Fractional Unit Root Tests Allowing for a Structural Change in Trend (Tables)

Table 1: The E¤ect of a Structural Break in Trend on the LM tests d0 LMT LMT; ^ d 0.5 0.55 0.6 0.65 0.7 T = 150 0.75 0.8 0.85 0.9 0.95 1 0.767 1 0.617 1 0.458 0.997 0.290 0.986 0.187 0.949 0.117 0.858 0.070 0.703 0.044 0.485 0.029 0.303 0.020 0.169 0.004 0.036 m Note: 5% nominal size tests are used. The DGP is speci…ed by yt = b Bt + ut and ut = ( d t )1t 1 with t i:i:d: N (0; 1) where d0 = m + d, d 2 [ 0:5; 0), and m = 1. The values of model parameters are set to 0 = 0:5; b = 0:5. The test LMT , designed for Model A1, ignores a structural break in the DGP, while the test LMT; ^ d , designed for Model A2, is well speci…ed with the estimate of the break date obtained from the dynamic regression (10). All results are obtained using 10,000 Monte Carlo replications. Table 2: Rejection Probabilities of the LMT test for Model A0 d0 0.5 0.55 0.6 0.65 0.7 1 1 1 0.997 0.983 1 1 1 1 1 0.75 0.8 0.85 0.9 0.95 1 T = 150 0.924 0.778 0.544 0.307 0.154 0.051 T = 500 1 0.997 0.920 0.608 0.260 0.056 m Note: 5% nominal size tests are used. The DGP is de…ned by yt = 1 + 1 t + ut , and ut = ( d t )1t 1 with t i:i:d: N (0; 1) where d0 = m + d, d 2 [ 0:5; 0], and m = 1. The values of model parameters are set to 1 = 1:72; 1 = 0:03. All results are obtained using 10,000 Monte Carlo replications. Table 3: Rejection Probabilities of the LMT;^ and LMT tests for Model A1 d0 0.5 0.55 0.6 0.65 0.7 LMT; ^ s LMT; ^ d LMT 1 1 1 1 1 1 1 1 1 0.989 0.997 0.997 LMT; ^ s LMT; ^ d LMT 1 1 1 1 1 1 1 1 1 1 1 1 0.75 T = 150 0.990 0.969 0.988 0.946 0.981 0.927 1 1 1 T = 500 1 1 1 0.8 0.85 0.9 0.95 1 0.897 0.839 0.772 0.765 0.655 0.541 0.589 0.445 0.309 0.410 0.263 0.151 0.255 0.134 0.052 0.999 0.996 0.997 0.970 0.932 0.925 0.802 0.674 0.616 0.494 0.330 0.250 0.252 0.118 0.055 m Note: 5% nominal size tests are used. The DGP is de…ned by yt = b Ct + ut , and ut = ( d t )1t 1 with t i:i:d: N (0; 1) where d0 = m + d, d 2 [ 0:5; 0], and m = 1. The values of model parameters are set to 0 = 0:5; b = 1. All results are obtained using 10,000 Monte Carlo replications. Table 4: Rejection Probabilities of the LMT;^ tests for Model A2 d0 0.5 0.55 0.6 0.65 LMT; ^ s LMT; ^ d LMT; ^ tr 1 0.998 1 1 0.994 0.999 1 0.982 0.999 0.998 0.951 0.997 LMT; ^ s LMT; ^ d LMT; ^ tr 1 1 1 1 1 1 1 1 1 1 1 1 0.7 0.85 0.9 0.95 1 T = 150 0.987 0.936 0.820 0.882 0.765 