QUESTION FOUR a) An analyst is using the regression model below to investigate the effects of the three independent variables, X1, X2 and X3 on the dependent variable Y: Y; = Bo + B,X11 + B2Xzi + B3X31 + ui Based on relevant theory, all three of the included independent variables should be relevant for explaining variation in the dependent variable. However, the analyst is concerned that there may be multicollinearity between some of the independent variables. To check this, they compute the sample correlation coefficients between each pair of independent variables: Table 1: Correlation coefficients Between Independent Variables X1 X2 1.00 0.29 0.86 0.29 1.00 0.32 0.86 0.32 1.00 X3 They also estimate the following three auxiliary regressions: (2) X2 =, +a, X2+Q_X +. X)=Y+Y.X.: +Y_X: +8 X=8, +8, X. +8X2+31 (3) The R2 values obtained for these three auxiliary regressions are 0.74 for model (2), 0.10 for model (3) and 0.75 for model (4). i) Based on the information above and any other relevant calculations, do you think the analyst is correct to be concerned about possible multicollinearity in model (1)? Clearly explain and justify your answer. (15 marks) ii) In situations in which multicollinearity is identified, in what ways could the model or analysis be changed to minimise the level of multicollinearity? Is it always correct to regard multicollinearity as a problem which needs to be solved in order to obtain reliable results from the model? Explain your answer. (18 marks) b) Briefly outline the principles or strategies that can be used when trying to select the most appropriate set of explanatory/independent variables for a particular regression model. Why is trying to identify the most suitable explanatory variables crucial in order to obtain reliable results? F - Distribution (a=0.05 in the Right Tail) Numerator Degrees of Freedom dt, df, 1 2 5 7 1 2 7 Denominator Degrees of Freedom 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60 120 161.45 18.513 10.128 7.7086 6.6079 5.9874 5.5914 5.3177 5.1174 4.9646 4.8443 4.7472 4.6672 4.6001 4.5431 4.49-40 4.4513 4.4139 4.3807 4.3512 4.3248 4.3009 4.2793 4.2597 4.2417 4.2252 4.2100 4.1960 4.1830 4.1709 4,0847 4.0012 3.9201 3.8415 199.50 19.000 9.5521 9.9443 5.7861 5.1433 4.7374 4.4590 4.2565 4.1028 3.9823 3.8853 3.8056 3.7389 3.6823 3.6337 3.5915 3.5546 3.5219 3.4928 3.4668 3.4434 3.4221 3.4028 3.3852 3.3690 3.3541 3.3404 3.3277 3.3158 3.2317 3.1504 3.0718 2.9957 215.71 19.164 9.2766 6.5914 5.4095 4.7571 4.3468 4.0662 3.8625 3.7083 3.5874 3.4903 3.4105 3.3439 3.2874 3.2389 3.1968 3.1599 3.1274 3.0984 3.0725 3.0491 3.0280 3.0088 2.9912 2.9752 2.9604 2.9467 2.9340 2.9223 2.8387 