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I am somewhat a beginner to time series, so please bear with me. I was taught in class that when you difference a series too much times, the variance will increase, but I am also taught that I should difference the series at lag 1 until the mean of the series on the plot is a horizontal line. Before anyone asks, my data is not seasonal, and I have already Box-Cox transformed it. The series at this point does NOT have constant variance, but the scope of the course does not cover transformations to make variance constant, so that's all I can do in terms of transformations.

Then, I differenced my series once at lag 1 and got an increasing mean line in the plot, so I differenced again at lag 1. This time, I got an almost horizontal line, but my variance more than doubled the variance of differencing only once. What's more important: almost 0 mean or lower variance? Also, is there anything else I can try?

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To begin with sometimes outliers are present when the series is high incorrectly suggesting a box-cox transformation. Transformations ala box -coxhttp://stats.stackexchange.com/questions/18844/when-and-why-to-take-the-log-of-a-distribution-of-numbers can be helpful and even hurtful. If The variance changes deterministically ( at fixed points ) in time consider Tsay's suggestion here http://docplayer.net/12080848-Outliers-level-shifts-and-variance-changes-in-time-series.html . If the variance of the errors change stochastically then and only then consider GARCH.

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  • $\begingroup$ Hmm I'm not sure what counts as an outlier in a time series, but my data only has 100 observations. I didn't see anything too out of the ordinary while going through all the observations. Also, I'm sorry, but the information that you're giving to me is just too much beyond my scope. I guess my question is: when differencing gives a near 0 mean (stationarity) but increases variance, should I keep the differencing? $\endgroup$ – mistersunnyd Nov 23 '17 at 2:53
  • $\begingroup$ if differencing increases the variance my intuition would be NOT to difference. Please post your 100 observations and I will try and help further. $\endgroup$ – IrishStat Nov 23 '17 at 11:09
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The purpose of "differencing the series at lag 1" is to make the series weakly stationary (with time-invariant mean and variance) by removing stochastic trend caused by unit root in the series. In other words, constant mean and variance is the important output that you should expect from this transformation. On the other hand, the magnitude of mean and variance can be changed by shifting and/or scaling.

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