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Durbin watson tables
- 1. 1 Appendix A Durbin-Watson Significance Tables The Durbin-Watson test statistic tests the null hypothesis that the residuals from an ordinary least-squares regression are not autocorrelated against the alternative that the residuals follow an AR1 process. The Durbin-Watson statistic ranges in value from 0 to 4. A value near 2 indicates non-autocorrelation; a value toward 0 indicates positive autocorrelation; a value toward 4 indicates negative autocorrelation. Because of the dependence of any computed Durbin-Watson value on the associated data matrix, exact critical values of the Durbin-Watson statistic are not tabulated for all possible cases. Instead, Durbin and Watson established upper and lower bounds for the critical values. Typically, tabulated bounds are used to test the hypothesis of zero autocorrelation against the alternative of positive first-order autocorrelation, since positive autocorrelation is seen much more frequently in practice than negative autocorrelation. To use the table, you must cross-reference the sample size against the number of regressors, excluding the constant from the count of the number of regressors. The conventional Durbin-Watson tables are not applicable when you do not have a constant term in the regression. Instead, you must refer to an appropriate set of Durbin-Watson tables. The conventional Durbin-Watson tables are also not applicable when a lagged dependent variable appears among the regressors. Durbin has proposed alternative test procedures for this case. Statisticians have compiled Durbin-Watson tables from some special cases, including: „ Regressions with a full set of quarterly seasonal dummies. „ Regressions with an intercept and a linear trend variable (CURVEFIT MODEL=LINEAR). „ Regressions with a full set of quarterly seasonal dummies and a linear trend variable.
- 2. 2 Appendix A In addition to obtaining the Durbin-Watson statistic for residuals from REGRESSION, you should also plot the ACF and PACF of the residuals series. The plots might suggest either that the residuals are random, or that they follow some ARMA process. If the residuals resemble an AR1 process, you can estimate an appropriate regression using the AREG procedure. If the residuals follow any ARMA process, you can estimate an appropriate regression using the ARIMA procedure. In this appendix, we have reproduced two sets of tables. Savin and White (1977) present tables for sample sizes ranging from 6 to 200 and for 1 to 20 regressors for models in which an intercept is included. Farebrother (1980) presents tables for sample sizes ranging from 2 to 200 and for 0 to 21 regressors for models in which an intercept is not included. Let’s consider an example of how to use the tables. In Chapter 9, we look at the classic Durbin and Watson data set concerning consumption of spirits. The sample size is 69, there are 2 regressors, and there is an intercept term in the model. The Durbin- Watson test statistic value is 0.24878. We want to test the null hypothesis of zero autocorrelation in the residuals against the alternative that the residuals are positively autocorrelated at the 1% level of significance. If you examine the Savin and White tables (Table A.2 and Table A.3), you will not find a row for sample size 69, so go to the next lowest sample size with a tabulated row, namely N=65. Since there are two regressors, find the column labeled k=2. Cross-referencing the indicated row and column, you will find that the printed bounds are dL = 1.377 and dU = 1.500. If the observed value of the test statistic is less than the tabulated lower bound, then you should reject the null hypothesis of non-autocorrelated errors in favor of the hypothesis of positive first-order autocorrelation. Since 0.24878 is less than 1.377, we reject the null hypothesis. If the test statistic value were greater than dU, we would not reject the null hypothesis. A third outcome is also possible. If the test statistic value lies between dL and dU, the test is inconclusive. In this context, you might err on the side of conservatism and not reject the null hypothesis. For models with an intercept, if the observed test statistic value is greater than 2, then you want to test the null hypothesis against the alternative hypothesis of negative first-order autocorrelation. To do this, compute the quantity 4-d and compare this value with the tabulated values of dL and dU as if you were testing for positive autocorrelation. When the regression does not contain an intercept term, refer to Farebrother‚Äôs tabulated values of the ‚Äúminimal bound,‚Äù denoted dM (Table A.4 and Table A.5), instead of Savin and White‚Äôs lower bound dL. In this instance, the upper bound is
- 3. 3 Durbin-Watson SignificanceTables the conventional bound dU found in the Savin and White tables. To test for negative first-order autocorrelation, use Table A.6 and Table A.7. To continue with our example, had we run a regression with no intercept term, we would cross-reference N equals 65 and k equals 2 in Farebrother’s table. The tabulated 1% minimal bound is 1.348.
- 4. 4 Appendix A Table A-1 Models with an intercept (from Savin and White) Durbin-Watson Statistic: 1 Per Cent Significance Points of dL and dU k’*=1 k’=2 k’=3 k’=4 k’=5 k’=6 k’=7 k’=8 k’=9 k’=10 n dL dU dL dU dL dU dL dU dL dU dL dU dL dU dL dU dL dU dL dU 6 0.390 1.142 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 7 0.435 1.036 0.294 1.676 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 8 0.497 1.003 0.345 1.489 0.229 2.102 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 9 0.554 0.998 0.408 1.389 0.279 1.875 0.183 2.433 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 10 0.604 1.001 0.466 1.333 0.340 1.733 0.230 2.193 0.150 2.690 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 11 0.653 1.010 0.519 1.297 0.396 1.640 0.286 2.030 0.193 2.453 0.124 2.892 ----- ----- ----- ----- ----- ----- ----- ----- 12 0.697 1.023 0.569 1.274 0.449 1.575 0.339 1.913 0.244 2.280 0.164 2.665 0.105 3.053 ----- ----- ----- ----- ----- ----- 13 0.738 1.038 0.616 1.261 0.499 1.526 0.391 1.826 0.294 2.150 0.211 2.490 0.140 2.838 0.090 3.182 ----- ----- ----- ----- 14 0.776 1.054 0.660 1.254 0.547 1.490 0.441 1.757 0.343 2.049 0.257 2.354 0.183 2.667 0.122 2.981 0.078 3.287 ----- ----- 15 0.811 1.070 0.700 1.252 0.591 1.465 0.487 1.705 0.390 1.967 0.303 2.244 0.226 2.530 0.161 2.817 0.107 3.101 0.068 3.374 16 0.844 1.086 0.738 1.253 0.633 1.447 0.532 1.664 0.437 1.901 0.349 2.153 0.269 2.416 0.200 2.681 0.142 2.944 0.094 3.201 17 0.873 1.102 0.773 1.255 0.672 1.432 0.574 1.631 0.481 1.847 0.393 2.078 0.313 2.319 0.241 2.566 0.179 2.811 0.127 3.053 18 0.902 1.118 0.805 1.259 0.708 1.422 0.614 1.604 0.522 1.803 0.435 2.015 0.355 2.238 0.282 2.467 0.216 2.697 0.160 2.925 