2.7581 2.6802 2.6049 224,58 19.247 9.1172 6.3882 5.1922 4.5337 4.1203 3.8379 3.6331 3.4780 3.3567 3.2592 3.1791 3.1122 3.0556 3.0069 2.9647 2.9277 2.8951 2.8661 2.8401 2.8167 2.7955 2.7763 2.7587 2.7426 2.7278 2.7141 2.7014 2.6896 2.6060 2.5252 2.4472 2.3719 230.16 19.296 9.0135 6.2561 5.0503 4.3874 3.9715 3.6875 3.4817 3.3258 3.2039 3.1059 3.0254 2.9582 2.9013 2.8524 2.8100 2.7729 2.7401 2.7109 2.6848 2.6613 2.6400 2.6207 2.6030 2.5868 2.5719 2.5581 2.5454 2.5336 2.4495 2.3683 2.2899 2.2141 233.99 19.330 8.9406 6.1631 4.9503 4.2839 3.8660 3.5806 3.3738 3.2172 3.0946 2.9961 2.9153 2.8477 2.7905 2.7413 2.6987 2.6613 2.6283 2.5990 2.5727 2.5491 2.5277 2.5082 2.4904 2.4741 2.4591 2.4453 2.4324 2.4205 2.3359 2.2541 2.1750 2.0986 236.77 19.353 8.8867 6.0942 48759 4.2067 3.7870 3.5005 3.2927 3.1355 3.0123 2.9134 2.8321 2.7642 2.7066 2.6572 2.6143 2.5767 2.5435 2.5140 24876 2.4638 2.4422 2.4226 2.4047 2.3883 2.3732 2.3593 2.3463 2.3343 2.2490 2.1665 2.0868 2.0096 238.88 19.371 8.8452 6,0410 4.8183 4.1468 3.7257 3.4381 3.2296 3.0717 2.9480 2.8486 2.7669 2.6987 2.6408 2.5911 2.5480 2.5102 2.4768 2.4471 2.4205 2.3965 2.3748 2.3551 2.3371 2.3205 2.3053 2.2913 2.2783 2.2662 2.1802 2.0970 2.0164 1.9384 240.54 19.385 8.8123 6.9988 4.7725 4,0990 3.6767 3.3881 3.1789 3.0204 2.8962 2.7964 2.7144 2.6458 2.5876 2.5377 2.4943 2.4563 2.4227 2.3928 2.3660 2.3419 2.3201 2.3002 2.2821 2.2655 2.2501 2.2360 2.2229 2.2107 2.1240 2.0401 1.9588 1.8799 TABLE of CRITICAL VALUES for STUDENT'S t DISTRIBUTIONS Column headings denote probabilities (a ) above tabulated values. d.f. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 60 80 100 120 140 160 180 200 250 inf 0.40 0.325 0.289 0.277 0.271 0.267 0.265 0.263 0.262 0.261 0.260 0.260 0.259 0.259 0.258 0.258 0 258 0.257 0.257 0.257 0.257 0.257 0.256 0.256 0.256 0.256 0.256 0256 0.256 0.256 0.256 0.256 0.255 0.255 0.255 0.255 0.255 0.255 0.255 0 255 0.255 0.254 0.254 0.254 0.254 0.254 0.254 0.254 0.254 0.254 0.253 0.25 1.000 0.816 0.765 0.741 0.727 0.718 0.711 0.706 0.703 0.700 0.697 0.695 0.694 0.692 0.691 0.690 0.689 0.688 0.688 0.687 0.686 0.686 0.685 0.685 0.684 0.684 0.684 0.683 0.683 0.683 0.682 0.682 0.682 0.682 0.682 0.681 0.681 0.681 0.681 0.681 0.679 0.678 0.677 0.677 0.676 0.676 0.676 0.676 0.675 0.674 0.10 3.078 1.886 1.638 1.533 1.476 1.440 1.415 1.397 1.383 1.372 1.363 1.356 1.350 1.345 1.341 1337 1.333 1.330 1.328 1.325 1.323 1.321 1.319 1.318 1.316 1.315 1.314 1.313 1.311 1.310 1.309 1.309 1.308 1.307 1.306 1.306 1.305 1.304 1.304 1.303 1.296 1 292 1.290 1.289 1.288 1.287 1.286 1.286 1.285 1 282 0.05 6.314 2.920 2.353 2.132 2.015 1.943 1.895 1.860 1.833 1.812 1.796 1.782 1.771 1.761 1.753 