Purchasing Power Parity (PPP), Sweden before and after EURO times

Transcription

1 School of Economics and Managemen Purchasing Power Pariy (PPP), Sweden before and afer EURO imes - Uni Roo Tes - Coinegraion Tes Masers hesis in Saisics - Spring 2008 Auhors: Mansoor, Rashid Smora, Ami Paul Advisor: Prof. Panagiois Manalos

2 Table of conens. Absrac Acknowledgemen Chaper : Inroducion Problem Discussion Purpose Daa Srucure of he hesis Chaper 2: An economeric Evaluaion of Purchasing Power Pariy 2.. Real Exchange Rae Saionariy of Real Exchange Rae.0 6. Chaper 3: Uni Roos 3.. Inroducion Augmened Dickey Fuller Tes Phillips Perron s Tes Chaper 4: Coinegraion 4.. Inroducion Tes for coinegraion Residual Based Tes Tes using Eigen values 4 8. Chaper 5: Graphical Presenaion of Real Exchange Rae 5. For he Period [Jan990-Dec998] For he Period [Jan999-Dec2007] For he Period [Jan990-Dec2007] Chaper 6: Applicaion 6. Applicaion of Uni Roo Tes o daa Applicaion of Coinegraion Tes Residual Based Tes for he period [Jan 990-Dec998] for he Period [Jan999-Dec2007] for he period [Jan990-Dec2007]..26 2

3 6.2.2 Johansen Tes of coinegraion for he Period[Jan990-Dec998] for he period[jan999-dec2007] for he Period[Jan990-dec2007] Conclusion Fuure Work References. 32 3

4 Absrac This hesis presens an Economeric Evaluaion of Purchasing Power Pariy. Uni Roo Tes and coinegraion Tess are used o examine he issue of he Purchasing Power Pariy for Sweden for he Periods [Jan 990-Dec 999], [Jan 999-Dec 2007] and [Jan990-Dec2007]. The resul of Uni Roo ess failed o find evidence in favour of Purchasing Power Pariy in all he hree periods. However he resul of Johansen es of coinegraion finds evidence in favour of Purchasing Power Pariy in he long-run. Keywords: Coinegraion, Purchasing Power Pariy, Uni Roo. 4

5 Acknowledgemens A special word of hanks goes o he members of our families for heir suppor and encouragemen hrough ou our MSc. Special appreciaion goes o our fahers for heir uncondiional love and financial suppor. 5

6 Chaper. Inroducion Purchasing Power Pariy involves he sudy of Exchange Raes and Prices Level across counries. While sudying he Economeric Evaluaion of he Purchasing Power Pariy, wo imporan ypes of issues arise in he mind. Firs issue is concerned wih he very absence or presence of he PPP. To check he absence and presence of PPP, we sudy he saionary and non-saionary of he real exchange rae. If he real exchange Rae is saionary, i means ha PPP holds, oherwise i does no. For his purpose we use he Augmen Dickey Fuller Tes for he Uni Roo. We se he null Hypohesis ha he real Exchange Rae is non-saionary, which means he absence of PPP agains he alernaive hypohesis ha he Real Exchange Rae is Saionary, which means he presence of PPP. We also use he Phillips Perron s Tes for srucural changes because someimes he real exchange rae suffers from srucural breaks. If he PPP does no hold hen he second issue arises, ha is he equilibrium relaionship of he variables, which is he presence of PPP in he long-run. To check he presence of PPP in he long run we use coinegraion Tes. Here we use wo ype of coinegraion es () Residual Based Tes (2) Johansen Tes The mehods are applied o he exchange raes beween Sweden and USA, and heir corresponding consumer price indices. 6

