Statistical Office of the European Communities / Directorate General Statistics
16 April 2007
TF-SAQNA-19
Task Force on Seasonal Adjustment
of Quarterly National Accounts
Second Meeting
Agenda Item 6
______
Chain-Linking and Seasonal Adjustment
Order of Chain-Linking, Benchmarking and Adjustment
Draft contribution to a paper on chain-linking by Sven Öhlén
Statistics Sweden 14 April 2007
Sven Öhlén
Draft
Order of Chain-Linking Benchmarking and Seasonal Adjustment
Abstract
This paper addresses the combined issues of chain-linking (CL) benchmarking (BM) and seasonal adjustment (SA). The purpose of CL is to make comparisons over time in NA aggregates in times of changing relative prices and volumes. The purpose of BM is to adjust the CL quarterly aggregates to revised yearly totals in ANA, without rewriting the history of dynamics in QNA. The purpose of seasonal adjustment is to justify intra-year comparisons when the time series depend on seasonality and working-day/calendar effects and also on special sources of short term and/or 'external' variability, i.e., outliers. Comparisons between periods of arbitrary dates and lengths should be consistent. In a broad sense, all methods are about maximizing 'utility of statistics' from a users point of view. The mean is to separate 'true signals' from different sources of not relevant variability. In this paper we focus on the order of procedures. It could not be taken for granted that the order of procedures does not have any effect on e.g., estimated growth rates in QNA.
Key words :National accounts, chain-linking, annual overlap, over-the-years, quarterly overlap, benchmarking, model-based seasonal adjustment,
Contents
0. Introduction 2
1. Approach 2
2. Model-based Seasonal Adjustment 4
3. Benchmarking 5
4. A tentative Discussion of the Order of Procedures 7
4.1 Non-linearity of Procedures and Additivity 7
4.2 Seasonality and the Order of Procedures 8
4.3 Maintenance of SA and the Order of Procedures 9
4.4 Choice of Chain-linking Method and Seasonality 9
5. Some Empirical Findings on the Order of Procedures 10
5.1 An Extended Example from the IMF-manual 10
5.2 . Examples from real Data 14
6. Summary and Suggestions 19
0. Introduction
In the task-force in February, one delegate raised the question 'Does the order of seasonal adjustment and chain-linking matter?' in terms of estimating e.g., relative growth rates (GR). The purpose of this paper is to start a discussion of this issue which will be called 'Order of Procedures (OP)'. The reader is supposed to be familiar with chain-linking (CL), benchmarking (BM) and model-based seasonal adjustment (SA). The paper is organized as follows. In the next chapter, we introduce some notations for clarity and simplifying the discussion on order of procedures. Chapter 2 draws the attention to some important properties of model-based seasonal adjustment followed by an outline of benchmarking. The discussion of order is given in chapter 4 followed by some empirical findings. The empirical results shown are restricted to the order of CL and SA.
1. Approach
We consider CL, BM and SA as transformations of original data to the final data through the combined scheme
(1) à CL()àBM()àSA()
within an input/output context. is the input in CL, which produces the output which is input in BM with output , and finally transformed to by SA. The scheme (1) is a simplification in a wider class of possible sequences of the order of procedures in NA. BM could be and is sometimes used in many phases of the production cycle in the NA. That is, the chain of operations could be extended to 'cycles' as
(1:1) à CL()àBM()àSA()
where intermediate or 'preliminary final' results are used e.g., in conciliations, changing original data, followed by a new cycle until consistency of 'all parts' of the NA. In what follows, we do not consider this extended use of the methods[1].
We also have users of statistics. The total satisfaction from all users is considered to be measurable within some kind of total satisfaction of ,based on utility U, i.e.,
User utility:
(1:2) U=U()
Producing costs money, C according to
Production costs:
(1.3) C=C(à CL()àBM()àSA())
The cost of producing depends on the number of production processes, such as CL, SA and BM and of course on the complexity of the methods. Complex methods are difficult to implement and costs extra money and also in e.g., maintenance. In what follows, we assume positive marginal costs of including one more procedure, i.e.,
(1.4) C(SA,CL,BM,SA)>C(CL,BM,SA)
(1) can also be written
(2) ,
where F is the notation for the combined procedure
(3) F=F(CL,BM,SA)
If there are three methods of CL, ten for BM, and say 100 options for SA, F will transform original data to 3000 different output values with some kind of intermediate results . With these notations, a change of order of the procedures could e.g., be
(4)
If the results are independent of the order of procedures for all F.
In what follows, we use the simplified notation
(5) ,
It is well known that different choices of CL produce different output from this step alone. That is also the case for BM and SA. What we would like to know, is if
(6) =
are equal or for any order
(7) order 1
order 2
order 3
order 4
order 5
order 6
.
That is, given the same original data in current prices, there are six schemes of CL, BM and SA. If we choose AO, the Min D4 –method (see below) of BM and model-based SA with TRAMO/SEATS with a multiplicative ARIMA(011)(011) with trading-days regression with an international calendar, there are still six ways of producing the results depending on the order of procedures. Before going to the issue of order, we draw the attention to some important features of model-based seasonal adjustment and benchmarking.
2. Model-based Seasonal Adjustment
Very few users of SA series do understand the nature of seasonal adjustment. Many have the idea of true seasonal components because it could be associated with 'real things' like weather and observed consumer behaviour of e.g., consumption. Everyone with some experience in seasonal adjustment know that there is no ‘true’ seasonal adjusted value. For a particular time there is a distribution of SA values depending on many things as software, statistical method, numerical algorithms and the crucial choice of a model for the series. In order to illustrate the uncertainty of the SA series stemming from different choices of the ARIMA-model, we show graph 2.1 below. It shows the annualised quarterly changes of the SA values for Swedish GDP based on 50 different ARIMA-models for 1993 – 2001. In the first place we can observe that the choice of the ARIMA-model is important for all times[2] not only at the end of the series. The range is about two per cent
Fig. 1
Annualised Quarterly Growth Rates for Swedish GDP.