19 0.928 1.133 0.835 1.264 0.742 1.416 0.650 1.583 0.561 1.767 0.476 1.963 0.396 2.169 0.322 2.381 0.255 2.597 0.196 2.813 20 0.952 1.147 0.862 1.270 0.774 1.410 0.684 1.567 0.598 1.736 0.515 1.918 0.436 2.110 0.362 2.308 0.294 2.510 0.232 2.174 21 0.975 1.161 0.889 1.276 0.803 1.408 0.718 1.554 0.634 1.712 0.552 1.881 0.474 2.059 0.400 2.244 0.331 2.434 0.268 2.625 22 0.997 1.174 0.915 1.284 0.832 1.407 0.748 1.543 0.666 1.691 0.587 1.849 0.510 2.015 0.437 2.188 0.368 2.367 0.304 2.548 23 1.017 1.186 0.938 1.290 0.858 1.407 0.777 1.535 0.699 1.674 0.620 1.821 0.545 1.977 0.473 2.140 0.404 2.308 0.340 2.479 24 1.037 1.199 0.959 1.298 0.881 1.407 0.805 1.527 0.728 1.659 0.652 1.797 0.578 1.944 0.507 2.097 0.439 2.255 0.375 2.417 25 1.055 1.210 0.981 1.305 0.906 1.408 0.832 1.521 0.756 1.645 0.682 1.776 0.610 1.915 0.540 2.059 0.473 2.209 0.409 2.362 26 1.072 1.222 1.000 1.311 0.928 1.410 0.855 1.517 0.782 1.635 0.711 1.759 0.640 1.889 0.572 2.026 0.505 2.168 0.441 2.313 27 1.088 1.232 1.019 1.318 0.948 1.413 0.878 1.514 0.808 1.625 0.738 1.743 0.669 1.867 0.602 1.997 0.536 2.131 0.473 2.269 28 1.104 1.244 1.036 1.325 0.969 1.414 0.901 1.512 0.832 1.618 0.764 1.729 0.696 1.847 0.630 1.970 0.566 2.098 0.504 2.229 29 1.119 1.254 1.053 1.332 0.988 1.418 0.921 1.511 0.855 1.611 0.788 1.718 0.723 1.830 0.658 1.947 0.595 2.068 0.533 2.193 30 1.134 1.264 1.070 1.339 1.006 1.421 0.941 1.510 0.877 1.606 0.812 1.707 0.748 1.814 0.684 1.925 0.622 2.041 0.562 2.160 31 1.147 1.274 1.085 1.345 1.022 1.425 0.960 1.509 0.897 1.601 0.834 1.698 0.772 1.800 0.710 1.906 0.649 2.017 0.589 2.131 32 1.160 1.283 1.100 1.351 1.039 1.428 0.978 1.509 0.917 1.597 0.856 1.690 0.794 1.788 0.734 1.889 0.674 1.995 0.615 2.104 33 1.171 1.291 1.114 1.358 1.055 1.432 0.995 1.510 0.935 1.594 0.876 1.683 0.816 1.776 0.757 1.874 0.698 1.975 0.641 2.080 34 1.184 1.298 1.128 1.364 1.070 1.436 1.012 1.511 0.954 1.591 0.896 1.677 0.837 1.766 0.779 1.860 0.722 1.957 0.665 2.057 35 1.195 1.307 1.141 1.370 1.085 1.439 1.028 1.512 0.971 1.589 0.914 1.671 0.857 1.757 0.800 1.847 0.744 1.940 0.689 2.037 36 1.205 1.315 1.153 1.376 1.098 1.442 1.043 1.513 0.987 1.587 0.932 1.666 0.877 1.749 0.821 1.836 0.766 1.925 0.711 2.018 37 1.217 1.322 1.164 1.383 1.112 1.446 1.058 1.514 1.004 1.585 0.950 1.662 0.895 1.742 0.841 1.825 0.787 1.911 0.733 2.001 38 1.227 1.330 1.176 1.388 1.124 1.449 1.072 1.515 1.019 1.584 0.966 1.658 0.913 1.735 0.860 1.816 0.807 1.899 0.754 1.985 39 1.237 1.337 1.187 1.392 1.137 1.452 1.085 1.517 1.033 1.583 0.982 1.655 0.930 1.729 0.878 1.807 0.826 1.887 0.774 1.970 40 1.246 1.344 1.197 1.398 1.149 1.456 1.098 1.518 1.047 1.583 0.997 1.652 0.946 1.724 0.895 1.799 0.844 1.876 0.749 1.956 45 1.288 1.376 1.245 1.424 1.201 1.474 1.156 1.528 1.111 1.583 1.065 1.643 1.019 1.704 0.974 1.768 0.927 1.834 0.881 1.902 50 1.324 1.403 1.285 1.445 1.245 1.491 1.206 1.537 1.164 1.587 1.123 1.639 1.081 1.692 1.039 1.748 0.997 1.805 0.955 1.864 55 1.356 1.428 1.320 1.466 1.284 1.505 1.246 1.548 1.209 1.592 1.172 1.638 1.134 1.685 1.095 1.734 1.057 1.785 1.018 1.837 60 1.382 1.449 1.351 1.484 1.317 1.520 1.283 1.559 1.248 1.598 1.214 1.639 1.179 1.682 1.144 1.726 1.108 1.771 1.072 1.817 65 1.407 1.467 1.377 1.500 1.346 1.534 1.314 1.568 1.283 1.604 1.251 1.642 1.218 1.680 1.186 1.720 1.153 1.761 1.120 1.802 70 1.429 1.485 1.400 1.514 1.372 1.546 1.343 1.577 1.313 1.611 1.283 1.645 1.253 1.680 1.223 1.716 1.192 1.754 1.162 1.792 75 1.448 1.501 1.422 1.529 1.395 1.557 1.368 1.586 1.340 1.617 1.313 1.649 1.284 1.682 1.256 1.714 1.227 1.748 1.199 1.783 80 1.465 1.514 1.440 1.541 1.416 1.568 1.390 1.595 1.364 1.624 1.338 1.653 1.312 1.683 1.285 1.714 1.259 1.745 1.232 1.777 85 1.481 1.529 1.458 1.553 1.434 1.577 1.411 1.603 1.386 1.630 1.362 1.657 1.337 1.685 1.312 1.714 1.287 1.743 1.262 1.773 90 1.496 1.541 1.474 1.563 1.452 1.587 1.429 1.611 1.406 1.636 1.383 1.661 1.360 1.687 1.336 1.714 1.312 1.741 1.288 1.769 95 1.510 1.552 1.489 1.573 1.468 1.596 1.446 1.618 1.425 1.641 1.403 1.666 1.381 1.690 1.358 1.715 1.336 1.741 1.313 1.767 100 1.522 1.562 1.502 1.582 1.482 1.604 1.461 1.625 1.441 1.647 1.421 1.670 1.400 1.693 1.378 1.717 1.357 1.741 1.335 1.765 150 1.611 1.637 1.598 1.651 1.584 1.665 1.571 1.679 1.557 1.693 1.543 1.708 1.530 1.722 1.515 1.737 1.501 1.752 1.486 1.767 200 1.664 1.684 1.653 1.693 1.643 1.704 1.633 1.715 1.623 1.725 1.613 1.735 1.603 1.746 1.592 1.757 1.582 1.768 1.571 1.779 *k’ is the number of regressors excluding the intercept
- 5. 5 Durbin-Watson SignificanceTables k’*=11 k’=12 k’=13 k’=14 k’=15 k’=16 k’=17 k’=18 k’=19 k’=20 n dL dU dL dU dL dU dL dU dL dU dL dU dL dU dL dU dL dU dL dU 16 0.060 3.446 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 17 0.084 3.286 0.053 3.506 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 18 0.113 3.146 0.075 3.358 0.047 3.557 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 19 0.145 3.023 0.102 3.227 0.067 3.420 0.043 3.601 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 20 0.178 2.914 0.131 3.109 0.092 3.297 0.061 3.474 0.038 3.639 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 21 0.212 2.817 0.162 3.004 0.119 3.185 0.084 3.358 0.055 3.521 0.035 3.671 ----- ----- ----- ----- ----- ----- ----- ----- 22 0.246 2.729 0.194 2.909 0.148 3.084 0.109 3.252 0.077 3.412 0.050 3.562 0.032 3.700 ----- ----- ----- ----- ----- ----- 23 0.281 2.651 0.227 2.822 0.178 2.991 0.136 3.155 0.100 3.311 0.070 3.459 0.046 3.597 0.029 3.725 ----- ----- ----- ----- 24 0.315 2.580 0.260 2.744 0.209 2.906 0.165 3.065 0.125 3.218 0.092 3.363 0.065 3.501 0.043 3.629 0.027 3.747 ----- ----- 25 0.348 2.517 0.292 2.674 0.240 2.829 0.194 2.982 0.152 3.131 0.116 3.274 0.085 3.410 0.060 3.538 0.039 3.657 0.025 3.766 26 0.381 2.460 0.324 2.610 0.272 2.758 0.224 2.906 0.180 3.050 0.141 3.191 0.107 3.325 0.079 3.452 0.055 3.572 0.036 3.682 27 0.413 2.409 0.356 2.552 0.303 2.694 0.253 2.836 0.208 2.976 0.167 3.113 0.131 3.245 0.100 3.371 0.073 3.490 0.051 3.602 28 0.444 2.363 0.387 2.499 0.333 2.635 0.283 2.772 0.237 2.907 0.194 3.040 0.156 3.169 0.122 3.294 0.093 3.412 0.068 3.524 29 0.474 2.321 0.417 2.451 0.363 2.582 0.313 2.713 0.266 2.843 0.222 2.972 0.182 3.098 0.146 3.220 0.114 3.338 0.087 3.450 30 0.503 2.283 0.447 2.407 0.393 2.533 0.342 2.659 0.294 2.785 0.249 2.909 0.208 3.032 0.171 3.152 0.137 3.267 0.107 3.379 31 0.531 2.248 0.475 2.367 0.422 2.487 0.371 2.609 0.322 2.730 0.277 2.851 0.234 2.970 0.193 3.087 0.160 3.201 0.128 3.311 32 0.558 2.216 0.503 2.330 0.450 2.446 0.399 2.563 0.350 2.680 0.304 2.797 0.261 2.912 0.221 3.026 0.184 3.137 0.151 3.246 33 0.585 2.187 0.530 2.296 0.477 2.408 0.426 2.520 0.377 2.633 0.331 2.746 0.287 2.858 0.246 2.969 0.209 3.078 0.174 3.184 34 0.610 2.160 0.556 2.266 0.503 2.373 0.452 2.481 0.404 2.590 0.357 2.699 0.313 2.808 0.272 2.915 0.233 3.022 0.197 3.126 35 0.634 2.136 0.581 2.237 0.529 2.340 0.478 