7 . Problem Discussion The concep of Purchasing Power Pariy was given by Karl Gusav Cassel, a Swedish Economis and Professor a Sockholm Universiy. The heory is based on he law of one price, which saes ha in an ideal marke which is efficien, he cos of idenical goods is same. Differen counries use differen currencies, and ideally he exchange rae beween any wo currencies should be ailored in such ha i equalises he price of idenical goods in he wo counries. If his happens, hen PPP holds..2 Purpose The purpose of his hesis is o check for he presence or absence of PPP before and afer he inroducion of Euro currency (999). Firsly, we check for he saionary of he Real Exchange Rae before and afer he esablishmen of euro zone and if he real exchange rae is no saionary hen we check for he equilibrium relaionship beween variables and see if here are signs for he presence of PPP in he long run..3 Daa We have used he Exchange Raes beween Sweden and USA and he Consumer Price Indices (CPI) of boh he counries. The Exchange Rae daa is aken from oanda.com. We ook reciprocal of he exchange rae of USA o Sweden o obain he nominal exchange rae of Sweden o USA. The daa for CPI of Sweden is aken from he websie of Saisiska Cenralbyrån and ha of USA is aken from he US Deparmen of Labor, Bureau of Labor Saisics. The daa is aken from January 990-December 998 and January 999- December 2007 o accoun for boh pos euro and pre euro imes. 7

8 .4 Srucure of he hesis This hesis consiss of six chapers. A brief overview of each chaper is as follows. Chaper- This inroducory chaper has he following aims: ) A general overview of uni roo es and coinegraion ess regarding PPP. 2) A hisoric and economic overview of PPP. 3) Abou he acquisiion of daa. Chaper-2 This chaper consis of wo secions. In he firs secion we have explain he concep of Real exchange Rae and also derived an equaion of he Real Exchange Rae. In he second secion we have explain he concep of STATIONARITY and we have also explain he echniques which give us rough idea abou he saioariy of Real Exchange Rae and absence or presence of Purchasing Power Pariy. Chaper-3 This chaper give an overview of he concep of Uni Roo es. i consis of hree secion.in he firs secion we have explain he concep of Uni Roo.In he second secion we have explain he mehod of Augmened Dickey Fuller es, Hypohesis Seing and decision Rules. In he hird secion we have explain he Phillips Perron s Tes for srucural change. Chaper-4 This chaper gives us he idea of coinegraion Analysis Regarding Purchasing Power Pariy. We have explained he concep of coinegraion. we have also explain he wo coinegraion Tess ha are Residual Based Tes and Tes using Eigen values ha is Johansen es of coinegraion. We have also explained he mehod of hese wo ess. Chaper-5 In his chaper we have checked he saionary/non-saionary of he Real exchange rae hrough Graphical Presenaion. We have used line graphs and correlogram o check he saionary/non-saionariy of Real Exchange Rae. Chaper-6 In his chaper differen uni roo ess and coinegraion es are applied o check he absence or presence of purchasing power pariy. 8

9 Chaper 2. An economeric Evaluaion of Purchasing Power Pariy 2. Real Exchange Rae: Real Exchange Rae is a funcion of he nominal exchange rae and he raio of he relaive price level beween wo counries. Nominal exchange rae provides some informaion regarding measures of he value of one currency in erm of anoher while price levels provides informaion relaed o he cos of he baske of commodiies for any given counry. in oher words Real Exchange Rae = Le Price level of foreign counry Nominal Exchange Rae x Price level of domesic counry r / $ =Real Exchange Rae for Sweden and Unied Sae of America N / $ =Nominal Exchange Rae P $ = Price level of Unied Sae of America P =Price level of Sweden Then we can wrie r P$ = () P / $ N / $ * Taking Log of equaion () on boh sides we obain log r / $ logn / $ + log P$ = log P (2) Now PPP implies ha Nominal Exchange Rae is equal o he difference beween in he price level beween counries ha is log N / $ = log P log P$ + e Where e represens shor erm deviaion from PPP e = log N / $ + log P$ log P 9

10 Here shor run deviaion from PPP ha is e equal o e = log r $ = logn / $ + log P$ / log P log r / $ ha is Now if e is equal o zero hen i means ha PPP holds. $ Le log r =, logn = N, log P = P and log P = hen / $ r / $ $ P r = N $ + P P In he applied work and refers o he naional price indices of USA and Sweden P $ P respecively in relaive o a base year. Long run PPP is said o be hold if he r sequence is saionary. 2.2 STATIONARITY A ime-series variables ha posses a consan mean and a consan variance over ime and he auocorrelaion funcion ha depends solely on he lengh of he expressed lags is known as saionary ime series. If a ime series variable saisfy he following properies hen i is said o be covariance or weakly saionary: () E( Y ) = μ, =,2,3... ha is Time Independen Mean [consan for all ] 2 (2) Var( ) = σ Time Independen Variance [consan for all ] Y (3) Cov( Y, Y ) = γ s Consan for all and - So if he Process is covariance Saionary, all he variance are he same and all he covariance depends on he difference beween and -s. PPP heory saes ha if he real exchange rae possesses he above properies hen i will be saionary. The following echniques can be used o ge a rough idea of a saionary series. 0

