Background Facts on Economic Statistics

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1 Background Facs on Economic Saisics 004:13 Revised Tesing for Normaliy and ARCH An Empirical Sudy of Swedish GDP Revisions Deparmen of Economic Saisics

2 The series Background facs presens background maerial for saisics produced by he Deparmen of Economic Saisics a Saisics Sweden. Produc descripions, mehodology repors and various saisic compilaions are exampels of background maerial ha give an overview and faciliae he use of saisics. Publicaions in he series Background facs on Economic Saisics 001:1 Offenlig och priva verksamhe saisik om anordnare av välfärdsjänser 1995, 1997 och :1 Forskar kvinnor mer än män? Resula från en arbesidsundersökning rikad ill forskande och undervisande personal vid universie och högskolor år : Forskning och uveckling (FoU) i föreag med färre än 50 ansällda år :3 Föreagsenheen i den ekonomiska saisiken 00:4 Saisik om privaiseringen av välfärdsjänser En sammansällning från SCB:s saisikkällor 003:1 Effeker av minskad dealjeringsgrad i varunomenklauren i Inrasa från KN8 ill KN6 003: Consequences of reduced grade in deail in he nomenclaure in Inrasa from CN8 o CN6 003:3 SAMU. The sysem for co-ordinaion of frame populaions and samples from he Business Regiser a Saisics Sweden 003:4 Projek med anknyning ill projeke Saisik om den nya ekonomin. En karläggning av uvecklingsprojek och uppdrag 003:5 Developmen of Alernaive Mehods o Produce Early Esimaes of he Swedish Foreign Trade Saisics 003:6 Övergång från SNI 9 ill SNI 00: Underlag för a bedöma effeker av idsseriebro 003:7 Sveriges indusriprodukionsindex Tidsserieanalys The Swedish Indusrial Producion Index Time Series Analysis 003:8 Cross-counry comparison of prices for durable consumer goods: Pilo sudy washing machines 003:9 Monhly leading indicaors using he leading informaion in he monhly Business Tendency Survey 003:10 Priva drif av offenlig finansierade välfärdsjänser. En sammansällning av saisik 003:11 Säsongrensning av Naionalräkenskaperna Översik 003:1 En illämpning av TRAMO/SEATS: Den svenska urikeshandeln :13 A noe on improving impuaions using ime series forecass 003:14 Definiions of goods and services in exernal rade saisics Coninued on inside of he back cover! These publicaions and ohers can be ordered from: Saisics Sweden, Publicaion Services, SE ÖREBRO, Sweden phone or fax You can also purchase our publicaions a our Saisics Shop: Karlavägen 100, Sockholm, Sweden

3 004:13 Revised Tesing for Normaliy and ARCH An Empirical Sudy of Swedish GDP Revisions Saisics Sweden 004

4 Producer Inquiries Saisics Sweden Deparmen for Economic Saisics Lars-Erik Öller, Saisics Sweden and Deparmen of Saisics, Sockholm Universiy Undergraduae Thesis, 10 poins, Spring 004 Auhors Supervisor Rober Bosröm Frida Tomberg Lars-Erik Öller 004 Saisics Sweden ISSN Prined in Sweden SCB-Tryck, Örebro MILJÖMÄRKT Trycksak

5 3 Conens 1 Inroducion... 6 Mehod and heory Normaliy Saisical Conceps Skewness Kurosis Normaliy ess Anderson-Darling Normaliy Tes Jarque-Bera Tes of Normaliy Kolmogorov-Smirnov Tes Ryan-Joiner Tes Tesing for ARCH Engle s LM es for ARCH The Squared Ljung-Box Tes Daa Resuls and analysis Skewness and kurosis ess The normaliy ess The ARCH es Conclusions References... Appendices... 3 Appendix I Glossary... 3 Appendix II ARCH models... 5 Appendix III GARCH models... 6 Appendix IV LJUNG-BOX es... 7

6 4 Absrac A revision is defined as a difference beween a final and a preliminary figure. In his hesis we invesigae Swedish revisions of all he expendiure componens of Gross Domesic Produc (GDP). We use quarerly daa for he years o check GDP revisions for normaliy, skewness, kurosis and ARCH behavior. Four ess for normaliy are used: Anderson-Darling, Jarque-Bera, Kolmogorov-Smirnov and Ryan-Joiner. The resuls are ha many of he revisions are non-normal. We also find ha he revisions are skewed and hick-ailed bu hey do no conain any ARCH effecs. Keywords: Normaliy Tess, ARCH, GDP Revisions, Skewness, Kurosis

7 5 Acknowledgemen During he course of his sudy, we have had he pleasure of alking o many people whose knowledge and paience have made his paper possible. We would herefore like o express our sincere graiude o he following people and organizaions. Inernaional Mr Jorge Mina, Head of Quaniaive Research, RiskMerics Group, New York, USA. Mr Thomas Ta, Quaniaive Research, RiskMerics Group, London, UK. Ms Karin Bosröm, Quaniaive Research, RiskMerics Group, London, UK. Naional Mr Jörgen Säve-Söderbergh, Docoral Suden, Deparmen of Saisics, Sockholm Universiy, for fruiful discussions. Mr Alfred Kanis, Senior Lecurer, Deparmen of Economics, Sockholm Universiy, for valuable ideas and advice. Ms Pera Jansson, Saisics Sweden, who augh us TRAMO/SEATS. Mr Håkan Släman, Compuer Assisan, Deparmen of Saisics, Sockholm Universiy, who helped us ge access o he compuers and programs. Special Thanks The librarians a he Royal Insiue of Technology Library, Sockholm Universiy Library, Sockholm School of Economics Library and Saisics Sweden Library. Ms Jenny Axelsson and Mr David Ringkvis for consrucive opinions a he opposiion seminar. Our Supervisor, Adjunc Professor Lars-Erik Öller, Saisics Sweden and Deparmen of Saisics, Sockholm Universiy. Sockholm Spring 004 Rober Bosröm Frida Tomberg