0.606 0.981 0.922 0.794 0.610 0.409 0.587 0.387 0.246 0.369 0.211 0.130 0.205 0.077 0.039 0.072 T = 500 1 0.995 0.974 0.893 1 0.997 0.904 0.717 0.917 0.567 0.426 0.591 0.229 0.165 0.244 0.063 0.048 0.071 1 0.997 1 0.75 0.8 m Note: 5% nominal size tests are used. The DGP is de…ned by yt = b Bt + ut , and ut = ( d t )1t 1 with t i:i:d: N (0; 1) where d0 = m + d, d 2 [ 0:5; 0], and m = 1. The values of model parameters are set to 0 = 0:5; b = 1. All results are obtained using 10,000 Monte Carlo replications. Table 5: Rejection Probabilities of the LMT;^ tests for Model A3 d0 0.5 0.55 0.6 0.65 0.7 LMT; ^ s LMT; ^ d LMT; ^ tr 1 1 1 1 1 1 1 0.999 1 0.998 0.995 0.996 0.990 0.977 0.981 LMT; ^ s LMT; ^ d LMT; ^ tr 1 1 1 1 1 1 1 1 1 1 1 1 1 0.999 1 0.75 0.8 0.85 0.9 0.95 1 T = 150 0.953 0.857 0.921 0.806 0.926 0.798 0.679 0.616 0.587 0.480 0.419 0.380 0.305 0.258 0.218 0.175 0.128 0.083 T = 500 1 0.995 0.996 0.974 0.999 0.993 0.924 0.851 0.899 0.659 0.561 0.571 0.350 0.259 0.233 0.156 0.101 0.067 m Note: 5% nominal size tests are used. The DGP is de…ned by yt = b Bt + ut , and ut = ( d t )1t 1 with t i:i:d: N (0; 1) where d0 = m + d, d 2 [ 0:5; 0], and m = 1. The values of model parameters are set to 0 = 0:5; b = 1. All results are obtained using 10,000 Monte Carlo replications. Table 6: Rejection Probabilities of LMT;^ and LMT for Model A1 with short-run dynamics 0.5 0.55 0.6 0.65 d0 0.75 0.7 0.8 0.85 0.9 0.95 1 T = 150 -0.5 LM ^ T; s LM ^ T; d LMT 1 1 1 0.999 0.999 0.999 0.997 0.997 0.994 0.987 0.986 0.974 0.965 0.962 0.928 0.894 0.891 0.813 0.776 0.769 0.639 0.618 0.598 0.442 0.449 0.429 0.259 0.305 0.277 0.137 0.187 0.168 0.059 0 LM ^ T; s LM ^ T; d LMT 0.993 0.991 0.986 0.986 0.981 0.969 0.967 0.955 0.930 0.927 0.904 0.851 0.862 0.824 0.728 0.774 0.711 0.572 0.656 0.582 0.411 0.528 0.435 0.267 0.424 0.321 0.163 0.329 0.224 0.102 0.244 0.143 0.046 0.3 LM ^ T; s LM ^ T; d LMT 0.946 0.922 0.894 0.899 0.863 0.811 0.839 0.788 0.707 0.767 0.683 0.576 0.671 0.573 0.445 0.553 0.441 0.303 0.450 0.337 0.209 0.365 0.245 0.133 0.272 0.168 0.077 0.215 0.115 0.054 0.158 0.069 0.026 0.6 LM ^ T; s LM ^ T; d LMT 0.558 0.456 0.361 0.484 0.370 0.277 0.385 0.274 0.184 0.296 0.197 0.121 0.234 0.144 0.079 0.163 0.086 0.042 0.127 0.060 0.029 0.091 0.039 0.018 0.061 0.024 0.013 0.048 0.020 0.013 0.029 0.010 0.007 0.8 LM ^ T; s LM ^ T; d LMT 0.096 0.044 0.019 0.074 0.033 0.016 0.049 0.019 0.011 