Caution
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Time Series Analysis
Oren Bochman
October 23, 2024
time series, stationarity, strong stationarity, weak stationarity, lag, autocorrelation function (ACF), partial autocorrelation function (PACF), smoothing, trend, seasonality, differencing operator, back shift operator, moving average
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---
date: 2024-10-23
title: "Homework Stationarity, the ACF and the PACF - M1L1HW1"
subtitle: Time Series Analysis
description: "This lesson we will define the AR(1) process, Stationarity, ACF, PACF, differencing, smoothing"
categories:
- Bayesian Statistics
keywords:
- time series
- stationarity
- strong stationarity
- weak stationarity
- lag
- autocorrelation function (ACF)
- partial autocorrelation function (PACF)
- smoothing
- trend
- seasonality
- differencing operator
- back shift operator
- moving average
---
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::: {#exr-1 }
1. $Y_t - Y_{t-1} = e_t - 0.8e_{t-1}$ How is this process written using backshift operator notation (B)?
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## Solution
- [ ] $B Y_t = (1-0.8B)e_t$
- [ ] $B(Y_{t} - Y_{t-1}) = 0.8Be_t$
- [X] $(1-B)Y_t = (1-0.8B)e_t$
- [ ] None of the above
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::: {#exr-2 }
2. Which of the following plots is the most likely to correspond to a realization of a stationary time series process?
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## Solution
- [ ] 
- [X] 
- [ ] 
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3. If $\{Y_t\}$ is a strongly stationary time series process with finite first and second moments, the following statements are true:
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## Solution
- [ ] {Y_t} is weakly or second order stationary
- [X] The variance of $Y_t$, $\mathbb{V}ar(Y_t)$, changes over time
- [ ] The expected value of $Y_t$, $\mathbb{E}(Y_t)$, does not depend on $t$
- [X] $\{Y_t\}$ is a Gaussian process
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4. If $\{Y_t\}$ is weakly or second order stationary with finite first and second moments, the following statements are true:
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## Solution
- [ ] $\{Y_t\}$ is also strongly stationary
- [ ] None of the above
- [x] If $\{Y_t\}$ is a Gaussian process then $\{Y_t\}$ is strongly stationary
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5. Which of the following moving averages can be used to remove a period $d=8$ from a time series?
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## Solution
- [ ] $\frac{1}{8} \left( y_{t-4} + y_{t-3} + y_{t-2} + y_{t-1} + y_t + y_{t+1} + y_{t+2} + y_{t+3} + y_{t+4} \right)$
- [x] $\frac{1}{8} y_{t-4} + \frac{1}{4} \left( y_{t-3} + y_{t-2} + y_{t-1} + y_t + y_{t+1} + y_{t+2} + y_{t+3} \right) + \frac{1}{8} y_{t+4}$
- [ ] $\frac{1}{8} \sum_{j=-8}^8 y_{t-k}$
- [ ] $\frac{1}{2} \left( y_{t-4} + y_{t-3} + y_{t-2} + y_{t-1} + y_t + y_{t+1} + y_{t+2} + y_{t+3} + y_{t+4} \right)$
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6. Which of the following moving averages can be used to remove a period $d=3$ from a time series?
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## Solution
- [ ] $\frac{1}{2} (y_{t-1} + y_t + y_{t+1})$
- [ ] $(y_{t-3} + y_{t-2} + y_{t-1} + y_t + y_{t+1} + y_{t+2} + y_{t+3})$
- [x] $\frac{1}{3}(y_{t-1} + y_{t} + y_{t+1})$
- [ ] None of the above
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