8 6 1 Inroducion In his Secion we are presening he background of he Gross Domesic Produc (GDP) revisions, followed by he purpose and finally he ouline of his sudy. The need for reliable forecass of he macroeconomic developmen has increased wih ime and he need for daa likewise. Saisics Sweden (SCB) assembles daa from all areas of he Swedish economy and compiles hem ino he Naional Accouns. The sum of i all is he Gross Domesic Produc (GDP). The Naional Accouns have become a very imporan source for hose who work wih macroeconomic analysis. The daa ha are used in forecasing need o be correc and published early for he forecas o be useful and ineresing. More reliable bu old figures may have los heir relevance when a decision has o be made. In order o achieve puncualiy, early saisical figures are published as preliminary informaion ha is evenually revised when more informaion becomes available. The revisions are measured as he difference beween final and preliminary growh figures. I has o be kep in mind ha one can never be sure ha a revised figure really is more relevan and accurae han he preliminary one because all published figures of GDP are esimaes. Revisions can also be a moral maer. Jus because he revisions are small i is no sure ha he preliminary figures were ha accurae. The saisician who made he revisions may no have been careful enough. Anoher saisician goes o grea lenghs o find errors and his resuls in a big revision. Preliminary figures are ofen based on sample esimaes ha are revised laer when more figures become available. A lo of differen ypes of error will appear in he revisions, for many of hem we canno easily consruc numerical measures of reliabiliy. One ype of error is model assumpion errors. An ordinary model assumpion is ha he disribuion is normal, bu he normal disribuion only exiss in heory and he real disribuions are only approximaely normal and someimes hey are no even ha. Öller and Hansson (00) have checked Swedish revisions of he componens of GDP beween he years 1980 o 1998 o see if hey conain bias. One of heir aims was o expose he shorcomings of he saisical producion process so ha i would be possible o see if improvemens could be made. They found ha he revisions are generally posiively biased which means ha he preliminary figures are lower hen he revised figures in general. They also came o he conclusions ha he frequency disribuions of revisions are hick-ailed and skewed, bu hey never esed heir revision disribuions. Tha is wha we inend o do in his essay. Nilsson and Rosander (003) picked up where Öller and Hansson (00) lef. They checked for anoher ype of sysemaic errors, auocorrelaion in he revision ime series. They came o he conclusion ha he revisions for some variables were auocorrelaed and hey firs hough ha i would be possible o esimae models ha can make he preliminary figures beer. Models could be specified and esimaed, bu hey could no be used o improve preliminary figures, due o lack of final figures for year -1. We are going o check he revisions in Öller and Hansson (00) for normaliy. A convenien way o do his is o look a a hisogram, or a frequency disribuion. A hisogram says much abou locaion, spread, non-normaliy and ouliers. Bu o ge more deailed informaion some esing mus be done. Harvey and Newbold (00) invesigaed he disribuional properies of individual and consensus ime series macroeconomic forecas errors, using daa from Survey of Professional Forecasers. The degree of auocorrelaion and he presence of ARCH in he consensus errors were also esed. They found srong evidence of lepokuric (see Secion.1.3) forecas errors as well as evidence of skewness (see Secion.1.),

9 suggesing ha an assumpion of forecas error normaliy was inappropriae. Many of he forecass error series were found o have non-zero mean. They also found widespread evidence of consensus error ARCH. We are going o es for skewness and kurosis wih he same measures as he ones Harvey and Newbold (00) used. Furhermore we are going o check for ARCH behavior in he GDP revisions. In Secion we discuss mehodology and he heoreical framework. The heory for he saisical measures normaliy, skewness and kurosis are discussed followed by he normaliy ess. Afer ha he heory for he ARCH es is presened. The daa are described in Secion 3. In Secion 4 he empirical resuls and heir analysis are presened and Secion 5 gives an overall summary of he sudy. 7

10 8 Mehod and heory In his Secion he procedures of he sudy are presened. The saisical measures: normaliy, skewnes and kurosis are presened and he heory of he normaliy ess is discussed. Afer ha follows he heory of he ARCH es. A revision is defined as he difference beween a final and a preliminary growh figure. Preliminary figures of he quarerly Naional Accouns are published days afer he quarer has expired. Since preliminary figures are revised several imes one mus decide which o choose as he final figures. I is imporan o remember ha boh he preliminary and he final figures are esimaes because one can never find he exac figures for he variables. Even hough he final figures are esimaes we expec hem o be more rue han he preliminary ones, alhough his is no always he case. We are going o sudy he revisions of all expendiure componens of GDP separaely. The revisions are: Privae Consumpion, Governmen Consumpion, Cenral Governmen Consumpion, Local Governmen Consumpion, Invesmens, Change in Invenories, Expors, Expors of Goods, Expors of Services, Impors, Impors of Goods, Impors of Services and GDP. We will es for skewness and kurosis using Excel o calculae he figures. I is ineresing o know if i is skewness or kurosis, or boh ha make he disribuions non-normal. The program package in Miniab conains criical values for hree ess: Anderson- Darling, Kolmogorov-Smirnov and Ryan-Joiner. We also did a Jarque-Bera es of normaliy using he formula in Secion.. o calculae he JB saisic. Excel was used o calculae he JB saisic using he skewness and kurosis formulas from Harvey and Newbold (00). We hough a priori ha he resuls of all our ess would sae ha many of he revisions of he componens of he GDP are no normally disribued. Finally he revisions were esed for heeroskedasiciy. Two ess were used, Lagrange muliplier (LM) es for Auoregressive Condiional Heeroskedasiciy (ARCH) and he squared Ljung-Box es saisic. To decide which model we should use when esing for ARCH in he revisions we used he program package TRAMO/ SEATS, which is an Auoregressive Inegraed Moving Average-model (ARIMA) based mehod. A common assumpion in many ime series echniques is ha he daa are saionary. This is a necessary condiion for he ime series o be considered as an adequae ARIMA model. A saionary process has he propery ha he mean, variance and auocorrelaion srucure do no change over ime. Anoher assumpion is he concep whie noise, which means ha he revision erm is independen and normally disribued 1..1 Normaliy.1.1 Saisical Conceps The normal disribuion, or he Gaussian disribuion, is paramoun in saisics. I is a family of disribuions of he same general form, differing only in heir locaion and scale parameers, ha is he mean and he sandard deviaion. Many measuremens have approximae normal disribuions. The reason why he normal disribuion is used so ofen is ha i has several good mahemaical feaures. 1 For furher reading abou hese condiions he reader is referred o Bowerman and O Connel (1993).

