I am working with a financial time series (monthly frequency) and the raw data is not stationary according to ADF, KPSS. I then apply deflation (accounting for inflation), log transformation (to make an exponential trend linear) and lastly take the 1st differences. This series is not stationary.

When running the ACF/PACF on the first differences, I receive the following plot:

enter image description here

Which kind of suggests seasonality at 11 and 22 lags (This pattern was not visible before 1st differences). Does this imply I should apply another difference, now with lag 11 and potentially 22 to remove the seasonality?

EDIT: Thanks for the answers. The link to text data is here.


2 Answers 2


The answer is no because you may have injected this phenomenon as a result of transforming the data in an unwarranted fashion ... see the Slutsky Effect where a linear (weighted ) combinations of i.i.d. values leads to a series with auto-correlative structure . Slutsky http://www-history.mcs.st-andrews.ac.uk/Biographies/Slutsky.html Effect ... Unnecesaary differencing can INJECT variability. Consider the variance of a random process that is differenced OR unnecessarily filtered http://mathworld.wolfram.com/Slutzky-YuleEffect.html

Non-stationarity is a symptom with possibly many causes. One cause is a shift in the mean at one or more points in time. Another possible cause is a change in parameters at one or more points in time. Another cause is a deterministic change in error variance at one or more points in time. Prof. Spyros Makridakis wrote an article http://www.insead.edu/facultyresearch/research/doc.cfm?did=46900 of the danger of using differencing to render a series stationary.

When (and why) should you take the log of a distribution (of numbers)? discusses when you should take a power transform i.e. to decouple the relationship between the Expected Value and the variance of the model's residuals.

You may be injecting structure via unwarranted transformations ( differencing is a transformation) .

Simply adjusting for a contemporaneous series (inflation) may be incorrect as the Y variable may be impacted by changes in the X variable or lags of the X variable. This is why we build SARMAX models https://autobox.com/pdfs/SARMAX.pdf.

Why don't you post your original data in a csv format and I and others may be able to help .


I took your 132 monthly values into AUTOBOX ( a piece of software that I have helped to develop ) and automatically developed a useful model . It has a number of advanced features that can be helpful.

Here is the data enter image description here which clearly suggests that as the series gets higher the variability increases. An even "truer" statement is that the variance changes at one point in time (around period 54) and not pervasively suggesting that a Weighted least Squares would be more appropriate than a Log Transform . This will be found via the TSAY test described here https://onlinelibrary.wiley.com/doi/abs/10.1002/for.3980070102 with an excerpt here enter image description here

The TSAY test shown here enter image description here led to a first difference model (nearly second differences as suggested by the ar coefficients nearly summing to 1.0 ) here enter image description here with 9 pulses/shocks and a positive level shift (intercept change) at period 68.

The model in more detail is here enter image description here and here enter image description here

The Actual , Fit and Forecast graph is here enter image description here with MOnte-Carlo generated simulations leading to these forecasts and limits enter image description here

The role of statistics is to separate the data into signal and noise thus the litmus test is "did the equation generate a suitable noise process" . I would say a loud "Yes" .

Here is the plot of the model's residuals enter image description here with this acf enter image description here

In summary a useful model requires that the data be treated for non-constant variance by employing Weighted Least Squares effectively discounting the values 54-132 . The arima model is (2,1,0)(0,0,0)12 with a constant and 1 level shift along with 9 pulses.

It can help to see a segment of the augmented data matrix with the pulses and level shift where the columns represent the latent deterministic structure that was "scraped" from the data . enter image description here

Hope this helps you and the list better ( partially ) understand the extraction of signal from data. No seasonality is detected with the data given .

  • $\begingroup$ Thanks for a really detailed answer. I added a link to data. $\endgroup$
    – abu
    Jun 9, 2019 at 21:03
  • $\begingroup$ Thank you for all the work and references! I will go over them in detail now. $\endgroup$
    – abu
    Jun 10, 2019 at 10:38
  • $\begingroup$ "a Weighted least Squares would be more appropriate than a Log Transform" Does this mean you are using WLS to change multiplicative data into additive data? Do you have a reference for this? $\endgroup$
    – Frank
    Jun 10, 2019 at 23:04
  • $\begingroup$ See onlinelibrary.wiley.com/doi/abs/10.1002/for.3980070102 and in particular this excerpt $\endgroup$
    – IrishStat
    Jun 11, 2019 at 1:44
  • $\begingroup$ see stats.stackexchange.com/questions/412230/… for the excerpt where the w's are developed inversely related to the cahnge in variance . Portion of the data is multlplied by a constant ...the resultant data can then be modeled with an additive model $\endgroup$
    – IrishStat
    Jun 11, 2019 at 1:54

The answer is no, there is no problem of seasonality and autocorrelation here.

ACF and PACF charts use mostly 95% confidence intervals. This means, that typically 5% of values happens to be outside this interval - even when process do not show any autocorrelation or partial autocorrelation. Such things just happen.

Also, seasonal series tend to have different ACF functions - they tend to have form of weaves as you can observe in this question.

  • $\begingroup$ These plots do exhibit precisely those wave patterns you mention, apparently contradicting your conclusion. $\endgroup$
    – whuber
    Jun 9, 2019 at 21:06
  • $\begingroup$ I would not agree, that differentiated series differs significantly from the white noise, which also may be tested. ACF patterns for seasonal data should be much stronger. Besides - it is financial data. It rarely shows any patterns due to arbitration. $\endgroup$
    – cure
    Jun 9, 2019 at 21:18
  • $\begingroup$ How much stronger would "much" stronger be? You seem to arguing in a circular fashion: because you don't expect the series to exhibit seasonality, you cannot agree that the PACF and ACF show evidence of seasonality! $\endgroup$
    – whuber
    Jun 9, 2019 at 21:21
  • $\begingroup$ Significantly different from white noise. $\endgroup$
    – cure
    Jun 9, 2019 at 21:42

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