theory and application of benchmarking in business surveys susie fortier and benoit quenneville...
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Theory and Application of Benchmarking in Business Surveys
Susie Fortier and Benoit QuennevilleStatistics Canada -TSRAC
ICES – June 2007
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Content
Introduction NotationBenchmarking methodsTimeliness issue
Implied forecasts and annual growth ratesOther uses:
Seasonally adjusted dataLinking problem
Conclusions
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Introduction
Main references
Dagum, E.B. and Cholette, P. (2006) Benchmarking, Temporal Distribution, and Reconciliation Methods for Time Series, New York: Springer-Verlag, Lecture Notes in Statistics 186.
Bloem, A. M., R. J. Dippelsman, and N. Ø. Mæhel (2001): Quarterly National Accounts Manual, Concepts, Data Sources and Compilation. International Monetary Fund, Washington DC.
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Introduction
Benchmarking :
Combining a series of high-frequency data with a series of less frequent data into a consistent time series.
Monthly/Quarterly Annual
Explicit information about the short-term movement
Reliable information on the overall level and long-term movement
The “indicator” series The “benchmarks”
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Introduction
Issues in Benchmarking :
Preserve period to period movement of the indicator (monthly/quarterly) series while simultaneously attaining the level of the benchmarks (annual).
Consider the timeliness of the benchmarks.
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IntroductionExample of a quarterly series
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IntroductionA quarterly series with its auxiliary source
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IntroductionTimeliness issue
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IntroductionBenchmarked series
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Notation
Methodological details :
Indicator (monthly/quarterly) Benchmarks (annual)
DATA mySeries;
INPUT @01 year 4.
@06 period 1.
@08 value;
CARDS;
2000 1 1851
2000 2 2436
2000 3 3115
2000 4 2205
2001 1 1987
…
;
RUN;
DATA myBenchmarks;
INPUT @01 startYear 4.
@06 startPeriod 1.
@08 endYear 4.
@13 endPeriod 1.
@15 value;
CARDS;
2000 1 2000 4 10324
2001 1 2001 4 10200
…
;
RUN;
Ttss t ,,1),( Mmaa m ,,1),(
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Notation
Methodological details :
With binding benchmarking, the benchmarked series is
such that
Ttt ,,1),ˆ(ˆ
Mmammt
t ,,1,ˆ
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Notation
A bias parameter can be estimated and used to pre-adjust the indicator series:
A bias corrected series is obtained as:
m mt
m mtt
mm sa
b1
bss tt *
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Notation
Alternatively, the bias can be expressed in terms of a ratio:
The bias corrected series is then:
bss tt *
m mtt
mm
s
ab
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Notation
Bias correction is a preliminary adjustment to reduce, on average, the discrepancies between the two sources of data.
Useful for periods not covered by benchmarks.
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Notation
Effect of the Bias Correction (ratio)
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Methods : Pro-rating
A simple way to respect the constraints
is to use
This is the well-known formula for pro-rating.
mt for s
as
mt
mt
t
t
,ˆ*
*
Mmammt
t ,,1,
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Methods : Pro-ratingBenchmarked series with pro-rating
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Methods : Pro-ratingBI ratio with pro-rating
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Methods : Pro-ratingGrowth rates with pro-rating
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Methods : Pro-ratingGrowth rates with pro-rating
DATEIndicator
SeriesBenchmarked
SeriesGrowth Rate in
Indicator Series (%)Growth Rate in
Benchmarked Series (%)
2000-01
1851 1989.15 . .
2000-02
2436 2617.81 31.60 31.60
2000-03
3115 3347.48 27.87 27.87
2000-04
2205 2369.57 -29.21 -29.21
2001-01
1987 1945.42 -9.89 -17.90
2001-02
2635 2579.86 32.61 32.61
2001-03
3435 3363.12 30.36 30.36
2001-04
2361 2311.60 -31.27 -31.27
2002-01
2183 2059.05 -7.54 -10.93
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Methods : Proportional DentonBenchmarked series with Prop. Denton
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Methods : Proportional DentonBI ratio with Prop. Denton
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Methods : Proportional DentonGrowth rates with Prop. Denton
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Methods : Proportional DentonGrowth rates with Prop. Denton
DATEIndicator
SeriesBenchmarked
SeriesGrowth Rate in
Indicator Series (%)Growth Rate in
Benchmarked Series (%)
2000-01
1851 1989.15 . .
2000-02
2436 2617.81 31.60 30.86
2000-03
3115 3347.48 27.87 26.18
2000-04
2205 2369.57 -29.21 -30.86
2001-01
1987 1945.42 -9.89 -12.66
2001-02
2635 2579.86 32.61 29.04
2001-03
3435 3363.12 30.36 27.62
2001-04
2361 2311.60 -31.27 -32.14
2002-01
2183 2059.05 -7.54 -8.16
.
31.60
27.87
-29.21
-17.90
32.61
30.36
-31.27
-10.93
Pro-rating
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Based on Dagum and Cholette (2006).
