Conference on Seasonality, Seasonal Adjustment and their implications for Short-Term Analysis and Forecasting 10-12 May 2006 Benchmarking in X-12-ARIMA and at Statistics Canada Benoit Quenneville Susie Fortier Zhao-Guo Chen Edith Latendresse
Benchmarking in X-12-ARIMA and at Statistics Canada Benoit Quenneville, Susie Fortier, Zhao-Guo Chen and Edith Latendresse Statistics Canada Eurostat Symposium on Seasonal Adjustment 10-12 May 2006
Dagum, E.B. and Cholette, P. (2006). Benchmarking, Temporal Distribution and Reconciliation Methods for Time Series Data. New York: Springer-Verlag, Lecture Notes in Statistics #186. Tommaso Di Fonzo, Università di Padova, Italy Guy Huot and Kim Chiu, Statistics Canada
Content Benchmarking � � Parameters and illustration � Estimation � Proc Benchmarking � Illustration X-12-ARIMA FORCE spec � � Target and start arguments � Illustration Guidelines � Future developments �
Benchmarking Adjustment of the level of a sub-annual series using � auxiliary annual benchmarks. Issues: � Preserve movement in the sub-annual series as much as � possible Timeliness of annual benchmarks � Parameters: � λ λ ∈ λ = but usually or ( I R 0 , 1 2 1 ) � ρ ≤ ρ ≤ ( 0 1 ) � Bias parameter option or X-12-ARIMA target argument � X-12-ARIMA start argument �
Benchmarking : pro-rating λ = ρ = , 0 1 2 4,5 Billions 4,0 3,5 3,0 2,5 2,0 1,5 1998 1999 2000 2001 2002 2003 Indicator Series Benchmarked Series
Benchmarking : suggested default values λ = ρ = = 3 1 , 0 . 729 0 . 9 4,5 Billions 4,0 3,5 3,0 2,5 2,0 1,5 1998 1999 2000 2001 2002 2003 Indicator Series Benchmarked Series
− y y Benchmarking : Q to Q changes − t t 1 y − t 1 Date t Indicator Benchmarked Benchmarked λ = ρ = λ = ρ = 1 , 0 . 729 1 , 0 2 1998.Q2 31.1% 31.1% 32.3% 1998.Q3 29.0% 29.0% 28.4% 1998.Q4 -29.8% -29.8% -31.5% 1999.Q1 -9.2% -17.1% -12.6% 1999.Q2 30.9% 30.9% 27.1% 1999.Q3 32.2% 32.2% 29.6% 1999.Q4 -31.1% -31.1% -31.5% 2000.Q1 -9.0% -11.1% -9.0%
Benchmarking : BI-ratios Benchmarked to Indicator series ratios 1.10 λ = ρ = = 3 1 , 0 . 729 0 . 9 1.05 1.00 0.95 0.90 0.85 1998 1999 2000 2001 2002 2003 B/I ( λ =0.5, ρ =0) B/I ( λ =1, ρ =0.729)
Benchmarking : bias parameter λ = ρ = = 3 1 , 0 . 729 0 . 9 4.5 illions B 4.0 3.5 3.0 2.5 2.0 1.5 1998 1999 2000 2001 2002 2003 2004 2005 Indicator Series Benchmarked Series(no bias) Benchmarked Series(bias)
Benchmarking : bias parameter BI-ratios with and without bias correction 1,10 1,05 1,00 0.964 0,95 0,90 0,85 1998 1999 2000 2001 2002 2003 2004 2005 BI ratio( λ =1; ρ =0.729; no bias correction) BI ratio ( λ =1; ρ =0.729; with bias correction)
Benchmarking: Rho Parameter Illustration of the Effect of the Rho Parameter 1 .1 5 1 .1 0 1 .05 1 .00 0.95 0.90 0.85 0.80 1 998 1 999 2000 2001 2002 2003 2004 2005 Rho=0 Rho=0.2^3 Rho=0.4^3 Rho=0.6^3 Rho=0.8^3 Rho=0.9^3 Rho=0.99^3 Rho=1 Bias=0.964
Benchmarking : estimation ∑ a = y ( y ) the quarterly series m ∗ t = = ⋅ with b m and y b y ∑∑ = a ( a ) the annual series y m t ∈ m t m With correction for bias , the benchmarked series is the solution to: Minimize 2 2 ⎧ ⎫ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ − θ − θ − θ * * * ⎜ ⎟ ⎪ ⎜ ⎟ ⎜ ⎟ ⎪ y T y y ∑ − ρ + − ρ 2 ) − − ( 1 1 1 t t t 1 t 1 ⎨ ⎬ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ λ λ λ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ * * * ⎪ ⎪ y y y = t 2 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎩ ⎭ − 1 t t 1 ∑ θ = a Subject to: t m ∈ t m
Benchmarking : estimation ( ) ⎧ − ′ 1 + − ≤ ρ < * * y V J V a Jy 0 1 ˆ θ = e d ⎨ ( ) + − ρ = * * y W a Jy 1 ⎩ Large matrix inversion is required to compute W .
