barma: Fit Beta Autoregressive Moving Average (BARMA) Models via...

In betaARMA: Beta Autoregressive Moving Average Models

Fit Beta Autoregressive Moving Average (BARMA) Models via Maximum Likelihood

Description

Fits a Beta Autoregressive Moving Average (BARMA) model to time series data valued in (0, 1) using Maximum Likelihood Estimation (MLE). The function performs complete model estimation including parameter estimation, hypothesis testing infrastructure, and model diagnostics.

Usage

barma(y, ar = NA, ma = NA, link = "logit", xreg = NULL)

Arguments

y	A time series object ('ts') with values strictly in the open interval (0, 1). Must have at least \max(p,q) + 1 observations.
ar	A numeric vector specifying autoregressive (AR) lags (e.g., 'c(1, 2)' for AR(2)). Set to 'NA' for models without AR component. Default is 'NA'.
ma	A numeric vector specifying moving average (MA) lags (e.g., '1' for MA(1)). Set to 'NA' for models without MA component. Default is 'NA'.
link	The link function connecting the mean \mu_t to the linear predictor \eta_t. One of: '"logit"' (default): g(x) = \log(x/(1-x)) '"probit"': g(x) = \Phi^{-1}(x) '"cloglog"': Complementary log-log link '"loglog"': Log-log link
xreg	A matrix or data frame of external regressors (covariates), optional. Must have the same number of rows as 'y'. If provided, its columns are included in the linear predictor with associated coefficients in 'beta'.

Details

Fit Beta Autoregressive Moving Average (BARMA) Models

This function fits the BARMA(p,q) model as proposed by Rocha & Cribari-Neto (2009, with erratum 2017). It serves as the main wrapper for the optimization process, calling specialized helper functions for likelihood computation, gradient calculation, and Fisher Information Matrix estimation.

**Model Specification**: The BARMA model is defined as:

g(\mu_t) = \alpha + X_t\beta + \sum_{i=1}^{p} \varphi_i (g(y_{t-i}) - X_{t-i}\beta) + \sum_{j=1}^{q} \theta_j \epsilon_{t-j}

where y_t | F_{t-1} \sim Beta(\mu_t\phi, (1-\mu_t)\phi), g(\cdot) is the link function, and F_{t-1} is the information set at time t-1.

**Model Types** (specified via 'ar' and 'ma' arguments):

• **BARMA(p,q):** Both 'ar' and 'ma' are specified.

• **BAR(p):** Only 'ar' is specified; set 'ma = NA'.

• **BMA(q):** Only 'ma' is specified; set 'ar = NA'.

**External Regressors**: Covariates can be included via 'xreg'. The model becomes a regression BARMA, where the mean depends on current covariates and lagged responses/errors.

**Optimization**: The function uses the BFGS quasi-Newton algorithm via optim with analytic gradients for efficiency. Initial values are obtained from the 'start_values()' function.

**Implementation Notes**: - The conditional log-likelihood is computed, conditioning on the first m = \max(p,q) observations (burn-in period). - The 2017 Erratum corrections are implemented for correct handling of moving average components in the score vector and Fisher Information Matrix. - All computations are vectorized where possible for efficiency.

Value

An object of class '"barma"' containing:

coef	Named vector of all estimated parameters, ordered as: alpha, AR parameters, MA parameters, beta parameters, phi.
vcov	The variance-covariance matrix of the estimators, computed as the inverse of the observed Fisher Information Matrix.
model	A summary table with coefficients, standard errors, z-statistics, and p-values for hypothesis tests H_0: \theta_i = 0.
fitted	Fitted conditional mean values as a 'ts' object (NA-padded for burn-in period).
muhat	Alias for 'fitted' (fitted mean values).
etahat	Estimated linear predictor values (full vector, NA-padded).
errorhat	Estimated errors on predictor scale (full vector, 0-padded).
loglik	The conditional log-likelihood at the MLE.
fisher_info_mat	The observed Fisher Information Matrix.
conv	Convergence code from 'optim' (0 = success).
alpha, beta, varphi, theta, phi	Individual parameter estimates.
start_values	Initial parameter values used in optimization.
call	The original function call.
opt	Raw output object from 'optim()' call.

Note

The original version of this function was developed by Fabio M. Bayer (Federal University of Santa Maria, [e-mail]). It has been substantially modified and improved by Everton da Costa, with suggestions and contributions from Francisco Cribari-Neto.

Author(s)

Everton da Costa (Federal University of Pernambuco, [e-mail]); Francisco Cribari-Neto (Federal University of Pernambuco, [e-mail])

References

Rocha, A.V., & Cribari-Neto, F. (2009). Beta autoregressive moving average models. TEST, 18(3), 529-545. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/s11749-008-0112-z")}

Rocha, A.V., & Cribari-Neto, F. (2017). Erratum to: Beta autoregressive moving average models. TEST, 26, 451-459. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/s11749-017-0528-4")}

See Also

simu_barma for simulation, loglik_barma for likelihood computation, score_vector_barma for gradient computation, fim_barma for Fisher Information Matrix

Examples

  # Example 1: Fit a BAR(1) model
  set.seed(2025)
  y_sim_bar <- simu_barma(
    n = 250,
    alpha = 0.0,
    varphi = 0.6,
    phi = 25.0,
    link = "logit",
    freq = 12
  )

  # Fit the model
  fit_bar <- barma(y_sim_bar, ar = 1, link = "logit")

  # View results
  summary(fit_bar)
  coef(fit_bar)

  # Example 2: Fit a BARMA(1,1) model
  set.seed(2025)
  y_sim_barma <- simu_barma(
    n = 250,
    alpha = 0.0,
    varphi = 0.6,
    theta = 0.3,
    phi = 25.0,
    link = "logit",
    freq = 12
  )

  # Fit ARMA structure
  fit_barma <- barma(y_sim_barma, ar = 1, ma = 1, link = "logit")
  summary(fit_barma)

  # Example 3: BARMA(1,1) model with harmonic seasonal regressors
  hs <- sin(2 * pi * seq_along(y_sim_barma) / 12)
  hc <- cos(2 * pi * seq_along(y_sim_barma) / 12)

  # Create regressor matrix
  X <- cbind(hs = hs,
             hc = hc)

  fit_barma_xreg <- barma(
    y_sim_barma,
    ar = 1, ma = 1,
    link = "logit", xreg = X
  )
  summary(fit_barma_xreg)

betaARMA documentation built on March 29, 2026, 5:08 p.m.