Forecasting Behavior with Age- Period-Cohort...
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Forecasting Behavior with Age- Period-Cohort Models How APC Predicted the US Mortgage Crisis, but Also Does So Much More - 2700
Transcript of Forecasting Behavior with Age- Period-Cohort...
Forecasting Behavior with Age-Period-Cohort ModelsHow APC Predicted the US Mortgage Crisis, but Also Does So Much More - 2700
About the presenterDr. Breeden has been designing and deploying forecasting systems since 1994. He co-founded Strategic Analytics in 1999, where he led the design of advanced analytic solutions including the invention of Dual-time Dynamics. He currently runs Prescient Models, which focuses on portfolio and account-level forecasting solutions for lifetime value assessment, account management, and stress testing.
Dr. Breeden has created models through the 1995 Mexican Peso Crisis, the 1997 Asian Economic Crisis, the 2001 Global Recession, the 2003 Hong Kong SARS Recession, and the 2007-2009 US Mortgage Crisis and Global Financial Crisis. These crises have provided Dr. Breeden with a rare perspective on crisis management and the analytics needs of executives for strategic decision-making.
He has published over 40 academic articles and 6 patents. The second edition of his book “Reinventing Retail Lending Analytics: Forecasting, Stress Testing, Capital, and Scoring for a World of Crises” was published by Riskbooks in 2014.
Dr. Breeden received separate BS degrees in mathematics and physics in 1987 from Indiana University. He earned a Ph.D. in physics in 1991 from the University of Illinois studying real-world applications of chaos theory and genetic algorithms.
Talk Outline What are Age-Period-Cohort models?
What’s the hidden story of the US Mortgage Crisis?
Where else do APC models apply?
Are there implementation details we need to know?
What are Age-Period-Cohort models?Definitions
Lexis Diagrams Between 1869 and 1875, Zuener,
Brasche, Becker and Lexis invented a way to look at mortality rates by separating the data into year of birth (cohort) and year of death (period). Age is of course the age at death (period –cohort).
In the 1960s and 70s, a rich literature developed in demography and epidemiology on how to estimate functions of age, period, and cohort using aggregate data like in the Lexis diagram.
Age Period Cohort (APC) Models Given an origination date (vintage), the age of the account is
calendar date – vintage date, .
Functions of age , vintage , and time are then estimated. For binomially distributed events, this is
The functions are most commonly estimated parametrically via splines or nonparametrically. Generalized Linear Model (GLM) estimation for splines, and Bayesian, Partial Least Squares, or Ridge Regression are used for the nonparametric functions.
logp(a,v, t)
1- p(a,v, t)
æ
èç
ö
ø÷= F(a)+G(v)+H (t)
F(a) G(v) H (t)
a = t -v
What’s the hidden story of the US Mortgage Crisis?
Use Case
Joint research with José Canals-Cerdá, Federal Reserve Bank of
Philadelphia, [email protected].
Age-Period-Cohort Decomposition
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0 12 24 36 48 60 72 84 96 108 120
60-8
9 D
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Rate
Superprime Prime Subprime
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tiv
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Vintage Date
Superprime Prime Subprime
F(a)
G(v)
H(t) Environment by state
Credit Quality by FICO
Lifecycle by FICO
logp(a,v, t)
1- p(a,v, t)
æ
èç
ö
ø÷ = F(a)+G(v)+H (t)
Bayesian
APC Estimation
Adjusting for the Environment The environment function can reveal impacts for which no predictive
factors are available.
The example below shows Hurricane Katrina’s impact on mortgage delinquency. Later results are normalized any environmental impacts, like those shown here.
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CA FL LA MS TX
Loan-level Modeling We use an APC decomposition initial step so that we capture all of
the lifecycle and environment variation, and so that we control the linear trend ambiguity in age, vintage, time models. (More on this later.)
Then we keep the lifecycle and environment as fixed offsets in a GLM score using quarterly performance data.
