Empirical Size of Asymptotic and Bootstrap Tests

Table B-1: Empirical size of asymptotic and bootstrap tests, T ( , ) Te st =1 M o del 1: =c 200 (0 ,0 ) (0 .5 ,0 ) (0 ,0 .5 ) 400 (0 ,0 ) (0 .5 ,0 ) (0 ,0 .5 ) [m = 0, 5% nominal size] = 0:5 M o del 2: M o del 3: c M o del 1: = 0:7 M o del 2: M o del 3: c M o del 1: M o del 2: M o del 3: c 1 1=3 3 1=3 3 0 10 1 1=3 3 1=3 3 0 10 1 1=3 3 1=3 3 0 10 H1 .0 6 .3 9 .2 1 .2 6 .1 1 .4 9 .3 2 .0 5 .1 5 .1 4 .1 0 .0 8 .2 8 .1 5 .0 7 .1 8 .1 5 .1 4 .1 0 .3 1 .1 9 H2 .0 4 .3 3 .3 5 .0 7 .1 4 .4 1 .3 8 .0 3 .1 5 .1 4 .0 7 .0 5 .2 3 .1 5 .0 5 .2 1 .1 9 .0 9 .0 7 .2 5 .2 0 Hmax .0 5 .4 4 .3 7 .1 7 .1 6 .5 6 .4 6 .0 4 .1 9 .1 6 .1 0 .0 8 .3 1 .1 7 .0 6 .2 4 .1 9 .1 4 .0 9 .3 4 .2 2 H1 .0 6 .0 6 .0 4 .0 6 .0 3 .0 6 .0 6 .0 5 .0 3 .0 4 .0 4 .0 4 .0 3 .0 3 .0 8 .0 6 .0 4 .0 7 .0 5 .0 4 .0 3 H2 .0 6 .0 5 .0 6 .0 6 .0 6 .0 5 .0 6 .0 4 .0 2 .0 1 .0 5 .0 3 .0 3 .0 2 .0 7 .0 3 .0 3 .0 6 .0 4 .0 3 .0 4 Hmax .0 6 .0 5 .0 5 .0 6 .0 5 .0 5 .0 5 .0 5 .0 3 .0 2 .0 4 .0 3 .0 3 .0 3 .0 8 .0 5 .0 3 .0 6 .0 4 .0 4 .0 3 H1 .0 6 .4 2 .2 4 .2 7 .1 2 .5 1 .3 3 .0 7 .1 8 .1 6 .1 3 .1 1 .3 1 .1 7 .0 9 .2 2 .1 8 .1 6 .1 3 .3 1 .1 9 H2 .0 5 .3 3 .3 6 .0 7 .1 6 .4 4 .3 5 .0 3 .1 9 .1 9 .0 5 .0 7 .2 3 .1 8 .0 5 .2 4 .2 2 .0 9 .0 9 .2 6 .2 1 Hmax .0 6 .4 6 .3 9 .1 7 .1 8 .5 8 .4 5 .0 5 .2 3 .2 0 .1 1 .1 0 .3 3 .2 0 .0 8 .2 9 .2 4 .1 4 .1 2 .3 6 .2 4 H1 .0 6 .0 6 .0 5 .0 6 .0 5 .0 5 .0 6 .0 8 .0 4 .0 4 .0 5 .0 5 .0 4 .0 4 .1 0 .0 5 .0 5 .0 6 .0 6 .0 4 .0 4 H2 .0 8 .0 7 .0 7 .0 5 .0 7 .0 6 .0 6 .0 4 .0 2 .0 3 .0 4 .0 3 .0 3 .0 3 .0 7 .0 4 .0 4 .0 7 .0 5 .0 4 .0 3 Hmax .0 7 .0 7 .0 7 .0 6 .0 7 .0 6 .0 6 .0 7 .0 4 .0 2 .0 4 .0 4 .0 3 .0 3 .0 9 .0 4 .0 4 .0 6 .0 4 .0 4 .0 4 H1 .0 5 .3 5 .2 3 .2 1 .1 2 .4 8 .3 2 .0 3 .1 2 .1 4 .0 8 .0 7 .2 7 .1 3 .0 9 .2 0 .1 8 .1 6 .1 2 .3 1 .1 9 H2 .0 5 .2 6 .2 9 .0 6 .1 0 .3 7 .3 1 .0 1 .0 7 .0 8 .0 2 .0 2 .1 7 .1 2 .0 4 .1 6 .1 5 .0 7 .0 7 .2 3 .1 5 Hmax .0 6 .3 9 .3 2 .1 4 .1 2 .5 3 .3 9 .0 3 .1 3 .1 4 .0 7 .0 7 .2 7 .1 5 .0 8 .2 4 .2 3 .1 4 .1 1 .3 3 .2 1 H1 .0 8 .0 6 .0 5 .0 6 .0 4 .0 5 .0 6 .0 4 .0 2 .0 3 .0 2 .0 3 .0 2 .0 2 .1 0 .0 4 .0 6 .0 5 .0 5 .0 4 .0 5 H2 .0 6 .0 5 .0 4 .0 6 .0 5 .0 6 .0 4 .0 1 .0 1 .0 1 .0 2 .0 1 .0 2 .0 1 .0 6 .0 2 .0 2 .0 5 .0 3 .0 2 .0 3 Hmax .0 6 .0 6 .0 4 .0 6 .0 4 .0 5 .0 5 .0 3 .0 1 .0 2 .0 2 .0 2 .0 2 .0 2 .0 9 .0 3 .0 3 .0 5 .0 4 .0 2 .0 3 H1 .0 5 .4 0 .2 4 .2 5 .1 3 .5 3 .3 5 .0 6 .1 8 .1 7 .1 4 .1 0 .3 2 .2 2 .0 7 .2 0 .1 8 .1 4 .1 2 .3 5 .2 2 H2 .0 5 .3 8 .4 4 .0 9 .1 8 .4 5 .4 2 .0 6 .2 1 .2 0 .0 9 .0 8 .3 0 .2 5 .0 6 .2 4 .2 5 .1 1 .1 0 .3 2 .2 6 Hmax .0 5 .5 0 .4 5 .1 9 .2 1 .6 1 .5 1 .0 6 .2 2 .2 2 .1 4 .1 1 .3 6 .2 4 .0 7 .2 5 .2 4 .1 5 .1 3 .4 0 .2 7 H1 .0 4 .0 5 .0 6 .0 4 .0 4 .0 5 .0 5 .0 7 .0 5 .0 5 .0 6 .0 5 .0 4 .0 6 .0 8 .0 6 .0 5 .0 7 .0 5 .0 5 .0 6 H2 .0 5 .0 5 .0 6 .0 5 .0 5 .0 5 .0 5 .0 7 .0 3 .0 3 .0 5 .0 4 .0 3 .0 4 .0 8 .0 4 .0 3 .0 7 .0 4 .0 4 .0 5 Hmax .0 4 .0 4 .0 5 .0 4 .0 5 .0 5 .0 5 .0 6 .0 5 .0 5 .0 5 .0 4 .0 4 .0 4 .0 8 .0 6 .0 4 .0 8 .0 5 .0 4 .0 6 H1 .0 5 .4 1 .2 5 .2 6 .1 3 .5 3 .3 6 .0 7 .2 2 .2 0 .1 6 .1 3 .3 4 .2 2 .0 9 .2 2 .2 2 .1 8 .1 5 .3 7 .2 3 H2 .0 6 .3 8 .4 2 .0 9 .1 7 .4 5 .4 1 .0 7 .2 5 .2 7 .1 3 .1 2 .3 3 .2 7 .0 9 .3 0 .3 0 .1 4 .1 3 .3 5 .3 0 Hmax .0 6 .4 9 .4 5 .1 8 .1 9 .6 1 .5 0 .0 8 .2 6 .2 6 .1 7 .1 5 .3 9 .2 7 .1 0 .2 8 .2 9 .2 0 .1 8 .4 2 .3 0 H1 .0 5 .0 4 .0 5 .0 4 .0 5 .0 4 .0 6 .0 7 .0 6 .0 7 .0 8 .0 7 .0 5 .0 7 .1 0 .0 8 .0 7 .0 9 .0 8 .0 5 .0 7 H2 .0 5 .0 6 .0 5 .0 6 .0 5 .0 5 .0 5 .0 9 .0 4 .0 4 .0 9 .0 5 .0 5 .0 5 .1 1 .0 6 .0 5 .1 0 .0 6 .0 4 .0 6 Hmax .0 5 .0 5 .0 5 .0 5 .0 5 .0 4 .0 5 .0 8 .0 6 .0 6 .0 8 .0 7 .0 4 .0 6 .1 0 .0 8 .0 6 .0 9 .0 7 .0 4 .0 6 H1 .0 6 .3 7 .2 5 .2 3 .1 4 .5 3 .3 4 .0 4 .1 7 .1 5 .1 1 .0 9 .3 2 .2 0 .1 4 .2 8 .2 6 .2 1 .1 7 .3 8 .2 9 H2 .0 6 .3 7 .3 7 .0 9 .1 5 .4 4 .3 8 .0 4 .1 8 .1 6 .0 8 .0 7 .2 9 .2 2 .1 4 .3 3 .3 0 .1 8 .1 7 .3 5 .3 2 Hmax .0 6 .4 8 .4 0 .1 8 .1 7 .6 0 .4 7 .0 5 .1 9 .1 8 .1 2 .1 1 .3 5 .2 3 .1 6 .3 6 .3 4 .2 3 .2 1 .4 2 .3 5 H1 .0 6 .0 5 .0 7 .0 4 .0 7 .0 5 .0 6 .0 5 .0 6 .0 4 .0 5 .0 4 .0 4 .0 4 .1 5 .1 0 .0 9 .1 0 .1 0 .0 7 .0 8 H2 .0 6 .0 5 .0 5 .0 5 .0 5 .0 5 .0 4 .0 5 .0 2 .0 2 .0 4 .0 3 .0 3 .0 2 .1 5 .0 5 .0 5 .1 2 .0 8 .0 5 .0 6 Hmax .0 6 .0 5 .0 5 .0 4 .0 5 .0 4 .0 4 .0 5 .0 5 .0 4 .0 5 .0 3 .0 4 .0 3 .1 6 .0 9 .0 9 .1 1 .1 0 .0 5 .0 7 Table B-2: Size-adjusted power of bootstrap tests, [m = 1, T DGP Test =c Model 1: 1 1=3 Model 2: 3 1=3 3 Model 3: c 0 10 = Model 1: 1 1=3 1 2 3 1=3 1 2 0 10 .97 .03 .84 .16 .96 .08 .35 .61 .04 .51 .04 .68 .04 .08 H2 .60 .03 .17 .15 .36 .04 .08 .16 .05 .12 .05 .14 .03 .04 Hmax .93 .03 .63 .11 .89 .04 .15 .41 .04 .27 .03 .47 .03 .05 H1 .97 .04 .84 .28 .95 .26 .53 .61 .04 .46 .06 .65 .10 .22 H2 .57 .05 .21 .25 .45 .10 .22 .16 .05 .12 .07 .18 .06 .08 Hmax .91 .05 .67 .23 .89 .19 .38 .43 .04 .30 .05 .48 .08 .13 H1 .99 .77 .22 .96 .69 .14 .56 .85 .45 .15 .70 .29 .07 .25 H2 .69 .29 .03 .74 .12 .07 .12 .28 .17 .04 .38 .06 .05 .06 Hmax .98 .68 .06 .94 .37 .08 .28 .73 .37 .05 .66 .11 .05 .10 H1 1.0 .77 .30 .94 .76 .32 .71 .85 .41 .21 .62 .39 .18 .42 H2 .71 .35 .06 .71 .23 .16 .31 .30 .18 .05 .36 .10 .09 .16 Hmax .98 .69 .10 .93 .53 .24 .58 .74 .35 .07 .57 .17 .13 .29 H1 1.0 .05 .86 .69 .99 .18 .65 1.0 .83 .33 .99 .93 .24 .75 = 0:2; 2 = 0:9; = 0:7 1 = 0:9; 2 = 0:2; = 0:7 H2 .93 .03 .19 .59 .67 .06 .15 .94 .30 .05 .90 .45 .08 .20 Hmax .99 .04 .76 .62 .98 .12 .41 1.0 .79 .11 .98 .81 .16 .53 H1 1.0 .06 .76 .69 .96 .31 .71 1.0 .78 .33 .95 .89 .35 .77 H2 .88 .04 .23 .60 .71 .15 .41 .93 .32 .06 .82 .59 .18 .46 Hmax .99 .03 .70 .67 .95 .24 .61 1.0 .72 .17 .95 .83 .28 .69 = 0:7 H1 1.0 .07 1.0 .94 1.0 .41 .84 1.0 .07 .96 .24 .99 .13 .49 H2 1.0 .03 .65 .75 .95 .12 .26 .75 .04 .44 .16 .64 .07 .12 Hmax 1.0 .03 1.0 .92 1.0 .27 .63 .99 .05 .88 .20 .97 .08 .24 H1 1.0 .07 1.0 .95 1.0 .51 .84 1.0 .05 .95 .33 .99 .31 .62 H2 .99 .05 .69 .78 .96 .32 .57 .69 .04 .45 .23 .67 .15 .28 Hmax 1.0 .06 1.0 .93 1.0 .42 .75 .99 .05 .87 .26 .98 .22 .43 H1 1.0 1.0 .44 1.0 .99 .45 .91 1.0 .94 .21 1.0 .74 .25 .70 H2 1.0 .74 .05 .99 .49 .14 .34 .84 .57 .04 .93 .14 .10 .17 Hmax 1.0 1.0 .13 1.0 .97 .32 .79 1.0 .90 .06 .99 .45 .17 .40 H1 1.0 1.0 .53 1.0 .99 .56 .92 1.0 .93 .31 .99 .78 .40 .78 H2 1.0 .78 .06 .99 .67 .39 .67 .83 .59 .05 .91 .24 .23 .41 Hmax 1.0 1.0 .20 1.0 .96 .48 .84 1.0 .90 .11 .99 .54 .29 .64 H1 1.0 .39 1.0 1.0 1.0 .45 .95 1.0 1.0 .83 1.0 1.0 .50 .97 1 3 3 = 0:7 = 0:5 400 Model 3: c H1 1 3 = 0:5, 5%] Model 2: = 0:5 200 0 1 = 0, = 0:2; 2 = 0:9; = 0:7 1 = 0:9; 2 = 0:2; = 0:7 H2 1.0 .09 .91 .99 1.0 .20 .62 1.0 .95 .15 1.0 .98 .24 .69 Hmax 1.0 .15 1.0 1.0 1.0 .39 .87 1.0 1.0 .42 1.0 1.0 .43 .94 H1 1.0 .45 1.0 1.0 1.0 .54 .93 1.0 1.0 .83 1.0 1.0 .56 .95 H2 1.0 .17 .91 .99 1.0 .42 .82 1.0 .92 .24 1.0 .98 .44 .84 Hmax 1.0 .27 1.0 1.0 1.0 .49 .90 1.0 1.0 .57 1.0 1.0 .52 .92 B-6 Table B-3: Size-adjusted power of bootstrap tests under di¤erent change points and volatility intensity, [Model 1, DGP 1, m = 1, = = 0, = 0:5] T 200 0 1 0.2 0.3 0.5 0.6 0.7 0.9 400 0.2 0.3 0.5 0.6 0.7 0.9 Test Model 1: 1/5 1/3 1/2.5 1/1.5 1/1.1 H1 .05 .05 .04 .07 .28 H2 .04 .04 .04 .05 .09 Hmax .05 .04 .05 .06 .20 H1 .05 .04 .05 .15 .61 H2 .04 .04 .04 .08 .20 Hmax .04 .05 .04 .11 .46 H1 .04 .05 .06 .56 .94 H2 .04 .04 .04 .24 .55 Hmax .05 .04 .05 .44 .89 H1 .08 .17 .27 .87 .98 H2 .03 .04 .07 .41 .70 Hmax .04 .08 .12 .77 .95 H1 .33 .51 .61 .92 .96 H2 .19 .24 .27 .58 .77 Hmax .23 .32 .44 .86 .94 H1 .31 .39 .44 .58 .62 H2 .30 .31 .32 .43 .47 Hmax .29 .33 .37 .55 .60 H1 .06 .06 .06 .30 .88 H2 .05 .05 .05 .08 .38 Hmax .05 .05 .05 .16 .74 H1 .06 .06 .06 .73 1.0 H2 .05 .05 .05 .26 .73 Hmax .05 .05 .05 .55 .98 H1 .06 .08 .20 .99 1.0 H2 .05 .05 .06 .78 .99 Hmax .05 .05 .10 .97 1.0 H1 .58 .84 .94 1.0 1.0 H2 .07 .18 .31 .93 1.0 Hmax .30 .65 .83 1.0 1.0 H1 .96 .99 1.0 1.0 1.0 H2 .67 .75 .82 .98 1.0 Hmax .88 .96 .98 1.0 1.0 H1 .84 .88 .90 .93 .93 H2 .78 .79 .80 .84 .86 Hmax .80 .84 .87 .93 .93 B-7 Table B-4: Size-adjusted power of bootstrap tests, [m = 1, T DGP Test =c Model 1: 1 1=3 Model 2: 3 1=3 3 Model 3: c 0 10 = 0:5, Model 1: 1 1=3 1 2 3 1=3 1 2 0 10 .82 .06 .56 .09 .80 .06 .20 .39 .06 .33 .06 .48 .04 .07 H2 .35 .05 .11 .10 .21 .05 .07 .11 .05 .08 .05 .11 .03 .04 Hmax .67 .06 .29 .06 .62 .05 .09 .25 .05 .15 .04 .29 .03 .05 H1 .82 .06 .53 .15 .75 .16 .32 .39 .05 .25 .06 .43 .06 .13 H2 .30 .05 .10 .17 .26 .10 .16 .10 .05 .07 .08 .11 .06 .08 Hmax .63 .06 .29 .13 .59 .12 .22 .22 .05 .12 .07 .26 .05 .09 H1 .95 .40 .22 .81 .50 .10 .38 .74 .26 .17 .51 .26 .06 .21 H2 .45 .16 .04 .49 .10 .07 .10 .21 .11 .05 .28 .05 .05 .06 Hmax .90 .30 .06 .75 .21 .06 .17 .55 .20 .07 .45 .09 .05 .09 H1 .96 .41 .24 .75 .49 .24 .53 .70 .21 .18 .43 .28 .15 .33 H2 .46 .14 .05 .53 .16 .14 .27 .22 .09 .04 .28 .09 .09 .14 Hmax .89 .32 .08 .75 .29 .15 .40 .53 .16 .06 .42 .13 .10 .21 H1 .91 .04 .63 .29 .89 .07 .31 .99 .67 .38 .89 .81 .16 .55 = 0:2; 2 = 0:9; = 0:7 1 = 0:9; 2 = 0:2; = 0:7 H2 .44 .03 .22 .25 .31 .04 .09 .64 .34 .09 .66 .32 .08 .15 Hmax .78 .03 .53 .21 .81 .05 .15 .97 .62 .20 .88 .61 .11 .30 H1 .81 .03 .47 .29 .71 .16 .36 .95 .44 .40 .76 .76 .23 .57 H2 .34 .02 .14 .27 .27 .08 .16 .45 .17 .03 .48 .25 .13 .26 Hmax .69 .03 .38 .27 .64 .11 .23 .89 .42 .16 .75 .61 .17 .42 = 0:7 H1 1.0 .07 1.0 .64 1.0 .26 .70 .98 .08 .92 .13 .97 .09 .33 H2 .96 .04 .64 .41 .89 .10 .19 .49 .04 .37 .11 .46 .06 .10 Hmax 1.0 .05 .97 .57 1.0 .19 .42 .93 .05 .75 .11 .90 .07 .15 H1 1.0 .08 .99 .71 1.0 .43 .74 .96 .07 .90 .17 .96 .26 .48 H2 .91 .05 .63 .44 .89 .27 .45 .46 .05 .38 .15 .48 .14 .20 Hmax 1.0 .06 .97 .63 1.0 .35 .61 .89 .06 .76 .13 .92 .18 .30 H1 1.0 1.0 .34 1.0 .94 .34 .84 1.0 .90 .21 .98 .55 .20 .58 H2 .96 .67 .05 .98 .29 .12 .27 .64 .45 .04 .78 .12 .08 .13 Hmax 1.0 .98 .12 1.0 .80 .21 .61 .99 .81 .07 .98 .29 .12 .25 H1 1.0 .99 .37 1.0 .92 .50 .87 .99 .87 .25 .97 .62 .34 .68 H2 .96 .68 .06 .97 .41 .33 .55 .63 .39 .05 .76 .16 .19 .34 Hmax 1.0 .97 .14 1.0 .83 .41 .76 .98 .77 .09 .96 .37 .25 .52 H1 1.0 .15 .99 .99 1.0 .33 .88 1.0 .98 .68 1.0 1.0 .41 .94 1 3 3 = 0:7 = 0:5 400 Model 3: c H1 1 3 = 0:5] Model 2: = 0:5 200 0 1 = 0, = 0:2; 2 = 0:9; = 0:7 1 = 0:9; 2 = 0:2; = 0:7 H2 1.0 .05 .86 .93 .99 .12 .36 1.0 .92 .17 1.0 .89 .19 .47 Hmax 1.0 .07 .99 .98 1.0 .26 .70 1.0 .98 .38 1.0 1.0 .35 .83 H1 1.0 .15 .97 .97 .99 .44 .84 1.0 .95 .72 1.0 1.0 .52 .92 H2 .99 .05 .81 .89 .98 .31 .63 .99 .81 .19 .99 .90 .36 .73 Hmax 1.0 .07 .95 .96 .99 .39 .78 1.0 .94 .47 1.0 .99 .45 .87 B-8 Table B-5: Size-adjusted power of bootstrap tests, [m = 1, T DGP Test =c Model 1: 1 1=3 Model 2: 3 1=3 3 Model 3: c 0 10 = 0, Model 1: 1 1=3 1 2 3 1=3 1 2 0 10 .84 .03 .85 .12 .90 .07 .26 .53 .03 .51 .03 .63 .03 .08 H2 .35 .02 .07 .13 .14 .03 .04 .13 .03 .05 .08 .07 .01 .01 Hmax .71 .02 .56 .09 .73 .04 .10 .37 .04 .23 .03 .39 .02 .03 H1 .83 .04 .83 .21 .89 .24 .45 .49 .04 .45 .08 .60 .12 .22 H2 .34 .04 .12 .20 .31 .09 .20 .11 .04 .07 .08 .10 .05 .09 Hmax .69 .03 .65 .18 .80 .16 .31 .33 .04 .30 .06 .46 .07 .14 H1 .93 .75 .15 .87 .48 .08 .40 .74 .44 .14 .58 .23 .05 .21 H2 .47 .12 .02 .47 .10 .04 .06 .23 .07 .03 .25 .05 .04 .04 Hmax .85 .63 .03 .82 .27 .04 .19 .59 .33 .04 .52 .09 .04 .07 H1 .91 .77 .19 .86 .56 .28 .58 .66 .41 .12 .53 .29 .16 .34 H2 .49 .18 .05 .43 .23 .13 .26 .23 .09 .04 .20 .10 .07 .15 Hmax .84 .66 .09 .83 .41 .20 .47 .56 .33 .05 .49 .19 .10 .24 H1 .95 .03 .70 .40 .95 .12 .47 .97 .64 .15 .93 .69 .13 .56 = 0:2; 2 = 0:9; = 0:7 1 = 0:9; 2 = 0:2; = 0:7 H2 .72 .01 .07 .44 .39 .04 .07 .77 .10 .01 .69 .27 .04 .08 Hmax .90 .02 .60 .35 .90 .07 .25 .94 .57 .04 .91 .48 .08 .30 H1 .93 .04 .48 .41 .87 .25 .58 .95 .44 .11 .82 .66 .26 .60 H2 .61 .01 .11 .37 .45 .09 .29 .70 .08 .03 .51 .39 .12 .34 Hmax .86 .02 .40 .40 .83 .18 .48 .91 .41 .05 .81 .58 .20 .52 = 0:7 H1 1.0 .05 1.0 .64 1.0 .37 .68 .97 .05 .97 .19 .99 .14 .39 H2 .90 .02 .51 .46 .85 .07 .14 .55 .03 .22 .16 .39 .04 .06 Hmax .99 .03 1.0 .61 1.0 .26 .45 .90 .03 .90 .17 .94 .07 .17 H1 1.0 .06 1.0 .69 1.0 .48 .74 .94 .05 .93 .27 .98 .31 .50 H2 .84 .04 .61 .54 .84 .27 .41 .44 .05 .25 .20 .47 .14 .20 Hmax .99 .03 1.0 .65 1.0 .41 .62 .88 .03 .86 .23 .94 .22 .35 H1 1.0 1.0 .28 1.0 .87 .38 .84 .99 .96 .16 .99 .54 .21 .56 H2 .95 .60 .04 .95 .32 .09 .19 .70 .31 .02 .73 .13 .06 .10 Hmax 1.0 1.0 .08 1.0 .70 .26 .61 .97 .90 .04 .98 .32 .13 .27 H1 1.0 1.0 .37 1.0 .90 .54 .85 .98 .92 .21 .98 .59 .36 .66 H2 .93 .66 .04 .94 .51 .34 .55 .63 .36 .04 .74 .19 .20 .33 Hmax 1.0 1.0 .13 1.0 .80 .46 .75 .95 .87 .07 .97 .40 .28 .52 H1 1.0 .19 .99 .95 1.0 .38 .87 1.0 .99 .49 1.0 .98 .39 .92 1 3 3 = 0:7 = 0:5 400 Model 3: c H1 1 3 = 0:5] Model 2: = 0:5 200 0 1 = 0:5, = 0:2; 2 = 0:9; = 0:7 1 = 0:9; 2 = 0:2; = 0:7 H2 1.0 .05 .90 .93 .98 .12 .38 1.0 .89 .06 1.0 .87 .14 .42 Hmax 1.0 .08 .99 .94 1.0 .31 .72 1.0 .99 .17 1.0 .95 .29 .79 H1 1.0 .23 .97 .95 1.0 .44 .86 1.0 .95 .47 1.0 .98 .49 .93 H2 .99 .10 .82 .92 .98 .34 .67 1.0 .78 .11 .99 .91 .36 .73 Hmax 1.0 .11 .97 .94 1.0 .41 .80 1.0 .95 .23 1.0 .96 .44 .87 B-9 Table B-6: Size-adjusted power of bootstrap tests, [m = 2, T DGP Test =c Model 1: 1 1=3 Model 2: 3 1=3 3 Model 3: c 0 10 = = 0, Model 1: 1 1=3 0 1 Model 2: 3 1=3 = 0:5 200 4 5 4 5 Model 3: c 3 0 10 .38 .12 .43 .14 .38 .05 .13 .27 .02 .26 .07 .24 .04 .07 H2 .96 .69 .59 .91 .81 .14 .34 .73 .24 .25 .51 .37 .06 .12 Hmax .95 .54 .56 .72 .74 .07 .31 .66 .15 .23 .24 .32 .04 .11 H1 .36 .11 .42 .16 .44 .13 .20 .25 .03 .25 .07 .29 .08 .12 H2 .95 .62 .61 .89 .83 .26 .57 .68 .22 .25 .42 .44 .11 .25 Hmax .94 .51 .58 .73 .77 .21 .51 .63 .12 .24 .22 .40 .09 .21 H1 .68 .64 .79 .73 .85 .13 .24 .34 .34 .38 .40 .43 .07 .11 H2 .46 .05 .06 .40 .14 .05 .04 .08 .04 .03 .11 .04 .02 .03 Hmax .53 .51 .47 .65 .63 .09 .11 .15 .23 .13 .34 .24 .05 .06 H1 .73 .65 .82 .72 .87 .28 .41 .34 .35 .38 .40 .45 .15 .22 H2 .60 .17 .10 .54 .26 .15 .26 .16 .06 .04 .18 .08 .07 .11 Hmax .65 .59 .52 .68 .68 .23 .31 .23 .28 .17 .35 .25 .11 .15 H1 .83 .78 .91 .82 .92 .24 .33 .65 .46 .79 .41 .72 .13 .32 = 0:2; 2 = 0:9; = 0:7 1 = 0:9; 2 = 0:2; = 0:7 H2 .86 .25 .21 .78 .49 .03 .07 .99 .85 .79 .95 .92 .25 .60 Hmax .84 .69 .77 .79 .83 .16 .15 .98 .74 .81 .83 .88 .17 .55 H1 .77 .74 .88 .75 .90 .29 .44 .50 .38 .69 .34 .64 .20 .27 H2 .85 .40 .26 .76 .58 .20 .38 .95 .65 .62 .85 .83 .27 .64 Hmax .81 .70 .78 .74 .84 .25 .37 .91 .56 .70 .68 .75 .21 .51 = 0:7 H1 .49 .30 .60 .26 .50 .17 .19 .46 .20 .44 .22 .43 .09 .15 H2 1.0 .99 .97 1.0 1.0 .42 .80 .99 .85 .75 .97 .93 .19 .46 Hmax 1.0 .98 .97 1.0 1.0 .35 .77 .98 .75 .74 .89 .91 .13 .39 H1 .46 .27 .59 .28 .54 .22 .26 .44 .19 .48 .22 .46 .14 .22 H2 1.0 .99 .96 1.0 1.0 .56 .91 .99 .79 .73 .95 .94 .33 .69 Hmax 1.0 .99 .97 1.0 .99 .45 .85 .98 .71 .71 .87 .89 .24 .57 H1 1.0 .99 1.0 .99 1.0 .47 .64 .89 .86 .92 .91 .94 .22 .35 H2 .99 .28 .47 .94 .86 .07 .16 .68 .05 .11 .50 .30 .04 .04 Hmax .99 .94 .96 .99 1.0 .34 .33 .76 .64 .67 .86 .84 .14 .15 H1 .99 .98 1.0 .99 1.0 .66 .77 .91 .84 .92 .89 .95 .41 .51 H2 .98 .51 .49 .95 .82 .44 .52 .70 .13 .14 .58 .34 .22 .27 Hmax .99 .96 .97 .98 .99 .55 .61 .77 .71 .68 .86 .84 .31 .34 H1 1.0 1.0 1.0 1.0 1.0 .59 .79 .90 .94 .99 .86 .97 .35 .59 1 6 = 0:8] = 0:7 = 0:5 400 0 2 H1 1 6 = 0:3, = 0:2; 2 = 0:9; = 0:7 1 = 0:9; 2 = 0:2; = 0:7 H2 1.0 .82 .91 1.0 1.0 .19 .48 1.0 1.0 1.0 1.0 1.0 .54 .95 Hmax 1.0 .99 1.0 1.0 1.0 .51 .59 1.0 1.0 1.0 1.0 1.0 .47 .93 H1 1.0 .99 1.0 1.0 1.0 .70 .84 .79 .88 .96 .73 .91 .46 .60 H2 1.0 .89 .88 1.0 .99 .53 .72 1.0 1.0 .98 1.0 1.0 .61 .93 Hmax 1.0 .99 1.0 1.0 1.0 .64 .77 1.0 .98 .98 1.0 .99 .55 .87 B-10 Table B-7: Size-adjusted power of bootstrap tests, [m = 2, T DGP Test =c Model 1: 1 1=3 Model 2: 3 1=3 3 = 0:5, Model 3: c 0 10 Model 1: 1 1=3 0 1 = 0, Model 2: 3 1=3 = 0:5 200 4 5 4 5 Model 3: c 3 0 10 .22 .04 .31 .08 .29 .04 .08 .19 .04 .19 .06 .19 .06 .06 H2 .89 .43 .49 .74 .65 .10 .22 .58 .20 .23 .36 .32 .06 .09 Hmax .85 .31 .45 .46 .58 .07 .19 .51 .12 .21 .15 .28 .06 .08 H1 .21 .05 .29 .08 .28 .09 .11 .16 .03 .17 .06 .20 .06 .08 H2 .86 .37 .42 .74 .69 .19 .41 .48 .15 .18 .31 .32 .10 .20 Hmax .80 .30 .38 .49 .56 .13 .29 .40 .10 .15 .14 .26 .07 .14 H1 .51 .51 .56 .58 .67 .11 .19 .28 .30 .29 .35 .33 .07 .12 H2 .24 .08 .06 .23 .10 .05 .05 .05 .06 .06 .09 .04 .03 .04 Hmax .31 .40 .26 .51 .44 .08 .10 .10 .21 .12 .27 .16 .05 .07 H1 .54 .48 .56 .54 .62 .24 .30 .29 .29 .27 .30 .30 .15 .17 H2 .26 .12 .08 .34 .14 .15 .20 .10 .07 .05 .14 .06 .07 .10 Hmax .35 .42 .27 .54 .41 .19 .23 .15 .24 .11 .30 .16 .11 .12 H1 .68 .68 .67 .71 .72 .18 .25 .36 .07 .62 .07 .51 .07 .14 = 0:2; 2 = 0:9; = 0:7 1 = 0:9; 2 = 0:2; = 0:7 H2 .45 .19 .06 .43 .18 .04 .05 .93 .47 .69 .76 .80 .15 .35 Hmax .52 .59 .33 .65 .53 .13 .12 .90 .32 .67 .46 .72 .09 .30 H1 .50 .52 .53 .53 .60 .21 .26 .21 .03 .45 .05 .40 .11 .11 H2 .41 .18 .07 .41 .23 .14 .18 .67 .26 .42 .54 .63 .17 .38 Hmax .41 .48 .34 .53 .46 .16 .18 .63 .17 .44 .33 .53 .12 .22 = 0:7 H1 .33 .14 .42 .14 .35 .12 .13 .33 .08 .36 .14 .32 .07 .10 H2 1.0 .96 .94 1.0 .99 .29 .64 .97 .62 .63 .92 .86 .13 .31 Hmax 1.0 .92 .93 .98 .99 .24 .59 .96 .51 .62 .74 .80 .11 .27 H1 .28 .12 .40 .15 .38 .19 .18 .29 .09 .35 .12 .34 .12 .16 H2 1.0 .94 .94 1.0 .99 .50 .83 .97 .53 .62 .88 .84 .27 .53 Hmax 1.0 .91 .90 .98 .97 .38 .72 .94 .45 .59 .66 .74 .21 .42 H1 .98 .96 .98 .97 .99 .37 .53 .75 .76 .77 .79 .84 .19 .28 H2 .93 .16 .29 .76 .68 .06 .11 .39 .06 .07 .28 .17 .05 .04 Hmax .95 .86 .89 .96 .97 .25 .28 .53 .54 .44 .74 .65 .11 .11 H1 .97 .94 .99 .97 1.0 .58 .68 .75 .75 .76 .79 .82 .36 .43 H2 .89 .26 .34 .80 .64 .38 .43 .43 .10 .10 .34 .21 .20 .23 Hmax .92 .85 .90 .95 .97 .48 .53 .54 .54 .50 .74 .65 .28 .29 H1 1.0 1.0 1.0 .99 1.0 .50 .69 .59 .55 .92 .41 .81 .21 .31 1 6 = 0:8] = 0:7 = 0:5 400 0 2 H1 1 6 = 0:3, = 0:2; 2 = 0:9; = 0:7 1 = 0:9; 2 = 0:2; = 0:7 H2 .99 .59 .56 .97 .87 .13 .20 1.0 .98 .99 1.0 1.0 .40 .86 Hmax .99 .98 .97 .99 1.0 .40 .42 1.0 .93 .99 .99 1.0 .33 .80 H1 .99 .98 .99 .98 1.0 .63 .75 .38 .43 .83 .27 .68 .31 .31 H2 .97 .63 .58 .92 .88 .43 .55 1.0 .92 .95 .99 .99 .52 .86 Hmax .98 .96 .96 .98 .99 .55 .62 1.0 .77 .92 .91 .95 .44 .72 B-11 Table B-8: Size-adjusted power of bootstrap tests, [m = 2, T DGP Test =c Model 1: 1 1=3 Model 2: 3 1=3 3 = 0, Model 3: c 0 10 Model 1: 1 0 1 = 0:5, 1=3 Model 2: 3 1=3 = 0:5 200 4 5 4 5 Model 3: c 3 0 10 .40 .15 .38 .15 .37 .05 .13 .34 .06 .23 .11 .24 .03 .08 H2 .89 .47 .36 .77 .60 .09 .24 .63 .18 .16 .46 .28 .05 .09 Hmax .86 .38 .36 .56 .54 .05 .21 .61 .13 .16 .26 .25 .03 .08 H1 .40 .14 .39 .18 .46 .13 .20 .27 .06 .23 .09 .27 .07 .13 H2 .87 .47 .46 .77 .73 .20 .48 .55 .17 .18 .36 .35 .09 .21 Hmax .86 .36 .42 .65 .69 .15 .38 .51 .11 .18 .23 .33 .06 .16 H1 .27 .33 .56 .33 .57 .08 .13 .15 .20 .33 .20 .29 .04 .06 H2 .26 .03 .02 .22 .07 .02 .02 .06 .02 .02 .07 .03 .02 .01 Hmax .25 .22 .28 .27 .31 .04 .05 .08 .12 .11 .15 .11 .03 .02 H1 .26 .35 .55 .31 .56 .17 .19 .15 .22 .31 .18 .30 .10 .13 H2 .34 .09 .07 .33 .16 .13 .21 .13 .05 .03 .14 .06 .07 .10 Hmax .30 .30 .34 .29 .40 .13 .16 .14 .16 .17 .16 .18 .08 .09 H1 .35 .40 .71 .37 .69 .11 .13 .75 .51 .70 .50 .70 .11 .32 = 0:2; 2 = 0:9; = 0:7 1 = 0:9; 2 = 0:2; = 0:7 H2 .54 .11 .11 .46 .24 .02 .03 .95 .64 .54 .86 .76 .15 .35 Hmax .50 .31 .46 .33 .49 .07 .06 .93 .57 .56 .71 .75 .10 .32 H1 .25 .28 .53 .22 .56 .16 .17 .51 .37 .54 .36 .58 .17 .30 H2 .54 .16 .16 .40 .29 .11 .26 .80 .38 .39 .61 .53 .17 .45 Hmax .40 .25 .43 .24 .46 .13 .18 .71 .40 .46 .50 .59 .17 .39 = 0:7 H1 .52 .35 .59 .29 .53 .16 .21 .47 .24 .44 .23 .44 .08 .16 H2 1.0 .94 .85 .99 .97 .31 .63 .98 .63 .53 .91 .81 .14 .34 Hmax 1.0 .91 .86 .98 .96 .23 .59 .96 .56 .53 .80 .78 .10 .30 H1 .47 .34 .60 .31 .57 .26 .30 .38 .21 .40 .23 .42 .17 .23 H2 1.0 .92 .88 .99 .98 .51 .82 .94 .59 .53 .87 .78 .31 .58 Hmax 1.0 .89 .86 .98 .96 .42 .74 .90 .49 .46 .73 .69 .25 .47 H1 .77 .80 .95 .80 .95 .37 .37 .53 .57 .79 .62 .77 .20 .21 H2 .85 .09 .24 .68 .54 .04 .07 .42 .02 .05 .28 .15 .03 .03 Hmax .85 .63 .82 .77 .88 .25 .17 .43 .36 .49 .54 .55 .11 .07 H1 .71 .82 .93 .82 .94 .52 .46 .46 .59 .76 .58 .76 .33 .30 H2 .81 .25 .28 .77 .59 .38 .41 .40 .08 .07 .42 .21 .21 .25 Hmax .81 .68 .81 .80 .87 .47 .41 .42 .42 .47 .55 .59 .27 .22 H1 .89 .91 .99 .87 .98 .42 .41 .97 .96 .98 .91 .98 .33 .69 1 6 = 0:8] = 0:7 = 0:5 400 0 2 H1 1 6 = 0:3, = 0:2; 2 = 0:9; = 0:7 1 = 0:9; 2 = 0:2; = 0:7 H2 .98 .50 .68 .93 .92 .09 .23 1.0 .99 .96 1.0 .99 .42 .84 Hmax .98 .82 .96 .89 .97 .33 .29 1.0 .98 .97 .99 .99 .36 .82 H1 .80 .88 .97 .82 .98 .49 .53 .82 .89 .94 .83 .92 .46 .64 H2 .98 .63 .68 .95 .93 .42 .57 1.0 .94 .90 .99 .97 .52 .87 Hmax .96 .80 .94 .87 .97 .45 .54 .99 .92 .93 .96 .96 .49 .79 B-12 Table B-9: Break selection probabilities, [ = T 400 0 0 1= 1; m 0 2 0 DGP 0 /( 1; 0.5 0 1 1 = 0:5 1 2 3 0.5 0.5 (0.2, 0.9) (0.9, 0.2) 2 0 1 = 0:3; 0 2 = 0:8 4 0.5 5 0.5 6 (0.2, 0.9) (0.9, 0.2) 600 0 0 1 0.5 0 1 1 = 0:5 1 0.5 2 0.5 3 (0.2, 0.9) (0.9, 0.2) 2 0 1 = 0:3; 0 2 = 0:8 Model 1: 2) 1 4 0.5 5 0.5 6 (0.2, 0.9) (0.9, 0.2) = 0, = :10] Model 2: Model 3: c =c 1 1=3 3 1=3 3 0 10 Pc .91 .91 .91 .92 .90 .90 .91 Po .09 .09 .09 .08 .10 .10 .10 Pc .86 .91 .90 .88 .89 .91 .90 Po .14 .09 .10 .12 .11 .10 .10 Pc .90 .07 .91 .82 .89 .39 .68 Po .10 .04 .09 .14 .11 .15 .14 Pc .88 .87 .21 .88 .79 .47 .73 Po .12 .12 .12 .12 .20 .14 .15 Pc .89 .42 .87 .88 .89 .46 .80 Po .11 .08 .13 .12 .11 .13 .13 Pc .88 .82 .62 .88 .85 .48 .81 Po .12 .18 .14 .12 .15 .13 .12 Pc .86 .70 .46 .85 .70 .27 .59 Po .10 .17 .16 .11 .13 .11 .14 Pc .83 .07 .08 .39 .28 .18 .36 Po .15 .08 .06 .12 .18 .09 .10 Pc .89 .08 .11 .84 .73 .24 .57 Po .12 .07 .09 .12 .16 .09 .11 Pc .90 .72 .62 .89 .84 .32 .75 Po .09 .23 .20 .10 .12 .12 .11 Pc .89 .88 .89 .89 .89 .88 .89 Po .11 .12 .11 .11 .11 .12 .11 Pc .87 .89 .88 .88 .88 .91 .89 Po .13 .11 .12 .12 .12 .09 .11 Pc .88 .11 .89 .87 .89 .49 .80 Po .12 .05 .11 .13 .11 .17 .11 Pc .89 .88 .37 .89 .86 .51 .79 Po .11 .12 .21 .11 .14 .17 .15 Pc .85 .71 .84 .85 .85 .56 .86 Po .15 .17 .17 .15 .15 .12 .11 Pc .83 .82 .75 .86 .82 .55 .85 Po .17 .18 .22 .14 .18 .14 .12 Pc .86 .76 .64 .86 .78 .34 .69 Po .11 .20 .15 .12 .12 .14 .16 Pc .87 .05 .08 .76 .53 .27 .51 Po .13 .07 .08 .13 .22 .10 .13 Pc .86 .33 .30 .85 .84 .33 .69 Po .14 .12 .13 .15 .16 .11 .13 Pc .90 .83 .78 .91 .89 .38 .81 Po .10 .17 .18 .09 .11 .14 .13 B-13 Table B-10: Break selection probabilities [Model 1, DGP 5 and 6, m = 2, T 400 DGP 5 /( 1; 0.2 0.3 0.5 0.8 0.9 6 (0.2, 0.9) 0.2 0.3 0.5 0.8 0.9 600 5 0.5 0.2 0.3 0.5 0.8 0.9 6 (0.2, 0.9) = 0] Model 1: 2) 0.5 = 0.2 0.3 0.5 0.8 0.9 1/3 1/2.5 1/1.5 1/1.1 1 1.1 1.5 2.5 3 Pc .47 .54 .76 .83 .84 .84 .84 .63 .44 Po .13 .15 .17 .16 .15 .14 .14 .30 .45 Pc .11 .11 .62 .83 .84 .85 .80 .50 .35 Po .06 .07 .14 .15 .14 .14 .16 .37 .46 Pc .07 .06 .66 .84 .84 .82 .66 .08 .08 Po .08 .10 .14 .14 .15 .16 .17 .08 .06 Pc .68 .77 .87 .86 .84 .80 .61 .18 .15 Po .15 .13 .12 .13 .15 .17 .19 .08 .06 Pc .89 .88 .87 .85 .84 .83 .78 .69 .66 Po .09 .11 .12 .14 .15 .15 .17 .17 .16 Pc .86 .85 .89 .89 .90 .90 .89 .72 .55 Po .12 .14 .11 .11 .11 .10 .11 .27 .42 Pc .18 .41 .88 .89 .89 .90 .88 .67 .50 Po .07 .09 .11 .11 .11 .10 .12 .29 .40 Pc .08 .28 .88 .88 .90 .89 .88 .31 .11 Po .07 .09 .12 .12 .10 .11 .11 .12 .09 Pc .80 .86 .88 .89 .88 .89 .86 .46 .32 Po .14 .12 .12 .11 .12 .11 .12 .12 .08 Pc .90 .90 .90 .89 .89 .88 .88 .86 .86 Po .09 .10 .10 .11 .11 .12 .12 .13 .13 Pc .78 .80 .85 .86 .87 .87 .88 .74 .58 Po .15 .16 .15 .14 .13 .13 .12 .25 .41 Pc .09 .12 .83 .86 .87 .87 .88 .71 .54 Po .05 .06 .15 .14 .13 .13 .12 .27 .40 Pc .05 .07 .86 .88 .86 .85 .83 .13 .07 Po .07 .08 .13 .12 .14 .15 .16 .10 .08 Pc .82 .86 .87 .87 .86 .86 .76 .25 .18 Po .15 .13 .13 .14 .14 .14 .21 .13 .09 Pc .88 .89 .88 .86 .87 .87 .85 .81 .81 Po .12 .11 .12 .14 .14 .14 .15 .18 .18 Pc .84 .85 .86 .85 .86 .86 .86 .80 .67 Po .16 .15 .15 .15 .14 .14 .14 .20 .33 Pc .57 .76 .86 .86 .86 .86 .87 .79 .67 Po .12 .15 .14 .14 .14 .14 .13 .21 .32 Pc .32 .68 .85 .86 .85 .86 .88 .64 .30 Po .11 .15 .15 .14 .15 .14 .12 .14 .13 Pc .84 .85 .86 .86 .86 .87 .86 .74 .56 Po .16 .16 .14 .14 .14 .14 .14 .15 .12 Pc .86 .86 .86 .86 .85 .86 .86 .87 .85 Po .14 .14 .14 .14 .15 .14 .14 .14 .15 B-14 Table B-11: Model selection probabilities between pure mean shift and persistence break [ = :10] T 200 DGP ; Model 1: 2 Model 2: Model 3: c 1/3 3 1/3 3 0 10 =1 .87 .32 .83 .38 .84 .31 .63 =3 .86 .84 .86 .88 .84 .48 .77 1 = 0:5 .97 .05 .94 .35 .98 .33 .62 2 = 0:5 .98 .94 .26 .99 .68 .42 .78 0 1 = 0:2; 1 = 0:9; 2 = 0:9 1.00 .12 .93 .89 .99 .47 .86 2 = 0:2 1.00 .93 .48 .99 .96 .51 .91 .87 .62 .87 .74 .89 .48 .82 0 =1 =3 .89 .86 .87 .89 .86 .63 .88 1 = 0:5 1.00 .06 1.00 .96 1.00 .59 .87 2 3 600 1 1 3 400 = = 1.00 1.00 .54 1.00 .99 .64 .95 2 = 0:9 1.00 .65 1.00 1.00 1.00 .62 .97 2 = 0:2 1.00 1.00 .96 1.00 1.00 .61 .99 .90 .75 .88 .84 .90 .51 .87 = 0:5 1 = 0:2; 1 = 0:9; 0 =1 =3 .90 .87 .87 .87 .89 .69 .89 1 = 0:5 1.00 .15 1.00 1.00 1.00 .64 .93 2 = 0:5 1.00 1.00 .85 1.00 1.00 .70 .98 3 1 = 0:2; 1 = 0:9; 2 = 0:9 1.00 .99 1.00 1.00 1.00 .70 .98 2 = 0:2 1.00 1.00 1.00 1.00 1.00 .69 1.00 B-15 Supplement C: Comparison with the Recursive Bootstrap In order to highlight the advantages of employing the proposed bootstrap schemes A and B, we now provide a comparison with the fully recursive bootstrap schemes. The recursive counterpart of scheme A entails replacing step 3 in scheme A with the recursion (1) yt (1) yt (1) 1 = yt + PlT j=1 j (1) j (1) yt + ut ; t = lT + 2; :::; T (C.1) = yt ; t = 1; :::; lT + 1 while the recursive counterpart of scheme B involves replacing step 3 in scheme B with the recursion (0) yt (0) yt (0) 1 = e c + e yt + PelT j=1 = 0; t = 1; :::; e lT + 1 (0) j e j yt (0) + ut ; t = lT + 2; :::; T (C.2) Since the bootstrap data obtained from (C:1) and (C:2) are serially correlated, conditional on the original data, the bootstrap statistics will now need to be adjusted by including lagged …rst di¤erences in the estimated regression as in the construction of the statistics based on the original data fyt g. The lag length is again chosen