Financial Econometrics, MFE 2024-25
Practical Work 1
Individual Assignment
Let X1, . . . , Xn be an iid (independently and identically distributed) sequence. The density function of Xi
is given by
for every i = 1, . . . , n. You can assume that this distribution is correctly specified. Let γ0 be the unknown true value of γ. We are also given the information that
Finally, we define X = (X1, . . . , Xn), the collection of the random variables X1, . . . , Xn.
1. Find ℓ(γ; X), the joint log-likelihood function of X1, . . . , Xn.
2. Calculate
3. Find the maximum likelihood estimator of γ0.
4. Find the asymptotic distribution of the likelihood estimator .
5. Suppose that we know that γ0 = 3. Use the asymptotic distribution and the variance you calculated in the previous part to find approximations for E[] and Var[] when n = 1000.