# ARMA.studentt.ff: VGLTSMs Family Functions: Generalized autoregressive moving... In VGAMextra: Additions and Extensions of the 'VGAM' Package

## 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

 ```1 2 3 4 5 6 7 8 9``` ``` 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 `Reg`ressand? 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

y[t] = β + β y[t - 1] + … + β[p] y_[t - p] + e[t]+ φ e[t - 1] + … φ[q] e[t - q],

where e[t] are distributed as a shift-scaled Student–t with ν degrees of freedom, i.e., e[t] ~ t(ν[t], μ[t], σ[t]). This family functions estimates the location (mu[t]), scale (σ[t]) and degrees of freedom (ν[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 η = (μ, σ, log log ν)^T, where log σ and ν 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

`ARIMAXff`, `studentt`, `vglm`.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31``` ```### 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 <- 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) ```