0.035 0.013 0.012 0.025 0.014 0.018 0.017 0.012 0.026 0.014 0.015 0.033 0.010 0.022 0.041 0.012 0.024 0.042 0.011 0.027 0.038 0.008 0.036 0.026 -0.5 LM ^ T; s LM ^ T; d LMT 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0.999 0.999 1 0.979 0.974 0.980 0.856 0.829 0.830 0.581 0.537 0.502 0.305 0.262 0.215 0.129 0.100 0.066 0 LM ^ T; s LM ^ T; d LMT 1 1 1 1 1 1 1 1 1 1 1 1 0.999 0.999 0.999 0.992 0.985 0.979 0.936 0.894 0.858 0.781 0.695 0.595 0.576 0.451 0.330 0.403 0.273 0.168 0.263 0.143 0.058 0.3 LM ^ T; s LM ^ T; d LMT 1 1 1 1 1 1 1 1 1 0.998 0.996 0.995 0.979 0.967 0.957 0.920 0.869 0.837 0.777 0.684 0.611 0.575 0.447 0.376 0.394 0.261 0.205 0.257 0.162 0.124 0.155 0.074 0.051 0.6 LM ^ T; s LM ^ T; d LMT 0.984 0.977 0.974 0.956 0.935 0.927 0.880 0.844 0.833 0.736 0.673 0.670 0.559 0.487 0.497 0.348 0.297 0.316 0.195 0.171 0.199 0.114 0.105 0.135 0.056 0.065 0.083 0.033 0.045 0.058 0.016 0.023 0.028 0.8 LM ^ T; s LM ^ T; d LMT 0.249 0.202 0.217 0.148 0.133 0.159 0.075 0.084 0.126 0.040 0.067 0.120 0.024 0.066 0.142 0.015 0.069 0.157 0.008 0.084 0.171 0.007 0.091 0.163 0.005 0.101 0.134 0.006 0.107 0.102 0.004 0.103 0.043 T = 500 Note: 5% nominal size tests are used. The DGP is de…ned by yt = and t i:i:d: N (0; 1) where d0 = m + d, d 2 [ 0:5; 0]. We set replications. m( d )1 + ut and ut = L) t = t t t 1 with (1 = 1. All results are obtained using 10,000 Monte Carlo b Ct b Table 7: Rejection Probabilities of LMT;^ for Model A2 with short-run dynamics d0 0.75 0.5 0.55 0.6 0.65 0.7 0.8 0.85 0.9 0.95 1 -0.5 LM ^ T; s LM ^ T; d LM ^ 1 0.993 1 0.999 0.982 0.996 0.996 0.955 0.990 0.985 0.908 0.969 0.949 0.816 0.926 0.861 0.680 0.822 0.713 0.521 0.671 0.515 0.350 0.478 0.354 0.233 0.335 0.215 0.136 0.203 0.100 0.054 0.098 0 LM ^ T; s LM ^ T; d LM ^ 0.991 0.912 0.988 0.978 0.851 0.965 0.943 0.779 0.931 0.885 0.668 0.868 0.795 0.562 0.774 0.652 0.425 0.645 0.507 0.311 0.496 0.364 0.205 0.373 0.268 0.153 0.280 0.189 0.104 0.205 0.097 0.046 0.113 0.3 LM ^ T; s LM ^ T; d LM ^ 0.907 0.691 0.868 0.848 0.602 0.792 0.761 0.502 0.696 0.662 0.409 0.594 0.533 0.311 0.467 0.429 0.230 0.369 0.319 0.163 0.271 0.239 0.115 0.198 0.170 0.076 0.141 0.129 0.056 0.109 0.071 0.027 0.055 0.6 LM ^ T; s LM ^ T; d LM ^ 0.436 0.230 0.421 0.360 0.176 0.344 0.282 0.130 0.269 0.215 0.092 0.211 0.157 0.058 0.156 0.119 0.044 0.118 0.083 0.028 0.082 0.061 0.020 0.062 0.050 0.019 0.054 0.038 0.016 