What is the Asymptotic Distribution for the Test Statistic - Economics Assignment Help

Assignment Task:

Task:

1. [There are three parts (a through c) to this question.] Suppose you have two random samples, each of size n, possibly from different populations. They contain the data, [y1,X1], [y2,X2], respectively, where yj is an n ×1 vector and Xj is a n × K matrix, j = 1, 2. For each sample, we assume that the Linear Regression Model (LRM) holds, i.e., , y X j jj j = + β ε ( ) , E ε j j X 0 = Xj is of full rank, and ( ) 2 , 1, 2. E j εε X jj j ′ = = σ I Let bj denote the OLS estimator of , 1, 2. j β j = a) What is the limiting distribution of n (( )( ) b b 21 21 −− − β β ) as n → ∞ and the asymptotic distribution of 2 1 ( ) b b− ? [Hint: See Appendix D.3 in Greene.] b) What is the appropriate test statistic for H0: 2 1 , β β k k = where βjk is the coefficient on variable xjk in kth column of Xj , j = 1, 2.

c) What is the asymptotic distribution for the test statistic in b) under the null hypothesis, H0? 2 2. [There are 7 parts (a through g) to this question.]

Suppose we have a random sample of size n with observations for the random variable X, i.e., the data is: X1, X2, …, Xn. We know that X has a bounded distribution, with a lower bound of 0 and an upper bound of θ, where 1 , < <∞ θ i.e., X ∈[0, ]. θ Further, the distribution of X either one of the following two distributions: Uniform, or Triangular, with mode c∈( ) 0,θ . The following is a graph of the probability density function (pdf), denoted by f(x), of the uniform distribution with bounds [a, b] is : and the graph of the pdf of the triangular distribution, denoted by g(x), with bounds [a,b] and mode c, is :

The objectives of this quesiton are to determine some properties of the data on X and to try to determine whether the correct distribution for X is the uniform or triangular distribution. 3 (a) Write out the mathematical expression for the probability density function (pdf) for X, assuming a uniform distribution with bounds [ ] 0,θ , i.e., write out f(x). (b) Write out the mathematical expression for the probability density function (pdf) for X, assuming a triangular distribution with bounds [ ] 0,θ , i.e., write out g(x). (c) Let µ f and µg denote the means of X when the true distributions are uniform and triangular, respectively. What is µ f equal to? What is µg equal to? Under what conditions would µ µ g f = ? In answering this question, provide the expressions for µ f and µg in terms of the parameters of the uniform and triangular distributions noted above and the pdfs you derived in parts (a) and (b).

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