ARIMA Modeling via Gaussian Maximum Likelihood
DESCRIPTION:
Returns a list representing a univariate ARIMA model estimated by Gaussian maximum likelihood.
USAGE: 
arima.mle(x, model, n.cond=<<see below>>, xreg=NULL,  ...)
REQUIRED ARGUMENTS:
x:
a univariate time series or a vector.
Missing values (NA) are allowed.
model:
a list specifying an ARIMA model. If a simple ARIMA model is specified (i.e., only one component), then model should be a list with component names from: order, period, ar, ma, ndiff, ar.opt, ma.trans, and ma.opt. See the DETAILS section for further details.
OPTIONAL ARGUMENTS:
n.cond:
the number of observations on which to condition the likelihood. n.cond must be at least P+D (the default), where P and D are the orders of the expanded autoregressive and differencing polynomials.
xreg:
a univariate or multivariate time series or a vector or matrix with univariate time series per column. These will be used as additive regression variables.
...:
additional arguments can be passed to nlmin.
VALUE:
a list representing the result of the fitted model:
model:
the same as the input model with the estimated coefficients substituted in the components ar and ma.
var.coef:
the variance-covariance matrix for the autoregressive and moving average coefficients.
loglik:
-2 times the log likelihood of the model (up to a constant factor).
aic:
Akaike's information criteria, which is loglik plus 2 times the number of parameters fit.
sigma2:
the estimated innovations variance.
n.used:
the number of observations used to compute the likelihood.
n.cond:
the number of observations on which the likelihood is conditioned.
reg.coef:
the estimated regression coefficients (returned if xreg is provided).
converged:
if the optimizer has apparently successfully converged to a minimum, then converged is TRUE; otherwise converged is FALSE.
conv.type:
a character string describing the type of convergence.
series:
a character string giving the name of x, including transformations.
reg.series:
a character string giving the name of xreg, including transformations (only returned if xreg is provided).
DETAILS:
SPECIFYING A MODEL. The components of the model argument should be some or all of: order, period, ar, ma, ndiff, ar.opt, ma.trans, and ma.opt. order is a vector of length 3 specifying the order of an ARIMA(p,d,q); the elements of the vector are (p, d, q), where p and q are the order of the autoregressive and moving average operators and d is the number of differences. period specifies the period of the ARIMA operators; e.g., a seasonal operator for monthly data would have a period of 12 (the default is 1). ar and ma are vectors of initial values for autoregressive and moving average coefficients respectively (if not provided, the initial values to the optimizer are set to 0). ndiff is an alternative way to specify the number of differences. A model can be specified using either order or ar, ma, and ndiff, or both (but they must agree if both are specified). The model doesn't explicitly allow for a non-zero mean of the series, though the estimation is invariant to the mean of the data for many of the possible models; if you estimate an AR model, you must subtract the mean yourself or use the xreg argument in order to get sensible estimates.

If ma.trans is TRUE (the default), the moving average coefficients will be transformed before passing them to the optimizer, ensuring invertibility of the model. ar.opt and ma.opt are logical vectors of length p and q respectively. If the ith element is TRUE, then the optimizer will optimize over the ith autoregressive or moving average coefficient. By default, the vectors are TRUE. This option is useful to fit models in which some low order coefficients are set to 0.

If a multiplicative ARIMA model is specified (i.e., more than one component), then model is a list of lists. Each list represents a component of the multiplicative ARIMA model, and is specified in the same manner as above (i.e., the components of each list are ar, ma, ndiff, order, period, ma.trans, ar.opt, ma.opt).

THE ALGORITHM. The likelihood is conditioned on n.cond observations. If no missing values are present, the likelihood is computed by forming the Choleski decomposition of the covariance matrix of the process (see Ansley, 1979). If missing values are present, then the likelihood is computed using the Kalman filter applied to a state space representation of the likelihood. The state space representation is the same as that used by Kohn and Ansley (1986). The method of initializing the Kalman filter recursions is that given by Bell and Hillmer (1987). The regression parameters are concentrated out of the likelihood, as in Kohn and Ansley (1985). The autoregressive and moving average parameters are transformed to ensure stationarity and invertibility as in Jones (1980).

WARNING:
Unlike the ar function, the mean of the series will not be estimated by arima.mle unless you use the xreg argument. arima.mle assumes a zero mean series.
REFERENCES:
Ansley, C. F. (1979). An algorithm for the exact likelihood of a mixed autoregressive-moving average process. Biometrika 66, 59-65.

Bell, W. and Hillmer, S. (1987). Initializing the Kalman filter in the nonstationary case. Research Report CENSUS/SRC/RR-87/33, Statistical Research Division, Bureau of the Census, Washington, DC, 20233.

Jones, R. H. (1980). Maximum likelihood fitting of ARMA models to time series with missing observations. Technometrics 22, 389-395.

Kohn, R. and Ansley, C. F. (1985). Efficient estimation and prediction in time series regression models. Biometrika 72, 694-697.

Kohn, R. and Ansley, C. F. (1986). Estimation, prediction, and interpolation for ARIMA models with missing data. Journal of the American Statistical Association 81, 751-761.

The chapter "Analyzing Time Series" of the S-PLUS Guide to Statistical and Mathematical Analysis.

SEE ALSO:
ar , arima.diag , arima.filt , arima.forecast , nlmin , sabl .
EXAMPLES: 
# Simulate an MA(2) series and fit the series using Gaussian
# Maximum Likelihood
ma <- arima.sim(model=list(ma=c(-.5, -.25)))
ma.fit <- arima.mle(ma, model=list(ma=c(-.5, -.25)))


# Fit a Box-Jenkins (0,1,1)x(0,1,1)12 Airline model to the ship data
# Use zeros as the starting values for the optimizer
al.mod <- list(list(order=c(0,1,1)), list(order=c(0,1,1), period=12))
fit <- arima.mle(ship, model=al.mod)