Testing Jointly for Structural Changes in the Error Variance and Coefficients of a Linear Regression Model (Tables)

ε = 0.10 na = 1 ma = 1 ma = 2 ∗ Table 1: Asympotic critical values of the upper bound of the sup LR4,T test ¢ ¡ −1 ∗ (the entries are quantiles x such that P (na + ma ) sup LR4 ≤ x ≥ α) ε = 0.15 na = 2 ma = 1 ma = 2 na = 1 ma = 1 ma = 2 na = 2 ma = 1 ma = 2 ε = 0.20 na = 1 na = 2 ma = 1 ma = 2 ma = 1 ε = 0.25 na = 1 ma = 1 U D max LR∗4 M =N =2 ε = 0.10 ε = 0.15 ε = 0.20 q α 1 .90 .95 .975 .99 6.59 7.63 8.54 9.79 6.34 7.12 7.78 8.73 6.32 7.10 7.75 8.70 6.20 6.83 7.44 8.17 6.21 7.18 8.12 9.24 5.75 6.49 7.17 7.98 5.72 6.46 7.23 8.00 5.46 6.13 6.71 7.45 5.83 6.79 7.70 8.83 5.19 5.93 6.56 7.42 5.18 5.89 6.70 7.52 5.48 6.43 7.42 8.56 7.18 8.03 8.81 10.00 6.61 7.51 8.32 9.42 6.15 7.05 7.87 8.95 2 .90 .95 .975 .99 7.88 8.87 9.85 11.12 7.96 8.78 9.52 10.55 7.18 7.94 8.69 9.52 7.41 8.03 8.69 9.52 7.45 8.45 9.45 10.73 7.31 8.12 8.91 9.90 6.54 7.36 8.02 8.93 6.66 7.33 7.88 8.73 7.10 8.12 9.08 10.27 6.72 7.52 8.34 9.31 6.01 6.77 7.50 8.33 6.70 7.72 8.69 9.94 8.47 9.37 10.32 11.47 7.93 8.88 9.77 10.96 7.39 8.42 9.40 10.54 3 .90 .95 .975 .99 8.98 10.06 11.08 12.43 9.34 10.23 10.98 12.01 7.93 8.72 9.43 10.33 8.44 9.11 9.75 10.53 8.53 9.52 10.61 11.87 8.63 9.51 10.30 11.30 7.30 8.07 8.80 9.67 7.63 8.31 8.98 9.80 8.09 9.11 10.18 11.50 7.94 8.77 9.59 10.50 6.70 7.50 8.25 9.09 7.67 8.75 9.73 10.89 9.73 10.66 11.48 12.66 9.09 10.08 10.93 12.19 8.55 9.48 10.41 11.64 4 .90 .95 .975 .99 9.96 11.10 12.17 13.50 10.60 11.51 12.30 13.36 8.54 9.38 10.13 11.07 9.32 10.05 10.72 11.59 9.51 10.54 11.61 13.08 9.90 10.83 11.62 12.62 7.87 8.73 9.47 10.42 8.56 9.30 9.98 10.73 9.09 10.14 11.17 12.67 9.17 10.01 10.89 11.90 7.31 8.14 8.91 9.76 8.66 9.73 10.87 12.33 10.88 11.85 12.81 13.99 10.24 11.19 12.20 13.39 9.64 10.66 11.53 12.84 5 .90 .95 .975 .99 10.94 12.14 13.22 14.47 11.81 12.76 13.68 14.66 9.19 10.00 10.74 11.77 10.21 10.99 11.63 12.50 10.45 11.66 12.72 14.06 11.03 12.01 12.89 14.13 8.53 9.33 10.09 11.15 9.41 10.13 10.82 11.67 9.99 11.20 12.28 13.56 10.36 11.33 12.22 13.29 7.94 8.75 9.54 10.52 9.56 10.73 11.93 13.23 12.07 13.06 13.99 15.16 11.33 12.38 13.38 14.50 10.70 11.84 12.86 13.95 ∗ Table 2: Size and power of the sup LR1,T (na = 1, ε), U D max LR1,T and CU SQ tests in a dynamic model with GARCH(1,1) errors √ (DGP: yt = c + αyt−1 + et , et = ut ht , with ut ∼ i.i.d. N(0, 1), ht = τ 1 + τ 2 