 0.5) + (2u - 1)I(u > 0.5)$

References

Pumi G, Prass TS, Souza RR (2021). “A dynamic model for double bounded time series with chaotic driven conditional averages.” Scandinavian Journal of Statistics, 48(1), 68–86. doi:10.1111/sjos.12439.

See Also

BTSR.functions: sim, extract and fit functions for BTSR models

BTSR.parent.models: sim, extract and fit functions for parent models

get.defaults: Retrieve default arguments for BTSR package functions

Examples

#########################################################################
#
#   Example of usage of BARC.sim, BARC.extract and BARC.fit
#
#########################################################################

#------------------------------------------------------------
# Generating a sample from a BARC model
#------------------------------------------------------------
set.seed(1234)
m1 <- BARC.sim(
  coefs = list(nu = 15, theta = 0.85, u0 = pi / 4),
  linkg = "linear",
  linkh = "linear",
  configs.linkh = list(ctt = 0.6),
  n = 100,
  complete = TRUE
)

plot.ts(m1$yt)
lines(m1$mut, col = "red")

#------------------------------------------------------------
#  Extracting the conditional time series given yt and
#  a set of parameters
#------------------------------------------------------------

e1 <- BARC.extract(
  yt = m1$yt,
  map = 4,
  coefs = list(nu = 15, theta = 0.85),
  fixed.values = list(u0 = pi / 4),
  linkg = "linear",
  linkh = "linear",
  configs.linkh = list(ctt = 0.6),
  llk = TRUE,
  sco = TRUE,
  info = TRUE
)

#----------------------------------------------------
# comparing the simulated and the extracted values
#----------------------------------------------------
cbind(head(m1$mut), head(e1$mut))

#---------------------------------------------------------
# the log-likelihood, score vector and information matrix
# score vector and information matrix are obtained
# numerically.
#---------------------------------------------------------
e1$sll
e1$score
e1$info.Matrix

#------------------------------------------------------------
#  Fitting a BARC model. Assuming only alpha fixed.
#------------------------------------------------------------
f1 <- BARC.fit(
  yt = m1$yt,
  map = 4,
  configs.linkh = list(ctt = 0.6),
  start = list(nu = 10, theta = 0.6, u0 = 0.5),
  lower = list(nu = 0, theta = 0, u0 = 0),
  upper = list(theta = 1, u0 = 1),
  fixed.values = list(alpha = 0),
  control = list(iprint = -1, method = "Nelder-Mead")
)

coefficients(f1)

plot.ts(m1$yt)
lines(f1$fitted.values[, "mut"], col = "red")

#------------------------------------------------------------
#  Out-of-sample forecast
#------------------------------------------------------------
pred <- predict(f1, nnew = 5)
pred$forecast

Description

These generic functions can be used to simulate, extract components and fit any model of the class btsr. See ‘Details’ below.

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.

Usage

btsr.sim(model, ...)

btsr.extract(model, ...)

btsr.fit(model, ...)

Arguments

Details

A detailed description of the general structure (mathematical formulation) of BTSR models, associated to the btsr class, is presented in Section ‘The BTSR Structure’ of btsr-package. Particular models are discussed in arguments.model.

All functions are compatible with the new format for the arguments, introduced in version 1.0.0. and the previous format.

• The function btsr.sim is used to generate random samples from any BTSR models.

• The function btsr.extract allows the user to extract all conditional time series, the log-likelihood, and the vectors and matrices used to calculate the score vector and the information matrix associated to a given set of parameters. This function can be used by any user to create an objective function that can be passed to optimization functions not available in BTSR Package. At this point, there is no other use for which this function was intended.

• The function btsr.fit fits a BTSR model to a given univariate time series. For now, available optimization algorithms are "L-BFGS-B" and "Nelder-Mead". Both methods accept bounds for the parameters. For "Nelder-Mead", bounds are set via parameter transformation.

For compatibility with previous versions of the package, all functions associated to parent models (e.g. BARFIMA) are still available (see BTSR.parent.models). Also, analogous functions are available for parent models with time varying $\nu$ (e.g. BARFIMAV). The list of arguments and default values for these specific functions can be accessed using the function get.defaults.

Particular models (e.g. BETA, BARMA) share the same arguments as the parent model, however, some arguments can have different default values (see the documentation for shared arguments for details). Information on the parent model can be obtained using the function BTSR.model.defaults.

Value

By default, the simulation function return the simulated time series yt. If complete = TRUE, it returns a list with the following components

• model: string with the name of the model.

• yt: the simulated time series $Y_t$

• mut: the conditional mean $\mu_t$

• etat: the linear predictor(s)
  For all models where $\nu$ is constant over time, returns $\eta_{1t} = g_{11}(\mu_t)$
  For models with time varying $\nu$, returns a matrix whose columns are $\eta_{1t} = g_{11}(\mu_t)$ and $\eta_{2t} = g_{21}(\vartheta_t)$.

