arima
ARIMA Modelling of Time Series
Fit an ARIMA model to a univariate time series.
 Keywords
 ts
Usage
arima(x, order = c(0L, 0L, 0L), seasonal = list(order = c(0L, 0L, 0L), period = NA), xreg = NULL, include.mean = TRUE, transform.pars = TRUE, fixed = NULL, init = NULL, method = c("CSSML", "ML", "CSS"), n.cond, SSinit = c("Gardner1980", "Rossignol2011"), optim.method = "BFGS", optim.control = list(), kappa = 1e6)
Arguments
 x
 a univariate time series
 order
 A specification of the nonseasonal part of the ARIMA model: the three integer components $(p, d, q)$ are the AR order, the degree of differencing, and the MA order.
 seasonal
 A specification of the seasonal part of the ARIMA
model, plus the period (which defaults to
frequency(x)
). This should be a list with componentsorder
andperiod
, but a specification of just a numeric vector of length 3 will be turned into a suitable list with the specification as theorder
.  xreg
 Optionally, a vector or matrix of external regressors,
which must have the same number of rows as
x
.  include.mean
 Should the ARMA model include a mean/intercept term? The
default is
TRUE
for undifferenced series, and it is ignored for ARIMA models with differencing.  transform.pars
 logical; if true, the AR parameters are
transformed to ensure that they remain in the region of
stationarity. Not used for
method = "CSS"
. Formethod = "ML"
, it has been advantageous to settransform.pars = FALSE
in some cases, see alsofixed
.  fixed
 optional numeric vector of the same length as the total
number of parameters. If supplied, only
NA
entries infixed
will be varied.transform.pars = TRUE
will be overridden (with a warning) if any AR parameters are fixed. It may be wise to settransform.pars = FALSE
when fixing MA parameters, especially near noninvertibility.  init
 optional numeric vector of initial parameter
values. Missing values will be filled in, by zeroes except for
regression coefficients. Values already specified in
fixed
will be ignored.  method
 fitting method: maximum likelihood or minimize conditional sumofsquares. The default (unless there are missing values) is to use conditionalsumofsquares to find starting values, then maximum likelihood. Can be abbreviated.
 n.cond
 only used if fitting by conditionalsumofsquares: the number of initial observations to ignore. It will be ignored if less than the maximum lag of an AR term.
 SSinit
 a string specifying the algorithm to compute the
statespace initialization of the likelihood; see
KalmanLike
for details. Can be abbreviated.  optim.method
 The value passed as the
method
argument tooptim
.  optim.control
 List of control parameters for
optim
.  kappa
 the prior variance (as a multiple of the innovations variance) for the past observations in a differenced model. Do not reduce this.
Details
Different definitions of ARMA models have different signs for the AR and/or MA coefficients. The definition used here has
$$X_t= a_1X_{t1}+ \cdots+ a_pX_{tp} + e_t + b_1e_{t1}+\cdots+ b_qe_{tq} $$
and so the MA coefficients differ in sign from those of SPLUS.
Further, if include.mean
is true (the default for an ARMA
model), this formula applies to $X  m$ rather than $X$. For
ARIMA models with differencing, the differenced series follows a
zeromean ARMA model. If am xreg
term is included, a linear
regression (with a constant term if include.mean
is true and
there is no differencing) is fitted with an ARMA model for the error
term.
The variance matrix of the estimates is found from the Hessian of the loglikelihood, and so may only be a rough guide.
Optimization is done by optim
. It will work
best if the columns in xreg
are roughly scaled to zero mean
and unit variance, but does attempt to estimate suitable scalings.
Value

A list of class
 coef
 a vector of AR, MA and regression coefficients, which can
be extracted by the
coef
method.  sigma2
 the MLE of the innovations variance.
 var.coef
 the estimated variance matrix of the coefficients
coef
, which can be extracted by thevcov
method.  loglik
 the maximized loglikelihood (of the differenced data), or the approximation to it used.
 arma
 A compact form of the specification, as a vector giving the number of AR, MA, seasonal AR and seasonal MA coefficients, plus the period and the number of nonseasonal and seasonal differences.
 aic
 the AIC value corresponding to the loglikelihood. Only
valid for
method = "ML"
fits.  residuals
 the fitted innovations.
 call
 the matched call.
 series
 the name of the series
x
.  code
 the convergence value returned by
optim
.  n.cond
 the number of initial observations not used in the fitting.
 nobs
 the number of “used” observations for the fitting,
can also be extracted via
nobs()
and is used byBIC
.  model
 A list representing the Kalman Filter used in the
fitting. See
KalmanLike
.