We include typical origination scoring factors and then test for the inclusion of vintage fixed effects (dummy variables) .
logpi(a,v, t)
1- pi(a,v, t)
æ
èç
ö
ø÷= offset F(a)+H (t)( ) + c0 + c jxij
j=1
ns
å + gvv=1
nv
å
gv
Scoring Factors
The origination scoring factors and coefficients were typical for 1st lien, fixed rate mortgages.
We excluded all of the exotic products, e.g. Option-ARMs. We’re analyzing traditional mortgage products.
Table4:OutputCoefficientsfromtheGLMAnalysisofMortgageDelinquency.
variables Coef. t-val Variables(cont.) Coef. t-val
Intercept 2.268 67.83 SourceChannel
JumboLoan -0.128 -16.57 source1 0
Documentation
source2 0.217 63.27
Fulldocumentation 0
source7 0.100 30.25
Lowdocumentation 0.103 25.48 sourceT 0.240 38.63
Nodocumentation -0.030 -5.16 sourceU 0.437 32.19
Documentationunknown 0.135 40.61 Occupancy
FicoatOrigination
Owner 0
upto540 0
Non-owner -0.109 -17.75
540to580 -0.188 -17.33 OtherorUnknown -0.187 -17.84
580to620 -0.435 -44.66 PMI
620to660 -0.807 -85.86 No 0
660to700 -1.373 -145.22 Yes 0.119 32.06
700to740 -1.956 -202.66 Unknown 0.170 37.92
740to780 -2.671 -264.46 Term
780to820 -3.380 -283.96 0to120 0
820+ -3.623 -52.71 120to180 0.144 10.39
LoantoValue
180to240 0.407 27.38
0to0.75 0
240to360 0.593 44.05
0.75to0.8 0.157 40.09 360+ 0.640 36.25
0.8to0.85 0.221 46.80 Purpose
0.85to0.9 0.247 42.01 Purchase 0
0.9to0.95 0.262 42.53 Refinance -0.001 -0.41
0.95to1 0.305 46.79 purposeU -0.462 -64.29
1to1.13 0.285 36.81 purposeZ 0.090 5.08
DTI 0.007 73.35
Note:Themodelspecificationincludesalsoquarterlyvintagedummiesthatarenotexplicitlyreportedinthistable.
APC Vintage Function versus Scoring Vintage Dummies
The APC vintage function measures the net impact to log-odds of default for the variation in credit risk by vintage.
The scoring vintage dummies measure the vintage residual after scoring factors.
By comparing, we see that only half of the vintage variation is explainable by scoring factors.
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Vintage Date
AVT Vintage Function Origination Score Vintage Fixed Effects
FRB SLOOS Survey – Underwriting Standards The Federal Reserve publishes a Senior Loan Officer Opinion
Survey (SLOOS).
Self-reported changes in underwriting standards show a correlation of ρ =0.4 ± 0.4 to the vintage fixed effects, with the wrong sign.
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Vintage Date
Net Percentage of Domestic Respondents Tightening Standards for Mortgage Loans* All
Net Percentage of Domestic Respondents Tightening Standards for Mortgage Loans* Prime
Score Vintage Fixed Effects
FRB SLOOS Survey – Consumer Loan Demand The same senior loan officers in the same survey report on consumer
demand for loans.
Consumer demand has a correlation of ρ = -0.7 ± 0.3 to the vintage effects.
When consumer demand is high, the loans are good – consumer risk appetite is a real driver of credit risk.
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Net Percentage of Domestic Respondents Reporting Stronger Demand for Mortgage Loans* All
Net Percentage of Domestic Respondents Reporting Stronger Demand for Mortgage Loans* Prime
Score Vintage Fixed Effects
Drivers of Consumer Demand Historically, SLOOS-reported consumer mortgage demand is highly
correlated to the 24-month change in mortgage interest rates.
When offered interests fall over a sustained period, consumer demand rises.
