Transcription of Deterministic or Stochastic Trend? - Hedibert
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Deterministic or Stochastic Trend? - Hedibert
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 : random walk + driftTimey1050100150200020406080100 TimeResiduals050100150200 2024 Noise doesn't look white05101520
2t with deterministic trends Even after removing a determinist trend from y 1t, the residuals still behave like a random walk. On the other hand, y 2t is de nitely trend-stationary. Modeling y1 with DT Time y1 0 50 100 150 200 0 20 40 60 80 Time Residuals 0 50 100 150 200-6-4-2 0 2 4 Noise doesn't look white 0 5 10 15 20 0.0 0.2 0.4 0.6 0.8 1.0 ...
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