Lloyds TSB. Derek Hull, John Adam & Alastair Jones

Transcription

1 Forecasting Bad Debt by ARIMA Models with Multiple Transfer Functions using a Selection Process for many Candidate Variables Lloyds TSB Derek Hull, John Adam & Alastair Jones

2 INTRODUCTION: No statistical processes in place 8 unsecured lending portfolio models many candidate explanatory variables variables of short timespan - forecasts for 12 months data had to be gathered, discussed, understood and verified process to check data - inputting twice and transferring Microsoft Excel 97 spreadsheet to SAS dataset

3 MODELLING METHOD: Box-Jenkins ARIMA (AutoRegressive Integrated Moving- Average) allows fitting of models comprising combinations of previous values of the series plus lagged values of explanatory series relatively easy to understand well established & documented very flexible: PC SAS/ETS allows models to be built iteratively, retaining previous versions for comparison purposes. Found no evidence of current or previous academic research for: Bad Debt being forecasted ARIMA used with large numbers of explanatory variables

4 DETAILS of SELECTION PROCESS (1): Macro written using GPLOT Procedure to graph all candidate data series This highlighted possible: seasonality interventions shortness of series missing values Subjective measure of a variable s usefulness Number of candidate variables was reduced

5 DETAILS of the SELECTION PROCESS (2): Macro written to test & correct for non-stationarity ARIMA requires the series to be stationary constant mean over time constant variance over time The Macro incorporated these SAS/ETS Macros: %LOGTEST» tests if logarithmic transformation is required %DFTEST» tests the order of differencing required if series is nonstationary %BOXCOXAR» gives optimal Box-Cox power transformation

6 DETAILS of the SELECTION PROCESS (3): Macro written to test for white noise series random error series uncorrelated over time likely to have little use as explanatory variables The Macro, used the ARIMA Procedure with these options on the IDENTITY statement:» ESACF» SCAN» MINIC no recommended models indicate might be just white noise described in SAS/ETS Software: Changes and Enhancements for Release 6.12» also the test Q-statistics no significant sample autocorrelations indicate white noise

7 DETAILS of the SELECTION PROCESS (4): EXPAND Procedure used to fill in missing values there were few missing values however, these voids needed to be filled to allow process to execute

8 DETAILS of the SELECTION PROCESS (5): Univariate models for candidate variables evaluated use of IDENTIFY & ESTIMATE statements within ARIMA procedure diagnostic information to test model viability» absolute t ratios for parameter estimates should be >2» the smaller Akaike s Information Criterion (AIC) is, the better the fit» similarly for the standard error estimate» correlation between parameter estimates should be <0.9» autocorrelation results on residuals should be non-significant

9 Example of an autocorrelation plot, used to indicate the relationship of each series with itself Lag Covariance Correlation Std ******************** ********** *** ******** **** ** ***** ** ***** *********** ******** * ***** ****** ** ** "." marks two standard errors

10 Example of a partial autocorrelation plot, another means of indicating the relationship of each series with itself Lag Correlation ********** ********** * ** * *** **** ***** ****** *** *** * ** **.

11 Examples of model diagnostics illustrating how to determine whether a model is statistically viable Approx. Parameter Estimate Std Error T Ratio Lag MU AR1, AR1, AR1, Constant Estimate = Variance Estimate = Std Error Estimate = AIC = SBC = Number of Residuals= 40

12 Examples of model diagnostics cont. Correlations of the Estimates Parameter MU AR1,1 AR1,2 AR1,3 MU AR1, AR1, AR1, ARIMA Procedure Autocorrelation Check of Residuals To Chi Autocorrelations Lag Square DF Prob

13 DETAILS of the SELECTION PROCESS (6): The need to pre-whiten» no simple cross-correlation» fit ARIMA model to input series» white noise left» filter input & Bad Debt series with model» then cross-correlate Macro written to cross-correlate each input with Bad Debt» avoid feedback models» build model with strongest relationship» re-iterate process

14 Jul-9 9 O ct Bad Debt: actuals and forecasts O ct-93 Jan-94 A p r-9 4 Jul-9 4 O ct-94 Jan-95 A p r-9 5 Jul-9 5 O ct-95 Jan-96 A p r-9 6 Jul-9 6 O ct-96 Jan-97 A p r-9 7 Jul-9 7 O ct-97 Jan-98 A p r-9 8 Jul-9 8 O ct-98 Jan-99 A p r-9 9 Date Bad Debt F0=p(1,3) best univariate F1=6$(2)tvar43 F1A =q1 6$(2)tvar43 F2=6$(2)tvar43 0$(1)tvar361 F2A =p1 6$(2)tvar43 0$(1)tvar361 Jul-9 3 Jan-93 A p r-9 3 A m ount

15 PROBLEMS: Lack of precision with forecast estimates large confidence limits for 12 months ahead text book examples also large! Confidence limits for the total year ahead simulation? Collapse data using EXPAND procedure Number of candidate variables very high Time consuming manually adding variables several branches business considerations long lags required

16 CONCLUSIONS: Need to check univariate forecasts of candidate variables beforehand significant correlation with Bad Debt reasonable time lag for this relationship narrow confidence limits

17 RECOMMENDATIONS: Use above method + principal components analysis phase?» reduce dimensionality» individual components linearly independent STATESPACE procedure Bad Debt & Collections Inflows jointly forecast needs more investigation similar problems to ARIMA may be more automatic

18 RECOMMENDATIONS contd.: Use Stepwise Multiple Regression with REG procedure more what if than time series» each candidate variable gives several lagged variables» set minimum lag value» order of any autocorrelation tested by high order Durbin-Watson statistics within AUTOREG procedure any autocorrelation correction needed» No automatic way of forecasting but scenario/problem more what if now more variables could be permitted less weight on a few variables avoid overfitting more automatic