11 The sandard normal disribuion has mean zero and sandard deviaion one. For all normal disribuions, he densiy funcion is symmeric abou is mean value. A saisic probabiliy disribuion can be described by is differen momens. If a discree random variable is defined by X and he summaion index is denoed by k, he p x (k) is equal o he probabiliy ha X assumes he value k, which leads o ha he differen momens can be wrien as follow The firs momen is he popoulaion mean and is defined as E ( X ) = kp x ( k) = µ (1) k The second momen is wrien as E( X ) = k px ( k) = Var( X ) + ( E( X )) () k Momens of higher order can be wrien by he general formula r r E( X ) = k p ( k) r = 1,,3,... (3) k x In his hesis we are ineresed in he hird and fourh momens of he disribuion, ha is he skewness and kurosis respecively which will be discussed in nex Secions [Kleinbaum e al. (1998)]. As noed above, a probabiliy disribuion can be summarized in erms of he momens of is disribuion. Several popular ess for normaliy focus on measuring skewness and kurosis, which are higher momens of he probabiliy disribuion. We will use he same measures ha Harvey and Newbold (00) used in heir sudy, which are bias correced Skewness The erm skewness in his repor refers o he hird momen. Skewness is a measure of he degree of asymmery of he probabiliy disribuion and is defined by Skewness = n n r r 1 = 1 sr ( n )( n ) 3 where r is he mean of he revisions, s r is he sample sandard deviaion and n is he sample size. When a disribuion is symmerical abou he mean, like he normal disribuion, skewness is zero. A disribuion having posiive skewness is skewed o he righ. For such a disribuion, he ail falls off o he righ. Likewise, a negaively skewed disribuion is skewed o he lef [Kleinbaum e al. (1998)]. An illusraion of righ and lef skewed disribuions is displayed in Figure 1. (4)

12 10 Figure 1 Righ skewed, lef skewed and symmerical disribuions Source: Gujarai (00)..1.3 Kurosis The erm kurosis in his hesis refers o he fourh momen. Kurosis measures he peakness or fa-ailedness versus flaness or shor-ailedness of he probabiliy disribuion and may be compued as Kurosis = n( n + 1) ( n 1)( n )( n 3) 3( n 1) ( n )( 3) 4 n r r = 1 sr n +3 (5) A normal disribuion has a sandardized kurosis value of hree. This definiion of sandardized kurosis is no universally used. Someimes he sandardized kurosis is defined o be he preceding sandardized kurosis value minus hree. Wih such a definiion, he sandardized kurosis value of a normal disribuion is equal o zero. Peaked disribuions, which have posiive kurosis, are referred o as lepokuric while fla disribuions, which have negaive kurosis, are referred o as playkuric. Neural disribuions, like he normal, are referred o as mesokuric. Figure shows an illusraion of lepokuric and playkuric disribuions. Figure Lepokuric, Playkuric and Mesokuric disribuions Source: Gujarai (00). A common noaion is ha a disribuion wih posiive kurosis, a sandardized kurosis value larger han hree, is more peaked han he corresponding normal disribuion, while one wih negaive kurosis, a sandardized kurosis value less han hree is flaer. Kaplansky (1945) proves ha his is no always he case.

13 Finally, he reader should bear in mind ha skewness and kurosis saisics are highly variable in small samples and hence are ofen difficul o inerpre. However, since we have eighy observaions in our sudy, hese measures should be reasonably sable [Hogg and Tanis (001)]. 11. Normaliy Tess..1 Anderson-Darling Normaliy Tes The Anderson-Darling saisic A measures how well he daa follow a paricular disribuion. The beer he disribuion fis he daa, he smaller his saisic will be. I is a modificaion of he Kolmogorov-Smirnov es (see Secion..3) and gives more weigh o he ails han he Kolmogorov-Smirnov es. The Anderson-Darling es makes use of a specific disribuion in calculaing criical values. This has he advanage of allowing a more sensiive es and he disadvanage ha criical values mus be calculaed for each disribuion [Sephens (1974)]. The Anderson-Darling normaliy es is a Normal Probabiliy Plo (NPP), which makes use of normal probabiliy paper, a specially designed graph paper. On he horizonal axis he values of he variable of ineres is ploed, and on he verical he expeced value of his variable if i would be normally disribued. Therefore, if his variable is in fac from he normal populaion, he NPP will be approximaely a sraigh line, as shown in Figure 3 [Gujarai (00)]. Miniab calculaes he A saisic using he weighed squared disance beween he fied line of he probabiliy plo ha is based on he chosen disribuion, using eiher maximum likelihood or leas squares esimaes, and he nonparameric sep funcion. Under he underlying null hypohesis, H 0 : he revisions are approximaely normally disribued, he assumpion of normaliy will be rejeced when he A saisic is greaer han he criical value, which is equivalen o rejecing he null hypohesis if he observed p-value is smaller han he chosen significance level [D Augosino and Sephens (1986)]. Figure 3 Normal Probabiliy Plo (NPP) y 99, , x The Anderson-Darling es saisic is defined as A where = n S (6) n (i 1) S = [ ln F( zi ) + ln(1 F( zn+ 1 i ))] (7) n i= 1

14 1 F is he assumed normal disribuion wih he assumed or sample esimaed parameers ( ˆ, µ ˆ σ ), z i is he ih sored, sandardized, sample value, n is he sample size, ln is he naural logarihm and subscrip i runs from i o n. Among he ess based on he empirical disribuion funcion, Anderson-Darling ends o be more effecive in deecing deparures in he ails of he disribuion. In pracice, if deparures from normaliy a he ails were he major concern, many saisicians would use Anderson-Darling as heir firs choice. However, you need big samples o be able o say somehing abou he ails... Jarque-Bera Tes of Normaliy The JB es, originally suggesed by Jarque and Bera (1987), is probably he mos commonly used es of normaliy. They developed a Lagrange Muliplier (LM) es of he null hypohesis agains he wo-sided alernaive hypohesis, which is equivalen o esing he null hypohesis of normaliy. The LM es, or he JB es, uses he following es saisic ( K 3) S JB = n + (8) 6 4 where n is he sample size, S is he skewness value from (4), and K is he kurosis value from (5). For a normally disribued variable, S is equal o zero and K is equal o hree. Therefore, he JB es of normaliy is a es of he join hypohesis ha S and K are zero and hree, respecively. In ha case he value of he JB saisic is expeced o be zero. Under he null hypohesis ha he revisions are normally disribued, Jarque and Bera showed ha asympoically he JB saisic given in (8) follows he chi-square disribuion wih wo degrees of freedom, χ [ ]. If he compued p-value of he JB saisic is sufficienly low, which will happen if he value of he saisic is differen from zero, one can rejec he hypohesis ha he revisions are normally disribued [Gujarai (00)]. Noe ha skewness and kurosis can be esed separaely using chi-square disribuion wih one degree of freedom, each. This es needs a big sample size o give appropriae resuls, which is no a problem in our case...3 Kolmogorov-Smirnov Tes This es considers he goodness of fi beween a hypohesized disribuion funcion and an empirical disribuion funcion. The empirical disribuion funcion is given here in erms of he order saisics. Le y 1 < y <... < y n be he observed values of he order saisics of a random sample x 1, x,..., x n of size n. When no wo observaions are equal, he empirical disribuion funcion is defined by 0, x < y1 Fn ( x) = k / n, yk x < yk + 1, k = 1,,..., n 1 (9) 1, x yn The empirical disribuion funcion has a jump of magniude 1/n occurring a each observaion. F n (x) is he fracion of sample observaions ha are less han or equal o x. Because of he convergence of he empirical disribuion funcion o he heoreical disribuion funcion, i makes sense o consruc a goodness of fi es based on he closeness of he empirical funcion and a hypohesized disribuion funcion, say F n (x) and F 0 (x). The Kolmogorov-Smirnov saisic D n is defined by [ F ( x) F ( ) ] D = sup 0 x (10) n x n

