On Structural Breaks and Nonstationary Fractional. Intergration in Time Series

Transcription

1 European Journal of Business and Managemen ISSN (Paper) ISSN -839 (Online) Vol 4, No.5, 01 On Srucural Breaks and Nonsaionary Fracional Inergraion in Time Series Olanrewaju I. Shiu 1 OlaOluwa S. Yaya * Raphael A. Yemian 3 13 Deparmen of Saisics, Universiy of Ibadan, Nigeria * of he corresponding auhor: os.yaya@ui.edu.ng Absrac The growh of an economy is deermined largely by he growh of is Gross Domesic Produc (GDP) over ime. However, GDP and some economic series are characerized by nonsaionariy, srucural breaks and ouliers. Many aemps have been made o analyze hese economic series assuming uni roo process even in he presence of changes in he mean level wihou considering possible fracional inegraion. This paper aims a examining he srucural breaks and nonsaionariy in he GDP series of some seleced African counries wih a view o deermining he influence of srucural breaks on he level of saionariy of hese series. These series are found o be nonsaionary wih some evidence of long memory. They were found o experience one or more breaks over he years and his may be due o insabiliy in he governmen and economic policies in he seleced African counries. The measure of relaive efficiency shows ha auoregressive fracional inegraed moving average (ARFIMA) models is beer han he corresponding auoregressive inegraed moving average (ARIMA) models for he series considered in his sudy. Keywords: fracional inegraion, gross domesic produc, srucural breaks 1. Inroducion Economic growh for many counries is majorly deermined by he counry s Gross Domesic Produc (GDP). Among African counries Souh Africa is raed as he riches counry because of her highes value of GDP each year. For his reason, i is sensible o sudy he paern in which his is realized over he years bearing in mind ha he series are usually nonsaionary. Mos researches in economic ime series have concenraed on he behavior of oher economic measures and model are fied o he series bu fewer aricles have considered GDP. Economic and financial ime series ofen display properies such as breaks, heeroscedasiciy, missing values, ouliers, nonlineariy jus o menion a few. Of much imporance in ime series is he srucural break or mean shif which affec he level of saionariy in he series. Quie a number of aricles have shown ha break in srucure of he series may cause a saionary series I ( 0) o be fracionally inegraed (Granger and Hyung, 004; Ohanissian e al., 008). In he conex of nonsaionary series, here are fewer aricles o show he effec of breaks in he series. Chivillon (004) in he discussion paper on A 40

