Comparing Automatic Modeling Procedures of TRAMO andX-12-ARIMA, an Update

Kathy McDonald-Johnson, U.S. Census Bureau

Catherine Hood, Catherine Hood Consulting

Brian Monsell, U.S. Census Bureau

Chak Li, U.S. Census Bureau

ICES III June 2007

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Acknowledgments

• Agustín Maravall

• Víctor Gómez

• Rita Petroni

• James Gomish

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Update

• Similar comparisons in the past, especially Farooque, Hood, and Findley (2001)

»X-12-ARIMA chose models of similar quality to TRAMO models

»X-12-ARIMA perhaps better at identifying trading day effects than TRAMO

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Update, Similar Approach

• We used a similar approach to that of Farooque, Hood, and Findley (2001), but we used improved versions of both programs

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Outline

• Background on Automatic Modeling

• Methods

• Results

»Actual time series

»Simulated time series

• Conclusions

Background

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Automatic ARIMA Modeling

• X-11-ARIMA from Statistics Canada, Dagum

»Picks best model from list

• TRAMO (Time series Regression with ARIMA noise, Missing observations and Outliers) from the Bank of Spain, Gómez and Maravall

»Multiple steps to obtain a model

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TRAMO Automatic Modeling

• Gómez and Maravall (2000) gives description

• FORTRAN code from Gómez and Maravall provides additional detail

»Generously provided to U. S. Census Bureau for X-12-ARIMA Version 0.3 development

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X-12-ARIMA Version 0.3

• Retains pick model method

»Pickmdl specification

• Adds step-through method based on the TRAMO method

»Automdl specification

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X-12-ARIMA Comparisons

• Dent, Hood, McDonald-Johnson, and Feldpausch (2005) compared the step-through method to the pick model method

»Models of similar quality

»Step-through method more flexible

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X-12-ARIMA's Automatic Transformation Selection

• Identification with the transform specification

• Fit a default model with the log transformation and with no transformation»Usually the airline model (0 1 1)(0 1 1) from Box

and Jenkins (1976)

• The model is chosen whose maximum likelihood value is larger»Likelihood of the log data is adjusted to be a

likelihood of the untransformed data

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•By default, slight bias toward the log transformation

, ,ˆ ˆ 1Y YUntransformed N Log NL L

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X-12-ARIMA's Automatic Regression Selection

• Trend Constant

»Identification with the step-through method, automdl specification

• Outliers

»Identification with the outlier specification

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X-12-ARIMA's Automatic Regression Selection

Trading-Day Effect

Easter Effect

• Identification with the regression specification, aictest argument

• Test uses the AICC

»No bias (user can set bias)

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AICC

• AIC Corrected (for sample size)

• Note: As N gets larger, AICC approaches the AIC

Npp

LAICC NN 11

2ˆ2

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Trading-Day Effects

• User specifies type

»Flow (cumulative)

»Stock (inventory)

• X-12-ARIMA compares AICC with and without the effect

»No bias (user can set bias)

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Easter Effects

• Default tests for Easter effects of length 1, 8, and 15 days»User can specify length

• X-12-ARIMA compares four AICC values»No effect vs. each of the three different

length effects

»No bias (user can set bias)

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Modeling Diagnostics

• Ljung-Box Q»Goodness-of-fit diagnostics (Ljung and

Box 1978)

• Spectrum of the model residuals»Diagnostic indicating seasonal or trading

day effects remaining in the model residuals

»Trading day frequencies defined in Cleveland and Devlin (1980)

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Ljung-Box Q

• Based on sample autocorrelation of the regARIMA model residuals

• Residuals should behave like white noise

• Each Ljung-Box Q statistic of positive degrees of freedom has a corresponding p value

• An individual lag fails if the p value for the Q statistic for the lag is less than 0.05

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Ljung-Box Q Failure

For this study

• If seven or more of the first 12 lags failor

• If 13 or more of the first 24 lags failor

• If lag 12 fails

Then the model fails according to this diagnostic

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Spectrum of the Model Residuals

• Diagnostic indicating strength at frequencies of interest

• Visually significant peaks at seasonal or trading day frequencies indicate possible model problems

