F70TS2 – Time Series Exercises 4
1. In the notes it is shown that, under given conditions, the sample mean x ̄ obtained from a realisation of a stochastic process is asymptotically normal. Is this property important? Why? Assume that x ̄ = 1.56 is obtained from a realisation of a time series with absolutely summable γ(k). Furthermore, we also obtained its asymptotic variance Var(x ̄) ≈ 0.01. What is the approximate 95% confidence interval of the unknown expectation μ?
For a large sample you can use the approximation of the normal quantile Z0.025 = 1.96 ≈ 2.
2. Assume that x ̄ is calculated from a realisation x1,…,x900 of the following causal stationary
ARMA models with unknown mean:
(a) Xt −μ = 0.5(Xt−1 −μ)+0.2(Xt−2 −μ)+0.4εt−1 +εt, (b) Xt − μ = 0.2(Xt−1 − μ) + 0.6εt−1 + 0.3εt−2 + εt,
(c) Xt −μ=−0.6(Xt−1 −μ)+0.2εt−1 +εt,
where εt are i.i.d. N (0, 1) random variables. Calculate the asymptotic Var(x ̄) in each case and compare your results in all cases. What general conclusions can be drawn?
3. Three causal stationary AR models with unknown mean are given below. (a) Xt −μ=0.7(Xt−1 −μ)+εt,
(b) Xt −μ=−0.3(Xt−1 −μ)+εt,
(c) Xt −μ=0.45(Xt−1 −μ)+0.3(Xt−2 −μ)+εt,
where εt are i.i.d. N(0,1) random variables. Let Yt denote an i.i.d. process with unknown mean and the same variance as Xt, i.e. Var(Yt) = Var(Xt) = γ(0). Calculate γ(0) for each of the models above. Given data x1, …, x400 and y1, …, y400, you can obtain x ̄ and y ̄. You should calculate the asymptotic Var(x ̄) for each model and compare them with the corresponding Var(y ̄) = 1 γ(0). Comment on your results.
400
4. Suppose you have calculated the first 20 sample autocorrelations ρˆ(k) for k = 1, …, 20, from a time series with n = 400 observations. Assume that you know the underlying process {Xt} is stationary. You want to check whether Xt could be independent. What are your conclusions in the following cases: a) |ρˆ(k)| > 0.1 for at least one k, and b) |ρˆ(k)| < 0.1 for all k? Why are the condition ‘at least one’ and the bound 0.1 used? Assume now that you calculated the first 40 sample autocorrelations ρˆ(k) for k = 1, ..., 40, from a time series with n = 1600 observations. How should you formulate and answer similar questions?
1