Module: Principles of Financial Econometrics I Lecturer: Dr Baboo M Nowbutsing

Transcription

1 BSc (Hons) Finance II/ BSc (Hons) Finance wih Law II Modle: Principles of Financial Economerics I Lecrer: Dr Baboo M Nowbsing Topic 10: Aocorrelaion Serial Correlaion

2 Oline 1. Inrodcion. Cases of Aocorrelaion 3. OLS Esimaion 4. BLUE Esimaor 5. Conseqences of sing OLS 6. Deecing Aocorrelaion Serial Correlaion

3 1. Inrodcion Aocorrelaion occrs in ime-series sdies when he errors associaed wih a given ime period carry over ino fre ime periods. For example, if we are predicing he growh of sock dividends, an overesimae in one year is likely o lead o overesimaes in scceeding years. Serial Correlaion

4 1. Inrodcion Times series daa follow a naral ordering over ime. I is likely ha sch daa exhibi inercorrelaion, especially if he ime inerval beween sccessive observaions is shor, sch as weeks or days. Serial Correlaion

5 1. Inrodcion We expec sock marke prices o move or move down for several days in sccession. In siaion like his, he assmpion of no ao or serial correlaion in he error erm ha nderlies he CLRM will be violaed. We experience aocorrelaion when E ( i j ) 0 Serial Correlaion

6 1. Inrodcion Someimes he erm aocorrelaion is sed inerchangeably. However, some ahors prefer o disingish beween hem. For example, Tinner defines aocorrelaion as lag correlaion of a given series wihin iself, lagged by a nmber of imes nis whereas serial correlaion is he lag correlaion beween wo differen series. We will se boh erm simlaneosly in his lecre. Serial Correlaion

7 1. Inrodcion There are differen ypes of serial correlaion. Wih firs-order serial correlaion, errors in one ime period are correlaed direcly wih errors in he ensing ime period. Wih posiive serial correlaion, errors in one ime period are posiively correlaed wih errors in he nex ime period. Serial Correlaion

8 . Cases of Aocorrelaion 1. Ineria - Macroeconomics daa experience cycles/bsiness cycles.. Specificaion Bias- Exclded variable Appropriae eqaion: Y 1 X 3 X 3 4 X 4 Esimaed eqaion Y 1 X 3 X 3 Esimaing he second eqaion implies v 4 X 4 v Serial Correlaion

9 Serial Correlaion. Cases of Aocorrelaion 3. Specificaion Bias- Incorrec Fncional Form v X X Y 3 1 X Y 1 v X 3

10 . Cases of Aocorrelaion 4. Cobweb Phenomenon In agriclral marke, he spply reacs o price wih a lag of one ime period becase spply decisions ake ime o implemen. This is known as he cobweb phenomenon. Ths, a he beginning of his year s planing of crops, farmers are inflenced by he price prevailing las year. Serial Correlaion

11 . Cases of Aocorrelaion 5. Lags Consmpion 1 Consmpion The above eqaion is known as aoregression becase one of he explanaory variables is he lagged vale of he dependen variable. If yo neglec he lagged he resling error erm will reflec a sysemaic paern de o he inflence of lagged consmpion on crren consmpion. 1 Serial Correlaion

12 . Cases of Aocorrelaion 6. Daa Maniplaion Y Y 1 X Y X v This eqaion is known as he firs difference form and dynamic regression model. The previos eqaion is known as he level form. Noe ha he error erm in he firs eqaion is no aocorrelaed b i can be shown ha he error erm in he firs difference form is aocorrelaed. X Serial Correlaion

13 . Cases of Aocorrelaion 6. Nonsaionariy When dealing wih ime series daa, we shold check wheher he given ime series is saionary. A ime series is saionary if is characerisics (e.g. mean, variance and covariance) are ime varian; ha is, hey do no change over ime. If ha is no he case, we have a nonsaionary ime series. Serial Correlaion

