Deterministic or Stochastic Trend? - Hedibert

1 Deterministic or Stochastic Trend? Let us consider two of the simplest versions: Deterministic trend (DT) :yt= t+ tStochastic trend (ST) :yt= +yt 1+ t,where tis white noise with variance 2(= 1, for simplicity) andy0= 0 (also for simplicity).It is easy to see thatEDT(yt) =EST(yt) = tbutVDT(yt) = 1andVDT(yt) = with respect to all information up to timet= DT and ST time seriesTime050100150200020406080100120y(t ) = *t + rby(t) = + y(t 1) + rb2 How to modely1tandy2t?Even withn= 100 one can argue that the trend of{y2t} looks more Deterministic than the trend of{y1t}.Time050100150200020406080100Y1Y2 3 Modely1tandy2twith Deterministic trendsEven after removing a determinist trend fromy1t, the residuals stillbehave like a random walk. On the other hand,y2tis y1 with DTTimey1050100150200020406080 TimeResiduals050100150200 6 4 2024 Noise doesn't look +FACP => random ACFM odeling y2 with DTTimey2050100150200020406080100 TimeResiduals050100150200 2 1012 Noise looks +FACP => white noise5101520 ACF4 Modely1tandy2twith Stochastic trendsAfter fitting a random walk plus drift fory1t, the residuals behavelike a white noise, : random walk + driftTimey1050100150200020406080 TimeResiduals050100150200 2 10123 Noise looks white05101520 +FACP => white noise5101520 ACFy2.

2 Random walk + driftTimey1050100150200020406080100 TimeResiduals050100150200 2024 Noise doesn't look white05101520 +FACP => not white noise5101520 ACFF itting a random walk plus drift fory2t(which is trend-stationary),induces an MA(1) behavior in the trend stationary,yt= t+ tthenyt 1= (t 1) + t 1and yt= +vtwherevt= t t 1, such thatE(vt) = 0,V(vt) = 2 andCov(vt,vt 1) =Cov( t t 1, t 1 t 2) = V( t) = 1andCov(vt,vt h) = 0, forh>1. Therefore, the 1st orderautocorrelation is (1) =Cov(vt,vt 1)V(vt)= trend-stationary:IStochastic trend fit: residuals with MA(1) trend fit: residuals are white difference-stationary:IStochastic trend fit: residuals are white trend fit: residuals are random : ALWAYS check the residuals!7