11 ) Wih he help of graphical represenaion: A plo of non-saionary series produces a line wih definie upward or downward rend wih he passage of ime. On he oher side he saionary ime series does no produce such ype of line 2) Observing correlogram of auocorrelaion funcion: For a saionary ime-series, he ACFs end o zero raher quickly while for a nonsaionary he ACFs are suffered from linear decline Chaper 3 - Uni Roo Tes: 3. Inroducion A formal es ha can help us in knowing ha he given series conains a rend and also give us he informaion ha he rend is deerminisic or sochasic. This ype of es is known as Uni Roo Tes. here are several ype of Uni Roo Tes. The es which we use here is he Augmen Dickey Fuller (ADF) es. This es provides a formal es for non-saionary in he ime series daa. The basic idea behind he Augmen Dickey Fuller equaion is o es for he presence of Uni Roo in he coefficien of lagged variables. If he coefficien of a lagged variable shows a value of one, hen he equaion show he sign of non-saionary. 3.. Augmened Dickey-Fuller es (Enders, Waler) To formally es for he presence of a uni roo in he real exchange rae, Augmened Dickey- Fuller es of he form given below is carried ou: Δr = a o + γ r The null Hypohesis for he es is + β 2Δr + β 3Δr 2 + β 4Δr Η 0 = γ = 0 Uni Roo Problem Decision rule:

12 If -saisic > ADF criical value => no rejec null hypohesis i.e. uni roo exiss If -saisic<adf criical value => rejec null hypohesis, i.e. uni roo does no exis. Here -saisic is he saisic used in he ADF es. If he null Hypohesis is acceped, we assume ha here is a uni roo and difference he daa before running a regression. If he null is rejeced, he daa are saionary and can be used wihou differencing Perron s Tes for Srucural Change: I have also applied Perron s es for uni Roo. Perron s (989) goes on o develop a formal es for a uni roo in he presence of a srucural change a he ime period a = τ + Model under he null Hypohesis: Y D + e = a0 + Y + μ p Model under he alernaive Hypohesis: Y a + a + D + e = 0 2 μ 2 L Where D p = if = τ + and Zero oherwise. And DL is level Dummy variable ha is if > τ and Zero oherwise. Decision Rule: Rejec he null hypohesis of he uni roo if he calculaed value of he - saisics is greaer han he criical values. 2

13 Chaper-4 Coinegraion: 4. Inroducion Now we move oward coinegraion Tes of Purchasing Power Pariy. If a series needs o be difference d imes before i becomes saionary hen i conains d uni roos and is said o inegraed of order d ha is I(d). I is necessary for he coinegraion es ha he order of inegraion of all he variables in he long run will be he same. The order of inegraion is he number of imes, a ime series variables mus be difference for i o become saionary. According o coinegraion mehod if PPP hold hen he sum of he Nominal Exchange Rae and Price level of Unied Sae ha is ( N + Sweden ( P ) sequence. $ P ) will be coinegraed wih Price Level of Le suppose Y = N + P $ Then PPP asser ha exis a linear combinaion of he form Y α + P + e such ha is saionary and he coinegraion vecor such haα =.where is he residuals of he regression equaion. = 0 α e e 4.2 -Tesing for coinegraion: (Enders, Waler) 4.2. Residual Based Tes he Engle-Granger Mehodology (procedure) -Tes for a Uni Roo on Residuals - ADF Type coinegraion Tes - Phillips Perron s Tes -Seps for Residual Based Tes Sep- Tes he variable for heir order of coinegraion.in he s sep we deermine he order of inegraion. Here I will use he Augmened Dickey-Fuller Tes and Phillips Perron s Tes o deermine he order of inegraion. Sep-2 3

