Lecture Twelve 1 Outline Failure Time Analysis Linear Probability Model Poisson Distribution 2 Failure Time Analysis Example Duration of Expansions Issue does the probability of an expansion ending depend on how long it has lasted Exponential distribution assumes the answer since the hazard rate is constant Weibull distribution allows a test to be performed 3 Part II Failure Time Analysis Exponential survival function hazard rate Weibull Exploratory Data Analysis Lab Seven 4 Duration of Post War Economic Expansions in Months 5 Trough Oct 1945 Oct 1949 May 1954 April 1958 Feb 1961 Nov 1970 March 1975 July 1980 Nov 1982 March 1991 Peak Nov 1948 July 1953 August 1957 April 1960 Dec 1969 Nov 1973 January 1980 July 1981 July 1990 March 2000 Duration 37 45 39 24 106 36 58 12 92 120 6 Estimated Survivor Function for Ten Post War Expansions 7 Duration 0 12 24 36 37 39 45 58 92 106 120 Ending 0 1 1 1 1 1 1 1 1 1 1 At Risk 10 10 9 8 7 6 5 4 3 2 1 F t 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 1 Survivor 1 0 9 0 8 0 7 0 6 0 5 0 4 0 3 0 2 0 1 0 8 Figure 2 Estimated Survivor Function for Post War Expansions 1 2 Survivor Function 1 0 8 0 6 0 4 0 2 0 0 20 40 60 80 100 120 140 Duration in Months 9 Figure 3 Exponential Trendline Fitted to Estimated Survivor Function 1 S t e 0 9 t 0 8 0 0217x y 1 1972e 2 R 0 9533 Survivor Function 0 7 0 6 0 5 0 4 0 3 0 2 0 1 0 0 20 40 60 80 100 120 Duration in Months 10 Figure 4 Constrained Expontial trendline Fitted to Estimated Survivor Function 1 0 9 0 8 Survivor Function 0 7 0 019x y e 2 R 0 9313 0 6 0 5 0 4 0 3 0 2 0 1 0 0 20 40 60 80 100 120 Duration in Months 11 Exponential Distribution Hazard rate ratio of density function to the survivor function h t f t S t measure of probability of failure at time t given that you have survived that long for the exponential it is a constant h t exp t exp t Duration 0 12 24 36 37 39 45 58 92 106 120 Ending 0 1 1 1 1 1 1 1 1 1 1 At Risk Inter Haz 10 0 10 0 1000 9 0 1111 8 0 1250 7 0 1429 6 0 1667 5 0 2000 4 0 2500 3 0 3333 2 0 5000 1 1 0000 Interval hazard rate ending at risk 13 Cumulative Hazard Function In general t H t h u du 0 t For the exponential H t du t 0 Duration 0 12 24 36 37 39 45 58 92 106 120 Ending 0 1 1 1 1 1 1 1 1 1 1 At Risk 10 10 9 8 7 6 5 4 3 2 1 Inter Haz Cum Hazard 0 0 0 1000 0 1000 0 1111 0 2111 0 1250 0 3361 0 1429 0 4790 0 1667 0 6456 0 2000 0 8456 0 2500 1 0956 0 3333 1 4290 0 5000 1 929 1 0000 2 929 15 Cumulative Hazard Function Postwar Expansions 3 5 3 y 0 0223x 0 2422 2 R 0 9288 Cumulative Hazard 2 5 2 1 5 1 0 5 0 0 20 40 60 80 100 120 140 0 5 Duration in Months 16 Cumulative Hazard Function Postwar Expansions 3 5 3 y 0 0192x 2 R 0 9015 Cumulative Hazrd 2 5 2 1 5 1 0 5 0 0 20 40 60 80 100 120 140 Duration in Months 17 Weibull Distribution F t 1 exp t exp t S t ln S t t h t f t S t f t dF t dt exp t t h t t if h t constant if h t is increasing function if h t is a decreasing function Weibull Distribution Cumulative Hazard Function H t 1 t ln H t ln 1 ln t 19 Cumulative Hazard Function Post War Expansions Cumulative Hazard 10 1 10 100 1000 0 1 Duration in Months 20 Log Log Plot of Cumulativ e Hazard Function Vs Duration Post War Expansions 2 LNCUMHAZ 1 0 1 2 3 2 0 2 5 3 0 3 5 4 0 4 5 5 0 LNDUR 21 Dependent Variable LNCUMHAZ Method Least Squares Sample 2 11 Included observations 10 Variable Coefficient Std Error t Statistic Prob LNDUR C 1 436662 5 920740 0 103558 0 403303 13 87303 14 68061 0 0000 0 0000 R squared 0 960092 Mean dependent var Adjusted R squared 0 955103 S D dependent var S E of regression 0 220022 Akaike info criterion Sum squared resid 0 387276 Schwarz criterion Log likelihood 2 066628 F statistic Durbin Watson stat 1 210695 Prob F statistic 0 409591 1 038386 0 013326 0 047191 192 4609 0 000001 22 Is Beta More Than One H0 beta 1 HA beta 1 and hazard rate is increasing with time i e expansions are more likely to end the longer they last t 1 437 1 0 104 4 20 23 Conclude Economic expansions are at increasing risk the longer they last the business cycle is not dead so much for the new economics maybe Karl Marx was right capitalism is an inherently unstable system subject to cycles 24 Lab Seven Source Wayne Nelson Applied Life data Analysis 1982 John Wiley Diesel Generators hours to fan failure indicates running time i e still running whenlast observed Hours Ending At Risk Interval Interval Hazard Rate Cumulative Hazard Rate 450 460 1150 1150 1560 1600 1660 1850 1850 1850 1850 1850 2030 2030 2030 2070 2070 2080 2200 25 Source Wayne Nelson Applied Life data Analysis 1982 John Wiley Diesel Generators hours to fan failure indicates running time i e still running when last observ Hours Ending At Risk Interval Interval Hazard Rate Cumulative Hazard Rate 450 1 70 450 460 68 1150 2 68 700 1150 1560 65 1600 1 65 450 1660 63 1850 62 1850 61 1850 60 1850 59 1850 58 2030 57 2030 56 2030 55 2070 2 55 470 2070 2080 1 53 10 26 Source Wayne Nelson Applied Life data Analysis 1982 John Wiley Diesel Generators hours to fan failure indicates running time i e still running whenlast observed Hours Ending At Risk Interval Interval Hazard Rate Cumulative Hazard Rate 450 1 70 450 0 0143 0 0143 460 68 1150 2 68 700 0 0294 0 0437 1150 1560 65 1600 1 65 450 0 0154 0 0591 1660 63 1850 62 1850 61 1850 60 1850 59 1850 58 2030 57 2030 56 2030 55 2070 2 55 470 0 0364 0 0955 2070 2080 1 53 10 0 0189 0 1143 2200 51 …
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