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Significant Spectrum Peaks

• A spectral peak is considered to be significant if it

»Reaches a height beyond the median height of all the frequency measures

»Are taller than nearest neighbors by a visually significant amount

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Significant Spectrum Peaks

For this study, any significant peak at

»seasonal frequencies one, two, three, for or five cycles per year and

»At either of the two trading-day frequencies

indicates model failure according to this diagnostic

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Spectrum Diagnostic Information

• Graphical form»Output file line printer graph

»Higher resolution graph

• Text form»Log file

»Diagnostics file

• Failure warnings listed onscreen when X-12-ARIMA runs

Methods

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Automatic Modeling Settings

• Test for log transformation

• Automatic regARIMA model identification

• Automatic outlier detection

• Test for

»Usual trading day

»Leap year

»Easter effects

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Settings for X-12-ARIMA

• We expected some quarterly effects (higher autocorrelation three months apart), so we chose the maximum nonseasonal model order (maximum p, q) to be three»Default is two

• We chose to prefer balanced models to have an approach more like the TRAMO procedure»Default is not to prefer balanced models

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Model Choices

• Ran TRAMO, X-12-ARIMA to identify transformation, model

• Hard-coded results into X-12-ARIMA input specification files

• Compared diagnostics from X-12-ARIMA

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Clarification

• "TRAMO model" results are from X-12-ARIMA runs

»Initial TRAMO runs determined the transformation and model choices

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Changes to Models

• Used X-12-ARIMA outlier set

• If any Easter regressor chosen, used X-12-ARIMA Easter effect of eight days

Actual Time Series

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457 U. S. Census Bureau Series

• U. S. Building Permits

• Manufacturing

• Retail Sales

• Import/Export data

Descriptions available at

www.census.gov/cgi-bin/briefroom/BriefRm

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Transformation Choice

• TRAMO and X-12-ARIMA agreed for 91% (417) of the series

• 40 series differed»85% (34/40) TRAMO chose log and X-12-

ARIMA chose no transformation

»15% (6/40) X-12-ARIMA chose log and TRAMO chose no transformation

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Transformation Choice

• Transformation choice is fundamental

• We did not want to favor one program’s transformation over the other

• We dropped the 40 series of disagreement from further comparisons

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Full Model Agreement(of 417 Series)

• 30% (124) of the regARIMA models agreed

»Any length Easter considered match

• 293 series to compare diagnostics

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ARIMA Model Agreement(of 293 Series)

• 24% (70) of the ARIMA models agreed, showing differences only in the chosen regression effects

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Easter Effects

• 76% (222) Easter effect agreement»13% (37) both programs chose an Easter effect

»63% (185) neither program chose an Easter effect

• 24% (71) Easter effect disagreement»24% (70) X-12-ARIMA chose Easter and TRAMO

did not

»0.3% (1) TRAMO chose Easter and X-12-ARIMA did not

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Why Does X-12-ARIMA Include Easter Effects More Often?

• TRAMO checks for an Easter effect of one length

• X-12-ARIMA checks for three different lengths

• Do more possible regressors raise the chance of including an Easter effect?

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Are the Easter Effects Appropriate?

• These economic series could indeed have Easter effects, but the results for X-12-ARIMA show Easter effects to be more prevalent than we would have expected

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Trading-day Effects

• 57% (166) trading day agreement»24% (70) both programs chose trading-day

effects

»33% (96) neither program chose trading-day effects

• 43% (127) trading day disagreement»35% (104) X-12-ARIMA chose trading-day effects

and TRAMO did not

»8% (23) TRAMO chose trading-day effects and X-12-ARIMA did not

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Appropriate Trading-day Effects

• Under specific conditions, we can evaluate whether a trading-day effect was missed

»One model includes a trading-day effect but the other does not

»The model with a trading-day effect has no spectrum peak at either trading-day frequency, but the model without a trading-day effect results in a peak at one or both of the trading-day frequencies

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Trading Day Omitted

• 22% (64) of the series had this omission problem»20% (60) TRAMO omission

»1% (4) X-12-ARIMA omission

• Using a binomial distribution, the probability of seeing 60 out of 64 failures for one method if the probability of failure were equally 0.5 for each method is less than 0.01

