I'm estimating an ARMA(2,2) model from generated data using gretl's exact likelihood. When I forecast 10 periods out from the end of the sample, the first 2 forecast standard errors are as I would expect. However, after this the standard error of the prediction is larger than the standard error of my series. I was under the impression that this cannot be.

On Thu, Aug 25, 2011 at 9:21 PM, Allin Cottrell <cottrell(a)wfu.edu&gt; wrote: > > On Thu, 25 Aug 2011, Skipper Seabold wrote: > >> I'm estimating an ARMA(2,2) model from generated data using gretl's >> exact likelihood. When I forecast 10 periods out from the end of the >> sample, the first 2 forecast standard errors are as I would expect. >> However, after this the standard error of the prediction is larger >> than the standard error of my series. I was under the impression that >> this cannot be. > > "This cannot be" only for a one-step ahead forecast. For an h-step > ahead forecast the error variance is \sigma^2 for the series times > (1 + \psi_1^2 + \psi_2^2 + ... + \psi_{h-1}^2) where the \psi's > are the coefficients of the infinite MA representation. [...] Thanks Allin, I realize all of the above, and it's how I came to ask the question. But I guess I'm still misunderstanding or being dense.

Consider the following R code (apologies, I'm still mainly a point and click guy with gretl) data <- read.csv('y_arma.csv', header=FALSE) mod <- arima(data$V1, order=c(2,0,2), include.mean=FALSE) ma_rep <- ARMAtoMA(mod$coef[1:2],mod$coef[3:4], lag.max=10) ma_rep <- c(1, ma_rep) serr <- sqrt(mod$sigma2 * cumsum(ma_rep^2)) print(serr) [1] 1.054555 1.448353 1.494564 1.634206 1.693062 1.699978 1.700106 1.701582 [9] 1.702436 1.702578 1.702578

On Fri, 26 Aug 2011, Allin Cottrell wrote: > I'll do some more checking tomorrow to confirm that the fix is > right, but thanks for spotting this! Confirmed, the fix looks ok to me. I'm attaching a zip file containing Skipper's dataset and a small script performing the relevant computations.