# VGLTSMs Family Functions: Generalized integrated regression with order--$(p, q)$ ARMA errors A VLTSMff for dynamic regression. Estimates regression models with order--$(p, d, q)$ ARIMA errors by maximum likelihood. ```r ARIMAX.errors.ff(order = c(1, 1, 1), zero = "var", # optionally, "mean". order.trend = 0, include.int = TRUE, diffCovs = TRUE, xLag = 0, include.currentX = TRUE, lvar = "loglink", lmean = "identitylink") ``` ## Arguments - `order`: The usual $(p, d, q)$ integer vector as in, e.g., `ARIMAXff`. By default, an order--$(p, q)$ ARMA model is fitted on the errors, whlist $d$ is the degree of differencing on the response. - `zero`: What linear predictor is modelled as intercept--only? See `zero` and `CommonVGAMffArguments` for further details. - `order.trend`: Non--negative integer. Allows to incorporate a polynomial trend of order `order.trend` in the forecast mean function. - `include.int`: Logical. Should an intercept (int) be included in the model for $y[t]$? Default is `TRUE`. See below for details. - `diffCovs`: Logical. If `TRUE` (default) the order--`d` difference of the covariates is internally computed and then incorporated in the regression model. Else, only the current values are included. - `xLag`: Integer. If entered, the covariates, say $x[t]$ are laggeg (up to order `xLag`) and then embedded in the regression model. See below for further details. - `include.currentX`: Logical. If `TRUE`, the **actual** values, $x[t]$, are included in the regression model. Else, this is ignored and only the lagged c("$\n$", "$ x[t - 1], \\ldots, x[t - xLag]\n$", "$ $") will be included. - `lvar, lmean`: Link functions applied to conditional mean and the variance. Same as `uninormal`. ## Details The generalized linear regression model with ARIMA errors is another subclass of VGLTSMs (Miranda and Yee, 2018). For a univariate time series, say $y[t]$, and a $p$--dimensional vector of covariates $x[t]$ covariates, the model described by this VGLTSM family function is $$ y_t = \boldsymbol{\beta}^T \boldsymbol{x}_t + u_t,y[t] = \beta^T x_t + u[t], $$ $$ u_t = \theta_1 u_{t - 1} + \cdots + \theta_p u_{t - p} + z_t +\phi_1 z_{t - 1} + \cdots + \phi_1 z_{t - q}.u[t] = \theta[1] u[t - 1] + \ldots +\theta[p] u[t - p] + z[t] +\phi[1] z[t - 1] + \ldots +\phi[q] u[t - q]. $$ The first entry in $x_t$ equals 1, allowing an intercept, for every $t$. Set `include.int = FALSE` to set this to zero, dimissing the intercept. Also, if `diffCovs = TRUE`, then the differences up to order `d` of the set $x[t]$ are embedded in the model for $y[t]$. If `xLag`$> 0$, the lagged values up to order `xLag` of the covariates are also included. The random disturbances $z[t]$ are by default handled as $N(0, \sigma[z]^2)$. Then, denoting $\Phi[t]$ as the history of the process $(x[t + 1], u[t])$ up to time $t$, yields $$ E(y_t | \Phi_{t - 1}) = \boldsymbol{\beta}^T \boldsymbol{x}_t +\theta_1 u_{t - 1} + \cdots + \theta_p u_{t - p} +\phi_1 z_{t - 1} + \cdots + \phi_1 z_{t - q}.E(y[t] | \Phi[t - 1]) = \beta^T x[t] +\theta[1] u[t - 1] + \ldots +\theta[p] u[t - p] +\phi[1] z[t - 1] + \ldots +\phi[q] u[t - q]. $$ Denoting c("$\n$", "$\\mu[t] = E(y[t] | \\Phi[t - 1]), $") the default linear predictor for this VGLTSM family function is $$ \boldsymbol{\eta} = ( \mu_t, \log \sigma^2_{z})^T.\eta = (\mu[t], log \sigma[t]^2)^T. $$ ## 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 `d = 0` in `order`, then `ARIMAX.errors.ff` will perform as `ARIMAXff`. ## See Also `ARIMAXff`, `CommonVGAMffArguments`, `uninormal`, `vglm`. ## Examples ```r ### Estimate a regression model with ARMA(1, 1) errors. ## Covariates are included up to lag 1. set.seed(20171123) nn <- 250 x2 <- rnorm(nn) # One covariate sigma2 <- exp(1.15); theta1 <- 0.5; phi1 <- 0.27 # True coefficients beta0 <- 1.25; beta1 <- 0.25; beta2 <- 0.5 y <- numeric(nn) u <- numeric(nn) z <- numeric(nn) u[1] <- rnorm(1) z[1] <- rnorm(1, 0, sqrt(sigma2)) for(ii in 2:nn) { z[ii] <- rnorm(1, 0, sqrt(sigma2)) u[ii] <- theta1 * u[ii - 1] + phi1 * z[ii - 1] + z[ii] y[ii] <- beta0 + beta1 * x2[ii] + beta2 * x2[ii - 1] + u[ii] } # Remove warm-up values. x2 <- x2[-c(1:100)] y <- y[-c(1:100)] plot(ts(y), lty = 2, col = "blue", type = "b") abline(h = 0, lty = 2) ## Fit the model. ARIMAX.reg.fit <- vglm(y ~ x2, ARIMAX.errors.ff(order = c(1, 0, 1), xLag = 1), data = data.frame(y = y, x2 = x2), trace = TRUE) coef(ARIMAX.reg.fit, matrix = TRUE) summary(ARIMAX.reg.fit, HD = FALSE) # Compare to arima() # arima() can't handle lagged values of 'x2' by default, but these # may entered at argument 'xreg'. arima(y, order = c(1, 0, 1), xreg = cbind(x2, c(0, x2[-150]))) ```

ARIMAX.errors.ff function