arima0 ARIMA Modelling of Time Series – Preliminary Version

Description

Fit an ARIMA model to a univariate time series, and forecast from the fitted model.

Usage

arima0(x, order = c(0, 0, 0),
       seasonal = list(order = c(0, 0, 0), period = NA),
       xreg = NULL, include.mean = TRUE, delta = 0.01,
       transform.pars = TRUE, fixed = NULL, init = NULL,
       method = c("ML", "CSS"), n.cond, optim.control = list())

## S3 method for class 'arima0'
predict(object, n.ahead = 1, newxreg, se.fit = TRUE, ...)

Arguments

x	a univariate time series
order	A specification of the non-seasonal part of the ARIMA model: the three 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 components order and period, but a specification of just a numeric vector of length 3 will be turned into a suitable list with the specification as the order.
xreg	Optionally, a vector or matrix of external regressors, which must have the same number of rows as x.
include.mean	Should the ARIMA model include a mean term? The default is TRUE for undifferenced series, FALSE for differenced ones (where a mean would not affect the fit nor predictions).
delta	A value to indicate at which point ‘fast recursions’ should be used. See the ‘Details’ section.
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".
fixed	optional numeric vector of the same length as the total number of parameters. If supplied, only NA entries in fixed will be varied. transform.pars = TRUE will be overridden (with a warning) if any ARMA parameters are fixed.
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 sum-of-squares. Can be abbreviated.
n.cond	Only used if fitting by conditional-sum-of-squares: the number of initial observations to ignore. It will be ignored if less than the maximum lag of an AR term.
optim.control	List of control parameters for optim.
object	The result of an arima0 fit.
newxreg	New values of xreg to be used for prediction. Must have at least n.ahead rows.
n.ahead	The number of steps ahead for which prediction is required.
se.fit	Logical: should standard errors of prediction be returned?
...	arguments passed to or from other methods.

Details

Different definitions of ARMA models have different signs for the AR and/or MA coefficients. The definition here has

X[t] = a[1]X[t-1] + … + a[p]X[t-p] + e[t] + b[1]e[t-1] + … + b[q]e[t-q]

and so the MA coefficients differ in sign from those of S-PLUS. Further, if include.mean is true, this formula applies to X-m rather than X. For ARIMA models with differencing, the differenced series follows a zero-mean ARMA model.

The variance matrix of the estimates is found from the Hessian of the log-likelihood, and so may only be a rough guide, especially for fits close to the boundary of invertibility.

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.

Finite-history prediction is used. This is only statistically efficient if the MA part of the fit is invertible, so predict.arima0 will give a warning for non-invertible MA models.

Value

For arima0, a list of class "arima0" with components:

coef	a vector of AR, MA and regression coefficients,
sigma2	the MLE of the innovations variance.
var.coef	the estimated variance matrix of the coefficients coef.
loglik	the maximized log-likelihood (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 non-seasonal and seasonal differences.
aic	the AIC value corresponding to the log-likelihood. Only valid for method = "ML" fits.
residuals	the fitted innovations.
call	the matched call.
series	the name of the series x.
convergence	the value returned by optim.
n.cond	the number of initial observations not used in the fitting.

For predict.arima0, a time series of predictions, or if se.fit = TRUE, a list with components pred, the predictions, and se, the estimated standard errors. Both components are time series.

Fitting methods

The exact likelihood is computed via a state-space representation of the ARMA process, and the innovations and their variance found by a Kalman filter based on Gardner et al (1980). This has the option to switch to ‘fast recursions’ (assume an effectively infinite past) if the innovations variance is close enough to its asymptotic bound. The argument delta sets the tolerance: at its default value the approximation is normally negligible and the speed-up considerable. Exact computations can be ensured by setting delta to a negative value.

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 also constrained to be invertible during optimization by the same transformation if transform.pars is true. Note that the MLE for MA terms does sometimes occur for MA polynomials with unit roots: such models can be fitted by using transform.pars = FALSE and specifying a good set of initial values (often obtainable from a fit with transform.pars = TRUE).

Missing values are allowed, but any missing values will force delta to be ignored and full recursions used. Note that missing values will be propagated by differencing, so the procedure used in this function is not fully efficient in that case.

Conditional sum-of-squares 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 log-likelihood’ 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.

Note

This is a preliminary version, and will be replaced by arima.

The standard errors of prediction exclude the uncertainty in the estimation of the ARMA model and the regression coefficients.

The results are likely to be different from S-PLUS'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.

References

Brockwell, P. J. and Davis, R. A. (1996). Introduction to Time Series and Forecasting. Springer, New York. Sections 3.3 and 8.3.

Gardner, G, Harvey, A. C. and Phillips, G. D. A. (1980). Algorithm AS 154: An algorithm for exact maximum likelihood estimation of autoregressive-moving average models by means of Kalman filtering. Applied Statistics, 29, 311–322. doi: 10.2307/2346910.

Harvey, A. C. (1993). Time Series Models. 2nd Edition. Harvester Wheatsheaf. Sections 3.3 and 4.4.

Harvey, A. C. and McKenzie, C. R. (1982). Algorithm AS 182: An algorithm for finite sample prediction from ARIMA processes. Applied Statistics, 31, 180–187. doi: 10.2307/2347987.

Jones, R. H. (1980). Maximum likelihood fitting of ARMA models to time series with missing observations. Technometrics, 22, 389–395. doi: 10.2307/1268324.

See Also

arima, ar, tsdiag

Examples

## Not run: arima0(lh, order = c(1,0,0))
arima0(lh, order = c(3,0,0))
arima0(lh, order = c(1,0,1))
predict(arima0(lh, order = c(3,0,0)), n.ahead = 12)

arima0(lh, order = c(3,0,0), method = "CSS")

# for a model with as few years as this, we want full ML
(fit <- arima0(USAccDeaths, order = c(0,1,1),
               seasonal = list(order=c(0,1,1)), delta = -1))
predict(fit, n.ahead = 6)

arima0(LakeHuron, order = c(2,0,0), xreg = time(LakeHuron)-1920)
## Not run: 
## presidents contains NAs
## graphs in example(acf) suggest order 1 or 3
(fit1 <- arima0(presidents, c(1, 0, 0), delta = -1))  # avoid warning
tsdiag(fit1)
(fit3 <- arima0(presidents, c(3, 0, 0), delta = -1))  # smaller AIC
tsdiag(fit3)
## End(Not run)