Purpose: Further understanding moving average processes and the auto-correlation functions.

# Introduction

The process $$y_t$$ is a moving average process based on standard normal white noise terms $$e_t$$ as defined below:

$y_t = \frac{1}{2}e_t + \frac{1}{2}e_{t-1} \qquad \mbox{where} \quad \mathbb{E}(e_t) = 0 \quad \mbox{and} \quad Var(e_t) = 1 = \sigma^2_e$

# Question 1

1 Point

What is the unconditional mean $$\mathbb{E}(y_t)$$?

# Question 2

1 Point

What is the variance $$\mbox{Var}(y_t)$$?

# Question 3

1 Point

What is the auto-covariance at one lag $$\mbox{Cov}(y_t,y_{t-1})$$?

# Question 4

0.5 Points

What is the auto-covariance at two lags $$\mbox{Cov}(y_t,y_{t-2})$$?

# Question 5

0.5 Points

What is the auto-covariance at $$\ell>2$$ lags $$\mbox{Cov}(y_t,y_{t-\ell})$$?

# Question 6

1 Point

What is the auto-correlation at one lag $$\mbox{Corr}(y_t,y_{t-1})$$?

# Question 7

0.5 Points

What is the autocorrelation at two lags $$\mbox{Corr}(y_t,y_{t-2})$$?

# Question 8

0.5 Points

What is the autocorrelation at $$\ell>2$$ lags $$\mbox{Corr}(y_t,y_{t-\ell})$$?

# A Random Walk

A random walk $$\{Z_t\}$$ is given by

\begin{align} Z_1 &= e_1 \\ Z_2 &= e_1 + e_2 \\ Z_3 &= e_1 + e_2 + e_3 \\ \vdots \end{align}

or in other words

\begin{align} Z_t = Z_{t-1} + e_t \end{align}

again with white noise $$e_t$$ such that $$\mathbb{E}(e_t) = 0$$ and $$Var(e_t) = 1 = \sigma^2_e$$,

# Question 9

1 Point

What is the unconditional mean (expectation) $$\mathbb{E}(Z_t)$$?

# Question 10

1 Point

What is the variance $$\mbox{Var}(Z_t)$$?

1. $$t$$
2. $$s$$
3. $$\sqrt{t/s}$$
4. $$\sqrt{s/t}$$
5. $$t^2$$
6. $$s^2$$

# Question 11

1 Point

What is the autocovariance between observation $$t$$ and observation $$s$$ $$\mbox{Cov}(Z_t,Z_{s}) = \gamma_{t,s} \quad \mbox{for} \quad 0\leq t\leq s$$?

1. $$t$$
2. $$s$$
3. $$\sqrt{t/s}$$
4. $$\sqrt{s/t}$$
5. $$t^2$$
6. $$s^2$$

# Question 12

1 Point

What is the autocorrelation $$\mbox{Cor}(Z_t,Z_{s}) = \frac{\gamma_{t,s}}{\sqrt{\gamma_{t,t}\cdot\gamma_{s,s}}} \quad \mbox{for} \quad 0\leq t\leq s$$?

1. $$t$$
2. $$s$$
3. $$\sqrt{t/s}$$
4. $$\sqrt{s/t}$$
5. $$t^2$$
6. $$s^2$$