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The Trench matrix inversion algorithm is used to compute the exact concentrated loglikelihood function.
TrenchLoglikelihood(r, z)The loglikelihood concentrated over the parameter for the innovation variance is returned.
autocovariance or autocorrelation at lags 0,...,n-1, where n is length(z)
time series data
A.I. McLeod
The concentrated loglikelihood function may be written Lm(beta) = -(n/2)*log(S/n)-0.5*g, 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)).
McLeod, A.I., Yu, Hao, Krougly, Zinovi L. (2007). Algorithms for Linear Time Series Analysis, Journal of Statistical Software.
DLLoglikelihood
#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")
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