[4] (a) Define a random walk and explain whether or not it is stationary. (b) Consider the time series given pointwise,

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4 A Define A Random Walk And Explain Whether Or Not It Is Stationary B Consider The Time Series Given Pointwise 1 (53.05 KiB) Viewed 37 times

[4] (a) Define a random walk and explain whether or not it is stationary. (b) Consider the time series given pointwise, by y = 1, for all t, where {2,} is a Gaussian second order stationary time series model with mean 0 and autocovari- ance k = Cov(It, It+k). Find the mean and autocovariance of {y} expressed in terms of .. Note that for Gaussian random variables you can use Isserlis Theorem given by: E(x₁x₁x₁) = E(x₁x₂)E(TRT₁) + E(X₁TR)E(151₁) +E(1₁x)E(TIR). [8] (c) Consider a time series {a} modelled by a stationary ARCH(1) where the errors are normally distributed. i. Write down this model in terms of parameters (0,0₁) and find the con- ditional second moment given by E(a? F-1) where F-1 is the information available up to time t - 1. [3] ii. Find the unconditional second moment of {a} given by E(a?). [3] iii. By using Isserlis' Theorem or otherwise, show that the conditional fourth moment of {a} is given by E(a F-1)=3(00+ a₂a²_1)². [3] iv. Using B1(c)iii. or otherwise, show that the unconditional fourth moment of {a} is given by E(a) 3a²(1+0₁) (1-₁)(1-3a²) [5] = v. As the fourth moment of {a} must be positive, find an additional condition that the ARCH(1) parameters must satisfy. [1] vi. Show that the kurtosis of {a}, which for zero-mean variables is given by K= E(a) [E(af) is greater than 3, and comment on the significance of this result. [3]