AR(1)(I) ◆NOW, 1)E(y)=0 forall (2)Var()=, forall (3)Co(,y) for all t v and cove, y-s are finite if and only if o< 1, which is the stationarity requirement for an ar(1) process
11 AR(1) (II) Now, Var(yt ) and cov(yt , yt-s ) are finite if and only if |1 | < 1, which is the stationarity requirement for an AR(1) process. . 1 (3) ( , ) . 1 (2) ( ) (1) ( ) 0 . 2 1 1 2 2 1 2 Cov y y for all t Var y for all t E y for all t s t t s t t e e − = − = = −
AR(1)(V) Special case: P It is a random walk process. Now ∑ 0 ◆Thus )E(=0 for all t (2)Var()=to s for all B)Cov,Ds=t-sof for all t 12
12 AR(1) (IV) Special Case: 1 = 1 It is a “random walk” process. Now, Thus, . t t 1 t y = y +e − − = = − 1 0 . t j t t j y e (3) ( , ) | | . (2) ( ) . (1) ( ) 0 . 2 2 Cov y y t s for all t Var y t for all t E y for all t t t s t t e e = − = = −
AR(1)(V) ◆ Consider y, is a homogeneous non-stationary series The number of times that the original series must be differenced before a stationary series results is called the order of integration 13
13 AR(1) (V) Consider, yt is a homogeneous non-stationary series. The number of times that the original series must be differenced before a stationary series results is called the order of integration. . 1 t t t t y y y = e = − −
Theoretical autocorrelation Function(tac)( Autoregressive(AR) Processes Consider an ar(1) process without drift v,=,y-lt8 Recall that (1)E(=0 forall 2 (2)Var( =,e=ro forall t (3)Cov(, y_= forall 14
14 Theoretical Autocorrelation Function (TAC) (I) Autoregressive (AR) Processes Consider an AR(1) process without drift : Recall that . 1 (3) ( , ) . 1 (2) ( ) (1) ( ) 0 . 2 1 1 2 2 0 1 2 Cov y y for all t Var y for all t E y for all t s s t t s t t e e = − = = − = = − . t 1 t 1 t y = y +e −
Theoretical autocorrelation Function (TAc)(D) The autocorrelation function at lag k is fork=0,1,2, So for a stationary ar(1)process, the TAC dies down gradually as k increases 15
15 Theoretical Autocorrelation Function (TAC) (II) The autocorrelation function at lag k is So for a stationary AR(1) process, the TAC dies down gradually as k increases. . 0,1,2,... 1 0 k k k for k = = =