VEŽBA 4: JEDINIČNI KOREN I KOINTEGRACIJA U VREMENSKIM SERIJAMA PRIVREDE SRBIJE
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1 EKONOMSKI FAKULTET UNIVERZITETA U BEOGRADU MASTER: MEĐUNARODNA EKONOMIJA PREDMET: EKONOMETRIJA 1-M GODINA: 010/11. PREDAVAČ: ZORICA MLADENOVIĆ VEŽBA 4: JEDINIČNI KOREN I KOINTEGRACIJA U VREMENSKIM SERIJAMA PRIVREDE SRBIJE Datoteka vezba4.wf1 sadrži tri vremenske serije privrede Srbije. To su: - logaritmovana vrednost indeksa cena na malo (oznaka: prices) - logaritmovana vrednost nominalnog deviznog kursa, koji je korigovan za evropsku inflaciju (oznaka: ex) i - logaritmovana vrednost ukupnog izvoza (oznaka: x) Period analize: januar 00 maj 006 (ukupno 53 podatka). 1. Ispitati validnost teorije o paritetu kupovne snage u periodu: januar 00 maj 006. Proveravamo da li je vremenska serija realni devizni kurs stacionarna ili poseduje jedinični koren. Realni devizni kurs (oznaka: rex) dobijamo na sledeći način: ex-prices. Za potrebe dalje analize obrazovaćemo prvu i drugu diferencu realnog deviznog kursa: o drex=rex-rex(-1) o ddrex=drex-drex(-1) Realni devizni kurs (rex) Prva diferenca realnog deviznog kursa U cilju preliminarne analize može se pogledati ponašanje obične i parcijalne autokorelacione funkcije serije realni devizni kurs (Quick/Series Statistics/Correlogram/Ukucati ime serije/izabrati broj docnji i opciju nivo serije). Uočavamo da ocene običnih autokorelacionih koeficijenata sporo opadaju tokom vremena i da je samo parcijalni autokorelacioni koeficijent na prvoj docnji značajno različit od nule. Budući da ocene koeficijenata opadaju od vrednosti koja je znatno manja od 1, ne možemo sa sigurnošću tvrditi da je serija nestacionarna. Test jediničnog korena (DF): U prvom koraku testiramo hipotezu H 0 : REX~ I(1) protiv H 1 : REX~ I(0). Ocenjujemo regresiju: Dependent Variable: DREX Sample (adjusted): 00M0 006M05 Included observations: 5 after adjustments 1
2 C E E REX(-1) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) Vrednost DF test-statistike je t-odnos ocene uz rex(-1): τ t = Međutim, DF test-statistika nema t- raspodelu. Potrebno je koristiti poseban skup kritičnih vrednosti. U konkretnom slučaju odgovarajuća kritična vrednost na nivou značajnosti 5% je Ta kritična vrednost dobija se prema formuli: τ t t = T T = = gde je T obim uzorka (videti Mladenović i Nojković, 008, str. 61). Pre nego što donesemo konačnu odluku potrebno je proveriti da li u modelu postoji autokorelacija. U tom cilju analiziramo korelogram reziduala ocenjenog modela (u okviru prozora jednačine: View Residual Tests/Correlogram - Q statistics/ukucati broj ocnji). Zaključujemo da ne postoji autokorelacija, tako da nema potrebe za korekcijom polaznog modela. Prihvatamo za tačnu hipotezu H 0 : REX~ I(1) da serija poseduje bar jedan jedinični koren. U drugom koraku testiramo hipotezu H 0 : REX~ I() protiv H 1 : REX~ I(1), da bismo utvrdili tačan broj jediničnih korena. Budući da se hipoteze mogu zapisati na sledeći način H 0 : REX~ I(1), H 1 : REX~ I(0), proveravamo da li jedinični koren postoji u prvoj diferenci serije. Stoga ocenjujemo model oblika: Dependent Variable: DDREX Sample (adjusted): 00M03 006M05 Included observations: 51 after adjustments C E E DREX(-1) R-squared Mean dependent var.74e-05 Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) U ovom modelu ne postoji autokorelacija, tako da je relevantna vrednost DF testa: τ t =-5.81< Zaključak: realni devizni kurs poseduje jedan jedinični koren. Kako realni devizni kurs nije stacionaran, to u razmatranom periodu nominalni devizni kurs nije formiran prema paritetu kupovne snage.