15 D n is he absolue maximum difference beween he empirical and he hypohesized disribuion funcion. An illusraion of his is displayed in Figure 4. The Kolmogorov-Smirnov saisic D n is used o es he hypohesis, H 0 : F(x)=F 0 (x), he daa follow a specified disribuion, agains all alernaives, H 1 : F(x) separaed from F 0 (x), he daa do no follow he specified disribuion where F 0 (x) is some specified disribuion funcion. We will accep H 0 if he empirical disribuion funcion F n (x) is sufficienly close o F 0 (x), ha is, if he value of D n is sufficienly small. H 0 is rejeced if he observed value of D n is larger han he criical value seleced from a able where his criical value depends upon he desired significance level and sample size [Hogg and Tanis (001)]. Figure 4 Maximum difference beween F n (x) and F 0 (x) 13 F n (X) F 0 (X) max An aracive feaure of his es is ha he disribuion of he D n es saisic iself does no depend on he underlying disribuion funcion being esed. Anoher advanage is ha i is an exac es. Despie hese advanages, he Kolmogorov- Smirnov es has limiaions: (i) i only applies o coninuous disribuions, (ii) i ends o be more sensiive near he cenre of he disribuion han a he ails, (iii) he disribuion mus be fully specified. Tha is, if locaion, scale, and shape parameers are esimaed from he daa, he criical region of he Kolmogorov-Smirnov es is no longer valid, i mus be deermined by simulaion. Due o limiaions (ii) and (iii) above, many analyss prefer o use he Anderson- Darling es for normaliy insead [Chakravari e al. (1967)]. X..4 Ryan-Joiner Tes The Ryan-Joiner es, which is similar o Shapiro-Wilk es, is based on regression and correlaion 3. The es ends o work well in idenifying a disribuion as no normal when he disribuion under consideraion is skewed. I is less discriminaing when he underlying disribuion is a -disribuion and non-normaliy is due o kurosis. We can use he Ryan-Joiner saisic R p o es he hypohesis, H 0 : he daa {x 1,..., x n } are a random sample of size n from a normal disribuion, H 1 : he daa are a random sample from some oher disribuion. The es saisic R p is he correlaion beween he daa and he normal scores. 3 For more informaion abou Shapiro-Wilk es he reader is referred o he original Shapiro and Wilk (1965) paper.

16 14 If he daa are a sample from a normal disribuion hen he NPP, plo of normal scores agains he daa, will be close o a sraigh line. The correlaion R p will be close o one and if he daa are sampled from a non-normal disribuion hen he plo will exhibi some degree of curvaure, resuling in a smaller correlaion R p. Small values of R p are herefore regarded as srong evidence agains H 0. The Ryan-Joiner es is given by he formula for he correlaion coefficien, namely ( ( Yi Yi Y )( bi b) R p = (11) Y ) ( b b) Since b = 0, R p can be simplified o ( i ( Yi Y ) bi R p = (1) Y Y) b i i Y i is he ordered observaions in a sample of size n and b i is he ph percenage poin of he sandard normal disribuion, ha is, b 1 i = Φ ( p i ). The saisic R p can be used o provide an indicaion of how non-normal he revisions are. This will be paricularly rue wih larger samples. The es has he desirable feaure of linking ogeher a graphical display of he daa wih a simple, objecive es saisic. Some may objec o he use of he erm correlaion coefficien since he b i are no random variables. However, given any se of poins in he plane, one can use he correlaion coefficien associaed wih hose poins as a descripive measure of how close hey are o a sraigh line. In his sense, R p can be hough of as a correlaion coefficien. Since R p does no arise from sampling a bivariae disribuion, i is no he same as he usual correlaion coefficien [Ryan e al. (1976)]. Ryan e al. (1976) have compared he power of R p and he Shapiro-Wilk saisic W. The ess show ha overall here is lile difference beween he powers of he wo ess for mos of he alernaives. The only appreciable difference is ha for exremely shor-ailed disribuions like he uniform and riangular, W has more power han R p, while for heavy-ailed disribuions like he Cauchy and conaminaed normal, R p does slighly beer..3 Tesing for ARCH To es if he revisions are heeroskedasic we used wo differen ess, Engle s LM es for ARCH and he squared Ljung-Box es..3.1 Engle s LM es for ARCH The original Lagrange muliplier (LM) es for ARCH proposed by Engle (198) is very simple o compue, and relaively easy o derive. Under he null hypohesis i is assumed ha he model is a sandard dynamic regression model, which can be wrien as y = β + ε (13) x where x is a se of weakly exogenous and lagged dependen variables and ε is a Gaussian whie noise process Ψ 1 ε ~ ( 0, σ ) N (14)

