Midterm_Review (1)

E test yt gyt 1 wnt where g b 1 yt test

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: = b*y(t-1) - y(t-1) + wn(t) or ∆ y(t) = (b -1)*y(t-1) + wn(t) or so we could regress ∆ y(t) on y(t-1) and so test the coefficient for y(t-1), i.e. test ∆ y(t) = g*y(t-1) + wn(t), where g = (b-1) y(t) test null: (b-1) = 0 against (b-1)< 0 21 21 Unit Roots i.e. test b=1 against b<1 This would be a simple t-test except for a This problem. As b gets closer to one, the distribution of (b-1) is no longer distributed as Student’s t distribution as Dickey and Fuller simulated many time Dickey series with b=0.99, for example, and looked at the distribution of the estimated coefficient g ˆ 22 22 III. Lab Five 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 1.05 -0.66 0.32 =0.71 31 31 32 32 33 33 34 34 Unit Roots Dickey Dickey and Fuller tabulated these simulated results into Tables simulated In specifying Dickey-Fuller tests there are In three formats: no constant-no trend, constant-no trend, and constant-trend, and three sets of tables. three 35 35 Unit Roots Example: Example: the price of gold, weekly data, January 1992 through December 1999 January 36 36 450 Trace of the Weekly Closing Price of Gold 400 350 300 250 1/06/92 12/06/93 11/06/95 10/06/97 9/06/99 37 37 60 Series: GOLD Sample 1/06/1992 12/27/1999 Obs erv ations 417 50 Mean Median Max imum Minimum Std. Dev . Skewnes s Kurtosis 40 30 20 10 J arque-Bera Probability 3 45.7320 3 52.5000 4 14.5000 2 53.8000 4 1.31026 - 0.514916 1 .999475 3 5.82037 0 .000000 0 260 280 300 320 340 360 380 400 38 38 39 39 40 40 41 41 42 42 Simulated Student's t-Distribution, 415 Degrees of Freedom 0.5 TDENS 0.4 0.3 0.2 0.1 5% 0.0 -4 -2 ­ 1.65 0 2 4 TVAR 43 43 Dickey-Fuller Tests The The price of gold might vary around a “constant”, for example the marginal cost of production of PG(t) = MCG + RW(t) = MCG + WN(t)/[1-Z] RW(t) WN(t)/[1-Z] PG(t) - MCG = RW(t) = WN(t)/[1-Z] RW(t) [1-Z][PG(t) - MCG ] = WN(t) WN(t) [PG(t) - MCG ] - [PG(t-1) - MCG ] = WN(t) [P WN(t) [PG(t) - MCG ] = [PG(t-1) - MCG ] + WN(t) [P WN(t) 44 44 Dickey-Fuller Tests Or: [PG(t) - MCG ] = b* [PG(t-1) - MCG ] + WN(t) b* WN(t) PG(t) = MCG + b* PG(t-1) - b*MCG + WN(t) PG(t) = (1-b)*MCG + b* PG(t-1) + WN(t) (t-1) subtract PG(t-1) subtract (t-1) PG(t) - PG(t-1) = (1-b)*MCG + b* PG(t-1) - PG(t(t-1) 1) + WN(t) or ∆ PG(t) = (1-b)*MCG + (1-b)* PG(t-1) + WN(t) or Now there is an intercept as well as a slope 45 45 46 46 47 47 Augmented Dickey- Fuller Tests 48 48 ARTWO’s and Unit Roots Recall the edge of the triangle of stability: b2 = 1 – b1 , so for stability b1 + b2 < 1 so x(t) = b1 x(t-1) + b2 x(t-2) + wn(t) x(t-1) x(t-2) Subtract x(t-1) from both sides x(t) – x(-1) = (b1 – 1)x(t-1) + b2 x(t-2) + wn(t) 1)x(t-1) x(t-2) Add and subtract b2 x(t-1) from the right side: x(t) – x(-1) = (b1 + b2 - 1) x(t-1) - b2 [x(t-1) - x(t-2)] + 1) wn(t) wn(t) Null hypothesis: (b1 + b2 - 1) = 0 1) Alternative hypothesis: (b1 + b2 -1)<0 -1)<0 49 49 Lab Five 50 50 2005 Midterm 51 51 52 52 Fig. 2.1: Debt Service Payments as a Percent of Disposable Personal Income 16 14 12 2004.3 Percent 10 1993.1 8 6 4 2 0 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 Quarter 53 53 54 54 55 55 56 56 57 57 58 58 59 59 60 60 61 61 F ig ur e 4 .1 M o nt hly C hang e in C ap aci t y U t il iz at io n, T o t al ind ust r y, 19 6 7.0 2 - 2 0 0 5.0 3 2 1 0 -1 -2 -3 -4 D at e 62 62 63 63 64 64 65 65 66 66 67 67 68 68 69 69 70 70 71 71...
View Full Document

Ask a homework question - tutors are online