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Theory and Application of Benchmarking in Business Surveys. Susie Fortier and Benoit Quenneville Statistics Canada -TSRAC ICES – June 2007. Content. Introduction Notation Benchmarking methods Timeliness issue Implied forecasts and annual growth rates Other uses:
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Theory and Application of Benchmarking in Business Surveys Susie Fortier and Benoit Quenneville Statistics Canada -TSRAC ICES – June 2007
Content • Introduction • Notation • Benchmarking methods • Timeliness issue • Implied forecasts and annual growth rates • Other uses: • Seasonally adjusted data • Linking problem • Conclusions
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.
Introduction Benchmarking : • Combining a series of high-frequency data with a series of less frequent data into a consistent time series.
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.
Introduction Example of a quarterly series
Introduction A quarterly series with its auxiliary source
Timeliness issue Introduction
Introduction Benchmarked series
Notation Methodological details :
Notation Methodological details : • With binding benchmarking, the benchmarked series is such that
Notation • A bias parameter can be estimated and used to pre-adjust the indicator series: • A bias corrected series is obtained as:
Notation • Alternatively, the bias can be expressed in terms of a ratio: • The bias corrected series is then:
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.
Notation Effect of the Bias Correction (ratio)
Methods : Pro-rating • A simple way to respect the constraints is to use This is the well-known formula for pro-rating.
Methods : Pro-rating Benchmarked series with pro-rating
Methods : Pro-rating BI ratio with pro-rating
Methods : Pro-rating Growth rates with pro-rating
Methods : Pro-rating Growth rates with pro-rating
Methods : Proportional Denton Benchmarked series with Prop. Denton
Methods : Proportional Denton BI ratio with Prop. Denton
Methods : Proportional Denton Growth rates with Prop. Denton
Methods : Proportional Denton Growth rates with Prop. Denton
Main method • 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 Forillon • Software Demo
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
Main method : Formula • Solution when : Solution: “Regression-based” model from Dagum & Cholette
Main method : Formula • Solution when : Solution: where W is the T x M upper-right corner matrix from :
Main method : Formula • We can obtain pro-rating with the general formula with and : minimise under gives
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 Dentonmethod
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 Dentonmethod with positive data!
Main method : Effect of • 3 parameters at play: • : model adjustment parameter • : “smoothing” parameter • bias (implied with ) subject to
Main method : Effect of bias Benchmarking without bias ( )
Main method : Effect of bias Benchmarking with bias ( )
Main method : Effect of bias Benchmarking without bias ( )
Main method : Effect of bias Benchmarking with bias ( )
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)
Timeliness issues • 2 implicit forecasts for 2006: • Enhanced benchmarking method with explicitforecasts
Timeliness issues • One possibility for explicit forecast: • Use the annual growth rate from the indicator series on the last known benchmark.
Timeliness issues With explicit forecast ( )
Timeliness issues With explicit forecast ( )
Timeliness issues With ″recent″ bias( , bias=0.94)
Timeliness issues With ″recent″ bias( , bias=0.94)
Timeliness issues Minimize revision?
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.
Methods • Other methods • Other 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?
Syntax : PROC Benchmarking PLEASE SEE SOFTWARE DEMO !! PROCBENCHMARKING BENCHMARKS=myBenchmarks SERIES=mySeries OUTBENCHMARKS=outBenchmarks OUTSERIES=outSeries OUTGRAPHTABLE=outGraph RHO=0.729 LAMBDA=1 BIASOPTION=3; RUN;
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