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Evaluating generalised calibration / Fay-Herriot model in CAPEX
Tracy Jones, Angharad Walters, Ria Sanderson and Salah Merad (Office for National Statistics)
Overview
• Introduction
• Generalised calibration estimation
• Fay-Herriot model
• Conclusions and further work
Introduction
• Quarterly survey of capital expenditure (Capex)– Sample size – 28,000– Stratified by industry and size– Main user is National Accounts– Many zeros and some very large values
• Aim to reduce costs and respondent burden – Reduce the sample size whilst maintaining quality
• Investigated two strategies– Calibration estimation in cut-off sampling– Fay-Herriot model
Current cut-off sampling
• Not sample businesses with < 20 employees
• G-weights adjusted to account for this
Sampled (20-299)
Fully enumerated (300+)
Not sampled (<20)
Extension of cut-off sampling
• Extend to a cut-off of < 50 employees
• Sample size reduced by about 9,000• Reduce bias introduced through cut-off sampling
Sampled (50-299)
Fully enumerated (300+)
Not sampled (<50)
Relationship between acquisitions and employment
Direct calibration
• Find set of weights wi such that:
distance (d,w) is minimised while
• Solution
Xx si
iiw
i
iT
ii cFdw
x
Generalised calibration (Deville 2002)
Xx
si
ii
iT
i c
xFd
Xxz si
iiT
iFd
The set of calibration equations
Can be generalised to yield the set of equations
Generalised calibration
• In context of cut-off sampling, Haziza et al
(2010) assumed a linear function F of the form
• And obtained weights
Esi
iii
iii d
-dwTE xz
zXX ˆ1~
iT
iTF z1z
Applying generalised calibration
• Cut-off set deterministically based on employment
• Consider two auxiliary variables:–x well correlated with variable of interest
• employment from the business register
–z well correlated with probability of being above the cut-off• turnover from the business register
Generalised calibration estimation
• 2008 sample data – Bands 2 to 4
• 3 Estimates – Ratio estimate using full sample data – Ratio estimate with extended g-weight adjustment– Generalised calibration estimate
• Relative difference compared to ratio estimate using full sample data
Results
Relative difference (in %) compared to ratio estimate using full sample data
Period Ratio estimate with g-weight adjustment for
band 2
Generalised calibration estimate
Q1 2008 3.3% 32.1%
Q2 2008 5.9% 41.6%
Q3 2008 7.8% 32.8%
Q4 2008 8.1% 37.0%
Industries with largest contribution to total acquisitions in size-bands 2-4
Summary – Extension of cut-off sampling
• Adjusted g-weights method performs better overall
• Generalised calibration estimation does not consistently improve on simple method in any industry
• Residual relationship between x and p
Fay-Herriot model
• Combine direct estimate with synthetic estimate
• Fay-Herriot aggregate level model fitted to obtain synthetic estimator
,direct i i i ix u e
i=1, 2, …,m
Fay-Herriot model - BLUP
)ˆ( ,2
2
idirectu
ui
V
iiidirectiicomb x ˆ1ˆˆ,,
Fay-Herriot model
• 2008 sample data
• Two variables - total acquisitions and total disposals
• Auxiliary variables for Fay-Herriot model – VAT turnover and expenditure
• Scaled estimates and auxiliary variables using the total number of employees
• Fitted mixed model
Plot of Residuals against Predicted (mixed model with no transformation)
Transformation
• Transformation needed
• Implementation of BLUP becomes complicated– noted by Chandra and Chambers, 2006
,log logdirect i i i ix u e
Plot of Residuals against Predicted(mixed model)
Plot of Residuals against Predicted(linear model without random effects)
Back transformation
• Used back transformation to obtain synthetic estimate (Chambers and Dorfman, 2003)
• Calculation of gamma - variance of random effects required back transformation
ie xisyn ee ˆ2/ˆ
,
2ˆ
Evaluating use of Fay-Herriot model
• Gamma very high – Gamma using back transformation may not be
suitable
• Investigated combined estimate using a fixed value for gamma
• Evaluation is via re-sampling – Reduced the sample size by 25% (about 6,000 units)– Repeated sub-sampling– Set gamma to 0.7 – Calculated a combined estimate
Evaluating use of Fay-Herriot model
• Estimated Bias and MSE of combined estimate
fullidirect
K
krunicomb
icomb KBias
k
,,1
,,
,
^ˆ
ˆˆ
Results
• Average of the direct estimates very similar to the direct estimate from the full sample
• Variance of the synthetic estimate is small
• Variance of combined estimate lower than variance of direct estimate from full sample
• Bias is high in most industries– Relative bias also large
• High bias ratio resulted in higher Mean Square Error in most divisions
Results – Acquisitions Q2 2008
Division
Percentage bias
(bands 2 to 4)
Percentage bias
(bands 2 to 5)
Bias ratio
Percentage
difference in MSE
Overall 21.0 5.9 7.1
20 3.5 1.9 0.3 -25
40 18.3 0.4 1.2 139
52 29.8 2.9 1.8 275
25 11.7 7.4 1.7 266
55 12.5 3.0 2.3 509
32 62.2 20.8 5.8 3371
15 28.7 5.5 5.3 2837
Conclusions and further work
• Cut-off sampling with g-weight adjustment performed best– Know this has bias
• More work to be done – Impact on growth– Modelling at unit level– Additional covariates– Alternative estimation methods
• Model-based direct approach (Chandra and Chambers, 2006)
Questions