using the BIC. Table C-1 reports the empirical size and size-adjusted power (only in the single break case, for brevity) of the recursive bootstrap tests (denoted with a superscript “r”) for = = 0. The procedure has accurate size in general with a tendency to under-reject in some cases. A power comparison with Table B-2 reveals that the recursive bootstrap tests are generally less powerful than the hybrid tests for DGP-1 and DGP-2 which contain an I(1) segment, in accordance with the discussion in Section 5. For DGP-3, the two approaches yield comparable power. The power gains are even more transparent if one were to a priori rule out the I(1) null hypothesis and hence apply the BP tests in isolation (see Tables C-2 and C-3). Overall, these …ndings favor the use of the proposed scheme over the recursive scheme in terms of its relative ability in detecting persistence change. C-1 Table C-1: Size and size-adjusted power of bootstrap recursive tests, [ = T DGP Test =c Model 1: 1 1=3 Model 2: 3 1=3 3 Model 3: c 0 10 Model 1: 1 1=3 0 3 1=3 2 0 2 .03 .05 .04 .05 .03 .06 .06 .04 .03 .04 .04 .03 .03 .03 .02 .03 .02 .02 .05 .03 .03 .02 .01 .04 .02 .03 .03 r Hmax .02 .04 .02 .05 .02 .05 .04 .04 .02 .02 .04 .02 .03 .03 = 0:7 H1r .95 .05 .74 .26 .92 .25 .51 .57 .04 .36 .06 .54 .08 .20 H2r .53 .04 .18 .25 .38 .11 .21 .15 .04 .07 .06 .14 .05 .07 r Hmax .89 .04 .59 .21 .84 .17 .37 .41 .04 .22 .05 .42 .07 .11 H1r H2r r Hmax .99 .74 .26 .92 .70 .30 .69 .79 .40 .16 .57 .28 .16 .39 .66 .29 .05 .65 .22 .14 .29 .26 .14 .04 .30 .09 .08 .12 .97 .65 .10 .90 .50 .23 .57 .69 .31 .53 .16 .12 .26 = 0:2; 2 = 0:9; = 0:7 1 .06 = 0:9; 2 = 0:2; = 0:7 H1r .99 .08 .82 .74 .97 .34 .72 1.0 .88 .33 .99 .91 .42 .82 H2r .87 .05 .32 .64 .73 .19 .43 .93 .45 .06 .89 .59 .24 .51 r Hmax .99 .04 .77 .69 .97 .30 .64 1.0 .85 .17 .99 .85 .36 .73 = 0:5 H1r .03 .04 .05 .04 .04 .05 .05 .05 .05 .04 .05 .05 .05 .06 H2r r Hmax .02 .03 .03 .02 .02 .04 .03 .06 .03 .03 .04 .04 .03 .04 .02 .02 .02 .04 .02 .05 .03 .05 .04 .04 .05 .04 .05 .05 H1r 1.0 .07 1.0 .94 1.0 .50 .85 1.0 .06 .91 .31 .98 .31 .61 H2r r Hmax .99 .03 .72 .77 .95 .30 .59 .68 .03 .42 .22 .60 .14 .28 1.0 .04 1.0 .91 1.0 .41 .76 .98 .03 .83 .23 .96 .22 .42 = 0:7 H1r H2r r Hmax 1.0 1.0 .48 1.0 .99 .58 .93 1.0 .92 .25 .99 .74 .39 .78 .99 .77 .06 .99 .65 .39 .68 .82 .54 .05 .88 .22 .22 .40 1.0 1.0 .20 1.0 .96 .50 .85 1.0 .89 .10 .99 .54 .29 .63 H1r 1.0 .47 1.0 1.0 1.0 .55 .94 1.0 1.0 .82 1.0 1.0 .60 .97 H2r r Hmax 1.0 .16 .96 .99 1.0 .46 .85 1.0 .96 .26 1.0 .98 .48 .87 1.0 .30 1.0 1.0 1.0 .52 .92 1.0 1.0 .55 1.0 1.0 .55 .93 1 3 10 .02 = 0:5 1 0 H2r =1 400 3 H1r 1 3 Model 3: c = 0:5 = 0:5 1 = 0:5, 5%] Model 2: =1 200 0 1 = 0, = 0:2; 2 = 0:9; = 0:7 C-2 1 = 0:9; 2 = 0:2; = 0:7 Table C-2: Empirical power of bootstrap recursive and non-recursive G1 tests, [m = 1, T DGP Test =c Model 1: 1 1=3 Model 2: 3 = 0:5 / 200 0.5 1 2 3 0.8 1 2 3 0.5 1 2 3 0.8 1 2 3 1 3 = 0:2; 0 2 = 0:9 10 Model 1: 1 1=3 Model 2: 3 = 0:7/ 1=3 1 Model 3: c 3 = 0:9; = 0:5] 2 0 10 = 0:2 Gr1 .89 .20 .60 .42 .82 .37 .62 .57 .16 .31 .23 .48 .27 .37 G1 .98 .72 .86 .88 .97 .63 .88 .90 .66 .64 .71 .83 .52 .73 Gr1 .94 .50 .43 .76 .66 .44 .72 .68 .26 .35 .45 .42 .32 .51 G1 .99 .82 .83 .96 .92 .65 .91 .93 .61 .74 .83 .79 .55 .79 Gr1 .90 .14 .63 .56 .83 .37 .65 .98 .60 .59 .87 .88 .49 .82 G1 .97 .42 .74 .85 .92 .48 .78 .99 .77 .78 .95 .96 .56 .90 Gr1 .62 .20 .36 .34 .55 .31 .45 .40 .17 .17 .24 .29 .23 .29 G1 .94 .74 .80 .81 .90 .61 .79 .83 .70 .62 .71 .77 .53 .67 Gr1 .74 .28 .40 .54 .49 .37 .55 .53 .15 .36 .29 .38 .27 .40 G1 .96 .75 .82 .91 .86 .62 .85 .91 .55 .78 .77 .77 .54 .76 Gr1 .64 .10 .37 .21 .58 .26 .44 .90 .38 .72 .74 .74 .43 .72 G1 .87 .38 .57 .60 .82 .43 .63 .96 .65 .84 .92 .88 .56 .83 = 0:5 / 400 1=3 Model 3: c 0 1 = 0, 2 = 0:9 Gr1 1.0 .24 .99 .86 1.0 .57 .86 .97 .16 .89 .40 .96 .42 .70 G1 1.0 .81 1.0 .99 1.0 .78 .97 .99 .70 .96 .91 .99 .67 .91 Gr1 1.0 .99 .61 1.0 .97 .68 .95 .99 .84 .47 .95 .72 .53 .82 G1 1.0 1.0 .95 1.0 1.0 .79 .98 1.0 .95 .87 .99 .95 .71 .94 Gr1 1.0 .36 .98 .99 1.0 .58 .95 1.0 .98 .85 1.0 1.0 .64 .97 G1 1.0 .75 .99 1.0 1.0 .63 .97 1.0 .99 .94 1.0 1.0 .67 .98 Gr1 .94 .20 .89 .47 .95 .45 .72 .74 .16 .62 .28 .74 .32 .50 G1 1.0 .75 .99 .93 1.0 .72 .92 .98 .70 .90 .80 .96 .64 .82 Gr1 .99 .83 .48 .96 .74 .53 .81 .90 .51 .41 .76 .53 .42 .65 G1 1.0 .98 .88 1.0 .97 .74 .95 .99 .87 .83 .97 .90 .65 .89 Gr1 .99 .11 .85 .60 .97 .44 .78 1.0 .84 .76 .98 .93 .54 .89 G1 1.0 .36 .93 .91 .99 .53 .85 1.0 .94 .87 .99 .98 .63 .92 1 = 0:2; C-3 = 0:7/ 1 = 0:9; 2 = 0:2 Table C-3: Empirical power of bootstrap recursive and non-recursive G1 tests, [m = 1, T DGP Test =c Model 1: 1 1=3 Model 2: 3 = 0:5 / 200 0.5 1 2 3 0.8 1 2 3 0.5 1 2 3 0.8 1 2 3 1 3 = 0:2; 0 2 = 0:9 10 Model 1: 1 1=3 Model 2: 3 = 0:7/ 1=3 1 = 0:5] Model 3: c 3 = 0:9; 0 1 2 0 10 = 0:2 Gr1 .92 .32 .84 .63 .91 .45 .75 .76 .28 .53 .44 .68 .33 .56 G1 .99 .74 .95 .90 .99 .64 .90 .93 .69 .74 .79 .88 .55 .82 Gr1 .95 .89 .46 .93 .70 .54 .82 .77 .62 .38 .72 .50 .41 .64 G1 .99 .96 .81 .98 .90 .66 .92 .93 .82 .72 .90 .77 .56 .80 Gr1 .97 .32 .79 .80 .95 .45 .80 .99 .78 .37 .96 .81 .45 .87 G1 .99 .51 .81 .91 .97 .49 .87 1.0 .78 .55 .97 .92 .50 .90 Gr1 .88 .58 .75 .75 .85 .54 .76 .82 .55 .56 .69 .77 .45 .71 G1 .96 .81 .75 .89 .91 .61 .86 .93 .77 .61 .83 .86 .54 .80 Gr1 .89 .87 .60 .90 .72 .54 .81 .83 .78 .54 .82 .61 .48 .71 G1 .96 .87 .78 .93 .87 .61 .87 .93 .83 .74 .88 .79 .54 .80 Gr1 .83 .37 .39 .69 .62 .31 .60 .80 .19 .35 .47 .66 .29 .56 G1 .82 .41 .37 .70 .59 .31 .60 .80 .17 .38 .46 .68 .31 .56 = 0:5 / 400 1=3 Model 3: c = 0, 2 = 0:9 Gr1 1.0 .34 1.0 .90 1.0 .63 .89 .97 .22 .93 .59 .97 .53 .78 G1 1.0 .82 1.0 .99 1.0 .78 .97 1.0 .73 .97 .94 .99 .70 .91 Gr1 1.0 1.0 .64 1.0 .96 .71 .97 .99 .95 .49 .99 .77 .60 .86 G1 1.0 1.0 .93 1.0 .99 .79 .98 1.0 .97 .85 1.0 .95 .70 .94 Gr1 1.0 .62 .99 .99 1.0 .61 .96 1.0 1.0 .75 1.0 .99 .62 .97 G1 1.0 .77 .99 1.0 1.0 .64 .97 1.0 1.0 .83 1.0 1.0 .64 .97 Gr1 .98 .53 .98 .83 .99 .67 .89 .94 .50 .91 .71 .96 .61 .83 G1 .99 .81 .97 .95 1.0 .74 .95 .98 .77 .91 .87 .98 .69 .89 Gr1 .99 .98 .66 1.0 .87 .70 .93 .94 .96 .60 .98 .73 .62 .86 G1 1.0 .98 .86 .99 .94 .75 .95 .98 .96 .80 .98 .87 .68 .90 Gr1 .99 .60 .82 .96 .96 .48 .85 .99 .68 .58 .95 .95 .47 .85 G1 .99 .65 .80 .96 .95 .48 .85 .99 .66 .65 .95 .95 .47 .84 1 = 0:2; C-4 = 0:7/ 1 = 0:9; 2 = 0:2 Supplement D: Additional Empirical Results A u s t r ia B e lg iu m 6 3 5 2 .5 Canada F in l a n d 2 Fr a n c e 5 1 .8 1 .6 4 1 .5 4 Ge rm a ny 2 1 .5 2 1 .4 1 0 1976 1996 1955 Ye a r 0 1975 1995 1956 Ye a r Gre e c e 1 0 .5 1 0 .5 0 1956 0 .5 2 Volatility Intensity 1 1 Volatility Intensity 2 1 .5 Volatility Intensity 3 Volatility Intensity Volatility Intensity Volatility Intensity 3 1996 1956 Ye a r It a ly 2 .5 5 4 2 4 0 1976 1996 1955 Ye a r Japan 5 1 0 .8 0 1976 1 .2 0 .6 1975 1995 1955 Ye a r Kore a 1995 Ye a r Lux e m bourg 8 1975 N e t h e r la n d s 3 4 2 .5 6 3 2 1 0 .5 0 1956 1996 1955 Ye a r Norw a y 1995 1956 2 4 Volatility Intensity 1 0 1976 1996 1956 Ye a r P o r tu g a l 5 2 1 .5 2 1 0 .5 0 1975 Ye a r 2 .5 4 1 0 1976 2 Volatility Intensity 1 Volatility Intensity 2 3 Volatility Intensity 1 .5 Volatility Intensity Volatility Intensity 3 0 1976 1996 1956 Ye a r S p a in 1996 1955 Ye a r Sw eden 3 0 1976 1975 1995 Ye a r S w it z e r la n d UK 2 4 4 1 .5 3 3 2 .5 2 0 .5 1 0 1956 Ye a r 1996 1956 0 .5 Volatility Intensity 1 1 2 1 2 1 0 .5 0 1976 1 .5 Volatility Intensity 2 Volatility Intensity 1 Volatility Intensity 3 Volatility Intensity Volatility Intensity 1 .5 0 1976 Ye a r 1996 1956 0 1976 1996 Ye a r 1956 0 1976 Ye a r 1996 1956 0 1976 Ye a r 1996 1956 1976 Ye a r Figure D-1: Nonparametric volatility estimates (% of the total) of OECD in‡ation rates. D-1 1996 A u s t r ia B e lg i u m Ca nada F in la n d Fr a n c e Ge rm a ny 1 1 1 1 1 1 0 .8 0 .8 0 .8 0 .8 0 .8 0 .8 0 .6 0 .6 0 .6 0 .6 0 .6 0 .6 0 .4 0 .4 0 .4 0 .4 0 .4 0 .4 0 .2 0 .2 0 .2 0 .2 0 .2 0 .2 0 0 0 0 .5 1 0 0 Gre e c e 0 .5 1 0 0 It a ly 0 .5 1 0 0 Japan 0 .5 1 0 0 Korea 0 .5 1 0 Lux e m bourg 1 1 1 1 1 0 .8 0 .8 0 .8 0 .8 0 .8 0 .8 0 .6 0 .6 0 .6 0 .6 0 .6 0 .6 0 .4 0 .4 0 .4 0 .4 0 .4 0 .4 0 .2 0 .2 0 .2 0 .2 0 .2 0 .2 0 0 0 .5 1 0 0 Norw a y 0 .5 1 0 0 P o r tu g a l 0 .5 1 0 0 S p a in 0 .5 1 0 0 Sw eden 0 .5 1 0 S w it z e r la n d 1 1 1 1 1 0 .8 0 .8 0 .8 0 .8 0 .8 0 .8 0 .6 0 .6 0 .6 0 .6 0 .6 0 .6 0 .4 0 .4 0 .4 0 .4 0 .4 0 .4 0 .2 0 .2 0 .2 0 .2 0 .2 0 .2 0 0 0 .5 1 0 0 0 .5 1 0 0 0 .5 1 0 0 0 .5 1 1 0 0 0 .5 Figure D-2: Estimated variance pro…le of OECD in‡ation rates. D-2 0 .5 UK 1 0 1 N e t h e r la n d s 1 0 0 .5 1 0 0 .5 1