Integrated ARMA Time Series Modelling

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Description

A collection and description of simple to use functions to model univariate autoregressive moving average time series processes, including time series simulation, parameter estimation, diagnostic analysis of the fit, and predictions of future values.

The functions are:

armaSim Simulates an artificial ARMA time series process,
armaFit Fits the parameters of an ARMA time series process,
print Print Method,
plot Plot Method,
summary Summary Method,
predict Forecasts and optionally plots an ARMA process,
fitted Method, returns fitted values,
coef|coefficients Method, returns coefficients,
residuals Method, returns residuals.

Usage

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armaSim(model = list(ar = c(0.5, -0.5), d = 0, ma = 0.1), n = 100, 
    innov = NULL, n.start = 100, start.innov = NULL, 
    rand.gen = rnorm, rseed = NULL, addControl = FALSE, ...) 

armaFit(formula, data, method = c("mle", "ols"), include.mean = TRUE, 
    fixed = NULL, title = NULL, description = NULL, ...)

## S4 method for signature 'fARMA'
show(object)

## S3 method for class 'fARMA'
plot(x, which = "ask", gof.lag = 10, ...)
## S3 method for class 'fARMA'
summary(object, doplot = TRUE, which = "all", ...)

## S3 method for class 'fARMA'
predict(object, n.ahead = 10, n.back = 50, conf = c(80, 95), 
    doplot = TRUE, ...)
    
## S3 method for class 'fARMA'
fitted(object, ...)
## S3 method for class 'fARMA'
coef(object, ...)
## S3 method for class 'fARMA'
residuals(object, ...) 

Arguments

addControl

[armaSim] -
a logical value. Should control parameters added to the returned series as a control attribute?

data

an optional timeSeries or data frame object containing the variables in the model. If not found in data, the variables are taken from environment(formula), typically the environment from which armaFit is called. If data is an univariate series, then the series is converted into a numeric vector and the name of the response in the formula will be neglected.

description

a character string which allows for a brief description.

doplot

[armaRoots] -
a logical. Should a plot be displayed?
[predict][summary] -
is used by the predict and summary methods. By default, this value is set to TRUE and thus the function calls generate beside written also graphical printout. Additional arguments required by underlying functions have to be passed through the dots argument.

fixed

[armaFit] -
is an optional numeric vector of the same length as the total number of parameters. If supplied, only NA entries in fixed will be varied. In this way subset ARMA processes can be modeled. ARIMA modelling supports this option. Thus for estimating parameters of subset ARMA and AR models the most easiest way is to specify them by the formulas x~ARIMA(p, 0, q) and x~ARIMA(p, 0, 0), respectively.

formula

[armaFit] -
a formula specifying the general structure of the ARMA form. Can have one of the forms x ~ ar(q), x ~ ma(p), x ~ arma(p, q), x ~ arima(p, d, q), or x ~ arfima(p, q). x is the response variable optionally to appear in the formula expression. In the first case R's function ar from the ts package will be used to estimate the parameters, in the second case R's function arma from the tseries package will be used, in the third case R's function arima from the ts package will be used, and in the last case R's function fracdiff from the fracdiff package will be used. The state space modelling based arima function allows also to fit ARMA models using arima(p, d=0, q), and AR models using arima(q, d=0, q=0), or pure MA models using arima(q=0, d=0, p). (Exogenous variables are also allowed and can be passed through the ... argument.)

gof.lag

[print][plot][summary][predict] -
the maximum number of lags for a goodness-of-fit test.

include.mean

[armaFit] -
Should the ARIMA model include a mean term? The default is TRUE, note that for differenced series a mean would not affect the fit nor predictions.

innov

[armaSim] -
is a univariate time series or vector of innovations to produce the series. If not provided, innov will be generated using the random number generator specified by rand.gen. Missing values are not allowed. By default the normal random number generator will be used.

method

[armaFit] -
a character string denoting the method used to fit the model. The default method for all models is the log-likelihood parameter estimation approach, method="mle". In the case of an AR model the parameter estimation can also be done by ordinary least square estimation, "ols".

model

[armaSim] -
a list with one (AR), two (ARMA) or three (ARIMA, FRACDIFF) elements . ar is a numeric vector giving the AR coefficients, d is an integer value giving the degree of differencing, and ma is a numeric vector giving the MA coefficients. Thus the order of the time series process is (F)ARIMA(p, d, q) with p=length(ar) and q=length(ma). d is a positive integer for ARIMA models and a numeric value for FRACDIFF models. By default an ARIMA(2, 0, 1) model with coefficients ar=c(0.5, -0.5) and ma=0.1 will be generated.

