On Wed, 20 Aug 2008, Alberto Monteiro wrote:
In the summary of the output of arma, there's a number Pr(>|t|), however, I
don't know what is its meaning - at least, it doesn't _seem_ to be a
Student's t distribution.
It is using asymptotic normality. There is no exact theory. Who
mentioned Student's t?
Reproducible test case:
x <- c(0.5, sin(1:9))
reg <- arma(x, c(1,0))
summary(reg)
<output>
Call:
arma(x = x, order = c(1, 0))
Model:
ARMA(1,0)
Residuals:
Min 1Q Median 3Q Max
-0.9217 -0.4915 0.2254 0.4580 0.7481
Coefficient(s):
Estimate Std. Error t value Pr(>|t|)
ar1 0.6089 0.2490 2.446 0.0145 *
intercept 0.0790 0.1815 0.435 0.6634
---
Signif. codes: 0 ?***? 0.001 ?**? 0.01 ?*? 0.05 ?.? 0.1 ? ? 1
Fit:
sigma^2 estimated as 0.3348, Conditional Sum-of-Squares = 2.68, AIC = 21.44
</output>
Now, 2.446 is 0.6089 / 0.2490, but 0.0145 is not
2 * (1 - pt(2.446, df = 7))
(I think there are seven degrees of freedom: the first value of
the series x is deterministic, and two degrees are lost in the
estimation of ar1 and intercept)
Why is the first value deterministic? This is not a conditional mle (see
the help page).
What am I misunderstanding?
BTW, a similar example:
x <- 1:10
y <- sin(x)
reg <- lm(y ~ x)
summary(reg)
will give a t-value for 'x' = 0.704 with P(>|t|) = 0.501,
which is 2 * (1 - pt(0.704, df=8))
Alberto Monteiro
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