Midterm_Review (1)

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

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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...
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