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UCSB ECON 240a - Probability Models

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Lecture ElevenOutlineBayesian ProbabilityJoint and Marginal ProbabilitiesFilling In Our FactsUsing Conditional ProbabilitySlide 7By Sum and By DifferenceFalse Positive ParadoxBayesian Probability By FormulaDuration ModelsDuration of Post-War Economic Expansions in MonthsSlide 13Slide 14Slide 15Estimated Survivor Function for Ten Post-War ExpansionsKaplan-Meyer Estimate of Survivor FunctionSlide 18Slide 19Exponential DistributionSlide 21Exponential Distribution (Cont.)Exponential Distribution(Cont.)Model Building20.2 Polynomial ModelsPolynomial Models with One Predictor VariableSlide 27Slide 28Polynomial Models with Two Predictor Variables20.3 Nominal Independent VariablesNominal Independent Variables; Example: Auction Car Price (II)Slide 32How Many Indicator Variables?Nominal Independent Variables; Example: Auction Car PriceExample: Auction Car Price The Regression EquationSlide 36Slide 37Nominal Independent Variables; Example: MBA Program Admission (MBA II)Nominal Independent Variables; Example: MBA Program Admission (II)Slide 4020.4 Applications in Human Resources Management: Pay-EquityHuman Resources Management: Pay-EquitySlide 43Slide 441Lecture ElevenProbability Models2Outline•Bayesian Probability•Duration Models3Bayesian Probability•Facts•Incidence of the disease in the population is one in a thousand•The probability of testing positive if you have the disease is 99 out of 100•The probability of testing positive if you do not have the disease is 2 in a 1004Joint and Marginal ProbabilitiesSick: S Healthy: HTest +Pr(+ S) Pr(+H)Pr(+)Test-Pr(-S) Pr(-H)Pr(-)Pr(S) Pr(H)5Filling In Our FactsSick: S Healthy: HTest +Test -Pr(s) =0.001Pr(H) =0.999Using Conditional Probability•Pr(+ H)= Pr(+/H)*Pr(H)= 0.02*0.999=.01998•Pr(+ S) = Pr(+/S)*Pr(S) = 0.99*0.001=.000997Filling In Our FactsSick: S Healthy: HTest +Pr(+S)=0.00099Pr(+H)=0.01998Test -Pr(s) =0.001Pr(H) =0.9998By Sum and By DifferenceFalse Positive Paradox•Probability of Being Sick If You Test +•Pr(S/+) ?•From Conditional Probability:•Pr(S/+) = Pr(S +)/Pr(+) = 0.00099/0.02097•Pr(S/+) = 0.0472Bayesian Probability By Formula•Pr(S/+) = Pr(S +)/Pr(+) = PR(+/S)*Pr(S)/Pr(+)•Where PR(+) = PR(+/S)*PR(S) + PR(+/H)*PR(H)•And Using our facts;Pr(S/+) = 0.99*(0.001)/[0.99*.001 + 0.02*.999]•Pr(S/+) = 0.00099/[0.00099+0.01998]•Pr(S/+) = 0.00099/0.02097 = 0.047211Duration Models•Exploratory (Graphical) Estimates– Kaplan-Meier•Functional Form Estimates–Exponential Distribution12Duration of Post-War Economic Expansions in Months13141516Estimated Survivor Function for Ten Post-War Expansions17Kaplan-Meyer Estimate of Survivor Function•Survivor Function = (# at risk - # ending)/# at risk18Duration # Ending # At Risk Survivor0 0 10 112 1 10 0.924 1 9 0.836 1 8 0.737 1 7 0.639 1 6 0.545 1 5 0.458 1 4 0.392 1 3 0.2106 1 2 0.1120 1 1 019Figure 2: Estimated Survivor Function for Post-War Expansions00.20.40.60.811.20 20 40 60 80 100 120 140Duration in MonthsSurvivor FunctionExponential Distribution•Density: f(t) = exp[ - t], 0 t •Cumulative Distribution Function F(t)•F(t) = •F(t) = - exp[- u] •F(t) = -1 {exp[- t] - exp[0]} •F(t) = 1 - exp[- t]•Survivor Function, S(t) = 1- F(t) = exp[- t]•Taking logarithms, lnS(t) = - t f (u)du 0t exp[ u]du0t0t21Postwar Expansionsy = -0.0217x + 0.1799R2 = 0.9533-2.5-2-1.5-1-0.500.50 20 40 60 80 100 120Duration (Months)Ln Survivor FunctionSo  Exponential Distribution (Cont.)•Mean = 1/ =•Memoryless feature: •Duration conditional on surviving until t = :•DURC( ) = = + 1/ •Expected remaining duration = duration conditional on surviving until time , i.e DURC, minus•Or 1/ , which is equal to the overall mean, so the distribution is memorylesst * f (t)dtt * f (t)dt / S()Exponential Distribution(Cont.)•Hazard rate or function, h(t) is the probability of failure conditional on survival until that time, and is the ratio of the density function to the survivor function. It is a constant for the exponential.•h(t) = f(t)/S(t) = exp[- t] /exp[- t] = 24Model Building•Reference: Ch 20 (Ch. 18, 8th Ed.)2520.2 Polynomial Models•There are models where the independent variables (xi) may appear as functions of a smaller number of predictor variables.•Polynomial models are one such example.26y = 0 + 1x1+ 2x2 +…+ pxp +  y = 0 + 1x + 2x2 + …+pxp +  Polynomial Models with One Predictor Variabley01x•First order model (p = 1)y = 0 + 1x + 2x2 + 2 < 02 > 0•Second order model (p=2) Polynomial Models with One Predictor Variabley = 0 + 1x + 2x2 + 3x3 + 3 < 0 3 > 0•Third order model (p = 3) Polynomial Models with One Predictor Variable•First order modely = 0 + 1x1 +  Polynomial Models with Two Predictor Variablesx1x2y2x2 + 1 < 01 > 0x1x2y2 > 02 < 03020.3 Nominal Independent Variables•In many real-life situations one or more independent variables are nominal.•Including nominal variables in a regression analysis model is done via indicator variables.•An indicator variable (I) can assume one out of two values, “zero” or “one”.1 if a first condition out of two is met0 if a second condition out of two is metI=1 if data were collected before 19800 if data were collected after 19801 if the temperature was below 50o0 if the temperature was 50o or more1 if a degree earned is in Finance0 if a degree earned is not in Finance31Nominal Independent Variables; Example: Auction Car Price (II)•Example 18.2 - revised (Xm18-02a)–Recall: A car dealer wants to predict the auction price of a car.–The dealer believes now that odometer reading and the car color are variables that affect a car’s price.–Three color categories are considered:•White•Silver•Other colorsNote: Color is a nominal variable.32•Example 18.2 - revised (Xm18-02b)I1 =1 if the color is white0 if the color is not whiteI2 =1 if the color is silver0 if the color is not silverThe category “Other colors” is defined by:I1 = 0; I2 = 0Nominal Independent Variables; Example: Auction Car Price (II)33•Note: To represent the situation of three possible colors we need only two indicator


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UCSB ECON 240a - Probability Models

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