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Table 1: Maximum relative asymptotic size losses over c {0, 1,..., 20} = 0 = 0.5 = S IR4 IR4 S IR4 IR4 S IR4 IR-4 0.18 0.00 0.07 0.00 0.27 0.03 0.00 0.81 0.-2 0.39 0.00 0.24 0.00 0.18 0.16 0.00 0.82 0.-1 0.57 0.00 0.29 0.00 0.10 0.22 0.00 0.72 0.-0.5 0.66 0.00 0.33 0.00 0.05 0.19 0.00 0.68 0.0 0.69 0.00 0.35 0.01 0.00 0.14 0.00 0.64 0.0.5 0.65 0.00 0.33 0.10 0.00 0.17 0.00 0.60 0.1 0.58 0.00 0.30 0.19 0.00 0.18 0.00 0.56 0.2 0.38 0.00 0.24 0.28 0.00 0.10 0.00 0.45 0.4 0.18 0.00 0.08 0.12 0.02 0.00 0.09 0.43 0.Table 2: Maximum relative asymptotic size gains over c {0, 1,..., 20} = 0 = 0.5 = S IR4 IR4 S IR4 IR4 S IR4 IR-4 0.00 0.15 0.09 0.21 0.00 0.19 0.45 0.00 0.-2 0.00 0.38 0.23 0.16 0.00 0.02 0.45 0.00 0.-1 0.00 0.47 0.32 0.18 0.10 0.00 0.42 0.00 0.-0.5 0.00 0.51 0.34 0.16 0.11 0.00 0.40 0.00 0.0 0.00 0.52 0.35 0.12 0.12 0.00 0.39 0.00 0.0.5 0.00 0.51 0.34 0.06 0.15 0.00 0.37 0.00 0.1 0.00 0.48 0.32 0.00 0.16 0.01 0.36 0.00 0.2 0.00 0.38 0.23 0.00 0.22 0.14 0.31 0.00 0.4 0.00 0.15 0.08 0.00 0.09 0.10 0.23 0.00 0.Table 3: Asymptotic critical values for Qi and Si, i = , at the significance level = 0.10 = 0.05 = 0.Q 6.93 8.04 10.Q 9.04 10.28 12.S 0.348 0.461 0.S 0.120 0.148 0.Table 4: Asymptotic critical values for s at the significance level = 0.10 = 0.05 = 0.-0.168 0.069 0.Table 5: Asymptotic scaling constants for intersection of rejections testing strategies at the significance level = 0.10 = 0.05 = 0.IR(Q, Q ) m 0.801 0.793 0.IR(Q, S) m 0.845 0.851 0. IR(Q, S ) m 0.897 0.894 0. IR(Q, Q, S, S ) m 0.571 0.551 0. IR(S, S ) m 0.576 0.554 0. 0.852 0.840 0.1.0 1.Q Q Q Q 0.0.S S S S 0.0.0.0.0.0.0.0.0 5 10 15 0 5 10 15 c c (a) = 0 (b) = 0.1.0 1.Q Q Q Q 0.8 0.S S S S 0.6 0.0.4 0.0.2 0.0.0 0.0 5 10 15 20 0 5 10 15 c c (c) = 1 (d) = -1/Figure 1: Asymptotic size and power for differen values of, T = T 1.0 1.Q Q Q Q UR UR 0.8 0. UR t UR t mmUR t UR t UR Dan J UR Dan J 0.6 0.0.4 0.0.2 0.0.0 0.0 5 10 15 20 0 5 10 15 c c (a) = 0 (b) = 1.0 1.Q Q Q Q UR 0.8 0.8 UR UR t UR t mmUR t UR t UR Dan J UR Dan J 0.6 0.0.4 0.0.2 0.0.0 0.0 5 10 15 20 0 5 10 15 