87 Homework Stationarity, the ACF and the PACF - M1L1HW1

Caution

Section omitted to comply with the Honor Code

--- 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 ---  ::::: {.content-visible unless-profile="HC"} ::: {.callout-caution} Section omitted to comply with the Honor Code ::: ::::: ::::: {.content-hidden unless-profile="HC"} ::: {#exr-1 } 1. $Y_t - Y_{t-1} = e_t - 0.8e_{t-1}$ How is this process written using backshift operator notation (B)? ::: ::: {.callout-tip .solution .column-screen-inset-right collapse="true"} ## 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 ::: ::: {#exr-2 } 2. Which of the following plots is the most likely to correspond to a realization of a stationary time series process? ::: ::: {.callout-tip .solution .column-screen-inset-right collapse="true"} ## Solution - [ ] ![](images/ts_plot_a.png) - [X] ![](images/ts_plot_b.png) - [ ] ![](images/ts_plot_c.png) ::: ::: {#exr-3 } 3. If $\{Y_t\}$ is a strongly stationary time series process with finite first and second moments, the following statements are true: ::: ::: {.callout-tip .solution .column-screen-inset-right collapse="true"} ## 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 ::: :::: {#exr-4 } 4. If $\{Y_t\}$ is weakly or second order stationary with finite first and second moments, the following statements are true: ::: ::: {.callout-tip .solution .column-screen-inset-right collapse="true"} ## 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 ::: ::: {#exr-5 } 5. Which of the following moving averages can be used to remove a period $d=8$ from a time series? ::: ::: {.callout-tip .solution .column-screen-inset-right collapse="true"} ## 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] $\frac{1}{8} y_{t-4} + \frac{1}{4} \left( y_{t-3} + y_{t-2} + y_{t-1} + y_t + y_{t+1} + y_{t+2} + y_{t+3} \right) + \frac{1}{8} y_{t+4}$ - [ ] $\frac{1}{8} \sum_{j=-8}^8 y_{t-k}$ - [ ] $\frac{1}{2} \left( y_{t-4} + y_{t-3} + y_{t-2} + y_{t-1} + y_t + y_{t+1} + y_{t+2} + y_{t+3} + y_{t+4} \right)$ ::: ::: {#exr-6 } 6. Which of the following moving averages can be used to remove a period $d=3$ from a time series? ::: ::: {.callout-tip .solution .column-screen-inset-right collapse="true"} ## 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 ::: :::::