Conference on Seasonality, Seasonal Adjustment and their - - PowerPoint PPT Presentation

10-12 May 2006

Benchmarking in X-12-ARIMA and at Statistics Canada

Benoit Quenneville Susie Fortier Zhao-Guo Chen Edith Latendresse

Conference on Seasonality, Seasonal Adjustment and their implications for Short-Term Analysis and Forecasting

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:
• Bias parameter option or X-12-ARIMA target argument
• X-12-ARIMA start argument

) 1 ( ≤ ≤ ρ ρ

) 1 2 1 , (

• r

usually but = ∈ λ λ λ R I

1,5 2,0 2,5 3,0 3,5 4,0 4,5

1998 1999 2000 2001 2002 2003 Billions Indicator Series Benchmarked Series

Benchmarking : pro-rating

,

2 1

= = ρ λ

1,5 2,0 2,5 3,0 3,5 4,0 4,5

1998 1999 2000 2001 2002 2003 Billions Indicator Series Benchmarked Series

3

9 . 729 . , 1 = = = ρ λ

Benchmarking : suggested default values

Benchmarking : Q to Q changes

Date t Indicator

Benchmarked Benchmarked 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%

1 1 − −

−

t t t

y y y ,

2 1

= = ρ λ 729 . , 1 = = ρ λ

Benchmarking : BI-ratios

3

9 . 729 . , 1 = = = ρ λ

Benchmarked to Indicator series ratios

0.85 0.90 0.95 1.00 1.05 1.10

1998 1999 2000 2001 2002 2003

B/I (λ=0.5, ρ=0) B/I (λ=1, ρ=0.729)

1.5 2.0 2.5 3.0 3.5 4.0 4.5 1998 1999 2000 2001 2002 2003 2004 2005 B illions Indicator Series Benchmarked Series(no bias) Benchmarked Series(bias)

3

9 . 729 . , 1 = = = ρ λ

Benchmarking : bias parameter

0,85 0,90 0,95 1,00 1,05 1,10 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)

BI-ratios with and without bias correction

0.964

Benchmarking : bias parameter

Benchmarking: Rho Parameter

Illustration of the Effect of the Rho Parameter

0.80 0.85 0.90 0.95 1 .00 1 .05 1 .1 1 .1 5 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

With correction for bias , the benchmarked series is the solution to: Minimize Subject to:

Benchmarking : estimation

y b y y a b a a y y

m m t t m m m t

⋅ = = = =

∗ ∈

∑∑ ∑

and with series annual the ) ( series quarterly the ) (

∑

= − − −

⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ − − ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ − + ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ − −

T t t t t t t t

y y y y y y

2 2 * 1 1 * 1 * * 2 * 1 1 * 1 2)

1 (

λ λ λ

θ ρ θ θ ρ

m m t t

a =

∑

∈

θ

Benchmarking : estimation

Large matrix inversion is required to compute W.

( ) ( )

⎩ ⎨ ⎧ = − + < ≤ − ′ + =

−

1 1 ˆ

* * * 1 *

ρ ρ θ Jy a W y Jy a V J V y

d e

[ ]

[ ]

J JV V C C V y C m t j j J

e d e e T j i j i e t t m t m

′ = Ω = = Ω ∝ ⎩ ⎨ ⎧ ∈ = =

= − ,..., 1 , * , ,

) ( diag else if 1 ρ

λ

Implied linear regression model for 0 ≤ ρ <1 Assumed model: Solution :

∑

∈

= + =

m t t m t t t t t

a e e c y θ θ ) 1 , ( ~ ;

*

Benchmarking : estimation

( )

* 1

ˆ Jy a V J V y

d e

− ′ + =

− *

θ

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

• ptional. 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

• utseries

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)
• Permprioradj

(Original series adjusted for 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

• f 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])}

• utlier{ …}

arima{…} forecast{…} x11{… save = D11} force{ lambda=1 rho=0.9 target=calendaradj type=regress save=SAA }

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 Jan-96 Jan-97 Jan-98 Jan-99 Jan-00 Jan-01 Jan-02 Jan-03 Jan-04 Millions D11 D11A (λ=1,ρ=0.9)

X-12-ARIMA Example

Differences between D11 and D11A

• 15,000
• 10,000
• 5,000

5,000 10,000

Jan-91 Jan-92 Jan-93 Jan-94 Jan-95 Jan-96 Jan-97 Jan-98 Jan-99 Jan-00 Jan-01 Jan-02 Jan-03 Jan-04

D11-D11A (λ=1,ρ=0.9)

X-12-ARIMA Example

Growth rates

• 2.0%
• 1.0%

0.0% 1.0% 2.0% 3.0% 4.0% Jan- 95 Apr- 95 Jul- 95 Oct- 95 Jan- 96 Apr- 96 Jul- 96 Oct- 96

Growth rate in D11 Growth rate in D11A

Example: Running sums

Differences between the sum of 12 consecutive months computed on D11A and on A18

• 40
• 20

20 40 60 80 100 120 140 160 Jan-91 Jan-92 Jan-93 Jan-94 Jan-95 Jan-96 Jan-97 Jan-98 Jan-99 Jan-00 Jan-01 Jan-02 Thousands

SumD11A-SumA18

Guidelines for X-12-ARIMA

• Use lambda = 0 for an additive decomposition model

and lambda = 1 for a multiplicative decomposition model.

• Use rho=0.9 (or 0.729 for quarterly series).
• Avoid using type=denton or type=regress with

rho=1. In general, benchmarked values in incomplete years are less accurate, and the program involves the inversion of a much larger matrix. Consequently, users are to expect a significant increase in computing time when using rho=1.

• An alternative to using rho=1 is thus to use rho=0.999.
• If the true autocorrelation satisfies ρ> 0.95, then use rho=

0.98.

Benchmarking

Future developments (in Forillon)

• Consider measurement errors from the infra-annual values

( related to matrix , i.e, C and ).

• Consider measurement errors from the annual values.
• Binding benchmarking even if the annual values are

subject to errors. ( Assume even if )

• Variance estimation on benchmarked values: θ

ˆ

V

e

Ω

) , ( ;

2 ε

σ ε θ ~

m m m t t m

ε a + = ∑

∈

2 = ε

σ

2 > ε

σ

e

V

Questions & comments Questions & commentaires

Benchmarking in X-12-ARIMA and at Statistics Canada Benoit.Quenneville@statcan.ca Susie.Fortier@statcan.ca (Proc Benchmarking) X12@census.gov (X-12-ARIMA)