MY-ASSIGNMENTEXPERT™可以为您提供 utstat.utoronto Stats531 Time series analysis时间序列分析的代写代考和辅导服务！
这是密歇根安娜堡的大学 时间序列分析的代写成功案例。
Stats531课程简介
Course description
This course gives an introduction to time series analysis using time domain methods and frequency domain methods. The goal is to acquire the theoretical and computational skills required to investigate data collected as a time series. The first half of the course will develop classical time series methodology, including auto-regressive moving average (ARMA) models, regression with ARMA errors, and estimation of the spectral density. The second half of the course will focus on state space model techniques for fitting structured dynamic models to time series data. We will progress from fitting linear, Gaussian dynamic models to fitting nonlinear models for which Monte Carlo methods are required. Examples will be drawn from ecology, economics, epidemiology, finance and elsewhere.
Prerequisites
Grading
Weekly homeworks (25%, due Tuesdays, in class).
A midterm exam (25%, in class on Thursday 2/25).
A midterm project analyzing a time series of your choice using methods covered in the first half of the course (15%, due Thursday 3/10).
A final project analyzing a time series of your choice using methods covered in the entire course (35%, due Thursday 4/28).
Discussion of homework problems is encouraged, but solutions must be written up individually. Direct copying is not acceptable.
Any material taken from any source, such as the internet, must be properly acknowledged. Unattributed copying from any source is plagiarism, and has potentially serious consequences.
Stats531 Time series analysis HELP（EXAM HELP， ONLINE TUTOR）
Prove that the process $y_t=z_t-z_{t-1}$, where $z_t=.9 z_{t-1}+a_t$ is stationary, remains stationary.
Prove that the above process can be written as $y_t=-.09 z_{t-2}+a_t-.1 a_{t-1}$
Calculate the inverse operator of $(1-.8 B)(1-B)$.
Justify whether the process $(1-.5 B)(1-.7 B)(1-.2 B) z_t=a_t$ is stationary and write it in its usual expression.
Calculate the theoretical coefficients of partial autocorrelation for the following AR(2) process: $z_t=$ $.7 z_{t-1}-.5 z_{t-1}+a_t$, where $a_t$ is white noise.
Prove that if a process is $\operatorname{AR}(1)$ if we then make the regression $z_t=\beta z_{t+1}+u_t$ we obtain $\widehat{\beta}=\phi$ and $\operatorname{var}\left(u_t\right)=\gamma_0\left(1-\phi^2\right)$, where $\gamma_0$ is the variance of the process.
Prove that if a process is $\operatorname{AR}(1)$ and we make the regression $z_t=\alpha z_{t-1}+\beta z_{t+1}+u_t$, we obtain $\widehat{\beta}=\widehat{\alpha}=\phi /\left(1+\phi^2\right)$ and $\operatorname{var}\left(u_t\right)=\gamma_0\left(1-\phi^2\right) /\left(1+\phi^2\right)$. Observe that the variance of the innovations is now less than that of an $\mathrm{AR}(1)$ and less than in above exercise and interpret this result.
Given the process of zero mean $z_t=(1-.7 B) a_t$ : (a) calculate the autocorrelation function; (b) write it as an $\operatorname{AR}(\infty)$ process.
Prove that the MA(1) processes $z_t=a_t-.5 a_{t-1}$ and $z_t=a_t-2 a_{t-1}$ have the same autocorrelation structure but that one is invertible and the other is not.
Prove that the two processes $z_t=a_t+.5 a_{t-1}$ and $z_t=.5 a_t+a_{t-1}$ are indistinguishable since they have the same variance and the same autocorrelation structure.
Given the MA(2) process $z_t=a_t-1.2 a_{t-1}+.35 a_{t-2}$ : (a) check whether it is invertible; (b) calculate its autocorrelation structure; (c) write it as an $\operatorname{AR}(\infty)$ process.
Given the model $z_t=5+.9 z_{t-1}+a_t+.4 a_{t-1}:$ (a) calculate its autocorrelation structure; (b) write it in $\mathrm{MA}(\infty)$ form; (c) write it in $\operatorname{AR}(\infty)$ form.
Given the process $\left(1-B+.21 B^2\right) z_t=a_t-.3 a_{t-1}:$ (a) check whether it is stationary and invertible:
(b) obtain the autocorrelation function; (c) obtain its $\operatorname{AR}(\infty)$ representation; (d) obtain its $\operatorname{MA}(\infty)$ representation.
Obtain the autocorrelation function of an $\operatorname{ARMA}(1,1)$ process writing it as an $\operatorname{MA}(\infty)$.
Prove that if we add two $\mathrm{MA}(1)$ processes we obtain a new MA(1) process with an MA parameter which is a linear combination of the MA parameters of the two processes, with weighs that are proportional to the quotients between the variances of the innovations of the summands related to the variance of the innovation of the sum process.
Prove that the sum and the difference of two stationary processes are stationary.
Prove that the model $z_t=a+b t+c t^2+a_t$, where $a_t$ is a white noise process, becomes a non-invertible stationary process when two differences are taken.
Prove that the autocorrelations of a random walk can be approximated by $\rho(t, t+k) \simeq 1-\frac{k}{2 t}$.
Simulate an ARIMA $(0,1,1)$ process using parameter values $\theta=.4, .7$ and .9 , and study the decay of the autocorrelation function of the process.
Simulate the process $\nabla^2 z_t=(1-.8 B) a_t$ and study the decay of the autocorrelation function of the process.
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