Example of a non-stationary series Time Series plot of dow-Jones Index 4000 3900 3800 3700 300/ 3500 1295887116145174203232261290 Time 11
11 Example of a non-stationary series 1 29 58 87 116 145 174 203 232 261 290 4000 3900 3800 3700 3600 3500 Time Time Series Plot of Dow-Jones Index
Stationarity Suppose y, follows an AR(1) process without drift Is y stationarity? ◆ Note that y=的y1=1+E 的(y,如1)+、85x +E11+的E12+3 +外y 12
12 Stationarity Suppose yt follows an AR(1) process without drift. Is yt stationarity? Note that o t t t t t t t t t t t y y y y 3 1 3 2 1 2 1 1 1 1 1 2 1 1 1 ........... ( ) = + + + + + = + + = + − − − − − −
Stationarity Without loss of generality, assume that yo=0. Then E(y =0 Assuming that t is large, 1. e, the process started a long time ago, then var( y provided that k ) It can also be shown that provided that the same condition is satisfied C0(=,)=口2 (1-2) Pi var(y,) 13
13 Stationarity Without loss of generality, assume that yo = 0. Then E(yt )=0. Assuming that t is large, i.e., the process started a long time ago, then It can also be shown that provided that the same condition is satisfied, , provided that | | 1. (1 ) var( ) 2 1 1 2 − = t y var( ) (1 ) cov( ) 2 1 1 2 1 t s s t t s y y y = − − =
Stationarity Special case: P It is a random walk process. Now ∑ 0 ◆Thus, ()E()=0 for all t (2)Var()=toe for all t B3)Cov(,y-s=(t-s)of for all t 14
14 Stationarity Special Case: 1 = 1 It is a “random walk” process. Now, Thus, . t t 1 t y = y + − − = = − 1 0 . t j t t j y 2 2 (1) ( ) 0 . (2) ( ) . (3) ( , ) ( ) . t t t t s E y for all t Var y t for all t Cov y y t s for all t − = = = −
Stationarity Suppose the model is an ar(2) without drift, i.e y=的y1+女2y1-2+E It can be shown that for y, to be stationary +2<1-的<1and2|<1 The key point is that ar processes are not stationary unless appropriate prior conditions are imposed on the parameters 15
15 Stationarity Suppose the model is an AR(2) without drift, i.e., It can be shown that for yt to be stationary, The key point is that AR processes are not stationary unless appropriate prior conditions are imposed on the parameters. yt =1 yt−1 +2 yt−2 + t 1 +2 1,2 −1 1and |2 |1