# Seeking certain type of ARIMA explanation

This may be hard to find, but I'd like to read a well-explained ARIMA example that

• uses minimal math

• extends the discussion beyond building a model into using that model to forecast specific cases

• uses graphics as well as numerical results to characterize the fit between forecasted and actual values.

My suggested reading for an intro to ARIMA modelling would be

Applied Time Series Analysis for the Social Sciences 1980 by R McCleary ; R A Hay ; E E Meidinger ; D McDowall

This is aimed at social scientists so the mathematical demands are not too rigorous. Also for shorter treatments I would suggest two Sage Green Books (although they are entirely redundant with the McCleary book),

The Ostrom text is only ARMA modelling and does not discuss forecasting. I don't think they would meet your requirement for graphing forecast error either. I'm sure you could dig up more useful resources by examining questions tagged with time-series on this forum as well.

• The McCleary book is wonderfully written, terse and a really good introduction. There's also some wonderful unintentional humour in the last chapter where they talk about high level languages like Fortran. Commented May 18, 2012 at 10:33

I will try and respond to the gentle urging of whuber to simply “respond to the question” and stay on topic. We are given 144 monthly readings of a series called “The Airline Series” . Box and Jenkins were widely criticized for providing a forecast that was wildly on the high side due to the “explosive nature” of a reverse logged transformation.

Visually we get the impression that the variance of the original series increases with the level of the series suggesting a need for a transformation. However we know that one the requirements for a useful model is that the variance of the “model errors” needs to be homogenous. No assumptions are necessary about the variance of the original series. They are identical if the model is simply a constant i.e. y(t)=u . As https://stats.stackexchange.com/users/2392/probabilityislogic stated so clearly in his response to Advice on explaining heterogeneity / heteroscedasticty “one thing which I always find amusing is this "non-normality of the data" that people worry about. The data does not need to be normally distributed, but the error term does”

Early work in time series often erroneously jumped to conclusions about unwarranted transformations. We will discover here that the remedial transformation for this data is to simply add three indicator dummy series to the ARIMA model reflecting an adjustment for three unusual data points. Following is the plot of the autocorrelation function suggesting a strong autocorrelation at lag 12 (.76) and at lag 1 (.948). Autocorrelations are simply regression coefficients in a model where y is the dependent variable being predicted by a lag of y.

!

The analysis above suggests that one model the first differences of the series and study that “residual series” which is identical to the first differences first for it’s properties.

This analysis reconfirms the idea that a strong seasonal pattern exists in the data that could be remedied or modeled by a model that contained two differencing operators .

This simple double differencing yields a set of residual a.k.a an adjusted series or loosely speaking a transformed series that evidences non-constant variance but the reason for the non-constant variance is the non-constant mean of the residuals.Here is a plot of the doubly differenced series , suggesting three anomalies at the end of the series. The Autocorrelation of this series falsely indicates that “all is well” and there might be a need for any Ma(1) adjustment. Care should be taken as there is a suggestion of anomalies in the data thus the acf is biased downwards. This is known as the “Alice in Wonderland Effect” i.e. accepting the null hypothesis of no evidented structure when that structure is being masked by a violation of one of the assumptions.

We visually detect three unusual points ( 117,135,136)

This step of detecting the outliers is called Intervention Detection and can be easily , or not so easily, programmed following the following the work of Tsay.

If we add three indicators to the model, we get

We can then estimate

And receive a plot of the residuals and the acf

This acf suggests that we add potentially two moving average coefficients to the model . Thus the next estimated model might be.

Yielding

One could then delete the non-significant constant and get a refined model :

We note that no power transformations were needed whatsoever to obtain a set of residuals that constant variance. Note that the forecasts are non-explosive.

In terms of a simple weighted sum , we have: 13 weights ; 3 non-zero and equal to (1.0.1,0.,-1.0)

This material was presented in a way that was non-automatic and consequentially required user interaction in terms of making modeling decisions.