1.746 1.740 1.734 1.729 1.725 1.721 1.717 1.714 1.711 1.708 1.706 1.703 1.701 1.699 1.697 1.696 1.694 1.692 1.691 1.690 1.688 1.687 1.686 1.685 1.684 1.671 1.664 1.660 1.658 1.656 1.654 1.653 1.653 1.651 1.645 0.04 7.916 3.320 2.605 2.333 2.191 2.104 2.046 2.004 1.973 1.948 1.928 1.912 1.899 1.887 1.878 1.869 1.862 1.855 1.850 1.844 1.840 1.835 1.832 1.828 1.825 1.822 1.819 1.817 1.814 1.812 1.810 1.808 1.806 1.805 1.803 1.802 1.800 1.799 1.798 1.796 1.781 1.773 1.769 1.766 1.763 1.762 1.761 1.760 1.758 1.751 0.025 0.02 0.01 0.005 0.0025 0.001 0.0005 12.706 15.894 31.821 63.656 127.321 318.289 636.578 4.303 4.849 6.965 9925 14.089 22.328 31,600 3.182 3.482 4.541 5.841 7.453 10.214 12.924 2.776 2.999 3.747 4.604 5.598 7.173 8.610 2.571 2.757 3.365 4.032 4.773 5.894 6.869 2.447 2.612 3.143 3.707 4.317 5.208 5.959 2.365 2.517 2.998 3.499 4.029 4.785 5.408 2.306 2.449 2.896 3.355 3.833 4.501 5.041 2.262 2.398 2.821 3.250 3.690 4.297 4.781 2.228 2.359 2.764 3.169 3.581 4.144 4.587 2.201 2.328 2.718 3.106 3.497 4.025 4.437 2.179 2.303 2.681 3.055 3.428 3.930 4.318 2.160 2.282 2.650 3.012 3.372 3.852 4.221 2.145 2.264 2.624 2.977 3.326 3.787 4.140 2.131 2.249 2.602 2.947 3.286 3.733 4.073 2.120 2.235 2.583 2 921 3.252 3.686 4015 2.110 2.224 2.567 2.898 3.222 3.646 3.965 2.101 2.214 2.552 2.878 3.197 3.610 3.922 2.093 2.205 2.539 2861 3.174 3.579 3.883 2.086 2.197 2.528 2.845 3.153 3.552 3.850 2.080 2.189 2.518 2.831 3.135 3.527 3.819 2.074 2.183 2.508 2.819 3.119 3.505 3.792 2.069 2.177 2.500 2.807 3.104 3.485 3.768 2.064 2.172 2.492 2.797 3.091 3.467 3.745 2.060 2.167 2.485 2.787 3 078 3.450 3.725 2.056 2.162 2.479 2.779 3.067 3.435 3.707 2.052 2.158 2.473 2.771 3.057 3.421 3.689 2.048 2.154 2.467 2.763 3.047 3.408 3.674 2.045 2.150 2.462 2.756 3.038 3.396 3.660 2.042 2.147 2.457 2.750 3030 3.385 3.646 2.040 2.144 2.453 2.744 3.022 3.375 3.633 2.037 2.141 2.449 2.738 3.015 3.365 3.622 2.035 2.138 2.445 2.733 3.008 3.356 3.611 2.032 2.136 2.441 2.728 3.002 3.348 3.601 2.030 2.133 2.438 2.724 2.996 3.340 3.591 2.028 2.131 2.434 2.719 2.990 3.333 3.582 2.026 2.129 2.431 2.715 2.985 3.326 3.574 2.024 2.127 2.429 2.712 2.980 3.319 3.566 2.023 2.125 2.426 2.708 2.976 3.313 3.558 2.021 2.123 2.423 2.704 2.971 3.307 3.551 2.000 2.099 2.390 2.660 2.915 3.232 3.460 1.990 2.088 2374 2639 2.887 3.195 3.416 1.984 2.081 2.364 2626 2.871 3.174 3.390 1.980 2.076 2.358 2.617 2.860 3.160 3.373 1.977 2.073 2.353 2611 2.852 3.149 3.361 1.975 2.071 2.350 2.607 2.847 3.142 3.352 1.973 2.069 2.347 2.603 2.842 3.136 3.345 1.972 2.067 2.345 2.601 2.838 3.131 3.340 1.969 2.065 2.341 2.596 2.832 3.123 3.330 1.960 2.054 2326 2576 2 807 3.090 3.290 MU ... -1 