[SOLVED] CS F70TS2 Time Series Exercises 4

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 (k). Furthermore, we also obtained its asymptotic variance Var(x ) 0.01. What is the approximate 95% confidence interval of the unknown expectation ?
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 = 0.5(Xt1 )+0.2(Xt2 )+0.4t1 +t, (b) Xt = 0.2(Xt1 ) + 0.6t1 + 0.3t2 + t,
(c) Xt =0.6(Xt1 )+0.2t1 +t,
where t 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 =0.7(Xt1 )+t,
(b) Xt =0.3(Xt1 )+t,
(c) Xt =0.45(Xt1 )+0.3(Xt2 )+t,
where t 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) = (0). Calculate (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 (0). Comment on your results.
400
4. Suppose you have calculated the first 20 sample autocorrelations (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) |(k)| > 0.1 for at least one k, and b) |(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 (k) for k = 1, …, 40, from a time series with n = 1600 observations. How should you formulate and answer similar questions?1