The crude birth rate (cbr) is defined as the ratio of the number of births in...

     

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The crude birth rate (cbr) is defined as the ratio of the number of births in a year over the population in the mid-year, expressed per 1000 population. The infant mortality rate (mort) is defined as the ratio of the number of deaths of infants under the age of one in a year over the number of live births in that year, expressed per 1000 live births. Data on these variables for 125 countries in the year 1985 can be used to run these two simple regressions: cbr=B,+Bmort+ & or cbra,+a, log(mort) +u where & and u are the errors in the respective equations. (1) (2) (a) Mukherjee et al. (Econometrics and Data Analysis for Developing Countries, 1998, where these data can be found) state, "Under conditions of poverty and ill-health, where infant mortality is high, it is the expected number of surviving children - that is, children who survive into adulthood - which guides fertility decisions". According to this reasoning, what is the expected sign of , or a,? (2 marks) (b) Given the scatter plots of cbr on mort and log(mort) below, which of the linear regressions above is likely to offer a better fit to the data? Why? (2 marks) MD 50- 40- 50- 10- MORT 120 100 200 40- 30- 20 10- 20 24 28 32 38 40 44 48 82 LOG(MORT) (c) Using data for all countries, least squares estimation of the two regressions above generated the results below. Can we compare the R measures in these two regressions? Explain. (2 marks) Regression 1: Dependent Variable cbr Estimate Std. Error t-statistic p-value Intercept 16.04092 0.98127 16.35 <0.001 mort 0.24299 0.01173 20.71 <0.001 R-squared = 0.78 Regression 2: Dependent Variable cbr Estimate Std. Error t-statistic p-value Intercept -15.33838 1.83720 -8.35 <0.001 log(mort) 12.50974 0.46485 26.91 R-squared 0.85 (d) Interpret the estimated coefficient of mort (B2) in the first table (Regression 1) above. Construct a 95% confidence interval for B. What does the hypothesis H, B = 0 mean? Can you reject this hypothesis using the 95% confidence interval for B? Note that the confidence interval must be used for the (8 marks) test. What is the significance level of this test? (e) Given the t-statistic (26.91) for a, in the second table (Regression 2), what is the approximate p-value? Give your reason. Use this value to test the hypothesis that Ha, =0 against the alternative H, : a, >0 at the 1% level of significance. (4 marks) (f) Use the estimates in the second table (Regression 2) to predict the crude birth rate (cbr) (2 marks) for a country with an infant mortality rate (mort) of 66. 2.5% Table 1: Cut-off Points for Student's t Distribution 2.5% The row labels are the degrees of freedom. The two-tail column headings give the probability that will exceed the tabulated value c in absolute value, i.e. P ( c), e.g. forv 7, P(t 2.365) = 0.05. -2.365 0 2.365 10% 0 1.638 The one-tail column headings give the probability that will simply exceed the tabulated value, c.g. for v = 3, P(t > 1.638) = 0.10. One-tail probability-values 0.10 0.05 0.025 0.01 Two-tail probability-values 0.005 0.001 0.0005 v (df) 0.20 0.10 0.05 0.02 0.01 0.002 0.001 