1 (t > [0.5T ]) + γe2t−1 + ρht−1 , h0 = τ 1 / (1 − γ − ρ), c = 0.5, τ 1 = 0.1, ρ = 0.2; ε = 0.15). T = 100 α = 0.2 γ = 0.3 γ = 0.1 γ = 0.5 α = 0.7 γ = 0.3 γ = 0.1 γ = 0.5 τ2 LR UDmax CUSQ LR UDmax CUSQ LR UDmax CUSQ LR UDmax CUSQ LR UDmax CUSQ LR UDmax CUSQ 0 0.05 0.1 0.15 0.2 0.3 0.059 0.171 0.396 0.593 0.744 0.902 0.059 0.167 0.373 0.575 0.725 0.888 0.029 0.158 0.354 0.574 0.693 0.851 0.083 0.165 0.307 0.432 0.542 0.741 0.086 0.171 0.307 0.409 0.542 0.738 0.039 0.103 0.232 0.349 0.446 0.626 0.098 0.151 0.224 0.312 0.415 0.535 0.099 0.155 0.228 0.312 0.408 0.540 0.042 0.082 0.136 0.199 0.270 0.370 0.066 0.164 0.383 0.591 0.741 0.897 0.061 0.158 0.367 0.573 0.723 0.887 0.029 0.149 0.356 0.564 0.684 0.856 0.078 0.147 0.300 0.425 0.534 0.724 0.084 0.149 0.297 0.414 0.534 0.724 0.038 0.100 0.232 0.330 0.441 0.624 0.097 0.137 0.218 0.307 0.384 0.534 0.092 0.140 0.224 0.308 0.385 0.534 0.039 0.080 0.138 0.201 0.259 0.376 T = 200 α = 0.2 γ = 0.3 γ = 0.1 γ = 0.5 α = 0.7 γ = 0.3 γ = 0.1 γ = 0.5 τ2 LR UDmax CUSQ LR UDmax CUSQ LR UDmax CUSQ LR UDmax CUSQ LR UDmax CUSQ LR UDmax CUSQ 0 0.05 0.1 0.15 0.2 0.3 0.049 0.315 0.709 0.918 0.980 0.997 0.044 0.311 0.692 0.910 0.977 0.996 0.034 0.335 0.751 0.928 0.979 0.997 0.058 0.217 0.446 0.672 0.780 0.910 0.060 0.202 0.431 0.648 0.764 0.903 0.035 0.203 0.455 0.649 0.764 0.878 0.064 0.129 0.263 0.404 0.510 0.682 0.063 0.123 0.249 0.384 0.497 0.662 0.045 0.110 0.225 0.345 0.456 0.601 0.055 0.311 0.702 0.918 0.981 0.997 0.056 0.303 0.682 0.912 0.980 0.997 0.036 0.332 0.734 0.923 0.981 0.998 0.061 0.208 0.442 0.648 0.777 0.903 0.064 0.202 0.428 0.641 0.766 0.898 0.034 0.205 0.448 0.643 0.763 0.877 0.060 0.122 0.257 0.386 0.496 0.676 0.061 0.115 0.241 0.370 0.489 0.654 0.040 0.100 0.222 0.335 0.441 0.606 ∗ Table 3: Size and power of the sup LR1,T (na , ε), U D max LR1,T and CU SQ tests with normal errors and two variance breaks (DGP: yt = et ; et ∼ i.i.d. N (0, 1 + θ1(T1v < t ≤ T2v ), T = 200) θ 0 0.25 0.5 0.75 1 1.25 1.5 2 2.5 3 T1v = [.3T ], T2v = [.6T ] ε = 0.10 ε = 0.15 na = 1 na = 2 UDmax na = 1 na = 2 UDmax 0.035 0.049 0.111 0.164 0.213 0.291 0.356 0.456 0.621 0.693 0.034 0.040 0.120 0.260 0.418 0.575 0.703 0.835 0.935 0.959 0.036 0.045 0.103 0.195 0.289 0.404 0.513 0.701 0.848 0.895 0.033 0.066 0.117 0.171 0.239 0.328 0.405 0.530 0.686 0.728 0.025 0.054 0.159 0.294 0.493 0.674 0.778 0.893 0.959 0.983 0.030 0.064 0.121 0.209 0.340 0.495 0.613 0.761 