Supplement 4/2005 to the Statistical Bulletin, November 2005

E U R O S Y S T E M

Seasonal Adjustment of Balance of Payments Statistics

Supplement 4/2005 to the Statistical Bulletin, November 2005

Banco de Portugal

E U R O S Y S T E M

Supplement 4|2005 to the Statistical Bulletin, November 2005

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Seasonal Adjustment of Balance of Payments Statistics

TABLE OF CONTENTS

1. INTRODUCTION ..................................................................................................................................................................................................

5

2. SEASONAL ADJUSTMENT RELEVANCE ............................................................................................................................................................

5

3. METHODOLOGY .................................................................................................................................................................................................

7

4. BRIEF DISCUSSION OF RESULTS ..................................................................................................................................................................... 10

ANNEXES

Annex A - Methodology .......................................................................................................................................................................................... 15

A.1 Time series and seasonal adjustment ................................................................................................................................................. 17

A.2 The X-12-ARIMA and TRAMO-SEATS programmes ........................................................................................................................... 21

A.3 Comparative analysis of the X-12-ARIMA and TRAMO-SEATS programmes ...................................................................................... 22

Annex B - Details of the models used in the seasonal adjustment of the information ................................................................................................. 25

Annex C - Table of the Statistical Bulletin with seasonally adjusted data ............................................................................................................. 29

REFERENCES ..................................................................................................................................................................................................... 33

SUPPLEMENTS TO THE STATISTICAL BULLETIN .......................................................................................................................................... 35

Banco de Portugal | Supplement 4/2005 to the Statistical Bulletin, November 2005 3


1. INTRODUCTION

The Banco de Portugal statutes stipulate that it is responsible for the compilation and publication of the country’s balance of payments statistics.

Such statistics constitute a systematic record of all transactions made by residents and non-residents of a given economy over a certain period of time. These include goods, services and income transactions, acquisition or disposal of financial claims on, and liabilities to, the rest of the world, and offsets to current economic values provided or acquired without a quid pro quo, such as donations and debt relief. Besides their intrinsic analytical value, the balance of payments statistics are essential for the compilation of the “rest of the world account” of the financial and non-financial national accounts integrated in the System of National Accounts, and this is important for a global and coherent appraisal of the economy.

In an increasingly global economic environment, foreign economic relations have assumed an ever greater importance. The balance of payments statistics have thus become an indispensable element of economic analysis and policy, notably in the assessment of the business sector financial situation and competitiveness.

The high level of the net external borrowing requirements of the Portuguese economy, highlighted by the balance of payments statistics, makes a systematic treatment of these statistics particularly relevant for assessing the economy’s long-term trends as well as short-term developments.

However, several items in the balance of payments statistics reveal that its monthly evolution is affected by what are generally regular and periodic fluctuations, i.e., of a seasonal nature. These can hinder the detection of other short-term trends and movements and consequently distort the economic analysis. With the purpose of improving the information and providing an additional analytical tool for users, the Banco de

Portugal has established a methodology for the removal of seasonal effects from statistical time series, above all from the current and capital accounts. The recommendations and experience of a number of international institutions such as the European Central Bank (ECB), Eurostat, and other National Central Banks have been taken into account. The aim of the present supplement is to present the aforementioned methodology as well as the main seasonally adjusted time series, which will henceforth be published regularly in the Statistical Bulletin.

The supplement is organised as follows: section 2 briefly discusses the relevance of the seasonal adjustment process; section 3 presents the methodology; and section 4 summarizes the main results. Some additional aspects of the study can be found in the Appendices. More specifically, Appendix A introduces the main methodological concepts of the time series analysis and, in particular, of the seasonal adjustment process. Appendix B goes through the details of the models used in the seasonal adjustment of the series; and Appendix C contains the table with the seasonally adjusted series of the current and capital accounts, to be published in the Statistical Bulletin. This table is included in chapter

C.1. “Current and Capital Accounts” and it provides information on the seasonally adjusted data for the current account total and its main components (goods, services, income and current transfers).

2. SEASONAL ADJUSTMENT RELEVANCE

It is well established that the time of the year has a significant effect on a set of balance of payments statistics. In fact there may be seasonal fluctuations large enough to conceal other trends and short-term movements and thus distort the economic analysis of short-term balance of payments developments.

Various factors lie at the root of seasonal fluctuations. The most important ones are natural factors (e.g., climate) and institutional factors (e.g., the right to holidays that tend to be taken in summer).

1 There are also others, those of a social, cultural or religious nature (e.g., Christmas,

Easter).

1 In Portugal, the annual wage of most workers is paid in 14 instalments, one per each month of the year, an additional instalment before the summer holidays

(the so called holiday wage) and another towards the end of the year (the so called Christmas wage).



In order to neutralise the impact of these factors, seasonally adjusted series are estimated, thus complementing the statistical information with an additional analytical tool.

To illustrate the importance of this new analytical tool, the diagram of both the raw series and the seasonally adjusted series of the

“Travel” balance is presented in Figure 1. Typically, the raw series shows higher values in July and August than in the adjacent months. However, this evolution is mainly due to seasonal fluctuations and is therefore not a very informative measure of the direction in which the series is moving. The seasonal adjustment gives a more precise idea of the trend. Moreover, due to the seasonal pattern, the impact of events such as EXPO 98 and EURO 2004 can hardly be perceived with a superficial inspection of the raw series. On the other hand, a look at the seasonally adjusted series shows the presence of a transient enhancement in the net exports of this type of service during these periods. This is bound to draw the analyst’s attention to the specific effects of these events.