14 If he resul of s sep indicae ha he variable are inegraed of order one hen esimae he long-run equilibrium relaion by regressing Y = N + P on ha is $ P Y α 0 + α P + e = Absolue PPP assers ha Y = P so his requires ha α = and α 0 0 = Sep-3 Check he residual of he equilibrium regression for saionary using DF es for Uni roo. To deermine ha he variables are coinegraed we denoe he Residual sequence from he equilibrium equaion by ê.so ê is he series of he esimaed Residuals of he long run relaionship. If he series of he esimaed Residuals are found o be saionary hen Y and P series will be coinegraed. Esimae he Auoregression of he form: Δeˆ n eˆ + β i Δeˆ + i= = β + ε If 2 < β < 0 we can conclude ha he Residuals series is saionary and and are inegraed of order one ha is C (, ). Y P Johansen Tes by Using Eigen Values Tes Procedure: In The Johansen Tes Procedure here are wo Tes Saisics: ) The Trace Saisic and 2) The Maximum Eigen Value Saisic ) Trace Saisic: λrace( r) = T ln( ˆ λ ) Where n i= r+ i λˆ = he esimaed values of he characerisic roos(also called Eigen values). T= he number of usable observaion Null Hypohesis: There is a mos r coinegraion relaion. Alernaive Hypohesis: There are m coinegraion relaionship (ha is series is saionary) r = 0,, 2.m- 4

15 Decision Rule: If he race Saisic is greaer han he given criical value hen rejec he null Hypohesis and conclude ha he series is saionary. (2) The Maximum Eigen Value Saisic= λ ( r, r + ) = T ln( ˆ λr ) max + Null Hypohesis: There is r coinegraion relaionship Alernaive Hypohesis: here are r+ coinegraion relaionship Decision Rule: If he Maximum Eigen value Saisic is greaer han he given criical value hen rejec he null Hypohesis and conclude ha he series is saionary. 5

16 Chaper 5 5.-Graphical Presenaion of Real Exchange Rae From our daa for period [Jan 990-dec 998] he real exchange rae for Sweden and USA are non-saionary. In he graph he real exchange rae of Sweden and USA shows he sronges upward and downward rends. -2. R_X We see ha he above graph of real exchange rae is like o have random walk paern, which random walk up and down in he line graph. Afer aking he firs difference he Real exchange Rae for he period [Jan 990-dec 998] becomes saionary. As we can observe from he line graph..0 DR_X

17 Correlogram of Real Exchange Rae for he Period [Jan990-Dec998] If we sudy he Correlogram for his period hen here is only one significan spike of PACFs and he ACFs are suffered from linear decline. Afer aking he firs difference he Correlogram looks like. 7

18 we see ha he real exchange rae is now saionary as shown no significan paerns in he graph of he Correlogram. 5.2-Graphical Presenaion of Real Exchange Rae. Similarly for he daa for he period [Jan 999-dec 2007] he line grape of he real exchange rae shows ha he real exchange rae afer he inroducion of Euro is no saionary. The Sweden and USA real exchange rae shows he down ward and upward rend from Jan 999 o Dec 2007.Similarly if we sudy he correlogrm, we will see ha The ACFs are suffered from linear decline and here is only one significan spike of PACFs.he resul is given in Table A R_X Afer aking he firs difference he Real Exchange Rae for he period [Jan 999-dec 2007] Look like. DR_X

19 Correlogram of Real exchange Rae for he Period [Jan999-Dec2007] From he above Correlogram we see ha he real exchange rae is no saionary during he period [Jan999-Dec2007] because he ACFs are suffered from linear decline and here is only one significan spike of PACFs.Afer aking he firs difference he correlogram look like as below. 9

20 Now he series is saionary as we see ha ACFs end o zero raher quickly. 5.3-Graphical Presenaion of Real Exchange Rae for he period [Jan990- Dec2007]. For his period he Graph of he real exchange rae is no saionary. The graph shows some upward and downward movemen during his period. Also here is some Srucural breaks during his period. If we look a he Correlogram we see ha he ACFs are suffered from linear decline and he is only one significan spike of PACFs. R_X Afer aking he firs diffence of he Real Exchange Rae. The Graph of he real exchange looks like. 20