0.882 0.919 CU SQ 0.025 0.031 0.059 0.085 0.124 0.147 0.197 0.276 0.375 0.430 T1v = [.2T ], T2v = [.8T ] ε = 0.10 ε = 0.15 na = 1 na = 2 UDmax na = 1 na = 2 UDmax 0.035 0.067 0.158 0.263 0.390 0.538 0.647 0.798 0.907 0.943 0.034 0.043 0.138 0.283 0.472 0.647 0.780 0.915 0.971 0.987 0.036 0.062 0.139 0.265 0.390 0.558 0.676 0.841 0.931 0.961 0.033 0.063 0.166 0.276 0.428 0.563 0.706 0.828 0.930 0.963 0.025 0.052 0.170 0.360 0.520 0.707 0.837 0.946 0.986 0.993 0.030 0.064 0.165 0.287 0.442 0.606 0.731 0.868 0.950 0.977 CU SQ 0.025 0.035 0.036 0.044 0.061 0.053 0.065 0.083 0.133 0.120 ∗ Table 4: Size of the sup LR2,T (ma = 1, na = 1, ε|n = 0, ma = 1) test with different trimming parameter ε in the case of normal errors (DGP: yt = μ1 + μ2 1(t > [0.5T ]) + et , et ∼ i.i.d. N (0, 1), μ1 = 0). T = 100 μ2 \ε 0 0.1 0.25 0.5 0.75 1 2 4 10 20 T = 200 0.1 0.15 0.2 0.25 0.1 0.15 0.2 0.25 0.045 0.038 0.037 0.037 0.043 0.034 0.030 0.034 0.038 0.031 0.042 0.028 0.039 0.035 0.047 0.031 0.023 0.032 0.033 0.030 0.030 0.033 0.034 0.036 0.046 0.031 0.028 0.031 0.032 0.035 0.023 0.030 0.030 0.033 0.041 0.031 0.028 0.027 0.031 0.027 0.039 0.045 0.034 0.031 0.044 0.034 0.041 0.034 0.038 0.040 0.032 0.046 0.034 0.025 0.033 0.027 0.029 0.026 0.033 0.034 0.030 0.036 0.035 0.029 0.035 0.020 0.028 0.024 0.025 0.023 0.031 0.037 0.030 0.027 0.031 0.017 0.029 0.026 0.022 0.021 ∗ Table 5: Power of the sup LR2,T (ma = 1, na = 1, ε|n = 0, ma = 1) test with different trimming parameter ε in the case of normal errors (DGP: yt = μ1 + μ2 1(t > [0.5T ]) + et , et ∼ i.i.d. N (0, 1 + θ1(t > [0.25T ])). T = 100 ε = 0.1 ε = 0.2 θ\μ2 0 0.1 0.5 2 4 10 20 0 0.1 0.5 2 4 10 20 0.25 0.5 0.75 1 1.25 1.5 2 3 4 0.063 0.101 0.150 0.237 0.270 0.388 0.533 0.760 0.887 0.046 0.094 0.162 0.233 0.300 0.379 0.519 0.771 0.876 0.047 0.089 0.133 0.218 0.319 0.378 0.496 0.771 0.865 0.056 0.090 0.168 0.212 0.293 0.419 0.557 0.779 0.892 0.065 0.099 0.177 0.222 0.353 0.417 0.556 0.830 0.908 0.063 0.096 0.181 0.244 0.362 0.448 0.598 0.843 0.909 0.053 0.101 0.178 0.242 0.327 0.398 0.559 0.802 0.916 0.056 0.091 0.168 0.270 0.318 0.443 0.592 0.827 0.921 0.040 0.092 0.174 0.285 0.323 0.431 0.586 0.823 0.910 0.043 0.097 0.160 0.226 0.335 0.435 0.558 0.825 0.920 0.049 0.077 0.176 0.225 0.316 0.425 0.588 0.822 0.924 0.045 0.096 0.177 0.231 0.375 0.448 0.602 0.857 0.927 0.044 0.091 0.176 0.236 0.383 0.462 0.620 0.863 0.943 0.047 0.101 0.171 0.235 0.321 0.445 