14 3. OLS Esimaion Y 1 X E ( i j ) 0 Assme ha he error erm can be modeled as follows: is known as he coefficien of aocovariance and he error erm saisfies he OLS assmpion. This Scheme is known as an Aoregressive (AR(1))process Serial Correlaion

15 3. OLS Esimaion Var( Cov( ) E(, s ) ) 1 E( s ) s 1 Cor (, ) s s ( ˆ x x x x x x Var ) n1 1 x x x x Serial Correlaion

16 Serial Correlaion 4. BLUE Esimaor Under he AR (1) process, he BLUE esimaor of β is given by he following expression. C x x y y x x n n GLS ) ( ) )( ( ˆ D x x Var n GLS 1 ) ( ) ˆ (

17 4. BLUE Esimaor The Gass Theorem provides only he sfficien condiion for OLS o be BLUE. The necessary condiions for OLS o be BLUE are given by Krshkal s heorem. Therefore, in some cases, i can happen ha OLS is BLUE despie aocorrelaion. B sch cases are very rare. Serial Correlaion

18 5. Conseqences of Using OLS OLS Esimaion Allowing for Aocorrelaion As noed, he esimaor is no more no BLUE, and even if we se he variance, he confidence inervals derived from here are likely o be wider han hose based on he GLS procedre. Hypohesis esing: we are likely o declare a coefficien saisically insignifican even hogh in fac i may be. One shold se GLS and no OLS. Serial Correlaion

19 5. Conseqences of Using OLS OLS Esimaion Disregarding Aocorrelaion The esimaed variance of he error is likely o overesimae he re variance Over esimae R-sqare Therefore, he sal and F ess of significance are no longer valid, and if applied, are likely o give seriosly misleading conclsions abo he saisical significance of he esimaed regression coefficiens. Serial Correlaion

20 5. Deecing Aocorrelaion Graphical Mehod There are varios ways of examining he residals. The ime seqence plo can be prodced. Alernaively, we can plo he sandardized residals agains ime. The sandardized residals is simply he residals divided by he sandard error of he regression. If he acal and sandard plo shows a paern, hen he errors may no be random. We can also plo he error erm wih is firs lag. Serial Correlaion

21 5. Deecing Aocorrelaion The Rns Tes- Consider a lis of esimaed error erm, he errors erm can be posiive or negaive. In he following seqence, here are hree rns. ( ) ( ) ( ) A rn is defined as ninerrped seqence of one symbol or aribe, sch as + or -. The lengh of he rn is defined as he nmber of elemen in i. The above seqence as hree rns, he firs rn is 6 minses, he second one has 13 plses and he las one has 11 rns. Serial Correlaion

22 5. Deecing Aocorrelaion The Rns Tes- Consider a lis of esimaed error erm, he errors erm can be posiive or negaive. In he following seqence, here are hree rns. ( ) ( ) ( ) A rn is defined as ninerrped seqence of one symbol or aribe, sch as + or -. The lengh of he rn is defined as he nmber of elemen in i. The above seqence as hree rns, he firs rn is 6 minses, he second one has 13 plses and he las one has 11 rns. Serial Correlaion

23 5. Deecing Aocorrelaion Define N: oal nmber of observaions N 1 : nmber of + symbols (i.e. + residals) N : nmber of symbols (i.e. residals) R: nmber of rns Assming ha he N 1 >10 and N >10, hen he nmber of rns is normally disribed wih: Serial Correlaion

24 5. Deecing Aocorrelaion Then, E( R) N 1 N N 1 If he nll hypohesis of randomness is ssainable, following he properies of he normal disribion, we shold expec ha Prob [E(R) 1.96 R R E(R) 1.96 R ] R N ( N ) (N 1) Hypohesis: do no rejec he nll hypohesis of randomness wih 95% confidence if R, he nmber of rns, lies in he preceding confidence inerval; rejec oherwise 1 N 1 ( N N N ) Serial Correlaion

