# Haziza

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```					  ESTIMATION IN THE
PRESENCE OF TAX DATA

David Haziza, Gordon Kuromi and Joana Bérubé
Université de Montréal & Statistics Canada

ICESIII
June 20, 2007

1
OUTLINE

• Introduction

• Current sampling design

• Current point estimators

• Alternative sampling design

• Alternative estimators

• Domain estimation

• When the tax variable is missing

2
TAX DATA PROGRAM

• Goal: To increase the use of tax data in business surveys in
order to

   reduce the respondents burden

   reduce costs

   potentially improve the quality of point estimators

3
TYPES OF VARIABLES
• We distinguish between 3 types of variables
   Financial survey variables (total revenue, total expenditure,
etc)
   Financial tax variables (total revenue, total expenditure, etc)
   Non-financial variables

• There is a direct link between the financial survey variables and
the financial tax variables
• No direct link between non-financial variables and tax variables
4
TAX DATA

• 3 type of tax data:

   T1 data: un-incorporated businesses (Unified Enterprise
Survey)

   T2 data: incorporated businesses (Unified Enterprise
Survey)

   GST data: both incorporated & un-incorporated (Monthly
surveys)

5
CURRENT SAMPLING DESIGN

• Stratification by Province, NAICS and Size

• 3 types of strata:

   Take-all strata (typically complex units)

   Take-some strata (simple and complex units)

   Take-none strata (simple units)

• Use of tax data is limited to take some strata (for simple units
only) and take-none strata

6
CURRENT SAMPLING DESIGN
STRATUM = PROVINCE x NAICS

Noneligible units                      Eligible units

 
seh (neh )

     
sneh (nneh )                    seh (neh )

 50%
 
sh (nh )
7
U h ( Nh )
CURRENT SAMPLING DESIGN

 The current design fits the imputation and estimation systems

 It is a two-phase sampling design
 The sample sizes for collection in both the eligible and non-
eligible strata are random variables         may increase the
variance of the estimators and add uncertainty to the collection
costs
 The use of tax data is limited to the first-phase sample.

8
FINANCIAL VARIABLES
• Survey variables: y  available only for the units in sh

• Tax variables: x  available for all the units in U h

• For many financial variables, there is a corresponding tax
variable

• We assume that both type of variables are known without
errors (measurement errors, nonresponse)

• These two assumptions are not satisfied in practice!

9
CURRENT TAX REPLACEMENT METHODS
• Model describing the relationship between x and y:
yi  f  xi    i
E  i   0,V  i    2ci

• Special cases:

   Direct tax replacement: f  xi   xi , ci  1      yi  xi   i

   Ratio type replacement: f  xi    xi , ci  xi   yi  xi   i

10
PREDICTED VALUES

ˆ
• yi : predicted value for yi

   Direct tax replacement: yi  xi
ˆ          (used in UES)

                                ˆ
Ratio type replacement: yi  Bxi
ˆ          (used in monthly surveys)

ˆ
• Estimate of B is obtained from the units in s.

11
NOT ONLY DIRECT TYPE REPLACEMENT?

• Considerable efforts have been made to standardize the
concepts and definitions between the tax variables and the
survey variables (Chart of Account compliance for T1 and T2)

• As a result, we expect that the model y  x   should be valid.
Sometimes, it is not because

   Difference in reporting of data and other issues (Jocelyn,
Mach et Pelletier, 2006)

   Difference in the reference period (GST data)
12
CURRENT POINT ESTIMATORS:
PREDICTION TYPE
• In the noneligible portion: Horvitz-Thompson estimator

• In the eligible portion: Prediction (or imputed) type estimator

   y is observed for all i in s

   y is used for i in s s
ˆ

   We have
ˆ s
YPRED   wi yi         wi yi
ˆ
is        is s

13
CURRENT POINT ESTIMATORS:
PREDICTION TYPE
 Similar to imputed estimators in the context of imputation
 They are simple and fit the current imputation and estimation
systems
 They fit the so-called micro approach for displaying the data

