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I removed trends and seasons from given time series and plotted the residual time series. I would like to know if there is any way that this plot could suggest that residual series is stationary? What are the key things I should look in time series plot to infer stationarity?

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From everywhere: A time series can be said to be stationary when the mean , standard deviation and auto-correlation is the same for all sub-intervals of time. If you have Pulses,Seasonal Pulses, Level Shifts or Local Time Trends this would be a violation of the stationarity of the mean. If the standard deviation changes over time for example dependent on the mean then this would be a violation of the assumption of a constant standard deviation. If the auto-correlation function changes over time then this might be an indication of time varying parameters BUT it could have other causes. If your visually descriptive plot confirms all of these issues then you are good to go otherwise you need to get involved with time series analysis and do some inferential analysis..

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If you removed the trend and seasonality then the remaining of the series must be stationary at least in mean. This means that the main thing to look at is heteroscedasticity, which can be tested with Engle's ARCH test and such

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  • $\begingroup$ Not necessarily as there may be level shifts OR simple Pulses which are not remedied by detrending. Before one over-complicates the solution it is always good practice to provide adjustments/transformations that are minimally sufficient much like a doctor providing an aspirin when appropriate and not recommending surgery for a headache. Furthermore if there is variance change it may be deterministic not stochastic. $\endgroup$ – IrishStat Feb 9 '15 at 21:08
  • $\begingroup$ It depends on de-trending. I often use filters, and they handle level shifts naturally. Pulses would be showing up as volatility. $\endgroup$ – Aksakal Feb 9 '15 at 21:10
  • $\begingroup$ pulses are not volatility as they are one-time events impacting the mean at that point. Untreated they incorrectly cause you to think that there is a variance change. $\endgroup$ – IrishStat Feb 9 '15 at 21:12
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    $\begingroup$ One time mean change is hard to interpret as anything but volatility, i'm afraid. $\endgroup$ – Aksakal Feb 9 '15 at 21:17
  • $\begingroup$ Volatility ( to my way of thinking ) represents the second momemt i.e.variability or dispersion . The pulse is a 1 period change in the mean and that's all , the level shift is a change in the mean that lasts for k periods, The seasonal pulse is an instantaneous change in the mean for observations s periods apart, the trend is an increase in the mean at a fixed rate for k periods. $\endgroup$ – IrishStat Feb 9 '15 at 21:29

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