21 DR_X Correlogram of he Real Exchange Rae for he Period [Jan990-Dec2007]: The real exchange rae he period[jan 990-Dec2007] look like Non-saionary because in he below Correlogram he ACFs are suffered from linear decline and here is only one significan spike of PACFs. 2

22 Afer aking he firs difference he Correlogram for real exchange rae for he period[jan990-dec2007] look like. Chaper 6 Applicaion 6.- Applicaion of Uni Roo Tess o Daa: In our daa I have applied boh ADF Tes and Phillips Perron s Tes. Long run PPP is said o hold if he Real Exchange Rae sequence is saionary. Here I have consruced he Real Exchange Rae for Sweden rading Parner ha is USA.he daa is divided ino hree periods [ Jan 990-dec 998] and [Jan 99-dec 2007] and [Jan990- Dec2007]. To ge he sequence of Real Exchange Rae (r ).I have Muliplied he Consumer Price Indices of USA o he Nominal Exchange Rae of Sweden o Foreign currency and hen divided by he Consumer Price Indices of Sweden. The Log of he consruced series is he Real Exchange Rae sequence (r ). Using he ADF es and he Monhly Daa of he synheic real krona/dollar Exchange Rae for he Period [Jan 990-dec 998], sugges ha he real exchange rae is Non-Saionary, sugges ha PPP no hold for he give period. In addiion I have also applied Perron s es which also find ha Real Exchange Rae for he period [Jan 990-dec 998] is Non-saionary, sugges ha PPP no hold for he given period. The Resul is given in TABLE- 22

23 Similarly using he ADF es, Perron s es and he monhly daa of synheic real krona/dollar Exchange Rae for he period [Jan 999-dec 2007] find ha he Real Exchange rae for he given period is Non-saionary, sugges PPP no hold for he given period. The Resul is given in TABLE-2 We have also applied he ADF es and Phillips Perron s Tes o he real Exchange rae for he period [Jan 990-Dec2007].These wo es give us he resul ha he real exchange rae during his period is also Non-saionary. The resul is given in Table-3. Afer aking he firs difference of he variable [Real Exchange Rae] for he period [ Jan 990-Dec 998] he null hypohesis of uni roo can be rejeced a %,5% and 0% level of significance using ADF and Phillips Perron s es. The resul is given in Table-4 Similarly afer aking he firs difference of he real exchange rae for he period [Jan 999-dec 2007] he null hypohesis of uni roo is rejeced a %, 5% and 0% level of significance using ADF and Phillips Perron s es. The resul is given in Table-5 and also he real exchange rae for he period [Jan990-Dec2007] become saionary afer aking he firs difference of he real exchange rae. Using he ADF es and he Phillips Perron s es he null hypohesis of he uni roo is rejeced a %, 5% and 0% level of significan. The resul is given in Table-6. Afer a brief sudy of hese hree periods ha are [Jan 990-Dec998], [Jan999-Dec2007] and [Jan990-dec2007] we reach a he decision ha Purchasing Power Pariy does no hold during hese periods. TABLE - Real Exchange Rae for he period(jan 990-dec 998) TEST T-saisic C.V % C.V 5% C.V 0% Null conclusion hypohesis ADF Accep Nonsaionary Perron s Accep Nonsaionary TABLE-2 Real Exchange Rae for he period(jan 99-dec 2007) TEST T-saisic C.V % C.V 5% C.V 0% Null conclusion hypohesis ADF Accep Nonsaionary 23

24 Perron s Accep Nonsaionary TABLE -3 Real Exchange Rae for he period(jan 990-dec 2007) TEST T-saisic C.V % C.V 5% C.V 0% Null conclusion hypohesis ADF Accep Nonsaionary Perron s Accep Nonsaionary Table-4 afer aking s difference Real Exchange Rae for he period(jan990-dec998) TEST T-saisic C.V % C.V 5% C.V 0% Null conclusion hypohesis ADF Rejec saionary Perron s Rejec saionary TABLE-5Afer s differen Real Exchange for he period(jan 99-dec 2007) TEST T-saisic C.V % C.V 5% C.V Null conclusion 0% hypohesis ADF Rejec saionary Perron s Rejec saionary TABLE-6Afer s differen Real Exchange for he period(jan 990-dec 2007) TEST T-saisic C.V % C.V 5% C.V Null conclusion 0% hypohesis ADF Rejec saionary Perron s Rejec saionary 6.2-Applicaion of coinegraion Tes o daa: Afer sudying he ADF es and Phillips Perron s Tes of Uni Roos we reach a he decision ha he real exchange are for he Periods [Jan 990-Dec998], [Jan 999-Dec2007] and 24

