# Given this time series, what statistical methods would be used for description and forecasting?

These static cumulative default rate tables and charts come from this public report published by a credit rating agency.

Basically, you take all the loans originated in a period of time (a "vintage") and track the cumulative amount of money due but not repayed by the borrowers as a percentage of the amount of money lent (the "cumulative default rate") for each period of time.

Each curve in the chart (each vintage) shows the behaviour of all the credits given by the bank during a given period of time (say, a month or a quarter). This mean:

• Observations included in all the curves are originated by the same bank using the same rules, therefore they share common characteristics and similar behaviour.
• The dataset analyzed includes all the loans originated by the bank. The observations include the whole population, they do not come from a sampling process.

Some visualization:

The dataset

Vintages Amount 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
1Q01 612,000 0.10 0.57 1.13 1.89 2.74 3.34 3.61 4.06 4.36 4.66 4.73 4.83 5.08 5.18 5.27 5.38 5.39 5.41 5.43 5.48 5.58
2Q01 700,000 0.09 0.64 0.99 1.77 2.58 3.03 3.46 3.64 4.01 4.28 4.32 4.47 4.73 4.82 4.97 5.05 5.07 5.10 5.23 5.32
3Q01 732,490 0.13 0.70 1.53 2.08 2.60 3.08 3.28 3.75 3.82 4.32 4.37 4.57 4.70 4.92 4.98 4.99 5.07 5.12 5.14
4Q01 793,984 0.23 0.85 1.73 2.19 2.72 3.31 3.79 3.96 4.40 4.66 4.78 4.92 5.13 5.21 5.37 5.38 5.49 5.57
1Q02 834,563 0.22 0.73 1.28 1.89 2.77 3.11 3.45 3.82 3.91 4.37 4.45 4.63 4.85 5.08 5.11 5.15 5.16
2Q02 904,485 0.14 0.65 1.45 2.17 2.72 3.20 3.81 4.08 4.38 4.43 4.54 4.91 5.08 5.11 5.15 5.18
3Q02 953,534 0.10 0.71 1.42 1.98 2.55 3.14 3.71 4.15 4.42 4.67 4.73 4.89 4.95 5.03 5.24
4Q02 980,124 0.13 0.64 1.13 1.61 1.98 2.23 2.71 3.03 3.30 3.68 3.77 3.82 3.87 4.08
1Q03 1,202,324 0.18 0.69 1.14 1.95 2.42 2.56 2.98 3.15 3.67 3.68 3.76 4.00 4.10
2Q03 1,014,485 0.16 0.59 1.39 1.90 2.23 2.58 2.92 3.46 3.68 3.74 3.87 3.99
3Q03 1,171,731 0.15 0.60 1.18 1.66 1.97 2.45 3.11 3.15 3.41 3.50 3.81
4Q03 1,312,338 0.13 0.61 1.13 1.44 2.11 2.33 2.85 2.87 3.01 3.05
1Q04 1,509,189 0.16 0.59 1.14 1.45 1.75 1.99 2.36 2.47 2.63
2Q04 1,569,557 0.13 0.58 1.09 1.46 1.85 2.19 2.28 2.44
3Q04 1,545,922 0.14 0.58 0.99 1.39 1.72 1.97 2.18
4Q04 1,514,485 0.18 0.63 0.94 1.46 1.81 1.84
1Q05 1,574,685 0.16 0.55 1.04 1.45 1.72
2Q05 1,578,485 0.10 0.38 0.74 0.90
3Q05 1,501,234 0.06 0.44 0.64
4Q05 1,472,187 0.07 0.41
1Q06 1,459,861 0.08


The chart

For better visualization, I add a chart from another dataset with more observations.

Since I'm not an expert in statistics, I'd like to know what kind of statistical methods I should study to be able to a) describe the behaviour of each vintage (for example, indicators to compare new credits versus old credits), and b) forecast the final cumulative default rate (the default rate at the end of the life of the credits) for the youngest vintages.

I know the I've only included very little information about the case, but I'm willing to do my research on any statistical method you would suggest as possibly useful.

