Covariance Matrix of MLE Parameters in an AR(p)
A direct method of computing the inverse of the covariance matrix of p successive observations in an AR(p) with unit innovation variance given by Siddiqui (1958) is implemented. This matrix, divided by n = length of series, is the covariance matrix for the MLE estimates in a regular AR(p).
SiddiquiMatrix(phi)
phi |
coefficients in a regular AR(p) |
Matrix, covariance matrix of MLE estimates
No check on whether the parameters are in the stationary region is done.
It has been shown a necessary and sufficient condition for the
parameters to be in the stationary region is that this matrix should
be positive-definite (Pagano, 1973).
But computationally it is probably better to test for stationarity
by using ARToPacf to transform to the PACF and then
check that the absolute value of all partial autocorrelations are
less than 1.
A.I. McLeod
Siddiqui, M.M. (1958) On the inversion of the sample covariance matrix in a stationary autoregressive process. Annals of Mathematical Statistics 29, 585-588.
Pagano, M. (1973), When is an autoregressive scheme stationary? Communications in Statistics A 1, 533-544.
#compute the inverse directly and by Siddiqui's method and compare: phi<-PacfToAR(rep(0.8,5)) A<-SiddiquiMatrix(phi) B<-solve(toeplitz(TacvfAR(phi, lag.max=length(phi)-1))) max(abs(A-B))
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