Lecture TwelveOutlineFailure Time AnalysisPart II: Failure Time AnalysisDuration of Post-War Economic Expansions in MonthsSlide 6Estimated Survivor Function for Ten Post-War ExpansionsSlide 8Slide 9Slide 10Slide 11Exponential DistributionSlide 13Cumulative Hazard FunctionSlide 15Slide 16Slide 17Weibull DistributionSlide 19Slide 20Slide 21Slide 22Is Beta More Than One?ConcludeSlide 25Slide 26Slide 27Slide 28Slide 29Slide 30Fan FailureSlide 32Part II: Linear Probability ModelLab SixSlide 35Slide 36Slide 37Effect of the Zeros on RegressionFirst Compare Tobit Slope to OLS SlopeSlide 40Slide 41Bernoulli Variable: BernSlide 43Slide 44Slide 45Slide 46Linear Probability ModelSlide 48Non-Linear Probability ModelsSlide 50Slide 51Part IV. Poisson Approximation to BinomialExample1Lecture Twelve2Outline•Failure Time Analysis•Linear Probability Model•Poisson Distribution3Failure 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 performed4Part II: Failure Time Analysis•Exponential–survival function–hazard rate•Weibull•Exploratory Data Analysis, Lab Seven5Duration of Post-War Economic Expansions in Months6Trough Peak DurationOct. 1945 Nov. 1948 37Oct. 1949 July 1953 45May 1954 August 1957 39April 1958 April 1960 24Feb. 1961 Dec. 1969 106Nov. 1970 Nov. 1973 36March 1975 January 1980 58July 1980 July 1981 12Nov. 1982 July 1990 92March 1991 March 2000 1207Estimated Survivor Function for Ten Post-War Expansions8Duration # Ending # At Risk F(t) Survivor0 0 10 0 112 1 10 0.1 0.924 1 9 0.2 0.836 1 8 0.3 0.737 1 7 0.4 0.639 1 6 0.5 0.545 1 5 0.6 0.458 1 4 0.7 0.392 1 3 0.8 0.2106 1 2 0.9 0.1120 1 1 1 09Figure 2: Estimated Survivor Function for Post-War Expansions00.20.40.60.811.20 20 40 60 80 100 120 140Duration in MonthsSurvivor Function10Figure 3: Exponential Trendline Fitted to Estimated Survivor Functiony = 1.1972e-0.0217xR2 = 0.953300.10.20.30.40.50.60.70.80.910 20 40 60 80 100 120Duration in MonthsSurvivor FunctiontetS)(11Figure 4: Constrained Expontial trendline, Fitted to Estimated Survivor Functiony = e-0.019xR2 = 0.931300.10.20.30.40.50.60.70.80.910 20 40 60 80 100 120Duration in MonthsSurvivor FunctionExponential 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(/)exp( tt13Duration # Ending # At Risk Inter. Haz.0 0 10 012 1 10 0.100024 1 9 0.111136 1 8 0.125037 1 7 0.142939 1 6 0.166745 1 5 0.200058 1 4 0.250092 1 3 0.3333106 1 2 0.5000120 1 1 1.0000Interval hazard rate=#ending/#at riskCumulative Hazard Function•In general:•For the exponential, tduuhtH0)()(ttdutH0)(15Duration # Ending # At Risk Inter. Haz. Cum. Hazard0 0 10 0 012 1 10 0.1000 0.100024 1 9 0.1111 0.211136 1 8 0.1250 0.336137 1 7 0.1429 0.479039 1 6 0.1667 0.645645 1 5 0.2000 0.845658 1 4 0.2500 1.095692 1 3 0.3333 1.4290106 1 2 0.5000 1.929120 1 1 1.0000 2.92916Cumulative Hazard Function: Postwar Expansionsy = 0.0223x - 0.2422R2 = 0.9288-0.500.511.522.533.50 20 40 60 80 100 120 140Duration in MonthsCumulative Hazard17Cumulative Hazard Function, Postwar Expansionsy = 0.0192xR2 = 0.901500.511.522.533.50 20 40 60 80 100 120 140Duration in MonthsCumulative HazrdWeibull Distribution•F(t) = 1 - exp[•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])/(t])/(exp[t19Weibull Distribution•Cumulative Hazard FunctionttHttHln])/1ln[()(ln)/1()(20Cumulative Hazard Function, Post-War Expansions0.111010 100 1000Duration in MonthsCumulative Hazard21-3-2-10122.0 2.5 3.0 3.5 4.0 4.5 5.0LNDURLNCUMHAZLog-Log Plot of Cumulativ e Hazard Function Vs. Duration, Post-War Expansions22Dependent Variable: LNCUMHAZMethod: Least SquaresSample: 2 11Included observations: 10Variable Coefficient Std. Error t-Statistic Prob. LNDUR 1.436662 0.103558 13.87303 0.0000C -5.920740 0.403303 -14.68061 0.0000R-squared 0.960092 Mean dependent var -0.409591 Adjusted R-squared 0.955103 S.D. dependent var 1.038386 S.E. of regression 0.220022 Akaike info criterion -0.013326 Sum squared resid 0.387276 Schwarz criterion 0.047191 Log likelihood 2.066628 F-statistic 192.4609 Durbin-Watson stat 1.210695 Prob(F-statistic) 0.00000123Is Beta More Than One?•H0: beta=1•HAA: beta>1, and hazard rate is increasing : beta>1, and hazard rate is increasing with time, i.e. expansions are more likely to with time, i.e. expansions are more likely to end the longer they lastend the longer they last•t = ( 1.437 - 1)/0.104 = 4.20t = ( 1.437 - 1)/0.104 = 4.2024Conclude•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 cycles25Source:Wayne Nelson, Applied Life data Analysis(1982) John WileyDiesel Generators, hours to fan failure, (+ indicates running time, i.e. still running whenlast observed)Hours # Ending # At Risk Interval Interval Hazard Rate Cumulative Hazard Rate450460+115011501560+16001660+1850+1850+1850+1850+1850+2030+2030+2030+2070207020802200+Lab Seven26Source:Wayne Nelson, Applied Life data Analysis(1982) John WileyDiesel Generators, hours to fan failure, (+ indicates running time, i.e. still running when last observed)Hours # Ending # At Risk Interval Interval Hazard Rate Cumulative Hazard Rate450 1 70 450460+ 681150 2 68 70011501560+ 651600 1 65 4501660+ 631850+ 621850+ 611850+ 601850+ 591850+ 582030+ 572030+ 562030+ 552070 2 55 47020702080 1 53 1027Source:Wayne Nelson, Applied Life data Analysis(1982) John WileyDiesel Generators, hours to fan failure, (+ indicates running time, i.e. still running whenlast observed)Hours # Ending # At Risk Interval Interval Hazard Rate Cumulative Hazard Rate450 1 70 450 0.0143 0.0143460+ 681150 2 68 700 0.0294 0.043711501560+ 651600 1 65 450 0.0154 0.05911660+ 631850+ 621850+