2.在整个问题中，{𝑍𝑡
}是一个IID白噪声过程，均值为0，

𝑧。
（a）讨论并提供证明，如何利用微分来使时间序列平稳

（b）零均值时间序列的自协方差生成函数（𝑋𝑡）
} 给出

Γ（𝑧）= -1
2个
𝑧−2
+1
4
𝑧−1
+ 3
2个
+1
4
𝑧− 1
2个
𝑧2

i）什么样的过程是{𝑋𝑡
}？
ii）假设白噪声方差𝜎2
𝑧= 1，计算剩余参数

iii）流程{𝑋𝑡
}可逆的？
[10分]
（c）让{𝑌𝑡
}表示固定的AR（1）进程。
i）确定{𝑌𝑡的光谱密度
}。
ii）绘制（c）（i）部分获得的光谱密度，为

iii）考虑过滤后的过程𝑊𝑡
=𝑌𝑡
− 1
2个
𝑌𝑡−1 − 1
3
𝑌𝑡-2。确定光谱
{𝑊𝑡的密度
}。
iv）在（c）（iii）部分中获得了哪种光谱密度的ARMA工艺？
[22分]

3.在整个问题中{𝑍𝑡
}是一个IID白噪声过程，均值为0，

𝑧。
（a）考虑程序
𝑋𝑡
= 𝜃（𝐵）𝑍𝑡

𝜃（𝐵）= 1 − 𝜃1𝐵 −…− 𝜃𝑞𝐵𝑞

i）计算{𝑋𝑡的自协方差函数𝛾（ℎ）
}。
ii）计算{𝑋𝑡的自相关函数（acf），𝜌（ℎ）
}。
[10分]
（b）巴特利特（Bartlett）的公式指出：
Var（𝑟𝑗）≈1
𝑛

∑𝑖 =-∞
{𝜌2
𝑖
+ 𝜌𝑖−𝑗𝜌𝑖 +𝑗− 4𝜌𝑖
𝜌𝑗𝜌𝑖 +𝑗+ 2𝜌2
𝑖
𝜌2
𝑗
}，

i）对于MA（2），对于每个≥≥1的情况，计算Bartlett的Var（𝑟𝑗）公式

ii）假设我们基于观察到的长度为200的时间序列进行了估算

𝑝值。
[18分]

2. Throughout this question {𝑍𝑡
} is an IID white noise process with mean 0 and
variance 𝜎2
𝑧.
(a) Discuss, giving proofs, how differencing can be used to make a time series stationary
when a linear or quadratic trend is present. [8 marks]
(b) The autocovariance generating function (AGF) of a zero-mean time series {𝑋𝑡
} is given
by
Γ(𝑧) = −1
2
𝑧−2
+ 1
4
𝑧−1
+ 3
2
+ 1
4
𝑧 − 1
2
𝑧2
.
i) What kind of process is {𝑋𝑡
}?
ii) Given that the white noise variance 𝜎2
𝑧 = 1, calculate the remaining parameters of
the process.
iii) Is the process {𝑋𝑡
} invertible?
[10 marks]
(c) Let {𝑌𝑡
} denote a stationary AR(1) process.
i) Determine the spectral density of {𝑌𝑡
}.
ii) Plot the spectral density obtained in part (c)(i), providing different plots for
fundamentally different cases.
iii) Consider the filtered process 𝑊𝑡
= 𝑌𝑡
− 1
2
𝑌𝑡−1 − 1
3
𝑌𝑡−2. Determine the spectral
density of {𝑊𝑡
}.
iv) What type of ARMA process has spectral density obtained in part (c)(iii)?
[22 marks]

3. Throughout this question {𝑍𝑡
} is an IID white noise process with mean 0 and
variance 𝜎2
𝑧.
(a) Consider the process
𝑋𝑡
= 𝜃(𝐵)𝑍𝑡
,
where 𝜃(𝐵) = 1 − 𝜃1𝐵 −…− 𝜃𝑞𝐵𝑞
.
i) Calculate the autocovariance function, 𝛾(ℎ) say, of {𝑋𝑡
}.
ii) Calculate the autocorrelation function (acf), 𝜌(ℎ) say, of {𝑋𝑡
}.
[10 marks]
(b) Bartlett’s formula states that
Var(𝑟𝑗) ≈ 1
𝑛

∑ 𝑖=−∞
{𝜌2
𝑖
+ 𝜌𝑖−𝑗𝜌𝑖+𝑗 − 4𝜌𝑖
𝜌𝑗𝜌𝑖+𝑗 + 2𝜌2
𝑖
𝜌2
𝑗
},
where 𝑟𝑗 is the sample acf at lag 𝑗 and 𝜌𝑗 is the theoretical acf at lag 𝑗.
i) Calculate Bartlett’s formula for Var(𝑟𝑗) for each 𝑗 ≥ 1 in the case of an MA(2)
process. Your answer, which should be given in as simple a form as possible,
should be expressed in terms of the 𝜌𝑗’s.
ii) Suppose that we have estimates based on an observed time series of length 200
given by 𝜃1 = 0.5, 𝜃2 = −0.6, 𝑟1 = 0.55, 𝑟2 = 0.45 and 𝑟3 = 0.4. Perform an
approximate test of the null hypothesis of an MA(2) model, stating any assumptions
that you make. Your answer should be expressed as an upper and lower bound for
a 𝑝-value.
[18 marks]

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