# VGLTSMs Family Functions: Generalized autoregressive moving average model with Student-t errors 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. ```r 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 $$ \theta(B)y_t = \phi(B) \varepsilon_t,y[t] = \beta[0] + \beta[1] y[t - 1] + \ldots + \beta[p] y_[t - p] + e[t]+\phi[1] e[t - 1] + \ldots \phi[q] e[t - q], $$ where $e[t]$ are distributed as a shift-scaled Student--$t$ with $\nu$ degrees of freedom, i.e., $e[t] ~ t(\nu[t], \mu[t], \sigma[t])$. 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 c("$\n$", "$\\eta = (\\mu, \\sigma, log log \\nu)^T,$") where $log \sigma$ and $\nu$ are intercept--only by default. ## Returns An object of class `"vglmff"` (see `vglmff-class`) to be used by VGLM/VGAM modelling functions, e.g., `vglm` or `vgam`. ## Author(s) Victor Miranda ## 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`. ## See Also `ARIMAXff`, `studentt`, `vglm`. ## Examples ```r ### 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) ```

ARMA.studentt.ff function