# Time Series - Is there increasing variance?

I can't figure out if this time plot has increasing variance or not. From my initial guess of the time plot, I would say no but I've taken the difference and it looks like variance is increasing slightly. I've subsequently taken a log transformation and differenced that also where the variance looks to be constant. I'm just wondering is the data variance increasing so much that a log transformation needs to be taken in order to start making the series stationary? I've attached all plots below. , ,

## 1 Answer

My eye tells me (marginally!) yes, but I would prefer actually testing for non-constant error variance via the TSAY test discussed here http://docplayer.net/12080848-Outliers-level-shifts-and-variance-changes-in-time-series.html . Note that the error variance can change at different points in time AS COMPARED to the error variance being proportional to the expected value as is discussed here When (and why) should you take the log of a distribution (of numbers)? .

Note well that untreated anomalies can often incorrectly guide the erroneous conclusion that variability of the error process is non-constant which is why Box and Jenkins incorrectly transformed/logged the Airline series. See a discussion here https://autobox.com/cms/index.php/blog and https://autobox.com/pdfs/vegas_ibf_09a.pdf illustrating the spurious correlarion between error variance and the expected value caused by a few anomalies in the most recent year.

Actually post your original observations and I will be more analytical/objective in my assessment of "constant error variance" .

• It is in fact airline passengers that I am reviewing. I'll have to take a look into the TSAY test as I've never heard of it, I'm still quite new to this subject. I have seen in a lot of examples that the log is taken but the increase in variance is often more pronounced. In my dataset I don't see any outliers later on that would cause the increase in variance. Commented Apr 14, 2020 at 17:39
• perhaps there are none ,,,,but that doesn't imply that there are no deterministic change points in error variance at one or more points in time. I definitely see more time series that have deterministic change points in error variance than those needing a power transform. AUTOBOX which I have helped to develop tests for both in it's tour de force. Commented Apr 14, 2020 at 18:04
• By the way my on-board computer suggests via my eyes that the error variance breakpoint is about at the end of 2013 or so. Why are you studying/examining the airline series ? Commented Apr 14, 2020 at 18:32
• journals.sagepub.com/doi/10.1177/0047287505279003 Chris Chatfield was not too complimentary about using logs for the airline series ... Kudos to my friend and mentor Chris C. Commented Apr 14, 2020 at 18:40