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SLOOS Mortgage Demand 30yr Mortgage Intere Rate, 24m log-ratio
Where else do APC models apply?Applications
Other Applications of APC Models Wine Forecasting
SETI@home
Dendrochronology
eCommerce customer lifetime value
HR
Sales
Fine Wine Value Forecasting Auctionforecast.com – Vincast
1.5 million auction prices over a 10 year period
Predicts wine price, starting with a Bayesian APC decomposition.
Wine is perfect for “vintage” analysis.
Vincast Decomposition – Chateau Lafite Rothschild
Prices actually drop
through the first 5
years from release.
The “Lafite Bubble”,
caused by a flurry of
interest from Chinese
investors.
Some vintages are
special.
Drivers of the Wine Market The environment function (market index) for auctions prices has
tracked Chinese wealth for the last decade.
SETI@home Member Analysis SETI@home is the Search for Extraterrestrial Intelligence signal
detection project.
We analyzed 2 years of member data, analyzing activity rates, number of cpus applied, and analysis return rates. Leading to member lifetime value estimates segmented by Country, OS, and software version.
As an example of the lifetime value analysis, we show load, equivalent to usage for a website or spend on a credit card.
Activity Decomposition activity(a,v, t) =activeaccounts(a,v, t)
activeaccounts(a = 0,v, t)
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Member Net Present Value Combining all factors, we predicted NPV for new members, allowing
the SETI@home managers to target their development by OS and country.
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co
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ala
nd
Pola
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ssia
n …
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Dendrochronology – Tree Ring Analysis Tree ring growth rates are driven by age of the tree, environmental
conditions, and growth conditions for the individual tree.
Species Number of Trees
Abies concolor (Gordon) Lindl. ex Hildebr 302
Abies lasiocarpa (Hook.) Nutt 3652
Abies magnifica A. Murray 241
Castanea dentata (Marsh.) Borkh 177
Juniperus occidentalis Hook. 1164
Juniperus virginiana L 630
Picea engelmannii Parry ex Engelm 1562
Pinus aristata Engelm 256
Pinus edulis Engelm 2075
Pinus ponderosa Douglas ex C. Lawson 4998
Pseudotsuga menziesii (Mirb.) Franco Quercus alba L 5324
Quercus alba L 2690
Tsuga canadensis (L.) Carr 1538
Tsuga mertensiana (Bong.) Carriere 1027
Tree Ring Decomposition – Lifecycles
log(w) = F(a)+G(v)+H(t)
Growth rate patterns
differ by species.
Tree Ring Growth Lifecycle Phases Tree ring growth goes through different phases. Nonparametric
estimation allows us to discover this.
Tree Ring Decomposition – Environment The environment functions for New Mexico and Arizona show
similar patterns through the centuries.
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Environmental Correlations Recent ring growth correlates well to rainfall, until the 1980s.
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Extrapolated NM Rainfall Series New Mexico Annual Precipitaiton
Climate change?
Predicting Historic Rainfall Levels Backward-extrapolating the rainfall correlation shows prehistoric
droughts, possibly explaining the fall of the Anasazi civilization.
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eCommerce, HR, Sales, etc. Any data set where a vintage can be defined and performance
tracked with time can be analyzed via APC
eCommerce: Usage or sales after initial registration. How is this driven by website design changes? Do night, workday, or weekend registrants have different lifetime value? Do new signup discounts hurt or help lifetime value?
HR: Employee attrition. Are employees stickier during recessions? Do hires during certain periods or policies stay longer? How do company policy changes affect retention?
Sales staff: Are new sales staff on track? How have product or pricing changes affected sales performance? Who are the true top performers adjusting for all this?
Store sales, etc. etc. etc.: Just look for the vintage…
Are there implementation details we need to know?
Implementation
Implementation Details Link Functions: Choose the correct distribution to match your data.
Estimators are readily available for binomial, Gaussian, and lognormal.
Estimation technique:
Spline estimation is available via proc transreg.
Bayesian estimation is available via proc genmod.
