R help - Aug 2008 - arma: what is the meaning of Pr(>|t|)?

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.

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)

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

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.
>
> Reproducible test case:
>   x <- c(0.5, sin(1:9))
>   reg <- arma(x, c(1,0))
>   summary(reg)
>   
Not quite. You forgot library(tseries).
> <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)
>
> What am I misunderstanding?
>
>   
That when at least some people say _asymptotic_ they effectively set the
df to Inf and use the normal distribution
> 2*pnorm(-2.446)
[1] 0.0144451

(Personally, I prefer to call the test statistic "z" in such cases,
but
traditions vary. Some use "t" for any quantity divided by its standard
error, others use it only when they intend to refer it to the t
distribution.)

I would expect that the "correct" df are not all that easy to come by;
with correlated data, the asymptotic theory is nontrival, and the exact
distribution of "t" is likely not a t distribution, although it sounds
likely that a t distribution with about 7 df would come closer than the
normal distribution. However, the whole thing is not really designed for
very short time series.


-- 
   O__  ---- Peter Dalgaard             ?ster Farimagsgade 5, Entr.B
  c/ /'_ --- Dept. of Biostatistics     PO Box 2099, 1014 Cph. K
 (*) \(*) -- University of Copenhagen   Denmark      Ph:  (+45) 35327918
~~~~~~~~~~ - (p.dalgaard at biostat.ku.dk)              FAX: (+45) 35327907

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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> 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