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）
Enders 2.6 Consider the second-order stochastic difference equation: $y_t=1.5 y_{t-1}-0.5 y_{t-2+}$ $\varepsilon_t$.
(a) Find the characteristic roots of the homogeneous equation.
(b) Demonstrate that the roots of $1-1.5 L+0.5 L^2$ are the reciprocals of your answer in part (a)
(c) Given initial conditions for $y_0$ and $y_l$, find the solution for $y_t$, in terms of the current and past values of the $\left{\varepsilon_t\right}$ sequence.
(d) Find the forecast function for $y_{T+s}$, (i.e., find the solution for any value of $y_{T+s}$, given the values of $y_T$ and $y_{T-1}$ ).
(e) Find $E y_t, E y_{t+1}, \operatorname{var}\left(y_t\right), \operatorname{var}\left(y_{t+1}\right)$, and $\operatorname{cov}\left(y_{t+1}, y_t\right)$.
Enders 2.8 The file entitled SIM_2.csv contains the simulated data sets used in chapter 2 of Enders (2010). The first series, denoted $Y 1$, contains the 100 values of the simulated $A R(I)$ process used in Section 7 of Chapter 2. Use this series to perform the following tasks (Note: Due to differences in data handling and rounding, your answers need only approximate those presented here):
(a) Plot the sequence against time. Does the series appear to be stationary?
(b) Verify that the first 12 coefficients of the ACF and the $\mathrm{PACF}^1$ are as follows:
ACF
$1: 0.7390 .5840 .471 \quad 0.389 \quad 0.344 \quad 0.335$
7: $\quad 0.2970 .3250 .2690 .201 \quad 0.1890 .082$
PACF
7: $-0.017 \quad 0.144-0.100-0.065 \quad 0.070-0.204$
Ljung-Box Q-Statistics: $Q(8)=177.58, Q(16)=197.84, Q(24)=201.28$.
[Enders 2.9] The second column in file SIM_csv contains the 100 values of the simulated $A R M A(I, 1)$ process used in Section 2.7 of Enders (2010). This series is entitled Y2. Use this series to perform the following tasks (Note.. Due to differences in data handling and rounding, your answers need only approximate those presented here): (To facilitate comparisons all models should be estimated over the same sample. The final model can of course be re-estimated over the entire sample).
(a) Plot the sequence against time. Does the series appear to be stationary? Verify that the first 12 coefficients of the ACF and the PACF are as follows:
ACF
$$
\begin{array}{lllllll}
\text { 1: } & -0.834 & 0.597 & -0.440 & 0.350 & -0.319 & 0.332 \
\text { 7: } & -0.337 & 0.317 & -0.276 & 0.179 & -0.084 & 0.038
\end{array}
$$
PACF
$$
\begin{array}{lrrrrrr}
1: & -0.834 & -0.328 & -0.194 & -0.015 & -0.140 & 0.849 \
7: & 0.000 & 0.010 & 0.017 & -0.199 & -0.084 & -0.021
\end{array}
$$
[Enders 2.10 ] The third column in file SIM_2.csv contains the 100 values of the simulated $A R(2)$ process used in Section 2.7. This series is entitled $Y 3$. Use this series to perform the following tasks (Note: Due to differences in data handling and rounding, your answers need only approximate those presented here):
(a) Plot the sequence against time. Verify the ACF and the PACF coefficients reported in Section 2.7. Compare the sample ACF and PACF to the those of a theoretical $A R(2)$ process.
(b) Estimate the series as an $A R(I)$ process. You should find that the estimated $A R(I)$ coefficient and the $t$-statistic are
$$
y_t=\underset{(5.24)}{0.467} y_{t-1}+\varepsilon_t
$$
Why is an AR(I) model inadequate?
(c) Could an ARMA $(1,1)$ process generate the type of sample ACF and PACF found in part a? Estimate the series as an $A R M A(1,1)$ process. You should obtain:
\begin{tabular}{|l|c|c|c|c|}
\hline \multicolumn{5}{|l|}{ Included observations: 99 } \
\hline Coefficient & Estimate & Standard Error & $t$-statistic & Probability \
\hline$\alpha_1$ & 0.183 & 0.159 & 1.15 & 0.2523 \
$\beta_1$ & 0.510 & 0.140 & 3.64 & 0.0004 \
\hline
\end{tabular}
Use the Ljung-Box Q-Statistics to show that the ARMA(1,1) model is inadequate.
(d) Estimate the series as an $\operatorname{AR}(2)$ process to verify the results reported in the text.
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