QUESTION FOUR a) An analyst is using the regression model below to investigate the effects of the three independent variables, X1, X2 and X3 on the dependent variable Y: Y; = Bo + B,X11 + B2Xzi + B3X31 + ui Based on relevant theory, all three of the included independent variables should be relevant for explaining variation in the dependent variable. However, the analyst is concerned that there may be multicollinearity between some of the independent variables. To check this, they compute the sample correlation coefficients between each pair of independent variables: Table 1: Correlation coefficients Between Independent Variables X1 X2 1.00 0.29 0.86 0.29 1.00 0.32 0.86 0.32 1.00 X3 They also estimate the following three auxiliary regressions: (2) X2 =, +a, X2+Q_X +. X)=Y+Y.X.: +Y_X: +8 X=8, +8, X. +8X2+31 (3) The R2 values obtained for these three auxiliary regressions are 0.74 for model (2), 0.10 for model (3) and 0.75 for model (4). i) Based on the information above and any other relevant calculations, do you think the analyst is correct to be concerned about possible multicollinearity in model (1)? Clearly explain and justify your answer. (15 marks) ii) In situations in which multicollinearity is identified, in what ways could the model or analysis be changed to minimise the level of multicollinearity? Is it always correct to regard multicollinearity as a problem which needs to be solved in order to obtain reliable results from the model? Explain your answer. (18 marks) b) Briefly outline the principles or strategies that can be used when trying to select the most appropriate set of explanatory/independent variables for a particular regression model. Why is trying to identify the most suitable explanatory variables crucial in order to obtain reliable results? F - Distribution (a=0.05 in the Right Tail) Numerator Degrees of Freedom dt, df, 1 2 5 7 1 2 7 Denominator Degrees of Freedom 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60 120 161.45 18.513 10.128 7.7086 6.6079 5.9874 5.5914 5.3177 5.1174 4.9646 4.8443 4.7472 4.6672 4.6001 4.5431 4.49-40 4.4513 4.4139 4.3807 4.3512 4.3248 4.3009 4.2793 4.2597 4.2417 4.2252 4.2100 4.1960 4.1830 4.1709 4,0847 4.0012 3.9201 3.8415 199.50 19.000 9.5521 9.9443 5.7861 5.1433 4.7374 4.4590 4.2565 4.1028 3.9823 3.8853 3.8056 3.7389 3.6823 3.6337 3.5915 3.5546 3.5219 3.4928 3.4668 3.4434 3.4221 3.4028 3.3852 3.3690 3.3541 3.3404 3.3277 3.3158 3.2317 3.1504 3.0718 2.9957 215.71 19.164 9.2766 6.5914 5.4095 4.7571 4.3468 4.0662 3.8625 3.7083 3.5874 3.4903 3.4105 3.3439 3.2874 3.2389 3.1968 3.1599 3.1274 3.0984 3.0725 3.0491 3.0280 3.0088 2.9912 2.9752 2.9604 2.9467 2.9340 2.9223 2.8387 