Generalization of many well-known methods:Pro-rating
Denton (and proportional Denton)
Implemented at Statistics Canada with a user-defined SAS procedure: PROC BENCHMARKING
Project ForillonForillon
Software Demo
Main method
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Main method : Formula
The benchmarked series can be obtained as the solution of a minimization problem.
For given parameters and find the values that minimize the following function of :
subject to
Ttt ,,1,ˆ
T
tt
tt
t
tt
s
s
s
s
s
s
2
2
*1
1*
1
*
*2
*1
1*12 )1(
Mmammt
t ,,1,
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*1*ˆ JsaVJVs de
Solution when :
Solution:
Main method : Formula
1
“Regression-based”model from
Dagum & Cholette
JJVV
CCV
sC
mtjjJ
ed
ee
Tjiji
e
t
tmtm
,...,1,
*
,,
)(diag
else 0
if1
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Solution when :
Solution:
where W is the T x M upper-right corner matrix from :
Main method : Formula
W
WI
IJ
CC
J
JCC T
M 0
0
0
111111
**ˆ JsaWs
mult.) (Lagrange W
matrixIdentity 1,
MM
MMIM
TTji
1
whereelse0
1when 1
when 1
, ij
ij
ji
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Main method : FormulaWe can obtain pro-rating with the general formula with and :minimise
under
gives
Mmammt
t ,,1,
T
tt
tt
t
tt
s
s
s
s
s
s
2
2
*1
1*
1
*
*2
*1
1*12 )1(
21 0
T
tt
tt
s
s
1
2
*
*
T
tt
tt
s
s
1
2
*
*
mts
as
mt
mt
t
t
for ,ˆ*
*
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Main method : Effect of
Consider the case where and .
The function to be minimized under the constraints
which aims at preserving the period-to-period change in the original series.
Modified DentonModified Denton method
T
tt
tt
t
tt
s
s
s
s
s
sf
2
2
*1
1*
1
*
*2
*1
1*12 )1(
Mmammt t ,,1,
T
ttttt ssf
2
2
1*
1*
0 1
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Main method : Effect of
Consider the case where and .
The function to be minimized under the constraints
which seeks to minimize the change in the ratios (not to preserve the growth rates but a fairly close
approx). Variant of Proportional Proportional DentonDenton method
T
tt
tt
t
tt
s
s
s
s
s
sf
2
2
*1
1*
1
*
*2
*1
1*12 )1(
Mmammt t ,,1,
T
t t
t
t
t
ssf
2
2
*1
1*
1 1
with positive data!
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Main method : Effect of
3 parameters at play: : model adjustment parameter : “smoothing” parameter bias (implied with )
subject to
T
tt
tt
t
tt
s
s
s
s
s
s
2
2
*1
1*
1
*
*2
*1
1*12 )1(
Mmammt
t ,,1,
*ts
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Main method : Effect of
Illustration of the effect of the rho parameter(BI ratios)
0.80
0.85
0.90
0.95
1.00
1.05
1.10
1.15
2000 2001 2002 2003 2004 2005 2006 2007
Rho=0 Rho=0.2 3̂ Rho=0.4 3̂ Rho=0.6 3̂ Rho=0.8 3̂
Rho=0.9 3̂ Rho=0.99 3̂ Rho=1 Bias=0.964
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Main method : Effect of bias
Benchmarking without bias ( )39.0,1
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Main method : Effect of bias
Benchmarking with bias ( )39.0,1
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Main method : Effect of bias
Benchmarking without bias ( )39.0,1
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Main method : Effect of bias
Benchmarking with bias ( )39.0,1
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Timeliness issues
Adjustments for periods without benchmarks:
Benchmarked series give an implicit forecast for the unknown annual values.
The better the forecast, the lesser the revision!
Proportional Denton (ρ=1, λ=1) Benchmarking with bias (ρ=0.93, λ=1)
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Timeliness issues
2 implicit forecasts for 2006:
Enhanced benchmarking method with explicitexplicit forecasts
Year Benchmark Indicator Benchmarked
(bias)
Benchmarked
(prop Denton)
2004 11,582
4.37%
11,891
1.98%
11,582
4.37%
11,582
4.37%
2005 11,092
-4.23%
12,399
4.27%
11,092
-4.23%
11,092
-4.23%
2006 n/a
n/a
12,196
-1.64%
11,352
2.35%
10,689
-3.64%
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Timeliness issues
One possibility for explicit forecast:Use the annual growth rate from the indicator series on the last known benchmark.
Year Benchmark Indicator Benchmarked
(bias)
Benchmarked
(prop Denton)
2004 11,582
4.37%
11,891
1.98%
11,582
4.37%
11,582
4.37%
2005 11,092
-4.23%
12,399
4.27%
11,092
-4.23%
11,092
-4.23%
2006 10,910
-1.64%
12,196
-1.64%
11,352
2.35%
10,689
-3.64%
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Timeliness issues
With explicit forecast ( )1,1
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Timeliness issues
With explicit forecast ( )1,1
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Timeliness issues
With ″recent″ bias( , bias=0.94)39.0,1
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Timeliness issues
With ″recent″ bias( , bias=0.94)39.0,1
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Timeliness issues
Minimize revision?
