yulewalker(vec [, inverse:T]), vec a REAL vector or matrix. |

yulewalker(Rho) computes a REAL vector of autoregressive (AR) coefficients phi[1], phi[2], ... phi[p] of a p-th order AR time series whose autocorrelations r[1], r[2], ... r[p], are in REAL vector Rho, where p = nrows(Rho). The values of phi satisfy the Yule-Walker equations rho[j] = sum(phi[k]*rho[j-k],j=1,...,p), k = 1,...,p, with rho[0] = 1, rho[-j] = rho[j]. When Rho is a matrix, yulewalker(Rho) computes AR coefficients from each column separately, that is, yulewalker(Rho)[,j] = yulewalker(Rho[,j]), j = 1,...,ncols(Rho). If Rho is a generalized matrix (at most two dimensions >= 1), yulewalker(Rho) = yulewalker(matrix(Rho)) (see 'matrices', matrix()). When any column of Rho is not a valid autocorrelation function, that is, if the implied Toeplitz correlation matrix is not positive definite, yulewalker() prints a warning message, sets the element in the result where the violation occurred to the most extreme value possible and any subsequent elements to zero. For instance, yulewalker(vector(-.3, -.9,.5)) returns the result vector(-.6, -1, 0). A typical usage is Cmd> rhohat <- autocor(y, 10) # compute 1st 10 autocorrelations Cmd> phihat <- yulewalker(rhohat) phihat is a vector of length 10 containing the coefficients of the autoregressive series whose first 10 autocorrelations are the same as rhohat. Thus yulewalker() provides a method-of-moments way to estimate the parameters of an AR (autoregressive) model. When time series y is in fact a normal stationary AR series of order length(rhohat), these estimates are asymptotically equivalent to maximum likelihood estimates. NOTE: autocor() is a macro in file tser.mac. Type help(autocor) for details. yulewalker(Phi,inverse:T) computes auto correlations corresponding to autoregressive coefficients in the columns of REAL vector or matrix Phi. Effectively yulewalker(Phi,inverse:T) is the inverse function to yulewalker() in that yulewalker(yulewalker(Rho),inverse:T) should the same as Rho, except for rounding error. One important usage is Cmd> rho <- yulewalker(padto(Phi,n), inverse:T) where n >= nrows(Phi). This computes the first n autocorrelations of the autoregressive series with autoregression coefficients in Phi. See also padto(), partacf(), toeplitz().

Gary Oehlert 2003-01-15