c c (c) = 2 (d) = -1/Figure 2: , T = T 1.0 1.Q Q S S 0.8 0.IR IR IR s IR s 0.6 0.0.4 0.0.2 0.0.0 0.0 1 2 3 4 5 6 0 1 2 3 4 5 (a) c = 2.5, mean case (b) c = 5, trend case 1.0 1.Q Q S S 0.8 0.IR IR IR s IR s 0.6 0.0.4 0.0.2 0.0.0 0.0 1 2 3 4 5 6 0 1 2 3 4 5 (c) c = 5, mean case (d) c = 10, trend case 0.10 0.Q Q 0.25 S S 0.IR IR IR s IR s 0.0.0.0.0.0.0.0.00 0.0 1 2 3 4 5 6 0 1 2 3 4 5 (e) c = 10, mean case (f) c = 15, trend case Figure 3: Asymptotic size for different initial values 1.0 1.0 1.S S S 0.8 IR4 0.8 IR4 0.8 IRIR 4 IR 4 IR 0.6 0.6 0.0.4 0.4 0.0.2 0.2 0.0.0 0.0 0.0 5 10 15 20 0 5 10 15 20 0 5 10 15 c c c (a) = -4 (b) = -2 (c) = -1.0 1.0 1.S S S 0.8 IR4 0.8 IR4 0.8 IRIR 4 IR 4 IR 0.6 0.6 0.0.4 0.4 0.0.2 0.2 0.0.0 0.0 0.0 5 10 15 20 0 5 10 15 20 0 5 10 15 c c c (d) = -0.5 (e) = 0 (f) = 0.1.0 1.0 1.S S S 0.8 IR4 0.8 IR4 0.8 IRIR 4 IR 4 IR 0.6 0.6 0.0.4 0.4 0.0.2 0.2 0.0.0 0.0 0.0 5 10 15 20 0 5 10 15 20 0 5 10 15 c c c (g) = 1 (h) = 2 (i) = Figure 4: Asymptotic size and power for stationarity tests, = 1.0 1.0 1.S S S 0.8 0.8 0.IR4 IR4 IRIR 4 IR 4 IR 0.6 0.6 0.0.4 0.4 0.0.2 0.2 0.0.0 0.0 0.0 5 10 15 20 0 5 10 15 20 0 5 10 15 c c c (a) = -4 (b) = -2 (c) = -1.0 1.0 1.S S S 0.8 0.8 0.IRIR4 IRIR IR 4 IR 0.6 0.6 0.0.4 0.4 0.0.2 0.2 0.0.0 0.0 0.0 5 10 15 20 0 5 10 15 20 0 5 10 15 c c c (d) = -0.5 (e) = 0 (f) = 0.1.0 1.0 1.S S S 0.8 0.8 0.IRIR4 IRIR IR 4 IR 0.6 0.6 0.0.4 0.4 0.0.2 0.2 0.0.0 0.0 0.0 5 10 15 20 0 5 10 15 20 0 5 10 15 c c c (g) = 1 (h) = 2 (i) = Figure 5: Asymptotic size and power for stationarity tests, = 0.1.0 1.0 1.S S S 0.8 0.8 0.IRIR4 IRIR IR 4 IR 0.6 0.6 0.0.4 0.4 0.0.2 0.2 0.0.0 0.0 0.0 5 10 15 20 0 5 10 15 20 0 5 10 15 c c c (a) = -4 (b) = -2 (c) = -1.0 1.0 1.S S S 0.8 0.8 0.IR4 IR4 IRIR 4 IR 4 IR 0.6 0.6 0.0.4 0.4 0.0.2 0.2 0.0.0 0.0 0.0 5 10 15 20 0 5 10 15 20 0 5 10 15 c c c (d) = -0.5 (e) = 0 (f) = 0.1.0 1.0 1.S S S 0.8 IR4 0.8 0.IR4 IRIR 4 IR 4 IR 0.6 0.6 0.0.4 0.4 0.0.2 0.2 0.0.0 0.0 0.0 5 10 15 20 0 5 10 15 20 0 5 10 15 c c c (g) = 1 (h) = 2 (i) = Figure 6: Asymptotic size and power for stationarity tests, =

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