Final Exam 2006 - Dec 14 2006 ECON 240A-1 Final L Phillips...

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Dec. 14, 2006 ECON 240A- 1 L. Phillips Final Answer all five questions . They are weighted equally. 1. (30) For a regression using ordinary least squares, OLS, y = b 0 + b 1 x 1 + b 2 x 2 …+ b n x n + e, we make certain assumptions about the properties of the error term, e. a. List five assumptions about e i. ____________________ ii. _____________________ iii. ______________________ iv. ______________________ v. ______________________ b. For one-way analysis of variance, using regression of a quantitative variable against binary dummy explanatory variables (zero/one) we used one of these assumptions to interpret the meaning of the regression coefficients, b 0 , b 1 etc. Which assumption did we use? _________________ c. Which assumption is frequently violated in time series regressions? _________________ d. Explain the difference between homoskedasticity and heteroskedasticity. ______________________________________________________ e. One can obtain estimates of the OLS parameters by minimizing the sum of squared residuals with respect to each regression parameter without making any assumptions about the error term e. So why are these assumptions about the error term important? __________________________________________ _________________________________________________________________ _ 2. (30) The number of days spent recovering from a heart attack was studied for a random sample of 300 patients in the US. The duration of days recovering was used to calculate the Kaplan-Meier estimates of (1) the hazard function, (2) the cumulative hazard function, and (3) the survivor function, as displayed in Table 2-1. These Kaplan-Meier
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Dec. 14, 2006 ECON 240A- 2 L. Phillips Final estimates for the hazard rate and the cumulative hazard rate are plotted in Figures 2-1 and 2-2. Table 2-1: Kaplan-Meier Estimates of Days Recovering from a Heart Attack, US US days # ending # at risk interval hazard rate cumulative hazard rate ratio Survivor Function (# ending/# at risk) (# at risk - # ending)/# at risk 8 1 300 0.0033 0.0033 0.997 0.997 9 1 299 0.0033 0.0066 0.997 0.994 12 4 298 0.0134 0.0201 0.987 0.980 13 1 294 0.0034 0.0235 0.997 0.977 14 4 293 0.0137 0.0371 0.986 0.964 15 10 289 0.0346 0.0717 0.965 0.930 16 3 279 0.0108 0.0825 0.989 0.920 17 8 276 0.0290 0.1115 0.971 0.894 18 8 268 0.0299 0.1413 0.970 0.867 19 13 260 0.0500 0.1913 0.950 0.824 20 12 247 0.0486 0.2399 0.951 0.784 21 11 235 0.0468 0.2867 0.953 0.747 22 14 224 0.0625 0.3492 0.938 0.700 23 13 210 0.0619 0.4111 0.938 0.657 24 9 197 0.0457 0.4568 0.954 0.627 25 16 188 0.0851 0.5419 0.915 0.574 26 19 172 0.1105 0.6524 0.890 0.510 27 11 153 0.0719 0.7243 0.928 0.473 28 15 142 0.1056 0.8299 0.894 0.423 29 16 127 0.1260 0.9559 0.874 0.370 30 11 111 0.0991 1.0550 0.901 0.333 31 11 100 0.1100 1.1650 0.890 0.297 32 12 89 0.1348 1.2998 0.865 0.257 33 13 77 0.1688 1.4686 0.831 0.213 34 15 64 0.2344 1.7030 0.766 0.163 35 6 49 0.1224 1.8255 0.878 0.143 36 10 43 0.2326 2.0580 0.767 0.110 37 11 33 0.3333 2.3914 0.667 0.073 38 6 22 0.2727 2.6641 0.727 0.053 39 5 16 0.3125 2.9766 0.688 0.037 40 2 11 0.1818 3.1584 0.818 0.030 41 1 9 0.1111 3.2695 0.889 0.027 42 1 8 0.1250 3.3945 0.875 0.023 43 4 7 0.5714 3.9659 0.429 0.010 44 1 3 0.3333 4.2993 0.667 0.007 47 2 2 1.0000 5.2993 0.000 0.000
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Dec. 14, 2006 ECON 240A- 3 L. Phillips Final
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Dec. 14, 2006 ECON 240A- 4 L. Phillips Final Figure 2-1: Interval Hazard Rate, Duration: Days Recovering from Heart Attack Before Returning To Work, US 0.0000 0.2000 0.4000 0.6000 0.8000 1.0000 1.2000 0 5 10 15 20 25 30 35 40 45 50 # of Days hazard rate h
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Dec. 14, 2006 ECON 240A- 5 L. Phillips Final Figure 2-2: Cumulative Hazard Rate, Duration: Days Recovering from Heart Attack Before Returning To Work, US y = 0.1223x - 2.0201 R 2 = 0.8507 -2 -1 0 1 2 3 4 5 6 0 5 10 15 20 25 30 35 40 45 50 Days Cumulative Hazard The exponential distribution is often used for duration studies. The density function, f(t), for the exponential is f(t) = λ e - λ t , where the reciprocal of λ would be the mean recovery time. The cumulative distribution function, F(t), or probability the recovery time lasts up to time t* is F(t*) = 1 - e - λ t* . The survivor function, S(t*), i.e. the probability that recovery time is longer than t*, is S(t*) = 1 –F(t*) = e - λ t* . The hazard rate,
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