Figure 1. Balance of the “Travel” item

700

600

500

400

300

200

100

0

1996 1997 1998 1999 2000

Raw series

2001 2002

Seasonally adjusted series

2003 2004 2005

The interpretation of the joint balance of the Current and Capital Accounts, which reflects the net external borrowing requirements of the economy, constitutes another example of the usefulness and analytical value of the seasonally adjusted series. In this context, the raw and seasonally adjusted series of the Current and Capital Account balances as a percentage of GDP are presented in the next figure. As can be observed, the raw series gives the impression that the Current and Capital Account deficit widened considerably from the first to the second quarter of 2005. However, as can be deduced from the corresponding seasonally adjusted series, part of this stems from a seasonal effect. In point of fact, the adjusted deficit for the second quarter of 2005 is only marginally greater than the estimated deficit for the first quarter.

6 Banco de Portugal | Supplement 4/2005 to the Statistical Bulletin, November 2005


Figure 2. Combined current and capital account balance (in value and as a percentage of GDP)

0

- 1

- 2

- 3

- 4

- 5

- 6

- 7

- 8

- 9

- 10

2002

2

2002

3

2002

4

2003

1

2003

2

2003 2003

3 4

2004

1

2004

2

2004

3

2004

4

2005 2005

1 2

Raw series Seasonally adjusted series

3. METHODOLOGY

The methodological approach adopted in the seasonal adjustment of certain balance of payments statistics is based on the estimation of ARIMA 2 models for the statistical time series of both Current and Capital Accounts. For each series, an ARIMA model is selected and estimated by taking into account a set of statistical criteria, such as the stability and the relatively small number of estimated parameters, and the quality of the estimation. The estimated models are then used for the decomposition of the time series into their components and, consequently, for the identification of the respective seasonal components. It should be noted that this identification is of a probabilistic nature and may be revised in accordance with any revision of the raw information. Thus, the seasonally adjusted series should not be regarded as a substitute of the raw series but as complementary information for economic analysis.

The seasonal adjustment is made on a monthly basis in agreement with the production and dissemination frequency of the balance of payments statistics. It should be noted that the results from a monthly adjustment might not be exactly equal to those that would be obtained from a quarterly adjustment of the same information. In the quarterly adjustment, the three monthly values of the series under study are added up and then seasonally adjusted. This summation of the monthly values means that the quarterly series are not as variable as the monthly series. Monthly adjustment is therefore generally a more demanding process than quarterly adjustment. Nevertheless, given the need for information to be available before the end of each quarter, a monthly frequency has been adopted. The same option has been taken by the ECB, Eurostat and by a majority of National Central Banks.

2 The ARIMA models (Auto Regressive Integrated Moving Average) are univariate stochastic models frequently applied in the seasonal adjustment of time series. For a brief presentation of these models, see Appendix A, Section A.1.



It is important to stress that the estimation for the models is carried out from the raw series corrected of outliers 3 and calendar effects 4 , preventing any distortion caused by these effects on the identification of the seasonal component. Subsequently, the respective seasonal component is removed from each original series. Therefore, the resultant seasonally adjusted data continue to include the outliers and the calendar effects present in the raw series.

The set of seasonally adjusted data must be produced and made available every month in good time, so it became pertinent to adopt software programmes specially suited for time series seasonal adjustment. The two most popular programmes in this domain are X-12-ARIMA and TRAMO-SEATS, 5 which are used by Eurostat and the ECB. Both TRAMO-SEATS and X-12-ARIMA rely on ARIMA modelling of time series, but they follow two different methods for seasonal adjustment: X-12-ARIMA is an empirical programme based on ad-hoc rules that take into account certain characteristics shared by many economic time series; TRAMO-

-SEATS is the combination of the programmes TRAMO (Time Series Regression with ARIMA Noise, Missing Observations, and

Outliers) and SEATS (Signal Extraction in ARIMA Time Series), and is based on signal extraction techniques applied to ARIMA models.

As there is no consensus on the most adequate programme for the seasonal adjustment of economic time series, a comparative analysis of the two programmes was carried out.

6 It was concluded that the procedure applied by TRAMO-SEAT, supported by a statistical theory, was more complete and versatile than X-12-ARIMA. Nevertheless, it was observed that SEATS may induce residual seasonality in series that, in conventional tests, have no statistically significant seasonal effects. To get over this limitation, an initial step was included in the methodological procedure, making use of X-12-ARIMA tests to evaluate the statistical significance of the seasonal component of each time series.

The methodological approach adopted thus combines the two programmes in an articulated way. As a first stage, X-12-ARIMA is used to test for the presence of seasonal effects. Once the presence of seasonality is confirmed, TRAMO-SEATS is then used in the selection and estimation of the ARIMA model and in the production of the seasonally adjusted data.

Furthermore, the seasonal adjustment procedure entails the evaluation of both the global statistical quality of the estimated ARIMA model and the adjustment quality. The latter includes tests for the statistical significance of the estimated parameters and for stationarity and invertibility conditions. The adjustment quality is verified by an analysis of the respective residuals.

7 In those cases in which the seasonal effects remain present in the adjusted series, the possible causes must be identified and corrected. This process may imply a new ARIMA model estimation.

To guarantee arithmetical consistency between the seasonally adjusted data, the outstanding amount for each balance item is obtained by an indirect method, 8 i.e., the adjusted amounts are obtained from the difference between the seasonally adjusted credits and debits. Similarly, the Current Account credit and debit data are also obtained by the indirect method, as is the sum of the

3

4

5

6

7

8

An outlier may be understood as an observation with a clearly anomalous value in comparison with the remaining data. These observations are also designated as strange, extreme or aberrant observations and may be due to measurement errors or atypical behaviour. For more information on outliers, see Appendix A,

Section A.1.

