1
Due STA244
1/08/2003
Homework 1/15/2003.
Please provide concise, neatly written or typed solutions. All work should be your own and not copied from other texts or sources. Do feel free to discuss questions with me, the TA, others in class, or post a question for clarication on the Course Info Discussion Board. 1. Assume that we have a sample of size n where Y i = 0 + 1 Xi + and the errors
i i
are iid N (0, 2 ).
(a) Find the maximum likelihood estimator of 2 , 2 . (b) Under the assumption of normal errors as above, nd the E( 2 ). (c) Is 2 an unbiased estimate of 2 ? If not, nd an unbiased estimate of 2 . 2. Invariance Suppose that we recode the data above so that Yi = a + bYi and Xi = c + dXi for known a, b, c, and d. Consider the model where Yi s are independent normal random variables with mean 0 + Xi 1 and variance 2 . (a) What is the relationship between the mles 0 and 0 ? (b) What is the relationship between the mles 1 and 1 ? (c) What is the relationships between 2 and 2 ? (d) Are hypothesis tests 1) that the intercept equals zero versus non-zero and 2) that the slope is zero versus non-zero the same or dierent? 3. Consistency Recall that the variance of 1 is 2 /
n i=1 (Xi
X)2 .
(a) Under what conditions on the values of X will this variance approach zero as the sample size n goes to innity? (b) Construct a sequence of values (X1 , X2 , . . .) such that as the sample size goes to innity, the variance of 1 does not approach zero. What does this imply about convergence of 1 ? 4. In the simple linear regression as model, in Exercise (1), (a) show that the correlation between 0 and 1 is (0 , 1 ) = X Var(1 ) Var(0 )
How can the values of X be chosen 1) so that the correlation is arbitrarily close to 0 and 2) be chosen so that the correlation is close to +1 or -1?
(b) In your own words, describe what (0 , 1 ) means. (c) Let Zi = Xi X be the predictor variable centered to have average zero. An alternative parameterization of the regression model is to have Yi = + 1 Zi + i , where = 0 + 1 X. Write Var() as a function of n and 2 , and nd the value 1 ) for this model. of (, 5. Consider the simple linear regression model in exercise 1) with residuals ei = Yi Yi i = 0 + 1 Xi . and tted values Y (a) Prove that ei = 0 and Yi = Yi . (b) Show that the sample covariance between e and Y equals zero, and hence the residuals and tted values are uncorrelated. Hint: it is useful to write Yi = + 1 (Xi X) . You ...