Applied Regression Analysis - open to bidding

4 Questions ranging from simple linear regression to multiple regression to use of ANOVA. . Given observations (xi, yi) for i = 1, 2, ...n, consider the regression model y = β + βx + e , e ∼ N(0, σ2) (note that they share the same β). i) Find the estimators for β and σ2, i.e. βˆ and σˆ2. ii) Find the mean and variance of βˆ. iii) Find the mean of σˆ2. iv) What can you say about βˆ when n → ∞? (Hint: What’s the variance of βˆ when n → ∞?) v) Does the fitted regression line pass through (x ̄,y ̄)? Is the sum of residuals equals zero? 3. Use the dataset [login to view URL], consider simple linear regression with Wt as dependent variable (response) and Ht as independent variable (predictor). i) Test the effect of Ht on Wt using a t-test. ii) Test the effect of Ht on Wt using an ANOVA table with F-test. iii) Find the 95% confidence interval for E(Wt|Ht = 160). Draw it. iv) Find the 90% simultaneous confidence band for E(Wt|Ht). Draw it. v) Find the 99% prediction interval for a new observations at Ht = 160. vi) Draw a residual plot. Is the regression a good fit? 4. Let  1   Y1   2   3 −1 0  X=1. Y=Y2 . E(Y)=−1. Var(Y)=−1 1 0. 1 Y3 5 001 (i) Find X(X′X)−1X′, (X′X)−1X′Y , X(X′X)−1X′Y and Y ̄ , in terms of Y1, Y2, Y3. (ii) Base on (i), is “Taking sample mean” the same as ”Regression using constant term only”? (iii) Let H = X(X′X)−1X′. Express Y ′(1 − H)Y in terms of Y1, Y2, Y3. Then find E(Y ′(1 − H)Y ). (iv) Using the fact that E(X′AX) = E[tr(X′AX)] = tr[AE(XX′)], find E(Y ′(I − H)Y ). (v) Repeat (iii) and (iv) using V ar(Y ) = I3. 5. Let Y = (21,25,19,34,36,36,24,10)′, X1 = (3,9,4,3,7,9,4,1)′, X2 = (3, 9, 4, 3, 7, 9, 4, 2)′ . Consider a multiple linear regression with an intercept term. (i) Using matrix methods find a)(X′X)−1, b) βˆ, c)ˆe, d)H = X(X′X)−1X′, e) SYY, f) SSReg, g)RSS, h)σˆ2,i)Yˆ, j) R2. (ii) Find the variance-covariance matrix of βˆ in terms of σ. Note that σ is an unknown parameter. How do you estimate this matrix from the data? (iii) What is the variance-covariance matrix of β (not βˆ)? (iv) Find the mean of σˆ2 in terms of σ. (v) Estimate V ar(Yˆ ). Are Yis independent? Are Yˆis independent? (vi) Write an ANOVA table to test the effect of the regression. (vii) At the point (X1, X2) = (12, −2), estimate E(Y |X1, X2) and a) 90% confidence interval for the fitted value of (X1,X2). b) 95% prediction interval for a new observation of (X1,X2). Will you trust the estimates? Why? Clear set of questions will be provided upon acceptance of project.