17 15 where Ψ 1 denoes he available informaion se. Because he null hypohesis, H 0 : here are no ARCH errors, is easily esimaed, he LM es is a naural choice. The alernaive hypohesis, H 1 : is ha he condiional error variance is given by an ARCH (q) process 1 [Bollerslev e al. (1994)]. We examine wo serial dependence properies of ineres, he exen o which he revisions are auocorrelaed and wheher hey exhibi ARCH-ype behavior. The order of auocorrelaion presen in a given revision ime series is found by TRAMO/SEATS. Tesing for ARCH in he revisions is performed using he sandard Engle (198) es. Firs he regression of he preferred ARIMA-model is esimaed for observaions = q + 1, q +,..., T and he sample residuals εˆ are saved. Nex sep is o regress he squared residuals ε on a consan and q lagged values of he squared residuals, ε 1,..., ε q ˆ ε + v (15) = ˆ ω + α ˆ 1ε 1 + α ˆ ε α ˆ qε q for = 1,,,T. The sample size T imes he squared coefficien of deerminaion from he regression of (15) hen converges o a chi-square disribuion, χ [ q] wih q degrees of freedom. There is evidence o rejec he null hypohesis if he es saisic exceeds a criical value, which means ha he revisions are acually heeroskedasic [Hamilon (1994)]. R.3. The Squared Ljung-Box Tes If no significan auocorrelaion can be found by he Ljung-Box es he conclusion is ha here is no linear srucure in he revisions 5. A dependence can however exis according o underlying non-linear srucure. Mcleod and Li (1983) showed ha he Ljung-Box es saisic has he same disribuion as he squared Ljung-Box saisic, which can be used as an indicaion of non-lineariy, ha is, heeroskedasiciy. This can also be an indicaion ha he specified model suffers from ARCH-effecs. A slow decline of he auocorrelaion funcion (ACF) of he squared residuals suggess ha a GARCH (1,1) process may be suiable for describing he revisions 6. Tha is, a low order ARCH process may no fully capure he ime-varying volailiy in he daa. The problem is ha in fac, he LM es for GARCH (1,1) is jus he same as he LM es for ARCH (1), which proposes a locally mos powerful es for ARCH and GARCH. Since i is found ha he GARCH (1,1) is ofen a superior model and is surely more parsimoniously parameerised 7, one would like a es, which is more powerful for his alernaive 8. We sugges ha when quarerly daa are being used, a fourh order process may be appropriae. However, insead of a general fourh order process, we sugges ha only he residuals in corresponding quarers of each year should be correlaed, ha is 4, 8, 1 and so on. 1 See Appendix II for an overview of his process. 5 See Appendix IV for more deails abou his es. 6 See Appendix III for more deails. 7 Parsimoniously means ha i is desirable wih as few parameers as possible in he model, because ha gives more sable and safe esimaed forecass. 8 See Bollerslev e al. (1994).

18 16 3 Daa We use quarerly daa of GDP revisions for he years from Öller and Hansson (00). The preliminary and revised figures are given in percenage change from he same quarer las year and are given in consan prices. The daa are neiher seasonally nor working day adjused. The revised, also called final, quarerly figures are published in December +, his is he ime when he firs revised annual accouns are published. The same choice was made as Öller and Hansson (00) who argued ha: By choosing + we ry o avoid, as much as possible, revisions ha are due o changes in definiions or mehods. We are going o use daa from 1980 o 1999 for he revisions Privae Consumpion and GDP. For he oher eleven revisions we are going o use daa for he period 1984 quarer wo o 1999 quarer four. These revisions have a lo of missing values in he beginning and we prefer unbroken series. The revision Invenories has one missing value in year 1990 quarer one for which we have subsiued he mean.

19 4 Resuls and analysis This Secion presens he empirical resuls of our sudy and is divided ino four pars, he resuls of he skewness and kurosis ess, he resuls of he normaliy ess, he resuls of he ARCH es and a las he resuls of he squared Ljung-Box es Skewness and kurosis ess In Table 1 he resuls of he skewness and kurosis ess are presened. The value of he firs should be close o zero and of he oher hree, for he GDP revisions o be considered normally disribued. Table 1 Values of skewness and kurosis Revisions of he componens of GDP Skewness (S) Kurosis (K) Privae Consumpion Governmen Consumpion Cenral Governmen Consumpion Local Governmen Consumpion Invesmens Change in Invenories Expors Expors of Goods Expors of Services Impors Impors of Goods Impors of Services GDP P-values are given in parenheses (0.999) (0.770) (0.458) (0.3) (0.09) (0.894) (0.85) 0.54 (0.614) (0.189) (0.664) (0.386) (0.15) (0.574).775 (0.096).676 (0.10) (0.037) 6.90 (0.01) 8.97 (0.003).603 (0.107) 8.96 (0.004) (0.046) (0.000) (0.048) (0.041) (0.000) (0.056) Privae Consumpion and Change in Invenories have skewness values near zero, which indicaes ha hey are close o a symmeric disribuion. Their kurosis values also indicae ha hey are close o being normally disribued. Governmen Consumpion is due o is skewness and kurosis values also prey close o a normal disribuion. Overall he revisions seem o be more posiively skewed han negaively, which means ha hey are more righ skewed hen lef skewed. They also have more kurosis values larger han hree hen less han hree, which indicaes ha he revisions seem o be more lepokuic han playkuric. Almos all revisions are saisically significan, due o heir p-values. Impors of Services and Expors of Services have he wors skewness and kurosis values, indicaing ha hese variables are no nearly normally disribued.