n

[armaSim] -
an integer value setting the length of the series to be simulated (optional if innov is provided). The default value is 100.

n.ahead, n.back, conf

[print][plot][summary][predict] -
are presetted arguments for the predict method. n.ahead determines how far ahead forecasts should be evaluated together with errors on the confidence intervals given by the argument conf. If a forecast plot is desired, which is the default and expressed by doplot=TRUE, then n.back sets the number of time steps back displayed in the graph.

n.start

[armaSim] -
gives the number of start-up values discarded when simulating non-stationary models. The start-up innovations will be generated by rand.gen if start.innov is not provided.

object

[summary][predict] -
is an object of class fARMA returned by the fitting function armaFit and serves as input for the summary, and predict methods. Some methods allow for additional arguments.

rand.gen

[armaSim] -
is the function which is called to generate the innovations. Usually, rand.gen will be a random number generator. Additional arguments required by the random number generator rand.gen, usually the location, scale and/or shape parameter of the underlying distribution function, have to be passed through the dots argument.

rseed

[armaSim] -
the random number seed, by default NULL. If this argument is set to an integervalue, then the function set.seed(rseed) will be called.

start.innov

[armaSim] -
is a univariate time series or vector of innovations to be used as start up values. Missing values are not allowed.

title

a character string which allows for a project title.

which

[plot][summary] -
if which is set to "ask" the function will interactively ask which plot should be displayed. This is the default value for the plot method. If which="all" is specified all plots will be displayed. This is the default setting for the summary method. On the other hand, if a vector of logicals is specified, then those plots will be displayed for which the elements of the vector are set to TRUE.

x

[print][plot] -
is an object of class fARMA returned by the fitting function armaFit and serves as input for the predict, print, print.summary, and plot methods. Some methods allow for additional arguments.

...

additional arguments to be passed to the output timeSeries. (charvec, units, ...)

Details

AR - Auto-Regressive Modelling:

The argument x~ar(p) calls the underlying functions ar.mle or ar.ols depending on the method's choice. For definiteness, the AR models are defined through

\code{(x[t] - m) = a[1]*(x[t-1] - m) + … + a[p]*(x[t-p] - m) + e[t]}

Order selection can be achieved through the comparison of AIC values for different model specifications. However this may be problematic, as of the methods here only ar.mle performs true maximum likelihood estimation. The AIC is computed as if the variance estimate were the MLE, omitting the determinant term from the likelihood. Note that this is not the same as the Gaussian likelihood evaluated at the estimated parameter values. With method="yw" the variance matrix of the innovations is computed from the fitted coefficients and the autocovariance of x. Burg's method allows for two alternatives method="burg1" or method="burg2" to estimate the innovations variance and hence AIC. Method 1 is to use the update given by the Levinson-Durbin recursion (Brockwell and Davis, 1991), and follows S-PLUS. Method 2 is the mean of the sum of squares of the forward and backward prediction errors (as in Brockwell and Davis, 1996). Percival and Walden (1998) discuss both.
[stats:ar]

MA - Moving-Average Modelling:

The argument x~ma(q) maps the call to the argument x ~ arima(0, 0, q).

ARMA - Auto-Regressive Moving-Average Modelling:

The argument x~arma(p,q) maps the call to the argument x~arima(p, 0, q).

ARIMA - Integrated ARMA Modelling:

The argument x~arima() calls the underlying function arima from R's ts package. For definiteness, the AR models are defined through

\code{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.
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. The exact likelihood is computed via a state-space 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. For a differenced process the non-stationary 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 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 from arima: The functions parse their arguments to the original time series functions available in R's time series library ts.
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.
[stats:arima]

ARFIMA/FRACDIFF Modelling:

The argument x~arfima() calls the underlying functions from R's fracdiff package. The estimator calculates the maximum likelihood estimators of the parameters of a fractionally-differenced ARIMA (p,d,q) model, together (if possible) with their estimated covariance and correlation matrices and standard errors, as well as the value of the maximized likelihood. The likelihood is approximated using the fast and accurate method of Haslett and Raftery (1989). Note, the number of AR and MA coefficients should not be too large (say < 10) to avoid degeneracy in the model.
The optimization is carried out in two levels: an outer univariate unimodal optimization in d over the interval [0,.5], and an inner nonlinear least-squares optimization in the AR and MA parameters to minimize white noise variance.
[fracdiff:fracdiff]

Value

armaFit
returns an S4 object of class "fARMA", with the following slots:

call

the matched function call.

data

the input data in form of a data.frame.

description

allows for a brief project description.

fit

the results as a list returned from the underlying time series model function.

method

the selected time series model naming the applied method.

formula

the formula expression describing the model.

parameters

named parameters or coefficients of the fitted model.

title

a title string.