2.444 0.430 2.550 0.383 2.655 0.339 2.761 0.297 2.865 0.257 2.969 0.221 3.071 36 0.658 2.113 0.605 2.210 0.554 2.310 0.504 2.410 0.455 2.512 0.409 2.614 0.364 2.717 0.322 2.818 0.282 2.919 0.244 3.019 37 0.680 2.092 0.628 2.186 0.578 2.282 0.528 2.379 0.480 2.477 0.434 2.576 0.389 2.675 0.347 2.774 0.306 2.872 0.268 2.969 38 0.702 2.073 0.651 2.164 0.601 2.256 0.552 2.350 0.504 2.445 0.458 2.540 0.414 2.637 0.371 2.733 0.330 2.828 0.291 2.923 39 0.723 2.055 0.673 2.143 0.623 2.232 0.575 2.323 0.528 2.414 0.482 2.507 0.438 2.600 0.395 2.694 0.354 2.787 0.315 2.879 40 0.744 2.039 0.694 2.123 0.645 2.210 0.597 2.297 0.551 2.386 0.505 2.476 0.461 2.566 0.418 2.657 0.377 2.748 0.338 2.838 45 0.835 1.972 0.790 2.044 0.744 2.118 0.700 2.193 0.655 2.269 0.612 2.346 0.570 2.424 0.528 2.503 0.488 2.582 0.448 2.661 50 0.913 1.925 0.871 1.987 0.829 2.051 0.787 2.116 0.746 2.182 0.705 2.250 0.665 2.318 0.625 2.387 0.586 2.456 0.548 2.526 55 0.979 1.891 0.940 1.945 0.902 2.002 0.863 2.059 0.825 2.117 0.786 2.176 0.748 2.237 0.711 2.298 0.674 2.359 0.637 2.421 60 1.037 1.865 1.001 1.914 0.965 1.964 0.929 2.015 0.893 2.067 0.857 2.120 0.822 2.173 0.786 2.227 0.751 2.283 0.716 2.338 65 1.087 1.845 1.053 1.889 1.020 1.934 0.986 1.980 0.953 2.027 0.919 2.075 0.886 2.123 0.852 2.172 0.819 2.221 0.789 2.272 70 1.131 1.831 1.099 1.870 1.068 1.911 1.037 1.953 1.005 1.995 0.974 2.038 0.943 2.082 0.911 2.127 0.880 2.172 0.849 2.217 75 1.170 1.819 1.141 1.856 1.111 1.893 1.082 1.931 1.052 1.970 1.023 2.009 0.993 2.049 0.964 2.090 0.934 2.131 0.905 2.172 80 1.205 1.810 1.177 1.844 1.150 1.878 1.122 1.913 1.094 1.949 1.066 1.984 1.039 2.022 1.011 2.059 0.983 2.097 0.955 2.135 85 1.236 1.803 1.210 1.834 1.184 1.866 1.158 1.898 1.132 1.931 1.106 1.965 1.080 1.999 1.053 2.033 1.027 2.068 1.000 2.104 90 1.264 1.798 1.240 1.827 1.215 1.856 1.191 1.886 1.166 1.917 1.141 1.948 1.116 1.979 1.091 2.012 1.066 2.044 1.041 2.077 95 1.290 1.793 1.267 1.821 1.244 1.848 1.221 1.876 1.197 1.905 1.174 1.943 1.150 1.963 1.126 1.993 1.102 2.023 1.079 2.054 100 1.314 1.790 1.292 1.816 1.270 1.841 1.248 1.868 1.225 1.895 1.203 1.922 1.181 1.949 1.158 1.977 1.136 2.006 1.113 2.034 150 1.473 1.783 1.458 1.799 1.444 1.814 1.429 1.830 1.414 1.847 1.400 1.863 1.385 1.880 1.370 1.897 1.355 1.913 1.340 1.931 200 1.561 1.791 1.550 1.801 1.539 1.813 1.528 1.824 1.518 1.836 1.507 1.847 1.495 1.860 1.484 1.871 1.474 1.883 1.462 1.896 *k’ is the number of regressors excluding the intercept
- 6. 6 Appendix A Table A-2 Models with an intercept (from Savin and White) Durbin-Watson Statistic: 5 Per Cent Significance Points of dL and dU k’*=1 k’=2 k’=3 k’=4 k’=5 k’=6 k’=7 k’=8 k’=9 k’=10 n dL dU dL dU dL dU dL dU dL dU dL dU dL dU dL dU dL dU dL dU 6 0.610 1.400 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 7 0.700 1.356 0.467 1.896 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 8 0.763 1.332 0.559 1.777 0.367 2.287 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 9 0.824 1.320 0.629 1.699 0.455 2.128 0.296 2.588 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 10 0.879 1.320 0.697 1.641 0.525 2.016 0.376 2.414 0.243 2.822 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 11 0.927 1.324 0.758 1.604 0.595 1.928 0.444 2.283 0.315 2.645 0.203 3.004 ----- ----- ----- ----- ----- ----- ----- ----- 12 0.971 1.331 0.812 1.579 0.658 1.864 0.512 2.177 0.380 2.506 0.268 2.832 0.171 3.149 ----- ----- ----- ----- ----- ----- 13 1.010 1.340 0.861 1.562 0.715 1.816 0.574 2.094 0.444 2.390 0.328 2.692 0.230 2.985 0.147 3.266 ----- ----- ----- ----- 14 1.045 1.350 0.905 1.551 0.767 1.779 0.632 2.030 0.505 2.296 0.389 2.572 0.286 2.848 0.200 3.111 0.127 3.360 ----- ----- 15 1.077 1.361 0.946 1.543 0.814 1.750 0.685 1.977 0.562 2.220 0.447 2.471 0.343 2.727 0.251 2.979 0.175 3.216 0.111 3.438 16 1.106 1.371 0.982 1.539 0.857 1.728 0.734 1.935 0.615 2.157 0.502 2.388 0.398 2.624 0.304 2.860 0.222 3.090 0.155 3.304 17 1.133 1.381 1.015 1.536 0.897 1.710 0.779 1.900 0.664 2.104 0.554 2.318 0.451 2.537 0.356 2.757 0.272 2.975 0.198 3.184 18 1.158 1.391 1.046 1.535 0.933 1.696 0.820 1.872 0.710 2.060 0.603 2.258 0.502 2.461 0.407 2.668 0.321 2.873 0.244 3.073 19 1.180 1.401 1.074 1.536 0.967 1.685 0.859 1.848 0.752 2.023 0.649 2.206 0.549 2.396 0.456 2.589 0.369 2.783 0.290 2.974 20 1.201 1.411 1.100 1.537 0.998 1.676 0.894 1.828 0.792 1.991 0.691 2.162 0.595 2.339 0.502 2.521 0.416 2.704 0.336 2.885 21 1.221 1.420 1.125 1.538 1.026 1.669 0.927 1.812 0.829 1.964 0.731 2.124 0.637 2.290 0.546 2.461 0.461 2.633 0.380 2.806 22 1.239 1.429 1.147 1.541 1.053 1.664 0.958 1.797 0.863 1.940 0.769 2.090 0.677 2.246 0.588 2.407 0.504 2.571 0.424 2.735 23 1.257 1.437 1.168 1.543 1.078 1.660 0.986 1.785 0.895 1.920 0.804 2.061 0.715 2.208 0.628 2.360 0.545 2.514 0.465 2.670 24 1.273 1.446 1.188 1.546 1.101 1.656 1.013 1.775 0.925 1.902 0.837 2.035 0.750 2.174 0.666 2.318 0.584 2.464 0.506 2.613 25 1.288 1.454 1.206 1.550 1.123 1.654 1.038 1.767 0.953 1.886 0.868 2.013 0.784 2.144 0.702 2.280 0.621 2.419 0.544 2.560 26 1.302 1.461 1.224 1.553 1.143 1.652 1.062 1.759 0.979 1.873 0.897 1.992 0.816 2.117 0.735 2.246 0.657 2.379 0.581 2.513 27 1.316 1.469 1.240 1.556 1.162 1.651 1.084 1.753 1.004 1.861 0.925 1.974 0.845 2.093 0.767 2.216 0.691 2.342 0.616 2.470 28 1.328 1.476 1.255 1.560 1.181 1.650 1.104 1.747 1.028 1.850 0.951 1.959 0.874 2.071 0.798 2.188 0.723 2.309 0.649 2.431 29 1.341 1.483 1.270 1.563 1.198 1.650 1.124 1.743 1.050 1.841 0.975 1.944 0.900 2.052 0.826 2.164 0.753 2.278 0.681 2.396 30 1.352 1.489 1.284 1.567 1.214 1.650 1.143 1.739 1.071 1.833 0.998 1.931 0.926 2.034 0.854 2.141 0.782 2.251 0.712 2.363 31 1.363 1.496 1.297 1.570 1.229 1.650 1.160 1.735 1.090 1.825 1.020 1.920 0.950 2.018 0.879 2.120 0.810 2.226 0.741 2.333 32 1.373 1.502 1.309 1.574 1.244 1.650 1.177 1.732 1.109 1.819 1.041 1.909 0.972 2.004 0.904 2.102 0.836 2.203 0.769 2.306 33 1.383 1.508 1.321 1.577 1.258 1.651 1.193 1.730 1.127 1.813 1.061 1.900 0.994 1.991 0.927 2.085 0.861 2.181 0.796 2.281 34 1.393 1.514 1.333 1.580 1.271 1.652 1.208 1.728 1.144 1.808 1.079 1.891 1.015 1.978 0.950 2.069 0.885 2.162 0.821 2.257 35 1.402 1.519 1.343 1.584 1.283 1.653 1.222 1.726 1.160 1.803 1.097 1.884 1.034 1.967 0.971 2.054 0.908 2.144 0.845 2.236 36 1.411 1.525 1.354 1.587 1.295 1.654 1.236 1.724 1.175 1.799 1.114 1.876 1.053 1.957 0.991 2.041 0.930 2.127 0.868 2.216 37 1.419 1.530 1.364 1.590 1.307 1.655 1.249 1.723 1.190 1.795 1.131 1.870 1.071 1.948 1.011 2.029 0.951 2.112 0.891 2.197 38 1.427 1.535 1.373 1.594 1.318 1.656 1.261 1.722 1.204 1.792 1.146 1.864 1.088 1.939 1.029 2.017 0.970 2.098 0.912 2.180 39 1.435 1.540 1.382 1.597 1.328 1.658 1.273 1.722 1.218 1.789 1.161 1.859 1.104 1.932 1.047 2.007 0.990 2.085 0.932 2.164 40 1.442 1.544 1.391 1.600 1.338 1.659 1.285 1.721 1.230 1.786 1.175 1.854 1.120 1.924 1.064 1.997 1.008 2.072 0.952 2.149 