Monthly economic series that are based on daily flows of trades or settlements can be significantly influenced by the moving structure of the calendar, i.e., by the fact that: (i) the total number of days in each month varies from 28 (in February in a non-leap year) to 31 days; (ii) the weekly cycle of a month and, consequently, the respective number of working days, varies yearly; (iii) the day of the week and even the month associated with certain holidays (in the case of Carnival and Easter) varies according to the year. For more details on this subject, see Appendix A, Section A.1.

A brief description of X-12-ARIMA and TRAMO-SEATS applications, developed by the U.S. Bureau of Census and by the Banco de España , respectively, can be found in Appendix A, Section A.2.

Information on the comparative analysis of X-12-ARIMA and TRAMO-SEATS will be found in Appendix A, Section A.3.

The most fundamental requirement of a seasonal adjustment, regarding quality, is that there remains no seasonal effect present in the seasonally adjusted series, i.e., that there is no residual seasonality. If the model adequately describes the time series under study, then the residuals should behave like a white noise process. A white noise process consists of a succession of non-autocorrelated random variables with a normal distribution with mean zero and constant variance.

The choice between direct and indirect seasonal adjustment methods is not trivial. On this subject, see Appendix A, Section A.1.

8 Banco de Portugal | Supplement 4/2005 to the Statistical Bulletin, November 2005

Seasonal Adjustment of Balance of Payments Statistics credits and debits of the seasonally adjusted component series. It should be noted, however, that the seasonally adjusted series obtained by this method are later checked for the presence of seasonal effects. Whenever the statistical tests disclosed the existence of residual seasonality in the seasonally adjusted time series obtained by the indirect method, then the direct method was chosen.

It is often taken for granted that the yearly sum of the seasonally adjusted series will correspond to that of the original series. This belief is based on the idea that seasonal movements do not affect the annual totals and that, as the purpose of the seasonal adjustment is to distribute the seasonal effects throughout the year, the sum of the seasonal components during the year must be equal to zero. However, this belief stems from the misconception that the seasonal fluctuations occur in a similar way year after year, i.e., that they are deterministic components of the time series. Actually, these components are of a stochastic nature. The time of the year when they appear and their amplitude vary from one year to another. These dynamics reflect the natural variability of the underlying seasonal factors.

Most seasonal adjustment programmes permit the enforcement of time consistency, i.e., of the agreement between the yearly totals of the raw and the seasonally adjusted series. However, in addition to its lack of scientific support, time consistency lowers the quality of the seasonal adjustment and consequently the quality of the information obtained.

Moreover, forcing the yearly sum of the seasonally adjusted data to concur with that of the original series would create a discontinuity at the beginning of each year, when the annual sum of the raw data is not yet available. On top of this, a prompt correction of this discontinuity by summing a scale constant is somewhat paradoxical, as this is tantamount to admitting the existence of a systematic bias in the seasonal adjustment.

Consequently, the methodology for seasonal adjustment of the series published by the Banco of Portugal does not mean that the annual sum of the monthly adjusted data must equal the annual sum of the monthly raw data. This is the procedure adopted by a growing number of institutions, such as Eurostat and the ECB.

It should be re-emphasized that the seasonally adjusted data are obtained through the use of probabilistic models. The structures of the estimated models (specification and values of the parameters) reflect the information available beforehand. However, additional observations are included every month and, occasionally or in a systematic way, revisions are made to previous observations. A question that naturally arises here is how these changes in the raw series should be incorporated in the seasonal adjustment procedure.

In this respect, given the flexibility of the software programmes and after evaluating other more conservative alternatives, it was decided that the parameters of the selected models should be re-estimated 9 every month, incorporating the newly released observations of the raw series.

This, in turn, entails revision of the adjusted data. The specifications of the ARIMA models will be re-evaluated 10 every year in April during the annual revision of the balance of payments statistics, in particular of the Current Account. Nevertheless, this re-evaluation can be undertaken in other periods, whenever an appreciable change in the underlying information occurs. The aim of this procedure is to ensure that the seasonal adjustment incorporates the most recent available information and to avoid a significant revision, needed if the value of the parameters were fixed for the twelve months.

9 During the re-estimation process, the model’s structure (i.e., the number of and the relations between the parameters) remains unchanged. The parameters are simply reestimated (updated) by taking into account new available information.

10 The re-evaluation process includes a re-estimation of the parameters and also, whenever necessary, a re-specification of the model’s structure. In other words, a new identification of the model is carried out by using the new available information.



4. BRIEF DISCUSSION OF THE RESULTS

The application of the aforementioned methodology to the balance of payments series in the period from January 1996 to September 2005 confirmed the existence of seasonality, mainly in Current Account items. As for the Capital Account, due to the instability of its seasonal component, the original values were kept. The models’ specifications used for the seasonal adjustment are presented in Appendix B. As previously mentioned, Appendix C contains the table with the adjusted series that will henceforth be published monthly in the Statistical Bulletin.

The following table displays the information available for 2005. The data relate to the three-month period ending in each month, in order to smooth the natural irregularity associated with monthly information.

