Hi k9queen!
These are the answers to your questions.
a) We know that the equation for the trend line of quarterly demand
for Jaguar is the following:
Y = 10 + 3X
where X indicates the index of each quarter. I'm assuming here, in
order for the rest of the question to make sense, that this trend line
was calculated over deseasonalized data. Thus this trend line implies
that in the first quarter of the last year (X=0), deseasonalized
demand was approximately 10+3*0=10; in the 2nd quarter (X=1),
deseasonalized demand was approximately 10+3*1=13, and so on.
In order to find the deseasonalized demand for the next year, we must
first know the the numbers that correspond to each of next year's
quarters. This is easy to do:
This year, 1st quarter: 4
This year, 2nd quarter: 5
This year, 3rd quarter: 6
This year, 4th quarter: 7
Next year, 1st quarter: 8
Next year, 2nd quarter: 9
Next year, 3rd quarter: 10
Next year, 4th quarter: 11
So the numbers are 8, 9, 10 and 11. Therefore, following the trend
line, deseasonalized demand for each quarter of the next year is
simply:
Demand next year, 1st quarter: 10 + 3*8 = 34
Demand next year, 2nd quarter: 10 + 3*9 = 37
Demand next year, 3rd quarter: 10 + 3*10 = 40
Demand next year, 4th quarter: 10 + 3*11 = 43
As you can see, the fact that the equation is 10+3X implies that
deseasonalized demand increases by 3 each quarter.
We must know seasonalize the data. Seasonal indices show the effect
the time of the year has on sales. So for example, we can see that in
the 1st quarter of each year, sales are usually substantially lower
than sales in the 3rd quarter of each year. In order to seasonalize
the data, we must simply multiply each trend forecast by its seasonal
index. So we get:
Demand next year, 1st quarter: 34*0.8 = 27.2
Demand next year, 2nd quarter: 37*1.0 = 37
Demand next year, 3rd quarter: 40*1.3 = 52
Demand next year, 4th quarter: 43*0.9 = 38.7
Note how despite the fact that the trend line indicates that demand is
always increasing (by 3 each quarter), we have that in the 4th quarter
sales are lower then in the 3rd one. This is so because of the
seasonal effect: apparently, for some reason, sales in the 3rd quarter
are higher than normal, and in the 4th quarter they are lower than
normal.
b) There are many techniques that can be used in order to
deseasonalize the data, some of them quite complicated. An easy way to
do it is using centered moving averages (CMA). The procedure can be
found from slide 13 in the following Powerpoint presentation
(right-click on the link and choose "Save Target as...").
Forecasting
http://www.rpi.edu/~hollid/classes/statom2/Lecture03-Forecasting--Part%202.ppt
The idea is the following: compute the CMA for each quarter and
compare the actual value for demand each quarter with this average.
Divide each actual value by its corresponding centered moving average.
The point of doing this is that you're comparing how different is the
current quarter from the ones "around" it (through the CMA). If the
actual value is higher than the CMA, it means that there is a positive
seasonal effect in the current quarter. For example, mall sales tend
to be substantially higher in the 4th quarter because of Christmas
shopping. You would see that the actual value for sales in the 4th
quarter is higher than its corresponding CMA, because the CMA takes
the average among months that didn't have the Christmas effect.
So, back to the procedure. After dividing each value by its
corresponding CMA you will have several indices. Finally, you define
the seasonal index of quarter 1 as the average among the indices you
got for all 1st quarters; the seasonal index of quarter 2 as the
average among the indices you got for all 2nd quarters, etc. These are
the seasonal indices. A seasonal index greater than 1 implies that
demand that quarter is higher than "normal", while an index smaller
than 1 implies the opposite. Please also refer to a previous question
I answered for you.
Quantitiative Stats
http://answers.google.com/answers/threadview?id=271030
(note that in this question, seasonal indices are transformed to
percentages. The idea is the same, we're only using different units. A
seasonal index of 0.8 would correspond to an index of 80 in that
question)
c) The forecast for the next year was already computed in question
(a). Please let me know through a clarification request if you're
interested in computing the forecast for another year (though I'm sure
you could do it yourself!).
Google search strategy
seasonal adjustment "centered moving average"
://www.google.com.ar/search?hl=es&ie=UTF-8&oe=UTF-8&q=seasonal+adjustment+%22centered+moving+average%22&meta=
seasonal adjustment
://www.google.com.ar/search?hl=es&ie=UTF-8&oe=UTF-8&q=seasonal+adjustment&meta=
I hope this helps! If you have any doubt regarding my answer, please
don't hesitate to request a clarification.
Best wishes!
elmarto |
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