Instructions: Answer all 4 questions. For the Statistics questions, I need to see the do-files and a log file demonstrating that your do-file/program actually works.
1. In the file takehome.dat posted in the DATA folder on Blackboard, 100 iid observations on a certain random variable are provided. You may assume that moments of any order exists for this random variable. You may not assume that the underlying distribution is normal (in fact, it isn’t). Using large sample theory, construct 95% confidence intervals for the a) mean (μ), b) variance (σ2) and c) the coefficient of variation (μ/σ).
2. Suppose the three random variables x, w, ε be jointly normally distributed with zero means and unit variances. Let also E(xε) = .15, E(xw) = .75 and E(wε) = 0. Finally, let y = βx + ε.
(a) Assume that we have n observations on y, x, w. Write down the expression for the OLS estimator for β and that for a 2SLS estimator for β in the simplest possible forms.
(b) Let true β = 1. For 500 replications, and for different sample sizes compute the mean bias and mean MSE (mean squared error) for both estimators (use n = 10, 20, 30 ... etc. until you begin to see a pattern). Comment on what you find.
(c) Also compute the percentage of time the 2SLS estimator is closer to the true parameter than the OLS estimator. Based on your simulations, recommend what sample size to use with what estimator.
3. Often we have to choose between two competing models, which do not have the following feature: there is one large model where under the null hypothesis, some parameters are presumed to be zero whereas under the alternative at least one of them is non-zero.
Consider two competing models:
Ho : y = Xβ + uo
and
H1 : y = Zγ + u1
where the variables contained in the X and Z matrices are not identical (but some variables may be common to both).
A test that has been proposed in the literature in such a situation is as follows. Regress y on the variables in Z, and collect the predicted value in a variable called yˆ. Now run a regression where y is regressed on the variables in X plus ˆy. The null is deemed rejected if the coefficient on ˆy turns out significant via a standard t-test.
(a) Provide as much rigorous justification as you can for the asymptotic validity of this test.
(b) Let ct denote log of real consumption and yt denote log of real disposable income. Let Model 1 be given by:
ct = α + βct−1 + γ0yt + γ1yt−1 + ut
and Model 2 given by
?ct = α' + β'Δct−1 + γ'?yt + γ1'?yt−1 + ut
How do you think these models differ qualitatively?
(c) Using data from 1929-2015 from the FRED database, test each model against the other.
4. Write a program which can generate for any square, symmetric, positive-definite matrix A, a square root matrix B such that A = BB. Make sure that if any of the required properties of the input matrix A is not true, your program alerts the user to that fact.