0.594 0.838 0.940 T = 200 ε = 0.1 ε = 0.2 θ\μ2 0 0.1 0.5 2 4 10 20 0 0.1 0.5 2 4 10 20 0.25 0.5 0.75 1 1.25 1.5 2 3 4 0.052 0.175 0.311 0.485 0.648 0.771 0.918 0.990 0.998 0.066 0.177 0.352 0.506 0.643 0.771 0.907 0.996 1.000 0.066 0.153 0.340 0.469 0.660 0.773 0.928 0.992 1.000 0.077 0.204 0.361 0.518 0.716 0.821 0.933 0.996 1.000 0.084 0.178 0.382 0.553 0.716 0.827 0.962 0.999 1.000 0.092 0.207 0.369 0.529 0.717 0.842 0.942 0.998 1.000 0.090 0.219 0.365 0.567 0.741 0.821 0.955 0.996 1.000 0.063 0.205 0.383 0.551 0.695 0.834 0.943 0.997 1.000 0.067 0.188 0.385 0.566 0.685 0.813 0.943 0.998 1.000 0.059 0.165 0.364 0.529 0.694 0.824 0.953 0.996 1.000 0.073 0.216 0.376 0.542 0.729 0.852 0.950 0.996 1.000 0.074 0.185 0.384 0.585 0.745 0.851 0.972 0.999 1.000 0.067 0.199 0.385 0.574 0.760 0.871 0.961 0.999 1.000 0.071 0.212 0.381 0.599 0.770 0.851 0.973 0.998 1.000 ∗ Table 6: Size of the sup LR3,T (ma = 1, na = 1, ε|m = 0, na = 1) test with different trimming parameter ε in the case of normal errors (DGP: yt = μ1 + et , et ∼ i.i.d. N (0, 1 + θ1(t > [0.5T ]), μ1 = 0). T = 100 T = 200 θ\ε 0.1 0.15 0.2 0.25 0.1 0.15 0.2 0.25 0 0.1 0.25 0.5 0.75 1 2 4 10 20 0.043 0.050 0.042 0.044 0.039 0.033 0.046 0.030 0.034 0.046 0.053 0.053 0.042 0.024 0.039 0.043 0.045 0.054 0.043 0.039 0.051 0.033 0.042 0.038 0.037 0.045 0.039 0.035 0.030 0.027 0.031 0.037 0.023 0.038 0.033 0.027 0.022 0.020 0.027 0.027 0.042 0.027 0.034 0.036 0.043 0.029 0.038 0.038 0.037 0.032 0.041 0.035 0.044 0.035 0.038 0.044 0.032 0.032 0.035 0.039 0.039 0.033 0.039 0.035 0.040 0.042 0.029 0.030 0.031 0.030 0.036 0.026 0.040 0.028 0.034 0.029 0.013 0.014 0.015 0.012 (DGP: yt = c + αyt−1 + et , et ∼ i.i.d. N (0, 1 + θ1(t > [0.5T ]), c = 0, α = 0.5). T = 100 T = 200 θ\ε 0.1 0.15 0.2 0.25 0.1 0.15 0.2 0.25 0 0.1 0.25 0.5 0.75 1 2 4 10 20 0.069 0.057 0.057 0.050 0.055 0.065 0.047 0.052 0.050 0.040 0.066 0.060 0.055 0.058 0.055 0.055 0.066 0.053 0.063 0.065 0.066 0.062 0.055 0.048 0.057 0.051 0.062 0.039 0.050 0.059 0.055 0.056 0.049 0.043 0.046 0.042 0.045 0.025 0.026 0.024 0.049 0.044 0.039 0.051 0.043 0.044 0.043 0.030 0.043 0.048 0.043 0.047 0.044 0.044 0.036 0.053 0.040 0.051 0.038 0.038 0.050 0.048 0.053 0.050 0.036 0.045 0.040 0.031 0.034 0.034 0.042 0.039 0.035 0.035 0.034 0.028 0.027 0.017 0.018 0.025 ∗ Table 7: Power of the sup LR3,T (ma = 1, na = 1, ε|m = 0, na = 1) test with different trimming parameter ε in the case of