Use a $5^{th}-$order moving average smoother to smooth the following. In each case plot the smoothed data as an overlay on the original data:

$($a$)$ bitcoin

$($b$)$ wtcrude $2020$

Using the tswge ts object dfw $.$ mon, do the following:

$($a$)$ Use the Base $R$ window function to create a ts object consisting of the monthly temperatures from January $2000$ through December $2021.$ Name the file dfw$.$mon. $2000.$

$($b$)$ Apply a $5^{th}-$order moving average smoother to the data in dfw$.$ mon. $2000.$ Plot the monthly data and overlay the smoothed data. Describe the appearance of the smoothed data.

$($c$)$ Apply an additive decomposition to dfw$.$mon. $2000$ using the techniques in Section $2\mathrm{.}1\mathrm{.}2\mathrm{.}1$ and plot the four plots as in Figure $2\mathrm{.}10.$

$($d$)$ Base $R$ has a function called decompose that performs an additive or multiplicative decompostion. Issue the following commands:

dfw$.$ decomp$=$decompose $($dfw$.$ mon $.2000,$ type$=\text{'}$ additive $\text{'})$

plot $($dfw$.$ decomp$)$

Compare the plots with those you produced in $($c$).$

Consider the following three tswge datasets.

$(1)$ us $.$ retail: quarterly US retail sales $($in $millions$)$ from the fourth quarter of $1999$ through the second quarter of $2021.$

$(2)$ NSA: Monthly Total Vehicle Sales $($NSA$)$ in the United States from January $1976$ through July $2021.$

$(3)$ NAICS: Monthly US total sales in retail trade and food services in billions of dollars from Jan $2000$ through December $2019.$

Exploring Time Series Data $73$

For each of these datasets:

$($a$)$ Plot the data.

$($b$)$ Discuss behavior of the time series.

$($c$)$ Decompose the data $($additive or multiplicative $-$ you decide$).$

$($d$)$ These data are not seasonally adjusted. Seasonally adjust the data using either $(2\mathrm{.}8)$ or $(2\mathrm{.}9)$ and plot your results.

Answer $1,2$ and $4$ a$-$d