1 3.078 6.314 12.706 31.821 63.656 318.289 636.578 2 1.886 2.920 4.303 6.965 9.925 22.328 31.600 3 1.638 2.353 3.182 4.541 5.841 10.214 12.924 4 1.533 2.132 2.776 3.747 4.604 7.173 8.610 1.476 2.015 2.571 3.365 4.032 5.893 6.869 6 1.440 1.943 2.447 3.143 3.707 5.208 5.959 7 1.415 1.895 2.365 2.998 3.499 4.785 5.408 8 1.397 1.860 2.306 2.896 3.355 4.501 5.041 9 1.383 1.833 2.262 2.821 3.250 4.297 4.781 10 1.372 1.812 2.228 2.764 3.169 4.144 4.587 11 1.363 1.796 2.201 2.718 3.106 4.025 4.437 12 1.356 1.782 2.179 2.681 3.055 3.930 4.318 1.350 1.771 2.160 2.650 3.012 3.852 4.221 1.345 1.761 2.145 2.624 2.977 3.787 4.140 1.341 1.753 2:131 2.602 2.947 3.733 4.073 1.337 1.746 2.120 2.583 2.921 3.686 4.015 1.333 1.740 2.110 2.567 2.898 3.646 3.965 1.330 1.734 2.101 2.552 2.878 3.610 3.922 1.328 1.729 2.093 2.539 2.861 3.579 3.883 1.325 1.725 2.086 2.528 2.845 3.552 3.850 22222222224828 1.323 1.721 2.080 2.518 2.831 3.527 3.819 1.321 1.717 2.074 2.508 2.819 3.505 3.792 1.319 1.714 2.069 2.500 2.807 3.485 3.767 1.318 1.711 2.064 2.492 2.797 3.467 3.745 1.316 1.708 2.060 2.485 2.787 3.450 3.725 26 1.315 1.706 2.056 2.479 2.779 3.435 3.707 1.314 1.703 2.052 2.473 2.771 3.421 3.690 1.313 1.701 2.048 2.467 2.763 3.408 3.674 1.311 1.699 2.045 2.462 2.756 3.396 3.659 1.310 1.697 2.042 2.457 2.750 3.385 3.646 1.303 1.684 2.021 2.423 2.704 3.307 3.551 1.296 1.671 2.000 2.390 2.660 3.232 3.460 120 1.289 1.658 1.980 2.358 2.617 3.160 3.373 1.282 1.645 1.960 2.326 2.576 3.090 3.290 Table 2: 5% Cut-off points for the F Distribution 2.76 Tail area =0.05 F3,60 8 m=degrees of freedom in the numerator n = degrees of freedom in the deniminator Example: Pr(F3,60 2.76) = 0.05 771 1 2 3 4 5 9 7 8 9 10 72 10 4.96 4.10 3.71 3.48 3.33 3.22 3.14 3.07 3.02 2.98 11 4.84 3.98 3.59 3.36 3.20 3.09 3.01 2.95 2.90 2.85 12 4.75 3.89 3.49 3.26 3.11 3.00 2.91 2.85 2.80 2.75 13 4.67 3.81 3.41 3.18 3.03 2.92 2.83 2.77 2.71 2.67 14 4.60 3.74 3.34 3.11 2.96 2.85 2.76 2.70 2.65 2.60 15 4.54 3.68 3.29 3.06 2.90 2.79 2.71 2.64 2.59 2.54 16 4.49 3.63 3.24 3.01 2.85 2.74 2.66 2.59 2.54 2.49 17 4.45 3.59 3.20 2.96 2.81 2.70 2.61 2.55 2.49 2.45 18 4.41 3.55 3.16 2.93 2.77 2.66 2.58 2.51 2.46 2:41 19 4.38 3.52 20 4.35 3.49 3.10 3.13 2.90 2.74 2.63 2.54 2.48 2.42 2.38 2.87 2.71 2.60 2.51 2:45 2.39 2.35 21 4.32 3.47 3.07 2.84 2.68 2.57 2.49 2.42 2.37 2.32 22 4.30 3.44 3.05. 2.82 2.66 2.55 2.46 2.40 2.34 2.30 23 4.28 3.42 3.03 2.80 2.64 2.53 2.44 2.37 2.32 2.27 24 4.26 3.40 3.01 2.78 2.62 2.51 2.42 2.36 2.30 2.25 25 4.24 3.39 2.99 2.76 2.60 2.49 2.40 2.34 2.28 2.24 26 4.23 3.37. 2.98 2.74 2,59 2.47 2.39 2.32 2.27 2.22 27 4.21 3.35 2.96 2.73 2.57 2.46 2.37 2.31 2.25 2.20 28 4.20 3.34 2.95 2.71 2.56 2.45 2.36 2.29 2.24 2.19 29 4.18 3.33 2.93 2.70 2.55 2.43 2.35 2.28 2,22 2.18 30 4.17 3.32 2.92 2.69 2.53 2.42 2.33 2.27 2.21 2.16 40 4.08 3.23 2.84 2.61 2.45 2.34' 2.25 2.18 2.12 2.08 50 4.03 3.18 2.79 2.56 2.40 2.29 2.20 2.13 2.07 2.03 60 4.00 3.15 2.76 2.53 2.37 2.25 2.17 2.10 2.04 1.99 70 3.98 3.13 2.74 2.50 2.35 2.23 2.14 2.07 2.02 1.97 80 3.96 3.11 2.72 2.49 2.33 2.21 2.13 2.06 2.00 1.95 90 3.95 3.10 2.71 2.47 2.32 2.20 2.11 2.04 1.99 1.94 100 3.94 3.09 2.70 2,46 2.31 2.19 2.10 2.03 1.97 1.93 120 3.92 3.07 2.68 2.45 2.29 2.17 2.09 2.02 