Question

Problem 1. Let (X1, ...., Xn) be an i.i.d random sample with X; ~ U[0, 2a), and (Y1, ..., Yn) be an i.i.d random sample with Y; ~ Exp ( 1. Find E[X;], E[X3], E[Y/] and E[Y;?). 2. Notwithstanding the actual distributions of the random samples, suppose the modeller believes that they are i.i.d draws from a U (0, 2a distribution. Find the (simple) method of moments estimator â. 3. Let n = 1000. Generate random samples (X1, ..., X1000) and (Y1, ..., Y1000). For each random sample, estimate â using the (simple) method of moments, assuming the data are drawn from a Uniform Uſ0, 2a) distribution. 4. Suppose instead we try to fit the data to two moments (E[X] and E[X²]) rather than one, using GMM. As before, suppose the modeller believes that the data is drawn from a U10, 2a distribution. Find the GMM estimator GMM, using the weighting matirx W =1. 5. Using your answer to the previous part, find the efficient (2-stage) GMM estimator. 6. For each sample, conduct a J test for over-identification. Comment on the model fit for each sample.

Only Questions 4,5 and 6

a=5

Answer 1

Solution. 1. X; v 0 [o, 9 43 forse sex ola Laa 1 o, e.w. ECX:): see a z foxy dx - soka de 22 122 4.to TES) = a Eftie). S.24 da 2 . TER; 2) = de Yi ~ Exp (ta) 1. o, pow. Elli) E. (i) = &(ti) . So z ECiDa S. 22. d(a) x elo do

2. Finding Method of moment estimator- et & =(x, 7. . , x n ) be a random sample from . U[o, ad].. Eltila 2 in 1 E comparing . t with Eltil- we have Un £ tis a n 1 ax

3 crevenite two reindom samples 2 = 106, R., Noo) & 7 = (2, L., using any statistical software. oo) EXI)e Elli): a 4