Justifying the trend component in a time series?

Question

I am working on a time series forecasting problem ,I used Dickey–Fuller test to check the stationary of the time series and the p value is 0.000835 , hense I rejected the null hypothesis and assumed that it's a stationary time series.

I decomposed the time series and the trend component does not have any particular pattern.plot-

I want to ask how can we justify that a time series has any particular trend, Is this decomposition enough to ensure and guarantee that this time series has no trend ?

Plot of rolling mean-

Plot of rolling variance-

Answer 1

The Augmented Dickey–Fuller test is UNIT ROOT test - NOT a stationarity test. The null-hypothesis on this test is that the data have been generated by a restricted AR model containing a unit-root. Rejecting this hypothesis DOES NOT mean the series is stationary. It just means that there is enough evidence to allow you rejecting the hypothesis of a very specific form of NON-stationarity (i.e. unit root). Evidence is assessed contrastively i.e. v.s. an unrestricted AR model. Rejecting the hypothesis that an animal is a chicken with an alternative being a pig doesn't mean the animal might not be a horse! The examined animal might contrastively not look like a chicken (e.g. because it has 4 legs) but that doesn't make it a pig.

Now look at your decomposition. You could see with a bare eye data don't look stationary. Trend as well as variance change with time. The decomposition does not support your conclusion. It supports the opposite view. There both parametric as well as non-parametric ways to check changes in the trend of the series. For example (see following poster about Detecting Changes in the Mean ):

Answer 2

visually there is an apparent autoregressive process and a change in model error variance .. thus non-stationarity is the call.