UNIVERSITETET I OSLO ØKONOMISK INSTITUTT Eksamen i: ECON5/45 Elemenær økonomeri Exam: ECON5/45 Inroducory Economerics Eksamensdag: Onsdag. mai 9 Sensur kunngjøres: Fredag. juni 9 Dae of exam: Wednesday, May, 9 Grades will be given: Friday, June, 9 Tid for eksamen: kl. 9: : Time for exam: 9: a.m. : noon Oppgavesee er på 8 sider The problem se covers 8 pages English version on page Tillae hjelpemidler: Alle ryke og skrevne hjelpemidler, sam kalkulaor er illa Resources allowed: All wrien and prined resources, as well as calculaor, is allowed Eksamen blir vurder eer ECTS-skalaen. A-F, der A er bese karaker og E er dårligse såkaraker. F er ikke beså. The grades given: A-F, wih A as he bes and E as he weakes passing grade. F is fail. I denne oppgaven skal vi sudere eerspørselen eer bensin ved hjelp av daa fra USA. Daasee er kvaralsdaa for perioden. kvaral 959 il 4. kvaral 99, og innholder variablene: Y ) - bensinugifer per capia i fase priser, ) - realpris per gallon bensin en gallon er ca. 4,5 lier), ) - realinnek per capia og ) - kjørelengde mål i miles) per gallon. Våre analyser er baser på logarimisk ransformasjon av disse variable, dvs. Y ) = lny ), ) = ln ), ) = ln ) og endelig ) = ln ). Vi ar ugangspunk i regresjonen ) Y ) = β + β ) + β ) + β ) + ε ) =,,,..., 8 der resleddene ε ) anas å ilfredssille beingelsene: E ε )) =, Var ε )) = σ, Cov ε ), ε h)) = for h Spørsmål Hvordan vil du olke regresjonskoeffisienene β og β i regresjonsligningen )? Uskrif viser resulaene når regresjon ) anvendes på vår daamaeriale.
Spørsmål Gi en kor og presis olkning av resulaene slik de fremgår av Uskrif. Resulaene fra Uskrif kan yde på a resleddene ε ) ikke ilfredssiller sandardbeingelsene slik de ble presiser ovenfor. For å undersøke dee spesifiseres ε ) som en sasjonær AR) prosess: ) ε ) = ρε ) + ν ) ρ < der ν ) er en hel ilfeldig prosess som oppfyller beingelsene: E ν )) =, Var ν )) = σ ν, Cov ν ), ν h)) = h. Fra Uskrif kan vi ulede de esimere resleddene ˆ ε ) der ˆ ε ) = Y) Yˆ ). Uskrif viser resulaene av regresjonen ) ˆ ε ) = ρ ˆ ε ) + u ) =,, 4,..., 8 Spørsmål a) Tes nullhypoesen H : ρ = mo alernaive H A : ρ > ved å bruke esen il Durbin-Wason med signifikansnivå α =. 5. Du finner de relevane abeller vedlag) b) Tes denne hypoesen når du ar ugangspunk i esimae ρˆ fra Uskrif, og du benyer samme signifikansnivå som ovenfor. Ved sore daase anall observasjoner er sor) er de vanlig å omgå probleme med auokorrelere resledd ved å ersae ε ) med ˆ ε ) der ˆ ε ) ilfredssiller ligningen ) ovenfor. Vi anar a denne approksimasjonen er illa for vår daase, slik a regresjon ) kan reformuleres il 4) Y ) = β + β ) + β ) + β ) + ρ ˆ ε ) + u ) =,,..., 8 Resulaene for denne regresjonen er vis i Uskrif. Spørsmål 4 Bruk opplysinger i Uskrif il å ese nullhypoesen om a eerspørselen eer bensin har innekselasisie lik mo den alernaive hypoesen a den er forskjellig fra. Velg signifikansnivå α =. 5 Daasee dekker oljekrisen høsen 7 og vineren 74 som medføre øke bensinpriser. Anagelsen om a regresjonskoeffisienene i 4) er konsane for hele perioden kan derfor være vilsom. For å undersøke dee har man esimer regresjonen 4) for periodene: fra. kvaral 959 il og med. kvaral 97, og fra. kvaral 97 il 4. kvaral 99. Resulaene av disse regresjonene er gi i Uskrif 4 og i Uskrif 5.
