Assignment #4 STA 457H1S/2202H1S Due Monday December 2, 2024 Instructions: Solutions to problems 1 and 2 are to be submitted on Quercus (PDF files only). You are strongly encouraged to do problems 3 and 4 but these are not to be submitted for grading. 1. Daily stock prices (adjusted for stock splits and dividends) for Barrick Gold (from January 12, 1995 to November 14, 2016) are given in the filw barrick.txt on Quercus; the data are already transformed by taking logs and you should analyze them on this scale. (a) Fit ARIMA(0,1,1) and ARIMA(0,1,2) models to the data. Do these models seem to fit the data adequately? Which model do you prefer and why? (You may want to do an ADF test to see if the time series needs to be differenced to make it stationary but the non-stationary should be quite clear.) (b) Using the residuals from your preferred model from part (a), fit ARCH(m) models for m = 1, 2, 3, 4, 5. Which model seems to be the best? (c) Repeat part (b), using GARCH(1, s) models for s = 1, 2, 3. Are any of these models an improvement over the best ARCH model from part (b)? Note: The R package fGarch contains the function garchFit, which can be used to fit both ARCH and GARCH models. 2. Consider a stationary bivariate process {(Xt, Yt)}. If γxy(s) = Cov(Xt, Yt+s) is the cross- covariance function then we can define the cross-spectral density function (cross-spectrum) as fxy(ω) = ∞∑ s=−∞ γxy(s) exp(2πιωs). The cross-spectrum can be complex-valued. (a) Show that the cross-covariance function can be recovered from the cross-spectrum as γxy(s) = ∫ 1 0 fxy(ω) exp(−2πιωs) dω. (b) Suppose we want to predict Yt using {Xt}: Ŷt = ∞∑ s=−∞ asXt−s where the as’s are chosen to minimize the prediction MSE E[(Yt − Ŷt)2]. Show that the as’s satisfy the equations ∞∑ s=−∞ asγx(u− s) = γxy(u) for u = 0,±1,±2, · · · where γx(s) is the autocovariance function of {Xt}. (c) Let Γ(ω) be the transfer function of the “optimal” coefficients {as} as defined in part (b): Γ(ω) = ∞∑ s=−∞ as exp(2πιωs). If {(Xt, Yt)} has cross-spectrum fxy(ω) and {Xt} has spectral density function fx(ω), deter- mine Γ(ω) in terms of fxy(ω) and fx(ω). (Hint: Multiply both sides of the equation in part (b) by exp(2πιωu) and sum over u from −∞ to ∞.) (d) Suppose that fx(ω) = 4(ω − 1/2)2 and fxy(ω) = (ω − 1/2)4. Use the method from Problem 3 of Assignment #1 to obtain the values of as for s = −20, · · · , 20. 3. Suppose that {εt} is a white noise process whose conditional variances σ2t = E(ε 2 t |εt−1, εt−2, · · ·) follow a GARCH(1,1) model: σ2t = α0 + α1ε 2 t−1 + β1σ 2 t−1 where α1, β1 ≥ 0 and α0 > 0. Assume that both {σ2t } and {ε2t} are stationary processes. (a) Show that E(σ2t ) = E(ε 2 t ) if both expectations are finite. (b) Show that α1 + β1 < 1 implies that E(σ 2 t ) is finite and E(σ 2 t ) = α0/(1− α1 − β1). (c) Find conditions on α1 and β1 so that E(σ 4 t ) is finite. (Hint: Note that σ 4 t = (α0+α1ε 2 t + β1σ 2 t−1) 2 and E(σ4t ) ≥ [E(σ2t )]2.) 4. Consider a VAR(p) process X t = Φ1X t−1 + · · ·+ ΦpX t−p + εt where Φ1, · · · ,Φp are k × k matrices. The k-variate process {X t} is stationary if det ( I − zΦ1 − z2Φ2 − · · · − zpΦp ) ̸= 0 for all z with |z| ≤ 1. (a) Suppose that {aTX t} is a stationary (univariate) AR(p) process for some a ̸= 0. What can be said about a? (b) Suppose that k = p = 2 and define X t = Φ1X t−1 + Φ2Xt−2 + εt where Φ1 =  1/2 0 0 1/2  Φ2 =  3/8 −1/8 −1/8 3/8  (i) Show that each component of X t is integrated, that is, non-stationary with stationary first differences. (ii) Find a vector a such that {aTX t} is stationary. 51作业君版权所有