Ieraioal Mahemaical Forum Vol. 6 0 o. 5 595-604 Iferece of he Secod Order Auoregressive Model wih Ui Roos Ahmed H. Youssef Professor of Applied Saisics ad Ecoomerics Isiue of Saisical Sudies ad Research Cairo Uiversi Ahmed Ami El-Sheih Associaed Professor of Applied Saisics ad Ecoomerics Isiue of Saisical Sudies ad Research Cairo Uiversi aham0@ahoo.com Mohamed Khalifa Ahmed Issa Maser sude of Applied Saisics ad Ecoomerics Isiue of Saisical Sudies ad Research Cairo Uiversi Absrac I his paper ordiar leas squares (OLS mehod will be used o esimae he parameers of auo-regressive model of order wo ad he properies of he esimaed parameers of AR ( have bee sudied. Also closed form of he variace of he esimaed parameers has bee derived. Kewords: secod order auoregressive Ui roos esimaors ubiasedess ad lieari. - Iroducio Ma ad Wald (94 have proved ha he basic properies of OLS esimaes do o chage i he case of large sample. Box ad Jeis (97 developed a mehod for aalzig saioar uivariae ime series daa. The imporace ad geeral aure of he ARIMA approach o ime series aalsis are discussed. The ovel coribuios of his mehod ad limiaios are explaied. Prerequisies of Box Jeis models are defied ad explored. Differe pes of o-saioar ime series are elaboraed.
596 A. H. Youssef A. Ami El-Sheih ad M. Kh. A. Issa Dice ad Fuller (979 foud a represeaio for he ui roo disribuio usig simulaio. The abulaed various ui roo disribuios ha ca be used o perform ui roo ess. Le he firs order auoregressive process AR( be defied as: α + ρ + ε... where ε is a sequece of idepede ideicall disribued radom variables wih mea zero ad variaceσ. The values of α ρ ad deermied he aure of he ime series. If ρ - he radom wal is said o displa drif Paula e al (994. Also he iroduced he leas square esimaor for ρ which ae he form: ρ ( ( ( ( ( - Esimae of he Parameers of AR ( I his secio he parameers of AR ( will be esimaed usig he OLS mehod. Le α + ρ + ρ + ε 4 ( whereε ad he values of α ρ ρ ad are defied as above. The OLS esimaor for( α ρ ρ will be obaied b rewriig equaio ( as follows: ε α ρ ρ 4... Le S ( α ρ ρ ε ( α ρ ρ ( B differeiaig equaio ( wih respec o α ρ ad ρ he followig esimaors will be obaied: ˆ α ρ ρ (4 (0 ( (
Auoregressive model wih ui roos 597 where ( (0 ad ( ρ ( ( ( ( (. ( ( ( ( ( ( ( (. ( ( ( ( ( ( (5 ρ ca be rewrie as: ρ Where a ( ( ( a ab (6 ( bb a ( ( ( ( b ( ( ( ( ad b ( ( ( (
598 A. H. Youssef A. Ami El-Sheih ad M. Kh. A. Issa ρ ( ( ( ( (. ( ( ( ( ( ( ( (. ( ( ( ( ( ( (7 Also ρ will ae he form: a ab ρ ( bb (8 The properies of he OLS mehod for he parameers of AR( will be iroduced i he followig secio. - Properies of The Leas Square Esimaor of AR ( Accordig o he codiio of Ma ad Wald (94 ad b usig equaios (5 ad (7 he ubiasedess ad lieari proper of ρ ad ρ will be proved as follows:. Lieari B usig equaio (6 ad (8 lieari of he leas squares esimaor i he case of a ui roo ca be proved as follows. Le a G ad ( bb a ( bb Ad a a G ad H ( b b ( b b (9 B subsiuig from equaio (9 i equaios (6 ad (8 o ge he resul; ˆ ρ G H b ˆ ρ G H b (0