20 18 4. The normaliy ess The resuls of he normaliy ess are presened in Table. We are going o explain how he resuls should be inerpreed and we will link he resuls o he heory. Table Normaliy ess for he GDP revisions Revisions of he componens of GDP Number of observaions Privae Consumpion 80 Governmen Consumpion 63 Cenral Governmen Consumpion 63 Local Governmen Consumpion 63 Invesmens 63 Change in Invenories 63 Expors 63 Expors of Goods 63 Expors of Services 63 Impors 63 Impors of Goods 63 Impors of Services 63 GDP 80 A saisics JB saisics Dn saisics Rp saisics 0.65 (0.687) (0.59) 0.8 (0.03).068 (<0.005) (0.015) (0.5) (0.09) 0.49 (0.301) (<0.005) (0.355) (0.01) 3.17 (<0.005) (0.468) (0.919) 0.35 (0.839) (0.00) (0.000) (0.000) (0.81) (0.000) (0.03) (0.000).496 (0.87) 9.66 (0.008) (0.000).801 (0.47) (>0.150) (>0.150) 0.11 (0.048) (<0.010) (0.138) (>0.150) (>0.150) (>0.150) (<0.010) (>0.150) (0.054) (<0.010) (>0.150) (>0.100) (>0.100) (0.035) (<0.010) (<0.010) (>0.100) (<0.010) (>0.100) (<0.010) (>0.100) (0.05) (<0.010) 0.99 (>0.100) P-values are given in parenheses. If he observed p-values of he saisics are more han 0.05 we canno rejec he null hypohesis ha he revisions are normally disribued. When esing for normaliy he A saisic should be small and is p-value large, his means ha he normal disribuion condiion is fulfilled a he chosen five percen level. Cenral Governmen Consumpion, Local Governmen Consumpion, Invesmens, Expors, Expors of Services, Impors of Goods and Impors of Services revisions are considered o be non-normal according o he Anderson-Darling es. None of he A saisics are close o zero and more han half of he revisions are no normal. The revisions Impors of Services, Local Governmen Consumpion and Expors of Services have he wors saisics and seem o be he revisions ha are mos far away from a normal disribuion. The A saisic for Privae Consumpion revision seems o be he bes, which means ha his is he one closes o a normal disribuion. We also wan he Jarque-Bera (JB) saisic o be small. The es gives he same resuls as he Anderson-Darling es for all he revisions. The hird es applied was he Kolmogorov-Smirnov es. For he null hypohesis o hold he saisic D n should be small. Cenral Governmen Consumpion, Local Governmen Consumpion, Expors of Services and Impors of Services revisions are according o he Kolmogorov-Smirnov es no saisfying he condiions for normaliy. The null hypohesis for Impors of Goods is nearly rejeced a he five percen level. The Ryan Joiner saisic R p should be as close o one as possible. We found ha he seven revisions ha did no saisfy he normaliy condiions of he Anderson-

21 Darling and he Jarque-Bera ess are also non-normal according o he Ryan Joiner es. We found ha barely half of he revisions are no normal according o he four ess above. We can also see ha all four ess choose he same bes and wors revisions, which means ha he ess are very concordan. The es ha produces slighly differen resuls is he Kolmogorov-Smirnov es. This es seems o be less sensible o non-normaliy han he ohers. The saisics for he revisions of Invesmens, Expors and Impors of Goods do no rejec he null hypohesis a he five percen level, which he oher ess do. This can be explained by he fac ha he Kolmogorov-Smirnov es ends o be more sensiive near he cenre of he disribuion han a he ails. This means ha he es does no deec all he non-normal behaviour in he ails. Revisions of Impors are close o be normal. Bu is wo componens, Impors of Goods and Impors of Services, have revisions ha are no even close o be normal. The explanaions can be ha he deviaions from normaliy in he wo revisions parly cancel The ARCH Tes Resuls of he auocorrelaion specificaion and ess for relaively low order ARCH (q=1,) are given in Table 3. Table 3 Auocorrelaion specificaion and ARCH ess for GDP revisions Revisions of he componens of GDP Auocorrealion specificaion* Privae Consumpion (0,0,1) Governmen Consumpion (0,0,1) Cenral Governmen Consumpion (0,0,1) Local Governmen Consumpion (0,0,1) Invesmens (1,0,0)s Expors (1,0,0) Expors of Services (1,0,0) (1,0,0) Impors (1,0,0) Impors of Goods (1,0,0) Impors of Services (0,0,1) ARCH (1) saisics 1 ARCH () saisics (0.64) (0.896) (0.513) (0.763) (0.999) (0.865) (0.999) (0.43) (0.094) (0.37) (0.671) (0.915) (0.340) (0.119) (0.639) (0.55) (0.486) (0.67) (0.77) (0.861) P-values are given in parenheses. 1 If he ARCH saisics are larger han we will rejec he null hypohesis, seing α equal o If he ARCH saisics are larger han we will rejec he null hypohesis, seing α equal o * s sands for seasonal AR(1). If he ARCH (1) saisics are larger han and he ARCH () saisics are larger han we will rejec he null hypohesis ha here are no ARCH effec. Looking a our resuls we can see ha here is no evidence of ARCH for any of he revisions considered, even if he null hypohesis for he revisions of Invesmens [ARCH (1)]

22 0 and Expors of Services [ARCH ()] are close o being rejeced a he five percen level. The revisions Change in Invenories, Expors of Goods and GDP shows no auocorrelaion and herefore we canno creae adequae models wih ARIMA for hem. The es saisics for hese revisions are insead given by he squared Ljung- Box o invesigae if hey have ARCH behaviour. An indicaion of ARCH is ha he residuals will be uncorrelaed bu he squared residuals will show auocorrelaion. In Table 4 he resuls of he squared Ljung-Box es is shown. Table 4 The squared Ljung-Box es for he GDP revisions Revisions of he componens of GDP Change in Invenories Expors of Goods GDP Q K (4) saisics (0.476) (0.71) 4.65 (0.01) Q K (8) saisics (0.174) (0.536) (0.33) Q K (1) saisics (0.154) 13.0 (0.80) (0.509) P-values are given in parenheses. 1 If he Q K saisics are larger han we will rejec he null hypohesis, seing α equal o If he Q K saisics are larger han we will rejec he null hypohesis, seing α equal o If he Q K saisics are larger han we will rejec he null hypohesis, seing α equal o The Q K saisics should be smaller han he criical values 7.815, and , respecively, for he null hypohesis o hold on he five percen level. The Q K saisics does no rejec he null hypohesis of homoscedasiciy for any of he revisions. Tha means ha he revisions Change in Invenories, Expors of Goods and GDP don have any ARCH effecs. Hence all revisions in his sudy can be regarded as homoscedasic.

