View source: R/TSfamilyARXff.ARMA.GARCH.R
ARXff  R Documentation 
Maximum likelihood estimation of the order–p autoregressive model (AR(p)) with covariates. Estimates the drift, standard deviation (or variance) of the random noise (not necessarily constant), and coefficients of the conditional–mean model.
ARXff(order = 1,
zero = c(if (nodrift) NULL else "ARdrift", "ARcoeff"),
xLag = 0,
type.EIM = c("exact", "approximate")[1],
var.arg = TRUE,
nodrift = FALSE,
noChecks = FALSE,
ldrift = "identitylink",
lsd = "loglink",
lvar = "loglink",
lARcoeff = "identitylink",
idrift = NULL,
isd = NULL,
ivar = NULL,
iARcoeff = NULL)
order 
The order (i.e., 'p') of the AR model, which is recycled if needed.
See below for further details.
By default, an autoregressive model of order 
zero 
Integer or character–strings vector.
Name(s) or position(s) of the parameters/linear predictors
to be modeled as interceptonly. Details at

xLag 
Same as 
type.EIM, var.arg, nodrift, noChecks 
Same as 
ldrift, lsd, lvar, lARcoeff 
Link functions applied to the drift,
the standar deviation (or variance) of the noise, and the
AR coefficients.
Same as Further details on

idrift, isd, ivar, iARcoeff 
Same as 
This family function describes an autoregressive model of orderp
with covariates (ARX(p)). It is a special case of the subclass VGLM–ARIMA
(Miranda and Yee, 2018):
Y_t  \Phi_{t  1} = \mu_t + \theta_{1} Y_{t  1} + \ldots +
\theta_p Y_{t  p} + \varepsilon_t,
where \boldsymbol{x}_t
a (possibly time–varying) covariate vector and
\mu_t = \mu^{\star} + \boldsymbol{\beta}^T \boldsymbol{x}_t
is a (time–dependent) scaled–mean, known as drift.
At this stage, conditional Gaussian white noise,
\varepsilon_t \Phi_{t  1}
is handled, in the form
\varepsilon_t  \Phi_{t  1} \sim
N(0, \sigma^2_{\varepsilon_t  \Phi_{t  1}}).
The distributional assumptions on the observations are then
Y_t  \Phi_{t  1} \sim
N(\mu_{t  \Phi_{t  1}},
\sigma^2_{\varepsilon_t  \Phi_{t  1}}),
involving the conditional mean equation for the ARX(p) model:
\mu_{t  \Phi_{t  1}} = \mu_t +
\boldsymbol{\beta}^T * \boldsymbol{x}_t
\theta_{1} Y_{t  1} + \ldots +
\theta_p Y_{t  p}.
\Phi_{t}
denotes the information
of the joint process
\left(Y_{t}, \boldsymbol{x}_{t + 1}^T \right)
,
at time t
.
The loglikelihood is computed by dARp
,
at each Fisher scoring iteration.
The linear predictor is
\boldsymbol{\eta} = \left(
\mu_t, \log \sigma^{2}_{\varepsilon_{t  \Phi_{t  1}}},
\theta_1, \ldots, \theta_p
\right)^T.
Note, the covariates may also intervene in the conditional
variance model
\log \sigma^{2}_{\varepsilon_{t  \Phi_{t  1}}}.
Hence, this family function
does not restrict the noise to be strictly white noise
(in the sense of constant variance).
The unconditional mean,
E(Y_{t}) = \mu
, satisfies
\mu \rightarrow \frac{\mu^{\star}}{1  (\theta_1 +
\ldots + \theta_p)}
when the process is stationary, and no covariates are involved.
This family function currently handles
multiple responses so that a matrix can be used as the response.
Also, for further details on VGLM/VGAM–link functions refer to
Links
.
Further choices for the random noise, besides Gaussian, will be implemented over time.
An object of class "vglmff"
(see vglmffclass
). The
object is used by VGLM/VGAM modelling functions, such as
vglm
or vgam
.
zero
can be either an integer vector
or a vector of character strings
specifying either the position(s) or name(s) (partially or not) of the
parameter(s) modeled as interceptonly. Numeric values can be set as usual
(See CommonVGAMffArguments
).
Character strings can be entered as per
parameter names in this family function, given by:
c("drift", "noiseVar" or "noiseSD", "ARcoeff")
.
Users can modify the zero
argument according to their needs.
By default, \mu_t
and the coefficients
\theta_1, \ldots, \theta_p
are intercept–only. That is,
\log \sigma^{2}_{\varepsilon_{t  \Phi_{t  1}}}
is modelled in terms of any explanatories
entered in the formula
.
Users, however, can modify this
according to their needs via
zero
. For example, set the covariates in the
drift
model, \mu_t
.
In addition, specific constraints
for parameters are handled through the function
cm.ARMA
.
If var.arg = TRUE
, this family function estimates
\sigma_{\varepsilon_t  \Phi_{t  1}}^2
.
Else, the
\sigma_{\varepsilon_t  \Phi_{t  1}}
estimate is returned.