2 European Journal of Business and Managemen ISSN (Paper) ISSN -839 (Online) Vol 4, No.5, 01 comparison of muli-sep GDP forecass for Souh Africa reviewed ha srucural breaks and uni roo occurred in Souh African s GDP over he las hiry years. Also, Romero-Ávila and De Olavide (009) considered uni roo hypohesis for per capia real GDP series in 46 African counries wih daa spreading from 1950 o 001 and found muliple srucural breaks. Srucural breaks is examined for expor, impor and GDP in Ehiopia using annual macroeconomic ime series from 1974 o 009 and he sudy shows ha he economy has suffered from srucural change in he sample periods 199, 1993 and 003 (Allaro e al., 011). Aly and Srazicich (011) considered he GDP of he Norh African counries and observed one or wo srucural breaks excep for Morocco where break was no observed. This sudy seeks o invesigae he sabiliy (saionariy) and/or change in he mean level (srucural breaks) over ime. We also invesigae he naure and ype of nonsaionariy ha may have been brough abou as a resul of srucural breaks in each series.. The GDP in African Counries The World s record in 005 shows ha Souh Africa was he riches counry among African counries wih GDP of $456.7 billion. This figure was followed by Egyp, Algeria, Morroco and Nigeria wih GDP of $95., 196, 18.3, and $71 billion leaving Nigeria as he fifh in he ranking. The sixh o 10 h counries were Sudan, Tunisia, Ehiopia, Ghana and Congo he Republic (hp:// Similar accoun repored in World Economic Oulook Daabase of Inernaional Moneary Fund (IMF, 009) shows ha Souh Africa sill mainained her posiion as he firs in 008 wih GDP of $76.8 billion, followed by Nigeria ($07.1 billion), Egyp ($16.6 billion), Algeria ($159.7 billion) and Libya ($89.9 billion). The nex counries in he ranking are Morroco, Angola, Sudan, Tunisia and Kenya. IMF (011) presened he 010 hisorical GDP daa wih similar repor on GDP wih Souh Africa having $54.0 billion of GDP, followed by Egyp ($497.8 billion), Nigeria ($377.9 billion), Algeria ($51.1 billion) and Morroco ($151.4 billion). Angola, Sudan, Tunisia, Libya and Ehiopia were in he sixh o 10 h wealh posiion in Africa. Comparaive analysis of he counry s wealh in 005, 008 and 010 shows ha Nigeria moved from he fifh (005) o second posiion in 008 and laer dropped o hird posiion in 010. This swerve in wealh of a counry as deermined by he GDP may be due o some governmen policies and poliical facors and herefore, here is need o sudy he paern in which hese series are realized over he years. Change in governmen policies and poliical insabiliy may cause a series o experience a sharp break and hese end o aler he disribuional paern of he series. As par of economeric modelling, we inroduce srucural breaks in form of mean shif in his work in order o examine possible breaks in he series and economeric ime series models are also applied o esablish our claim on nonsaionariy fracional inegraion of GDP series. 41

3 European Journal of Business and Managemen ISSN (Paper) ISSN -839 (Online) Vol 4, No.5, Mehodology The augmened Dickey Fuller (ADF) uni roo es is used o esablish nonsaionariy in he GDP series of each counry. Once he uni roo is insignifican, we esimae he fracional difference parameer. This is achieved by applying he mehod used in Shiu and Yaya (010) which sugges differencing he nonsaionary series of order d 0 as many number of imes o aain saionariy. Then, he fracional difference parameer is esimaed from he resuling saionary series. We hen apply differencing and adding back mehod of Velasco (005) o esimae he nonsaionary fracional difference parameer, d 0. Tha is, assuming he ime series X and aking he uni difference of he series n number of imes and his gives he uni difference order as u. We hen applied semi-parameric esimaion approach of described in Geweke and Porer-Hudak (1983) o esimae he saionary fracional difference parameer assumed o be 0.5 < d < 0.5. The esimae of nonsaionary fracional difference parameer is hen esimaed based on d0 = d + u (see Shiu and Yaya, 010). Srucural breaks can be visualized in he ime plo of he observed series as forms of nonlineariy and ouliers. However, his can be viewed more dearly from he plo of he differencing parameer d 0 agains he specified ime period. The laer mehod is more objecive and in line wih agreemen of Gil-Alana (008) and Gil-Alana e al. (011). The papers applied he non-parameric approach of Robinson (1994) and he same will be used in his paper. (1) y X B X u d0 = α + ; (1 ) =, = 1,,..., where y is he observed ime series, α is he inercep, d 0 is he fracional difference parameer and u is an I ( 0) process assumed o be a whie noise. When he differencing parameer ( ) d of a series is saionary fracional, 0.5 < d < 0.5 such a series is said o exhibi long memory. The appropriae model for such series is Auoregressive Fracional Moving Average (ARFIMA) model defined as, () d 0 y = φ + φ y + φ y φ y + ε + θ ε + θ ε θ ε p p 1 1 q q 4