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Ljung-Box Q Model Failures

• 24% (69) one model passed and the other model failed»17% (50) TRAMO model failed

»6% (19) X-12-ARIMA model failed

• Binomial probability that 50 of 69 failures would be from one method is less than 0.01

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Seasonal Spectrum Model Failures

• 14% (41) one model passed and the other model failed»8% (24) TRAMO model failed

»6% (17) X-12-ARIMA model failed

• Binomial probability of 24 of 41 failures being from one method is not significant at the 10% level, so there was no significant difference in the seasonal spectrum results

Simulated Time Series

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Airline Model Series (0 1 1)(0 1 1)

• 3,500 monthly series

• 15 years long

• Nonseasonal moving average coefficient 0.6

• Seasonal moving average coefficient 0.9

• Start date 1980 (arbitrary choice)

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No Model

• 0.6% (21) X-12-ARIMA did not choose a model

• TRAMO identified a model for each series

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Fully Correct Model Identification

• Airline model with no trading day or Easter effects

• 66% (2,305) TRAMO correct

• 72% (2,516) X-12-ARIMA correct

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Correct ARIMA Identification

• Also identified trading day or Easter effects

• 85% (2,978) TRAMO correct ARIMA

• 90% (3,159) X-12-ARIMA correct ARIMA

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Nonseasonal Differencing

• 99% (3,480) TRAMO chose nonseasonal difference of order 1

• 99% (3,466) X-12-ARIMA chose nonseasonal difference of order 1

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Seasonal Differencing

• 97% (3,378) TRAMO chose seasonal differencing of order 1

• 99% (3,470) X-12-ARIMA chose seasonal differencing of order 1

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Easter Effect Identification

• 4% (148) TRAMO chose Easter effect

• 11% (392) X-12-ARIMA chose Easter effect

• No Easter effect present

• Binomial probability is less than 0.01 that we would see such a difference assuming equal probabilities of selection

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Trading-day Effect Identification

• 13% (460) TRAMO chose trading-day effect

• 4% (138) X-12-ARIMA chose trading-day effect

• Binomial probability is less than 0.01 that we would see such a difference assuming equal probabilities of selection

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Conclusions

• X-12-ARIMA mistakenly chooses an Easter effect more often than TRAMO

• As noted in Farooque, Hood, and Findley (2001), X-12-ARIMA still seems to choose trading-day effects more appropriately than TRAMO

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Conclusions

• For known airline model simulations, X-12-ARIMA performed as well as TRAMO in identifying the ARIMA model

• X-12-ARIMA models performed as well as TRAMO when measured by the standard model diagnostics»Ljung-Box Q

»Spectrum of the model residuals

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Newer Version of X-12-ARIMA

• We now have an improved version of X-12-ARIMA and hope to rerun the model identification to see if there are any changes to these results

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Future Work

• Expand the study of simulated series to perform a more thorough evaluation of X-12-ARIMA’s new automatic modeling procedure using more varied models, model coefficients, regression effects, and series lengths

• Investigate how to improve the selection of the Easter effect

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Disclaimer

This report is released to inform interested parties of ongoing research and to encourage discussion of work in progress. Any views expressed on statistical, methodological, technical, or operational issues are those of the authors and not necessarily those of the U.S. Census Bureau.

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Much of the data analysis for this paper was generated using Base SAS® software, SAS/AF® software, and SAS/GRAPH® software, Versions 8 and 9 of the SAS System for Windows. Copyright © 1999-2003 SAS Institute Inc. SAS and all other SAS Institute Inc. product or service names are registered trademarks or trademarks of SAS Institute Inc., Cary, NC, USA.

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We used R to simulate the airline model time series. Additional analysis was performed using Microsoft® Excel 2000. Copyright © 1985-1999 Microsoft Corporation. We checked our own calculations of the binomial probabilities involving the actual data using the Binomial Calculator at onlinestatbook.com/java/binomialProb.html (home page at onlinestatbook.com), and we used it alone for the comparisons involving the simulated data.