Benchmarking : estimation Implied linear regression model for 0 ≤ ρ <1 ∈ ⎧ 1 if t m [ ] = = Assumed model: J j j ⎨ m , t m , t 0 else ⎩ = θ + * y c e ; e ~ ( 0 , 1 ) λ t t t t t ∝ * C diag ( y ) ∑ = θ t a [ ] m t − Ω = ρ i j ∈ t m = i , j 1 ,..., T e Solution : = Ω V C C e e ( ) ′ = V JV J ˆ − ′ 1 θ = * + − * y V J V a Jy d e e d
Benchmarking � How is it implemented? TSRAC version ( FORTRAN) programmed by K. Chiu � Forillon project from SDD ( A SAS procedure : proc � benchmarking) In X-12 (V0.3), the FORCE spec impose annual totals on � the seasonally adjusted series according to this methodology. ( note : benchmarks cannot be external and bias estimation parameter is replaced by the target argument)
Benchmarking : Implementation PROC BENCHMARKING Statement PROC BENCHMARKING <option(s)>; To do this Use this option Specify the input benchmarks data set BENCHMARKS= Specifies the input series data set SERIES= Specifies the output benchmarks data set OUTBENCHMARKS= Specifies the output series data set OUTSERIES= Specify Rho RHO= Specify Lambda LAMBDA= Specify the Bias Estimation option BIASOPTION=
Benchmarking : Implementation Options BENCHMARKS=SAS-data-set specifies the input SAS data set that contains the benchmarks. It is mandatory. Five numeric variables must be in this data set: STARTYEAR, STARTPERIOD, ENDYEAR, ENDPERIOD and VALUE. SERIES=SAS-data-set specifies the input SAS data set that contains the series to benchmark. It is mandatory. Three numeric variables must be in this data set: YEAR, PERIOD and VALUE.
Benchmarking : Implementation OUTBENCHMARKS=SAS-data-set names the output data set that contains the benchmarks used by the procedure. It is optional. Five numeric variables will be created in this data set: STARTYEAR, STARTPERIOD, ENDYEAR, ENDPERIOD and VALUE. OUTSERIES=SAS-data-set names the output data set that contains the benchmarked series. It is optional. Three numeric variables will be created in this data set: YEAR, PERIOD and VALUE.
Benchmarking : Implementation RHO=real number between 0 and 1 specifies Rho. It is mandatory. LAMBDA=real number specifies Lambda. It is mandatory. BIASOPTION= Bias Estimation Option specifies the Bias Estimation Option. It is mandatory. Valid values are 1, 2 and 3.
Benchmarking : Implementation � Example DATA myBenchmarks; INPUT @ 01 startYear 4. @ 06 startPeriod 1. @ 08 endYear 4. @ 13 endPeriod 1. @ 15 value; CARDS; 1998 1 1998 4 10.3 1999 1 1999 4 10.2 ; RUN ;
Benchmarking : Implementation Example � DATA mySeries; INPUT @ 01 year 4. @ 06 period 1. @ 08 value; CARDS; 1998 1 1.9 1998 2 2.4 1998 3 3.1 1998 4 2.2 1999 1 2.0 1999 2 2.6 1999 3 3.4 1999 4 2.4 2000 1 2.3 ; RUN ;
Benchmarking : Implementation � Example PROC BENCHMARKING BENCHMARKS=myBenchmarks SERIES=mySeries OUTBENCHMARKS=outBenchmarks OUTSERIES=outSeries RHO= 0.729 LAMBDA= 1 BIASOPTION= 3 ; RUN ;
Benchmarking : Implementation � Example NOTE: --- FORILLON System developed by Statistics Canada --- NOTE: PROCEDURE BENCHMARKING Version v1.01.001 NOTE: Created on Dec 5 2005 at 12:13:39 NOTE: Email: banff@statcan.ca NOTE: This procedure is for use at Statistics Canada only. NOTE: BENCHMARKS = WORK.MYBENCHMARKS NOTE: SERIES = WORK.MYSERIES NOTE: OUTBENCHMARKS = WORK.OUTBENCHMARKS NOTE: OUTSERIES = WORK.OUTSERIES NOTE: RHO = 0.729000000 NOTE: LAMBDA = 1.000000000 NOTE: BIASOPTION = 3 NOTE: BIAS (calculated) = 1.03
Benchmarking : Implementation � Example outseries Obs YEAR PERIOD VALUE 1 1998 1 2.04933 2 1998 2 2.60134 3 1998 3 3.33764 4 1998 4 2.31169 5 1999 1 2.02109 6 1999 2 2.55480 7 1999 3 3.29219 8 1999 4 2.33191 9 2000 1 2.26802
X-12-ARIMA FORCE spec (V03) � Bias parameter option is replaced with argument target , which specifies which series is used as the target for forcing the totals of the seasonally adjusted series. The choices are: � Original Caladjust (Calendar adjusted series) � (Original series adjusted for � Permprioradj permanent prior adjustment factors) Both (Original series adjusted for calendar and � permanent prior adjustment factors)
FORCE spec : start argument By default, the FORCE spec implies that the � calendar year totals in the SA = calendar year totals of the target series. Alternative starting period for the annual total can � be specified with start argument. Annual total starting at any other period other than � start may not be equal.
X-12-ARIMA FORCE spec series{… save = A18} transform{function=log} regression{ variables=(TD easter[8])} outlier{ …} arima{…} forecast{…} x11{… save = D11} force{ lambda=1 rho=0.9 target=calendaradj type=regress save=SAA }
Millions Example: Dept. stores sales 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Jan-91 Jan-92 Jan-93 Jan-94 Jan-95 D11 Jan-96 Jan-97 D11A ( λ =1, ρ =0.9) Jan-98 Jan-99 Jan-00 Jan-01 Jan-02 Jan-03 Jan-04
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