For this family function the order
is recycled. That is,
order
will be replicated up to the number of responses
given in the vglm
call is matched.
Values of the estimates may not correspond
to stationary ARs, leading to low accuracy in the MLE estimates,
e.g., values very close to 1.0.
Stationarity is then examined, via
checkTS.VGAMextra
,
if
noChecks = FALSE
(default)
and no constraint matrices are set
(See constraints
for further
details on this).
If the estimated model very close to be nonstationary, then
a warning
will be outlined.
Set noChecks = TRUE
to completely ignore this.
NOTE: Full details on these 'checks' are shown within the
summary()
output.
Victor Miranda and T. W. Yee
Madsen, H. (2008) Time Series Analysis. Florida, USA: Chapman & Hall(Sections 5.3 and 5.5).
Porat, B., and Friedlander, B. (1986) Computation of the Exact Information Matrix of Gaussian Time Series with Stationary Random Components. IEEE Transactions on Acoustics, Speech and Signal Processing. ASSp34(1), 118–130.
ARIMAXff
,
ARMAXff
,
MAXff
,
checkTS.VGAMextra
,
CommonVGAMffArguments
,
Links
,
vglm
,
set.seed(1)
nn < 150
tsdata < data.frame(x2 = runif(nn)) # A single covariate.
theta1 < 0.45; theta2 < 0.31; theta3 < 0.10 # Coefficients
drift < c(1.3, 1.1) # Two responses.
sdAR < c(sqrt(4.5), sqrt(6.0)) # Two responses.
# Generate AR sequences of order 2 and 3, under Gaussian noise.
# Note, the drift for 'TS2' depends on x2 !
tsdata < data.frame(tsdata, TS1 = arima.sim(nn,
model = list(ar = c(theta1, theta1^2)), rand.gen = rnorm,
mean = drift[1], sd = sdAR[1]),
TS2 = arima.sim(nn,
model = list(ar = c(theta1, theta2, theta3)), rand.gen = rnorm,
mean = drift[2] + tsdata$x2 , sd = sdAR[2]))
# EXAMPLE 1. A simple AR(2), maximizing the exact loglikelihood
# Note that parameter constraints are involved for TS1, but not
# considered in this fit. "rhobitlink" is used as link for AR coeffs.
fit.Ex1 < vglm(TS1 ~ 1, ARXff(order = 2, type.EIM = "exact",
#iARcoeff = c(0.3, 0.3, 0.3), # OPTIONAL INITIAL VALUES
# idrift = 1, ivar = 1.5, isd = sqrt(1.5),
lARcoeff = "rhobitlink"),
data = tsdata, trace = TRUE, crit = "loglikelihood")
Coef(fit.Ex1)
summary(fit.Ex1)
vcov(fit.Ex1, untransform = TRUE) # Conformable with this fit.
AIC(fit.Ex1)
##
# Fitting same model using arima().
##
(fitArima < arima(tsdata$TS1, order = c(2, 0, 0)))
# Compare with 'fit.AR'. True are theta1 = 0.45; theta1^2 = 0.2025
Coef(fit.Ex1)[c(3, 4, 2)] # Coefficients estimated in 'fit.AR'
# EXAMPLE 2. An AR(3) over TS2, with one covariate affecting the drift only.
# This analysis makes sense as the TS2's drift is a function ox 'x2',
# i.e., 'x2' affects the 'drift' parameter only. The noise variance
# (var.arg = TRUE) is estimated, as interceptonly. See the 'zero' argument.
##
# This model CANNOT be fitted using arima()
##
fit.Ex2 < vglm(TS2 ~ x2, ARXff(order = 3, zero = c("noiseVar", "ARcoeff"),
var.arg = TRUE),
## constraints = cm.ARMA(Model = ~ 1, lags.cm = 3, Resp = 1),
data = tsdata, trace = TRUE, crit = "log")
# True are theta1 < 0.45; theta2 < 0.31; theta3 < 0.10
coef(fit.Ex2, matrix = TRUE)
summary(fit.Ex2)
vcov(fit.Ex2)
BIC(fit.Ex2)
constraints(fit.Ex2)
# EXAMPLE 3. Fitting an ARX(3) on two responses TS1, TS2; interceptonly model with
# constraints over the drifts. Here,
# a) No checks on invertibility performed given the use of cm.ARMA().
# b) Only the drifts are modeled in terms of 'x2'. Then, 'zero' is
# set correspondingly.
##
# arima() does not handle this model.
##
fit.Ex3 < vglm(cbind(TS1, TS2) ~ x2, ARXff(order = c(3, 3),
zero = c("noiseVar", "ARcoeff"), var.arg = TRUE),
constraints = cm.ARMA(Model = ~ 1 + x2, lags.cm = c(3, 3), Resp = 2),
trace = TRUE, data = tsdata, crit = "log")
coef(fit.Ex3, matrix = TRUE)
summary(fit.Ex3)
vcov(fit.Ex3)
constraints(fit.Ex3)
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