4 European Journal of Business and Managemen ISSN (Paper) ISSN -839 (Online) Vol 4, No.5, 01 where d0 { } = u+ d and u = 1,,... depending on he number of he uni differences. The φi, θ j, i= 1,..., p; j = 1,..., q are he parameers in he model and i ε. disribued as i N ( 0,1) ε are he random process 5. Source of Daa The daa used in his sudy were he GDP per capia per person in curren US Dollar of 7 African counries from 1960 o 006. The annual daa were sourced from Inernaional Moneary Fund (IMF) daabase. The GDP daa is compued from he purchasing power pariy (PPP) of counries per capia, ha is he value of all final goods and services produced wihin a counry in a given year divided by he average or mid-year populaion for he same year. 6. Resuls and Discussion The ime plos of he GDP series for differen counries are shown in Figure 1 below. The daa used are given in US dollars in order o allow counry o counry comparison. Various ypes of movemens were noiced in he plos of GDP for hese counries. A general upward movemens were noiced from 1961 for he nex five years, followed by sharp increases from 1966 o 1976 or here abou. Thereafer, differen ypes of movemens were exhibied by differen counries for he res of he period under sudy. Mos of he counries experienced drops in he GDP which may be due o decrease in he values expressed in he counry s local currency. Nigeria for example experienced significan drops beween 1980 and

5 European Journal of Business and Managemen ISSN (Paper) ISSN -839 (Online) Vol 4, No.5, 01 Figure 1: Time Plos of GDP in African Counry wih figures given in US Dollars ALGERIA BENIN BOTSWANA BURKINA FASO BURUNDI CAMEROON CHAD CONGO COTE D'VOIRE EGYPT GABON GHANA KENYA LESOTHO LIBERIA MALAWI MALAYSIA MAURITANIA NIGER NIGERIA SENEGAL SIERRA LEONE SOUTH AFRICA SUDAN TOGO UGANDA ZAMBIA 44

6 European Journal of Business and Managemen ISSN (Paper) ISSN -839 (Online) Vol 4, No.5, 01 This significan drop experienced in Nigeria is also raced back o he behavior of Naira-US dollar exchange raes in he period under invesigaion. In ha case, he nominal GDP in Naira is given below in Figure, and his shows asronomical increase of GDP in he counry. Comparison of he wo plos of GDP for Nigeria shows ha exchange rae has effec on he counry s wealh. Figure : Time plo of Nigerian (nominal) GDP in millions of Naira NIGERIA 0,000,000 16,000,000 1,000,000 8,000,000 4,000, From Table 1, he ADF uni roo es shows ha all he series are nonsaionary a 5% level of significance. However, all he series aained saionariy afer he firs difference. The above shows ha he GDP series are inegraed of order one, I ( 1). This suggess ha he series can be modelled as ARIMA (p, d, q). Shiu and Yaya (009, 010) showed ha under cerain condiions, ARFIMA model may be beer han ARIMA model when nonsaionariy is esablished in a series. Table 1: Uni roo ess on GDP series Observed Series Firs Differenced Observed Series Firs Differenced Counries ADF Prob. ADF Prob. Counries ADF Prob. ADF Prob. Algeria Liberia Benin Malawi Boswana Malaysia Burkina Mauriania Burundi Niger Cameroon Nigeria Chad Senegal Congo Sierra Coe Souh Egyp Sudan Gabon Togo Ghana Uganda Kenya Zambia Lesoho

