1. Consider the following ARCH(1) model:yt = µ + utut = vtσt, vt is i.i.d. with mean 0 and variance 1σ2t = α0 + α1u2t−1.Please compute the following:(a). E[σ2t+1|Ωt](b). E[σ2t+2|Ωt](c). E[σ2t+3|Ωt](d). Derive a general formula for E[σ2t+s|Ωt] for any s ≥ 2.

2. Suppose that we have an AR(2) model as follows:yt = 0.1 + ϕ1yt−1 + ϕ2yt−2 + ut,where ut follows a white noise process with mean zero and variance σ2u = 0.2. Please computethe following:(a). Assume that ϕ1 and ϕ2 take values that make the model stationary. What is the meanof yt?(b). Compute E[yt+1|Ωt].(c). Compute E[yt+2|Ωt]

Which of the following is not true for ARCH(1) model (𝑎𝑎𝑡𝑡 = 𝜎𝜎𝑡𝑡 𝜀𝜀𝑡𝑡, 𝜎𝜎𝑡𝑡2 = 𝛼𝛼0 + 𝛼𝛼1 𝑎𝑎𝑡𝑡−12 ), where𝜀𝜀𝑡𝑡 has mean 0 and variance 1?a. mean 𝐸𝐸(𝑎𝑎𝑡𝑡) = 0b. variance 𝑉𝑉𝑎𝑎𝑟𝑟(𝑎𝑎𝑡𝑡) = 𝛼𝛼0/(1 − 𝛼𝛼1) if 0 < 𝛼𝛼1 < 1c. generates heavy tails under some parameter conditionsd. generates asymmetry between positive and negative prior returns

Complete the missing step from the lecture: for ε1 . . . , εn that are distributed i.i.d. N (0, σ2), for a pair of i ̸= j, calculate E[(εi −ε ̄)(εj −ε ̄)], where ε ̄ = 􏰆ni=1 εi/n.

1. Suppose X = (X1, . . . , Xn) is a random sample from the Gamma distribution withshape α and rate θ, i.e.,fX (x) = θαΓ(α)xα−1e−xθ, x > 0, α > 0, θ > 0.Each Xi has expectation E(X) = α/θ, variance Var(X) = α/θ2, and momentgenerating function MX (t) = (1 − t/θ)−α.(a) Assuming both α and θ to be unknown, write down the log likelihood functionℓ(θ, α; X) and the corresponding score functions ∂ℓ∂θ and ∂ℓ∂α .(b) Verify that each score function has zero expectation.(c) Assuming α to be known:(i) Calculate the Cramer Rao Lower Bound (CRLB) for the variance of anunbiased estimator of θ.(ii) Is there any unbiased estimator of θ whose variance attain the CRLB?(iii) Show thatS = α − 1nnXi=1 1Xiis an unbiased estimator for θ. What is the MVU estimator for θ?(iv) Identify a change of parameter η = η(θ) for which there exists an unbiasedestimator with variance attained the CRLB.(v) For the parameter in the previous part, identify the MVU estimator whosevariance attains the CRLB. Compute this variance

EF4822 Financial Econometrics Problem Set 2 Due: Right before the class next week. Please submit hard copies in groups and write down the names and EID of all group members. If R programming is used, please show the R commands. 1. Suppose that the simple return of a stock follows the model rt = 0.1 + at − 0.5at−2, where {at}T t=1 follows a white noise process with mean zero and variance σ2 a = 0.04. Please compute the following: (a). What is the mean of rt? (b). What is the variance of rt?