Rudi Seljak

1

Estimation of Standard Errors of Indices in the
Sampling Business Surveys

• Rudi Seljak
• Statistical Office of the Republic of Slovenia

2
Overview of the presentation

• Introduction of the problem
• Description of the simulation study
• Results of the study
• Conclusions

3
Concept of index numbers

• The concept of index numbers is a widely used
  concept in economy statistics, especially in the
  area of short-term statistics.
• Generally, the index number is a ratio of two or
  more quantities measured with the same unit.
• For the purposes of this study, we will consider
  the index as a ratio of two or more quantities
  measured in two different time points.

4
Different types of indices

• Aggregate approach
• Weighted average approach
• Chained index

5
Value index

• The value index is derived from the general form
  of the aggregate index
• If the values Vt ,V0 are estimated on the basis
  of the random sample, we can write the value
  index as a statistical estimator

6
Value index cont.

• Estimates are generally estimated from two
  different samples, which were selected from two
  different populations
• This fact causes departure from the classical
  estimation of a ratio problem

7
Target population of the study

• The basis for the simulation study were monthly
  data from tax authorities for enterprises from
  sections G-K of NACE classification for
  2004-2005.
• From these data we were able to get quite good
  approximation for turnover which is often the
  target variable in short-term surveys.
• The target population consisted of approximately
  18 000 units in each month.

8
Sampling design

• For the first month, a stratified simple random
  sample was selected.
• The second sample was selected by rotating part
  of the units out of the first sample and
  replacing them with the new units.
• For the rotation procedure the system of
  permanent random numbers was used.
• The stratification was done according to the
  2-digit NACE group and size class, determined on
  the basis of estimated turnover.

9
Sampling design cont.

• All the large enterprises were selected with
  certainty.
• The target statistics was the estimator of the
  turnover index
• ......sampling weights for current
  and base month
• ......turnover for selected units in
  current and base month

10
Simulation

• The object of the simulation study was to explore
  the sampling distribution of the estimator and to
  compare three different methods for variance
  estimation.
• The parameters in the simulation study were
• size of the samples
• time gap between the two selected months
• the rotation rate of the second sample

11
Methods for variance estimation

• Three methods for variance estimation were tested
  and compared. Two of them are based on the
  analytical approach. The third one uses the
  re-sampling approach.
• The basis for the analytical approaches was the
  Taylor linearization formula for the variance
  estimation of the estimator of the ratio
• Using SAS procedure SURVEYMEANS we were able to
  estimate and but the procedure
  doesnt provide the estimation of the sampling
  covariance.

12
Methods for variance estimation cont.

• With the first approach we estimated the sampling
  covariance by using the formula for the variance
  of the sum of two random variables
• With this approach we also had to construct the
  common weight for the sum. We used equation

13
Methods for variance estimation cont.

• With the second approach the sampling covariance
  was calculated directly by using formula for
  sampling covariance of estimates from two partly
  overlapping samples

14
Methods for variance estimation cont.

• With the third approach we used the well known
  Jackknife replication method.
• If we wanted to use the existing software, we had
  to merge two samples into one data set and then
  adequately adjust the data.
• The weights from both samples had to be
  composed into a common weight.
• Zero values were inserted for the units which
  were at a certain time point not in the sample.

15
Sampling distribution

• To explore the sampling distribution of the
  estimated index, we selected 10.000 partly
  overlapping pairs of samples, using the same
  sampling design.
• The changing parameters were the size of the
  samples, time lag between two time points and the
  rate of rotation in the second sample.

16
Sampling distribution cont.

• We expected to get the distribution that would be
  at least approximately normal.
• In most cases that was really the case and the
  histogram looked like the following one

17
Sampling distribution cont.

• We expected to get the distribution that would be
  at least approximately normal.
• In most cases that was really the case and the
  histogram looked like the following one

18
Sampling distribution cont.

• But in some cases the shape of normality
  disappeared and we have been faced with clear
  bimodal distribution.

19
Sampling distribution cont.

• But in some cases the shape of normality
  disappeared and we have been faced with clear
  bimodal distribution.

