# Autocorrelations to AR parameters Given autocorrelations at lags 1,...,n the AR parameters corresponding to the AR coefficients, partial autocorrelations (pacf) and standarized minimum-mean-square predictor variance (sigsqk) are computed. Can also be used as a test for valid acf sequence. ```r DLAcfToAR(r, useC = TRUE, PDSequenceTestQ = FALSE) ``` ## Arguments - `r`: autocorrelations starting at lag 1 - `useC`: TRUE, C-interface function used. Otherwise if FALSE calculations are done in R - `PDSequenceTestQ`: FALSE, an error message is given if the autocorrelation sequence in not pd otherwise test for pd ## Details This function is more general than the built-in `acf2AR` since it provides the pacf and standardized minimum-mean-square error predictors. The standardized minimum-mean-square error predictor variances are defined as the minimum-mean-square error predictor variance for an AR process with unit variance. So for a sufficiently high-order, an approximation to the innovation variance is obtained. The pacf may be used as an alternative parameterization for the linear time series model (McLeod and Zhang, 2006). ## Returns a matrix with 3 columns and length(r) rows is returned corresponding to the ar coefficients, pacf and sigsqk when PDSequenceTestQ = FALSE. Otherwise when PDSequenceTestQ = TRUE, the result is TRUE or FALSE according as the autocorrelation is a valid positive-definite sequence. ## References McLeod, A.I. and Zhang, Y. (2006). Partial autocorrelation parameterization for subset autoregression. Journal of Time Series Analysis, 27, 599-612. McLeod, A.I., Yu, Hao, Krougly, Zinovi L. (2007). Algorithms for Linear Time Series Analysis, Journal of Statistical Software. ## Author(s) A.I. McLeod ## See Also `acf2AR`, `ar` ## Examples ```r #Example 1: Yule-Walker estimates z<-log(lynx) p<-11 r<-(acf(z, lag.max=p, plot=FALSE)$acf)[-1] ans<-DLAcfToAR(r) #compare with built-in ar phiAR<-ar(z,aic=FALSE, order.max=p, method="yw")$ar #yet another way is to use acf2AR phi2<-(acf2AR(c(1,r)))[p,] cbind(ans,phiAR,phi2) # #Example 2: AR(1) illustration #For AR(1) case compare useC = T and F r<-0.9^(1:3) DLAcfToAR(r, useC=TRUE) DLAcfToAR(r, useC=FALSE) DLAcfToAR(r, useC=TRUE, PDSequenceTestQ=TRUE) DLAcfToAR(r, useC=FALSE, PDSequenceTestQ=TRUE) # #Example 3: test for valid tacf r<-c(0.8, rep(0,99)) DLAcfToAR(r, PDSequenceTestQ=TRUE) # #Example 4: Fractional-difference example #Hosking (1981), pacf, zeta[k]=d/(k-d) #we compare this numerically with our procedure `tacvfFdwn` <- function(d, maxlag) { x <- numeric(maxlag + 1) x[1] <- gamma(1 - 2 * d)/gamma(1 - d)^2 for(i in 1:maxlag) x[i + 1] <- ((i - 1 + d)/(i - d)) * x[i] x } n<-10 d<-0.4 r<-tacvfFdwn(d, n) r<-(r/r[1])[-1] HoskingPacf<-d/(-d+(1:n)) cbind(DLAcfToAR(r),HoskingPacf) # # Example 5: Determining a suitable MA approximation #Find MA approximation to hyperbolic decay series N<-10^4 #pick N so large that mmse forecast error converged r<-1/sqrt(1:N) out<-DLAcfToAR(r[-1]) InnovationVariance<-out[nrow(out),3] phi<-out[,1] psi<-ARMAtoMA(ar=phi, lag.max=N) Error<-r[1]-InnovationVariance*(1+sum(psi^2)) ```

DLAcfToAR function