ARMA.studentt.ff {VGAMextra}

VGLTSMs Family Functions: Generalized autoregressive moving average model with Student-t errors

Description

For an ARMA model, estimates a 3–parameter Student-t distribution characterizing the errors plus the ARMA coefficients by MLE usign Fisher scoring. Central Student–t handled currently.

Usage

            ARMA.studentt.ff(order = c(1, 0),
                             zero = c("scale", "df"),
                             cov.Reg = FALSE,
                             llocation = "identitylink",
                             lscale    = "loglink",
                             ldf       = "logloglink",
                             ilocation = NULL,
                             iscale = NULL,
                             idf = NULL)

Arguments

order

Two–entries vector, non–negative. The order $u$ and $v$ of the ARMA model.

zero

Same as studentt3.

cov.Reg

Logical. If covariates are entered, Should these be included in the ARMA model as a Regressand? Default is FALSE, then only embedded in the linear predictors.

llocation, lscale, ldf, ilocation, iscale, idf

Same as studentt3.

Details

The normality assumption for time series analysis is relaxed to handle heavy–tailed data, giving place to the ARMA model with shift-scaled Student-t errors, another subclass of VGLTSMs.

For a univariate time series, say y_t, the model described by this VGLTSM family function is

\theta(B)y_t = \phi(B) \varepsilon_t,

where \varepsilon_t are distributed as a shift-scaled Student–t with \nu degrees of freedom, i.e., \varepsilon_t \sim t(\nu_t, \mu_t, \sigma_t). This family functions estimates the location (\mu_t), scale (\sigma_t) and degrees of freedom (\nu_t) parameters, plus the ARMA coefficients by MLE.

Currently only centered Student–t distributions are handled. Hence, the non–centrality parameter is set to zero.

The linear/additive predictors are \boldsymbol{\eta} = (\mu, \log \sigma, \log \log \nu)^T, where \log \sigma and \nu are intercept–only by default.

Value

An object of class "vglmff" (see vglmff-class) to be used by VGLM/VGAM modelling functions, e.g., vglm or vgam.

Note

If order = 0, then AR.studentt.ff fits a usual 3–parameter Student–t, as with studentt3.

If covariates are incorporated in the analysis, these are embedded in the location–parameter model. Modify this through zero. See CommonVGAMffArguments for details on zero.

Author(s)

Victor Miranda

See Also

ARIMAXff, studentt, vglm.

Examples

### Estimate the parameters of the errors distribution for an
## AR(1) model. Sample size = 50

set.seed(20180218)
nn <- 250
y  <- numeric(nn)
ncp   <- 0           # Non--centrality parameter
nu    <- 3.5         # Degrees of freedom.
theta <- 0.45        # AR coefficient
res <- numeric(250)  # Vector of residuals.

y[1] <- rt(1, df = nu, ncp = ncp)
for (ii in 2:nn) {
  res[ii] <- rt(1, df = nu, ncp = ncp)
  y[ii] <- theta * y[ii - 1] + res[ii]
}
# Remove warm up values.
y <- y[-c(1:200)]
res <- res[-c(1:200)]

### Fitting an ARMA(1, 0) with Student-t errors.
AR.stut.er.fit <- vglm(y ~ 1, ARMA.studentt.ff(order = c(1, 0)),
                       data = data.frame(y = y), trace = TRUE)

summary(AR.stut.er.fit)
Coef(AR.stut.er.fit)


plot(ts(y), col = "red", lty = 1, ylim = c(-6, 6), main = "Plot of series Y with Student-t errors")
lines(ts(fitted.values(AR.stut.er.fit)), col = "blue", lty = 2)
abline( h = 0, lty = 2)