25 [Jan990-Dec2007] is Non-saionary and Purchasing Power Pariy does no hold for hese hree period. Now we wan o check he presence or absence of Purchasing Power Pariy in he long-run. For his purpose we will apply he below coinegraion ess. Afer sudying he consumer price indices of Sweden and USA, and he nominal exchange rae of Sweden, we reach a he decision ha all hese variables are inegraed of order one ha is I() because he firs difference of hese variables is saionary ha is I(0) Residual Based Tes: Y Y = α 0 + αp + e Absolue PPP assers ha = P so his requires ha α = and α 0 0 = Firs we have found he equilibrium regression before he inroducion of Euro zone ha is for he period [jan990-dec998].he Resul gives us he value of he coefficien ha is α and he sandards errors are given in able A-.we see from he able ha he value of he esimaed coefficien is very below from uniy. Similarly we have found he equilibrium regression afer he inroducion of he Euro zone ha is for he period [jan999-dec2007].in his case he value of he coefficien is very over uniy. for he Period[Jan990-Dec2007] we have find he Equilibrium regression and he esimaed value of he coefficien is approximaely close o uniy. The resul is given in Table A- Table A- Equilibrium Regression Period Period Period Jan990- dec998 Jan999- dec2007 Jan990- Dec2007 Esimaed α Esimaed α Esimaed α Sandard Error Sandard Error Sandard.2300 Error Now we wan o check he residual of he equilibrium regression for saionariy using Dickey-Fuller es(979). We esimae Auoregressive of he form Δeˆ n = β eˆ + β Δe + ε. ˆ i+ i= 25

26 If 2 < β < 0 we can conclude ha he Residuals series is saionary and and are inegraed of order one ha is C (, ). To check he residual of he equilibrium regression for he period [jan990-dec998], [Jan99-Dec2007]and [jan990-dec2007], we will use he Dickey-fuller es saisic and compare is value wih he criical values for he Engle-Granger coinegraion Tes. Null hypohesis: No coinegraion Y P For he period [Jan990-Dec998]: The Dikey-Fuller Tes saisic value ( ) is greaer han he criical values (-4.008,-3.398, ) a %, 5% and 0 % level of significance respecively so we can no rejec he null hypohesis of No coinegraion and conclude ha Y and P are no coinegraed. Table- A For he Period [Jan999-Dec2007]: The Dickey-Fuller Tes saisic value ( ) is greaer han he criical values (-4.008,-3.398, ) a %, 5% and 0 % level of significance respecively so we can no rejec he null hypohesis of No coinegraion and conclude ha Y and P are no coinegraed. Table- A For he Period [Jan990-Dec2007]: The Dickey-Fuller Tes saisic value ( ) is greaer han he criical values (-3.92,-3.50, ) a %, 5% and 0 % level of significance respecively so we can no rejec he null hypohesis of No coinegraion and conclude ha Y and P are no coinegraed. Table- A2. Table-A2 Residual Based Tes Dikey-Fuller Tes saisic Criical values for Engle-Granger coinegraion Tes(Two variables) Period -sa. values C.V % 5% 0% Jan990-Dec Jan 999-Dec

27 Jan990-Dec Johansen Tes of coinegraion For he period [Jan990-dec998] The resul of he Johansen es is quie sensiive o he lag lengh. So o selec he suiable Lag lengh for your coinegraion es we will firs find a suiable Var Model using he undifferenced daa and hen we will use he same Lag lengh for our coinegraion es. Selecion of appropriae Var model: TO selec an appropriae lag order for our VAR model, we have esimae a range of VARS wih o 8 lags and abulaed AIC and SIC values. The resul is as follow: Table -A3 NUMBER OF LAGS SCHWARZ Akaike informaion criera From he above able SCHWARZ has seleced lags VAR and AKAIKE has seleced four lag VAR. So we selec four lag VAR because if we selec one lag VAR hen heir some auocorrelaion problem a differen lags. We0 have checked auocorrelaion LM es for one LAG VAR.So we will use lag four for our coinegraion es. Here we use Eviews-6 Programme which auomaically selecs suiable lags for Johansen coinegraion es. Now we wan o procede o he Johansen Tes of coinegraion. -- Johansen Tes Johansen Tes Procedure: 27