2.7581 2.6802 2.6049 224,58 19.247 9.1172 6.3882 5.1922 4.5337 4.1203 3.8379 3.6331 3.4780 3.3567 3.2592 3.1791 3.1122 3.0556 3.0069 2.9647 2.9277 2.8951 2.8661 2.8401 2.8167 2.7955 2.7763 2.7587 2.7426 2.7278 2.7141 2.7014 2.6896 2.6060 2.5252 2.4472 2.3719 230.16 19.296 9.0135 6.2561 5.0503 4.3874 3.9715 3.6875 3.4817 3.3258 3.2039 3.1059 3.0254 2.9582 2.9013 2.8524 2.8100 2.7729 2.7401 2.7109 2.6848 2.6613 2.6400 2.6207 2.6030 2.5868 2.5719 2.5581 2.5454 2.5336 2.4495 2.3683 2.2899 2.2141 233.99 19.330 8.9406 6.1631 4.9503 4.2839 3.8660 3.5806 3.3738 3.2172 3.0946 2.9961 2.9153 2.8477 2.7905 2.7413 2.6987 2.6613 2.6283 2.5990 2.5727 2.5491 2.5277 2.5082 2.4904 2.4741 2.4591 2.4453 2.4324 2.4205 2.3359 2.2541 2.1750 2.0986 236.77 19.353 8.8867 6.0942 48759 4.2067 3.7870 3.5005 3.2927 3.1355 3.0123 2.9134 2.8321 2.7642 2.7066 2.6572 2.6143 2.5767 2.5435 2.5140 24876 2.4638 2.4422 2.4226 2.4047 2.3883 2.3732 2.3593 2.3463 2.3343 2.2490 2.1665 2.0868 2.0096 238.88 19.371 8.8452 6,0410 4.8183 4.1468 3.7257 3.4381 3.2296 3.0717 2.9480 2.8486 2.7669 2.6987 2.6408 2.5911 2.5480 2.5102 2.4768 2.4471 2.4205 2.3965 2.3748 2.3551 2.3371 2.3205 2.3053 2.2913 2.2783 2.2662 2.1802 2.0970 2.0164 1.9384 240.54 19.385 8.8123 6.9988 4.7725 4,0990 3.6767 3.3881 3.1789 3.0204 2.8962 2.7964 2.7144 2.6458 2.5876 2.5377 2.4943 2.4563 2.4227 2.3928 2.3660 2.3419 2.3201 2.3002 2.2821 2.2655 2.2501 2.2360 2.2229 2.2107 2.1240 2.0401 1.9588 1.8799 TABLE of CRITICAL VALUES for STUDENT'S t DISTRIBUTIONS Column headings denote probabilities (a ) above tabulated values. d.f. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 60 80 100 120 140 160 180 200 250 inf 0.40 0.325 0.289 0.277 0.271 0.267 0.265 0.263 0.262 0.261 0.260 0.260 0.259 0.259 0.258 0.258 0 258 0.257 0.257 0.257 0.257 0.257 0.256 0.256 0.256 0.256 0.256 0256 0.256 0.256 0.256 0.256 0.255 0.255 0.255 0.255 0.255 0.255 0.255 0 255 0.255 0.254 0.254 0.254 0.254 0.254 0.254 0.254 0.254 0.254 0.253 0.25 1.000 0.816 0.765 0.741 0.727 0.718 0.711 0.706 0.703 0.700 0.697 0.695 0.694 0.692 0.691 0.690 0.689 0.688 0.688 0.687 0.686 0.686 0.685 0.685 0.684 0.684 0.684 0.683 0.683 0.683 0.682 0.682 0.682 0.682 0.682 0.681 0.681 0.681 0.681 0.681 0.679 0.678 0.677 0.677 0.676 0.676 0.676 0.676 0.675 0.674 0.10 3.078 1.886 1.638 1.533 1.476 1.440 1.415 1.397 1.383 1.372 1.363 1.356 1.350 1.345 1.341 1337 1.333 1.330 1.328 1.325 1.323 1.321 1.319 1.318 1.316 1.315 1.314 1.313 1.311 1.310 1.309 1.309 1.308 1.307 1.306 1.306 1.305 1.304 1.304 1.303 1.296 1 292 1.290 1.289 1.288 1.287 1.286 1.286 1.285 1 282 0.05 6.314 2.920 2.353 2.132 2.015 1.943 1.895 1.860 1.833 1.812 1.796 1.782 1.771 1.761 1.753 