45 1.475 1.566 1.430 1.615 1.383 1.666 1.336 1.720 1.287 1.776 1.238 1.835 1.189 1.895 1.139 1.958 1.089 2.022 1.038 2.088 50 1.503 1.585 1.462 1.628 1.421 1.674 1.378 1.721 1.335 1.771 1.291 1.822 1.246 1.875 1.201 1.930 1.156 1.986 1.110 2.044 55 1.528 1.601 1.490 1.641 1.452 1.681 1.414 1.724 1.374 1.768 1.334 1.814 1.294 1.861 1.253 1.909 1.212 1.959 1.170 2.010 60 1.549 1.616 1.514 1.652 1.480 1.689 1.444 1.727 1.408 1.767 1.372 1.808 1.335 1.850 1.298 1.894 1.260 1.939 1.222 1.984 65 1.567 1.629 1.536 1.662 1.503 1.696 1.471 1.731 1.438 1.767 1.404 1.805 1.370 1.843 1.336 1.882 1.301 1.923 1.266 1.964 70 1.583 1.641 1.554 1.672 1.525 1.703 1.494 1.735 1.464 1.768 1.433 1.802 1.401 1.838 1.369 1.874 1.337 1.910 1.305 1.948 75 1.598 1.652 1.571 1.680 1.543 1.709 1.515 1.739 1.487 1.770 1.458 1.801 1.428 1.834 1.399 1.867 1.369 1.901 1.339 1.935 80 1.611 1.662 1.586 1.688 1.560 1.715 1.534 1.743 1.507 1.772 1.480 1.801 1.453 1.831 1.425 1.861 1.397 1.893 1.369 1.925 85 1.624 1.671 1.600 1.696 1.575 1.721 1.550 1.747 1.525 1.774 1.500 1.801 1.474 1.829 1.448 1.857 1.422 1.886 1.396 1.916 90 1.635 1.679 1.612 1.703 1.589 1.726 1.566 1.751 1.542 1.776 1.518 1.801 1.494 1.827 1.469 1.854 1.445 1.881 1.420 1.909 95 1.645 1.687 1.623 1.709 1.602 1.732 1.579 1.755 1.557 1.778 1.535 1.802 1.512 1.827 1.489 1.852 1.465 1.877 1.442 1.903 100 1.654 1.694 1.634 1.715 1.613 1.736 1.592 1.758 1.571 1.780 1.550 1.803 1.528 1.826 1.506 1.850 1.484 1.874 1.462 1.898 150 1.720 1.747 1.706 1.760 1.693 1.774 1.679 1.788 1.665 1.802 1.651 1.817 1.637 1.832 1.622 1.846 1.608 1.862 1.593 1.877 200 1.758 1.779 1.748 1.789 1.738 1.799 1.728 1.809 1.718 1.820 1.707 1.831 1.697 1.841 1.686 1.852 1.675 1.863 1.665 1.874 *k’ is the number of regressors excluding the intercept
- 7. 7 Durbin-Watson SignificanceTables k’*=11 k’=12 k’=13 k’=14 k’=15 k’=16 k’=17 k’=18 k’=19 k’=20 n dL dU dL dU dL dU dL dU dL dU dL dU dL dU dL dU dL dU dL dU 16 0.098 3.503 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 17 0.138 3.378 0.087 3.557 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 18 0.177 3.265 0.123 3.441 0.078 3.603 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 19 0.220 3.159 0.160 3.335 0.111 3.496 0.070 3.642 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 20 0.263 3.063 0.200 3.234 0.145 3.395 0.100 3.542 0.063 3.676 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 21 0.307 2.976 0.240 3.141 0.182 3.300 0.132 3.448 0.091 3.583 0.058 3.705 ----- ----- ----- ----- ----- ----- ----- ----- 22 0.349 2.897 0.281 3.057 0.220 3.211 0.166 3.358 0.120 3.495 0.083 3.619 0.052 3.731 ----- ----- ----- ----- ----- ----- 23 0.391 2.826 0.322 2.979 0.259 3.128 0.202 3.272 0.153 3.409 0.110 3.535 0.076 3.650 0.048 3.753 ----- ----- ----- ----- 24 0.431 2.761 0.362 2.908 0.297 3.053 0.239 3.193 0.186 3.327 0.141 3.454 0.101 3.572 0.070 3.678 0.044 3.773 ----- ----- 25 0.470 2.702 0.400 2.844 0.335 2.983 0.275 3.119 0.221 3.251 0.172 3.376 0.130 3.494 0.094 3.604 0.065 3.702 0.041 3.790 26 0.508 2.649 0.438 2.784 0.373 2.919 0.312 3.051 0.256 3.179 0.205 3.303 0.160 3.420 0.120 3.531 0.087 3.632 0.060 3.724 27 0.544 2.600 0.475 2.730 0.409 2.859 0.348 2.987 0.291 3.112 0.238 3.233 0.191 3.349 0.149 3.460 0.112 3.563 0.081 3.658 28 0.578 2.555 0.510 2.680 0.445 2.805 0.383 2.928 0.325 3.050 0.271 3.168 0.222 3.283 0.178 3.392 0.138 3.495 0.104 3.592 29 0.612 2.515 0.544 2.634 0.479 2.755 0.418 2.874 0.359 2.992 0.305 3.107 0.254 3.219 0.208 3.327 0.166 3.431 0.129 3.528 30 0.643 2.477 0.577 2.592 0.512 2.708 0.451 2.823 0.392 2.937 0.337 3.050 0.286 3.160 0.238 3.266 0.195 3.368 0.156 3.465 31 0.674 2.443 0.608 2.553 0.545 2.665 0.484 2.776 0.425 2.887 0.370 2.996 0.317 3.103 0.269 3.208 0.224 3.309 0.183 3.406 32 0.703 2.411 0.638 2.517 0.576 2.625 0.515 2.733 0.457 2.840 0.401 2.946 0.349 3.050 0.299 3.153 0.253 3.252 0.211 3.348 33 0.731 2.382 0.668 2.484 0.606 2.588 0.546 2.692 0.488 2.796 0.432 2.899 0.379 3.000 0.329 3.100 0.283 3.198 0.239 3.293 34 0.758 2.355 0.695 2.454 0.634 2.554 0.575 2.654 0.518 2.754 0.462 2.854 0.409 2.954 0.359 3.051 0.312 3.147 0.267 3.240 35 0.783 2.330 0.722 2.425 0.662 2.521 0.604 2.619 0.547 2.716 0.492 2.813 0.439 2.910 0.388 3.005 0.340 3.099 0.295 3.190 36 0.808 2.306 0.748 2.398 0.689 2.492 0.631 2.586 0.575 2.680 0.520 2.774 0.467 2.868 0.417 2.961 0.369 3.053 0.323 3.142 37 0.831 2.285 0.772 2.374 0.714 2.464 0.657 2.555 0.602 2.646 0.548 2.738 0.495 2.829 0.445 2.920 0.397 3.009 0.351 3.097 38 0.854 2.265 0.796 2.351 0.739 2.438 0.683 2.526 0.628 2.614 0.575 2.703 0.522 2.792 0.472 2.880 0.424 2.968 0.378 3.054 39 0.875 2.246 0.819 2.329 0.763 2.413 0.707 2.499 0.653 2.585 0.600 2.671 0.549 2.757 0.499 2.843 0.451 2.929 0.404 3.013 40 0.896 2.228 0.840 2.309 0.785 2.391 0.731 2.473 0.678 2.557 0.626 2.641 0.575 2.724 0.525 2.808 0.477 2.829 0.430 2.974 45 0.988 2.156 0.938 2.225 0.887 2.296 0.838 2.367 0.788 2.439 0.740 2.512 0.692 2.586 0.644 2.659 0.598 2.733 0.553 2.807 50 1.064 2.103 1.019 2.163 0.973 2.225 0.927 2.287 0.882 2.350 0.836 2.414 0.792 2.479 0.747 2.544 0.703 2.610 0.660 2.675 55 1.129 2.062 1.087 2.116 1.045 2.170 1.003 2.225 0.961 2.281 0.919 2.338 0.877 2.396 0.836 2.454 0.795 2.512 0.754 2.571 60 1.184 2.031 1.145 2.079 1.106 2.127 1.068 2.177 1.029 2.227 0.990 2.278 0.951 2.330 0.913 2.382 0.874 2.434 0.836 2.487 65 1.231 2.006 1.195 2.049 1.160 2.093 1.124 2.138 1.088 2.183 1.052 2.229 1.016 2.276 0.980 2.323 0.944 2.371 0.908 2.419 70 1.272 1.987 1.239 2.026 1.206 2.066 1.172 2.106 1.139 2.148 1.105 2.189 1.072 2.232 1.038 2.275 1.005 2.318 0.971 2.362 75 1.308 1.970 1.277 2.006 1.247 2.043 1.215 2.080 1.184 2.118 1.153 2.156 1.121 2.195 1.090 2.235 1.058 2.275 1.027 2.315 80 1.340 1.957 1.311 1.991 1.283 2.024 1.253 2.059 1.224 2.093 1.195 2.129 1.165 2.165 1.136 2.201 1.106 2.238 1.076 2.275 85 1.369 1.946 1.342 1.977 1.315 2.009 1.287 2.040 1.260 2.073 1.232 2.105 1.205 2.139 1.177 2.172 1.149 2.206 1.121 2.241 90 1.395 1.937 1.369 1.966 1.344 1.995 1.318 2.025 1.292 2.055 1.266 2.085 1.240 2.116 1.213 2.148 1.187 2.179 1.160 2.211 95 1.418 1.930 1.394 1.956 1.370 1.984 1.345 2.012 1.321 2.040 1.296 2.068 1.271 2.097 1.247 2.126 1.222 2.156 1.197 2.186 100 1.439 1.923 1.416 1.948 1.393 1.974 1.371 2.000 1.347 2.026 1.324 2.053 1.301 2.080 1.277 2.108 1.253 2.135 1.229 2.164 150 1.579 1.892 1.564 1.908 1.550 1.924 1.535 1.940 1.519 1.956 1.504 1.972 1.489 1.989 1.474 2.006 1.458 2.023 1.443 2.040 200 1.654 1.885 1.643 1.896 1.632 1.908 1.621 1.919 1.610 1.931 1.599 1.943 1.588 1.955 1.576 1.967 1.565 1.979 1.554 1.991 *K’ is the number of regressors excluding the intercept
