Question

We consider data in the following table, summarizing sales of a product (in thousands). For each of the questions, justify your answer.

Year	Quarter	Sales
2015	1 2 3 4	4.95 4.25 6.15 6.65
2016	1 2 3 4	5.85 5.35 6.95 7.55
2017	1 2 3 4	6.15 5.75 7.65 7.95

(3.1) Assuming the given time-series shows evidence of seasonality, Determine the estimate sales values. (5)

Answer 1

Here we will be considering the average method to obtain the estimate sales values,

To estimate the seasonal relatives, we are going to do it by averaging the demands each period, and dividing by the overall average.

Periods	2015	2016	2017	Average
Quarter 1	4.95	5.85	6.15	5.65
Quarter 2	4.25	5.35	5.75	5.1167
Quarter 3	6.15	6.95	7.65	6.9167
Quarter 4	6.65	7.55	7.95	7.3833
OVERALL AVERAGE				6.267

So we have found what the average demand is for Quarter 1 of a year, for Quarter 2, etc. If we divide these averages by the overall average, we get the following seasonal indices:

Periods	Quarter Average	Over-All Average	Seasonality Index
Quarter 1	5.65	6.267	= (5.65/6.267) = 0.9016
Quarter 2	5.1167	6.267	= (5.1167/6.267) = 0.8165
Quarter 3	6.9167	6.267	= (6.9167/6.267) = 1.1037
Quarter 4	7.3833	6.267	= (7.3833/6.267) = 1.1782

Now, to estimate the sales, we will de-seasonalize the individual sales numbers to see how sales per period go up or down.

The de-seasonalizing method uses the seasonal relatives we already created to try to get a picture of how much we’ve been growing over time. What is deseasonalizing? After we make a straight-line forecast of the future, we are going to multiply it by the seasonal indices to get a seasonalized forecast of the future. Deseasonalizing is basically the opposite: we are going to take the actual, seasonal data, and divide it by the seasonal factors to get something that looks more like a straight line. Then we are going to do a linear regression through this line. The deseasonalized demands should be a lot more like a straight line than the original data is, that is, it should generally show a more consistent 3 growth rate than we see with seasonality in it. When we do a linear regression through these deseasonalized points, the linear regression should give us a pretty good fit through the points.

Quarters	Seasonal Index	Deseasonal Index	Final Value
Quarter 1 - 2015	4.95	0.9016	= (4.95/0.9016) = 5.49
Quarter 2 - 2015	4.25	0.8165	= (4.25/0.8165) = 5.21
Quarter 3 - 2015	6.15	1.1037	= (6.15/1.1037) = 5.57
Quarter 4 - 2015	6.65	1.1782	= (6.65/1.1782) = 5.64
Quarter 1 - 2016	5.85	0.9016	= (5.85/0.9016) = 6.49
Quarter 2 - 2016	5.35	0.8165	= (5.35/0.8165) = 6.55
Quarter 3 - 2016	6.95	1.1037	= (6.95/1.1037) = 6.30
Quarter 4 - 2016	7.55	1.1782	= (7.55/1.1782) = 6.41
Quarter 1 - 2017	6.15	0.9016	= (6.15/0.9016) = 6.82
Quarter 2 - 2017	5.75	0.8165	= (5.75/0.8165) = 7.04
Quarter 3 - 2017	7.65	1.1037	= (7.65/1.1037) = 6.93
Quarter 4 - 2017	7.95	1.1782	= (7.95/1.1782) = 6.75

Now, we do a linear regression through these deseasonalized numbers :

We get an intercept of 5.24 and a slope of 0.16, and an R² value of 0.82.

Thus, multiplying the linear forecast we made by the seasonal factors to get a seasonalized estimated forecasts are:

Time Period	Estimated Trend	Estimated Forecast
1	5.400417291	4.86901623
2	5.557916104	4.538038499
3	5.715414918	6.308103445
4	5.872913731	6.919466958
5	6.030412545	5.43701995
6	6.187911358	5.052429624
7	6.345410172	7.003429207
8	6.502908985	7.661727366
9	6.660407799	6.005023671
10	6.817906612	5.566820749
11	6.975405426	7.698754968
12	7.132904239	8.403987774