Caution
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Time Series Analysis
Oren Bochman
November 5, 2024
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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---
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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::: {#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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1. [X] 
2. [ ] 
3. [ ] 
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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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- [ ] 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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