ARIMAX.errors.ff {VGAMextra}

VGLTSMs Family Functions: Generalized integrated regression with order–(p, q) ARMA errors

Description

A VLTSMff for dynamic regression. Estimates regression models with order–(p, d, q) ARIMA errors by maximum likelihood.

Usage

      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 \boldsymbol{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, \boldsymbol{x}_t, are included in the regression model. Else, this is ignored and only the lagged \boldsymbol{x}_{t - 1}, \ldots, \boldsymbol{x}_{t - xLag} 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 \boldsymbol{x}_t covariates, the model described by this VGLTSM family function is

y_t = \boldsymbol{\beta}^T \boldsymbol{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}.

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 \boldsymbol{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^2_z). 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}.

Denoting \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.

Value

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

Note

If d = 0 in order, then ARIMAX.errors.ff will perform as ARIMAXff.

Author(s)

Victor Miranda

See Also

ARIMAXff, CommonVGAMffArguments, uninormal, vglm.

Examples

### 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])))