Auoregressive model wih ui roos 599. Ubiasedess Assume ha ad (... ( ( ( B usig equaio ( he weighs ad Lemma (: The weighs (i (ii (iii (iv will be defied as: ( ad ( i is o-sochasic i i i 0 i i i have he followig properies: i Proof (i Sice i are assumed o be o-sochasic he i are o-sochasic (ii (iii i i 0 i i ( i i i ; ( i i ( i i (iv i i i ; i 0 ( (4
600 A. H. Youssef A. Ami El-Sheih ad M. Kh. A. Issa i i ( i ( i i i i ( i ( i ( i ( i B usig lemma ( equaios (5 ad (7 ca be rewrie as follows: (5. ρ. ad (6. ρ. Now ubiasedess proper of ρ will be prove as follows: B subsiuig from equaio ( i equaio (6 o ge ˆ ˆ ˆ ˆ ( α+ ρ + ρ+ ε ( α+ ρ + ρ+ ε. ˆ ρ. (7 Le I ( α + ˆ ρ + ˆ ρ + ε α + ˆ ρ + ˆ ρ + ε B usig lemma ( Le ˆ ρ + ˆ ρ + ε I (8
Auoregressive model wih ui roos 60 II ˆ ˆ ( α + ρ + ρ + ε. α ˆ ρ ˆ ρ ε B usig lemma ( II ˆ ρ ˆ ρ ε (9 B subsiuig from equaios ( ad ( i equaio (7 o ge ε ε ˆ ρ ρ+. (0 B aig he expecaio of equaio (0 he ubiasedess proper will be proved. Followig he same wa i ca be proved ha ρ is also a ubiased esimaor [Khalifa 0]. 4- The Variace of he Esimaors ˆρ Sice Ε [ ρ ] ˆ ρ he [ ˆ ] var( ˆ ρ Ε ρ ρ ( B subsiuig from equaio (0 i equaio ( o ge ε ε var( ˆ ρ Ε (. Le D. The
60 A. H. Youssef A. Ami El-Sheih ad M. Kh. A. Issa var( ˆ ρ Ε D ε ε var( ˆ ( (.( ρ Ε ε ε ε ε D + ( Lemma (: i ca be show ha: 0 ε Proof ε ( ˆ ˆ ˆ ˆ α ρ ρ Sice ˆ ˆ α + ˆ ρ ˆ + ρ The ε ( ˆ ˆ ˆ ˆ - ˆ ˆ α+ ρ + ρ α ρ ρ 0 ε (4 B he same wa i ca be prove ha : 0 ε B subsiuig from equaio (4 i equaio ( o ge var( ˆ ( ( ρ.( Ε ε ε D + (5 B aig he expecaio of equaio (5o obai var ( ρ ( D σ + σ
Auoregressive model wih ui roos 60 σ + σ var( ˆ ρ D (6 Equaio (6 ca be rewrie as: var( ˆ ρ σ C Where ε 4 σ ad C [ + ] D Followig he same seps var( ˆ ρ σ C Where [ + ] C D REFERENCES Box G. E. P. ad Jeis G. M. Time Series Aalsis: Forecasig ad Corol. Sa Fracisco. Holde Da p 8 (976. Dice D.A. ad Fuller W.A. Disribuio of he Esimaors For Auoregressive Time Series Wih A Ui Roo. Joural of he America Saisical Associaio 74 (979 47-4 Ma H. B. ad Wald A. O he Saisical Treame of he Liear Sochasic Differece Equaios. Ecoomerica (94 7-0. 4 Khalifa M. A. Some properies of he ui roo esimaors MSc hesis Isiue of Saisical Sudies ad ResearchCairo Uiversi (0
604 A. H. Youssef A. Ami El-Sheih ad M. Kh. A. Issa 5 Paula S. Farias G.G ad Fuller W. A Compariso of Ui Roo Tes Crieria. J. Bus. ad Ecoom. Sais (994 449-459 Received: April 0