 They are generally p-biased
 May be pm-biased if the tax replacement model is incorrectly
specified
14
DISPLAYING THE DATA: MICRO VS.
MACRO APPROACH
• We distinguish between two approaches for displaying the data:

(i) Micro approach: consists of reporting the observed y-values
as well as the predicted values (similar to an imputed file in the
context of item nonresponse)

(ii) Macro approach: consists of reporting the observed y-values
along with a calibration weight

• Currently, the micro approach is used
15
PREDICTION TYPE ESTIMATORS
Domain estimators
• Micro approach               potentially p-biased and pm-
biased
Unit      ~
y      wi
1       y1      
w1
2               
s 
y2     w2

              
Domain
n
         yn    wn 
n  1   ˆ
yn 1    
wn1
               
s s 
                
 
 n
        ˆ
yn       
wn                  16
ALTERNATIVE SAMPLING DESIGN
STRATUM = PROVINCE x NAICS

Noneligible units                  Eligible units
U neh ( N neh )                   U eh ( N eh )

seh (neh )
sneh (nneh )

17
U h ( Nh )
ALTERNATIVE SAMPLING DESIGN

 It is a single phase sampling design which simplifies the
estimation procedures, particularly variance estimation
 The sample sizes are known prior to sampling
 Full use of available tax data is now made

 The estimation systems need to be modified to fit the new
procedure

18
POINT ESTIMATION

• For Financial variables, we have 2 options:
   Tax/survey based framework: We simply use X        x
iU E
i   for
the eligible part and a design consistent estimator for the
noneligible part
   Survey based framework: We want to estimate Y       y
iU
i

Use design consistent estimators (calibration estimators such
as the GREG or optimal estimator) that make use of all the
available tax data (Monthly surveys)

19
GREG TYPE ESTIMATORS

• The GREG estimator is usually written as

YG   wiCAL yi
ˆ
is

• The GREG estimator fits the macro approach but it can
also fit the micro approach

YG   yi   wi  yi  yi 
ˆ     ˆ                ˆ
iU         is

20
GREG TYPE ESTIMATORS
Domain estimators
Micro approach                    asymptotically p-unbiased
Unit      ˆ
y      e y y
ˆ        wi
1          ˆ
y1      y1  y1
ˆ         w1
2          ˆ       y2  y2
ˆ
           y2                     w2
s 
                                            Domain
n
            ˆ
yn     yn  yn
ˆ        wn
n  1      ˆ
yn1        0           0


Us 

 N
           ˆ
yN           0          0
21
DOMAIN ESTIMATION

• Three situations are encountered in practice:
(i) The domain is identical with the model group
(ii) The domain is contained in the model group
(iii) The domain interesects more than one model groups

22
DOMAIN ESTIMATION

• Even if the prediction type estimators are pm-unbiased at the
model group level, they could be significantly biased if the
model prevailing at the domain level is different than the model
prevailing at the model group level

• The GREG type estimators are always asymptotically p-unbiased
at the domain level. However, they could be inefficient if the
model prevailing at the domain level is different than the model
prevailing at the model group level

23
DOMAIN ESTIMATION: MICRO vs.
MACRO

• Macro and micro approaches lead to identical estimators of
parameters at the model group level

• At the domain level, both approaches lead to different
estimators

• No definite comparison is possible but we expect that Yd micro 
ˆ
will perform better than Y  macro  if the domain size is small
ˆ d

24
WHEN THE TAX VARIABLE IS MISSING

•   In practice, the tax variable is subject to nonresponse and it is
imputed

•   Let z be a new variable defined as: x if the tax variable is
observed and x* if the tax variable is missing

•   Inference can be made conditional on z

25
FUTURE WORK

•   Find a compromise calibration weight if the macro approach
is used

•   For non-financial variables, find the best set of auxiliary
x*
variables and use it to calibrate

26
Pour plus d’informations, veuillez contacter/ for more

David.haziza@statcan.ca
(613) 951-5221

27

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