Partial least squares estimation is available via proc pls.
Ridge regression estimation is available via proc reg (w/ridge=
option).
Cross-terms: Test the decomposition to see if the age, vintage, and time functions are independent. If they are not, try segmentation.
Linear Trend Ambiguity Because a=t-v, there is an ambiguity in the allocation of linear trends.
Consider
These functions are measured on a discrete number of months. Therefore, they can be represented precisely with a polynomial through those points.
We can separate the constant, linear, and nonlinear terms as
lnr(a,v, t) = F a( ) +G v( )+H t( )+e a,v, t( )
tHttH
vGvvG
aFaaF
10
10
10
Constant Linear Nonlinear
0
a1
The Constant Terms Substituting the polynomial forms into the original expression, we
get
This shows that we have three constant terms, where only one can be estimated uniquely. This can be remedied by the following definition:
Rewriting gives:
tvatHtvGvaFatvar ,,),,(ln 101010
0
0
0
0
0000
tvatHtvGvaFatvar ,,),,(ln 1110
The Linear Terms Because of the relationship between age, vintage, and time:
or equivalently
We can write
With the definition
this becomes
vta
avt
tvatHavvGvaFatvar ,,)(),,(ln 1110
tvatHvGvaFatvar ,,)()(),,(ln 11110
0,, 1111111
lnr(v,a, t) = ¢a0 + ¢a1a+ ¢F a( )+ ¢b1v+ ¢G v( )+ ¢H t( )+e a,v, t( )
Linear Interpretation Because of the relationship between age, vintage, and time, we
cannot have three unique linear terms, only two.
Known from the Age Period Cohort literature, it means that we can never be certain of the magnitude of the linear trends in lifecycle, environment, or credit risk.
This result is true for all analysis of vintage data, regardless of the model used.
This problem is solved by making domain-specific assumptions.
Conclusions Age-Period-Cohort models are naturally suited to a
range of statistical problems, including many that are essential for today’s businesses.
APC models can be used together to create a complete picture of net present value.
APC models can be combined with other scoring or data mining techniques in order to answer critical account-level questions without loosing the long term impacts of vintage effects.
References Breeden, J.L. 2007. "Modeling Data with Multiple Time Dimensions", J of Computational Statistics and Data
Analysis, Vol. 51, Issue 9, pp. 4761-4785.
Breeden, J.L., L. Thomas, and J.W. McDonald III. 2008. “Stress-testing retail loan portfolios with dual-time dynamics,” The Journal of Risk Model Validation, 2(2), Summer, pp. 43-62.
Breeden, J.L., 2013. “Incorporating Lifecycle and Environment in loan-level forecasts and stress tests”, Proceedings of the Credit Scoring and Credit Control Conference XII, Edinburgh, 2013.
Breeden, J.L. 2014. Reinventing Retail Lending Analytics: Forecasting, Stress Testing, Capital, and Scoring for a World of Crises – Second Impression, Riskbooks.
Breeden, J.L. and J. Canals-Cerdá. 2016. “Consumer risk appetite, the Credit Cycle, and the Housing Bubble”, Working Papers, Research Department, Federal Reserve Bank of Philadelphia, No. 16-05.
Breeden, J.L. and L.C. Thomas. 2016. “Solutions to Specification Errors in Stress Testing Models”, to appear in Journal of the Operational Research Society, 2016, doi:10.1057/jors.2015.97.
Holford, T.R. 1983. “The Estimation of Age, Period and Cohort Effects for Vital Rates”, Biometrics, Vol. 39, No. 2 pp. 311-324
Schmid V.J. and L. Held. 2007. Journal of Statistical Software, Volume 21, Issue 8. “Bayesian Age-Period-Cohort Modeling and Prediction – BAMP”.
Yang, Y. and K. C. Land. 2013. Age-Period-Cohort Analysis: New Models, Methods, and Empirical Applications.New York: Chapman & Hall / CRC.