- 8. 8 Appendix A Table A-3 Models with no intercept (from Farebrother): Positive serial correlation Durbin-Watson One Per Cent Minimal Bound N K=0 K=1 K=2 K=3 K=4 K=5 K=6 K=7 K=8 K=9 K=10 K=11 K=12 K=13 K=14 K=15 K=16 K=17 K=18 K=19 K=20 K=21 2 0.001 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 3 0.034 0.000 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 4 0.127 0.022 0.000 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 5 0.233 0.089 0.014 0.000 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 6 0.322 0.175 0.065 0.010 0.000 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 7 0.398 0.253 0.135 0.049 0.008 0.000 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 8 0.469 0.324 0.202 0.106 0.038 0.006 0.000 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 9 0.534 0.394 0.268 0.164 0.086 0.031 0.005 0.000 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 10 0.591 0.457 0.333 0.223 0.136 0.070 0.025 0.004 0.000 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 11 0.643 0.515 0.394 0.284 0.189 0.114 0.059 0.021 0.003 0.000 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 12 0.691 0.568 0.451 0.341 0.244 0.161 0.097 0.050 0.018 0.003 0.000 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 13 0.733 0.617 0.503 0.396 0.298 0.212 0.139 0.083 0.043 0.015 0.002 0.000 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 14 0.773 0.662 0.552 0.448 0.350 0.262 0.185 0.121 0.072 0.037 0.013 0.002 0.000 ----- ----- ----- ----- ----- ----- ----- ----- ----- 15 0.809 0.703 0.598 0.496 0.400 0.311 0.232 0.163 0.107 0.063 0.032 0.011 0.002 0.000 ----- ----- ----- ----- ----- ----- ----- ----- 16 0.842 0.741 0.640 0.541 0.447 0.358 0.278 0.206 0.145 0.094 0.056 0.028 0.010 0.002 0.000 ----- ----- ----- ----- ----- ----- ----- 17 0.873 0.776 0.679 0.583 0.491 0.404 0.323 0.249 0.184 0.129 0.084 0.050 0.025 0.009 0.001 0.000 ----- ----- ----- ----- ----- ----- 18 0.901 0.808 0.715 0.623 0.533 0.447 0.366 0.292 0.225 0.166 0.116 0.075 0.044 0.023 0.008 0.001 0.000 ----- ----- ----- ----- ----- 19 0.928 0.839 0.749 0.660 0.572 0.488 0.408 0.333 0.265 0.204 0.150 0.105 0.068 0.040 0.020 0.007 0.001 0.000 ----- ----- ----- ----- 20 0.952 0.867 0.780 0.694 0.609 0.527 0.448 0.374 0.304 0.241 0.185 0.136 0.095 0.062 0.036 0.018 0.006 0.001 0.000 ----- ----- ----- 21 0.976 0.893 0.810 0.727 0.644 0.564 0.486 0.413 0.343 0.279 0.221 0.169 0.124 0.087 0.056 0.033 0.017 0.006 0.001 0.000 ----- ----- 22 0.997 0.918 0.838 0.757 0.677 0.599 0.523 0.450 0.381 0.316 0.257 0.203 0.155 0.114 0.079 0.051 0.030 0.015 0.005 0.001 0.000 ----- 23 1.018 0.942 0.864 0.786 0.709 0.632 0.558 0.486 0.417 0.352 0.292 0.237 0.187 0.143 0.104 0.073 0.047 0.027 0.014 0.005 0.001 0.000 24 1.037 0.964 0.889 0.813 0.738 0.664 0.591 0.520 0.452 0.387 0.327 0.270 0.219 0.172 0.131 0.096 0.067 0.043 0.025 0.013 0.004 0.001 25 1.056 0.984 0.912 0.839 0.766 0.693 0.622 0.553 0.486 0.421 0.361 0.304 0.251 0.203 0.160 0.122 0.089 0.062 0.040 0.023 0.012 0.004 26 1.073 1.004 0.934 0.863 0.792 0.722 0.652 0.584 0.518 0.454 0.394 0.336 0.283 0.233 0.189 0.148 0.113 0.083 0.057 0.037 0.022 0.011 27 1.089 1.023 0.955 0.886 0.817 0.749 0.681 0.614 0.549 0.486 0.426 0.368 0.314 0.264 0.218 0.176 0.138 0.105 0.077 0.053 0.034 0.020 28 1.105 1.040 0.974 0.908 0.841 0.774 0.708 0.643 0.579 0.517 0.457 0.400 0.345 0.294 0.247 0.204 0.164 0.129 0.098 0.071 0.050 0.032 29 1.120 1.057 0.993 0.929 0.864 0.798 0.734 0.670 0.607 0.546 0.487 0.430 0.376 0.324 0.276 0.232 0.191 0.154 0.120 0.091 0.067 0.046 30 1.134 1.073 1.011 0.948 0.885 0.822 0.759 0.696 0.635 0.574 0.516 0.460 0.405 0.354 0.305 0.260 0.217 0.179 0.144 0.113 0.086 0.062 31 1.147 1.088 1.028 0.967 0.905 0.844 0.782 0.721 0.661 0.602 0.544 0.488 0.434 0.383 0.334 0.288 0.244 0.205 0.168 0.135 0.106 0.080 32 1.160 1.103 1.044 0.985 0.925 0.865 0.805 0.745 0.686 0.628 0.571 0.516 0.462 0.411 0.362 0.315 0.271 0.230 0.193 0.158 0.127 0.100 33 1.173 1.117 1.060 1.002 0.944 0.885 0.826 0.768 0.710 0.653 0.597 0.542 0.489 0.438 0.389 0.342 0.298 0.256 0.218 0.182 0.149 0.120 34 1.185 1.130 1.075 1.018 0.961 0.904 0.847 0.790 0.733 0.677 0.622 0.568 0.516 0.465 0.416 0.369 0.324 0.282 0.243 0.206 0.172 0.141 35 1.196 1.143 1.089 1.034 0.978 0.923 0.867 0.811 0.755 0.700 0.646 0.593 0.541 0.491 0.442 0.395 0.350 0.308 0.268 0.230 0.195 0.163 36 1.207 1.155 1.102 1.049 0.995 0.940 0.886 0.831 0.777 0.723 0.669 0.617 0.566 0.516 0.467 0.421 0.376 0.333 0.292 0.254 0.218 0.185 37 1.217 1.167 1.116 1.063 1.010 0.957 0.904 0.850 0.797 0.744 0.692 0.640 0.590 0.540 0.492 0.446 0.401 0.358 0.317 0.278 0.241 0.207 38 1.228 1.178 1.128 1.077 1.026 0.974 0.921 0.869 0.817 0.765 0.713 0.663 0.613 0.564 0.516 0.470 0.425 0.382 0.341 0.302 0.265 0.230 39 1.237 1.189 1.140 1.090 1.040 0.989 0.938 0.887 0.836 0.785 0.734 0.684 0.635 0.587 0.540 0.494 0.449 0.406 0.365 0.325 0.288 0.252 40 1.247 1.200 1.152 1.103 1.054 1.004 0.954 0.904 0.854 0.804 0.754 0.705 0.657 0.609 0.562 0.517 0.473 0.430 0.388 0.349 0.311 0.275 45 1.289 1.247 1.204 1.160 1.116 1.071 1.026 0.981 0.936 0.890 0.845 0.800 0.755 0.710 0.666 0.623 0.581 0.539 0.499 0.459 0.421 0.384 50 1.325 1.287 1.248 1.208 1.168 1.128 1.087 1.046 1.004 0.963 0.921 0.880 0.838 0.797 0.756 0.715 0.675 0.636 0.597 0.559 0.521 0.485 55 1.356 1.321 1.286 1.250 1.213 1.176 1.139 1.101 1.063 1.025 0.987 0.948 0.910 0.872 0.833 0.796 0.758 0.721 0.684 0.647 0.611 0.576 60 1.383 1.351 1.319 1.285 1.252 1.218 1.183 1.149 1.114 1.078 1.043 1.008 0.972 0.936 0.901 0.865 0.830 0.795 0.760 0.725 0.691 0.657 65 1.408 1.378 1.348 1.317 1.286 1.254 1.222 1.190 1.158 1.125 1.092 1.059 1.026 0.993 0.960 0.927 0.894 0.861 0.828 0.795 0.762 0.730 70 1.429 1.401 1.373 1.345 1.316 1.286 1.257 1.227 1.197 1.166 1.136 1.105 1.074 1.043 1.012 0.981 0.950 0.919 0.888 0.857 0.826 0.795 75 1.448 1.423 1.396 1.369 1.342 1.315 1.287 1.260 1.231 1.203 1.174 1.146 1.117 1.088 1.058 1.029 1.000 0.971 0.941 0.912 0.883 0.854 80 1.466 1.442 1.417 1.392 1.367 1.341 1.315 1.289 1.262 1.236 1.209 1.182 1.155 1.127 1.100 1.072 1.045 1.017 0.989 0.962 0.934 0.907 85 1.482 1.459 1.436 1.412 1.388 1.364 1.340 1.315 1.290 1.265 1.240 1.214 1.189 1.163 1.137 1.111 1.085 1.059 1.033 1.006 0.980 0.954 90 1.497 1.475 1.453 1.431 1.408 1.385 1.362 1.339 1.315 1.292 1.268 1.244 1.220 1.195 1.171 1.146 1.121 1.097 1.072 1.047 1.022 0.997 95 1.510 1.490 1.469 1.448 1.426 1.405 1.383 1.361 1.338 1.316 1.293 1.271 1.248 1.225 1.201 1.178 1.155 1.131 1.108 1.084 1.060 1.037 100 1.523 1.503 1.483 1.463 1.443 1.422 1.402 1.381 1.359 1.338 1.317 1.295 1.273 1.251 1.229 1.207 1.185 1.162 1.140 1.118 1.095 1.072 150 1.611 1.598 1.585 1.571 1.558 1.544 1.530 1.516 1.502 1.488 1.474 1.460 1.445 1.431 1.416 1.402 1.387 1.372 1.357 1.342 1.327 1.312 200 1.664 1.654 1.644 1.634 1.624 1.613 1.603 1.593 1.582 1.572 1.561 1.551 1.540 1.529 1.519 1.508 1.497 1.486 1.475 1.434 1.453 1.442
