7-6 Stochastic Non-Stationarity Note that the model (1) could be generalised to the case where yt Is an explosive process: y=u+v+u whereφ>1 Typically, the explosive case is ignored and we use 1 to characterise the non-stationarity because o> 1 does not describe many data series in economics and finance o> 1 has an intuitively unappealing property: shocks to the system are not only persistent through time, they are propagated so that a given shock will have an Increasingly large influence. c Chris Brooks2002陈磊2004
© Chris Brooks 2002 陈磊2004 7-6 Stochastic Non-Stationarity • Note that the model (1) could be generalised to the case where yt is an explosive process: yt = + yt-1 + ut where > 1. • Typically, the explosive case is ignored and we use = 1 to characterise the non-stationarity because – > 1 does not describe many data series in economics and finance. – > 1 has an intuitively unappealing property: shocks to the system are not only persistent through time, they are propagated so that a given shock will have an increasingly large influence
7-7 Stochastic Non-stationarity The Impact of shocks To see this, consider the general case of an ar(1) with no drift y=oya+ Let o take any value for now. · We can write: y1=小y2+ut1 y42=y3+u Substituting into 3) yields: y=o(ov-2 +u-1+ Pv-2+ out Substituting again fory y=(φy3+u12)+φ1+u By-3+u-2+ouI+u, Successive substitutions of this type lead to y=yy+如n1+pun2+pu13+…+p1u1+ c Chris Brooks2002陈磊2004
© Chris Brooks 2002 陈磊2004 7-7 Stochastic Non-stationarity: The Impact of Shocks • To see this, consider the general case of an AR(1) with no drift: yt = yt-1 + ut (3) Let take any value for now. • We can write: yt-1 = yt-2 + ut-1 yt-2 = yt-3 + ut-2 • Substituting into (3) yields: yt = ( yt-2 + ut-1 ) + ut = 2yt-2 + ut-1 + ut • Substituting again for yt-2 : yt = 2 ( yt-3 + ut-2 ) + ut-1 + ut = 3 yt-3 + 2ut-2 + ut-1 + ut • Successive substitutions of this type lead to:* yt = T y0 + ut-1 + 2ut-2 + 3ut-3 + ...+ T-1u1 + ut
The Impact of shocks for 7-8 Stationary and Non-stationary series e have s cases 1.|小<1→y→>0asT>0 So the shocks to the system gradually die away. 2.小=1→y7=1VT So shocks persist in the system and never die away. we obtain: +∑ asT→)0 So just an infinite sum of past shocks plus some starting value ofy 3. >1. Now given shocks become more influential as time goes on, since if o>1,o3>o2>oetc c Chris Brooks2002陈磊2004
© Chris Brooks 2002 陈磊2004 7-8 The Impact of Shocks for Stationary and Non-stationary Series • We have 3 cases: 1. | | <1 T→0 as T→ So the shocks to the system gradually die away. 2. =1 T =1 T So shocks persist in the system and never die away. We obtain: as T→ So just an infinite sum of past shocks plus some starting value of y0 . 3. >1. Now given shocks become more influential as time goes on, since if >1, 3 > 2 > etc. = = + 0 0 i t ut y y
7-9 Detrending a non-stationary series Going back to our 2 characterisations of non-stationarity, the rw, with drift yt =u t y and the trend-stationary process y=a+ Bt+ The two will require different treatments to induce stationarity. The second case is known as deterministic non-stationarity and what is required is detrending去势 The first case is known as stochastic non-stationarity If we let 4t=y-y41=y2-Ly=(1-D)y then 4r=(1-D)y=p+ we have induced stationarity by "differencing once 单位根过程 c Chris Brooks2002陈磊2004
© Chris Brooks 2002 陈磊2004 7-9 Detrending a Non-stationary Series • Going back to our 2 characterisations of non-stationarity, the r.w. with drift: • yt = + yt-1 + ut (1) and the trend-stationary process yt = + t + ut (2) • The two will require different treatments to induce stationarity. The second case is known as deterministic non-stationarity and what is required is detrending去势. • The first case is known as stochastic non-stationarity. If we let yt = yt - yt-1 = yt - L yt =(1-L) yt then yt =(1-L) yt = + ut we have induced stationarity by “differencing once”. 单位根过程
7-10 Detrending a Series Using the Right Method Although trend-stationary and difference-stationary series are bothtrending"over time, the correct approach needs to be used in each case If we first difference the trend-stationary series, it would "remove the non-stationarity but at the expense on introducing an Ma(1)structure into the errors. p372 Conversely if we try to detrend a series which has stochastic trend. then we wil not remove the non stationarity. We will now concentrate on the stochastic non -stationarity model since deterministic non-stationarity does not adequately describe most series in economics or finance. c Chris Brooks2002陈磊2004
© Chris Brooks 2002 陈磊2004 7-10 Detrending a Series: Using the Right Method • Although trend-stationary and difference-stationary series are both “trending” over time, the correct approach needs to be used in each case. • If we first difference the trend-stationary series, it would “remove” the non-stationarity, but at the expense on introducing an MA(1) structure into the errors. p372 • Conversely if we try to detrend a series which has stochastic trend, then we will not remove the nonstationarity. • We will now concentrate on the stochastic non-stationarity model since deterministic non-stationarity does not adequately describe most series in economics or finance