1.96 1.91 00 3.84 3.00 2.60 2.37 2.21 2.10 2.01 1.94 1.88 1.83 ****** The crude birth rate (cbr) is defined as the ratio of the number of births in a year over the population in the mid-year, expressed per 1000 population. The infant mortality rate (mort) is defined as the ratio of the number of deaths of infants under the age of one in a year over the number of live births in that year, expressed per 1000 live births. Data on these variables for 125 countries in the year 1985 can be used to run these two simple regressions: cbr=B,+Bmort+ & or cbra,+a, log(mort) +u where & and u are the errors in the respective equations. (1) (2) (a) Mukherjee et al. (Econometrics and Data Analysis for Developing Countries, 1998, where these data can be found) state, "Under conditions of poverty and ill-health, where infant mortality is high, it is the expected number of surviving children - that is, children who survive into adulthood - which guides fertility decisions". According to this reasoning, what is the expected sign of , or a,? (2 marks) (b) Given the scatter plots of cbr on mort and log(mort) below, which of the linear regressions above is likely to offer a better fit to the data? Why? (2 marks) MD 50- 40- 50- 10- MORT 120 100 200 40- 30- 20 10- 20 24 28 32 38 40 44 48 82 LOG(MORT) (c) Using data for all countries, least squares estimation of the two regressions above generated the results below. Can we compare the R measures in these two regressions? Explain. (2 marks) Regression 1: Dependent Variable cbr Estimate Std. Error t-statistic p-value Intercept 16.04092 0.98127 16.35 <0.001 mort 0.24299 0.01173 20.71 <0.001 R-squared = 0.78 Regression 2: Dependent Variable cbr Estimate Std. Error t-statistic p-value Intercept -15.33838 1.83720 -8.35 <0.001 log(mort) 12.50974 0.46485 26.91 R-squared 0.85 (d) Interpret the estimated coefficient of mort (B2) in the first table (Regression 1) above. Construct a 95% confidence interval for B. What does the hypothesis H, B = 0 mean? Can you reject this hypothesis using the 95% confidence interval for B? Note that the confidence interval must be used for the (8 marks) test. What is the significance level of this test? (e) Given the t-statistic (26.91) for a, in the second table (Regression 2), what is the approximate p-value? Give your reason. Use this value to test the hypothesis that Ha, =0 against the alternative H, : a, >0 at the 1% level of significance. (4 marks) (f) Use the estimates in the second table (Regression 2) to predict the crude birth rate (cbr) (2 marks) for a country with an infant mortality rate (mort) of 66. 2.5% Table 1: Cut-off Points for Student's t Distribution 2.5% The row labels are the degrees of freedom. The two-tail column headings give the probability that will exceed the tabulated value c in absolute value, i.e. P ( c), e.g. forv 7, P(t 2.365) = 0.05. -2.365 0 2.365 10% 0 1.638 The one-tail column headings give the probability that will simply exceed the tabulated value, c.g. for v = 3, P(t > 1.638) = 0.10. One-tail probability-values 0.10 0.05 0.025 0.01 Two-tail probability-values 0.005 0.001 0.0005 v (df) 0.20 0.10 0.05 0.02 0.01 0.002 0.001 1 3.078 6.314 12.706 31.821 63.656 318.289 636.578 2 1.886 2.920 4.303 