0.043 0.027 0.012 0.030 0.8 LM ^ T; s LM ^ T; d LM ^ 0.057 0.022 0.040 0.040 0.012 0.026 0.038 0.013 0.024 0.038 0.017 0.025 0.044 0.025 0.034 0.053 0.032 0.039 0.061 0.044 0.049 0.063 0.047 0.053 0.058 0.044 0.051 0.044 0.034 0.038 0.028 0.023 0.024 -0.5 LM ^ T; s LM ^ T; d LM ^ 1 1 1 1 1 1 1 1 1 1 1 1 1 0.993 1 0.999 0.952 1 0.963 0.824 0.975 0.785 0.594 0.825 0.445 0.311 0.490 0.197 0.129 0.221 0.068 0.039 0.081 0 LM ^ T; s LM ^ T; d LM ^ 1 1 1 1 1 1 1 0.993 1 1 0.968 1 0.998 0.907 0.998 0.969 0.793 0.978 0.831 0.617 0.865 0.575 0.408 0.617 0.329 0.225 0.362 0.169 0.108 0.190 0.076 0.050 0.099 0.3 LM ^ T; s LM ^ T; d LM ^ 1 0.993 1 1 0.974 1 0.999 0.929 0.999 0.994 0.846 0.994 0.950 0.746 0.957 0.826 0.613 0.836 0.614 0.427 0.625 0.382 0.255 0.391 0.233 0.149 0.236 0.145 0.081 0.145 0.081 0.049 0.079 0.6 LM ^ T; s LM ^ T; d LM ^ 0.975 0.788 0.976 0.930 0.709 0.933 0.828 0.596 0.834 0.687 0.475 0.690 0.521 0.340 0.514 0.366 0.222 0.353 0.257 0.144 0.240 0.178 0.091 0.161 0.125 0.060 0.113 0.099 0.040 0.090 0.065 0.030 0.067 0.8 LM ^ T; s LM ^ T; d LM ^ 0.245 0.143 0.235 0.187 0.104 0.165 0.161 0.087 0.130 0.169 0.085 0.124 0.191 0.084 0.127 0.218 0.092 0.142 0.238 0.101 0.161 0.226 0.117 0.190 0.176 0.103 0.191 0.122 0.082 0.199 0.067 0.048 0.226 T = 150 T; tr T; tr T; tr T; tr T; tr T = 500 T; tr T; tr T; tr T; tr T; tr Note: 5% nominal size tests are used. The DGP is de…ned by yt = and t i:i:d: N (0; 1) where d0 = m + d, d 2 [ 0:5; 0]. We set replications. m( d )1 and ut = L) t = t t t 1 with (1 = 1. All results are obtained using 10,000 Monte Carlo b B t + ut b Table 8: Rejection Probabilities of LMT;^ for Model A3 with short-run dynamics d0 0.75 0.5 0.55 0.6 0.65 0.7 0.8 0.85 0.9 0.95 1 -0.5 LM ^ T; s LM ^ T; d LM ^ 1 1 1 0.999 0.999 0.996 0.996 0.996 0.989 0.986 0.983 0.970 0.952 0.946 0.925 0.880 0.871 0.832 0.749 0.738 0.686 0.561 0.549 0.491 0.403 0.390 0.339 0.272 0.258 0.219 0.149 0.139 0.096 0 LM ^ T; s LM ^ T; d LM ^ 0.992 0.991 0.983 0.980 0.973 0.963 0.946 0.939 0.919 0.897 0.881 0.848 0.813 0.786 0.740 0.697 0.670 0.609 0.569 0.537 0.473 0.441 0.401 0.343 0.344 0.303 0.250 0.262 0.229 0.184 0.183 0.149 0.095 0.3 LM ^ T; s LM ^ T; d LM ^ 0.919 0.899 0.877 0.868 0.835 0.809 0.785 0.747 0.710 0.695 0.646 0.607 0.587 0.518 0.483 0.488 0.425 0.381 0.377 0.307 0.274 0.302 0.236 0.208 0.230 0.167 0.158 0.183 0.121 0.119 0.137 0.083 0.069 0.6 LM ^ T; s LM ^ T; d LM ^ 0.481 0.407 0.385 0.396 0.317 0.298 0.318 0.241 0.242 0.251 0.173 0.175 0.191 0.125 0.127 0.150 0.088 0.092 0.115 0.063 0.071 0.086 0.046 0.051 0.072 0.032 0.042 0.055 0.026 0.032 0.039 0.016 0.021 0.8 LM ^ T; s LM ^ T; d LM ^ 0.082 0.045 0.040 0.059 0.028 0.030 0.052 0.023 0.030 0.039 0.015 0.026 0.036 0.014 0.031 0.033 0.014 0.042 0.032 0.016 0.044 0.033 0.019 0.050 0.027 0.024 0.044 0.024 0.026 0.039 0.019 0.030 0.026 -0.5 LM ^ T; s LM ^ T; d LM ^ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0.999 0.993 0.999 0.970 0.936 0.972 0.807 0.741 0.801 0.501 0.434 0.476 0.243 0.199 0.212 0.103 0.079 0.073 0 LM ^ T; s LM ^ T; d LM ^ 1 1 1 1 1 1 1 1 1 1 1 1 0.999 0.991 0.998 0.979 0.953 0.970 0.876 0.829 0.835 0.674 0.622 0.591 0.439 0.379 0.326 0.286 0.226 0.177 0.164 0.124 0.077 0.3 LM ^ T; s LM ^ T; d LM ^ 1 1 1 1 1 1 0.999 0.997 0.999 0.995 0.978 0.992 0.962 0.925 0.945 0.859 0.792 0.812 0.673 0.580 0.590 0.468 0.372 0.371 0.307 0.215 0.222 0.192 0.124 0.131 0.123 0.076 0.078 0.6 LM ^ T; s LM ^ T; d LM ^ 0.978 0.942 0.971 0.935 0.864 0.917 0.843 0.739 0.818 0.697 0.572 0.667 0.521 0.393 0.493 0.348 0.240 0.343 0.220 0.138 0.231 0.140 0.080 0.162 0.091 0.050 0.108 0.070 0.035 0.087 0.050 0.025 0.063 0.8 LM ^ T; s LM ^ T; d LM ^ 0.232 0.162 0.225 0.153 0.101 0.173 0.114 0.067 0.144 0.097 0.049 0.145 0.095 0.041 0.154 0.099 0.039 0.176 0.105 0.039 0.203 0.093 0.047 0.190 0.071 0.057 0.155 0.052 0.071 0.105 0.052 0.067 0.055 T = 150 T; tr T; tr T; tr T; tr T; tr T = 500 T; tr T; tr T; tr T; tr T; tr Note: 5% nominal size tests are used. The DGP is de…ned by yt = and t i:i:d: N (0; 1) where d0 = m + d, d 2 [ 0:5; 0]. We set replications. m( d )1 and ut = L) t = t t t 1 with (1 = 1. All results are obtained using 10,000 Monte Carlo b B t + ut b Table 9: Empirical Results for the Logarithmic Price Indices of the G7 Countries ADF Canada France Germany Italy Japan U.K. U.S.A. -0.16 (4) -1.47 (6) -1.59 (1) -0.61 (7) -2.50 (11) -1.47 (7) -0.44 (2) SW Break Date LMT; ^ tr LMT; ^ tr 19.40 28.19 10.78 16.37 18.86 16.50 15.75 1986 1985 1982 1986 1980 1983 1984 9.44 34.65 18.98 29.55 18.93 9.46 12.71 9.37 7.54 8.01 11.64 8.73 4.39 3.48 May [0.45] Nov [0.43] Oct [0.35] Jul [0.45] Nov [0.31] Nov [0.38] Jun [0.40] h0:13i h0:62i h0:41i h0:52i h0:42i h0:5i h0:38i Note: (a) The numbers in parentheses are the values of the autoregressive order selected by the BIC when constructing