Ch. 11 Binary Dependent Variable

Ch. 11 Binary Dependent Variable - BINARY DEPENDENDENT...

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1 | Binary BINARY DEPENDENDENT VARIABLE (Ch. 11) The recommended exercise questions from the textbook: Chapter 11: All except (11.10) and (11.11). [1] Motivation • What if my dependent variable is binary? • Coca Cola or Pepsi Cola • Whether to participate in labor force or not. • Union member or not. ( Y = 1 for union; Y = 0 for non-union) • Mortgage application accepted or not. • To be, or not to be.
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2 | Binary • Wish to estimate: • the probability of Y = 1 and determinants of the probability. • Our empirical example: • Is there any discrimination in mortgage loan to single people acceptance/denial? • Other things being equal, are some groups of applicants (say, group A) discriminated in mortgage loan acceptance?
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3 | Binary [2] Data • hmda.wf1. • A subset of the data compiled by FRB of Boston under HMDA (Home Mortgage Disclosure Act) • Data from the Boston area in 1990; pi_rat = P/I ratio; hse_inc = housing expense-to-income ratio; loan_val = loan-to-value ratio; ccred = consumer credit score; mcred = mortgage credit score; pubcred = public bad credit record; denpmi = denied mortgage insurance; selfemp = self-employed ; single = single ; hischl = high school diploma; probunmp = unemployment rate; condo = condominium; deny = deny; ltv_med = medium loan-to-value ratio (loan_val bet/w 0.8 and 0.95); ltr_high = high loan-to-value ratio (loan_val >= 0.95).
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4 | Binary
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5 | Binary [3] Bernoulli’s Distribution: (1) Bernoulli’s Distribution. Y is a random variable with pdf; p = Pr( Y= 1) and (1- p ) = Pr( Y =0). f ( y ) = p y (1- p ) 1- y f (1) = p and f (0) = 1- p . () 1 0 ( 1 ) y E Y yf y pp p       :b P r ( 1 ) ! E YY  • When the dependent variable is binary, 12 ( | , ,..., ) ( 1| , ,..., ) kk E YX X X P Y X X X . 22 2 2 2 var( ) ( ) [ ( )] (1 0 )) ) y Y yfy EY p p p p p     .
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6 | Binary (2) Estimation of p by MLE. • Let { Y 1 , . .. , Y n } be a random sample from a Bernoulli distribution: For example, Y = 1 for republicans and Y = 0 for democrats. • Likelihood function: 11 1 1 1 1 ( ) ( ) ( ) . .. ( ) (1 ) ... ) . nn n ni n i YY Lp f Y f Y f Y pp p p    • Log-Likelihood function:  () l n ln( ) )ln(1 ) ... ln( ) )ln(1 ) (. . . ) l n ( ) ( . . ) ) l n ( 1 ) ln( ) )ln(1 ). pL p Yp Y p Y p Y p p n p nY p n Y p  
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7 | Binary • Maximum Likelihood Estimator of p (say, ˆ ML p ) T h e p value that maximizes the log-likelihood function. S e t () (1 ) 0 1 n p nY n Y pp p   . ) ) 0 nY p n Y p  . [] 0 nY nY n nY p  . 0 nY np  pY ˆ ML !
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8 | Binary (3) Cases when Pr( Y = 1) changes depending on some X variables such as X 1 , . .. , X k .
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Ch. 11 Binary Dependent Variable - BINARY DEPENDENDENT...

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