25 5. Deecing Aocorrelaion The Drbin Wason Tes d n ( ˆ n 1 ˆ ˆ 1 ) I is simply he raio of he sm of sqared differences in sccessive residals o he RSS. The nmber of observaion is n-1 as one observaion is los in aking sccessive differences. Serial Correlaion

26 5. Deecing Aocorrelaion A grea advanage of he Drbin Wason es is ha based on he esimaed residals. I is based on he following assmpions: 1. The regression model incldes he inercep erm.. The explanaory variables are nonsochasic, or fixed in repeaed sampling. 3. The disrbances are generaed by he firs order aoregressive scheme. Serial Correlaion

27 5. Deecing Aocorrelaion 4. The error erm is assmed o be normally disribed. 5. The regression model does no inclde he lagged vales of he dependen an explanaory variables. 6. There are no missing vales in he daa. Drbin-Wason have derived a lower bond d L and an pper bond d U sch ha if he comped d lies oside hese criical vales, a decision can be made regarding he presence of posiive or negaive serial correlaion. Serial Correlaion

28 Serial Correlaion 5. Deecing Aocorrelaion Where B since -1 1, his implies ha 0 d ˆ ˆ ˆ 1 ˆ ˆ ˆ ˆ ˆ d ˆ 1 d 1 ˆ ˆ ˆ ˆ

29 5. Deecing Aocorrelaion If he saisic lies near he vale, here is no serial correlaion. B if he saisic lies in he viciniy of 0, here is posiive serial correlaion. The closer he d is o zero, he greaer he evidence of posiive serial correlaion. If i lies in he viciniy of 4, here is evidence of negaive serial correlaion Serial Correlaion

30 5. Deecing Aocorrelaion If i lies beween d L and d U / 4 d L and 4 d U, hen we are in he zone of indecision. The mechanics of he Drbin-Wason es are as follows: Rn he OLS regression and obain he residals Compe d For he given sample size and given nmber of explanaory variables, find o he criical d L and d U. Follow he decisions rle Serial Correlaion

31 5. Deecing Aocorrelaion Use Modified d es if d lies in he zone in he of indecision. Given he level of significance, H o : = 0 verss H 1 : > 0, rejec H o a level if d < d U. Tha is here is saisically significan evidence of posiive aocorrelaion. H o : = 0 verss H 1 : < 0, rejec H o a level if 4- d < d U. Tha is here is saisically significan evidence of negaive aocorrelaion. H o : = 0 verss H 1 : 0, rejec H o a level if d < d U and 4- d < d U. Tha is here is saisically significan evidence of eiher posiive or negaive aocorrelaion. Serial Correlaion

32 5. Deecing Aocorrelaion The Bresch Godfrey The BG es, also known as he LM es, is a general es for aocorrelaion in he sense ha i allows for 1. nonsochasic regressors sch as he lagged vales of he regressand;. higher-order aoregressive schemes sch as AR(1), AR ()ec.; and 3. simple or higher-order moving averages of whie noise error erms. Serial Correlaion

33 5. Deecing Aocorrelaion The Bresch Godfrey The BG es, also known as he LM es, is a general es for aocorrelaion in he sense ha i allows for 1. nonsochasic regressors sch as he lagged vales of he regressand;. higher-order aoregressive schemes sch as AR(1), AR ()ec.; and 3. simple or higher-order moving averages of whie noise error erms. Serial Correlaion

34 5. Deecing Aocorrelaion Consider he following model: Y 1 X p p H 0 o : 1... p Esimae he regression sing OLS Rn he following regression and obained he R-sqare Serial Correlaion

35 5. Deecing Aocorrelaion If he sample size is large, Bresch and Godfrey have shown ha (n p) R follow a chi-sqare If (n p) R exceeds he criical vale a he chosen level of significance, we rejec he nll hypohesis, in which case a leas one rho is saisically differen from zero. Serial Correlaion