For å a hensyn il a regresjonskoeffisienene kan være forskjellig for de o del-periodene innfører vi dummyvariabelen for perioden. k varal 959 il. k varal 97 D ) = for perioden. k varal 97 il 4. k varal 99 Dereer beregner vi regresjonen Y ) = β + γd ) + β) + β ˆ ) + β) + ρε ) + γd ) ) 5) + γ ˆ D ) ) + γd ) ) + γ4d ) ε ) + δ ) =,,4,...,8 der δ ) beegner sokasiske resledd. I denne regresjonsligningen beegner variablene D ) j ) produke av D ) og ) j =,,), og D ) ˆ ε ) er produke av D ) og ˆ ε ). Spørsmål 5 Den såkale Chow-esen for a regresjonskoeffisienene i de o del-periodene er like er ekvivalen med å ese nullhypoesen H : γ 4 = mo alernaive H A : mins en γ i i =,,,,4) Sill opp esobservaoren du vil benye for denne esen og bruk opplysninger i de vedlage uskrifer il å gjennomføre esen. Velg signifikansnivå α =. 5 Spørsmål 6 I Uskrifen 6 finner vi a summen av de kvadrere esimere resledd 8 RSS = ˆ δ ) = =.687887 De ilsvarende summene for de kvadrere esimere resledd i regresjonene for de o delperiodene finner vi i Uskrifene 4 og 5 og er henholdsvis.54747 og.6644. Vi ser a summen av disse blir.687887, dvs. nøyakig lik summen ˆ δ ) ovenfor. Hvordan vil du forklare dee resulae? 8 = j ENGLISH VERSION In his exercise we shall sudy he demand for perol using US quarerly daa. The daa se repors ime series daa for he period: from he firs quarer of 959 unil he fourh quarer of 99. The following variables are observed: Y ) - real perol expendiure per capia, ) - real price of perol per gallon, ) - real per capia income ) - miles per gallon. Our analyses are based on logarihmic ransformaions of hese variables, i.e. Y ) = lny ), ) = ln ), ) = ln ) and finally ) = ln ). We sar wih he specificaion ) Y ) = β + β ) + β ) + β ) + ε ) =,,,..., 8
where he random disurbances ε ) are assumed o saisfy he condiions: E ε )) =, Var ε )) = σ, Cov ε ), ε h)) = for h Quesion How will you inerpre he coefficiens β and β in regression )? Oupu shows he resuls when regression ) is applied o our daa. Quesion Give a shor and concise inerpreaion of he resuls appearing in Oupu. Resuls in Oupu make us doubful abou he assumpions above regarding he random disurbances ε ). In order o examine he properies of he disurbances more closely, we specify ε ) as a saionary AR) process ) ε ) = ρε ) + ν ) ρ < where ν ) is a purely random process saisfying he condiions: E ν )) =, Var ν )) = σ ν, Cov ν ), ν h)) = h. Having run he regression ) we derive he esimaed disurbances ˆ ε ) obained from ˆ ε ) = Y ) Yˆ ). Oupu shows he resuls of he regression: ) ˆ ε ) = ρ ˆ ε ) + u ) =,,4,..., 8 Quesion a) Tes he null hypohesis H : : ρ = agains he alernaive H A ρ > by using he Durbin-Wason es wih a level of significance α =. 5. The relevan ables are enclosed) b) Tes his hypohesis by uilizing he esimae ρˆ which you find in Oupu. Use he same level of significance as above. When he number of observaions are large i is cusomary o circumven he problem of auo-correlaed disurbances by replacing ε ) wih ˆ ε ) where ˆ ε ) saisfies equaion ) above. We assume ha his approximaion is valid in our applicaion, so ha regression ) can be reformulaed o 4) Y ) = β + β ) + β ) + β ) + ρ ˆ ε ) + u ) =,,..., 8 The resuls from his regression is shown in Oupu. 4
Quesion 4 Use he informaion you find in Oupu o es he null hypohesis ha he income elasiciy of he demand for perol is equal o agains he alernaive hypohesis ha i is differen from. Choose level of significance α =. 5. The so called oil crisis exending from auumn 7 ill spring 74 is covered by our daa se. This crisis caused large increases in he price of perol. Our implici assumpion ha he regression coefficiens in 4) are consan for he whole sample period migh herefore be doubful. In order o invesigae his issue we have run regression 4) for wo sub-periods: we firs applied regression 4) o he daa covering he period from he second quarer of 959 unil he second quarer of 97. The resul of his regression is given in Oupu 4. Then we applied regression 4) o he remaining period, i.e. from he hird quarer of 97 unil he fourh quarer of 99. The resuls from his regression is shown in Oupu 5. In order o ake care of he fac ha he regression coefficiens can be differen for he wo sub-periods we inroduce he dummy variable for