I've been trying to get comfy with arima() and associated functions
in the ts() package. I'm thinking seriously about using this
package, and R generally, in a 4th year intro time series course that
I'm teaching this autumn.
I have a couple of questions about arima:
(1) The help file says that residuals component of the value returned
by arima() consists of the ``standardized'' residuals. How
standardized? I would have thought that standardized residuals would
equal
(raw residuals) / sigma-hat
where sigma-hat is an estimate of the standard deviation of
the residuals. But if I do
flh <- arima(lh, order = c(3,0,0))
var(flh$residuals)
I get 0.1824616, not (something close to) 1 as I would have expected.
And flh$sigma2 is 0.1786603 --- which is pretty close to the
foregoing value given by var().
Also I note that tsdiag.Arima creates
rs <- object$resid
stdres <- rs/sqrt(object$sigma2)
i.e. it undertakes ``standardization''.
So it looks to me as if the residuals component is NOT
``standardized''? Am I correct?
(2) How is sigma2 calculated? The help says that it is
the MLE of sigma-squared (the variance of the innovations).
Can it be computed directly from the residuals? How?
(Since the whole MLE structure is very complicated, I'm
pretty sure the answer is ``No it can't'', but I'd like
a little more insight into what's going on here.)
(3) I tried
x <- rnorm(100)
fx1 <- arima(x,c(1,0,0),c(1,0,0))
fx2 <- arima(x,c(2,0,0))
Note that seasonality is not specified and so defaults to 1.
I would have thought that the coeficients of the second
model, say phi.21 and phi.22, would be related to the coefficients
of the first (``multiplicative'') model, say phi.1 and Phi.1,
by
phi.21 = phi.1 + Phi.1
phi.22 = -phi.1*Phi.1
(just multiplying out (1 - phi.1*x)(1 - Phi.1*x) and comparing
with 1 - phi.21*x - phi.22*x^2).
But I get
> fx2$coef
ar1 ar2 intercept
0.06465065 0.05896674 -0.10499616
> fx1$coef
ar1 sar1 intercept
0.03339341 0.03339341 -0.10562032
so phi.1 + Phi.1 = 0.06678682 --- close but no cigar, and
-phi.1*Phi.1 = -0.001115120
What am I missing/misunderstanding?
Thanks for any insight.
cheers,
Rolf Turner
rolf at math.unb.ca
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