Dayu Wu Applied Statistics Lecture Notes 3.n23p=7 4n2s=南 4 Violation of Regression Assumptions Heteroscedasticity 1.r,=1-mt-i 2t=是 Weighted Least Square 1.Loss Function:Q=∑，e2=∑，(--3)月 2.B=(XTWX)-XTWY Box-Cox 1.A≠0，y= 2.1=0,Y)=1ogy Autocorrelation 1.p=∑n2t-1/V∑2GV∑2- 2.DW-∑-2(e-e-1)2/∑-2≈2(1-)∈0,4刂 Outlier,High Leverage Point,and Influential Point 1.Outlier:big le,extreme y 2.High Leverage Point:extreme 3.Influential Point:result in different regression equations without it 4.extreme X:Cook's distance D. 2 5.extreme Y:e()= 5 Variable Selection 1.Full Model and Selected Model 2.Criteria:R,AIC,Cp 3.Forward,Backward,and Stepwise 6of6Dayu Wu Applied Statistics Lecture Notes 3. r12:3...p = √−∆12 ∆11∆22 4. r12:3 = √ r12−r13r23 (1−r 2 13)(1−r 2 12) 4 Violation of Regression Assumptions Heteroscedasticity 1. rs = 1 − 6 n(n2−1)Pd 2 i 2. t = √ √n−2rs 1−r 2 s Weighted Least Square 1. Loss Function: Q = Pwie 2 i = Pwi(yi − β0 − β1xi) 2 2. βbw = (XTW X) −1XTW Y Box-Cox 1. λ ̸= 0, Y (y) = Y λ−1 λ 2. λ = 0, Y (y) = log Y Autocorrelation 1. ρ = Pn t=2 ϵtϵt−1/ pPn t=2 ϵ 2 t pPn t=2 ϵ 2 t−1 2. DW = Pn t=2(et − et−1) 2/ Pn t=2 e 2 i ≈ 2(1 − ρb) ∈ [0, 4] Outlier, High Leverage Point, and Influential Point 1. Outlier: big |ei |, extreme yi 2. High Leverage Point: extreme xi 3. Influential Point: result in different regression equations without it 4. extreme X: Cook’s distance Di = e 2 i (p+1)σb2 hii (1−hii) 2 5. extreme Y: e(i) = ei 1−hii 5 Variable Selection 1. Full Model and Selected Model 2. Criteria: R2 a , AIC, Cp 3. Forward, Backward, and Stepwise 6 of 6