7 European Journal of Business and Managemen ISSN (Paper) ISSN -839 (Online) Vol 4, No.5, 01 Wih his in mind, we examined wheher or no all he series were acually I ( 0) or I ( d) where d is he fracional difference parameer for all he series. The resul is shown in Table 3. Counry Algeria Benin Boswana Table 3: Esimaes of Fracional Difference Parameer Burkina Faso Burundi Cameroon Chad Congo Côe D'Ivoire d ˆ Counry Egyp Gabon Ghana Kenya Lesoho Liberia Malawi Malaysia Mauriania d ˆ Sierra Souh Counry Niger Nigeria Senegal Sudan Togo Uganda Zambia Leone Africa d ˆ I can be observed ha all he series were no exacly of order one, I ( 1). 6.1 Invesigaion of Srucural Breaks The firs value corresponds o he esimae of d based on he sample wih he firs 35 observaions, ha is, from 1960 o 1994, hen he following one corresponds o he sample [ ], he nex. [ ] and so on ill he las one which corresponds o [ ] making 13 blocks of samples i.e , , , , , , , , , , , , The following figure displays for each counry he esimaes of differencing parameers d 0 along wih he 95% confidence band using he model in (1). Sable esimaes of d across he differen subsamples are observed in he GDP of Burundi, Cameroon, Gabon, Kenya, Liberia, Niger, Nigeria, Sierra Leone and Uganda. In Benin, Boswana, Lesoho and Souh Africa, we noice a decrease in he degree of inegraion abou he 10h esimae [003]. A sligh increase in he esimaed value of d abou he 10 h / 11 h esimae [003, 004] is observed in Algeria, Chad, Congo, Sudan and Zambia. In fac, in he above 10 counries, we observe a sharp increase abou he year 003. For anoher group, we observe a sligh decrease in he nd esimae [1995]. This group include Burkina Faso, Coe de Ivory, Senegal and Togo. For Ghana and Malawi, break is observed in he 8 h block [001]. 46

8 European Journal of Business and Managemen ISSN (Paper) ISSN -839 (Online) Vol 4, No.5,

9 European Journal of Business and Managemen ISSN (Paper) ISSN -839 (Online) Vol 4, No.5, Modelling of he Series To deermine he mos appropriae model for GDP series in he seleced counries in Africa, he ARIMA (p, d, q) and ARFIMA (p, d, q) modelling were carried ou on he series wih a view o measure he relaive efficiency (R.E) of he ARFIMA model over he ARIMA model. The resuls are shown in Table 4 and 5 below. Table 4: Esimaed Nonsaionary ARFIMA Models for he African GDP Series Algeria y = y y y y + ε Sk. = Ex. Kur. = ARCH = [0.1535] σ ARFIMA = σ ARIMA = 779 σ ARFIMA ARIMA = Benin y = y y y y y + ε Sk. = Ex. Kur. = ARCH =.5958 [0.0679] σ ARFIMA = σ ARIMA = σ ARFIMA ARIMA = Boswana y = y y y y y y y y + ε Sk. = Ex. Kur. = ARCH = [0.507] σ ARFIMA = σ ARIMA = σ ARFIMA ARIMA = Burkina Faso y = y y y y y + ε Sk. = Ex. Kur. =.311 ARCH = [0.5970] σ ARFIMA = σ ARIMA = σ ARFIMA ARIMA = Burundi y = y y y y y y y y + ε Sk. = 0.04 Ex. Kur. = ARCH = [0.4914] σ ARFIMA = σ ARIMA = σ ARFIMA ARIMA =

10 European Journal of Business and Managemen ISSN (Paper) ISSN -839 (Online) Vol 4, No.5, Cameroon y = y y y y y y y + ε Sk. = Ex. Kur. =.3534 ARCH = [0.466] σ ARFIMA = σ ARIMA = σ ARFIMA ARIMA = Chad y = y y y y y + ε Sk. = Ex. Kur. = ARCH = [0.6773] σ ARFIMA = σ ARIMA = σ ARFIMA ARIMA = Congo y =.8E y y y y y y y y + ε Sk. = Ex. Kur. = ARCH = 0.49 [0.7335] σ ARFIMA = σ ARIMA = σ ARFIMA ARIMA = Coe D'vore y = y y y + ε Sk. = Ex. Kur. = ARCH = [0.640] σ ARFIMA = σ ARIMA = σ ARFIMA ARIMA = Egyp y = y y 0.38 y + ε Sk. = Ex. Kur. = ARCH = [0.6604] σ ARFIMA = σ ARIMA = σ ARFIMA ARIMA = Gabon y = y 0.60 y y y y y y + ε Sk. = Ex. Kur. = ARCH = [0.6886] σ ARFIMA = σ ARIMA = σ ARFIMA ARIMA = Ghana = ( 1 ) y B X 49

