課程名稱︰迴歸分析 課程性質︰應數所數統組必修 課程教師︰鄭明燕 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰2016/11/7 考試時限（分鐘）：15:30~18:20 試題 : 1. Consider the simple linear regression model y =β+βx +ε, i 0 1 i1 i 2 with E(ε)=0,Var(ε)=σ, i=1,...,n,and ε's uncorrelated. i i i (a)(15 pts) Suppose β is known. Find the least squares estimator of β and 0 1 find the bias and variance of this estimator. ^ ^ ^ (b)(15 pts) Suppose we have fit the simple linear regression model y=β+βx 0 1 1 with β and β both unknown, but the true regression funciton is 0 1 E(y)=β+βx +βx . 0 1 1 2 2 ^ ^ Find the bias of β. Compare the variance of β with the variance of the 1 1 least squares estimator of β under the true model. 1 2. Consider two independent sets of observations y =β+βx +ε , i=1,...,n, 1i 0 1 1i 1i 2 where the ε 's are iid Normal(0,σ ), and 1i y =β+βx +ε , i=1,...,n, 2i 0 2 2i 2i 2 where the ε 's are iid Normal(0,σ ).Here, y 's and y 's are observations 2i 1i 2i of the same response variable and x 's and x 's are observations of the 1i 2i same covariate. (a)(5 pts) Find the least squares estimator of β based on the first set of 0 observations (x ,y ),...,(x ,y ). 11 11 1n 1n T (b)(5 pts) Find the least squares estimator of β=(β,β,β) based on the 0 1 2 2n observations, and find its mean vector and covariance matrix. (c)(10 pts) Compare biases and variances of the two estimators of β in (a) 0 and (b). 2 (d)(5 pts) Find an estimator for σ . (e)(10 pts) Derive a level-α test for the hypotheses H :β=β versus 0 1 2 H :β≠β. 1 1 2 3. Consider the multiple linear regression model y =β+βx +...+βx +ε, i=1,...,n, i 0 1 i1 k ik i where the ε's are uncorrelated each with zero mean and constant variance i 2 T T T T σ . Let β=(β,β,...,β) , y=(y ,...,y ) , ε=(ε,...,ε) and X=(x ,...x ) 0 1 k 1 n 1 n 1 n T with x =(1,x ,...,x ) , i=1,...,n. Then the above model can be written as i i1 ik y=Xβ+ε. Assume that the design matrix X is of rank k+1. (a)(5 pts) Write down the least squares problem for estimation of β and ^ find the solution, denoted by β. 2 ^ T ^ 2 (b)(10 pts) Let s =(y-Xβ) (y-Xβ)/(n-k-1). Show that s is an unbiased 2 estimator of σ . 2 ^ (c)(10) Let r ^ be the correlation coefficient between the y 's and y 's. yy i i 2 2 Show that r ^ = R . yy (d)(5 pts) Let H=(h ) be the hat matrix. Show that 0≦h ≦1, i=1,...,n. ij ii ^ (e)(10) Show that, for any constant (k+1)-vector l, l'β is a best linear unbiased estimator of l'β under the Gauss-Markov conditions. 4. Consider the reparameterization Y=Wα+ε of the model Y=Xβ+ε, where W=XC and C is nonsingular. (a)(5) Show that the hat matrices H and H for the two medels are the same. W X ^ ^ (b)(5) Suppose α and β are least squares estimators of α and β. Express ^ ^ α in terms of β. --

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