Current and Capital Accounts in 2005 (values for the three-month period ending in each month)

Current and capital accounts

Current account

Credit

Debit

Goods

Credit

Debit

Services

Credit

Debit

Income

Credit

Debit

Current transfers

Credit

Debit

Capital account

Mar Apr May

Raw data

Jun Jul Aug Sep

-3 082 -3 031 -3 657 -3 469 -2 458 -1 995 -2 106

-3 395 -3 395 -3 986 -3 750 -2 862 -2 374 -2 478

12 814 13 149 13 430 13 569 14 279 13 959 14 027

16 208 16 543 17 416 17 319 17 142 16 333 16 506

-3 790 -4 040 -4 435 -4 263 -3 926 -3 890 -3 933

7 518 7 628 7 747 7 777 7 936 7 298 7 392

11 307 11 668 12 182 12 040 11 862 11 188 11 325

536 650 838 957 1 255 1 532 1 692

2 434 2 612 2 878 3 006 3 300 3 674 3 836

1 897 1 963 2 040 2 048 2 045 2 142 2 144

- 668 - 801 -1 049 -1 059 - 991 - 750 - 737

1 391 1 415 1 412 1 425 1 435 1 411 1 419

2 059 2 216 2 461 2 484 2 425 2 161 2 157

527 797 661 615 799 734 500

1 472 1 493 1 393 1 361 1 608 1 576 1 380

945 696 733 746 809 842 880

313 364 329 281 404 379 372

EUR Milllions

Mar Apr

Seasonally adjusted data

May Jun Jul Aug Sep

-2 957 -2 798 -3 073 -3 018 -2 904 -2 950 -2 964

-3 270 -3 162 -3 402 -3 299 -3 308 -3 329 -3 336

13 237 13 375 13 370 13 428 13 480 13 540 13 744

16 506 16 537 16 772 16 727 16 788 16 869 17 080

-4 131 -4 165 -4 305 -4 174 -4 125 -4 064 -4 085

7 432 7 399 7 413 7 472 7 521 7 663 7 817

11 562 11 564 11 718 11 646 11 647 11 727 11 902

971 1 011 1 032 1 021 1 018 982 1 046

2 959 3 015 3 056 3 059 3 066 3 059 3 126

1 988 2 004 2 024 2 038 2 048 2 077 2 080

- 760 - 755 - 790 - 783 - 820 - 810 - 828

1 385 1 411 1 413 1 427 1 436 1 408 1 412

2 145 2 166 2 203 2 211 2 255 2 218 2 240

649 746 661 638 619 563 531

1 461 1 550 1 489 1 470 1 457 1 410 1 389

811 804 828 832 838 847 859

313 364 329 281 404 379 372

Taking original values into consideration, the deficit of the “Goods” balance increases between the three-month periods ending in

March and in May, it decreases until August and increases again in September. This evolution is confirmed by the adjusted series, despite a lower volatility, as can be seen in Figure 3. In broad terms, this behaviour reflects an evolution of the seasonally adjusted imports (goods debit), which is more moderate than that of the original series. The latter displays an increase of € 875 millions from the first three-month period to the three-month period ending in May and then a decrease by € 994 millions until August. In the three-month period ending in September, imports grow again (by € 137 millions). This behaviour is confirmed by the corresponding adjusted series, which have increased by € 175 millions.

1 0 Banco de Portugal | Supplement 4/2005 to the Statistical Bulletin, November 2005


Figure 3. Trade in Goods Balance (values for the three-month period ending in each month)

-3400

-3600

-3800

-4000

-4200

-4400

-4600

March April May June July August September


The adjusted series of the “Services” balance shows an increase until May and then a fall until August, followed by another rise in

September. This evolution contrasts with that of the original values, which increase from March to September, as can be seen in

Figure 4. As for the debit figures, both the raw and the adjusted series grow throughout the period. The same does not apply to the credit figures. Even though both the raw and the adjusted data increase all the time, the growth of credit figures is only moderate and is interrupted in August.

Figure 4. Services Balance (values for the three-month period ending in each month)

1800

1600

1400

1200

1000

800

600

400

200

0

March April

Raw series

May June July August


September

Regarding the two main service items, “Travel” and “Transportation,” the adjusted series of the latter balance decreases from May onwards with significant intensity in August and September. The seasonally adjusted series of the “Travel” balance increases throughout the period under review, except for a slight decrease in April, essentially due to credit developments (“Travel” earnings).

Banco de Portugal | Supplement 4/2005 to the Statistical Bulletin, November 2005 1 1


The “Income” deficit increased progressively until June (Figure 5) and decreased thereafter. Nevertheless, the seasonally adjusted series exhibits a more stable behaviour, though there is a slightly upward trend.

Figura 5. Income Balance (values for the three-month period ending in each month)

-600

-800

-1000

-1200

0

-200

-400

March April May June July August September


Finally, the seasonally adjusted “Current transfers” balance grows until April and falls in the following months (see Figure 6). The balance’s considerable growth in July ( € 184 millions) seems, therefore, to be the consequence of seasonal effects.

Figure 6. Current transfers balance (values for the three-month period ending in each month)

900

800

700

600

500

400

300

200

100

0

March April

Raw series

May June July August


September



To sum up, the table presented above indicates the net external borrowing requirements of the Portuguese economy as measured by the combined balance of Current and Capital Accounts. The raw series deficit deteriorates from March until May/June, and then improves consistently until August/September. On the other hand, the corresponding seasonally adjusted series evolution has a less pronounced amplitude and the values at the beginning and at the end of the period are not appreciably different. The main fact to be retained is the stability of the seasonally adjusted data of both the Current Account balance and of the combined balance of

Current and Capital accounts.


Annex A - Methodology




A.1 Time series and seasonal adjustment

A time series can be defined as a set of observations taken at successive periods of time. In general, a time series is seen as comprising three elements: (i) a trend-cycle component, indicative of the long-term movement of the series and of the oscillatory movement which, over the years, tends to be periodic; (ii) a seasonal component, responsible for short-term fluctuations, with a period of less than one year, remaining reasonably stable in terms of annual timing, direction and magnitude from year to year; and

(iii) an irregular or random component, the behaviour of which is unpredictable. The functional relationship between these components can take different forms, with the multiplicative and additive representations being the most prevalent.

Let and

Y

I t t

be the observed value of a time series at period t and denote by multiplicative model considers them to satisfy the relation Y t

= T t

× S t

T t

the trend-cycle component,

the irregular part. The additive model assumes the components to be related by the equation

× I t

.

