[R] arma: what is the meaning of Pr(>|t|)?

[Prof Brian Ripley]

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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>

-- 
Brian D. Ripley,                  ripley at stats.ox.ac.uk
Professor of Applied Statistics,  http://www.stats.ox.ac.uk/~ripley/
University of Oxford,             Tel:  +44 1865 272861 (self)
1 South Parks Road,                     +44 1865 272866 (PA)
Oxford OX1 3TG, UK                Fax:  +44 1865 272595