the ADF test; (b) the numbers in brackets [ ] denote the estimated break fractions; (c) the numbers in h i are the estimates of the AR coe¢ cient in the noise component. , , and denote a statistic signi…cant at the 10%, 5%, and 1% level, respectively. Figure 1: Null Rejection Probabilities in Model A2, λb = 0.5 [− + − : LMT,λ̂tr , − ∗ − : LMpT,λ̂ , −− : LMT,λb ] tr (a) d =1, T=150 (b) d =0.8, T=150 0 100 100 90 90 80 80 70 70 60 60 Power (%) Size (%) 0 50 40 50 40 30 30 20 20 10 10 0 −4 −2 0 βb 2 0 −4 4 100 90 90 80 80 70 70 60 60 50 40 20 20 10 10 0 βb 2 4 40 30 −2 2 50 30 0 −4 0 βb (d) d0=0.8, T=300 100 Power (%) Size (%) (c) d0=1, T=300 −2 4 0 −4 −2 0 βb 2 4 Figure 2: Null Rejection Probabilities in Model A3, λb = 0.5 [− + − : LMT,λ̂tr , − ∗ − : LMpT,λ̂ , −− : LMT,λb ] tr (c) d =1, T=300 (d) d =0.8, T=300 0 100 100 90 90 80 80 70 70 60 60 Power (%) Size (%) 0 50 40 50 40 30 30 20 20 10 10 0 −4 −2 0 βb 2 0 −4 4 100 90 90 80 80 70 70 60 60 50 40 20 20 10 10 0 βb 2 4 40 30 −2 2 50 30 0 −4 0 βb (d) d0=0.8, T=300 100 Power (%) Size (%) (c) d0=1, T=300 −2 4 0 −4 −2 0 βb 2 4 Figure 3: Null Rejection Probabilities in Model 2 where (1 − 0.3L)ηt = t , λb = 0.5 [− + − : LMT,λ̂tr , − ∗ − : LM∗p , −− : LMT,λb ]. T,λ̂ tr (a) d =1, T=150 (b) d =0.8, T=150 0 100 100 90 90 80 80 70 70 60 60 Power (%) Size (%) 0 50 40 50 40 30 30 20 20 10 10 0 −4 −2 0 βb 2 0 −4 4 100 90 90 80 80 70 70 60 60 50 40 20 20 10 10 0 βb 2 4 40 30 −2 2 50 30 0 −4 0 βb (d) d0=0.8, T=300 100 Power (%) Size (%) (c) d0=1, T=300 −2 4 0 −4 −2 0 βb 2 4 Figure 4: Null Rejection Probabilities in Model A3 where (1 − 0.3L)ηt = t , λb = 0.5 [− + − : LMT,λ̂tr , − ∗ − : LM∗p , −− : LMT,λb ]. T,λ̂ tr (a) d =1, T=150 (b) d =0.8, T=150 0 100 100 90 90 80 80 70 70 60 60 Power (%) Size (%) 0 50 40 50 40 30 30 20 20 10 10 0 −4 −2 0 βb 2 0 −4 4 100 90 90 80 80 70 70 60 60 50 40 20 20 10 10 0 βb 2 4 40 30 −2 2 50 30 0 −4 0 βb (d) d0=0.8, T=300 100 Power (%) Size (%) (c) d0=1, T=300 −2 4 0 −4 −2 0 βb 2 4 Figure 5: (log) CPI for the G7 Countries 5 5 4.5 4.5 4 Log(cpi) Log(cpi) 4 3.5 3.5 3 3 2.5 2.5 2 1986May (a) Canada 1985Nov (b) France 5 5 4.8 4.5 4.6 4 4.4 Log(cpi) Log(cpi) 4.2 4 3.8 3.6 3.5 3 2.5 3.4 2 3.2 3 1.5 1982Oct (c) Germany 1986Jul (d) Italy 5 5 4.8 4.5 4.6 4.4 4 Log(cpi) Log(cpi) 4.2 4 3.5 3.8 3 3.6 3.4 2.5 3.2 3 1980Nov (e) Japan 5 4.5 Log(cpi) 4 3.5 3 2.5 1984Jun (g) United States 2 1983Nov (f) United Kingdom