• nut: the conditional precision $\nu_t$ (only for models with time varying $\nu$)

• varthetat: the transformed time series $\vartheta_t = g_2(\nu_t)$. (only for models with time varying $\nu$)
  For BARFIMAV models, if g2 = "default", then varthetat is the conditional dispersion given by $\vartheta_t = (1+\nu_t)^{-1}$.

• error: the error term $e_{1t}$

• out.Fortran: the output from Fortran (if requested).

The extraction function returns a list with the following components

• model: string with the name of the model.

• yt: the simulated time series $Y_t$

• mut: the conditional mean $\mu_t$

• etat: the linear predictor(s)
  For models with $\nu$ fixed, returns $\eta_{1t} = g_{11}(\mu_t)$
  For models with time varying $\nu$, returns a matrix whose columns are $\eta_{1t} = g_{11}(\mu_t)$ and $\eta_{2t} = g_{21}(\vartheta_t)$.

• nut: the conditional precision $\nu_t$ (only for models with time varying $\nu$)

• varthetat: the transformed time series $\vartheta_t = g_2(\nu_t)$. (only for models with time varying $\nu$)
  For BARFIMAV models, if g2 = "default", then varthetat is the conditional dispersion given by $\vartheta_t = (1+\nu_t)^{-1}$.

• error: the error term $e_{1t}$ (depends on the argument error.scale)

• forecast: the out-of-sample forecast. (if requested)
  If $\nu_t$ is fixed: a vector with the predicted values for $\mu_t$ and $\eta_{1t}$
  If $\nu_t$ is time varying: a matrix the predicted values for $\mu_t$ and $\eta_{1t}$, $\nu_t$, $\vartheta_t$ and $\eta_{2t}$.

• xnew: the observations of the regressors observed/predicted values corresponding to the period of out-of-sample forecast. Only included if xreg is not NULL and nnew > 0.

• sll: the sum of the conditional log-likelihood (if requested)

• score: the score vector (if requested)

• info.Matrix.: the score vector (if requested)

• D, T, E, h: additional matrices and vectors used to calculate the score vector and the information matrix. (if requested)

• out.Fortran: FORTRAN output (if requested)

The fitting function returns a list with the following components.

• model: character; the same as the input argument.

• call: string with a complete description of the model, including the AR and MA order.

• n: the sample size used for estimation.

• series: the observed time series $Y_t$

• gyt: a vector or a matrix with the transformed time series $g_{11}(Y_t)$ and $g_{12}(Y_t)$. Only returns a matrix if the links $g_{11}$ and $g_{12}$ are not the same.

• xreg: a vector or matrix of regressors $\boldsymbol{X}_t$ (if included in the model).

• control: a list of control parameters.

• convergence: An integer code. 0 indicates successful completion. The error codes depend on the algorithm used.

• message: A character string giving any additional information returned by the optimizer (if any), or NULL.

• counts: an integer giving the number of function evaluations.

• start: the starting values used by the algorithm.

• coefficients: The best set of parameters found.

• fitted.values: in-sample forecast.
  If $\nu_t$ is fixed: a vector with the in-sample value of $\mu_t$.
  If $\nu_t$ is time varying: a matrix with the in-sample values of $\mu_t$, $\nu_t$ and $\vartheta_t$.

• etat: the linear predictor(s)
  For models with $\nu$ fixed, returns $\eta_{1t} = g_{11}(\mu_t)$
  For models with time varying $\nu$, returns a matrix whose columns are $\eta_{1t} = g_{11}(\mu_t)$ and $\eta_{2t} = g_{21}(\vartheta_t)$.

• error: the error term $e_{1t}$ (depends on the argument error.scale)

• residual: the observed values $Y_t$ minus the fitted values $\mu_t$. The same as the error term if error.scale = 0.

• forecast: the out-of-sample forecast. (if requested)
  If $\nu_t$ is fixed: a vector with the predicted values for $\mu_t$ and $\eta_{1t}$
  If $\nu_t$ is time varying: a matrix the predicted values for $\mu_t$ and $\eta_{1t}$, $\nu_t$, $\vartheta_t$ and $\eta_{2t}$.

• xnew: the observations of the regressors observed/predicted values corresponding to the period of out-of-sample forecast. Only included if xreg is not NULL and nnew > 0.

• sll: the sum of the conditional log-likelihood (if requested)

• score: the score vector (if requested)

• info.Matrix: the information matrix (if requested)

• link: the codes for the link functions (for summary purposes)

• configs: a list with the configurations passed to FORTRAN to fit the model. This information is used by the prediction function.

• out.Fortran: FORTRAN output (if requested).