Inference of the Second Order Autoregressive. Model with Unit Roots

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1 Ieraioal Mahemaical Forum Vol. 6 0 o 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.

2 596 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 ( (

3 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 ( ( ( (

4 598 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

5 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

6 600 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

7 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

8 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

9 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 ( Ma H. B. ad Wald A. O he Saisical Treame of he Liear Sochasic Differece Equaios. Ecoomerica ( Khalifa M. A. Some properies of he ui roo esimaors MSc hesis Isiue of Saisical Sudies ad ResearchCairo Uiversi (0

10 604 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 ( Received: April 0

OLS bias for econometric models with errors-in-variables. The Lucas-critique Supplementary note to Lecture 17

OLS bias for econometric models with errors-in-variables. The Lucas-critique Supplementary note to Lecture 17 OLS bias for ecoomeric models wih errors-i-variables. The Lucas-criique Supplemeary oe o Lecure 7 RNy May 6, 03 Properies of OLS i RE models I Lecure 7 we discussed he followig example of a raioal expecaios