Motel $5$ is interested in developing a model for estimating sales revenues. The file motel $5$ data.xlsx Download motel $5$ data.xlsxcontains sales revenue information for the motel along with the time period and advertising expenditures. $($Prior to this time frame, Motel $5$ spent $15($in $$1000$s$)$ on advertising in each quarter of $2019.)$ Note that there are two variables for advertising $$ Motel $5$ believes that advertising in this quarter can affect both this period$$s sales revenues as well as next period$$s sales revenues. Therefore, there are two variables representing advertising $($advertising and lagged advertising$).$ Use an alpha of $\mathrm{.}05$ if needed.

a$)(10$ points$)$ Run a multiple regression analysis to estimate sales based on advertising, advertising lagged by one period, and the linear trend indicated by the coded time period.

Interpret the slope for advertising.

Assuming that advertising for the quarters in $2025$ are $50,60,60,$ and $45($in $$1000$s$),$ predict sales revenues for those quarters.

b$)(10$ points$)$ Run a multiple regression analysis to estimate sales based on advertising, linear trend indicated by the coded time period, and the quarter category.

Interpret the slope for any one of the quarters.

Assuming that advertising for the quarters in $2025$ are $50,60,60,$ and $45($in $$1000$s$),$ predict sales revenues for those quarters.

c$)(4$ points$)$ Use Holts$$ method to forecast for the next four quarters. Optimize the smoothing constants.

d$)(4$ points$)$ Use Winters$$ method to forecast for the next four quarters. Optimize the smoothing constants.

e$)(5\mathrm{.}33$ points$)$ Which of the four forecasting methods considered would you recommend? Justify your recommendation quantitatively.Motel $5$ data

Note: All monetary values are in $$1000\text{'}$s

$\backslash$table$[[$Quarter$,$Year,Sales,Advertising$],[$Q$1,2020,1800,40],[$Q$2,2020,1320,30],[$Q$3,2020,2700,25],[$Q$4,2020,1575,50],[$Q$1,2021,2550,20],[$Q$2,2021,525,60],[$Q$3,2021,3750,15],[$Q$4,2021,1140,50],[$Q$1,2022,3450,30],[$Q$2,2022,1500,20],[$Q$3,2022,2355,70],[$Q$4,2022,3645,15],[$Q$1,2023,1980,45],[$Q$2,2023,2100,25],[$Q$3,2023,2835,80],[$Q$4,2023,4800,35],[$Q$1,2024,3300,40],[$Q$2,2024,2160,70],[$Q$3,2024,6000,90],[$Q$4,2024,6150,60]]$