| MAXff {VGAMextra} | R Documentation |
VGLTSMs Family Functions: Order–q Moving Average Model with covariates
Description
Estimates the intercept, standard deviation (or variance) of the random noise (not necessarily constant), and the conditional–mean model coefficients of an order–q moving average (MA) process with covariates (MAX(q)) by maximum likelihood estimation using Fisher scoring.
Usage
MAXff(order = 1,
zero = c(if (nomean) NULL else "Mean", "MAcoeff"),
xLag = 0,
type.EIM = c("exact", "approximate")[1],
var.arg = TRUE,
nomean = FALSE,
noChecks = FALSE,
lmean = "identitylink",
lsd = "loglink",
lvar = "loglink",
lMAcoeff = "identitylink",
imean = NULL,
isd = NULL,
ivar = NULL,
iMAcoeff = NULL)
Arguments
order |
The order 'q' of the MA model, recycled if needed.
By default |
zero |
Integer or character–string vector.
Same as |
xLag |
Same as |
type.EIM, var.arg |
Same as |
nomean |
Logical. |
noChecks |
Logical. Same as |
lmean, lsd, lvar, lMAcoeff |
Link functions applied to the mean (intercept),
the standard deviation or variance of the random noise, and
the MA coefficients (conditional–mean model).
Note, |
imean, isd, ivar, iMAcoeff |
Same as |
Details
Similar to ARXff,
this family function fits an order–q moving average model
with covariates (MAX(q)),
another special case of the class VGLM–ARIMA
(Miranda and Yee, 2018). Observations, Y_t, are
modelled as
Y_t | \Phi_{t - 1} = \mu_t + \phi_{1} \varepsilon_{t - 1} +
\ldots + \phi_q \varepsilon_{t - q} + \varepsilon_t,
where \mu_t is the (possibly time–dependent) intercept,
modelled as
\mu_t = \mu + \boldsymbol{\beta}^T \boldsymbol{x}_t, and
the errors are mean–zero Gaussian:
\varepsilon_t | \Phi_{t - 1} \sim
N(0, \sigma^2_{\varepsilon_t | \Phi_{t - 1}}).
The symbol
\Phi_{t} denotes the history
of the joint process
\left(Y_{t}, \boldsymbol{X}_{t + 1}^T \right),
at time t for a time–varying covariate
vector \boldsymbol{x}_t.
At each step of Fisher scoring, the exact log-likelihood
based on model above
is computed through dMAq.
The linear predictor by default is
\boldsymbol{\eta} = \left(
\mu_t, \log \sigma^{2}_{\varepsilon_{t | \Phi_{t - 1}}},
\phi_1, \ldots, \phi_q
\right)^T.
The unconditional mean of the process is simply
E(Y_{t}) = \mu, provided no covariates added.
This family function
is not restricted to the noise to be strictly white noise
(in the sense of constant variance).
That is, covariates may be incorporated in the linear predictor
\log \sigma^{2}_{\varepsilon_{t | \Phi_{t - 1}}}. Also, it handles
multiple responses so that a matrix can be used as the response.
For further details on VGLM/VGAM–link functions,
such as logitlink, refer to
Links.
Value
An object of class "vglmff"
(see vglmff-class) to be
used by VGLM/VGAM modelling functions, e.g.,
vglm or vgam.
Warning
By default, a moving-average model of order-1 is fitted.
If different, the order is recycled up to the number of responses
entered in the vglm \ vgam call has been matched.
Successful convergence depends on reasonably setting initial values. If
initial values computed by the algorithm are not adequate, make use of
the the optional initial values (imean, isd, etc.)
For constraints on the paramaters see
cm.ARMA.
Note
Further choices for the random noise, besides Gaussian, will be implemented over time.
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 intercept-only. For MAXff,
the parameters are placed and named as follows (by convention):
c("Mean", "noiseVar" or "noiseSD", "MAcoeff").
Users can modify the zero argument accordingly.
For simplicity, the second choice recommended.
See CommonVGAMffArguments
for further details on zero.