3 S obzirom na to da serija realnog deviznog kura nema izraženu komponentu trenda, testiranje je moguće ostvariti i prema modelu iz koga je izostavljen trend kao deterministička komponenta. Proveriti da se primenom τµ testa dolazi do istog rezultata o broju jediničnih korena. Alternativno, za primenu DF testa može se koristiti opcija: Quick/Series Statistics/Unit Root Test/Ukucati ime serije/izabrati test, determinističke komponente i broj docnji.. Odrediti broj jediničnih korena u vremenskoj seriji nominalni devizni kurs (ex) Nominalni devizni kurs (log) Prva diferenca logaritma nominalnog kursa (stopa rasta) Analiza korelograma: Ocene običnih autokorelacionih koeficijenata jako sporo opadaju tokom vremena. To opadanje počinje od vrednosti koja je bliska vrednosti 1. Samo je prvi parcijalni autokorelacioni koeficijent značajno različit od nule. Ovakvo ponašanje ocena autokorelacionih koeficijenata nominalnog deviznog kursa ukazuje na nestacionarnu prirodu analizirane serije. Test jediničnog korena: U prvoj fazi je ocenjen model oblika: Dependent Variable: DEX=EX-EX(-1) Sample (adjusted): 00M0 006M05 Included observations: 5 after adjustments C EX(-1) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) Korelogram reziduala ukazuje na to da u modelu postoji autokorelacija. Otuda je potrebno proširiti polazni model dodavanjem elemenata oblika: DEX(-1),, DEX(-K). Uključujemo prvo element DEX(-1): 3
4 Dependent Variable: DEX Sample (adjusted): 00M03 006M05 Included observations: 51 after adjustments C EX(-1) DEX(-1) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) Objašnjavajuća promenljiva, DEX(-1), je visoko statistički značajna. Njeno uključivanje bilo je dovoljno da se eliminiše autokorelacija. Vrednosti AIC i SC kriterijuma su manje od vrednosti AIC i SC u prethodnom modelu. Prema tome, ovo je model na osnovu koga ostvarujemo testiranje. Vrednost ADF(1) τ t test-statistike -.39 je veća od kritične vrednosti Zaključujemo da serija nije stacionarna, već da poseduje bar jedan jedinični koren. U drugoj fazi ispitujemo da li jedinični koren postoji u prvoj diferenci serije. Ocenjen je model oblika: Dependent Variable: DDEX=DEX-DEX(-1) Sample (adjusted): 00M03 006M05 Included observations: 51 after adjustments C E E DEX(-1) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) U ocenjenom modelu ne postoji autokorelacija. Zaključak: -3.99< Prva diferenca deviznog kursa jeste stacionarna vremenska serija, odnosno devizni kurs poseduje tačno jedan jedinični koren. 4
5 3. Pokazati da je izvoz (x) trend-stacionarna vremenska serija. Kako zaključujemo da serija ima izraženu sezonsku prirodu? 6.4 Ukupan izvoz (log) Dependent Variable: DX=X-X(-1) Included observations: 5 after adjustments C X(-1) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) τ t =-4.65<-3.50 Izvoz je trend-stacionarna vremenska serija. Alternativno, da bismo eliminisali autokorelaciju na 1 docnji, koja je rezultat sezonskih varijacija, u model ćemo uključiti sezonsku veštačku promenljivu za januar. Time se rezultat zaključivanja ne menja. Dependent Variable: DX Sample (adjusted): 00M0 006M05 Included observations: 5 after adjustments C X(-1) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic)
6 4. Proveriti validnost teorije o paritetu kupovne snage na osnovu kointegracione analize indeksa cena na malo i nominalnog deviznog kursa Potrebno je utvrditi da li je linearna kombinacija deviznog kursa i cena stacionarna ili ne. Ukoliko linearna kombinacija nije stacionarna, tada cene i kurs nisu kointegrisane vremenske serije. Obratno, ove promenljive su kointegrisane ako je njihova linearna kombinacija stacionarna. Postupak testiranja: Ocenjujemo primenom metoda ONK zavisnost kursa od cena Dependent Variable: EX Sample: 00M01 006M05 Included observations: 53 C PRICES R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) Proveravamo da li su reziduali stacionarni (U prozoru jednačine sačuvati reziduale na sledeći način: Proc/Make Residual Series/Ukucati naziv: rez) Faktički, ispitujemo da li u seriji reziduala postoji jedinični koren primenom DF testa. Kako se DF test primenjuje na reziduale, neophodno je koristiti posebne kritične vrednosti (Mladenović i Nojković, 008, str. 95). Pri tome, u model nije potrebno uvoditi determinističke komponente. Kao i prilikom primene standardnog DF testa, treba prethodno testirati postojanje autokorelacije. Dependent Variable: DREZ=rez-rez(-1) Method: Least Squares Included observations: 5 after adjustments REZ(-1) >-3.45 Reziduali nisu stacionarni. Kritična vrednost određena je prema Tabeli.1 na strani 95, Mladenović i Nojković (008): = = 3.45 T T Zaključak: Reziduali poseduju jedinični koren. To znači da linearna kombinacija cena i deviznog kursa nije stacionarna. Ne postoji dugoročna usklađenost između cena i kursa koja bi opravdala valjanost teorije o paritetu kupovne snage. 6
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