36 5. Deecing Aocorrelaion Poin o noe: The regressors inclded in he regression model may conain lagged vales of he regressand Y. In DW, his is no allowed. The BG es is applicable even if he disrbances follow a p h- order moving averages (MA) process, ha is is inegraed as follows: p p A drawback of he BG es is ha he vale of p, he lengh of he lag canno be specified as a priori. Serial Correlaion

37 5. Deecing Aocorrelaion Model Misspecificaion vs. Pre Aocorrelaion I is imporan o find o wheher aocorrelaion is pre aocorrelaion and no he resl of mis-specificaion of he model. Sppose ha he Drbin Wason es of a given regression model (wage-prodciviy) reveals a vale of This indicaes posiive aocorrelaion Serial Correlaion

38 5. Deecing Aocorrelaion However, cold his correlaion have arisen becase he model was no correcly specified? Time series model do exhibi rend, so add a rend variable in he eqaion. Serial Correlaion

39 5. Deecing Aocorrelaion The Mehod of GLS Y 1 X There are wo cases when (1) is known and () is no known Serial Correlaion

40 Serial Correlaion 5. Deecing Aocorrelaion When is known If he regression holds a ime, i shold hold a ime -1, i.e. Mliplying he second eqaion by gives Sbracing (3) from (1) gives X Y X Y ) ( ) 1 ( X X p Y Y ( 1 )

41 5. Deecing Aocorrelaion The eqaion can be Y * * X * * The error erm saisfies all he OLS assmpions Ths we can apply OLS o he ransformed variables Y* and X* and obain esimaion wih all he opimm properies, namely BLUE In effec, rnning his eqaion is he same as sing he GLS. Serial Correlaion

42 5. Deecing Aocorrelaion When is nknown, here are many ways o esimae i. Assme ha = +1 he generalized difference eqaion redces o he firs difference eqaion Y Y ( X X 1 ) ( 1 ) Y X Serial Correlaion

43 5. Deecing Aocorrelaion The firs difference ransformaion may be appropriae if he coefficien of aocorrelaion is very high, say in excess of 0.8, or he Drbin- Wason d is qie low. Maddala has proposed his rogh rle of hmb: Use he firs difference form whenever d< R. Serial Correlaion

44 5. Deecing Aocorrelaion There are many ineresing feares of he firs difference eqaion There is no inercep regression in i. Ths, yo have o se he regression hrogh he origin roine If however by misake one incldes an inercep erm, hen he original model has a rend in i. Ths, by inclding he inercep, one is esing he presence of a rend in he eqaion. Serial Correlaion

45 5. Deecing Aocorrelaion Anoher ineresing feare relaes o he saionariy propery. When =1, he error erm,, is nonsaionary, for he variances and covariances become infinie. When =1, he firs differenced becomes saionary, as i is eqal o q whie noise error erm. Serial Correlaion

46 5. Deecing Aocorrelaion Based on Drbin-Wason d saisic From he Drbin Wason Saisics, we know ha 1 d In reasonably large samples one samples one can obain rho from his eqaion and se i o ransform he daa as shown in he GLS. Serial Correlaion

47 5. Deecing Aocorrelaion Based on Drbin-Wason d saisic From he Drbin Wason Saisics, we know ha 1 d In reasonably large samples one samples one can obain rho from his eqaion and se i o ransform he daa as shown in he GLS. Serial Correlaion

48 5. Deecing Aocorrelaion Based on he error erms 1 1 Esimae he following eqaion ˆ ˆ.ˆ 1 v Serial Correlaion

49 5. Deecing Aocorrelaion Ieraive Procedre - We can esimae rho by sccessive approximaion, saring wih some iniial vale of rho. he Cochran-Orc ieraive procedre, he Cochran- Orc wo-sep procedre, he Drbin-Wason wosep procedre and he Hildreh-L scanning or search procedre. The mos poplar one is he Cochran-Orc ieraive procedre. Serial Correlaion