Y t

S t the seasonal component

= T t

+ S t

+ I t

, while the

Seasonal adjustment is the process of estimating and removing the seasonal effects from a time series in order to easily identify the non-seasonal features. Let tive one Y t

= Y t

S t

.

Y t

denote the seasonally adjusted value of Y t

, in the additive form and Y t

= Y t

− S t

in the multiplica-

The decision on which of the two models is more appropriate should be based on statistical tests. Nevertheless, the graphical analysis of the time series is generally indicative of the components’ functional relationship. If the amplitude of the seasonal component is roughly proportional to the trend level, the multiplicative model should be adopted. Instead, if the amplitude of the seasonal component is roughly constant over time, the additive model should be adopted. The majority of economic time series use the multiplicative model.

For the aforementioned reasons and in view of the tests performed, the Portuguese balance of payments statistics were seasonally adjusted using the multiplicative model.

An intrinsic feature of a time series is the fact that adjacent observations are generally inter-dependent. It is therefore necessary to develop techniques to analyse this dependency. This requires the use of probabilistic or stochastic models, as the majority of the dynamic phenomena in the real world are non-deterministic in nature.

The analysis of time series fundamentally comprises the specification of the stochastic process or model more likely to be the generator of the observed time series. From the univariate models used in the time series description, the ARIMA models, popularized by Box and Jenkins in

1970, continue to be the most widely studied and applied. In its general form, an ARIMA( p , d , q )( P , D , Q ) is defined by the equation

φ p

( ) ( )(

1 − B

) d

(

1 − B s

)

D

Y t

= c + θ ( ) ( ) ε where B is the backward shift operator, defined by B k x t

= x t − k

, with k an integer parameter; c is a constant; s is the seasonal period; d and D are the regular and seasonal differencing orders, respectively; and φ

ε

t

is the white-noise process 11 ; p

( )

=

(

1 + φ

1

B + K + φ p

B p ;

)

Φ

P

=

(

1 + Φ

1

B s + K + Φ

P

B s ×

)

θ q

=

(

1 + θ

1

B + K + θ q

B q

)

; Θ

Q

=

(

1 + Θ

1

B s + K + Θ

Q

B s × Q

)

.

11 The estimation of an ARIMA( p , d , q )( P , D , Q ) process presumes that the process is stationary and invertible, i.e., that the roots of φ p

, θ q

Φ

P

and Θ

Q are all out of the unit circle. For more details see, for example, Box and Jenkins (1970), Box et al . (1994), Hamilton (1994) and

Makridakis et al . (1998). On the white-noise definition, see footnote 8.

,



The modelling of time series can be biased by the presence of observations with atypical values, usually designated as outliers.

12 Outliers comprise specifically the permanent (LS – Level Shift ) or transitory (TC – Temporary Change ) structural changes in a series, as well as other exogenous or endogenous changes, designated in the literature as additive

(AO – Additive Outlier ) and innovational outliers (IO – Innovational Outlier ), respectively. The effects of the outliers LS, TC and

AO are independent of the time series underlying model specification. The effect of an innovational outlier consists of an initial shock that propagates in the subsequent observations with the model parameters. This type of outlier is related to events that produce breaks in the seasonal component of the time series. Examples of the effect that each type of outlier has on the observed time series are in Figure A.1.

Figure A.1. Outlier effects on the observed time series

1.2

1

0.8

0.6

0.4

0.2

0

0

Additive Outlier Effect (AO)

20 40 60 80 100 120

1.2

1

0.8

0.6

0.4

0.2

0

0

Level Shif Effect (LS)

20 40 60 80 100 120

1.2

1

0.8

0.6

0.4

0.2

0

0

Temporary Change Effect (TC)

20 40 60

700

600

500

400

300

200

100

0

0

Innovational Outlier Effect (IO)

Observed Time Series

20 40 60 80 100 120 80 100 120

Another important aspect that must be taken into account in the seasonal adjustment of time series is the moving structure of the calendar. This is designated as calendar effects and may hinder the comparison of different periods of the observed series or of short-term movements between series. In particular, for Portugal, special attention should be given to some national holidays. Out of a total of fourteen, there are three moveable holidays.

13

While the effects of the non-moving holidays on economic activity always impact on the same months, the day of the week varies every year. In respect to moving holidays, they may occur in different months and days of the week, according to the year. Some economic activities are also affected by the Easter period, which may occur either in March or in April. This effect, known as the

Easter effect, may induce serious distortions on seasonal component stability.

In order to improve seasonal adjustment quality, the original series are corrected for calendar effects whenever estimates are statistically significant. The correction is made by including regressors in the identified ARIMA model. Once it is identified, the Easter effect is corrected by using specific variables that reflect its impact on the different months.

14

12 For more information on outliers see, for example, Tsay (1986), Deutsh et al . (1990), Chen and Liu (1993), Chen and Liu (1993), De Jong and Penzer (1998) and Peña

(2001).

13 The national holidays on fixed dates are New Year’s Day (January 1), Carnation Revolution (April 25), Labour Day (May 1), National Day (June 10), Assumption of Mary

(August 15), Proclamation of the Republic (October 5), All Saints Day (November 1), Restoration of Independence (December 1), Immaculate Conception (December 8) and

Christmas Day (December 25). The moveable holidays are: Carnival, Good Friday, and Corpus Christi.

14 For details on the applied method, see Dosse and Planas (1996). For each one of the time series, the statistically significant calendar effects are indicated in Appendix B.



Another relevant issue in the production of seasonally adjusted series is the way the new information is incorporated in the model, i.e., the optimal frequency of review of the seasonal models/parameters/factors.

