TrenchLoglikelihood function

Loglikelihood function of stationary time series using Trench algorithm

The Trench matrix inversion algorithm is used to compute the exact concentrated loglikelihood function.

TrenchLoglikelihood(r, z)

Arguments

• r: autocovariance or autocorrelation at lags 0,...,n-1, where n is length(z)
• z: time series data

Details

The concentrated loglikelihood function may be written Lm(beta) = -(n/2)log(S/n)-0.5g, where beta is the parameter vector, n is the length of the time series, S=z'M z, z is the mean-corrected time series, M is the inverse of the covariance matrix setting the innovation variance to one and g=-log(det(M)).

Returns

The loglikelihood concentrated over the parameter for the innovation variance is returned.

References

McLeod, A.I., Yu, Hao, Krougly, Zinovi L. (2007). Algorithms for Linear Time Series Analysis, Journal of Statistical Software.

Author(s)

A.I. McLeod

See Also

DLLoglikelihood

Examples

#compute loglikelihood for white noise
z<-rnorm(100)
TrenchLoglikelihood(c(1,rep(0,length(z)-1)), z)

#simulate a time series and compute the concentrated loglikelihood using DLLoglikelihood and
#compare this with the value given by TrenchLoglikelihood.
phi<-0.8
n<-200
r<-phi^(0:(n-1))
z<-arima.sim(model=list(ar=phi), n=n)
LD<-DLLoglikelihood(r,z)
LT<-TrenchLoglikelihood(r,z)
ans<-c(LD,LT)
names(ans)<-c("DLLoglikelihood","TrenchLoglikelihood")