11 European Journal of Business and Managemen ISSN (Paper) ISSN -839 (Online) Vol 4, No.5, 01 y = y y y y y 0.66 y y + ε Sk. = Ex. Kur. = 0.61 ARCH = [0.7460] σ ARFIMA = σ ARIMA = σ ARFIMA ARIMA = Kenya y = y y y y y y + ε Sk. = Ex. Kur. = ARCH = 0.6 [0.8800] σ ARFIMA = σ ARIMA = σ ARFIMA ARIMA = Lesoho y = y y y y y y + ε Sk. = Ex. Kur. = ARCH = 1.64 [0.3156] σ ARFIMA = σ ARIMA = σ ARFIMA ARIMA = Liberia y = y y y y + ε Sk. = Ex. Kur. = ARCH =.433 [ ] σ ARFIMA = σ ARIMA = σ ARFIMA ARIMA = Malawi y = y y y y y y y y + ε Sk. = Ex. Kur. = ARCH = [0.531] σ ARFIMA = σ ARIMA = σ ARFIMA ARIMA = Malaysia y = y y y y y y y + ε Sk. = Ex. Kur. = ARCH = [0.531] σ ARFIMA = σ ARIMA = σ ARFIMA ARIMA = Mauriania y = y y y y y y y + ε

12 European Journal of Business and Managemen ISSN (Paper) ISSN -839 (Online) Vol 4, No.5, 01 Sk. = Ex. Kur. = ARCH =.6109 [0.0678] σ ARFIMA = σ ARIMA = σ ARFIMA ARIMA = Niger y = y y y y y y y + ε Sk. = Ex. Kur. = ARCH = [0.3949] σ ARFIMA = σ ARIMA = σ ARFIMA ARIMA = Nigeria y = y y y y y 0.61 y + ε Sk. = Ex. Kur. = ARCH = [0.159] σ ARFIMA = σ ARIMA = σ ARFIMA ARIMA = Senegal y = y y y y y y y + ε Sk. = Ex. Kur. = ARCH = [0.897] σ ARFIMA = σ ARIMA = σ ARFIMA ARIMA = Sierra Leone y = y y y y y y y + ε Sk. = Ex. Kur. = ARCH = [0.7998] σ ARFIMA = σ ARIMA = σ ARFIMA ARIMA = Souh Africa y = y 0.51 y y y y y y y + ε Sk. = Ex. Kur. = ARCH = [0.1601] σ ARFIMA = σ ARIMA = σ ARFIMA ARIMA = Sudan y = y y y y y + ε Sk. = Ex. Kur. = ARCH = [0.517] σ ARFIMA = σ ARIMA = σ ARFIMA ARIMA =

13 European Journal of Business and Managemen ISSN (Paper) ISSN -839 (Online) Vol 4, No.5, Togo y = y y y y y + ε Sk. = Ex. Kur. = ARCH =.3798 [0.0868] σ ARFIMA = σ ARIMA = σ ARFIMA ARIMA = Uganda y = y 0.86 y y y y y + ε Sk. = Ex. Kur. = ARCH = [0.366] σ ARFIMA = σ ARIMA = σ ARFIMA ARIMA = Zambia y = y y y + ε Sk. = Ex. Kur. =.454 ARCH = [0.5448] σ ARFIMA = σ ARIMA = σ ARFIMA ARIMA = The relaive efficiency beween he ARFIMA (p, d, q) and ARIMA (p, d, q) are given in Table 5. 5

