本次代写主要为数学时间序列相关的assignment
2.在整个问题中，{𝑍𝑡}是一个IID白噪声过程，均值为0，方差𝜎2𝑧。
（a）讨论并提供证明，如何利用微分来使时间序列平稳当存在线性或二次趋势时。 [8分]
（b）零均值时间序列的自协方差生成函数（𝑋𝑡）} 给出经过Γ（𝑧）= -12个𝑧−2+14𝑧−1+ 32个+14𝑧− 12个𝑧2。
i）什么样的过程是{𝑋𝑡}？
ii）假设白噪声方差𝜎2 𝑧= 1，计算剩余参数过程。
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，为根本不同的情况。
iii）考虑过滤后的过程𝑊𝑡=𝑌𝑡− 12个𝑌𝑡−1 − 13𝑌𝑡-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，方差𝜎2𝑧。
（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）≈1𝑛∞∑𝑖 =-∞{𝜌2𝑖+ 𝜌𝑖−𝑗𝜌𝑖 +𝑗− 4𝜌𝑖𝜌𝑗𝜌𝑖 +𝑗+ 2𝜌2𝑖𝜌2𝑗}，其中𝑟𝑗是滞后a的样本acf，𝑗是滞后𝑗的理论acf。
i）对于MA（2），对于每个≥≥1的情况，计算Bartlett的Var（𝑟𝑗）公式过程。您的答案应该以尽可能简单的形式给出，应该以𝜌𝑗表示。
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由by1 = 0.5，，2 = -0.6，𝑟1= 0.55，𝑟2= 0.45和𝑟3= 0.4给出。执行陈述任何假设的MA（2）模型原假设的近似检验你做的。您的答案应以上限和下限表示𝑝值。[18分]
2. Throughout this question {𝑍𝑡} is an IID white noise process with mean 0 and
variance 𝜎2𝑧.
(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
byΓ(𝑧) = −12𝑧−2+ 14𝑧−1+ 32+ 14𝑧 − 12𝑧2.
i) What kind of process is {𝑋𝑡}?
ii) Given that the white noise variance 𝜎2𝑧 = 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− 12𝑌𝑡−1 − 13𝑌𝑡−2. Determine the spectraldensity 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) ≈ 1𝑛∞∑ 𝑖=−∞{𝜌2𝑖+ 𝜌𝑖−𝑗𝜌𝑖+𝑗 − 4𝜌𝑖𝜌𝑗𝜌𝑖+𝑗 + 2𝜌2𝑖𝜌2𝑗}, 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]