时间序列分析模拟试题4

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诚实考试吾心不虚，公平竞争方显实力，考试失败尚有机会，考试舞弊前功尽弃。上海财经大学《时间序列分析》课程考试卷课程代码课程序号 20 —20 学年第一学期

一、Give a definition of second-order stationary. [10 marks]

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二、Re-write the 1st order autoregressive process:

XtXt1Yt

订

In term of Xt,Yt and the backwards operator B: BXtXt1. For what values of

a is the the stationary? [15 marks] 三、Re-write the 1st order moving average process:

Y XtYtb t

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In term of Xt,Yt and the backwards operator B: BYtYt

1. For what values of b is the the invertible? [15 marks] 四、What are the roots 1，2 of the characteristic equation

for

the AR(2) process:

Xt1Xt1

2Xt1t

what condition must 1，2 satisfy in order for the AR(2) process to be expressed as an infinite order MA process? [15 marks] 五、Derive the Yule-Walker equations for the AR(3) process

Xt1Xt12Xt13Xt2t

where t~N0,2.

(a) Express the Yule-Walker equation in the matrix form. [10 marks]

(b) Given sample value Ct of the covariance function t how would you estimate the parameters iof the AR(3) process. [10 marks]

ˆthat depends on a variance-covariance matrix (c) Write down the distribution of 

that must provide. [10 marks]

六、Given the observations

X1,X2,K,Xngenerated by the

process

Xtt1.80t10.81t2

where  is a real constant and t~N0,1. Find the exact distribution of the sample meanXn. Evaluate VarXn with n=10. Compare the VarX10 with the variance of the sample mean of a random sample of size 10 drawn from the distribution of X1.[15 marks]