normal errors (DGP: yt = μ1 + μ2 1(t > [0.25T ]) + et , et ∼ i.i.d. N (0, 1 + θ1(t > [0.5T ]), μ1 = 0). T = 100 ε = 0.1 μ2 \θ 0.1 0.25 0.5 0.75 1 4 ε = 0.2 0 0.1 0.5 2 4 10 20 0 0.1 0.5 2 4 10 20 0.050 0.096 0.349 0.670 0.901 1.000 0.050 0.092 0.351 0.663 0.899 1.000 0.055 0.092 0.340 0.651 0.892 1.000 0.058 0.082 0.300 0.580 0.853 1.000 0.059 0.078 0.263 0.538 0.821 1.000 0.057 0.074 0.255 0.503 0.799 1.000 0.059 0.080 0.245 0.485 0.785 1.000 0.050 0.117 0.353 0.702 0.930 1.000 0.049 0.115 0.350 0.696 0.929 1.000 0.043 0.110 0.334 0.692 0.929 1.000 0.034 0.088 0.305 0.625 0.901 1.000 0.031 0.077 0.283 0.586 0.866 1.000 0.037 0.077 0.283 0.586 0.866 1.000 0.030 0.077 0.243 0.544 0.811 1.000 T = 200 ε = 0.1 μ2 \θ 0.1 0.25 0.5 0.75 1 4 ε = 0.2 0 0.1 0.5 2 4 10 20 0 0.1 0.5 2 4 10 20 0.059 0.175 0.650 0.939 1.000 1.000 0.062 0.170 0.609 0.959 0.999 1.000 0.054 0.178 0.585 0.934 0.997 1.000 0.046 0.140 0.556 0.913 0.995 1.000 0.043 0.136 0.518 0.901 0.989 1.000 0.045 0.136 0.494 0.882 0.988 1.000 0.049 0.138 0.466 0.847 0.987 1.000 0.059 0.192 0.681 0.963 1.000 1.000 0.056 0.179 0.655 0.965 0.998 1.000 0.044 0.183 0.673 0.963 0.999 1.000 0.058 0.158 0.583 0.913 0.998 1.000 0.055 0.142 0.542 0.909 0.998 1.000 0.053 0.132 0.506 0.878 0.996 1.000 0.042 0.135 0.482 0.883 0.995 1.000 ∗ Table 8: Size of the sup LR4,T (ma , na ) and U D max LR4,T tests in the case of GARCH(1,1) errors √ (DGP: yt = et , et = ut ht , with ut ∼ i.i.d. N (0, 1), ht = τ 1 + γe2t−1 + ρht−1 , τ 1 = 1, ρ = 0.2, h0 = τ 1 / (1 − γ − ρ)) T=100 γ ma = na = 1 0.1 0.3 0.5 0.044 0.048 0.072 ε = 0.1 ma = 1, na = 2 ma = 2, na = 1 0.046 0.065 0.083 0.047 0.051 0.075 UDmax ma = na = 1 0.050 0.073 0.085 0.037 0.041 0.065 ε = 0.2 ma = 1, na = 2 ma = 2, na = 1 0.040 0.052 0.069 0.035 0.042 0.059 UDmax 0.046 0.055 0.061 T=200 γ ma = na = 1 0.1 0.3 0.5 0.034 0.032 0.039 ε = 0.1 ma = 1, na = 2 ma = 2, na = 1 0.035 0.041 0.044 0.034 0.035 0.041 UDmax ma = na = 1 0.041 0.043 0.051 0.036 0.036 0.040 ε = 0.2 ma = 1, na = 2 ma = 2, na = 1 0.034 0.037 0.040 0.037 0.031 0.024 UDmax 0.037 0.040 0.040 ∗ Table 9: Power of the sup LR4,T (ma , na ) and U D max LR4,T tests for DGPs with one break in coefficients and two breaks in variance (DGP: yt = μ1 + μ2 1(t > T c ) + et , et ∼ i.i.d. N (0, 1 + θ1(T1v < t ≤ T2v )), μ1 = 0, μ2 = θ, ε = 0.1) ma = 1 ma = 1 UDmax na = 1 na = 2 T c = T v1 = [.3T ], T v2 = [.6T ] ma = 1 ma = 1 UDmax na = 1 na = 2 T c = T2v = [.6T ], T v1 = [.3T ] θ ma = 1 ma = 1 UDmax na = 1 na = 2 T c = [.3T ], T v1 = [.5T ], T v2 = [.6T ] T = 100 0.25 0.5 0.75 1 1.25 1.5 0.081 0.263 0.576 0.854 0.980 1.000 0.069 0.263 0.560 0.860 0.974 1.000 0.090 0.280 0.586 0.857 0.976 0.997 0.091 0.314 0.655 0.892 0.988 0.998 0.082 0.280 0.631 0.902 0.985 0.999 0.097 0.313 0.643 0.896 0.984 1.000 0.083 0.262 0.592 0.874 0.982 0.999 0.069 0.233 0.570 0.861 0.974 0.997 0.25 0.5 0.75 1 1.25 1.5 0.119 0.552 0.925 1.000 1.000 1.000 0.124 0.561 0.929 1.000 1.000 1.000 0.129 0.569 0.925 1.000 1.000 1.000 0.156 0.633 0.961 1.000 1.000 1.000 0.138 0.622 0.958 1.000 1.000 1.000 0.159 0637 0.955 1.000 1.000 1.000 0.128 0.547 0.935 1.000 1.000 1.000 0.109 0.515 0.927 0.999 1.000 1.000 0.086 0.269 0.583 0.877 0.982 0.998 ma = 1 ma = 1 UDmax na = 1 na = 2 T c = [.5T ], T v1 = [.3T ], T v2 = [.6T ] ma = 1 ma = 1 UDmax na = 1 na = 2 T c = [.6T ], T v1 = [.3T ], T v2 = [.5T ] 0.089 0.320 0.687 0.895 0.986 1.000 0.085 0.294 0.661 0.906 0.983 1.000 0.097 0.326 0.691 0.918 0.987 1.000 0.092 0.318 0.648 0.890 0.983 1.000 0.079 0.281 0.628 0.886 0.987 0.999 0.100 0.315 0.650 0.888 0.987 0.999 0.152 0.642 0.968 1.000 1.000 1.000 0.153 0.645 0.976 1.000 1.000 1.000 0.158 0.656 0.971 1.000 1.000 1.000 0.142 0.628 0.966 1.000 1.000 1.000 0.134 0.593 0.956 0.999 1.000 1.000 0.149 0.624 0.962 1.000 1.000 1.000 T = 200 0.125 0.545 0.931 1.000 1.000 1.000 ∗ Table 10: Power of the sup LR4,T (ma , na ) and U D max LR4,T tests for DGPs with two breaks in coefficients and one break in variance c c (DGP: yt = μ1 + μ2 1(T1 < t ≤ T2 ) + et , et ∼ i.i.d. N (0, 1 + θ1(t > T v )), μ1 = 0, μ2 = θ, ε = 0.1). ma = 1 ma = 2 UDmax na = 1 na = 1 T1c = T v = [.3T ], T c2 = [.6T ] ma = 1 ma = 2 UDmax na = 1 na = 1 T1c = [.3T ], T c2 = T v = [.6T ] θ ma = 1 ma = 2 UDmax na = 1 na = 1 T1c = [.5T ], T c2 = [.6T ], T v = [.3T ] T = 100 0.25 0.5 0.75 1 1.25 1.5 0.064 0.107 0.238 0.412 0.566 0.707 0.085 0.181 0.405 0.688 0.849 0.935 0.081 0.164 0.352 0.612 0.800 0.902 0.076 0.140 0.294 0.512 0.709 0.854 0.080 0.194 0.479 0.809 0.958 0.995 0.091 0.177 0.423 0.750 0.923 0.987 0.051 0.085 0.137 0.237 0.299 0.409 0.051 0.092 0.161 0.278 0.436 0.564 0.25 0.5 0.75 1 1.25 1.5 0.079 0.275 0.590 0.844 0.964 0.995 0.117 0.421 0.814 0.976 0.996 1.000 0.097 0.379 0.774 0.963 0.992 1.000 0.083 0.330 0.688 0.919 0.991 1.000 0.104 0.490 0.913 0.997 1.000 1.000 0.113 0.447 0.870 0.994 1.000 