1.746 1.740 1.734 1.729 1.725 1.721 1.717 1.714 1.711 1.708 1.706 1.703 1.701 1.699 1.697 1.696 1.694 1.692 1.691 1.690 1.688 1.687 1.686 1.685 1.684 1.671 1.664 1.660 1.658 1.656 1.654 1.653 1.653 1.651 1.645 0.04 7.916 3.320 2.605 2.333 2.191 2.104 2.046 2.004 1.973 1.948 1.928 1.912 1.899 1.887 1.878 1.869 1.862 1.855 1.850 1.844 1.840 1.835 1.832 1.828 1.825 1.822 1.819 1.817 1.814 1.812 1.810 1.808 1.806 1.805 1.803 1.802 1.800 1.799 1.798 1.796 1.781 1.773 1.769 1.766 1.763 1.762 1.761 1.760 1.758 1.751 0.025 0.02 0.01 0.005 0.0025 0.001 0.0005 12.706 15.894 31.821 63.656 127.321 318.289 636.578 4.303 4.849 6.965 9925 14.089 22.328 31,600 3.182 3.482 4.541 5.841 7.453 10.214 12.924 2.776 2.999 3.747 4.604 5.598 7.173 8.610 2.571 2.757 3.365 4.032 4.773 5.894 6.869 2.447 2.612 3.143 3.707 4.317 5.208 5.959 2.365 2.517 2.998 3.499 4.029 4.785 5.408 2.306 2.449 2.896 3.355 3.833 4.501 5.041 2.262 2.398 2.821 3.250 3.690 4.297 4.781 2.228 2.359 2.764 3.169 3.581 4.144 4.587 2.201 2.328 2.718 3.106 3.497 4.025 4.437 2.179 2.303 2.681 3.055 3.428 3.930 4.318 2.160 2.282 2.650 3.012 3.372 3.852 4.221 2.145 2.264 2.624 2.977 3.326 3.787 4.140 2.131 2.249 2.602 2.947 3.286 3.733 4.073 2.120 2.235 2.583 2 921 3.252 3.686 4015 2.110 2.224 2.567 2.898 3.222 3.646 3.965 2.101 2.214 2.552 2.878 3.197 3.610 3.922 2.093 2.205 2.539 2861 3.174 3.579 3.883 2.086 2.197 2.528 2.845 3.153 3.552 3.850 2.080 2.189 2.518 2.831 3.135 3.527 3.819 2.074 2.183 2.508 2.819 3.119 3.505 3.792 2.069 2.177 2.500 2.807 3.104 3.485 3.768 2.064 2.172 2.492 2.797 3.091 3.467 3.745 2.060 2.167 2.485 2.787 3 078 3.450 3.725 2.056 2.162 2.479 2.779 3.067 3.435 3.707 2.052 2.158 2.473 2.771 3.057 3.421 3.689 2.048 2.154 2.467 2.763 3.047 3.408 3.674 2.045 2.150 2.462 2.756 3.038 3.396 3.660 2.042 2.147 2.457 2.750 3030 3.385 3.646 2.040 2.144 2.453 2.744 3.022 3.375 3.633 2.037 2.141 2.449 2.738 3.015 3.365 3.622 2.035 2.138 2.445 2.733 3.008 3.356 3.611 2.032 2.136 2.441 2.728 3.002 3.348 3.601 2.030 2.133 2.438 2.724 2.996 3.340 3.591 2.028 2.131 2.434 2.719 2.990 3.333 3.582 2.026 2.129 2.431 2.715 2.985 3.326 3.574 2.024 2.127 2.429 2.712 2.980 3.319 3.566 2.023 2.125 2.426 2.708 2.976 3.313 3.558 2.021 2.123 2.423 2.704 2.971 3.307 3.551 2.000 2.099 2.390 2.660 2.915 3.232 3.460 1.990 2.088 2374 2639 2.887 3.195 3.416 1.984 2.081 2.364 2626 2.871 3.174 3.390 1.980 2.076 2.358 2.617 2.860 3.160 3.373 1.977 2.073 2.353 2611 2.852 3.149 3.361 1.975 2.071 2.350 2.607 2.847 3.142 3.352 1.973 2.069 2.347 2.603 2.842 3.136 3.345 1.972 2.067 2.345 2.601 2.838 3.131 3.340 1.969 2.065 2.341 2.596 2.832 3.123 3.330 1.960 2.054 2326 2576 2 807 3.090 3.290 MU ... -1