As regards the parameters and the seasonal factor revision, i.e., re-estimation of the model, it is necessary to decide if this should be undertaken only when the model is re-evaluated, or if it should take place every month. In the first case, the seasonal factors are forecasted for the months up to the next re-evaluation. For the annual re-evaluation of the model, this means that the seasonal factors are forecasted for the forthcoming 12 months. In the second case, the model is re-estimated each month using all the information available at the current reference period. Based on the re-estimated model, the concurrent seasonal factors are then computed. In general, concurrent seasonal factors produce more accurate estimates of the final seasonally adjusted data.

Finally, it should be referred that when the time series stem from the aggregation of a set of other series, then the seasonal adjustment of the aggregated series may be made in a direct or in an indirect way. In the direct adjustment, the seasonally adjusted series are obtained by applying the seasonal adjustment procedure directly to the aggregated series. In the indirect adjustment, the adjusted series are obtained by combining the component series which have been adjusted separately.

The advantages and drawbacks of direct and indirect adjustments have been the object of debate amongst specialists (Ladiray and

Mazzi, 2003). A definite conclusion on which of these approaches is the best cannot be drawn from the available literature. Under most circumstances, the two adjustment procedures do not yield identical results. Only under very restrictive conditions do the results coincide. As an illustration of how direct and indirect seasonal adjustment may, legitimately, be very different, the “Book Fair” example, introduced by Maravall (2002), is described next.

Consider a country with several administrative regions. This country has an unusual pattern of book sales. Books are regularly sold throughout the year in each of the regions, but a very large number of books are sold at a book fair that is held every January. Each region takes its turn at hosting the book fair. Consequently, there is a huge boost to that region’s book sales in the month of January of the year in which it acts as host. The monthly time series graphs of book sales for the first three fairs are shown in Figure A.2.

Figure A.2. Monthly book sales time series (not seasonally adjusted)

Region 1

140

120

100

80

60

40

20

0

Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4

Region 3

40

20

0

80

60

140

120

100

Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4

Region 2

140

120

100

80

60

40

20

0

Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4

Whole Country

140

120

100

80

60

40

20

0

Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4

The direct seasonal adjustment of each of the series yields the seasonally adjusted series shown in Figure A.3.



Figure A.3. Monthly seasonally adjusted book sales time series

Region 1 (seasonally adjusted)

140

120

100

80

60

40

20

0

Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4


80

60

40

20

0

140

120

100

Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4


80

60

40

20

0

140

120

100

Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4

Whole Country (seasonally adjusted)

140

120

100

80

60

40

20

0

Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4

The comparison between the indirect seasonal adjustment, given by summing each of the regional seasonally adjusted series, and the direct adjustment of the national total highlights the difference between the two methods (Figure A.4.). While the direct seasonal adjustment interpreted the book fair peaks as seasonal and removed them, the indirect adjustment did not.

Figure A.4. Comparison of direct and indirect seasonal adjustments

Whole Country (direct)

140

120

100

80

60

40

20

0

Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4

Whole Country (indirect)

60

40

20

0

140

120

100

80

Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4

The situation described in the example can easily arise in real time series. Therefore, whenever there is evidence of a variety of seasonal features in a time series switching between its components, additivity should be questioned.



A.2 The X-12-ARIMA and TRAMO-SEATS programmes

X-12-ARIMA is a seasonal adjustment programme, developed by the U. S. Census Bureau, based on the well known X-11 and

X-11-ARIMA programmes.

15

X-12-ARIMA is an extension of X-11-ARIMA, whose details can be found in Findley et al . (1998). The major improvements in

X-12-ARIMA are the refinement of the ARIMA modelling process, the construction of new diagnostic tools, the development of a new routine for pre-adjustment of the data, and the inclusion of new seasonal filter options. X-12-ARIMA includes an automatic

ARIMA modelling procedure that is based on the AIC, AICC 16 and BIC 17 criteria and selects the best model from the following five: 18 ARIMA(0,1,1)(0,1,1) s

; ARIMA(0,1,2)(0,1,1) s

; ARIMA(2,1,0)(0,1,1) s

; ARIMA(0,2,2)(0,1,1) s

; and ARIMA(2,1,2)(0,1,1) s

.

Assuming a multiplicative model defined by Y t

=

T t

×

S t

×

I t

, where Y is the observed value of a time series, T is the trend-cycle component, S is the seasonal component and I is the irregular component, the decomposition procedure carried out by X-12-ARIMA is as follows:

1. An initial estimate of the trend ( T ’ ) is obtained using weighted moving averages of the observed series ( Y ).

2. This trend is removed from the observed series to obtain a “detrended” series.

3. Seasonal factors ( S’ ) are estimated from the “detrended” series by taking weighted moving averages for each group of months separately.

4. An estimate of irregular components is obtained by removing the seasonal component from the “detrended” series

( I’ = Y / (T’ x S’) ) .

5. Outliers are identified from the estimate of the irregular component ( I’ ) and the actual value is replaced by an imputed value

( Y ’ ).

6. A new trend estimate ( T ’ ) is obtained using Henderson moving averages of the modified original series.

7. Steps 2 to 6 are repeated (twice) to produce the trend (

T

), seasonal factors (

S

) and irregular (

I

) final components.

The TRAMO-SEATS programme, developed by Gómez and Maravall (1996, 1998, and 2001) at the Banco de España , is actually the combination of two programmes; TRAMO ( Time Series Regression with ARIMA Noise, Missing Observations, and Outliers ) and SEATS ( Signal Extraction in ARIMA Time Series ).

