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

[Prof Brian Ripley]

On Wed, 20 Aug 2008, Prof Brian Ripley wrote:

> 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).

It is a conditional SSq, so perhaps you meant 'held constant'?

>> 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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>> https://stat.ethz.ch/mailman/listinfo/r-help
>> PLEASE do read the posting guide 
>> http://www.R-project.org/posting-guide.html
>> and provide commented, minimal, self-contained, reproducible code.
>> 
>
> -- 
> 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
>

-- 
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