Re: [Gretl-users] Standard Errors of OOS ARIMA Forecast?
On Thu, Aug 25, 2011 at 9:21 PM, Allin Cottrell <cottrell(a)wfu.edu> 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.
In the limit, you take the limit of the above summation, which is
finite if the process is stationary, but is in general -- and pretty
obviously -- greater than 1.
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
Which is the same for the first two forecast errors in gretl, but not
afterwards, and indeed it seems to be approaching the standard error
of my series
var(data)^.5
V1
V1 1.704539
Thanks for any insight,
Skipper