TRAMO-SEATS originated from the “ARIMA-Model-Based” (AMB) approach to seasonal adjustment (Planas, 1997). This encompasses two phases: in the first, a model for the observed series is identified; based on this model, in the second stage, the different components (trend, seasonality and irregularity) are estimated by applying statistically founded signal extraction techniques (Burman,

1980). The development of the TRAMO-SEATS programme was based on work by Burman (1980), Hillmer and Tiao (1982), Bell and Hillmer (1984), and Maravall and Pierce (1987) made in the context of the seasonal adjustment of economic time series.

15 The X-11 method, developed by the U. S. Census Bureau in 1965, is an ad-hoc adjustment procedure that uses the Henderson moving average algorithm on the time series decomposition. Details on this method may be found in Shiskin, Young and Musgrave (1967). The major criticisms of the X-11 method are that: (i) it is not based on any statistical model; and (ii) the initial and final observations of the series are wasted due to the use of centred moving averages. These criticisms were at the origin of the X-11-ARIMA, developed by Dagum (1988) of the Statistics Canada . In broad terms, X-11-ARIMA incorporates into the X-11 program ARIMA models to forecast beyond the current series and backcast before the beginning of the series. Moreover, it provides new forms of diagnosis to assess the quality of the seasonal adjustment, as well as a set of statistical tests for the presence of seasonality. These tests encompass two types of seasonality: (i) stable seasonality, defined as the intra-annual variation which is the same each year; and (ii) moving seasonality related to intra-annual variation which slowly changes and evolves from one year to the next.

16 Akaike Information Criterion (AIC), defined by Akaike (1974), and Akaike Information Corrected Criterion (AICC), developed by Hurvich and Tsai (1989).

17 This criterion (Bayesian Information Criterion) is also known as Schwartz Criterion (SC) and Schwartz Bayesian Criterion (SBC). See definition in Schwarz (1978).

18 Let s denote the seasonal period. For monthly data, s = 12.



The TRAMO programme permits the identification and estimation of the ARIMA model that best describes the time series under study. The data pre-adjustment routine includes the production of optimal interpolators on the missing observations, the detection and correction for several types of outliers, and of the calendar effects. In the automatic modelling procedure, the ARIMA model selection follows the Hannan and Rissanen (1982) procedure and is based on the BIC criterion. The search is made over all the possible ARIMA models until ARIMA(3,2,3)(2,1,2) s

.

The decomposition of the observed series into its components (trend, seasonality and irregularity) is performed by the SEATS programme on the assumption that they are orthogonal and each one may be described by an ARIMA model. By resorting to signal extraction techniques applied to ARIMA models, the SEATS procedure decomposes the spectral density function of the model estimated by TRAMO into the spectral density functions of the various components.

The TRAMO-SEATS programme provides various diagnostic tools to assess the quality of the seasonal adjustment. The main differences between TRAMO-SEATS and other ad-hoc methods are its formal theoretical grounds and the adaptability of the filters used by SEATS to the features of the series under study.

A.3. Comparative analysis of the X-12-ARIMA and TRAMO-SEATS programmes

The Banco de Portugal put together a project aimed at defining a methodology to be applied by the Department of Statistics to the production of seasonally adjusted series. It involved the study of 164 balance of payments time series, half of which are credit series and the other half are debit series. The research covered the period from January 1996 to January 2005.

In view of the pressing need for harmonisation of seasonal adjustment procedures in the euro area, the decision was taken, as mentioned earlier, to choose one of the two adjustment programmes recommended by Eurostat. As the choice of the seasonal adjustment method is of the utmost importance in the definition of the methodology, a comparison of the X-12-ARIMA and TRAMO-

SEATS programmes was undertaken. The study used SAS software for the detailed econometric analysis of the different series, and this allowed for the validation of the results obtained by the two programmes.

The Banco de Portugal modelled the balance of payments statistics time series with the aid of the DEMETRA interface, developed by Eurostat.

This permits the use of the X-12-ARIMA and TRAMO-SEATS programmes in the same environment. The results obtained were analysed and compared and it was established that:

(i) The TRAMO-SEATS programme adjusted all the time series under study, whereas the X-12-ARIMA programme was only able to model 25 per cent of the series. This can be explained by the fact that the latter programme considers only a reduced set of models.

(ii) The same model was selected by both programmes only in 5 per cent of the series. It should be noted that the different results obtained are due to the fact that the procedures for the selection of the ARIMA model that best describes the underlying series vary according to the programme applied.

(ii) TRAMO-SEATS indicated that the ARIMA(0,1,1)(0,1,1)

12

process was adequate for a significant set of series. For the same series, the X-12-ARIMA rejected all of the models it took into consideration. These results are in line with the ideas put forward by the U. S. Census Bureau (Hood, 2002), according to which SEATS may induce residual seasonality when the original series displays no seasonality.

(iii) TRAMO-SEATS identified seasonality in a significantly higher number of series than the X-12-ARIMA programme. This result once again confirms the previous statement, highlighting the importance of the tests for the presence of seasonality undertaken by X-12-ARIMA and the problems that may arise with the use of TRAMO-SEATS in the modelling of time series without seasonality.

(iv) The two programmes entailed pre-adjustment routines with time series transformations and corrections of different number and degree, and this led to distinct results in some cases.



Concomitantly, the results obtained by the X-12-ARIMA and TRAMO-SEATS programmes were validated by a thorough econometric analysis of the series, supported on the SAS software, including: (i) a study of the estimated autocorrelation and partial autocorrelation functions; (ii) tests for the adequacy of the logarithmic transformation; (iii) tests for the presence of unit roots, i.e., for the need of time series differentiation; (iv) estimation of the models; and (v) evaluation of the statistical significance of the estimated parameters, verification of the stationarity and invertibility conditions, analysis of the correlation matrix of the parameter estimators and assessment of the quality of the adjustment by analysis of the corresponding residuals.

