the revenue data table: 2008_Quarter 2008_Revenue 2009_Quarter 2009_Revenue 2010
ID: 3222607 • Letter: T
Question
the revenue data table:
2008_Quarter 2008_Revenue 2009_Quarter 2009_Revenue 2010_Quarter 2010_Revenue 1 539 1 433 1 350 2 514 2 408 2 294 3 491 3 398 3 296 4 501 4 458 4 336 The accompanying data show the advertising revenue, in millions of dollars, for each quarter over a three-year period for a particular company. Use these data to complete parts a through e below Click the icon to view the advertising revenue data table b. Forecast the advertising revenue for each quarter in 2011 using multiplicative decomposition. The revenue forecast for the first quarter in 2011 is million. (Round to one decimal place as needed.) The revenue forecast for the second quarter in 2011 is million (Round to one decimal place as needed.) The revenue forecast for the third quarter in 2011 is million. (Round to one decimal place as needed.) The revenue forecast for the fourth quarter in 2011 is million (Round to one decimal place as needed.) c. Interpret the meaning of the seasonal components. Quarter Y has the highest revenue, followed by Quarters Y and Y n that order. The quarter with the highest revenue has more revenue than an average quarter during the year (Round to the nearest integer as needed.) d. Calculate the MAD for this forecast.Explanation / Answer
In a multiplicative model, we can express a time series by the multiplication of seasonality, trend and random components.
From the given quarterly data, let us find out the different components first. I have used R for better understanding.But the calculation can also be carried out in hand.
We used decompose function in R to divide each element into the data in three components. The formula to find the trend is
Trend(X(t)) =X(t-2)/8 + X(t-1)/4 + X(t)/4 + X(t+1)/4 + X(t+2)/4
After that, we will devide the Trend from original series and from their will find seasonality.
Below is the R code......................
library(forecast)
x=c(539,514,491,501,433,408,398,458,350,294,296,336)
tx=ts(x,start=c(2008,1),frequency = 4)
tx
plot(tx)
decompose_x = decompose(tx, "multiplicative")
plot(decompose_x)
decompose_x$seasonal
decompose_x$trend
decompose_x$random
##Modelling the Trend Part with Linear Regression#####
t=c(3:10)
x=c(498.000,471.500,446.625,429.625,413.875,389.250,362.250,334.250)
data1=data.frame(t,x)
model=lm(x~t,data=data1)
summary(model)
t_12=data.frame(c(13:16))
names(t_12)=c('t')
out=predict(model,newdata=t_12)
#### Multiply the Trend part with Quaterly Seasonality ######
seasonal=c(0.9737175,0.9201791,0.9797086,1.1263948)
output=out*seasonal
So, the answer of part b will be 265.5003 , 230.3036 , 223.2715, 231.4859
c.
Q1 has the highest revenue followed by Q4, Q2, and Q3
d.
MAD=mean(abs(output-mean(output)))=13.93
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