Properties of Autocorrelated Processes Economics 30331

Transcription

1 Properies of Auocorrelaed Processes Economics 3033 Bill Evans Fall 05 Suppose we have ime series daa series labeled as where =,,3, T (he final period) Some examples are he dail closing price of he S&500, quarerl per capia GDP, monhl unemplomen, weekl movie aendance, ec As we will show below, man ime series processes demonsrae auocorrelaion, ha is, here is some persisence from one period o he nex in he values For example, his monh s unemplomen rae is highl predicive of nex periods Likewise, he sock price for Apple oda is highl predicive of omorrow s prices A primar characerisic of auocorrelaed processes are wheher he are saionar or no The ime series is considered covariance saionar if he series has a finie second momen and exhibis hree characerisics i) Var( ) = consan for all ii) E[ ] = consan for all iii) Cov(, s)=cov( +h, s+h) where >s The firs characerisic sas ha he variance of he variable mus be finie for all values of ime The second sas ha he mean canno depend on ime The hird characerisic sas ha he covariance beween wo poins is onl a funcion of he disance beween he poins (-s) and no he poin we are considering (+h and s+h) We will pu aside for now he imporance of esablishing saionari bu in general, here are wo problems wih nonsaionar series Firs, if daa is non-saionar, he underling assumpions of our saisical ess are wrong so we canno do hings like -ess and f-ess Second, as we will show wih some simulaions, a regression of a non-saionar series on anoher non-saionar series ends o generae spurious correlaion and high Tpe I error raes We will discuss hese issues in some deail in class bu for now, we wan o focus on esablishing wheher a series is saionar or no When we ask wheher a series is saionar or no, we simpl check he hree crieria above and if one is violaed, he series is non-saionar Example : A moving average represenaion Suppose ha Where is an independen and idenicall disribued error, so E[ ] 0, Var( ), cov(, ) 0 and A graph of his process is below wih some random variables seleced for and assuming Noe ha his model looks a lo like he firs difference in monhl real per capia reail sales!

2 5 Moving average: α=05, epsilon ε~n(0,) Time Series of s Difference ln(per Capia Reail Sales) s Difference ln(per capia consumpion) Year:Monh -4 6 To check saionari, we need simpl go hrough he condiions above For ii) noe ha E [ ] 0 because E[ ] 0 and E[ ] 0 For condiion i), he variance is a lile harder -- Var( ) E[( E( )) ] E[ ] E[( ) ] E[ ] E[ ] E[ ] E[ ] ( ) For condiion iii), we noe ha cov(, ) E[( E[ ])( E[ ])] E[ ] because E[ ] E[ ] 0 Noe as well ha so cov(, ) E[ ] E[( )( )] E[ ] E[ ] You can show ha h cov(, h) for h> so he Cov(, s) is onl a funcion of x-s, no x or s Example : A linear ime rend Consider a linear model of he form 0 where E[ ] 0, Var( ) and cov(, ) 0 The ime series for quarerl populaion in he US monhl or quarerl real per capia GDP look like a linear ime rend

3 Populaion (x 000) GDP/Populaion (005$) Quarerl US Populaion (x 000) $50,000 Quarerl Real per Capia GDP (005$) $40, $30, $0, Year: quarer Noe from he sar ha $0, Year: quarer E[ ] E[ ] E[ ] and hence, he expeced value of is a funcion of ime so we violae condiion ii) above and hence, his is a nonsaionar series This model is called rends saionar because he model can be made saionar b de-rending Example 3: An AR() process The series for is auocorrelaed which means ha curren values are correlaed wih he pas The process can be ver complicaed or raher simple In his case, we will consider he simples ime of auocorrelaed process AR() auocorrelaion or order where he variable is onl correlaed wih a one-period lag Define he AR() process o be () A he sar, we have o make he assumpion ha I will become clear laer on wh we have o make his assumpion We will mainain man of he original assumpions abou he errors, namel ha E[ ] 0andVar[ ] for all and cov[, ] 0 The auocorrelaed process described in () and he assumpion ha means ha shocks o he ime series in one period will evenuall die ou in he series To demonsrae his poin, suppose here is some shock such ha is unusuall high How much of ha will persis ino he fuure? Noe ha he series for ime period + is defined as () If we make he assumpion ha E [ ] 0 and Var[ ] for all, hen (3) cov(, ) E[ ] E[( ) ] E[ ] E[ ] 3

4 Because (4) E[ ] and E[ ] =0, hen cov(, ) Noe ha (5) Corr(, ) cov(, ) 05 ( Var( ) Var( )) No consider somehing how long he shock persiss h periods in he fuure Wrie he definiion of has (6) h h h And noe ha we can wrie h as (7) h h h And subsiuing his ino (6), we ge (8) [ ] h h h h h h h h h Doing his again, we know ha (9) h h3 h So (8) can be wrien as (0) [ ] 3 h h h h h3 h h h h3 h h h If we coninue o make hese subsiuions, hen we will evenuall wrie (0) as () h h h h h h Looking a he covariance beween hand () cov(, ) E(, ) E[( ] h h h h h h h Noe ha [ h h E ] and E[ ] =0 for all m>0, hen m 4