• Hi IrishStat, it's me again. I loved your extensive example, but there are two passages that are a little obscure (at least to me): "The Autocorrelation of this series falsely indicates that “all is well” and there might be a need for any Ma(1) adjustment" and "This acf suggests that we add potentially two moving average coefficients to the model". Exactly what do you see in those ACF plots that makes you believe that? Don't they both look ok (almost all values are within the "blue lines")? Commented Feb 29, 2012 at 13:12
• :VBruder I think I was "wrong" with the statement "there might be a ....." In the second example there is evidence of "bad acf" at lag1 and lag 12 suggesting the potential need for the the t2o ma coefficients. You are over-believing these limits as boyh the acf91) and acf(12) are "dangerously close". You might contact me directly at my published email address available from my info. Commented Feb 29, 2012 at 14:55
• @Adam the three dummy variables play no role in forecasting as future values are all 0. Yes the presented error bounds are incorrect. We have addressed that flaw and now AUTOBOX presents increasing error bounds as time goes on. I am one of the developers of AUTOBOX. . Commented Apr 24, 2012 at 10:45
• @Adam This is equivalent to adjusting or cleansing the three values. They are not pulled out of the data , the values are just modified to reflect what should have been. Commented May 18, 2012 at 0:52
• Your first acf (lag 0)is 1.0 ( as it must be). The autobox acf graphs don't present lag 0. Your acf is for the doubly differenced series while the acf I presented was for the residuals from a reasonable model. Commented Jul 9, 2016 at 22:31

I tried to do that in chapter 7 of my 1998 textbook with Makridakis & Wheelwright. Whether I succeeded or not I'll leave others to judge. You can read some of the chapter online via Amazon (from p311). Search for "ARIMA" in the book to persuade Amazon to show you the relevant pages.

Update: I have a new book which is free and online. The ARIMA chapter is here.

I would recommend Forecasting with Univariate Box - Jenkins Models: Concepts and Cases by Alan Pankratz. This classic book has all the features that you asked for:

• uses minimal math
• extends the discussion beyond building a model into using that model to forecast specific cases
• uses graphics as well as numerical results to characterize the fit between forecasted and actual values.

The only disadvantage is it was printed in 1983 and might not have some recent developments. The publisher is coming with a 2nd edition in Jan 2014 with updates.

• I'd also recommend Alan Pankratz's other book: Forecasting with Dynamic Regression Models. Very similar material, but covers a bit more ground; albeit in less detail on the Box-Jenkins side of things. Great to hear that there's going to be a 2nd edition in Jan 2014! Commented Sep 23, 2013 at 18:15

An ARIMA model is simply a weighted average. It answers the double question;

1. How many period (k )should I use to compute a weighted average

and

1. Precisely what are the k weights

It answers the maiden's prayer to determine how to adjust to previous values ( and previous values ALONE ) in order to project the series ( which is really being caused by unspecified causal variables ) Thus an ARIMA model is a poor man's causal model .

• -1 This reply does not appear to respond to the question, which is looking for a "well-explained ...*example*."
– whuber
Commented Mar 29, 2011 at 22:02
• @whuber: The OP asked for an answer that "uses minimal math". My response detailed minimal math and was motivated to explain ARIMA models in common everyday words. This is never done as the math theory guys focus on the "high-end explanation" using polynomial , differencing operators , non-linear optimization etc. Commented Mar 29, 2011 at 22:59
• @Irish I agree with the motivation to keep the math down, especially when requested by the user. But this reply seems to answer a different question: "what is ARIMA". The specific nature of the original question also indicates the OP has a good idea of what ARIMA is and what it's good for; they want to see it in action. I bet you could easily contribute such a case study :-).
– whuber
Commented Mar 29, 2011 at 23:02
• :whuber: That would have been very easy for me to do and I might just do that. Commented Mar 29, 2011 at 23:15
• @Irish I look forward to seeing it. Moreover--this issue did not come up here, but it has come up in other places--such contributions are potentially more powerful, and more appreciated, ways of letting people know what you can do than many more overt forms of marketing are.
– whuber
Commented Mar 29, 2011 at 23:20