- 9. 9 Durbin-Watson SignificanceTables Table A-4 Models with no intercept (from Farebrother): Positive serial correlation Durbin-Watson Five Per Cent Minimal Bound N K=0 K=1 K=2 K=3 K=4 K=5 K=6 K=7 K=8 K=9 K=10 K=11 K=12 K=13 K=14 K=15 K=16 K=17 K=18 K=19 K=20 K=21 2 0.012 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 3 0.168 0.006 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 4 0.355 0.105 0.004 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 5 0.478 0.248 0.070 0.002 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 6 0.584 0.358 0.180 0.050 0.002 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 7 0.677 0.462 0.275 0.136 0.037 0.001 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 8 0.754 0.556 0.371 0.217 0.106 0.029 0.001 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 9 0.820 0.635 0.460 0.303 0.175 0.085 0.023 0.001 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 10 0.877 0.706 0.539 0.385 0.251 0.143 0.069 0.019 0.001 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 11 0.927 0.768 0.610 0.460 0.326 0.211 0.120 0.058 0.016 0.001 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 12 0.972 0.823 0.674 0.530 0.397 0.279 0.180 0.101 0.049 0.013 0.001 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 13 1.012 0.872 0.731 0.593 0.464 0.345 0.241 0.154 0.087 0.042 0.011 0.001 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 14 1.047 0.916 0.783 0.651 0.525 0.408 0.302 0.210 0.134 0.075 0.036 0.010 0.001 ----- ----- ----- ----- ----- ----- ----- ----- ----- 15 1.079 0.955 0.829 0.704 0.583 0.467 0.361 0.266 0.185 0.118 0.066 0.031 0.008 0.001 ----- ----- ----- ----- ----- ----- ----- ----- 16 1.109 0.992 0.872 0.752 0.635 0.523 0.418 0.322 0.237 0.164 0.104 0.058 0.028 0.007 0.000 ----- ----- ----- ----- ----- ----- ----- 17 1.136 1.024 0.911 0.797 0.684 0.575 0.472 0.376 0.288 0.211 0.146 0.093 0.052 0.025 0.007 0.000 ----- ----- ----- ----- ----- ----- 18 1.160 1.055 0.946 0.837 0.729 0.624 0.523 0.427 0.339 0.260 0.190 0.131 0.083 0.046 0.022 0.006 0.000 ----- ----- ----- ----- ----- 19 1.183 1.082 0.979 0.875 0.771 0.669 0.570 0.476 0.388 0.307 0.235 0.171 0.118 0.075 0.041 0.020 0.005 0.000 ----- ----- ----- ----- 20 1.204 1.108 1.010 0.910 0.810 0.711 0.615 0.523 0.436 0.354 0.280 0.213 0.156 0.107 0.067 0.037 0.018 0.005 0.000 ----- ----- ----- 21 1.224 1.132 1.038 0.942 0.846 0.751 0.657 0.567 0.481 0.400 0.324 0.256 0.195 0.142 0.097 0.061 0.034 0.016 0.004 0.000 ----- ----- 22 1.242 1.154 1.064 0.972 0.879 0.787 0.697 0.609 0.524 0.443 0.368 0.298 0.235 0.178 0.130 0.089 0.056 0.031 0.015 0.004 0.000 ----- 23 1.259 1.175 1.088 1.000 0.911 0.822 0.734 0.648 0.565 0.485 0.410 0.339 0.274 0.216 0.164 0.119 0.081 0.051 0.028 0.014 0.004 0.000 24 1.275 1.194 1.111 1.026 0.940 0.854 0.769 0.685 0.604 0.525 0.450 0.380 0.314 0.254 0.199 0.151 0.110 0.075 0.047 0.026 0.012 0.003 25 1.290 1.212 1.132 1.050 0.967 0.884 0.802 0.720 0.641 0.563 0.489 0.419 0.353 0.291 0.235 0.184 0.140 0.101 0.069 0.044 0.024 0.011 26 1.304 1.229 1.152 1.073 0.993 0.913 0.833 0.753 0.676 0.600 0.527 0.457 0.390 0.328 0.271 0.218 0.171 0.130 0.094 0.064 0.040 0.022 27 1.318 1.245 1.171 1.094 1.017 0.940 0.862 0.785 0.709 0.635 0.563 0.493 0.427 0.365 0.306 0.252 0.203 0.159 0.120 0.087 0.060 0.037 28 1.330 1.260 1.188 1.115 1.040 0.965 0.889 0.815 0.741 0.668 0.597 0.529 0.463 0.400 0.341 0.286 0.236 0.190 0.148 0.112 0.081 0.055 29 1.342 1.275 1.205 1.134 1.062 0.989 0.916 0.843 0.770 0.699 0.630 0.562 0.497 0.435 0.376 0.320 0.268 0.221 0.177 0.139 0.105 0.076 30 1.354 1.288 1.221 1.152 1.082 1.011 0.940 0.869 0.799 0.729 0.661 0.595 0.530 0.468 0.409 0.353 0.301 0.252 0.207 0.166 0.130 0.098 31 1.365 1.301 1.236 1.169 1.101 1.033 0.964 0.895 0.826 0.758 0.691 0.626 0.562 0.501 0.442 0.386 0.333 0.283 0.237 0.195 0.156 0.122 32 1.375 1.313 1.250 1.185 1.120 1.053 0.986 0.919 0.852 0.785 0.720 0.653 0.593 0.532 0.474 0.418 0.364 0.314 0.267 0.223 0.183 0.147 33 1.385 1.325 1.264 1.201 1.137 1.072 1.007 0.942 0.876 0.811 0.747 0.684 0.623 0.563 0.504 0.449 0.395 0.344 0.297 0.252 0.211 0.173 34 1.394 1.336 1.277 1.216 1.153 1.091 1.027 0.963 0.900 0.836 0.774 0.712 0.651 0.592 0.534 0.479 0.425 0.374 0.326 0.280 0.238 0.199 35 1.403 1.347 1.289 1.230 1.169 1.108 1.046 0.984 0.922 0.860 0.799 0.738 0.678 0.620 0.563 0.508 0.455 0.404 0.355 0.309 0.266 0.225 36 1.412 1.357 1.301 1.243 1.184 1.125 1.064 1.004 0.943 0.883 0.823 0.763 0.705 0.647 0.591 0.536 0.483 0.432 0.384 0.337 0.293 0.252 37 1.420 1.367 1.312 1.256 1.199 1.141 1.082 1.023 0.964 0.905 0.846 0.787 0.730 0.673 0.618 0.564 0.511 0.460 0.412 0.365 0.321 0.279 38 1.428 1.376 1.323 1.268 1.212 1.156 1.099 1.041 0.983 0.925 0.868 0.811 0.754 0.698 0.644 0.590 0.538 0.488 0.439 0.392 0.347 0.305 39 1.436 1.385 1.333 1.280 1.225 1.170 1.114 1.058 1.002 0.945 0.889 0.833 0.778 0.723 0.669 0.616 0.564 0.514 0.466 0.419 0.374 0.331 40 1.443 1.394 1.343 1.291 1.238 1.184 1.130 1.075 1.020 0.965 0.909 0.854 0.800 0.746 0.693 0.641 0.590 0.540 0.492 0.445 0.400 0.357 45 1.476 1.432 1.387 1.341 1.294 1.246 1.197 1.148 1.099 1.049 1.000 0.950 0.900 0.851 0.802 0.753 0.706 0.658 0.612 0.567 0.523 0.480 50 1.504 1.464 1.424 1.382 1.340 1.297 1.253 1.209 1.164 1.120 1.075 1.029 0.984 0.939 0.894 0.849 0.804 0.760 0.717 0.674 0.631 0.590 55 1.528 1.492 1.455 1.417 1.379 1.340 1.300 1.260 1.219 1.179 1.138 1.096 1.055 1.013 0.972 0.930 0.889 0.848 0.807 0.766 0.726 0.687 60 1.549 1.516 1.482 1.447 1.412 1.376 1.340 1.303 1.266 1.229 1.191 1.153 1.115 1.077 1.038 1.000 0.962 0.923 0.885 0.847 0.810 0.772 65 1.568 1.537 1.505 1.474 1.441 1.408 1.375 1.341 1.307 1.272 1.238 1.202 1.167 1.132 1.096 1.061 1.025 0.989 0.953 0.918 0.882 0.847 70 1.584 1.555 1.526 1.497 1.467 1.436 1.405 1.374 1.342 1.310 1.278 1.245 1.213 1.180 1.147 1.113 1.080 1.047 1.013 0.980 0.947 0.914 75 1.599 1.572 1.545 1.517 1.489 1.461 1.432 1.403 1.373 1.344 1.313 1.283 1.253 1.222 1.191 1.160 1.129 1.098 1.066 1.035 1.004 0.972 80 1.612 1.587 1.561 1.536 1.509 1.483 1.456 1.429 1.401 1.373 1.345 1.317 1.288 1.259 1.230 1.201 1.172 1.143 1.113 1.084 1.054 1.025 85 1.624 1.600 1.576 1.552 1.527 1.502 1.477 1.452 1.426 1.400 1.373 1.347 1.320 1.293 1.266 1.238 1.211 1.183 1.155 1.128 1.100 1.072 90 1.635 1.613 1.590 1.567 1.544 1.520 1.497 1.472 1.448 1.423 1.399 1.373 1.348 1.323 1.297 1.271 1.245 1.219 1.193 1.167 1.141 1.114 95 1.645 1.624 1.603 1.581 1.559 1.537 1.514 1.491 1.468 1.445 1.422 1.398 1.374 1.350 1.326 1.301 1.277 1.252 1.227 1.202 1.177 1.152 100 1.654 1.634 1.614 1.593 1.573 1.551 1.530 1.508 1.487 1.465 1.442 1.420 1.397 1.374 1.352 1.328 1.305 1.282 1.258 1.235 1.211 1.187 150 1.720 1.706 1.693 1.679 1.666 1.652 1.638 1.624 1.609 1.595 1.580 1.566 1.551 1.536 1.521 1.506 1.491 1.476 1.461 1.445 1.430 1.414 200 1.759 1.748 1.738 1.728 1.718 1.708 1.697 1.687 1.676 1.666 1.655 1.644 1.633 1.622 1.611 1.600 1.589 1.578 1.567 1.556 1.544 1.533