If no constraints are entered in the fitting process,
(e.g., via cm.ARMA)
this family function internally verifies by default
whether the estimated series
is invertible (since noChecks = FALSE).
To ignore this step, set noChecks = TRUE.
If the estimated MA process is non-invertible, MLE
coefficients will conform with the corresponding
invertible MA model.
Further details about these checks are shown within
the summary() output.
Author(s)
Victor Miranda and Thomas W. Yee.
References
Miranda, V. and Yee, T.W. (2018) Vector Generalized Linear Time Series Models. In preparation.
Madsen, H. (2008) Time Series Analysis Florida, USA: Chapman & Hall. (Sections 5.3 to 5.5).
Tsay, R. (2013)
An Introduction to Analysis of Financial data with R.
New Jersey, USA: Wiley Sections 2.2 to 2.4.
See Also
ARIMAXff,
ARXff,
checkTS.VGAMextra,
CommonVGAMffArguments,
Links,
vglm,
vgam.
Examples
set.seed(2)
nn <- 130
### Coefficients
phi1 <- 0.34; phi2 <- -1.19; phi3 <- 0.26
### Intercept
mu <- c(-1.4, 2.3)
### Noise standar deviations (Two responses)
sdMA <- c(sqrt(6.5), sqrt(4.0))
### A single covariate.
Xcov <- runif(nn)
# Generating two MA processes, TS1 and TS2, Gaussian noise.
# Note, the SD noise for TS2 is function of Xcov.
y1 <- mu[1] + arima.sim(nn,
model = list( ma = c(phi1, phi1^2)),
rand.gen = rnorm, sd = exp(sdMA[1]) )
y2 <- mu[2] + arima.sim(nn,
model = list( ma = c(phi1, phi2, phi3) ),
rand.gen = rnorm, sd = exp(Xcov + sdMA[2]) )
# OUR DATA
tsdata <- data.frame(x2 = Xcov , TS1 = y1, TS2 = y2)
#------------------------------------------------------------------------#
# 1. A simple MA(3) to compare with 'arima()'.
myfit0 <- vglm(TS1 ~ 1,
MAXff(order = 3, type.EIM = "exact",
var.arg = FALSE),
#constraints = cm.ARMA(Model = ~ 1,
# lags.cm = 2,
# Resp = 1),
data = tsdata, trace = TRUE)
Coef(myfit0)[c(3, 4, 1)]
fitArima <- arima(tsdata$TS1, order = c(0, 0, 2))
coef(fitArima)
AIC(myfit0); BIC(myfit0)
# ------------------------------------------------------------------------#
# 2. Estimate an MA(3), intercept-only, using initial values.
myfit <- vglm(TS2 ~ 1,
MAXff(order = 3, type.EIM = c("exact", "approx")[1],
# Optional initial values.
imean = 3,
iMAcoeff = c(0.3, -0.2, 0.25),
var.arg = TRUE),
data = tsdata, trace = TRUE)
Coef(myfit)
summary(myfit)
constraints(myfit)
#----------------------------------------#
# Same model fitted using arima()
#----------------------------------------#
fitArima <- arima(tsdata$TS2, order = c(0, 0, 3))
coef(fitArima)
#------------------------------------------------------------------------#
# 3. An MAX(3) with one covariate, testing its effect over the
# standard deviation of the Gaussian noise. Note the 'zero' argument.
myfit1 <- vglm(TS2 ~ x2,
# Or Multiple responses!
# cbind(TS1, TS2) ~ 1,
MAXff(order = 3, type.EIM = "exact", xLag = 1,
# Optional initial values:
# idev.mean = 1.4,
# iMAcoeff = c(2.3, -1.2, 0.25), isd = 1.6,
# NOTE THE ZERO ARGUMENT:
zero = c("Mean", "MAcoeff"),
var.arg = TRUE),
data = tsdata, trace = TRUE)
coef(myfit1, matrix = TRUE)
summary(myfit1)
vcov(myfit1)
constraints(myfit1)
#------------------------------------------------------------------------#
# Model above CANNOT be fitted using arima()
#------------------------------------------------------------------------#