28 () The Trace Saisic: λ race( r) = T λ race n i= r+ ln( ˆ λ ) i [ ln( λ ) + ln( λ ) + ln( )] ( 0) = T λ 2 = -03[ln (-0.985) +ln(-.052)+ln( )] = So he λ (0) = is greaer han he criical values [ , ] a 5% and race 0% level of significan respecively, so we rejec he null hypohesis of no coinegraion vecor and accep he alernaive hypohesis of one or more coinegraion vecor. Also λ ( ) = Which is less han he criical values [ , ] a 5% and 0% race level of significan respecively so we can no rejec he null hypohesis of no more han coinegraion vecors. 3 Similarly λ r, r ) = T ln( ˆ λ ) max ( + r+ λ 0,) = 03* ln( ˆ λ ) max ( λmax (0,) = -03*ln(-0.985) λ max (0,) = So he value of λ (0,) = is greaer han he criical values [ , 5.774] max so we rejec he null hypohesis of no coinegraion vecor and accep he alernaive hypohesis of one coinegraion vecor.similary λ (,2) is less han he criical max = values [.22480, ] a 5% and 0% level of significance respecively, so we accep he null hypohesis of one coinegraion vecor and rejec he alernaive hypohesis of wo coinegraion vecor. Afer sudying Johansen Tes of coinegraion, we reach a he decision ha here is only one coinegraion vecor and we can say ha for he period [Jan990-Dec 998] he coinegraion hold and find in favour of PPP. 28

29 Table-A4 Saisic λ and λ max Saisic race λ r =0 race saisic r r 2 λ max saisic r = 0 r = r =2 Null Hypohesis Alernaive Hypohesis r>0 r> r>2 r = r= 2 r=3 Eigen values Tessaisic value % criical value % criical value Null hypoh: Rejec Accep Accep Rejec Accep Accep Johansen Tes of coinegraion for he period [Jan999-dec2007]. The Trace Saisic: λ race( r) = T λ race n i= r+ ln( ˆ λ ) i [ ln( λ ) + ln( λ ) + ln( )] ( 0) = T λ 2 = -03[ln (-.282) +ln (-.23)+ln(-.0008)] = So he λ (0) = is greaer han he criical values [ , ] a 5% and race 0% level of significan respecively, so we rejec he null hypohesis of no coinegraion vecor and accep he alernaive hypohesis of one or more coinegraion vecor. Also λ ( ) = Which is greaer han he criical values [ , ] a 5% and race 0% level of significan respecively so we can rejec he null hypohesis of no more han coinegraion vecors. The resul is given in Table-A5 Similarly λ r, r ) = T ln( ˆ λ ) λ max ( + r+ 0,) = 03* ln( ˆ λ ) max ( λmax (0,) = -03*ln(-.2824) λ max (0,) = which is greaer han he criical values [ , 5.774] a 5% and 0% level of significance, so we rejec he null hypohesis of no coinegraion and accep he alernaive hypohesis of one coinegraion vecor. Similarly λ (,2) which is greaer han he criical values [.22480, ] max = a 5% and 0% level of significance so we rejec he null hypohesis of one coinegraion vecor and accep he alernaive hypohesis of wo coinegraion vecor. The resul is given in able-a5. Table-A5 λ and λ max Saisic race 3 29