In view of the above considerations, the TRAMO-SEATS programme was chosen for the ARIMA modelling of the time series. The next step consisted of the selection of the seasonal decomposition method.

After the identification of the model that best describes each of the time series, the seasonally adjusted series can be obtained by the ad-hoc method of X-12-ARIMA or by the signal extraction techniques of TRAMO-SEATS. It should be noted that the lack of statistical support of X-12-ARIMA significantly hinders the analyst’s ability to intervene not only in the adjustment process itself, but also in the statistical inference. Indeed, this method uses an empirical procedure based on ad-hoc rules which take into account features common to a vast set of economic series. On the contrary, TRAMO-SEATS, which is based on signal extraction techniques applied to the ARIMA model, respects the specific characteristics of each series. Furthermore, the statistical foundations of this latter method allow for the application of diagnostic tests and for inferences to be drawn from the data.

For the selection of one of the two methodologies, the Banco de Portugal gave special attention to the quality of the seasonal adjustment and to the minimization of seasonally adjusted series revisions.

The tests undertaken in the analysis of seasonal adjustment quality confirmed the quality of all the series obtained by both methodologies. A study of the seasonally adjusted series revisions was carried out subsequently with the aim of selecting the most adequate method for seasonal decomposition. For this purpose, and for operational reasons, only a restricted number of balance of payments series, regarded as relevant in this context, were considered.

The first step included the estimation of the aforementioned series by TRAMO-SEATS during the period from January 1996 to December

2003. The model for each of the series was fixed and the predictions of the seasonal factors and of the seasonally adjusted series were calculated for the period from January 2004 to January 2005. Subsequently, the period from January 1996 to January 2004 was considered as the estimation set and the parameters of the respective model (previously fixed) were re-estimated for each series. The predictions for the period from February 2004 to January 2005 were calculated. This procedure of adding an observation to the estimation set, re-estimating the parameters and computing new predictions, was applied iteratively until the estimation set coincided with the period from January 1996 to

December 2004.

An identical procedure was followed by applying X-12-ARIMA to the models identified by TRAMO-SEATS. For each of the methods, the percentage values of the mean and of the absolute mean for the following differences were calculated:

(i) difference between the use of predicted and concurrent seasonal factors, i.e., difference between the forecasts obtained from the predicted seasonal factors (forecasts made with the information available up to December 2003) and the estimates obtained with all the information available until then;

(ii) difference between the forecasts obtained from the predicted seasonal factors (forecasts made with the information available up to December 2003) and the estimates calculated once all the annual information is made available;

(iii) difference between the estimates obtained from the concurrent seasonal factors and those calculated once all the annual information is made available;

(iv) difference between estimated values considering concurrent seasonal factors and the respective monthly updates once an additional observation is included each month.

19

19 When using the concurrent seasonal factors, these are the values associated with the monthly reviews.



The results revealed that the X-12-ARIMA and TRAMO-SEATS programmes are fundamentally similar. However, taking as a measure the maximum value of the revisions mentioned in (iv), the X-12-ARIMA programme displayed a maximum value for the revisions much greater than TRAMO-SEATS. Since the use of concurrent seasonal factors generally produces more accurate estimates for the final seasonally adjusted data, together with the previously mentioned fact that X-12-ARIMA hinders the ability of the analyst to intervene in the adjustment process as well as in the statistical inferences, the TRAMO-SEATS programme was adopted for the seasonal decomposition of time series.

Annex


Annex B - Details of the models used in the seasonal adjustment of information




Table B.2. Identification and characterization of outliers in the published series

Identification and characterization of outliers b)

Goods

Credit

Debit

Services

Credit

Debit

Transportation

Credit

Debit

Travel

Credit

Debit

Income

Credit

Debit

Current Transfers

Credit

Debit

3

1 outliers outlier

1 outlier :

2 outliers :

3 outliers :

4 outliers :

3 outliers:

0 outliers:

2 outliers :

1 outlier :

1 outlier :

:

0 outliers .

: AO Aug1998, TC Mar2000 and LS Jan2001.

AO Dec2000.

AO Jun2004.

AO Jan1998 and AO Sep1999.

AO Nov1997, TC Jan1999 and TC Dec1999.

TC Jan1999, AO Jul2002, AO Feb2005 and AO Aug2005.

LS Feb1996, TC Jun1998 and AO Jun2004.

TC Nov2001 and AO Mar2003.

LS Jan2002.

AO May1999.

b) Notation: AO – Additive Outlier; LS – Level Shift; TC – Temporary Change; IO – Innovational Outlier.


Annex C - Table of the Statistical Bulletin with seasonally adjusted data




References

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SUPPLEMENTS TO THE STATISTICAL BULLETIN

1/1998 Statistical information on non-monetary financial institutions

2/1998 Foreign direct investment in Portugal: flows and stocks statistics for 1996 and stocks estimates for 1997

1/1999 New presentation of the balance of payments statistics

2/1999 Statistical information on Mutual Funds

1/2000 Portuguese direct investment abroad (available only in Portuguese)

1/2001 “Statistical balance sheet” and “Accounting balance sheet” of other monetary financial institutions

1/2005 A new source for monetary and financial statistics: the central credit register

2/2005 National financial accounts for the Portuguese economy

Methodological notes and statistical results for 2000-2004

3/2005 National financial accounts for the Portuguese economy

Statistics on financial assets and liabilities for 1999-2004

4/2005 Seasonal adjustment of balance of payments statistics

5/2005 Statistics on non-financial corporations from the central balance-sheet database


Supplement 4/2005 to the Statistical Bulletin, November 2005

Seasonal Adjustment of Balance of Payments Statistics

Supplement 4/2005 to the Statistical Bulletin, November 2005

Banco de Portugal