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GRA 6020 Multivariate Statistics; The Linear Probability model and The Logit Model (Probit) Ulf H. Olsson Professor of Statistics Binary Response Models y is a binary response var iable x' ( x1 , x2 ,......, xk ) is the full set of exp lanatory var iables Pr ob( y 1 | x) G( 0 1 x1 2 x2 ..... k xk ) G( 0 xβ) •The Goal is to estimate the parameters Ulf H. Olsson The Logit Model z e G( z) z 1 e •The Logistic Function •e ~ 2.71821828 ze ln( z ) Ulf H. Olsson The Logistic Curve G (The Cumulative Normal Distribution) Ulf H. Olsson The Logit Model G ( 0 1 x1 .... k xk ) 0 1 x1 .... k xk e 0 1 x1 .... k xk 1 e 1 ( ( 0 1 x1 .... k xk )) 1 e Ulf H. Olsson Logit Model for Pi y 1 or y 0; Pi Pr ob( yi 1) 1 ( ( 0 1 x1 .... k xk )) 1 e Pi 0 1 x1 .... k xk ln 1 Pi Ulf H. Olsson Simple Example Pi ln 0 1 x1 1 Pi Variables in the Equation Step a 1 addsc Constant B ,071 -5,898 S.E. ,017 1,018 Wald 17,883 33,588 df 1 1 Sig. ,000 ,000 Exp(B) 1,074 ,003 a. Variable(s) entered on step 1: addsc. Pi ln 5.898 0.071x1 1 Pi Ulf H. Olsson Simple Example Pi ln 5.898 0.071x1 1 Pi Pi ( 5.898 0.071x1 ) e 1 Pi x1 60 Pi 0.165 Ulf H. Olsson The Logit Model • Non-linear => Non-linear Estimation =>ML • Model can be tested, but R-sq. does not work. Some pseudo R.sq. have been proposed. • Estimate a model to predict the probability Ulf H. Olsson Binary Response Models • The magnitude of each effect j is not especially useful since y* rarely has a well-defined unit of measurement. • But, it is possible to find the partial effects on the probabilities by partial derivatives. • We are interested in significance and directions (positive or negative) • To find the partial effects of roughly continuous variables on the response probability: p( x) dG( z ) g ( 0 xβ) j ; where g ( z ) x j dz Ulf H. Olsson Introduction to the ML-estimator Let be the data matrix ( x1 , x2 ,......, xk ); where xi are vectors The Likelihood function is as a function of the unknown parameter vector : k f ( x1 , x2 ,......, xk , ) f ( xi , ) L( | X ) i 1 Ulf H. Olsson Introduction to the ML-estimator • The value of the parameters that maximizes this function are the maximum likelihood estimates • Since the logarithm is a monotonic function, the values that maximizes L are the same as those that minimizes ln L The necessary conditions for max imiz in g L( ) is ln L( ) 0 We denote the ML estimator ML L( ) L is the Likelihood function evaluated at Ulf H. Olsson Goodness of Fit 2 ln L 2 log likelihood •The lower the better (0 – perfect fit) 2(ln L2 ln L1 ) approximates a chi square df q(no. of exp lanatory var iables) •Some Pseudo R-sq. •The Wald test for the individual parameters Ulf H. Olsson The Wald Test If x N , , then ( x )' ( x ) is (d ) 1 2 H 0 : c( ) q, then under H 0 W (c( ) q) 'U (c( ) q) 1 is 2 (d ) Ulf H. Olsson Example of the Wald test • Consider a simple regression model y x H0 : 0 , we know | 0 | s( ) z or (t ) ; W ( 0 ) 'Var ( 0 ) ( 0 ) z 1 2 is (1) 2 Ulf H. Olsson