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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.