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 > > ______________________________________________ > R-help at r-project.org mailing list > 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