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Exam: Statistics for Econometrics January 7, 2014 Closed-Book Exam

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Exam: Statistics for Econometrics January 7, 2014 Closed-Book Exam Exam: Statistics for Econometrics January 7, 2014 Closed-book exam. Simple calculator allowed. The exam consists of 4 exer- cises. Question 1: 25pts; Question 2: 25pts; Question 3: 25pts; Question 4: 25pts. Time: 3 hours. Answers should be explained and clearly formulated. PLEASE GIVE ANSWERS IN ENGLISH. WRITE EVERY QUESTION ON A SEPARATE SHEET. WRITE CLEARLY! 1 Q1 Let X1;:::;Xn be a random sample from the EXP(1/θ)-distribution with density f(x; θ) = θe−θx (if x > 0), where θ > 0 is an unknown parameter. Without proof you may use the following results: 2 • EXi = 1/θ; Var(Xi) = 1/θ . 2 α α ν+2α ν • If Y ∼ χν, then for α > −ν: E(Y ) = 2 Γ 2 =Γ 2 . 6pts (i) • Determine the Maximum Likelihood Estimator (MLE) for θ. • Show that the MLE is equal to the Method of Moments Es- timator (MME) for θ. 5pts (ii) Determine a complete sufficient statistic for θ (provide also the arguments leading to your conclusion). 7pts (iii) Determine the UMVUE (provide sufficient arguments). 7pts (iv) • Determine the Fisher Information number I(θ). • Obtain the limiting distribution of the MLE and verify that the MLE is asymptotically efficient. Q2 Let X1;:::;Xn be a random sample from a normal distribution with density f(x; µ) = p1 exp{− 1 (x − µ)2g (thus, with mean µ and unit 2π 2 variance). 12pts (i) • Determine an appropriate pivotal quantity Q based on all observations X1;:::;Xn. • Use Q to determine a one-sided 95% confidence interval for µ when you want to test the null hypothesis H0 : µ ≤ 0 versus the alternative Ha : µ > 0. 13pts (ii) A Bayesian statistician has a sample of size n = 1 from a normal distribution: Xjµ ∼ N(µ, 1). Moreover, he considers a standard normal distribution as a prior for µ: µ ∼ N(0; 1) . • Show that the posterior distribution is normal as well with 1 mean X=2 and variance 2 . • Obtain the Bayes estimator for µ (mean squared error loss). • Calculate, using mean squared error loss, the risk function of the Bayes estimator for µ. • Calculate, using mean squared error loss, the Bayes risk of the Bayes estimator for µ. 2 Q3 Let X1;:::;Xn be a random sample from a distribution F with in- finitely many derivatives and let Fn denote the Empirical Distribution Function (EDF). Let 0 < h = hn ! 0 as n ! 1. To estimate the derivative f 0(0) of the density f = F 0 in zero, we use the following estimator: F (h) − F (0) F (0) − F (−h) f^0 (0) = n n − n n : n h2 h2 Hint: if you are not able to obtain the complete solution of a certain item, you are still allowed to use the required result in the subsequent items. For smooth functions g, the following Taylor expansion is known from your analysis courses: 0 1 00 2 1 000 3 1 0000 4 • g(y) = g(0) + g (0)y + 2! g (0)y + 3! g (0)y + 4! g (0)y + :::: 2 ^0 4pts (i) • Show that nh fn(0) is distributed as the difference of two binomial distributions. • Give the parameters of both binomial distributions. ^0 8pts (ii) • Determine E(fn(0)). ^0 0 • Prove: fn(0) is asymptotically unbiased for f (0). • Show, using a Taylor expansion around h = 0, that the leading ^0 0 f 000(0) 2 term of the bias expression Efn(0)−f (0) is given by 12 h . Fn(h)−Fn(0) Fn(0)−Fn(−h) 8pts (iii) • Determine Var( h2 ) and Var( h2 ). ^0 • TRUE or FALSE: Var(fn(0)) is the sum of these two vari- ances. Provide an intuitive argument (no calculations needed). ^0 • Without proof it is given that the leading term of Var(fn(0)) 2 is given by nh3 f(0). Determine the optimal bandwidth h from a local MSE perspective. −α ^0 5pts (iv) Let, for some α > 0, h = hn = n . For which values of α is fn(0) a MSE-consistent estimator of f 0(0)? 3 Q4 The information about the height (in cm) of 300 adults drawn randomly from a large population of adults is collected. Ten years ago, the mean height of adults in the population was 177 cm. The researcher wants to use a two-sided t-test in order to find out whether the mean height nowadays is different. She obtains the following SPSS output (and the table for t-distribution which might be useful for some questions): 100×γth Percentiles tγ of Student's t distribution with ν degrees of freedom. γ ν 0.942 0.95 0.975 299 1.574 1.650 1.968 5pts (i) Given the reported 99% confidence interval of the difference, can you meaningfully use the p-value in order to perform the test at the 5% significance level? Explain. If possible, draw a conclusion based on the p-value. 5pts (ii) Construct a 95% confidence interval. 5pts (iii) Now assume another researcher wants to test H0 : µ = 175 against Ha : µ 6= 175 at the 1% significance level based on the same tables. Calculate the p-value and obtain a conclusion. 4 Moreover, the researcher wants to apply the more robust Wilcoxon signed-rank test in order to test H0 : µ = 177 against Ha : µ 6= 177. The SPSS output looks as follows: 5pts (iv) Which assumption is needed in order to use the t-test that is not needed for the Wilcoxon signed-rank test? Which assumption is needed in order to apply the Wilcoxon test for the mean? 5pts (v) Given the results in the table, can it be concluded that the mean height nowadays is different from 177 (at the 5% significance level)? 5.
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