# theoretical autocovariance function (acvf) of ARMA The theoretical autocovariance function of ARMA(p,q) process is computed. This is more useful in some situations than the built-in R function `ARMAacf`. See Details. ```r tacvfARMA(phi = numeric(0), theta = numeric(0), maxLag = 1, sigma2 = 1) ``` ## Arguments - `phi`: ar parameters - `theta`: ma parameters - `maxLag`: acvf is computed at lags 0, ..., maxLag - `sigma2`: innovation variance ## Details The details of the autocovariance computation are given in McLeod (1975). In addition to this computation, we also test if the model is stationary-causal or not. The test, which is included directly in the function, uses the Durbin-Levison recursion to transform from the phi parameters to the pacf. See McLeod and Zhang (2006, eqn. (1)) for more details. Formally, the stationary-causal condition requires that all roots of the polynomial equation, $$ 1 - phi[1]*B -...- phi[p]*B^p = 0 $$ must lie outside the unit circle (Brockwell and Davis, 1991, Section 3.3). This function is included because it is necessary to demonstrate that in the case of ARMA models, TrenchInverse and the built-in R function predict.Arima produce equivalent results. See Example 1 in the documentation for `TrenchForecast` and the example discussed in McLeod, Yu and Krougly (2007, 3.2). ## Returns Vector of length maxLag containing the autocovariances at lags 0, ..., maxLag. But see Warning below. ## References P.J. Brockwell and R.A. Davis (1991) Time Series: Theory and Methods. Springer. A.I. McLeod (1975) Derivation of the theoretical autocovariance function of autoregressive-moving average models, Applied Statistics 24, 255-256. A.I. McLeod and Zhang, Y. (2006) Partial autocorrelation parameterizations for subset autoregression, Journal of Time Series Analysis, McLeod, A.I., Yu, Hao, Krougly, Zinovi L. (2007). Algorithms for Linear Time Series Analysis, Journal of Statistical Software. ## Note An error is returned if the model is not stationary-causal. ## Author(s) A.I. McLeod ## See Also `ARMAacf` ## Examples ```r #Example 1. Estimate the acvf of a fitted ARMA model #There are two methods but they give slighly different results, #general script, just change z, p, q, ML z<-sqrt(sunspot.year) n<-length(z) p<-9 q<-0 ML<-5 #for different data/model just reset above out<-arima(z, order=c(p,0,q)) phi<-theta<-numeric(0) if (p>0) phi<-coef(out)[1:p] if (q>0) theta<-coef(out)[(p+1):(p+q)] zm<-coef(out)[p+q+1] sigma2<-out$sigma2 rA<-tacvfARMA(phi, theta, maxLag=n+ML-1, sigma2=sigma2) rB<-var(z)*ARMAacf(ar=phi, ma=theta, lag.max=n+ML-1) #rA and rB are slighly different cbind(rA[1:5],rB[1:5]) # #Example 2. Compute Rsq for fitted ARMA model #Rsq = 1 - (series variance / innovation variance) #Again there are two methods but only the first method is guaranteed to #produce an Rsq which is non-negative! #Run last example and then evaluate the script below: RsqA <- 1 - rA/sigma2 RsqB <- 1 - rB/sigma2 # #Example 3. Test if model is stationary-causal or not. StationaryQ <- function(phi) tryCatch(is.vector(tacvfARMA(phi=phi)),error=function(e) FALSE ) StationaryQ(1.1) #AR(1) with phi=1.1 is not stationary-causal. #try with parameters from Example 1 above StationaryQ(phi) ```

tacvfARMA function