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My data is monthly CPI(https://fred.stlouisfed.org/series/CPIAUCNS). Here is the original plot: enter image description here

There is an exponential trend here, but variance isn't necessarily increasing with the trend. I tried a log transform, and here's what I got: enter image description here

The first 1/3 of the plot's variance increased, but the exponential trend is gone; however, neither the original plot nor the log transformed plot are normal. I've only learned log and Box-Cox transforms, so they're the only ones that I can use. Should I be transforming my data here? This data is the only variable that I'm using in my forecast.

mistersunnyd
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  • 1. You probably shouldn't do anything until your goals are clear? What are you trying to do with the series? 2. There's no point in considering normality when the mean is changing; you have different distributions at each time point. – Glen_b Nov 25 '17 at 22:46
  • 1. Sorry for not being clear. I'm trying to forecast using this series. – mistersunnyd Nov 25 '17 at 22:46
  • transformations are based upon the relationship between a model's errors and the expected value ...see https://stats.stackexchange.com/questions/18844/when-and-why-should-you-take-the-log-of-a-distribution-of-numbers for more – IrishStat Nov 25 '17 at 22:38
  • Thank you for answering, so are you saying that I should first find a model with the raw data, then look at the residuals, and then transform? Also, in your link, the answer stated that logging a non-linear model to make it linear would be justified (true for me), but it also stated that logging is only justified when variance increases with mean(not true for me). I'm a bit confused. – mistersunnyd Nov 25 '17 at 22:46
  • You should edit your question to include such information. Often with CPI people would look at modelling differences of the logs, though for some CPI sub series there may also be seasonality. Are you interested in forecasting only from the past history of this variable or are you planning to use other variables to help forecast it? – Glen_b Nov 25 '17 at 23:59
  • I am only using this variable to forecast. – mistersunnyd Nov 26 '17 at 00:18
  • Again you should edit your question to include that information. – Glen_b Nov 26 '17 at 00:39

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You should not try and forecast using this series as there are several regime changes in this series. You could model using breaks, but the wisdom of doing this is questionable. The reason is that the CPI is a consequence of things other than the CPI.

I am assuming from the nature of your question that you are not a professional economist. If you were, my answer would be very different. For a non-economist, I would suggest taking the problem into log space and presume the existence structural breaks. I would allow for up to ten and use either the AIC or BIC to choose among them. There are probably either five or six monetary regime changes depending on how you count the world post-2008. Your forecast would come from the last break only.

The issue here is that the result will be biased in a number of ways. First, the variability in other regimes is totally lost to the last break and yet this is part of the natural volatility and when making a prediction you do not know what national crisis may happen that could cause a new change. Second, least squares style regressions are biased in the sense that while unbiased in log-space they are biased in the raw data and the bias isn't actually small. Third, the likelihood function may be changing between regimes due to the possibility that the CPI is systematically shrinking in some sequences. Deflation would have a very different likelihood function than the one present under inflation. The likelihood would matter both because you would get fatter tails in inflation than deflation and because you are really seeing a large change in models.

If you are an undergraduate student, I would recommend starting the model in January 2009 as there are visually obvious changes in the monetary system as seen in the time series. I would borrow a copy of Friedman and Schwartz's A Monetary History of the United States 1867-1960 and read it. I would then use it with subsequent articles to justify starting the series at 2009. The post-1960 period is well documented.

Dave Harris
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    Thank you for answering. I'm actually doing this for a time series course not for economics, and the goal is to just fit a model and forecast. As long as the results are statistically sound, that is fine. – mistersunnyd Nov 26 '17 at 02:09
  • @mistersuunyd undergraduate, masters or doctoral – Dave Harris Nov 26 '17 at 02:25
  • I am an undergrad student. – mistersunnyd Nov 26 '17 at 02:30
  • Then you should learn a caution here. Just because a time series exists does not mean you should regress on it. The CPI is a highly refined measurement of a broader set of measurements. The broader set of measurements, as a whole, measure inflation and deflation. – Dave Harris Nov 26 '17 at 04:19
  • This is a problem because inflation and deflation have, at times, been planned. At other times, it is due to errors of estimation by banks as to the needed amount of money. And, at yet other times, it has been due to errors in thinking and planning by the national government. In addition, the central bank has planned a change in the rate of inflation rather than target the level itself. In the early period, it was designed to have a sinusoidal process in it. CPI is more like a sequence of functions pasted together. – Dave Harris Nov 26 '17 at 04:19
  • Finally, there are two sources of sinusoidal error, outside of the planned element from using a raw gold standard. The first is from actual seasonality. By seasonality, I mean summer, fall, winter, and spring. The series began when 80% of the economy was agriculture and it finishes with 2%. Seasons matter in terms of output. The second is the effect of additive errors. Additive errors will impose a sinusoidal process on its own. It may appear to be seasonality, but it is not. – Dave Harris Nov 26 '17 at 04:24