ECON 705 Midterm: 2016-705-Fall Econometrics Prelim Exam
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Friday august 12, 2016 time limit: 150 minutes. Instructions: (i) the exam consists of two parts. Let y = exp(x). (i) (6 points) suppose that x n ( , 2). Compute the expected value e[y ] and the variance v[y ]. (ii) (4 points) suppose that x1 and x2 are independent and have n (0, 1) distri- butions. De ne y1 = x1 + x2 and y2 = x1 x2. Use a change-of-variables argument to obtain the joint probability density for (y1, y2). Then compute the pdf for the marginal distribution of y1. Question 2: inference with two observations (23 points) X1, x2 iidn ( , 1) (i) (3 points) derive the likelihood function and the maximum likelihood estima- tor for this experiment. What is the sampling distribution of the maximum likelihood estimator? (ii) (3 points) derive a likelihood ratio (lr) test for the null hypothesis = . Question 3: inference for variance parameters (17 points)
