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

Deadline: Tuesday, November 7, 2017

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. \(s^2\)