20
The problem of bimodality

• The source of the bimodality is the distribution
  of the estimator of the total, either in the
  denominator or in the enumerator.
• In all the cases where the bimodality appeared at
  least one of the estimators of the totals was
  bimodally distributed.
• For now we couldnt find the exact reason for
  bimodality. It should be a subject of further
  investigation.

21
Comparison of the methods The case of normal
distribution

• In the case when the sampling distribution was
  (approx.) normal all the methods worked quite
  well.
• In the picture we will show the case when we
  fixed the months and the sample size and we were
  only changing the rotation rate.

22
Comparison of the methods The case of normal
distribution

• In the case when the sampling distribution was
  (approx.) normal all the methods worked quite
  well.
• In the picture we will show the case when we
  fixed the months and the sample size and we were
  only changing the rotation rate.

23
Comparison of the methods The case of normal
distribution

• In the case when the sampling distribution was
  (approx.) normal all the methods worked quite
  well.
• In the picture we will show the case when we
  fixed the months and the sample size and we were
  only changing the rotation rate.

24
Comparison of the methods The case of normal
distribution

• In the case when the sampling distribution was
  (approx.) normal all the methods worked quite
  well.
• In the picture we will show the case when we
  fixed the months and the sample size and we were
  only changing the rotation rate.

25
Comparison of the methods The case of normal
distribution

• In the case when the sampling distribution was
  (approx.) normal all the methods worked quite
  well.
• In the picture we will show the case when we
  fixed the months and the sample size and we were
  only changing the rotation rate.

26
Comparison of the methods The case of normal
distribution

• In the case when the sampling distribution was
  (approx.) normal all the methods worked quite
  well.
• In the picture we will show the case when we
  fixed the months and the sample size and we were
  only changing the rotation rate.

27
Comparison of the methods The case of bimodal
distribution

• In the case of bimodal sampling distribution the
  sampling variance was significantly higher.
• In the pictures we show the case where the
  sampling size was fixed to 4000, the rotation
  rate to 0.2 and we were changing the time lag
  between the months.

28
Comparison of the methods The case of bimodal
distribution

• In the case of bimodal sampling distribution the
  sampling variance was significantly higher.
• In the pictures we show the case where the
  sampling size was fixed to 4000, the rotation
  rate to 0.2 and we were changing the time lag
  between the months.

29
Comparison of the methods The case of bimodal
distribution

• In the case of bimodal sampling distribution the
  sampling variance was significantly higher.
• In the pictures we show the case where the
  sampling size was fixed to 4000, the rotation
  rate to 0.2 and we were changing the time lag
  between the months.

30
Comparison of the methods The case of bimodal
distribution

• In the case of bimodal sampling distribution the
  sampling variance was significantly higher.
• In the pictures we show the case where the
  sampling size was fixed to 4000, the rotation
  rate to 0.2 and we were changing the time lag
  between the months.

31
Comparison of the methods The case of bimodal
distribution

• In the case of bimodal sampling distribution the
  sampling variance was significantly higher.
• In the pictures we show the case where the
  sampling size was fixed to 4000, the rotation
  rate to 0.2 and we were changing the time lag
  between the months.

32
Comparison of the methods The case of bimodal
distribution

• In the case of bimodal sampling distribution the
  sampling variance was significantly higher.
• In the pictures we show the case where the
  sampling size was fixed to 4000, the rotation
  rate to 0.2 and we were changing the time lag
  between the months.

33
Conclusions

• The second analytical method performs slightly
  better but the advantage of the first method is
  that it requires less tailor-made programming.
• The Jackknife method slightly overestimates the
  variance but we judge this bias is due to the
  technical reasons of adjustment of the method and
  it could be decreased.
• The variability of the estimates is the lowest in
  the case of the JKK method.

34
Conclusions cont.

• The problem of bimodality should be further
  investigated in the future.
• Bimodal sampling distribution can cause serious
  instability in the procedure of variance
  estimation.
• Bimodality of the distribution should require
  different interpretation of the sampling
  variation.