Box and Jenkins introduced or rather codified a systematic approach to the identification of the model form. Estimation of non-linear models was well known, primarily due to Marquardt and others. Their straightforward approach to model formulation led to a rather simple model for the AIRLINE SERIES, some 144 monthly observations taken over 12 years time. However, after the actual values for the next 24 months became available, researchers found that their model had over-forecasted, clearly due to the rather expansive nature of log models. Logs had been found to be needed because it appeared that the variance of the series was proportional to the mean of the series, thus the log model decoupled this linkage, so to speak. Click here to view plot of original data.
The method of Box-Jenkins requires the analyst to internally validate/test various assumptions before proceeding to declare the modeling finished and conclude. One critical assumption often stated, but less practiced is that the mean of the errors is not significantly different from zero and is INVARIANT over time. To simply test the former and to assume that the latter assumption follows is a serious flaw. Outlier detection also known as intervention detection expressly looks at this issue. Early researchers were limited by the "expense" of computer time and to be truthful their ability to know the required tests. It wasn't until 1979 that any serious research was done on the issue of the effect and cure for pulses, seasonal pulses, level shifts and local time trends in the residuals and how it contaminated the identification, estimation and forecasting of time series. The statistician's name was I. Chang and she did her work under the direct tutelage of a master statistician, George Tiao who was, by the way, the first Phd student of George Box at Wisconsin.
As you are well aware the test for power transformations (BOX-COX) is not very powerful when you have other non-Gaussian issues. In particular, it is easy to show via simulation and I am sure analytically that unusual values in the series (untreated) cause the Box-Cox test to develop false positives, i.e. conclusions that a power transform has been proven etc.
I used AUTOBOX to analyze the ever-popular 144 observations and after a suitable guess as to the ARIMA structure had been developed the program proceeded to develop the likelihood ratio statistics for alternative hypothesis regarding the assumption that the mean of the errors from this tenative model was uniformly constant through the region. This is the test described by a number of researchers from I.Chang, Bell, Tiao through Tsay in many many places.
After adjusting for one unusual value ( a single pulse ) the conclusion that was developed was that the mean of the errors had a step change at two points in time ( t=54 and t=128 ). This is equivalent to concluding that the mean of the errors had two regime changes.
Subsequently, AUTOBOX tested for a regime change in the variance of an augmented model containing intervention series and concluded that the variance of the errors changed on or around time period 117 and specifically the F test was 2.02 implying that at that point the variance of the errors doubled (2.02). This test was described by Tsay in his JOF article article in 1988 (Vol 7 1-20).
Thus the conclusion is that the variance did change over time, but not in proportion to the mean but structurally. The variance changed by a multiple of two at a specific point in time. We may have sympathy with early researchers in not having access to high speed computers and the inventive work of Tsay and can only reflect that the body of work codified by Box and Jenkins continues to expand and become more robust, meaning more realistic and true to the observed data rather than some simple paradigm handed down as invariant. A level shift in a residuals series from a differenced model is in effect a change in the level of the original series. Changes in level for a series like this is in effect a change in the intercept.
Click here to view model and forecasts. This retrospective analysis of earlier research in the attempt to prove to the naysayers that the body of work by Box-Jenkins deserves to be improved upon not thrown away as some might have us do.
Click here to view plot of data and forecasts.