23 1 5 Conclusions There are several ess available when one wans o es daa for normaliy. We chose four of hem o use on he GDP revisions ha we were ineresed in. We have also checked for skewness and kurosis. We have found ha revisions of Privae Consumpion and Change in Invenories have skewness values near zero, indicaing ha hey are close o a symmeric disribuion. Their kurosis also indicaes ha hey are close o being normal. Revisions of Governmen Consumpion are prey close o a normal disribuion. Impors of Services and Expors of Services have he wors skewness and kurosis values indicaing ha hese revisions are no even nearly normally disribued. This is concordan wih he resuls in Öller and Hansson (00), which conains hisograms over he revision disribuions. We also found ha he revisions were more posiively skewed han negaively which also is concordan wih he resuls from Öller and Hansson (00). We can draw he conclusion ha more han half of he revisions are no normal according o he four ess ha we have used: Anderson-Darling, Jarque-Bera, Kolmogorov-Smirnov and Ryan-Joiner. We can also see ha all four ess chose he same bes (Privae Consumpion) and wors (Local Governmen Consumpion, Expors of Services and Impors of Services) revisions, which means ha he ess are very concordan. The es ha produces slighly differen resuls is he Kolmogorov-Smirnov es. I seems o be less sensiive o non-normaliy han he ohers. When we checked for he presence of ARCH effecs in he GDP revisions we used wo ess, he Lagrange Muliplier (LM) es and he squared Ljung-Box es. In TRAMO/SEATS we could idenify adequae models ha we used for he LM es for ARCH. We found ha he revisions Change in Invenories, Expors of Goods and GDP did no conain any auocorrelaion. Therefore we used he squared Ljung- Box es for hese revisions o invesigae if hey had ARCH behavior. None of he revisions conained any ARCH effec, bu he null hypoheses of homoscedasiciy for he revisions of Invesmens and Expors of Services were close o be rejeced a he five percen level. The Swedish GDP revisions for he years are skewed and hick-ailed and our figures show ha he revisions canno be described by a normal disribuion. We also found ha none of he revisions conained any ARCH effecs.

24 6 References Bollerslev, T. (1986). Generalized Auoregressive Condiional Heeroskedasiciy. Journal of Economerics, 31, Bollerslev, T., Engle, R. F. and Nelson, D. F. (1994). ARCH Models. I: R. F. Engle and D. L. McFadden (eds.), Handbook of Economerics, Vol. 4. Amserdam: Elsevier Science, Bowerman, B. L. and O Connel, R. T. (1993). Forecasing and Time Series an Applied Approach, 3 rd ed. Belmon: Duxbury Press. Chakravari, I. M., Laha, R. G. and Roy, J. (1967). Techniques of Compuaion Descripive Mehod and Saisical Inference. I: Handbook of Mehods of Applied Saisics, Vol. 1. New York: Wiley. D Augosino, R. B. and M. A. Sephens. (1986). Goodness-of-Fi Techniques. New York: Marcel Dekker. Engle, R. F. (198). Auoregressive Condiional Heeroskedasiciy wih Esimaes of he Variance of Unied Kingdom Inflaion. Economerica, 50, Gujarai, D. N. (00). Basic Economerics, 4 h ed. Boson: McGrawHill. Hamilon, J. D. (1994). Time Series Analysis. Princeon: Princeon Universiy Press. Harvey, D. I. and Newbold, P. (00). The Non-Normaliy of some Macroeconomic Forecass. Inernaional Journal of Forecasing, 19, Hogg, R. V. and Tanis, E. A. (001). Probabiliy and Saisical Inference, 6 h ed. Upper Saddle River: Prenice Hall. Jarque, C. M. and Bera, A. K. (1987). A Tess for Normaliy of Observaions and Regression Residuals. Inernaional Saisical Review, 55, Kaplansky, I. (1945). A Common Error Concerning Kurosis. Journal of he American Saisical Associaion, 40, Kleinbaum, D. G., Kupper, L. L., Muller, K. E. and Nizzan, A. (1998). Applied Regression Analysis and Oher Mulivariae Mehods, 3 rd ed. Pacific Grove: Duxbury Press. Ljung, G. M. and Box, G. E. P. (1978). On a Measure of a Lack of Fi in Time series Models. Biomerika, 65, Mcleod, A. I. and Li, W. K. (1983). Diagnosic Checking ARIMA Time Series Models Using Squared-Residual Auocorrelaion. Journal of Time Series Analysis, 4, Nilsson, J. and Rosander, O. (003). Tidsserieanalys av svenska BNP-revideringar Kan BNP-revideringarna prognosiseras med en ARIMA-modell?. Sockholm: Sockholm Universiy. Öller, L.-E. and Hansson, K.-G. (00). Revisions of Swedish Naional Accouns and an Inernaional Comparision. I: SOU 00:118. Sockholm: Frizes förlag. Ryan, A. T. Jr., Joiner, B. L. and Ryan, B. F. (1976). Miniab : Suden Handbook. Norh Sciuae: Duxbury Press. Shapiro, S. S. and Wilk, M. B. (1965). An Analysis of Variance Tes for Normaliy. Biomerika, 5, Sephens, M. A. (1974). EDF Saisics for Goodness of Fi and Some Comparisons. Journal of he American Saisical Associaion, 69,

25 3 Appendices Appendix I Glossary > bigger han < smaller han ^ α β bigger or equal o smaller or equal o an esimaor (usually maximum likelihood) or forecas value regression coefficien alpha and significance level regression coefficien bea α ( L) polynomial in he lag operaor L β ( L) polynomial in he lag operaor L χ [ q] chi-square disribuion wih q degrees of freedom ε µ he mean v σ σ variance sochasic error erm, usually called whie noise unpredicable, or innovaion, error sandard deviaion σ variance a ime n = 1 summaion over implici range summaion over range shown Ψ 1 he informaion se ω { } consan erm in ARCH specificaion sochasic process or sequence absolue value A b i is disribued as Anderson-Darling saisic he ph percenage poin of he sandard normal disribuion, ha is, 1 b i = Φ ( p i ) b d D n { } mean of he slope coefficien he degree of differencing Kolmogorov-Smirnov saisic E condiional expecaion of a variable given an oher E () expeced value of

26 4 F he cumulaive disribuion funcion of he specified disribuion in he Anderson-Darling es F n ( x) he empirical funcion in he Kolmogorov-Smirnov es F 0 ( x) he hypohesized disribuion funcion in he Kolmogorov-Smirnov es H 0 H 1 JB K K l L ln n n p NPP ( 0, σ ) null hypohesis alernaive hypohesis Jarque-Bera saisic kurosis he degrees of freedom in he Ljung-Box es lag l he lag operaor naural logarihm sample size he number of parameers Normal Probabiliy Plo N normal densiy funcion wih zero mean and variance p-value p p x (k) Q K Q K q r r l r R R p s r ( aˆ ) S Sup x T W Y i X z i probabiliy, or es-rejecion probabiliy he order of an auoregressive process he probabiliy ha X assumes he value k in a momen generaing funcion he Ljung-Box es saisic he squared Ljung-Box saisic he order of a moving average process he mean of he revisions he squared sample auocorrelaion of he residuals a lag l in a Ljung- Box es he revisions a ime he squared coefficien of deerminaion Ryan-Joiner saisic he esimaed sandard deviaion or sandard error of he revisions skewness sands for supremum, and is he maximum verical disance beween he graphs of F n (x) and F 0 (x) over he range of possible x values ime number of observaions in a ime series Shapiro-Wilk saisic he ordered observaions in a sample size n from a Ryan-Joiner es a random variable or a sochasic variable he ih sored, sandardized, sample value in he Anderson-Darling es σ

