marima {marima}

marima

Description

Estimate multivariate arima and arima-x models. Setting up the proper model for (especially) arima-x estimation can be accomplished using the routine 'define.model' that can assist in setting up the necessary autoregressive and moving average patterns used as input to 'marima'.

A more elaborate description of 'marima' and how it is used can be downloaded from:

http://www.imm.dtu.dk/~hspl/marima.use.pdf

Usage

marima(DATA = NULL, ar.pattern = NULL, ma.pattern = NULL, means = 1,
  max.iter = 50, penalty = 0, weight = 0.33, Plot = "none",
  Check = FALSE)

Arguments

DATA

time series matrix, dim(DATA) = c(kvar, n), where 'kvar' is the dimension of the time series and 'n' is the length of the series. If DATA is organized (n, kvar) (as a data.frame e.g.) it is automatically transposed in marima, and the user need not care about it. Also, and consequently, the output residuals and fitted values matrices are both organised c(kvar, n) at return from marima. The DATA is checked for completeness. Cases which include 'NA's or 'NaN's are initially left out. A message is given (on the console) and the active cases are given in the output object (...$used.cases). If DATA is a time series object it is transformed to a matrix and a warning is given ( if(is.ts(DATA)) DATA <- as.matrix(data.frame(DATA)) and a message is given (on the console).

ar.pattern

autoregressive pattern for model (see define.model). If ar.pattern is not specified a pure ma-model is estimated.

ma.pattern

moving average pattern for model (see define.model). If ma.pattern is not specified a pure ar-model is estimated. In this case the estimation is carried out by regression analysis in a few steps.

means

0/1 indicator vector of length kvar, indicating which variables in the analysis should be means adjusted or not. Default: means=1 and all variables are means adjusted. If means=0 is used, no variables are means adjusted.

max.iter

max. number of iterations in estimation (max.iter=50 is default which, generally, is more than enough).

penalty

parameter used in the R function 'step' for stepwise model reduction. If penalty=2, the conventional AIC criterion is used. If penalty=0, no stepwise reduction of model is performed. Generally 0<=penalty<=2 works well (especially penalty=1). The level of significance of the individual parameter estimates in the final model can be checked by considering the (approximate) 'ar.pvalues' and the 'ma.pvalues' calculated by marima.

weight

weighting factor for smoothing the repeated estimation procedure. Default is weight=0.33 which often works well. If weight>0.33 (e.g. weight=0.66) is specified more damping will result. If a large damping factor is used, the successive estimations are more cautious, and a slower (but safer) convergence (if possible) may result (max.iter may have to be increased to, say, max.iter=75.

Plot

'none' or 'trace' or 'log.det' indicates a plot that shows how the residual covariance matrix (resid.cov) develops with the iterations. If Plot= 'none' no plot is generated. If Plot= 'trace' a plot of the trace of the residual covariance matrix versus iterations is generated. If Plot='log.det' the log(determinant) of the residual covariance matrix (resid.cov) is generated. Default is Plot= 'none'.

Check

(TRUE/FALSE) results (if TRUE) in a printout of some controls of the call to arima. Useful in the first attemp(s) to use marima. Default=FALSE.

Value

Object of class marima containing:

N = N length of analysed series

kvar = dimension of time series (all random and non-random variables).

ar.estimates = ar-estimates

ma.estimates = ma-estimates

ar.fvalues = ar-fvalues (approximate)

ma.fvalues = ma-fvalues (approximate)

ar.stdv = standard devaitions of ar-estimates (approximate)

ma.stdv = standard deviations of ma-estimates (approximate)

ar.pvalues = ar.estimate p-values (approximate). If in the input data two series are identical or one (or more) series is (are) linearly dependent of the the other series the routine lm(...) generates "NA" for estimates t-values, p-values and and parameter standard deviations. In marima the corresponding estimates, F-values and parameter standard deviations are set to 0 (zero) while the p-value(s) are set to "NaN". Can happen only for ar-parameters.

