# DLResiduals: Prediction residuals In ltsa: Linear Time Series Analysis

## Description

The Durbin-Levison algorithm is used to compute the one-step prediction residuals.

## Usage

 `1` ```DLResiduals(r, z, useC = TRUE, StandardizedQ=TRUE) ```

## Arguments

 `r` vector of length n containing the autocovariances or autocorrelations at lags 0,...,n-1 `z` vector of length n, mean-corrected time series data `useC` if TRUE, the compiled C code is used, otherwise the computations are done entirely in R and much slower `StandardizedQ` TRUE, the residuals are divided by their standard deviation or FALSE, the raw prediction residuals are computed

## Details

If the model is correct the standardized prediction residuals are approximately NID(0,1) and are asymptotically equivalent to the usual innovation residuals divided by the residual sd. This means that the usual diagnotic checks, such as the Ljung-Box test may be used.

## Value

Vector of length n containing the residuals

A.I. McLeod

## References

W.K. Li (1981). Topics in Time Series Analysis. Ph.D. Thesis, University of Western Ontario.

McLeod, A.I., Yu, Hao, Krougly, Zinovi L. (2007). Algorithms for Linear Time Series Analysis, Journal of Statistical Software.

`DLLoglikelihood`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29``` ```# For the AR(1) the prediction residuals and innovation residuals are the same (except for # t=1). In this example we demonstrate the equality of these two types of residuals. # phi<-0.8 sde<-30 n<-30 z<-arima.sim(n=30,list(ar=phi),sd=sde) r<-phi^(0:(n-1))/(1-phi^2)*sde^2 e<-DLResiduals(r,z) a<-numeric(n) for (i in 2:n) a[i]=z[i]-phi*z[i-1] a<-a/sde ERR<-sum(abs(e[-1]-a[-1])) ERR # #Simulate AR(1) and compute the MLE for the innovation variance phi <- 0.5 n <- 2000 sigsq <- 9 z<-arima.sim(model=list(ar=phi), n=n, sd=sqrt(sigsq)) g0 <- sigsq/(1-phi^2) r <- g0*phi^(0:(n-1)) #comparison of estimate with actual e<-DLResiduals(r,z,useC=FALSE, StandardizedQ=FALSE) sigsqHat <- var(e) ans<-c(sigsqHat,sigsq) names(ans)<-c("estimate","theoretical") ans ```