Dayu Wu Applied Statistics Lecture Notes Confidential Interval 1.高~N(,) 2.a~N(,(信+)) 3.可N(风+A(日+)) 4.For example: mwP(品sga--1-a Then (1-a)IC is ( 3 Multiple Regression Basic Concepts 1x11…xp e1 2 1x212p i…m Bn) 2.Y=X8+e 3.G~N(0,o2),id 4.e N(0,aIn) Estimation:OLS 1.Note that:Xe=0. Thus T(Y-3)=0 We have XTY=XTxB. When det(xrx)≠0，B=(xrX)-1xTY. 2.9=x3=XXTX)-1XTY=HY 3.Hat matrix:H2=H,tr(H)=ha =p+1 4.e=Y-=(In -H) 5.Cou(e,e)=a2(In-H),Var(e;)=a2(1-ha) 6=∑ 4 of6Dayu Wu Applied Statistics Lecture Notes Confidential Interval 1. βb1 ∼ N(β1, σ 2 Lxx ) 2. βb0 ∼ N(β0,  1 n + x¯ 2 Lxx  σ 2 ) 3. yb ∼ N(β0 + β1x,  1 n + (x−x¯) 2 Lxx  σ 2 ) 4. For example: t = βb √1−β1 σb2/Lxx ∼ tn−2 Thus P     βb √1−β1 σb2/Lxx     ≤ t α 2 (n − 2) = 1 − α Then (1 − α) IC is  βb1 − √ σb Lxx t α 2 , βb1 + √ σb Lxx t α 2  3 Multiple Regression Basic Concepts 1.   y1 y2 . . . yn   =   1 x11 . . . x1p 1 x21 . . . x2p . . . . . . . . . . . . 1 xn1 . . . xnp     β1 β2 . . . βn   +   ϵ1 ϵ2 . . . ϵn   2. Y = Xβ + ϵ 3. ϵi ∼ N(0, σ2 ), iid 4. ϵ ∼ N(0, σIn) Estimation: OLS 1. Note that: Xϵ = 0. Thus X T (Y − Xβb) = 0. We have X TY = X TXβb. When det(X TX) ̸= 0, βb = (X TX) −1X TY. 2. Yb = Xβb = XXTX) −1X TY = HY 3. Hat matrix: H2 = H, tr(H) = Phii = p + 1 4. e = Y − Yb = (In − H)Y 5. Cov(e, e) = σ 2 (In − H), V ar(ei) = σ 2 (1 − hii) 6. σb 2 = 1 n−p−1 Pe 2 i 4 of 6