1.000 0.060 0.145 0.333 0.524 0.736 0.853 0.066 0.180 0.420 0.673 0.868 0.945 0.056 0.101 0.165 0.276 0.405 0.558 ma = 1 ma = 2 UDmax na = 1 na = 1 T1c = [.3T ], T c2 = [.6T ], T v = [.5T ] ma = 1 ma = 2 UDmax na = 1 na = 1 T1c = [.3T ], T c2 = [.5T ], T v = [.6T ] 0.073 0.141 0.282 0.498 0.684 0.835 0.080 0.194 0.468 0.770 0.933 0.987 0.087 0.178 0.416 0.720 0.905 0.980 0.061 0.098 0.187 0.302 0.399 0.520 0.065 0.150 0.358 0.677 0.876 0.964 0.080 0.134 0.307 0.577 0.806 0.934 0.086 0.335 0.681 0.891 0.983 0.999 0.115 0.486 0.895 0.992 1.000 1.000 0.114 0.453 0.852 0.986 1.000 1.000 0.075 0.224 0.480 0.700 0.861 0.944 0.088 0.378 0.818 0.982 0.999 1.000 0.096 0.340 0.769 0.960 0.999 1.000 T = 200 0.073 0.185 0.408 0.654 0.859 0.942 Table11: Finite sample performance of the specific to general sequential procedure to select the number of breaks in coefficients and variance (DGP: yt = μ1 + μ2 1(t > T c ) + et , et ∼ i.i.d. N (0, 1 + θ1(t > T v )), ε = 0.15, T = 200). m=n=0 μ2 = θ = 1 prob(m = 0, n = 0) prob(m = 0, n = 1) prob(m = 0, n = 2) prob(m = 1, n = 0) prob(m = 1, n = 1) prob(m = 1, n = 2) prob(m = 2, n = 0) prob(m = 2, n = 1) prob(m = 2, n = 2) μ2 = θ = 2 m=n=1 T c = [.25T ], T v = [.75T ] μ2 = θ = 1 μ2 = θ = 2 μ2 = 1, θ = 3 0.906 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.042 0.000 0.002 0.003 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.043 0.286 0.001 0.000 0.023 0.343 0.028 0.004 0.007 0.680 0.954 0.956 0.936 0.628 0.937 0.963 0.000 0.009 0.002 0.016 0.019 0.007 0.011 0.010 0.002 0.008 0.000 0.000 0.000 0.010 0.001 0.000 0.000 0.016 0.023 0.025 0.022 0.011 0.020 0.021 0.000 0.001 0.000 0.000 0.000 0.001 0.003 0.002 m=n=1 T c = T v = [.5T ] μ2 = θ = 1 μ2 = 1, θ = 3 prob(m = 0, n = 0) prob(m = 0, n = 1) prob(m = 0, n = 2) prob(m = 1, n = 0) prob(m = 1, n = 1) prob(m = 1, n = 2) prob(m = 2, n = 0) prob(m = 2, n = 1) prob(m = 2, n = 2) m=n=1 T c = [.5T ], T v = [.7T ] μ2 = 1, θ = 3 μ2 = 1, θ = 5 μ2 = 1 m = 1, n = 0 T c = [.5T ] μ2 = 2 m = 0, n = 1 T v = [.5T ] θ=2 μ2 = 3 θ=1 0.000 0.000 0.000 0.000 0.000 0.234 0.005 θ=3 0.000 0.003 0.029 0.000 0.000 0.000 0.706 0.924 0.924 0.000 0.002 0.000 0.000 0.000 0.013 0.027 0.031 0.240 0.000 0.931 0.935 0.934 0.009 0.000 0.000 0.729 0.917 0.039 0.038 0.038 0.035 0.040 0.041 0.008 0.034 0.000 0.000 0.000 0.002 0.003 0.003 0.005 0.000 0.028 0.023 0.024 0.001 0.000 0.000 0.014 0.017 0.002 0.004 0.004 0.000 0.001 0.001 0.001 