东北财经大学：《金融计量经济学导论》课程PPT教学课件（双语版）Chapter 7 Modelling long-run relationship in finance

7-1 Chapter 7 Modelling long-run relationshil in finance c Chris Brooks2002陈磊2004

© Chris Brooks 2002 陈磊2004 7-1 Chapter 7 Modelling long-run relationship in finance

7-2 1 Stationarity and Unit Root Testing 1. 1Why do we need to test for Non-Stationarity The stationarity or otherwise of a series can strongly influence its behaviour and properties -e.g. persistence of shocks will be infinite for nonstationary series Spurious regressions. If two variables are trending over time, a regression of one on the other could have a high rz even if the two are totally unrelated If the variables in the regression model are not stationary then it can be proved that the standard assumptions for asymptotic analysis will not be valid. In other words, the usual " t-ratios will not follow a t-distribution so we cannot validly undertake hypothesis tests about the regression parameters. c Chris Brooks2002陈磊2004

© Chris Brooks 2002 陈磊2004 7-2 1 Stationarity and Unit Root Testing • The stationarity or otherwise of a series can strongly influence its behaviour and properties - e.g. persistence of shocks will be infinite for nonstationary series • Spurious regressions. If two variables are trending over time, a regression of one on the other could have a high R2 even if the two are totally unrelated • If the variables in the regression model are not stationary, then it can be proved that the standard assumptions for asymptotic analysis will not be valid. In other words, the usual “t-ratios” will not follow a t-distribution, so we cannot validly undertake hypothesis tests about the regression parameters. 1.1Why do we need to test for Non-Stationarity?

Value of rl for 1000 Sets of Regressions of a Non- 7-3 stationary variable on another Independent non stationary Variable 200 160 120 80 0.25 0.50 0.75 c Chris Brooks2002陈磊2004

© Chris Brooks 2002 陈磊2004 7-3 Value of R2 for 1000 Sets of Regressions of a Non￾stationary Variable on another Independent Non￾stationary Variable

Value of t-ratio on Slope Coefficient for 1000 Sets 7-4 of regressions of a Non-stationary Variable on another Independent Non-stationary Variable 120 100 80 60 20 750-500-250 250500了50 c Chris Brooks2002陈磊2004

© Chris Brooks 2002 陈磊2004 7-4 Value of t-ratio on Slope Coefficient for 1000 Sets of Regressions of a Non-stationary Variable on another Independent Non-stationary Variable

7-5 1.2 Two types of Non-Stationarity Various definitions of non-stationarity exist In this chapter, we are really referring to the weak form or covariance stationarity There are two models which have been frequently used to characterise non-stationarity: the random walk model with drift: y=+y1+ and the deterministic trend process y,=a+Bt+ut where u is iid in both cases c Chris Brooks2002陈磊2004

© Chris Brooks 2002 陈磊2004 7-5 1.2 Two types of Non-Stationarity • Various definitions of non-stationarity exist • In this chapter, we are really referring to the weak form or covariance stationarity • There are two models which have been frequently used to characterise non-stationarity: the random walk model with drift: yt =  + yt-1 + ut (1) and the deterministic trend process: yt =  + t + ut (2) where ut is iid in both cases

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. 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 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