L1-Reg_part1.pdf - Quan%taNULLe Methods in Finance Lecture...

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Quan%ta%ve Methods in Finance Lecture 1 : Linear Regression Analysis: reminder Elena Dumitrescu ( [email protected] ) 1
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The Linear Regression Model: An Overview What is regression analysis? Deriva)on of OLS es)mators (simple and mul)ple regressions) Hypothesis tes)ng P-­૒values Probability distribu)on of OLS es)mators Test of significance Mul)ple hypothesis tes)ng Goodness of fit sta)s)cs OLS assump)ons Proper)es of OLS es)mators 2
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An example We collect the following data on Apple and on a market index Year, t Excess return = r XXX, t rf t Excess return on market index = rm t - rf t 1 17.8 13.7 2 39.0 23.2 3 12.8 6.9 4 24.2 16.8 5 17.2 12.3 3
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An example We collect the following data on Apple and on a market index Year, t Excess return = r XXX, t rf t Excess return on market index = rm t - rf t 1 17.8 13.7 2 39.0 23.2 3 12.8 6.9 4 24.2 16.8 5 17.2 12.3 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 Excess return on market portfolio Excess return on fund XXX 4
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Intuition – see on-line course General Regression: We minimize the sum of the squared distances between the dots and the fitted line (we minimize the residual sum of squares) Algebraically, we minimize , the residual sum of squares: min ( ) Estimation method known as OLS, Ordinary Least Squares 2 5 2 4 2 3 2 2 2 1 ˆ ˆ ˆ ˆ ˆ u u u u u + + + + = 5 1 2 ˆ t t u x y 10 8 6 4 2 0 1 3 0 5 2 7 4 6 5 y t = ! + " x t + u t
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Denote To minimise RSS with respect to (w.r.t.) and , set its derivatives to zero (1) (2) From (1), (3) Where we used that and Deriving the OLS Estimator α β ! RSS ! ˆ " = ! 2 ( y t ! ˆ " ! ˆ # x t ) = 0 t " ! RSS ! ˆ " = ! 2 x t ( y t ! ˆ # ! ˆ " x t ) = 0 t " 0 ˆ ˆ 0 ) ˆ ˆ ( = = t t t t t x T y x y β α β α = y T y t = x T x t RSS = ˆ u t = t ! ( y t " ˆ y t ) 2 = ( y t " ˆ ! " ˆ " x t ) 2 t ! t ! ! T y " T ˆ ! " T ˆ " x = 0 ! ˆ ! = y " ˆ " x 6
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