- 10. 10 Appendix A Table A-5 Models with no intercept (from Farebrother): Negative serial correlation Durbin-Watson Ninety Five Per Cent Minimal Bound N K=0 K=1 K=2 K=3 K=4 K=5 K=6 K=7 K=8 K=9 K=10 K=11 K=12 K=13 K=14 K=15 K=16 K=17 K=18 K=19 K=20 K=21 2 1.988 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 3 2.761 0.994 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 4 2.871 1.836 0.582 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 5 2.857 2.178 1.267 0.380 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 6 2.844 2.320 1.655 0.917 0.266 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 7 2.828 2.398 1.871 1.283 0.690 0.197 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 8 2.805 2.453 2.008 1.521 1.017 0.537 0.151 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 9 2.783 2.483 2.110 1.687 1.251 0.823 0.429 0.120 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 10 2.762 2.501 2.181 1.816 1.427 1.044 0.678 0.350 0.097 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 11 2.742 2.511 2.231 1.913 1.569 1.218 0.881 0.567 0.291 0.080 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 12 2.723 2.516 2.268 1.987 1.682 1.364 1.049 0.752 0.481 0.245 0.068 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 13 2.705 2.518 2.296 2.044 1.771 1.484 1.193 0.911 0.649 0.413 0.210 0.058 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 14 2.688 2.517 2.316 2.090 1.843 1.582 1.316 1.051 0.797 0.565 0.358 0.181 0.050 ----- ----- ----- ----- ----- ----- ----- ----- ----- 15 2.672 2.515 2.332 2.126 1.902 1.664 1.419 1.172 0.931 0.703 0.497 0.314 0.158 0.043 ----- ----- ----- ----- ----- ----- ----- ----- 16 2.657 2.512 2.344 2.155 1.950 1.732 1.506 1.276 1.049 0.829 0.624 0.439 0.277 0.139 0.038 ----- ----- ----- ----- ----- ----- ----- 17 2.644 2.508 2.353 2.179 1.990 1.789 1.580 1.367 1.153 0.944 0.743 0.557 0.391 0.246 0.124 0.034 ----- ----- ----- ----- ----- ----- 18 2.631 2.504 2.359 2.199 2.024 1.838 1.644 1.445 1.244 1.045 0.852 0.669 0.501 0.351 0.220 0.110 0.030 ----- ----- ----- ----- ----- 19 2.618 2.499 2.364 2.215 2.053 1.880 1.699 1.513 1.324 1.136 0.951 0.773 0.605 0.452 0.316 0.198 0.099 0.027 ----- ----- ----- ----- 20 2.607 2.494 2.368 2.228 2.077 1.916 1.747 1.573 1.395 1.216 1.040 0.868 0.704 0.550 0.410 0.286 0.179 0.090 0.025 ----- ----- ----- 21 2.596 2.489 2.370 2.239 2.098 1.947 1.789 1.625 1.457 1.289 1.120 0.955 0.796 0.644 0.502 0.373 0.260 0.162 0.081 0.022 ----- ----- 22 2.585 2.484 2.372 2.249 2.116 1.974 1.825 1.671 1.513 1.353 1.193 1.034 0.880 0.731 0.591 0.460 0.341 0.238 0.148 0.074 0.020 ----- 23 2.575 2.479 2.373 2.257 2.131 1.998 1.858 1.712 1.563 1.411 1.258 1.107 0.957 0.813 0.674 0.544 0.422 0.313 0.218 0.136 0.068 0.019 24 2.566 2.474 2.373 2.263 2.145 2.019 1.886 1.749 1.607 1.463 1.318 1.172 1.029 0.888 0.753 0.623 0.502 0.389 0.289 0.201 0.125 0.062 25 2.557 2.470 2.373 2.269 2.156 2.037 1.912 1.782 1.647 1.510 1.371 1.232 1.094 0.958 0.826 0.699 0.578 0.465 0.360 0.267 0.185 0.115 26 1.073 1.004 0.934 0.863 0.792 0.722 0.652 0.584 0.518 0.454 0.394 0.336 0.283 0.233 0.189 0.148 0.113 0.083 0.057 0.037 0.022 0.011 27 1.089 1.023 0.955 0.886 0.817 0.749 0.681 0.614 0.549 0.486 0.426 0.368 0.314 0.264 0.218 0.176 0.138 0.105 0.077 0.053 0.034 0.020 28 1.105 1.040 0.974 0.908 0.841 0.774 0.708 0.643 0.579 0.517 0.457 0.400 0.345 0.294 0.247 0.204 0.164 0.129 0.098 0.071 0.050 0.032 29 1.120 1.057 0.993 0.929 0.864 0.798 0.734 0.670 0.607 0.546 0.487 0.430 0.376 0.324 0.276 0.232 0.191 0.154 0.120 0.091 0.067 0.046 30 1.134 1.073 1.011 0.948 0.885 0.822 0.759 0.696 0.635 0.574 0.516 0.460 0.405 0.354 0.305 0.260 0.217 0.179 0.144 0.113 0.086 0.062 31 1.147 1.088 1.028 0.967 0.905 0.844 0.782 0.721 0.661 0.602 0.544 0.488 0.434 0.383 0.334 0.288 0.244 0.205 0.168 0.135 0.106 0.080 32 1.160 1.103 1.044 0.985 0.925 0.865 0.805 0.745 0.686 0.628 0.571 0.516 0.462 0.411 0.362 0.315 0.271 0.230 0.193 0.158 0.127 0.100 33 1.173 1.117 1.060 1.002 0.944 0.885 0.826 0.768 0.710 0.653 0.597 0.542 0.489 0.438 0.389 0.342 0.298 0.256 0.218 0.182 0.149 0.120 34 1.185 1.130 1.075 1.018 0.961 0.904 0.847 0.790 0.733 0.677 0.622 0.568 0.516 0.465 0.416 0.369 0.324 0.282 0.243 0.206 0.172 0.141 35 1.196 1.143 1.089 1.034 0.978 0.923 0.867 0.811 0.755 0.700 0.646 0.593 0.541 0.491 0.442 0.395 0.350 0.308 0.268 0.230 0.195 0.163 36 1.207 1.155 1.102 1.049 0.995 0.940 0.886 0.831 0.777 0.723 0.669 0.617 0.566 0.516 0.467 0.421 0.376 0.333 0.292 0.254 0.218 0.185 37 1.217 1.167 1.116 1.063 1.010 0.957 0.904 0.850 0.797 0.744 0.692 0.640 0.590 0.540 0.492 0.446 0.401 0.358 0.317 0.278 0.241 0.207 38 1.228 1.178 1.128 1.077 1.026 0.974 0.921 0.869 0.817 0.765 0.713 0.663 0.613 0.564 0.516 0.470 0.425 0.382 0.341 0.302 0.265 0.230 39 1.237 1.189 1.140 1.090 1.040 0.989 0.938 0.887 0.836 0.785 0.734 0.684 0.635 0.587 0.540 0.494 0.449 0.406 0.365 0.325 0.288 0.252 40 1.247 1.200 1.152 1.103 1.054 1.004 0.954 0.904 0.854 0.804 0.754 0.705 0.657 0.609 0.562 0.517 0.473 0.430 0.388 0.349 0.311 0.275 45 1.289 1.247 1.204 1.160 1.116 1.071 1.026 0.981 0.936 0.890 0.845 0.800 0.755 0.710 0.666 0.623 0.581 0.539 0.499 0.459 0.421 0.384 50 1.325 1.287 1.248 1.208 1.168 1.128 1.087 1.046 1.004 0.963 0.921 0.880 0.838 0.797 0.756 0.715 0.675 0.636 0.597 0.559 0.521 0.485 55 1.356 1.321 1.286 1.250 1.213 1.176 1.139 1.101 1.063 1.025 0.987 0.948 0.910 0.872 0.833 0.796 0.758 0.721 0.684 0.647 0.611 0.576 60 1.383 1.351 1.319 1.285 1.252 1.218 1.183 1.149 1.114 1.078 1.043 1.008 0.972 0.936 0.901 0.865 0.830 0.795 0.760 0.725 0.691 0.657 65 1.408 1.378 1.348 1.317 1.286 1.254 1.222 1.190 1.158 1.125 1.092 1.059 1.026 0.993 0.960 0.927 0.894 0.861 0.828 0.795 0.762 0.730 70 1.429 1.401 1.373 1.345 1.316 1.286 1.257 1.227 1.197 1.166 1.136 1.105 1.074 1.043 1.012 0.981 0.950 0.919 0.888 0.857 0.826 0.795 75 1.448 1.423 1.396 1.369 1.342 1.315 1.287 1.260 1.231 1.203 1.174 1.146 1.117 1.088 1.058 1.029 1.000 0.971 0.941 0.912 0.883 0.854 80 1.466 1.442 1.417 1.392 1.367 1.341 1.315 1.289 1.262 1.236 1.209 1.182 1.155 1.127 1.100 1.072 1.045 1.017 0.989 0.962 0.934 0.907 85 1.482 1.459 1.436 1.412 1.388 1.364 1.340 1.315 1.290 1.265 1.240 1.214 1.189 1.163 1.137 1.111 1.085 1.059 1.033 1.006 0.980 0.954 90 1.497 1.475 1.453 1.431 1.408 1.385 1.362 1.339 1.315 1.292 1.268 1.244 1.220 1.195 1.171 1.146 1.121 1.097 1.072 1.047 1.022 0.997 95 1.510 1.490 1.469 1.448 1.426 1.405 1.383 1.361 1.338 1.316 1.293 1.271 1.248 1.225 1.201 1.178 1.155 1.131 1.108 1.084 1.060 1.037 100 1.523 1.503 1.483 1.463 1.443 1.422 1.402 1.381 1.359 1.338 1.317 1.295 1.273 1.251 1.229 1.207 1.185 1.162 1.140 1.118 1.095 1.072 150 1.611 1.598 1.585 1.571 1.558 1.544 1.530 1.516 1.502 1.488 1.474 1.460 1.445 1.431 1.416 1.402 1.387 1.372 1.357 1.342 1.327 1.312 200 1.664 1.654 1.644 1.634 1.624 1.613 1.603 1.593 1.582 1.572 1.561 1.551 1.540 1.529 1.519 1.508 1.497 1.486 1.475 1.464 1.453 1.442