5 (3) cov(, ) h h Y Noe from (3) ha he covariance is a funcion of he disance beween and +h bu no, so crieria iii) for weak saionari is saisfied Recognizing he definiion of correlaion coefficiens (4) cov(, ) h Corr( h, ) ( Var( ) ( )) h h 05 h Var Example 4: A Highl persisen series he random walk Consider he AR() process and relax he assumpion ha The model hen can be rewrien as (5) This series is called a random walk and he series has a number of imporan properies Sock prices are hough o follow a random walk Given he highl persisen naure of he series, i is eas o demonsrae he hisor of as a funcion of he errors Noe ha he observaion for period - can be wrien as (6) And subsiuing his ino equaion (5), we can re-wrie he equaion as (7) Noing ha 3 we can re-wrie he equaion again as (6) 3 Doing his for all n observaions in he series, we can wrie he series for as (7) 0 Noe ha E[ ] E[ ] E[ ] E[ ] E[ ] E[ ] E[ ] E[ ] (8) 0 and because E[ ] 0for all, i mus be he case ha (9) E[ ] E[ 0] 5

6 For This simple resul has power implicaions abou he fuure predicions of Suppose we wan daa on and we wan he predicion of he variable someime in he fuure In one period, we can wrie (0) And noe ha because E[ ] 0 () E[ ] E[ ] Two periods in he fuure, we know ha () And because E[ ] E[ ] 0 (3) E[ ] E[ ] E[ ] Exending his o h periods in he fuure, i is eas o show ha (4) h h h And because E[ h] E[ h ] E[ h] E[ ] E[ ] 0hen E[ ] The bes predicion for an ime in he fuure is oda s h Anoher imporan proper of he series is he variance Noe havar[ ] and we will assume ha he errors are no auocorrelaed where cov[, h] 0for all h>0 Because is a linear combinaion of independen random variables and o is a fixed parameer, Var[ o ] 0 Therefore, given (4), i is he case ha (5) 0 Var( ) Var( ) Var( ) Var( ) Var( ) Var( ) Var( ) Var( ) (6) 0 Because he series is periods long (,,3 ) and Var[ ] for all and Var[ ] 0 o (7) Var( ) Noe hen ha lim Var( ) So his is a non-saionar series 6

7 Tesing for uni roos Tesing for a random walk is a lile difficul I is emping o simpl run a regression of on is lag, which is a pre good approximaion However, mos of he saisics associaed wih OLS models assume he models are saionar and herefore, as we approach a non-saionar model, he pical sandard ess we would calculae are now no longer valid Therefore, a whole se of oher saisical models have been produced o es wheher a model is a random walk or no Sar wih he basic AR model (8) Which is saionar so long as We are ineresed in esing he null H : 0 agains he alernaive H : To es his, subrac from boh sides ( ) (9) a Noe ha if, hen he coefficien on θ will equal 0 The ransformaion of he model from (8) o (9) allows us o proceed wih a hull H : 0agains he null 0 H : 0 0 The null is now ha he model is non-saionar and if we canno rejec he null, we canno evidence ha he model is a saionar process We can consruc he sandard -ess on ˆ bu his is no longer normall disribued in large samples Dicke and Fuller (979) have demonsraed wha he disribuion of his es saisic looks like under he null H : or H : Using he DJIA daa from class, we firs regress ln closing prices on a lag Noe ha he coefficien on he lag is ˆ When we ransform he model, we ge a coefficien on ˆ The esimae -saisic on his is -09 To ge he criical values for he null, we ask for hem b ping dfuller ln_close which produces he same resuls we have jus consruced and he criical values for he one-ailed es H : 0 0 The 5% criical value is -86 so we canno rejec he null he daa is a non-saionar 7

8 * es for random walk * run a regression of change ln(closing price) * on one period lag reg ln_close ln_close_ Source SS df MS Number of obs = F(, 4359) = Model Prob > F = Residual R-squared = Adj R-squared = Toal Roo MSE = ln_close Coef Sd Err P> [95% Conf Inerval] ln_close_ _cons es ln_close_== ( ) ln_close_ = F(, 4359) = 003 Prob > F = 0850 * now run model where null is ransformed ino 0 reg d_ln_close ln_close_ Source SS df MS Number of obs = F(, 4359) = 003 Model 8777e e-06 Prob > F = 0850 Residual R-squared = Adj R-squared = Toal Roo MSE = d_ln_close Coef Sd Err P> [95% Conf Inerval] ln_close_ _cons

9 * now ge dicke fuller es dfuller ln_close Dicke-Fuller es for uni roo Number of obs = Inerpolaed Dicke-Fuller Tes % Criical 5% Criical 0% Criical Saisic Value Value Value Z() MacKinnon approximae p-value for Z() = 0940 * ge he lag of he s difference gen d_ln_close_=d_ln_close[_n-] ( missing values generaed) * run a regression of he s difference on is lag reg d_ln_close d_ln_close_ Source SS df MS Number of obs = F(, 4358) = 6943 Model Prob > F = Residual R-squared = Adj R-squared = Toal Roo MSE = d_ln_close Coef Sd Err P> [95% Conf Inerval] d_ln_close_ _cons