ma.pvalues = ma.estimate p-values (approximate)

residuals = estimated residuals (for used.cases), leading values (not estimated values) are put equal to NA

fitted = estimated/fitted values for all data (including non random variables) (for used.cases), leading values (not estimated values) are put equal to NA

resid.cov = covariance matrix of residuals (including non random variables) (computed for used.cases)

data.cov = covariance matrix of (all) input data (for used.cases)

averages = averages of input variables

Constant = estimated model constant = (sum_i(ar[, , i])) x averages

call.ar.pattern = calling ar.pattern

call.ma.pattern = calling ma.pattern

out.ar.pattern = resulting ar.pattern (after possible model reduction)

out.ma.pattern = resulting ar.pattern (after possible model reduction)

max.iter = max no. of iterations in call

penalty = factor used in AIC model reduction, if penalty=0, no AIC model redukction is performed (default).

weight = weighting of successive residuals updating (default=0.33)

used.cases = cases in input which are analysed

trace = trace(random part of resid.cov)

log.det = log(det(random part of resid.cov))

randoms = which are random variables in problem?

one.step = one step ahead prediction (for time = N+1) based on whole series from obs. 1 to N. The computation is based on the marima residuals (as taken from the last regression step in the repeated pseudo-regression algorithm).

Source

The code is an R code which is based on the article (below) by Spliid (1983). A repeated (socalled) pseudo regression procedure is used in order to estimate the multivariate arma model.

References

Jenkins, G.M. & Alavi, A. (1981): Some aspects of modelling and forecasting multivariate time series, Journal of Time Series Analysis, Vol. 2, issue 1, Jan. 1981, pp. 1-47.

Madsen, H. (2008) Time Series Analysis, Chapmann \& Hall (in particular chapter 9: Multivariate time series).

Reinsel, G.C. (2003) Elements of Multivariate Time Series Analysis, Springer Verlag, 2$^nd$ ed. pp. 106-114.

Shumway, R.H. & Stoffer, D.S. (2000). Time Series Analysis and Its Applications, Springer Verlag, (4$^th$ ed. 2016).

Spliid, H.: A Fast Estimation Method for the Vector Autoregressive Moving Average Model With Exogenous Variables, Journal of the American Statistical Association, Vol. 78, No. 384, Dec. 1983, pp. 843-849.

Spliid, H.: Estimation of Multivariate Time Series with Regression Variables:

http://www.imm.dtu.dk/~hspl/marima.use.pdf

www.itl.nist.gov/div898/handbook/pmc/section4/pmc45.htm

Examples

# Example 1:
library(marima)
# Generate a 4-variate time series (in this example):
#
kvar<-4 ; set.seed(4711)
y4<-matrix(round(100*rnorm(4*1000, mean=2.0)), nrow=kvar)
# If wanted define differencing of variable 4 (lag=1)
# and variable 3 (lag=6), for example:
y4.dif<-define.dif(y4, difference=c(4, 1, 3, 6))
# The differenced series will be in y4.dif$y.dif, the observations
# lost by differencing being excluded.
#
y4.dif.analysis<-y4.dif$y.dif
# Give lags the be included in ar- and ma-parts of model:
#
ar<-c(1, 2, 4)
ma<-c(1)
# Define the multivariate arma model using 'define.model' procedure.
# Output from 'define.model' will be the patterns of the ar- and ma-
# parts of the model specified.
#
Mod <- define.model(kvar=4, ar=ar, ma=ma, reg.var=3)
arp<-Mod$ar.pattern
map<-Mod$ma.pattern
# Print out model in 'short form':
#
short.form(arp)
short.form(map)
# Now call marima:
Model <- marima(y4.dif.analysis, ar.pattern=arp, ma.pattern=map, 
                penalty=0.0)
# The estimated model is in the object 'Model':
#
ar.model<-Model$ar.estimates
ma.model<-Model$ma.estimates
dif.poly<-y4.dif$dif.poly  # = difference polynomial in ar-form.
# Multiply the estimated ar-polynomial with difference polynomial
# to compute the aggregated ar-part of the arma model:
#
ar.aggregated <- pol.mul(ar.model, dif.poly, L=12)
# and print everything out in 'short form':
#
short.form(ar.aggregated, leading=FALSE)
short.form(ma.model, leading=FALSE)