- 11. 11 Durbin-Watson SignificanceTables Table A-6 Models with no intercept (from Farebrother): Negative serial correlation Durbin-Watson Ninety Nine Per Cent Minimal Bound N K=0 K=1 K=2 K=3 K=4 K=5 K=6 K=7 K=8 K=9 K=10 K=11 K=12 K=13 K=14 K=15 K=16 K=17 K=18 K=19 K=20 K=21 2 1.999 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 3 2.951 0.999 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 4 3.221 1.967 0.586 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 5 3.261 2.462 1.359 0.382 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 6 3.235 2.682 1.878 0.983 0.268 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 7 3.198 2.776 2.177 1.459 0.740 0.198 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 8 3.166 2.817 2.347 1.776 1.158 0.576 0.153 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 9 3.133 2.837 2.448 1.983 1.465 0.937 0.460 0.121 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 10 3.101 2.847 2.514 2.121 1.684 1.224 0.773 0.375 0.098 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 11 3.071 2.847 2.560 2.220 1.842 1.441 1.035 0.647 0.312 0.081 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 12 3.043 2.843 2.592 2.294 1.961 1.607 1.244 0.885 0.549 0.263 0.069 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 13 3.017 2.836 2.612 2.349 2.054 1.737 1.410 1.082 0.764 0.471 0.225 0.059 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 14 2.992 2.828 2.626 2.391 2.127 1.842 1.544 1.244 0.948 0.666 0.409 0.195 0.051 ----- ----- ----- ----- ----- ----- ----- ----- ----- 15 2.969 2.818 2.635 2.423 2.185 1.928 1.656 1.379 1.104 0.837 0.585 0.358 0.170 0.044 ----- ----- ----- ----- ----- ----- ----- ----- 16 2.948 2.808 2.640 2.447 2.231 1.997 1.749 1.494 1.237 0.985 0.743 0.517 0.316 0.150 0.039 ----- ----- ----- ----- ----- ----- ----- 17 2.927 2.797 2.643 2.466 2.269 2.055 1.827 1.591 1.351 1.114 0.883 0.664 0.461 0.281 0.133 0.035 ----- ----- ----- ----- ----- ----- 18 2.908 2.787 2.644 2.480 2.299 2.102 1.893 1.675 1.451 1.227 1.007 0.796 0.597 0.413 0.251 0.119 0.031 ----- ----- ----- ----- ----- 19 2.890 2.776 2.643 2.492 2.324 2.142 1.948 1.746 1.538 1.327 1.118 0.915 0.721 0.539 0.372 0.226 0.107 0.028 ----- ----- ----- ----- 20 2.874 2.766 2.641 2.500 2.344 2.176 1.996 1.807 1.613 1.415 1.217 1.022 0.834 0.656 0.489 0.337 0.204 0.096 0.025 ----- ----- ----- 21 2.858 2.756 2.638 2.506 2.361 2.204 2.036 1.861 1.678 1.492 1.305 1.119 0.937 0.763 0.598 0.446 0.307 0.185 0.087 0.023 ----- ----- 22 2.842 2.746 2.635 2.511 2.375 2.228 2.071 1.907 1.736 1.561 1.384 1.207 1.032 0.862 0.700 0.548 0.408 0.280 0.169 0.080 0.021 ----- 23 2.828 2.736 2.631 2.515 2.387 2.249 2.102 1.947 1.786 1.621 1.454 1.285 1.118 0.954 0.796 0.645 0.504 0.374 0.257 0.155 0.073 0.019 24 2.814 2.727 2.627 2.517 2.396 2.267 2.128 1.983 1.831 1.675 1.516 1.356 1.196 1.038 0.884 0.736 0.596 0.465 0.345 0.237 0.143 0.067 25 2.801 2.717 2.623 2.518 2.404 2.282 2.151 2.014 1.871 1.723 1.572 1.420 1.267 1.115 0.966 0.821 0.683 0.552 0.430 0.319 0.218 0.132 26 2.789 2.709 2.618 2.519 2.411 2.295 2.171 2.042 1.906 1.766 1.623 1.478 1.331 1.186 1.042 0.901 0.765 0.635 0.512 0.399 0.295 0.202 27 2.777 2.700 2.614 2.519 2.416 2.306 2.189 2.066 1.938 1.805 1.669 1.530 1.390 1.250 1.111 0.975 0.842 0.714 0.592 0.477 0.371 0.274 28 2.766 2.692 2.609 2.519 2.421 2.316 2.205 2.088 1.966 1.839 1.710 1.577 1.444 1.309 1.176 1.043 0.914 0.788 0.667 0.553 0.445 0.346 29 2.755 2.684 2.604 2.518 2.425 2.325 2.219 2.107 1.991 1.871 1.747 1.621 1.493 1.364 1.235 1.107 0.981 0.858 0.739 0.625 0.517 0.416 30 2.745 2.676 2.600 2.517 2.428 2.332 2.231 2.125 2.014 1.899 1.781 1.660 1.537 1.414 1.290 1.166 1.044 0.924 0.807 0.695 0.587 0.485 31 2.735 2.668 2.595 2.515 2.430 2.339 2.242 2.140 2.035 1.925 1.812 1.696 1.579 1.460 1.340 1.221 1.102 0.986 0.872 0.761 0.654 0.552 32 2.725 2.661 2.590 2.514 2.432 2.344 2.252 2.155 2.053 1.948 1.840 1.729 1.616 1.502 1.387 1.272 1.157 1.043 0.932 0.823 0.718 0.617 33 2.716 2.654 2.586 2.512 2.433 2.349 2.260 2.167 2.070 1.970 1.866 1.759 1.651 1.541 1.430 1.319 1.208 1.097 0.989 0.882 0.779 0.678 34 2.707 2.647 2.581 2.510 2.434 2.353 2.268 2.179 2.086 1.989 1.889 1.787 1.683 1.577 1.470 1.363 1.255 1.148 1.042 0.938 0.836 0.737 35 2.699 2.640 2.576 2.508 2.435 2.357 2.275 2.189 2.100 2.007 1.911 1.813 1.713 1.611 1.507 1.404 1.299 1.196 1.093 0.991 0.891 0.794 36 2.690 2.634 2.572 2.506 2.435 2.360 2.281 2.199 2.113 2.023 1.931 1.837 1.740 1.642 1.542 1.442 1.341 1.240 1.140 1.041 0.943 0.847 37 2.683 2.627 2.567 2.503 2.435 2.363 2.287 2.207 2.124 2.038 1.950 1.859 1.765 1.670 1.574 1.477 1.379 1.282 1.184 1.088 0.992 0.898 38 2.675 2.621 2.563 2.501 2.435 2.365 2.292 2.215 2.135 2.052 1.967 1.879 1.789 1.697 1.604 1.510 1.416 1.321 1.226 1.132 1.039 0.947 39 2.667 2.615 2.559 2.499 2.435 2.367 2.296 2.222 2.145 2.065 1.982 1.898 1.811 1.722 1.632 1.541 1.450 1.358 1.266 1.174 1.083 0.993 40 2.660 2.609 2.555 2.496 2.434 2.369 2.300 2.229 2.154 2.077 1.997 1.915 1.831 1.746 1.659 1.570 1.482 1.392 1.303 1.213 1.124 1.036 45 2.628 2.583 2.535 2.484 2.430 2.374 2.315 2.253 2.190 2.124 2.056 1.986 1.914 1.841 1.767 1.691 1.614 1.537 1.459 1.381 1.302 1.224 50 2.600 2.559 2.516 2.471 2.424 2.374 2.323 2.269 2.214 2.157 2.098 2.037 1.975 1.911 1.847 1.781 1.714 1.646 1.578 1.509 1.439 1.370 55 2.575 2.538 2.500 2.459 2.417 2.373 2.327 2.280 2.231 2.180 2.128 2.075 2.020 1.964 1.907 1.849 1.790 1.730 1.669 1.608 1.546 1.484 60 2.553 2.519 2.484 2.448 2.409 2.370 2.329 2.286 2.242 2.197 2.151 2.103 2.054 2.004 1.954 1.902 1.849 1.796 1.742 1.687 1.631 1.576 65 2.534 2.503 2.470 2.437 2.402 2.366 2.329 2.290 2.250 2.210 2.168 2.125 2.081 2.036 1.990 1.944 1.896 1.848 1.799 1.750 1.700 1.650 70 2.516 2.487 2.458 2.427 2.395 2.361 2.327 2.292 2.256 2.219 2.181 2.142 2.102 2.061 2.020 1.977 1.934 1.891 1.846 1.802 1.756 1.710 75 2.500 2.473 2.446 2.417 2.387 2.357 2.325 2.293 2.260 2.226 2.191 2.155 2.118 2.081 2.043 2.005 1.965 1.926 1.885 1.844 1.802 1.760 80 2.486 2.461 2.436 2.408 2.380 2.352 2.323 2.293 2.262 2.231 2.198 2.165 2.132 2.098 2.063 2.027 1.991 1.954 1.917 1.879 1.841 1.803 85 2.473 2.449 2.425 2.399 2.374 2.347 2.320 2.292 2.263 2.234 2.204 2.174 2.143 2.111 2.079 2.046 2.012 1.979 1.944 1.909 1.874 1.838 90 2.460 2.438 2.415 2.391 2.367 2.342 2.317 2.291 2.264 2.237 2.209 2.181 2.152 2.122 2.092 2.061 2.030 1.999 1.967 1.935 1.902 1.869 95 2.449 2.428 2.406 2.384 2.361 2.338 2.314 2.289 2.264 2.239 2.212 2.186 2.159 2.131 2.103 2.075 2.046 2.016 1.986 1.956 1.926 1.895 100 2.438 2.418 2.398 2.377 2.355 2.333 2.310 2.287 2.264 2.240 2.215 2.190 2.165 2.139 2.113 2.086 2.059 2.031 2.003 1.975 1.946 1.917 150 2.363 2.349 2.336 2.322 2.308 2.294 2.279 2.265 2.250 2.235 2.220 2.204 2.188 2.173 2.156 2.140 2.124 2.107 2.090 2.073 2.056 2.039 200 2.317 2.307 2.296 2.286 2.276 2.265 2.255 2.244 2.233 2.222 2.211 2.200 2.189 2.177 2.166 2.154 2.142 2.131 2.119 2.106 2.094 2.082