p) process, R code" /> 96 Quiz: Spectral representation of the AR(p) – Bayesian Statistics

96 Quiz: Spectral representation of the AR(p)

Time Series Analysis

The AR(p) process, its state-space representation, the characteristic polynomial, and the forecast function

Coursera

notes

Bayesian Statistics

Autoregressive Models

Time Series

Author

Oren Bochman

Published

November 5, 2024

Keywords

time series, stability, order of an AR process, characteristic lag polynomial, autocorrelation function, ACF, partial autocorrelation function, PACF, smoothing, State Space Model, ARMA process, ARIMA, moving average, AR(p) process, R code

Caution

Section omitted to comply with the Honor Code

---
date: 2024-11-05
title: "Quiz: Spectral representation of the AR($p$)"
subtitle: Time Series Analysis
description: "The AR($p$) process, its state-space representation, the characteristic polynomial, and the forecast function"
categories: 
  - Coursera 
  - notes
  - Bayesian Statistics
  - Autoregressive Models
  - Time Series
keywords: 
  - time series
  - stability
  - order of an AR process 
  - characteristic lag polynomial
  - autocorrelation function
  - ACF
  - partial autocorrelation function
  - PACF
  - smoothing
  - State Space Model
  - ARMA process
  - ARIMA
  - moving average
  - AR($p$) process  
  - R code
---

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Section omitted to comply with the Honor Code
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::: {#exr-1}
Which is the most likely spectral density for the following AR(2) process,

$Y_t = 0.8Y_{t-1} - 0.3Y_{t-2} + \epsilon_{t}, \quad \epsilon_{t} \sim \mathcal{N}(0, 0.5)$
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::: {.solution}

1. [X] ![](images/c4/f1_sd1.png)
2. [ ] ![](images/c4/f1_sd3.png)
3. [ ] ![](images/c4/f1_sd4.png)

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::: {#exr-2}
Consider an AR(4) with two pairs of complex valued reciprocal roots with the following moduli and periods:

  - One pair of complex valued reciprocal roots has modulus 0.6 and period 12 (frequency $\omega=2\pi/12$)
  - One pair of complex valued reciprocal roots has modulus 0.8 and period 8 (frequency $\omega=2\pi/8$)

Then, the spectral density of this process has the following features:
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::: {.solution}

- [ ] The spectral density has two modes, one at frequency $\omega=2\pi/12$ and another one at frequency $\omega=2\pi/8$ with the highest peak at $\omega=2\pi/12$.
- [X] The spectral density has two modes, one at frequency $\omega=2\pi/12$ and another one at frequency $\omega=2\pi/8$ with the highest peak at $\omega=2\pi/8$.
- [ ] The spectral density has only one mode at frequency $\omega=2\pi/8$.
- [ ] The spectral density has only one mode at frequency $\omega=2\pi/12$.


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