6.965 9.925 22.328 31.600 3 1.638 2.353 3.182 4.541 5.841 10.214 12.924 4 1.533 2.132 2.776 3.747 4.604 7.173 8.610 1.476 2.015 2.571 3.365 4.032 5.893 6.869 6 1.440 1.943 2.447 3.143 3.707 5.208 5.959 7 1.415 1.895 2.365 2.998 3.499 4.785 5.408 8 1.397 1.860 2.306 2.896 3.355 4.501 5.041 9 1.383 1.833 2.262 2.821 3.250 4.297 4.781 10 1.372 1.812 2.228 2.764 3.169 4.144 4.587 11 1.363 1.796 2.201 2.718 3.106 4.025 4.437 12 1.356 1.782 2.179 2.681 3.055 3.930 4.318 1.350 1.771 2.160 2.650 3.012 3.852 4.221 1.345 1.761 2.145 2.624 2.977 3.787 4.140 1.341 1.753 2:131 2.602 2.947 3.733 4.073 1.337 1.746 2.120 2.583 2.921 3.686 4.015 1.333 1.740 2.110 2.567 2.898 3.646 3.965 1.330 1.734 2.101 2.552 2.878 3.610 3.922 1.328 1.729 2.093 2.539 2.861 3.579 3.883 1.325 1.725 2.086 2.528 2.845 3.552 3.850 22222222224828 1.323 1.721 2.080 2.518 2.831 3.527 3.819 1.321 1.717 2.074 2.508 2.819 3.505 3.792 1.319 1.714 2.069 2.500 2.807 3.485 3.767 1.318 1.711 2.064 2.492 2.797 3.467 3.745 1.316 1.708 2.060 2.485 2.787 3.450 3.725 26 1.315 1.706 2.056 2.479 2.779 3.435 3.707 1.314 1.703 2.052 2.473 2.771 3.421 3.690 1.313 1.701 2.048 2.467 2.763 3.408 3.674 1.311 1.699 2.045 2.462 2.756 3.396 3.659 1.310 1.697 2.042 2.457 2.750 3.385 3.646 1.303 1.684 2.021 2.423 2.704 3.307 3.551 1.296 1.671 2.000 2.390 2.660 3.232 3.460 120 1.289 1.658 1.980 2.358 2.617 3.160 3.373 1.282 1.645 1.960 2.326 2.576 3.090 3.290 Table 2: 5% Cut-off points for the F Distribution 2.76 Tail area =0.05 F3,60 8 m=degrees of freedom in the numerator n = degrees of freedom in the deniminator Example: Pr(F3,60 2.76) = 0.05 771 1 2 3 4 5 9 7 8 9 10 72 10 4.96 4.10 3.71 3.48 3.33 3.22 3.14 3.07 3.02 2.98 11 4.84 3.98 3.59 3.36 3.20 3.09 3.01 2.95 2.90 2.85 12 4.75 3.89 3.49 3.26 3.11 3.00 2.91 2.85 2.80 2.75 13 4.67 3.81 3.41 3.18 3.03 2.92 2.83 2.77 2.71 2.67 14 4.60 3.74 3.34 3.11 2.96 2.85 2.76 2.70 2.65 2.60 15 4.54 3.68 3.29 3.06 2.90 2.79 2.71 2.64 2.59 2.54 16 4.49 3.63 3.24 3.01 2.85 2.74 2.66 2.59 2.54 2.49 17 4.45 3.59 3.20 2.96 2.81 2.70 2.61 2.55 2.49 2.45 18 4.41 3.55 3.16 2.93 2.77 2.66 2.58 2.51 2.46 2:41 19 4.38 3.52 20 4.35 3.49 3.10 3.13 2.90 2.74 2.63 2.54 2.48 2.42 2.38 2.87 2.71 2.60 2.51 2:45 2.39 2.35 21 4.32 3.47 3.07 2.84 2.68 2.57 2.49 2.42 2.37 2.32 22 4.30 3.44 3.05. 2.82 2.66 2.55 2.46 2.40 2.34 2.30 23 4.28 3.42 3.03 2.80 2.64 2.53 2.44 2.37 2.32 2.27 24 4.26 3.40 3.01 2.78 2.62 2.51 2.42 2.36 2.30 2.25 25 4.24 3.39 2.99 2.76 2.60 2.49 2.40 2.34 2.28 2.24 26 4.23 3.37. 2.98 2.74 2,59 2.47 2.39 2.32 2.27 2.22 27 4.21 3.35 2.96 2.73 2.57 2.46 2.37 2.31 2.25 2.20 28 4.20 3.34 2.95 2.71 2.56 2.45 2.36 2.29 2.24 2.19 29 4.18 3.33 2.93 2.70 2.55 2.43 2.35 2.28 2,22 2.18 30 4.17 3.32 2.92 2.69 2.53 2.42 2.33 2.27 2.21 2.16 40 4.08 3.23 2.84 2.61 2.45 2.34' 2.25 2.18 2.12 2.08 50 4.03 3.18 2.79 2.56 2.40 2.29 2.20 2.13 2.07 2.03 60 4.00 3.15 2.76 2.53 2.37 2.25 2.17 2.10 2.04 1.99 70 3.98 3.13 2.74 2.50 2.35 2.23 2.14 2.07 2.02 1.97 80 3.96 3.11 2.72 2.49 2.33 2.21 2.13 2.06 2.00 1.95 90 3.95 3.10 2.71 2.47 2.32 2.20 2.11 2.04 1.99 1.94 100 3.94 3.09 2.70 2,46 2.31 2.19 2.10 2.03 1.97 1.93 120 3.92 3.07 2.68 2.45 2.29 2.17 2.09 2.02 1.96 1.91 00 3.84 3.00 2.60 2.37 2.21 2.10 2.01 1.94 1.88 1.83 ******