---
title: "HW 3"
author: "Name here"
date: "Due September 28, 2018"
output: html_document
---

Please use D2L to turn in both the PDF output and your R Markdown file. Include your code in line in the document.


#### Q1. (4 pts)

Show that if $\hat{\mu}_t = \alpha x_t + (1-\alpha) \hat{\mu}_{t-1}$ then this can be written as $\hat{\mu}_t = \alpha x_t + \alpha(1-\alpha)x_{t-1} + \alpha(1-\alpha)^2 x_{t-2} + \alpha(1-\alpha)^3 x_{t-3} \dots$

#### Q2. Explore Holt-Winters Method

##### a. (6 pts)
In lecture 6, we constructed a simulation that had the mean term evolve in time. Modify this code to include a seasonal term that evolves in time as well. The easiest way to do this is would be to have four seasons that repeat ever 4th observation. Include your code here and a figure of the resultant time series plot.

##### b. (4 pts)
On your simulated dataset, fit the HoltWinters function and summarize the results. The $\beta$ parameter should still be false in the function call.

##### c.  536 only (4 pts)
The Holt-Winters function in R and specifically `predict.HoltWinters` constructs a prediction interval. Summarize how these predictive intervals are calculated (note this may require some digging). Alternatively, if you'd prefer state the difficulty with understanding the predictive interval and sketch out how this model (just with a simple evolving mean - not the seasonality) can be constructed as a Dynamic Linear Model. See section 2.4 in the DLM book for guidance.