0.001 0.000 0.000 0.000 0.000 0.000 0.000 Note: prob(m = j, n = i) represents the probability of choosing j breaks in mean and i breaks in variance. The upper bounds for the coefficients and the variance breaks are set to M = 2 and N = 2. Table 12: Empirical results for the in‡ation rate a) Tests for structural changes in mean and/or variance supLR4;T U D max LR4;T ma = 1 ma = 2 ma = 3 M = 3; N = 3 na = 1 12.18 10.78 9.58 15.91 na = 2 15.27 13.33 11.81 na = 3 15.91 15.06 14.03 na na na na =0 =1 =2 =3 ma = 1 22.50 8.54 5.72 9.90 supLR3;T ma = 2 19.42 7.57 6.62 9.72 b) Tests for structural changes in mean U D max LR3;T supSeq9;T ma = 3 M =3 ma = 1 ma = 2 ma = 3 15.93 22.50 10.17 9.38 4.59 7.04 8.54 6.19 6.99 4.59 7.37 7.37 2.79 4.96 3.10 10.03 10.03 2.74 4.80 4.74 c) Tests for structural changes in variance supLR2;T U D max LR2;T supSeq10;T na = 1 na = 2 na = 3 N =2 na = 1 na = 2 na = 3 ma = 0 16.00 21.30 16.49 21.30 18.69 13.05 5.21 ma = 1 9.37 13.77 14.00 14.00 18.97 16.21 5.54 ma = 2 3.33 8.26 11.22 11.22 18.97 16.79 6.73 ma = 3 1.69 9.14 11.90 11.90 19.93 16.79 7.18 Notes: *,** and *** indicate signi…cance at the 10%, 5% and 1% levels, respectively. break dates 1982:1 break dates 1971:3 1971:3 1983:2 1982:1 2006:3 2006:3 Table 13: Empirical results for the real interest rate a) Tests for structural changes in mean and/or variance supLR4;T U D max LR4;T ma = 1 ma = 2 ma = 3 M = 3; N = 2 na = 1 8.34 4.66 7.50 11.44 na = 2 8.93 11.44 6.54 na = 0 na = 1 na = 2 ma = 1 14.66 8.42 8.17 supLR3;T ma = 2 25.75 25.75 25.71 b) Tests for structural changes in mean U D max LR3;T supSeq9;T ma = 3 M =3 ma = 1 ma = 2 ma = 3 20.60 25.75 27.86 7.63 3.33 24.08 25.75 25.82 6.20 2.99 21.57 25.71 25.48 6.87 3.33 break dates 1972:3 1972:3 1972:3 c) Tests for structural changes in variance supLR2;T U D max LR2;T supSeq10;T break dates na = 1 na = 2 N =2 na = 1 na = 2 15.96 30.03 17.05 5.89 1972:3 1981:2 ma = 0 30.03 ma = 1 21.70 12.02 21.70 4.25 6.36 1972:3 ma = 2 16.20 10.72 16.20 15.29 6.45 1964:3 1972:3 ma = 3 16.42 11.62 16.42 10.88 6.45 1966:4 1969:3 Notes: *,** and *** indicate signi…cance at the 10%, 5% and 1% levels, respectively. 1980:3 1980:3 1980:3 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 Figure 1: Annual change of the quarterly US in‡ation rate: 1959:1-2018:4 14 12 10 8 6 4 2 0 -2 -4 -6 -8 1961 1965 1969 1973 1977 1981 1985 Figure 2: US ex-post real interest rate: 1961:1-1986:3