"Arima"
with components:Note
The results are likely to be different from SPLUS's
arima.mle
, which computes a conditional likelihood and does
not include a mean in the model. Further, the convention used by
arima.mle
reverses the signs of the MA coefficients.
arima
is very similar to arima0
for
ARMA models or for differenced models without missing values,
but handles differenced models with missing values exactly.
It is somewhat slower than arima0
, particularly for seasonally
differenced models.
Fitting methods
The exact likelihood is computed via a statespace representation of
the ARIMA process, and the innovations and their variance found by a
Kalman filter. The initialization of the differenced ARMA process uses
stationarity and is based on Gardner et al (1980). For a
differenced process the nonstationary components are given a diffuse
prior (controlled by kappa
). Observations which are still
controlled by the diffuse prior (determined by having a Kalman gain of
at least 1e4
) are excluded from the likelihood calculations.
(This gives comparable results to arima0
in the absence
of missing values, when the observations excluded are precisely those
dropped by the differencing.) Missing values are allowed, and are handled exactly in method "ML"
. If transform.pars
is true, the optimization is done using an
alternative parametrization which is a variation on that suggested by
Jones (1980) and ensures that the model is stationary. For an AR(p)
model the parametrization is via the inverse tanh of the partial
autocorrelations: the same procedure is applied (separately) to the
AR and seasonal AR terms. The MA terms are not constrained to be
invertible during optimization, but they will be converted to
invertible form after optimization if transform.pars
is true. Conditional sumofsquares is provided mainly for expositional
purposes. This computes the sum of squares of the fitted innovations
from observation n.cond
on, (where n.cond
is at least
the maximum lag of an AR term), treating all earlier innovations to
be zero. Argument n.cond
can be used to allow comparability
between different fits. The ‘part loglikelihood’ is the first
term, half the log of the estimated mean square. Missing values
are allowed, but will cause many of the innovations to be missing. When regressors are specified, they are orthogonalized prior to
fitting unless any of the coefficients is fixed. It can be helpful to
roughly scale the regressors to zero mean and unit variance.
References
Brockwell, P. J. and Davis, R. A. (1996) Introduction to Time Series and Forecasting. Springer, New York. Sections 3.3 and 8.3.
Durbin, J. and Koopman, S. J. (2001) Time Series Analysis by State Space Methods. Oxford University Press.
Gardner, G, Harvey, A. C. and Phillips, G. D. A. (1980) Algorithm AS154. An algorithm for exact maximum likelihood estimation of autoregressivemoving average models by means of Kalman filtering. Applied Statistics 29, 311322.
Harvey, A. C. (1993) Time Series Models, 2nd Edition, Harvester Wheatsheaf, sections 3.3 and 4.4.
Jones, R. H. (1980) Maximum likelihood fitting of ARMA models to time series with missing observations. Technometrics 22 389395.
Ripley, B. D. (2002) Time series in R 1.5.0. R News, 2/2, 27. http://www.rproject.org/doc/Rnews/Rnews_20022.pdf
See Also
predict.Arima
, arima.sim
for simulating
from an ARIMA model, tsdiag
, arima0
,
ar
Examples
library(stats)
arima(lh, order = c(1,0,0))
arima(lh, order = c(3,0,0))
arima(lh, order = c(1,0,1))
arima(lh, order = c(3,0,0), method = "CSS")
arima(USAccDeaths, order = c(0,1,1), seasonal = list(order = c(0,1,1)))
arima(USAccDeaths, order = c(0,1,1), seasonal = list(order = c(0,1,1)),
method = "CSS") # drops first 13 observations.
# for a model with as few years as this, we want full ML
arima(LakeHuron, order = c(2,0,0), xreg = time(LakeHuron)  1920)
## presidents contains NAs
## graphs in example(acf) suggest order 1 or 3
require(graphics)
(fit1 < arima(presidents, c(1, 0, 0)))
nobs(fit1)
tsdiag(fit1)
(fit3 < arima(presidents, c(3, 0, 0))) # smaller AIC
tsdiag(fit3)
BIC(fit1, fit3)
## compare a whole set of models; BIC() would choose the smallest
AIC(fit1, arima(presidents, c(2,0,0)),
arima(presidents, c(2,0,1)), # < chosen (barely) by AIC
fit3, arima(presidents, c(3,0,1)))
## An example of ARIMA forecasting:
predict(fit3, 3)