27 Appendix II ARCH models The Auoregressive Condiional Heeroskedasiciy (ARCH) model was firs suggesed by Engle (198). This differed from earlier ime series and economeric models in ha i allowed for a ime dependan variance. The condiional variance may change over ime as a funcion of pas errors, leaving he uncondiional variance unchanged. Compuaional problems may arise when he polynomial presens a high order. To faciliae such compuaion, Bollerslev (1986) proposed a generalizaion of he ARCH model, he Generalized Auoregressive Condiional Heeroskedasiciy (GARCH) model. The simples and very useful ARCH model is { ε Ψ } = ω σ E 1 + αε 1 (16) where Ψ 1 denoes he informaion se, ypically including ε 1 and is enire hisory. This specificaion is called an ARCH (1) process. To ensure ha σ 0 irrespecive of ε 1 we need o impose ha ω > 0 andα 0. The ARCH (1) model says ha when a big shock happens in period 1 i is more likely ha ε has a large (absolue) value as well. Tha is, when ε 1 is large, he variance of he nex innovaion ε is also large. The specificaion in (16) does no imply ha he process for ε is non-saionary. I jus says ha he squared values ε and ε 1 are correlaed. The uncondiional variance of ε is given by { ε } = ω + αe{ ε } σ (17) = E 1 and has a saionary soluion ω σ = (18) 1 α provided ha 0 α < 1 The ARCH (1) model is easily exended o an ARCH (q) process, which we can wrie as ( L) ε, σ = ω + α1ε 1 + α ε α qε q = ω + α 1 (19) where α ( L) is a lag polynomial of order q 1. To ensure ha he condiional variance is non-negaive,ω and he coefficiens in ( L) saionariy i is also required ha α ( L) < 1. The effec of a shock j periods ago on α mus be non-negaive. For curren volailiy is deermined by he coefficienα j. In an ARCH (q) model, old shocks of more han q periods ago have no effec on curren volailiy. 5

28 6 Appendix III GARCH models In is general form, a GARCH (p,q) model can be wrien as p q + α jε j + j= 1 j= 1 σ = ω β σ (0) or ( L) ε β ( L) σ = + + ω α 1 1 j j σ, (1) β are lag polynomials. In pracice a GARCH (1,1) specificaion ofen performs very well. I can be wrien as Where α ( L) and ( L) σ, () = + + ω αε 1 βσ 1 which has only hree unknown parameers o be esimaed. Non-negaiviy of σ requires haω, α and β are non-negaive. If we define he surprise in squared shocks as v ε σ, he GARCH (1,1) process can be rewrien as ( α + β ) ε 1 + v βv 1 ε, (3) = ω + which shows ha he squared errors follow an ARMA (1,1) process. While he error v is uncorrelaed over ime, because i is a surprise erm, i does exhibi heeroskedasiciy. The roo of he auoregressive par isα + β, so ha saionariy requires ha α + β < 1. Values of α + β close o uniy imply ha he persisence in E ε = 1 = E σ 1 σ volailiy is high. Noing ha, under saionariy, { } { } uncondiional variance of ε can be wrien as 9 σ ω ασ + βσ or, he = + (4) ω σ = 1 α β (5) 9 The equaliy only holds if ε does no exhibi auocorrelaion.

29 Appendix IV LJUNG-BOX es One way o use he residuals o check he adequacy of he overall model is o examine a saisic ha deermines wheher he firs K sample auocorrelaions of he residuals, considered ogeher, indicae adequacy of he model. For his reason, i is ofen referred o as a pormaneau es. Ljung-Box es ha we have used for his sudy can be calculaed in he following way Q K = ( ) ( )( ) K r + ( l aˆ n d n d n d l ) (6) l= 1 where n is he sample size, d is he degree of (non seasonal) differencing used o ransform he original ime series values ino saionary ime series values and r l ( aˆ ) is he square of he r l ( aˆ ), he sample auocorrelaion of he residuals a lag l. Tha is, he sample auocorrelaion of residuals separaed by a lag of l ime unis. The modelling process is supposed o accoun for he relaionship beween he ime series observaions. If i does accoun for hese relaionships, he residuals should be small. The larger Q K is, he greaer he risk of auocorrelaed residuals. Hence a large value of Q K indicaes ha he model is inadequae [Bowerman and O Connell (1993)]. Under he null hypohesis ha he residuals are no correlaed he Q K will approximaely follow a chi-square disribuion. If Q K is greaer han χ [ α ]( K n p ) he null hypohesis will be rejeced on he significan level α and he model should be modified 10. This is equivalen o rejecing he null hypohesis if he observed p-value is less han α [Ljung and Box (1978)] K are he degrees of freedom and n p he number of parameers ha mus be esimaed in he model under consideraion.

30 004:01 Hjälpverksamhe. Avrapporering av projeke Sysemaisk hanering av hjälpverksamhe 004:0 Repor from he Swedish Task Force on Time Series Analysis 004:03 Minskad dealjeringsgrad i Sveriges officiella urikeshandelssaisik 004:04 Finansiell sparande i den svenska ekonomin. Uredning av skillnaderna i finansiell sparande Naionalräkenskaper, NR Finansräkenskaper, FiR Bakgrund jämförelser analys 004:05 Designuredning för KPI: Effekiv allokering av urvale för prismäningarna i buiker och jänsesällen. Examensarbee inom Maemaisk saisik uför på Saisiska cenralbyrån i Sockholm 004:06 Tidsserieanalys av svenska BNP-revideringar :07 Labor Qualiy and Produciviy: Does Talen Make Capial Dance? 004:08 Slurappor från projeke Uppsnabbning av den ekonomiska koridssaisiken 004:09 Bilagor ill slurapporen från projeke Uppsnabbning av den ekonomiska koridssaisiken 004:10 Förbäring av borfallsprocessen i Inrasa 004:11 PLÖS. Samordning av produkion, löner och sysselsäning 004:1 Ne lending in he Swedish economy. Analysis of differences in ne lending Naional accouns (NA) Financial accouns (FA). Background comparisons - analysis