14 European Journal of Business and Managemen ISSN (Paper) ISSN -839 (Online) Vol 4, No.5, 01 Table 5: Relaive Frequency of ARFIMA model over ARIMA model Break in nd subsample Break in 8 h subsample Break in 10 h /11 h subsamples No Break Counry R.E Counry R.E Counry R.E Counry R.E Burkina Ghana Algeria Burundi Faso Coe Malawi Chad Cameroon 0016 D ivoire Senegal Congo Togo Sudan Zambia Gabon Benin Kenya Boswana Liberia Lesoho Niger Souh Nigeria Africa Sierra Leone Uganda One would have expeced ha he R.E for hose counries wih sable esimaes of d across he differen subsamples would have R.E = 1. This is no so because he series also exhibi long memory in heir saionary processes. 5. Conclusion We have considered he dynamics of GDP of some African counries in his paper using he economeric ime series modelling approach. This approach involved sudying he propery of he series via esing for occasional breaks. The approach proposed in Robinson (1994) was used o examine he breaks in he series and one or more breaks were observed in some counries. Some of hese counries are poor and heir GDPs end o rise and fall as expressed in dollars, hough he value may rise asronomically as in he case of Nigeria when expressed in Naira. Finally, we applied he nonsaionary ARFIMA models on he 7 series considered in he paper and found ha he GDP series are acually nonsaionary. The ARFIMA models are found o perform beer ha he corresponding ARIMA models and hese resuls follow ha of Shiu and Yaya (009, 010). References 53

15 European Journal of Business and Managemen ISSN (Paper) ISSN -839 (Online) Vol 4, No.5, 01 Alaro, H. B., Kassa, B. and Hundie, B. (011). A ime series analysis of srucural break ime in he macroeconomic variables in Ehiopia. African Journal of Agriculural Research, 6(): Aly, H. Y. and Srazicich, M. C. (011). Which counries in Norh Africa can resis he global Recession? An empirical invesigaion. Proceeding from he Annual conference on Poliics and Economic Developmen. Chevillon, G. (004). A comparison of Muli-sep GDP forecass for Souh Africa. Deparmen of Economics Discussion paper series. No. 1, pp Gil-Alana, L.A. (008) Fracional inegraion and srucural breaks a unknown periods of ime, Journal of Time Series Analysis 9, Gil-Alana, L. A., Shiu, O. I. and Yaya, O. S. (011). Long memory, Srucural breaks and Mean shifs in he Inflaion raes in Nigeria. Navarra Cener for Inernaional Developmen, Universidad de Navarra, Spain. Working Paper No. 04/011. Granger, C.W.J. and N. Hyung (004) Occasional srucural breaks and long memory wih an applicaion o he S&P 500 absolue sock reurns. Journal of Empirical Finance 11, Geweke, J. and Poer-Hudak, S. (1983): The Esimaion and Applicaion of Long memory Time series models, Journal of Time Series Analysis, 4: IMF (009). "World Economic Oulook Daabase, Released 1 s Ocober, 009. IMF (011). IMF Hisrorical GDP (PPP) Daa. Ohanissian, A., Russell, J. R. and Tsay, R.S. (008) True or spurious long memory? A new es, Journal of Business Economics and Saisics 6, Robinson, P.M., 1994, Efficien ess of nonsaionary hypoheses, Journal of he American Saisical Associaion 89, Romero-Ávila, D. and De Olavide, P. (009). Muliple Breaks, Terms of Trade Shocks and he Uni-Roo Hypohesis for African Per Capia Real GDP. World Developmen, 37: Shiu, O.I. and Yaya, O.S. (009). Measuring Forecas Performance of ARMA and ARFIMA Models: An Applicaion o US Dollar/UK Pound Foreign Exchange Rae. European Journal of Scienific Research 3 ():

16 European Journal of Business and Managemen ISSN (Paper) ISSN -839 (Online) Vol 4, No.5, 01 Shiu, O. I. and Yaya, O. S. (010): On he Auoregressive Fracional Uni Inegraed Moving Average (ARFUIMA) Process. Proceedings of he Nigerian Saisical Associaion. pp Velasco, C. (005). Semiparameric Esimaion of Long-memory Models. Deparmen of Economics Universidad Carlos III de Madrid, Spain. 55

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