Trend or no trend? I have a quarterly time series and test for stationarity with an augmented Dickey-Fuller test using R.
adf.test(myseries)
# returns
# Dickey-Fuller = -3.9828, Lag order = 4, p-value = 0.01272
# alternative hypothesis: stationary 

so the H0 is rejected. I tried to validate this intuitively and regressed the same series on a linear trend.
x<- 104:1
fit.1<-lm(myseries~x)
summary(fit.1)
#returns
# x      0.024  1.31e-05 ***

Even though a simple linear model is not so appropriate here and the intercept is large (around 80), there seems to be a slight downwards trend over time, which is in line with my thoughts after looking at the initial data. So do I get the adf.test wrong or is the trend just to small to be discovered? 
Besides I used
plot(stl(myseries,"per"))

and ended up with a graph which sidebars suggested that trend and remainder were the main components driving the data, while seasonal influence was negligible. I saw that stl() uses Local Polynomial Regression Filtering and got a rough idea how that works (still I wonder why smoothed trends of Hadley's ggplot2 package looked that different even though it uses the same method by default).
So summing up I got:
- adf finding no evidence for a trend
- a slight downwards trends "detected" by eyeballing and the naive approach
- loess decomposition stating that the trend has strong influence (by the relation of its bars in the plot)
So what can I learned from this? Probably I do have a terminology problem here, because the former two seem to address time trends while the latter address some other trend I cannot fully grasp yet. Maybe my question is just: Can you help me to understand the trend extracted by loess? And how is it related to smoothed / filtered stuff like HP-Filter or Kalman Smoothing (if there is a relationship and similarity does not only occur in my case)?
 A: The answer to your first question is no. If the null hypothesis of unit root is rejected, the alternative in its most general form is stationary series with time trend. Here is the example:
> rr <- 1+0.01*(1:100)+rnorm(100)
> plot(rr)
> adf.test(rr)

    Augmented Dickey-Fuller Test

data:  rr 
Dickey-Fuller = -4.1521, Lag order = 4, p-value = 0.01
alternative hypothesis: stationary 

Message d'avis :
In adf.test(rr) : p-value smaller than printed p-value

So your findings are consistent with ADF test: there is no unit root, but there is a time trend.
A: Larry Bretthorst's extended phd will greatly help you I think.  You should take the discrete fourier transform of the data.  This will give you a look at your series in the frequency domain.  Trend is represented by low frequency.  Ultimate modeling book.  It's 200 pages, but well worth it - includes computer code to implement the methods
A: The ADF test has weak power and, as Dmitrij Celov mentioned, you should probably also check the results of PP and KPSS tests. If you find that your results are on the margin of detecting a unit root, it's possible your series is fractionally integrated. I would also check ACF and PACF plots of the series, looking for slow decay patterns. Generally, if you find that ADF test and Phillips-Perron reject the null of a unit root, but that the KPSS and ACF/PACF plots demonstrate some statistically-significant persistence through several lags, this may be strong evidence for fractional integration.