Bias Explicit forecast
(based on indicator)
Will change annual growth rate of indicator series
Preserve annual growth rate of indicator when nothing else is available
Could be ″infected″ with non-representative historical data
Annual discrepancies based only on one year
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Methods : Summary so far!
Summary of methods presented:Pro-rating
Denton (and proportional Denton)
Regression-based (Dagum and Cholette)with or without bias correction
Denton with explicit forecast
Results from all of the above can be obtained by PROC BENCHMARKING.
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Methods
Other methodsOther numerical methods revolve around different minimisation functions.
Statistical model-based approaches
See annex 6.1 in Bloem, Dippelsman, and Mæhel (2001) for variants and references
See also Chen and Wu (2006) for link between numerical, regression based and signal extraction methods.
Future version of PROC benchmarking?
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Syntax : PROC Benchmarking
PLEASE SEE SOFTWARE DEMO !!PLEASE SEE SOFTWARE DEMO !!
PROC BENCHMARKING
BENCHMARKS=myBenchmarks
SERIES=mySeries
OUTBENCHMARKS=outBenchmarks
OUTSERIES=outSeries
OUTGRAPHTABLE=outGraph
RHO=0.729 LAMBDA=1 BIASOPTION=3;
RUN;
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In SAS Enterprise Guide®(Demo)
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Other uses : Seasonal adjustment
Seasonally adjusted series can be required to ″match″ given annual totals :
System of National Accounts (typical cases)
X-12-ARIMA version 0.3+FORCE spec (table D11 A)With argument Type=regress : same methodology as PROC BENCHMARKING
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Other uses : Seasonal adjustment
X-12-ARIMA V0.3
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)
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X-12-ARIMA V0.3
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.
Other uses : Seasonal adjustment
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Other uses : Seasonal adjustment
X-12-ARIMA V0.3 : example specseries{… save = A18}
transform{function=log}
regression{ variables=(TD easter[8])}
outlier{ …}
arima{…}
forecast{…}
x11{… save = D11}
force{ type=regress
lambda=1
rho=0.9
target=calendaradj
save=SAA }
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Other uses : Seasonal adjustment
Canadian Department Stores Sales SA (D11) and SA with forced annual totals (D11 A)
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
Mill
ions
D11 D11A (λ=1,ρ=0.9)
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Other uses : Seasonal adjustment
Canadian Department Stores Sales
Differences between D11 and D11A
-15,000
-10,000
-5,000
0
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)
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Other uses : Seasonal adjustment
Canadian Department Stores Sales
Growth rates
-2.0%
-1.0%
0.0%
1.0%
2.0%
3.0%
4.0%
Jan- 95 A
pr-
95 Jul-
95
Oct
-95 Ja
n- 96 Apr
-96 Ju
l-96
Oct
-96
Growth rate in D11 Growth rate in D11A
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Other uses : Seasonal adjustment
Annual total starting at any other period other than start may not be equal.
Differences between the sum of 12 consecutive months computed on D11A and on A18
-40-20
020406080
100120140160
Jan-
91
Jan-
92
Jan-
93
Jan-
94
Jan-
95
Jan-
96
Jan-
97
Jan-
98
Jan-
99
Jan-
00
Jan-
01
Jan-
02
Th
ou
san
ds
SumD11A-SumA18
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Other uses : Linking (bridging)
Linking segments of time series with different levels or ranges.
Used to minimize breaks caused by survey redesign, reclassification, change in concept…
Challenges:Estimation of the potential break (parallel run, forecasting, backcasting, …)
Preserve data coherence.
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Other uses : Linking (bridging)
Can usually be achieved with PROC BENCHMARKING:
If the two segments overlap (if not, use a model to extend one of the two segments)
With proper identification of “anchor” points as benchmarks
The smoothing parameter can gradually “bridge” the gap between the two levels
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Other uses : Linking (bridging)
Two segments of a series
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Other uses : Linking (bridging)
Adjusted as a level shift (λ=1, ρ=0.9, bias)
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Other uses : Linking (bridging)
Adjusted as a level shift (λ=1, ρ=0.9, bias)
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Other uses : Linking (bridging)
Adjusted as a gradual level shift (λ=1, ρ=0.9, no bias)
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Other uses : Linking (bridging)
BI ratio for a gradual level shift (λ=1, ρ=0.9, no bias)
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Conclusions
Summary :Many numerical methods can be achieved through PROC BENCHMARKINGDifferent uses of benchmarking
Future developments in PROC BENCHMARKINGSimplify the use of explicit forecastsImprove bias estimationEnhance batch processing (VAR and BY statements)Include more options provided in Dagum and Cholette (2006): more generalised autocorrelation structure of the residuals, measurement errors in the input series, variance estimation of the results.
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