## University of California, Los Angeles Department of Statistics Statistics 100C Instructor: Nicolas Christou Exam 1 26 April 2013 Name: Problem 1 (25 points) Consider the simple regression model yi = 0 + 1xi + i with E(i) = 0, var(i) = 2, and cov(i; j) = 0. Answer the following questions: a. Find cov(i; ^1). b. If 0 = 4 what is the least squares estimate of 1? c. What is the variance of the estimate of part (b)? d. Is the estimate of part (b) unbiased? Problem 2 (25 points) Answer the following questions: a. Consider the simple regression model yi = 0 + 1xi + i with E(i) = 0, var(i) = 2, and cov(i; j) = 0. Show that ^0 is BLUE (it has the smallest variance among all the linear unbiased estimators of 0). b. Consider the model of part (a). Find cov(ei; ^ Yi). c. Consider the simple regression model through the origin yi = 1xi + i with E(i) = 0, var(i) = 2, and cov(i; j) = 0. Show that Pn i=1 xiei where ei = Yi ? ^ Yi. d. Consider the simple regression model yi = 0 + 1xi + i with E(i) = 0, var(i) = 2xi, and cov(i; j) = 0. Also, x is nonrandom. Is the assumption of constant variance satised in the following model? Please explain. Yi p xi = 0 p xi + 1 xi p xi + i p xi : Problem 3 (25 points): Answer the following questions: a. Consider the model yi = 0 +1xi +i. Assume that E(i) = 0, var(i) = 2, and cov(i; j) = 0. Suppose we rescale the x values as x = x ? , and we want to t the model yi = 0 + 1xi + i. Find the least squares estimates of 0 and 1 . b. Refer to the model yi = 0 + 1xi +i of part (a). Find the SSE of this model and compare it to the SSE of the model yi = 0 + 1xi + i. What is your conclusion? c. Consider the simple regression model yi = 0 + 1xi + i, with E(i) = 0, var(i) = 2, and cov(i; j) = 0. Show that ESY Y = (n ? 1)2 + 2 1SXX, where SY Y = Pn i=1(yi ? y)2 and SXX = Pn i=1(xi ? x)2. d. Refer to the model of part (c). Find cov(i; ei). Problem 4 (25 points) Suppose that a simple linear regression of miles per gallon (Y ) on car weight (x) has been performed on 32 observations. The least squares estimates are ^0 = 68:17 and ^1 = ?1:112, with se = 4:281. Other useful information: x = 30:91 and P32 i=1(xi ? x)2 = 2054:8. Answer the following questions: a. Construct a 95% condence interval for 1. b. Construct a 95% condence interval for 2. c. What is the value of R2? d. Construct a condence interval for 30 ? 21 ? 50.

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