Time series - Principle of parsimony to choose value of 'd'

[Bill SD]

Hi,
The solution to Q14.3(i) (at end of Chapter 14 Time Series 2, in the 2019 version of material) mentions the two criteria to determine the parameter 'd' -the smallest non-negative integer to consider a series to be stationary: 1) Sample autocorrelations decaying to one; and 2) minimising the sample standard deviation.

The solution concludes that: "If there is any conflict between the two criteria then we should use the principle of parsimony in choosing the value for d."
I understand that parsimony generally means to represent a series with as few parameters as possible but how does it apply here -would one set the value of 'd' to lowest possible value? (ie. if criteria#1 implies d=1 but criteria#2 implies d=2; then should conclude d=1 because of parsimony)

Appreciate this question may have since disappeared from the CMP but will anyway help me to clarify this principle.

 

[Andrew Martin]

Hi Bill

Good question! I'm not actually sure what the 'best' approach is when there is a conflict between these two criteria. If the minimum variance occurs after differencing n times but this differenced series still has a slowly decaying positive SACF, then I think I would be quite skeptical of the possible stationarity of this differenced series.

On the other hand, if the slow positive decay in the SACF is removed after differencing n times but the minimum variance occurs after say n + 1 differences, then I think that I would be more inclined to believe that the series after differencing n times is possibly stationary.

From a practical perspective, we can always plot the differenced series (after differencing various numbers of times) and see whether there appear to be any issues (eg does there still appear to be a non-constant mean?).

An alternative approach that isn't mentioned in the CR is performing hypothesis tests. This is what is used in some functions that perform automatic model selection. An example in R is using the PP.test() function, which is covered in the removing trends section of the PBOR time series chapter. Given that this function is not in the CR, I think I would only use it as a supplement to any exam answer and not use it on its own. Even in practice, if the PP.test() output suggests differencing for example, then we should always check the results by considering the SACF / SPACF and a plot of the series before and after differencing (ie not just use the PP.test() outcome without checking).

In terms of the exam, if this comes up in Paper B, then I think I would primarily be checking plots of the series, the first differenced series and so on as necessary, along with their SACFs / SPACFs to see at what point stationarity appears plausible. The variances could also then be checked to see if this aligns with the graphs. You could then also use PP.test() if you wanted, to supplement your answer. I believe this has come up only once in recent years, in the September 2019 Paper B exam (Q1 part (iv)). In this exam, the examiners' solution only considers plots and the SACF and SPACF, without considering the variances at all. Indeed, the examiners' report also indicates that students using a variance approach were only awarded partial credit.

If this were to appear in a Paper A exam, then obviously we can't plot the different series or use PP.test(). If there were a conflict between the criteria then I think I would focus on whether the slow decay in the SACF appears to have been removed. As I mentioned above, if the slow positive decay in the SACF is removed after differencing n times but the minimum variance occurs after say n + 1 differences, then I think that I would be more inclined to believe that the series after differencing n times is possibly stationary. You can always supplement you answer by talking about how in practice we could plot the data and also perform hypothesis tests for example.

Hope this helps!

Andy