Note

There is nothing really new in this package. The benefit you will get with this collection is, that all functions have a common argument list with a formula to specify the model and presetted arguments for the specification of the algorithmic method. For users who have already modeled GARCH processes with R/Rmetrics and SPlus/Finmetrics, this approach will be quite natural.

The function armaFit allows for the following formula arguments:

x ~ ar() autoregressive time series processes,
x ~ ma() moving average time series processes,
x ~ arma() autoregressive moving average processes,
x ~ arima() autoregressive integrated moving average processes, and
x ~ arfima() fractionally integrated ARMA processes.

For the first selection x~ar() the function armaFit() uses the AR modelling algorithm as implemented in R's stats package.

For the second x~ma(), third x~arma(), and fourth selection x~arima() the function armaFit() uses the ARMA modelling algorithm also as implemented in R's stats package.

For the last selection x~arfima() the function armaFit() uses the fractional ARIMA modelling algorithm from R's contributed fracdiff package.

Note, that the AR, MA, and ARMA processes can all be modelled by the same algorithm specifying the formula x~arima(p,d,q) in the proper way, i.e. setting d=0 and choosing the orders of p and q as zero in agreement with the desired model specification.

Alternatively, one can still use the functions from R's "stats" package: arima.sim that simulates from an ARIMA time series model, ar, arima, arima0 that fit an AR, ARIMA model to an univariate time series, predict that forecasts from a fitted model, and tsdiag that plots time-series diagnostics. No function from these packages is masked, modified or overwritten.

The output of the print, summary, and predict methods have all the same style of format for each time series model with some additional algorithm specific printing. This makes it easier to interpret the results obtained from different algorithms implemented in different functions.

For arfima models the following methods are not yet implemented: plot, fitted, residuals, predict, and predictPlot.

Author(s)

M. Plummer and B.D. Ripley for ar functions and code,
B.D. Ripley for arima and ARMAacf functions and code,
C. Fraley and F. Leisch for fracdiff functions and code, and
Diethelm Wuertz for the Rmetrics R-port.

References

Brockwell, P.J. and Davis, R.A. (1996); Introduction to Time Series and Forecasting, Second Edition, Springer, New York.

Durbin, J. and Koopman, S.J. (2001); Time Series Analysis by State Space Methods, Oxford University Press.

Gardner, G, Harvey, A.C., Phillips, G.D.A. (1980); Algorithm AS154. An algorithm for exact maximum likelihood estimation of autoregressive-moving average models by means of Kalman filtering, Applied Statistics, 29, 311–322.

Hannan E.J. and Rissanen J. (1982); Recursive Estimation of Mixed Autoregressive-Moving Average Order. Biometrika 69, 81–94.

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, 20, 389–395.

Percival, D.P. and Walden, A.T. (1998); Spectral Analysis for Physical Applications. Cambridge University Press.

Whittle, P. (1963); On the fitting of multivariate autoregressions and the approximate canonical factorization of a spectral matrix. Biometrika 40, 129–134.

Haslett J. and Raftery A.E. (1989); Space-time Modelling with Long-memory Dependence: Assessing Ireland's Wind Power Resource (with Discussion), Applied Statistics 38, 1–50.

Examples

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## armaSim -
   # Simulation:
   x = armaSim(model = list(ar = c(0.5, -0.5), ma = 0.1), n = 1000)
   
## armaFit -
   # Estimate the Parameters:
   fit = armaFit(~ arma(2, 1), data = x)
   print(fit)
      
## summary -
   # Diagnostic Analysis:
   par(mfrow = c(2, 2), cex = 0.7)
   summary(fit, which =  "all")
   
## plot - 
   # Interactive Plots:
   # par(mfrow = c(1, 1))
   # plot(fit)
   
## predict -
   # Forecast 5 Steps Ahead:
   par(mfrow = c(1, 1))
   predict(fit, 5)
   
## armaFit -
   # Alternative Calls:
   TS = MSFT
   armaFit(formula = diff(log(Close)) ~ ar(5), data = TS)
   armaFit(Close ~ ar(5), data = returns(TS, digits = 12))
   TS.RET = returns(TS, digits = 12)
   armaFit(Close ~ ar(5), TS.RET)
   armaFit(Close ~ ar(5), as.data.frame(TS.RET))
   armaFit(~ ar(5), as.vector(TS.RET[, "Close"]))
   armaFit(~ ar(5), as.ts(TS.RET)[, "Close"])
   attach(TS.RET)
   armaFit(Close ~ ar(5))
   detach(TS.RET)