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MATH454/554: Project I

TO BE HANDED IN BY MONDAY 27/11/2017 (WEEK 8), 10:00.

This project will contribute 16% towards the final mark.

Submission: Upload the pdf of your answer and your R code file to the Moodle site. Your R code should be as .r or .txt file so that it can be copied and pasted to run. Submit also a printed copy of your report (no need for a printed copy of the R code), together with a plagiarism cover sheet, to the MSc submissions pigeon hole. Please write your student ID on your answers, not your name.

An AR(1) time series model satisfies $Y_{0}=0$ and for $t=1,2,\ldots$ ,

Y_{t}=\alpha Y_{t-1}+\epsilon_{t},

where $\{\epsilon_{t}\}$ ’s are independent and identically distributed according to $N(0,\sigma^{2})$ .

Suppose that we observe data $y_{0}=0,y_{1},\ldots,y_{n}$ from the AR(1) process.

1.

Write down the log-likelihood function for $\mbox{\boldmath$\theta$}=(\alpha,\sigma^{2})$ given $\mathbf{y}=(y_{0},y_{1},\ldots,y_{n})$ , $\ell(\mbox{\boldmath$\theta$};\mathbf{y})$ . [2]
2.

Derive the maximum likelihood estimates of $\alpha$ and $\sigma^{2}$ given $(y_{0},y_{1},\ldots,y_{n})$ . [3]
Hint: Compute $\hat{\alpha}$ first and then insert $\hat{\alpha}$ into the computation of $\hat{\sigma^{2}}$ .

Suppose that observation $y_{k}$ is missing for some $1\leq k<n$ .

3.

Show that the conditional distribution of $y_{k}$ given $\mathbf{y}_{-k}$ (the vector $\mathbf{y}$ excluding $y_{k}$ ), $\alpha$ and $\sigma^{2}$ is [2]

$y_{k}|y_{k-1},y_{k+1},\alpha,\sigma^{2}\sim N\left(\frac{\alpha(y_{k-1}+y_{k+1% })}{\alpha^{2}+1},\frac{\sigma^{2}}{\alpha^{2}+1}\right).$

Hint: If $X$ is a random variable with pdf $f_{X}(x)\propto\exp(-(x^{2}-2\mu x)/(2\phi^{2}))$ then $X\sim N(\mu,\phi^{2})$ .
4.

Write down $Q(\mbox{\boldmath$\theta$},\mbox{\boldmath$\theta$}^{\ast})=\mathbb{E}_{\theta% ^{\ast}}[\ell(\mbox{\boldmath$\theta$};\mathbf{y}_{-k},Y_{k})|\mathbf{y}_{-k}]$ and identify the moments of $y_{k}$ which are required to compute $Q(\mbox{\boldmath$\theta$},\mbox{\boldmath$\theta$}^{\ast})$ . [2]
5.

Write an R routine to compute the MLE’s of $\theta$ given $\mathbf{y}_{-k}$ . [5]
6.

Apply your routine to ardata.txt, where observation 15 is missing. (Note that observation 15 is set equal to 0 in the data file.) Report the MLE’s of $\alpha$ and $\sigma^{2}$ . [2]