SA and working-day adjusted data based on
50 ARIMA-models[3]
To every method of seasonal adjustment there are several kinds of uncertainty about the estimated seasonal adjusted value, the trend, or any unobserved component of the time series. X-12-ARIMA and TRAMO/SEATS produces different seasonal adjusted values for the same time series even when the same ARIMA-model is used. From a users point of view, that is very depressing because many users have the idea of ’true’ seasonal adjusted value. Corresponding results can be shown for all unobserved components for a particular time series, e.g. the trend, the working-day effects, and possibly estimated outliers. That is, the estimated unobserved components depend on the choice of the ARIMA-model. Every time we change the ARIMA-model for the series, we will revise the estimated unobserved components. Behind the original data for Fig. 1, we have used the particular order of procedures, CLàBMàSA. Changing CL and BM would certainly change this distribution of growth rates. That is, the variability of estimated growth rates depends on CL, BM and SA. Does the order of procedures count?
3. Benchmarking
Benchmarking is used in national accounts and also in seasonal adjustment to attain consistency in several ways. In NA it is used to attain consistency between revised yearly totals in ANA and preliminary quarterly accounts QNA. In SA it can be used to force yearly SA values to correspond to the original yearly values. In what follows, we focus on how BM is used in NA and possible implications from BM in general terms and/or implications from particular BM-methods and seasonal adjustment. The details of BM go far away from the scope of the TF. Therefore, we will not go into details on this issue. Before the discussion of the order of procedures, we will show some results on the issue of choice of ARIMA-model, type of bias of the preliminary QNA and software. [4] First we give an outline of BM-principles in lines with Denton 1971.
Benchmarking Principles
Let x be preliminary QA, z revised QA, X preliminary ANA and Z revised ANA. , , and B=Z-X the yearly bias. The idea of Denton, is that the quarterly revised values should be chosen that they do not distort the history of 'dynamics' of the preliminary series . This is done by minimizing different distance measures like where bold face indicates vector of corresponding values. This measure is called MinD4[5]. It is minimizing the changes of the quarterly revision errors. The BM-problem is given as follows. Minimize the particular choice of a distance function like D4 subject to the yearly restrictions . In most cases, there is an exact solution, which distributes the yearly bias B to the preliminary quarterly values . One way of distributing B, is in equal amounts, i.e., . A different way is to distribute the bias it in proportion to , i.e., .
Benchmarking and Seasonal Adjustment
The properties of BM are well known. The effects from BM upon SA are to a large extent unknown. For instance, the question 'does BM' change the dynamics of the series such that we have to search for a new model for the time series. Another issue is about possible differences between TRAMO/SEATS and X-12-ARIMA. In table 3.1 below[6], we show the proportion of cases, when BM does not change the ARIMA-model used with automatic model-selection. Two different schemes of yearly bias has been used, uniformly bias and when bias in increasing from Q1 to Q4.[7] Distributing the bias in equal amounts is called UA (Uniformly and Additive), in equal proportion to the preliminary values UM (Uniformly and multiplicative) in the table.
Table 3.1. Proportion of cases where the ARIMA-model for the preliminary series is not changed by benchmarking.
Uniform bias:
TRAMO
/SEATS / 0.37 / 0.37 / 0.23
X-12-A / 0.43 / 0.43 / 0.26
Increasing bias:
TRAMO
/SEATS / 0.43 / 0.43 / 0.43
X-12-A / 0.43 / 0.71 / 0.71
In 37 per cent of cases of BM with the method UA, there is no need to change the ARIMA-model in the case of uniformly distributed bias when TRAMO/SEATS is used. The corresponding for X-12-ARIMA is 43 per cent. For TRAMO/SEATS, the normal result from BM is that we have to change the ARIMA-model. X-12-ARIMA seem to be more robust in the case of increasing bias and the methods UM and MinD4.
As a summary of this chapter, we should expect that even when BM is used for the preservation of the old dynamics of the preliminary series, the rule is that BM changes the series so much that there is a need to reexamine the ARIMA-model used for the preliminary series
4. A tentative Discussion of the Order of Procedures
In chapter 2, the question was raised, whether or not the output from the six schemes of order would be the same, i.e., if the output . In practice, there would certainly be some differences. If so, what are their magnitude?
This question is at present almost impossible to answer from a theoretical point of view. We can use simulations and/or we can empirically investigate the sensitivity for particular data and different schemes of order. In this chapter, we give some general comments on the order followed by a few empirical examples.
4.1 Non-linearity of Procedures and Additivity
Chain-linking introduces non-additive elements and non-consistency in the system of NA. For simplicity, CL could be considered as a non-linear method or transformation of original data. BM could be made with linear or even non-linear methods[8]. Seasonal adjustment is a linear procedure but the estimation of the parameters of the models is highly non-linear[9]. Putting non-linear methods together, could certainly have un-expected results. Therefore, there seems to be reason to avoid non-linear methods as far as possible within the context of CL-BM- and SA because the combined effects will be unpredictable. When relative prices are changing, we can not avoid chain-linking. Thus, we can not avoid non-linear elements. However, we can avoid non-linear BM methods by complying to simple methods in lines with Denton and the the recommendations given by the IMF-manual of 2001[10]. Next, it seems a good idea to separate the production processes and to avoid multiple production steps as indicated by the scheme
à CL()àBM()àSA()