30 Saisic λ race saisic Null Hypohesis r =0 r r 2 Alernaive Hypohesis r>0 r> r>2 Eigen values Tessaisic value % criical value % criical value Null hypoh: Rejec Rejec Accep λ max saisic r = 0 r = r =2 r = r= 2 r= Rejec Rejec Accep Afer sudying he Johansen es of coinegraion for he period [Jan999-Dec2007] we reach a he decision ha here are wo coinegraion relaionship which suppor he presence of Purchasing Power Pariy in he long-run Johansen Tes of coinegraion for he period [Jan990-dec2007]: The Trace Saisic: λ race( r) = T λ race n i= r+ ln( ˆ λ ) i [ ln( λ ) + ln( λ ) + ln( )] ( 0) = T λ 2 = -2*[ln (-.994) +ln (-.088)+ln ( )] = So he λ (0) = is greaer han he criical values [ , 2.777] a 5% and 0% race level of significan respecively, so we rejec he null hypohesis of no coinegraion vecor and accep he alernaive hypohesis of one or more coinegraion vecor. Similarly λ r, r ) = T ln( ˆ λ ) λ max ( + r+ 0,) = 2* ln( ˆ λ ) max ( λmax (0,) = -2*ln(-0.994) 3 λ max (0,) = So he value of λ max (0,) = is greaer han he criical values [ , 2.777]. 30

31 So we rejec he null hypohesis of no coinegraion vecor and accep he alernaive hypohesis of one coinegraion vecor. The resul is given in Table-A6 Table-A6 Saisic λ race saisic λ and λ max Saisic race Null Hypohesis r =0 r r 2 Alernaive Hypohesis r>0 r> r>2 Eigen values E-06 Tessaisic value % criical value % criical value Null hypoh: Rejec Accep Accep λ max saisic r = 0 r = r =2 r = r= 2 r= E Rejec Accep Accep Afer sudying he Johansen es of coinegraion we reach a he decision ha here is only one coinegraion relaionship beween variables and here is a sign of he presence of Purchasing Power Pariy in he long-run. Conclusion: This hesis is concern wih he absence or presence of purchasing power pariy for Sweden before and afer he inroducion of Euro. The Sweden is aken as domesic counry and unied sae of America is aken as foreign counry. To check he absence or presence of he purchasing power pariy wo differen es mehods are used. The firs mehod is concern wih he uni roo ess. In his mehod o check he absence or presence of PPP we check he saioariy and Non-saionary of he real exchange rae. If he real exchange rae is saionary, i mean PPP hold oherwise do no hold. For his purpose we have used wo differen ypes of uni roo ess ha ADF es and Phillips Perron s es. Boh of hese es has failed o find evidence in favour of purchasing power pariy. The oher es mehod which we have used is he coinegraion es. The resul of his es is encouraging and provides evidence of Purchasing Power Pariy in he long run. The residual based es and he Johansen es of coinegraion are used. The Johansen Trace saisic es and maximum Eigen value saisic es are used. Boh he ess have rejeced he null hypohesis of no coinegraion and hese ess provide evidence in favour of Purchasing Power pariy in he long-run. 3

32 Using hese es we found ha here is only one coinegraion relaionship for he period [Jan990-Dec998], wo coinegraion relaionships among variables for he period [Jan999- Dec2007] and one coinegraion relaionship among variables for he period [Jan990- Dec2007]. I can be herefore concluded ha here is srong long run relaion among he exchange raes and he price indices and he hree variables will say close o each oher in he long run. Fuure work: Exchange raes consiue a very cardinal componen of global economy which suppors he inernaional ransacion beween corporaes, naion saes and individuals. Exchange raes measure he value of one currency agains he oher. I is considered very imporan by respecive governmens, banks and oher financial insiuions which are involved exensively ino he inernaional scale ransacions. The global rading volume per day is in he range of rillions of US dollars and even small flucuaion in he exchange rae can have big effec on he financial ransacions. Thus i is very imporan o undersand he mechanism of he exchange raes, and is volailiy. This hesis can be exended in fuure o cover modelling of exchange raes volailiy using he ARCH and GRACH models. 32

33 References: ) Perron, P., (989), The Grea Crash, he Oil Price Shock and he Uni Roo Hypohesis, Economerica, 57, ) David A. Dickey and Wayne A. Fuller(June, 979) Disribuion of he Esimaors for auoregressive Time Series Wih a Uni Roo, Journal of he American Saisical Associaion, Vol. 74, No. 366, pp ) Enders Waler, Applied Economeric Timeseries, Wiley, 2 nd Ediion,