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)
r |
autocovariance or autocorrelation at lags 0,...,n-1, where n is length(z) |
z |
time series data |
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)).
The loglikelihood concentrated over the parameter for the innovation variance is returned.
A.I. McLeod
McLeod, A.I., Yu, Hao, Krougly, Zinovi L. (2007). Algorithms for Linear Time Series Analysis, Journal of Statistical Software.
#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")Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.