• I think question is more appropriate for the Quantitative Finance StackExchange forum rather than this one. Jan 1, 2014 at 22:45
• @rocinante I'll wait and see if somebody has any suggestion about this, otherwise I'll try again on quantitative finance. Ty! Jan 2, 2014 at 0:23
• Fair enough. For what it's worth, question (a) and question (b) are two entirely different things and can't really be summarized in an answer because entire books are written on the subject. You don't need statistical methods as much as you need a book on credit instruments. If you have not done so already, get some books on credit instruments. The Frank Fabozzi series and the Wiley Finance series are pretty good. Jan 2, 2014 at 0:56

This 'triangular' data structure (where a new 'diagonal' can be observed each time period) occurs in many contexts.

Some examples

• Age-period-cohort data is of this form. A few links:

A report by Bendix Carstensen is probably a decent place to read about these models:

http://publichealth.ku.dk/sections/biostatistics/reports/2006/ResearchReport06-1.pdf/

He has a 2007 paper as well. I'll try to find the reference.

http://www.stata-journal.com/sjpdf.html?articlenum=st0211

Kupper, Janis, Karmous, & Greenberg (1985),
Statistical age-period-cohort analysis: a review and critique.
J Chronic Dis. 1985;38(10):811-30.
http://www.ncbi.nlm.nih.gov/pubmed/4044767

• Davison and Hinkley (1997) analyze the AIDS reporting data set from De Angelis and Gilks (1994), which is count data of this general form.

De Angelis, D. and Gilks, W.R. (1994) Estimating acquired immune deficiency syndrome accounting for reporting delay. Journal of the Royal Statistical Society, A, 157, 31–40.

Davison, A.C. and Hinkley, D.V. (1997) Bootstrap Methods and Their Application. Cambridge University Press.

• Claims- (Loss-) reserving data is of this form; there are many articles on the various kinds of analysis used for this data (though I regard many of the more popular approaches as badly misguided for several reasons)

The tendency to cumulate the data is common in this application; unfortunately the consequences of the particular kinds of dependence this induces are often ignored, and the impact this has on the ability to identify calendar-period changes is also often ignored.

• Bond yield curve data is of similar form

(there are numerous other kinds of data that come in this form or an equivalent form.)

I suggest you be cautious when analyzing the cumulated data - it induces dependence along the vintages and obscures potential calendar-period effects that may be very important; on the other hand, if analyzed as a proportion of the origination amount, the variance structure may require you to deal with it as cumulative.

One possibility is taking it back to the per-period data as dollar amounts with origination amount as exposure.

they do not come from a sampling process.

This is a common issue, especially, with time series data - but you can regard it as a sample path from a process about which you wish to make inference. The data can be seen as resulting from a probabilistic process which (if the history were re-run) would have turned out differently.

• @rocinante It begins by discussing the particular structure of the data, which many people are not especially familiar with, relating it to problems which they may have heard of. Such a structure presents particular issues in the analysis which are common to all of those areas (such as, for example, the fact that there are three 'time' directions, but such that any two of them determine the third). This has been (to my mind) most clearly elucidated in the age-period-cohort framework but it applies to all the application areas I listed. ...(ctd) Jan 2, 2014 at 5:28
• (ctd) ... ANY attempt to model or even describe these data needs to (explicitly or implicitly) be aware of, and preferably take this structure into account; failure to do so can lead to difficulties which are already well understood and have some good approaches to solutions. Your argument would be like claiming that a bank can't use a Poisson process to model the number of loan defaults because biostatisticians use it. I've made no particular attempt to introduce any particular methods yet, but any of the areas I have raised do use methods that may be relevant ... (ctd) Jan 2, 2014 at 5:30
• (ctd) ... to the analysis of this data; to exclude ideas from consideration because the application area is different would be ludicrous. As for the charge that it's not a complete answer to the question -- indeed it isn't (yet). Jan 2, 2014 at 5:34
• @rocinante "the repeated editing of your comment to alter your answer ... against forum rules". Uh... you get a few minutes to edit for a good reason. If me editing my comments to clarify my intent bothers you, perhaps you can hold off responding until those (very) few minutes elapse, and respond at the end of that time. It's not like I'm throwing insults and then deleting it or something. If you can point to a rule that is broken by editing a comment, I'd love to read it. Jan 2, 2014 at 5:37
• @rocinante as my answer currently stands, it's intended as a partial response to "I'd like to know what kind of statistical methods I should study", to which I so far give some pointers. I think it does respond to that part of the question even though it's not a complete answer to the entire question. Jan 2, 2014 at 5:47