I don't understand results of
x <- c(1,1,2,3,5,8,13,21,34,55)
x(n) = 1* x(n-1) + 1 * x(n-2)
You may use
ar.ols to estimate this non-stationary series.
From the docs,
ar.ols fits the general AR model to a possibly non-stationary and/or multivariate system of series x.
Example with your data:
ar.ols(x,demean=FALSE) Call: ar.ols(x = x, demean = FALSE) Coefficients: 1 2 1 1