Econ 120B Ramu RamanathanWinter 1998 First Midterm (20%)Your name ______________________________ Your Student Id. __________________DO NOT TURN THE PAGE UNTIL EVERYONE HAS RECEIVEDTHE EXAM AND YOU ARE GIVEN THE SIGNAL TO START.ALSO, YOU MUST STOP WRITING WHEN YOU ARE ASKEDTO DO SO (YOU WILL BE GIVEN A 2 MINUTE WARNING).TEN POINTS WILL BE DEDUCTED FOR EACH MINUTE OFEXTRA TIME IT TAKES YOU TO STOP WRITING.If you use a pencil, you forfeit the right to complain about gradingUNLESS YOU PICK UP THE EXAM FROM THE TA FROMHIS/HER OFFICE AND LOOK AT THE GRADING BEFORELEAVING THE OFFICE.Make sure that all pages (1 through 4) are there. The maximumnumber of points for the exam is 100. Read the questions carefullyand make sure that you do not misunderstand them. If you get stucksomewhere, don’t waste time but move on.I CONSIDER CHEATING AS A VERY SERIOUS MATTER ANDWILL GIVE AN F IN THE COURSE TO ANY ONE CHEATINGAND ALSO REFER HIM/HER TO THE DEAN FORDISCIPLINARY ACTION.- 2 -I.a. (12 points) The continuous random variable X has the uniform distribution with the constantdensity function f(x) = 1, 0 ≤≤ x ≤≤ 1. Using the fact that∫∫ xn dx = xn+1/(n+1) and the definitionof expected values, derive E(x), E(x2), E(x3), and E(x4). Note that the integrals go from 0 to 1.b. (14 points) Use the above derivations and compute Var(X), Var(Y), and Cov(X,Y) where Y =X2.c. (16 points) Next compute the coefficient of correlation between X and Y (use the fact that√√MMMM MM15= 3.87) and show that it is not equal to 1 even though there is an exact relation between X andY.- 3 -II. (16 points)Let X and Y be two random variables with E(X) = E(Y) = 0, Var(X) = σσx2, Var(Y) = σσy2, andCov(X,Y) = σσxy. Now make the transformations U = X + Y and V = X - Y. Derive Cov(U,V) and thecondition under which U and V will be uncorrelated.III.Suppose the true model is Yt= ββ Xt+ut, that is, αα = 0. I construct an alternative estimator of ββ as ββ˜=Y____/X____, where Y____= (1/n)ΣΣ Ytand similarly for X____.a. (20 points) Compute the expected value of ββ˜and check whether it is unbiased or not. Be sureto state assumptions needed for your proof (you will lose points if you state unnecessaryassumptions).- 4 -b. (10 points) Without any derivations explain why ββ˜is inferior to the OLS estimator of ββ,clearly defining what you mean by "inferior."IV.In the simple linear regression model, utis the random error term with E(ut) = 0. The subscript trefers to a typical observation and n is the size of the sample. For each of the following equationsstate whether it is correct or not. Explain why or why not (note: the explanations are fairly sim-ple).a. (6 points)t=1ΣΣnut=0.b. (6 points)t=1ΣΣnuˆt= 0, where uˆtis the sample residual Yt −− ααˆ −−