I thought up this method to make data stationary for time series modeling with Arima. Does this method make any sense or is it completely flawed?

For stationary data we need a constant mean and variance.

Step 1: Partition the data into n sets, and calculate the mean and variance on each the n partitions. New mean and variance equals the mean and variance of the first partition.

Step 2: Transform the mean of each partition by subtracting or adding a constant to the set of points so that the new mean will equal the mean of partition 1.

Step 3: Find a scaling factor by setting the variance of the first partition equal to the variance of the second partition. Next, multiply each data point in the second partition by that scaling factor. This should set the variance of partition two to equal partition one.

Step 4: Fit Arima to this transformed data, forecast one time step, and perform the inverse operation of the last partition onto the forecasted value.

If I do this transform on every partition, the mean and variance will all be the same as the first partition. If the time step is small, the transform should be approximatly valid for the new predicted value.

Would this approximation be valid/converge to the true solution as data points and partitions increase, and the time step decreases?

If you think it's valid, why? Why Wouldn't the transform mess up the Arima fit? If not valid, why not? Why does this transform mess up the Arima fit? By how much will this transformation mess up the fit?



1 Answer 1


One way to assess your method is to actually employ it to a series that has actually been identified as having a changing error variance. I suggest that you write a script to test out your suggested procedure and in this way you can answer your own question as to the validity of your suggestion for an example time series. As stated your question is unanswerable by me without actually following your steps.

Here is a monthly series (100 values) that was simulated and then analyzed (without any knowledge of how it had been created )to extract a useful equation.

106.42180000 106.52310000 107.16100000 107.54770000 108.58580000 109.06040000 108.39190000 109.03420000 108.62970000 109.17610000 109.26100000 109.74330000 109.91110000 110.36590000 111.09530000 111.67710000 112.61290000 112.78900000 112.53360000 112.69260000 112.05640000 112.80140000 113.56820000 114.03360000 113.92130000 114.52090000 115.07210000 115.58770000 116.49890000 116.22230000 116.28600000 115.96300000 115.38920000 115.88420000 116.45560000 116.39630000 116.34920000 117.14230000 117.83130000 117.47930000 118.29500000 118.95320000 119.16990000 118.56290000 118.31370000 118.53630000 118.63620000 119.05070000 118.40140000 119.80770000 120.62600000 120.52950000 120.69500000 121.01230000 121.55450000 121.79610000 121.58630000 122.71950000 123.24040000 122.59030000 118.79360000 119.09070000 118.06250000 118.59590000 119.74000000 116.67910000 117.91960000 117.74500000 120.00620000 123.05880000 123.45650000 120.86730000 120.38030000 121.27480000 122.91580000 123.49020000 124.80820000 122.34040000 123.16010000 117.72260000 114.77890000 119.70050000 113.90680000 113.08350000 113.36290000 114.74080000 120.17530000 122.50790000 124.68600000 123.03420000 126.68660000 124.53100000 123.18900000 125.37530000 121.16230000 118.61020000 123.20670000 120.97600000 124.86020000 123.50470000

enter image description here with enter image description hereequation here incorporating a error variance change at or about period 49 enter image description here

The variance change was detected using TSAY's procedure enter image description here . The residual plot is here enter image description here with accompanying acf suggesting sufficiency enter image description here while the acf of the original series is here enter image description here . Finally the Actual , Fit and Forecast is here enter image description here

Please post your results from your suggested approach and compare them with what is presented here. Additionally in another question you might actually simulate your own time series , present it and your results in order to substantiate your approach .

I have added a snapshot of the augmented data matrix (periods 65-100) to depict/illustrate the form of the 5 deterministic series that were identified enter image description here

  • 1
    $\begingroup$ Thank you for the answer. I will try it on your data, give me 1-2 days to get it done. Your post says generalized linear model, but the equation of the model looks like a differenced Arima(5,1,?). What was the model you used to fit the data? If GLM, what type of exponential distribution did you use, and why did you choose it? Also, do you know any documentation I can read on the variance adjusted residual weight procedure mentioned above? $\endgroup$
    – Frank
    Commented Jun 18, 2019 at 13:41
  • 1
    $\begingroup$ the model in pdq notation (0,1,0)(1,0,0)12 is a regularly differenced model with a seasonal autoregessive coefficient AND 5 suporting deterministic input series (3 pulses and two seasonal pulses ) with a deterministic positive variance at period 48. It is a Weighted Least Square Model thus a GLS model . docplayer.net/… contains the variance change detection material. $\endgroup$
    – IrishStat
    Commented Jun 18, 2019 at 13:59
  • $\begingroup$ I realized now that what I have proposed is nothing more than stabilizing the variance with a log transform, subtracting a best fit line, and fitting Arima to the errors. I thought it would be easier to do this in a partition form, after studying more I realized there is no point. Thanks for your help again! $\endgroup$
    – Frank
    Commented Jun 20, 2019 at 0:20

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