he period : from he sec ond quarer of 959 unil he second quarer of 97 D ) = for he period : from he hird quarer of 97 unil he fourh quarer of 99 Then we run he regression Y ) = β + γd ) + β) + β ˆ ) + β) + ρε ) + γd ) ) 5) + γ ˆ D ) ) + γd ) ) + γ4d ) ε ) + δ ) =,,4,...,8 where δ ) denoes he random disurbances. In his regression equaion he variables D ) j ) denoe he produc of D ) and ) for j =,,), and D ) ˆ ε ) is he produc of D ) and ˆ ε ). j Quesion 5 The so called Chow-es for esing wheher he regression coefficiens for he wo sub-periods are equal is equivalen o esing he null hypohesis: H : γ 4 = agains he alernaive H A : a leas one γ i i =,,,,4). Define he es saisic you will use for his es. Then use informaion you find in he enclosed Oupus o carry ou he esing procedure. Choose level of significance α =. 5. Quesion 6 In Oupu 6 we can see ha he sum of he squared residuals 8 RSS = ˆ δ ) =.687887. = The corresponding sums of he squared residuals in he regressions for he wo sub-periods are given in Oupu 4 and Oupu 5, and are, respecively,.54747 and.6644. We observe ha he sum of hese numbers is.687887, i.e. equal o sum 8 = ˆ δ ) calculaed above.. How will you explain his resul? 5
Uskrif /Oupu EQ ) Modelling Y by OLS using idsrekkedaa.xls) The esimaion sample is: o 8 Coefficien Sd.Error -value -prob Consan -.5454.7 -.9. -.856.98 -.6..998547.54 64.8. -.588.79-9.8. sigma.694 RSS.49944989 R^.97774 F,4) = 477 [.]** DW =.74 no. of observaions 8 no. of parameers 4 meany ) -7.76 vary ).4 Uskrif /Oupu EQ ) Modelling εˆ by OLS using idsrekkedaa.xls) The esimaion sample is: o 8 Coefficien Sd.Error -value -prob ε.6679.695 9.5. ˆ sigma.545 RSS.7657 DW. no. of observaions 7 no. of parameers mean ε ) ) -.897 var ε ) ).9795 Uskrif /Oupu EQ 4) Modelling Y by OLS using idsrekkedaa.xls) The esimaion sample is: o 8 Coefficien Sd.Error -value -prob Consan -.54.977-6.4. -.65.8594-5.9..99964.9 8.. -.57948.67-7.9. ε.6759.78 8.9. ˆ sigma.5694 RSS.49678 R^.987 F4,) = 766 [.]** DW. no. of observaions 7 no. of parameers 5 meany) -7.764 vary).94 6
Uskrif 4/Oupu 4 EQ 5) Modelling Y by OLS using idsrekkedaa.xls) The esimaion sample is: o 58 Coefficien Sd.Error -value -prob Consan.85695.5579.5. -.97.67 -.9.59.798646.5 5.. -.78.58-5.8. ε.89868.46.98.5 ˆ sigma.6 RSS.54747 R^.99455 F4,5) = 7 [.]** DW.74 no. of observaions 57 no. of parameers 5 meany) -7.846 vary).7476 Uskrif 5/Oupu 5 EQ 6) Modelling Y by OLS using idsrekkedaa.xls) The esimaion sample is: 59 o 8 Coefficien Sd.Error -value -prob Consan -.9774.664-6.48. -.5997.5 -.9..5958.8 4.6. -.7965.5-6.. ε.7956.65.6. ˆ sigma.5865 RSS.6644 R^.85658 F4,65) = 94.4 [.]** DW.8 no. of observaions 7 no. of parameers 5 meany) -7.6958 vary).57655 Uskrif 6/Oupu 6 EQ 7) Modelling Y by OLS using idsrekkedaa.xls) The esimaion sample is: o 8 Coefficien Sd.Error -value -prob Consan -.9774.5-7.5. D 4.784.984 5.6. -.5997.6-5...5958.48 4.95. -.7965.446-7.. ˆ ε.7956.97.5. D.4698.894.48.68 D.7989.67..9 7
D -.466.499 -.57. D ε.7.55.478.96 ˆ sigma.649 RSS.687887 R^.98775 F9,7) = 48 [.]** DW.8 no. of observaions 7 no. of parameers meany) -7.764 vary).94 8
Table 5 Criical Values for he Durbin-Wason Tes: 5% Significance Level a K= K= K=4 K=5 K=6 K=7 K=8 K=9 K=O K=l T d* d d* d d* d d* d d* d* d* d'" d* d d* d* d* d; d* di, L L L L L u L u L L u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ll7.75.46.657.79.58.5 7.6.469.4.556.6.65.84.75.4.86.95.974.845.9.767.6.69.4.66.47 8.8.476.55.56.8.65.4.747.8.85.95.958.874.7.798.88.7.9.65.4 9.4.48.7.56.98.65.4.74.5.84.975 9.44.9.5.86.64.75.78.68.96.5.489.84.567.4.65.4.79.7.8.998.9.96.4.854.4.78.5.7.6.6.496.97.57.9.65.6.75.9.85..9.95.8.879..8.6.74. "K refers o he number of columns in, including he consan enn.
... : Ta b I e 5 coninued) K= K= K=4 K=5 K=6 K=7 K=8 K=9 K= K=lI T d* d* d* d* d* d* d* d* d* d* d* d* L d;] L d'u L u L d u L d'u L u L d U L d U L d U L d U.7.5.9.574.44.65.77.7.l9.89.4.99.97.4.94..86..769.6.8.58..577.58.65.9.7.7.8.6.9.994.99.97.85.86.8.795.8 4.9.54..58.7.65.8.78.44.88.8.89.5.979.95.69.885.6.8.57 5.4.59.4.584.8.65..76.6.8.97.884.4.967.97.54.98.44.845.6 6.4.55.54.587.95.654.6.74.75.799.4.877.5.957.99.4.9.7.868.6 7.49.5.64.59.7.655.49.7.9.795..87.7.948 LOll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