7-11 Sample plots for various Stochastic Processes :a White noise process 3 c Chris Brooks2002陈磊2004
© Chris Brooks 2002 陈磊2004 7-11 Sample Plots for various Stochastic Processes: A White Noise Process -4 -3 -2 -1 0 1 2 3 4 1 40 79 118 157 196 235 274 313 352 391 430 469
7-12 Random walk and a random walk with drift Random walk Random Walk with Drift 分的m235656他 10 c Chris Brooks2002陈磊2004
© Chris Brooks 2002 陈磊2004 7-12 Random Walk and a Random Walk with Drift -20 -10 0 10 20 30 40 50 60 70 1 19 37 55 73 91 109 127 145 163 181 199 217 235 253 271 289 307 325 343 361 379 397 415 433 451 469 487 Random Walk Random Walk with Drift
7-13 A Deterministic Trend process 30 25 20 15 14079118157196235274313352391430469 c Chris Brooks2002陈磊2004
© Chris Brooks 2002 陈磊2004 7-13 A Deterministic Trend Process -5 0 5 10 15 20 25 30 1 40 79 118 157 196 235 274 313 352 391 430 469
Autoregressive Processes with 7-14 differing values of o(o, 0.8, 1) Phi= 1 Phi=0.8 Phi=0 5310515720926131336541745215735679w838597989 -15 c Chris Brooks2002陈磊2004
© Chris Brooks 2002 陈磊2004 7-14 Autoregressive Processes with differing values of (0, 0.8, 1) -20 -15 -10 -5 0 5 10 15 1 53 105 157 209 261 313 365 417 469 521 573 625 677 729 781 833 885 937 989 Phi=1 Phi=0.8 Phi=0
7-15 1.3 Definition of Non-Stationarity Consider again the simplest stochastic trend model: y=y-1+u or Ayu We can generalise this concept to consider the case where the series contains more than one unit root. That is we would need to apply the first difference operator, 4, more than once to induce stationarity. Definition If a non-stationary series, y, must be differenced d times before it becomes stationary, then it is said to be integrated of order d we write y, (d) So ify -I(d then Ay(0) An i(O)series is a stationary series An i(1) series contains one unit root, e.g. y,=v1+ut c Chris Brooks2002陈磊2004
© Chris Brooks 2002 陈磊2004 7-15 1.3 Definition of Non-Stationarity • Consider again the simplest stochastic trend model: yt = yt-1 + ut or yt = ut We can generalise this concept to consider the case where the series contains more than one “unit root”. That is, we would need to apply the first difference operator, , more than once to induce stationarity. Definition If a non-stationary series, yt must be differenced d times before it becomes stationary, then it is said to be integrated of order d. We write yt I(d). So if yt I(d) then dyt I(0). An I(0) series is a stationary series An I(1) series contains one unit root, e.g. yt = yt-1 + ut