artfima (version 1.5)

artfima: MLE for ARTFIMA model

Description

Maximum likelihood estimation of the ARTFIMA model as well as the edge cases ARIMA and ARFIMA. Exact MLE and Whittle approximate MLE are implemented.

Usage

artfima(z, glp = c("ARTFIMA", "ARFIMA", "ARIMA"), arimaOrder = c(0, 0, 0),  likAlg = c("exact", "Whittle"),  fixd = NULL, b0 = NULL,  lambdaMax = 3, dMax = 10)

Arguments

z

time series data

glp

general linear process type: ARTFIMA, ARFIMA or ARMA.

arimaOrder

c(p,D,q), where p is the AR order, D is the regular difference parameter and q is the MA order.

likAlg

"exact" or "Whittle" or "Whittle2"

fixd

only used with ARTFIMA, default setting fixd=NULL means the MLE for the parameter d is obtained other if fixed=d0, where d0 is a numeric value in the interval (-2, 2) the d parameter in ARTFIMA is fixed at this value while the remaining parameters are estimated.

b0

initial estimates - use only for high order AR models. See Details and Example.

lambdaMax

ARTFIMA boundard setting - upper limit for lambda

dMax

ARTFIMA boundard setting - absolute magnitude for d. See Note and Example

Value

print method.

Details

The ARFIMA and ARIMA are subsets or edge-cases of the ARTFIMA model. The likelihood and probability density function for these models is defined by the multivariate normal distribution. The log-likelihood, AIC and BIC are comparable across models. When the Whittle MLE algorithm is used, the final log-likelihood is obtained by plugging this estimates into the exact log-likelihood.

The argument b0 is provided for fitting for fitting high order AR models with ARTFIMA. That is ARTFIMA(p,0,0) when p is large. This fitting is best done by fitting values with p=1,2,...,pmax. For p>1, set b0 equal to c(ans$b0, 0), where ans is the output from artfima for the p-1 order model. An example is given below. This technqiue is used by bestModels with q=0 and p>3.

References

McLeod, A.I., Yu, Hao and Krougly, Z. (2007). Algorithms for Linear Time Series Analysis: With R Package. Journal of Statistical Software 23/5 1-26.

See Also

bestModels

Examples

artfima(Nile) #Nile is a built in dataset in R
artfima(Nile, likAlg = "exact")
#
#fitting a high-order AR using recursion
## Not run: 
# #This may take 3 to 6 hours if exact MLE used!
# #But Whittle MLE doesn't work properly for this example!!
#  data(SB32)
#  z <- SB32
#  likAlg <- "exact"
#  pmax <- 30
#  startTime <- proc.time()[3]
#  ic <- matrix(numeric(0), ncol=3, nrow=pmax+1)
#  out <- artfima(z, arimaOrder=c(0,0,0), likAlg=likAlg)
#  ic[1, 1] <- out$aic
#  ic[1, 2] <- out$bic
#  ic[1, 3] <- out$LL
#  b1 <- c(out$b0, 0)
#  for (i in 1:pmax) {
#   out <- artfima(z, arimaOrder=c(i,0,0), b0=b1, likAlg=likAlg)
#   b1 <- c(out$b0, 0)
#   ic[i+1, 1] <- out$aic
#   ic[i+1, 2] <- out$bic
#   ic[i+1, 3] <- out$LL
#  }
#  endTime <- proc.time()[3]
#  (totTime <- endTime-startTime)
#  plot(0:pmax, ic[,1], xlab="AR order", ylab="AIC", pch=20, col="blue")
#  indBest <- which.min(ic[,1])
#  pBest <- indBest-1
#  icBest <- ic[indBest,1]
#  abline(h=icBest, col="brown")
#  abline(v=pBest, col="brown")
#  plot(0:pmax, ic[,2], xlab="AR order", ylab="BIC", pch=20, col="blue")
#  indBest <- which.min(ic[,2])
#  pBest <- indBest-1
#  icBest <- ic[indBest,2]
#  abline(h=icBest, col="brown")
#  abline(v=pBest, col="brown")
#  plot(0:pmax, ic[,3], xlab="AR order", ylab="log-lik", pch=20)
#  ## End(Not run)#end dontrun
#
#setting new boundary limit
## Not run: 
# data(SB32)
# #ARTFIMA(1,0,2) - MLE for d on boundar, dHat = 10
# artfima(SB32, arimaOrder=c(1,0,2))
# #note:
# #log-likelihood = -10901.14, AIC = 21816.29, BIC = 21862.41
# #Warning: estimates converged to boundary!
# #mean     -0.5558988 8.443794e-02
# #d         9.9992097 1.396002e-05
# #lambda    2.9304658 8.050071e-02
# #phi(1)    0.9271892 6.862294e-03
# #theta(1)  0.8440911 1.709824e-02
# #theta(2) -0.3650004 2.744227e-02
# #
# #now reset upper limit dMax and lambdaMax
# #NOTE - there is only a very small improvement in the log-likelihood
# artfima(SB32, arimaOrder=c(1,0,2), lambdaMax=20, dMax=40)
# #ARTFIMA(1,0,2), MLE Algorithm: exact, optim: BFGS
# #snr = 4.665, sigmaSq = 3.38228734331338
# #log-likelihood = -10900.56, AIC = 21815.12, BIC = 21861.25
# #               est.    se(est.)
# #mean     -0.5558988  0.08443794
# #d        27.0201256 36.94182328
# #lambda    3.9412050  1.38296970
# #phi(1)    0.9276901  0.00676589
# #theta(1)  0.8342879  0.01715041
# #theta(2) -0.3644787  0.02691869
# ## End(Not run)