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Midterm_Review (1)

# One for example starting at artwo0 at for time zero

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Unformatted text preview: of one period. one For example, starting at ARTWO(0) at For time zero, ARTWO(2) = b2 ARTWO(0) , time ignoring the white noise shocks, and ARTWO(4) = b2 ARTWO(2) = ARTWO(4) b 2 ARTWO(0) and for stability -1<b <1 9 Triangle of Stable Parameter Triangle Space Space 1 b2 b1 = 0 ­1 10 10 Triangle of Stable Parameter Triangle Space: Boundary points 1 b2 ­1 b1 = 0 b2 = 1 – b1 If b1 = 0, b2 = 1, If b1 = 1, b2 =0, If b2 = ­1, b1 =2 +1 ­1 Draw a horizontal line through (0, ­1) for (b1, b2) 11 11 Triangle of Stable Parameter Triangle Space: (0, 1) b2 (­1, 0) (1, 0) b1 = 0 (0, ­1) Draw a line from the vertex, for (b1=0, b2=1), though the end points for b1, i.e. through (b1=1, b2= 0) and (b1=­1, b2=0), 12 12 Triangle of Stable Parameter Triangle Space: (0,1) b2 = 1 ­ b1 b2 (­1, 0) (1, 0) b1 = 0 ­1 (2, ­1) Note: along the boundary, when b1 = 0, b2 = 1, when b1 = 1, b2 = 0, and when b2 = ­1, b2 = 2. 13 13 Triangle of Stable Parameter Triangle Space: (0, 1) b2 (­1, 0) b1<0 b2>0 b1<0 b2<0 b1>0 b2>0 b1 = 0 (1, 0) b1>0 b2<0 (0, ­1) 14 14 Is the behavior different in each Is Quadrant? (0, 1) II (­1, 0) b1<0 b2>0 b1<0 b2<0 III b2 b1>0 b2>0 I b1 = 0 (0, ­1) (1, 0) b1>0 b2<0 IV 15 15 We could study with simulation 16 16 Is the behavior different in each Is Quadrant? (0, 1) II (­1, 0) b1= ­0.3 b2= 0.3 b1= ­0.3 b2= ­0.3 III b2 b1= 0.3 I b2= 0.3 b1 = 0 (0, ­1) (1, 0) b1= 0.3 b2= ­0.3 IV 17 17 Simulation Sample 1 1000 Sample 1 1000 Genr wn=nrnd Sample 1 2 Genr artwo =wn Sample 3 1000 Genr artwo = 0.3*artwo(-1)+0.3*artwo(-2) + Genr wn wn Sample 1 1000 18 18 II. Augmented Dickey-Fuller Tests 19 19 Part II: Unit Roots First First Order Autoregressive or RandomWalk? RandomWalk? y(t) = b*y(t-1) + wn(t) y(t) = y(t-1) + wn(t) 20 20 Unit Roots y(t) = b*y(t-1) + wn(t) we could test the null: b=1 against b<1 instead, subtract y(t-1) from both sides: y(t) - y(t-1)...
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