Question

Deseasonalize the data by calculating the centered moving average. Plot the deseasonalized data on the graph created in (a). Calculate the seasonal index for each quarter. Write a short explanation of why the results make sense.

Period	Sales
Mar-01	26.84
Jun-01	26.95
Sep-01	32.21
Dec-01	27.30
Mar-02	26.87
Jun-02	29.82
Sep-02	32.34
Dec-02	29.28
Mar-03	30.07
Jun-03	32.40
Sep-03	35.41
Dec-03	32.15
Mar-04	31.84
Jun-04	35.08
Sep-04	37.80
Dec-04	33.46
Mar-05	33.85
Jun-05	36.73
Sep-05	38.09
Dec-05	32.98
Mar-06	32.93
Jun-06	35.61
Sep-06	39.76
Dec-06	35.32
Mar-07	36.61
Jun-07	39.95
Sep-07	42.69
Dec-07	37.94
Mar-08	38.76
Jun-08	41.57
Sep-08	45.94
Dec-08	42.25
Mar-09	43.75
Jun-09	48.29
Sep-09	54.43
Dec-09	48.11
Mar-10	51.11
Jun-10	53.33
Sep-10	59.71
Dec-10	51.81
Mar-11	52.00
Jun-11	53.71
Sep-11	56.72
Dec-11	49.36
Mar-12	49.30
Jun-12	54.63
Sep-12	55.71
Dec-12	50.43
Mar-13	51.99
Jun-13	56.97
Sep-13	58.61
Dec-13	53.02
Mar-14	57.75
Jun-14	59.87
Sep-14	60.80
Dec-14	56.28
Mar-15	57.67
Jun-15	63.86
Sep-15	66.57
Dec-15	62.44
Mar-16	64.57
Jun-16	71.73
Sep-16	74.31
Dec-16	67.86

Answer 1

Data is given: Quarter-wise for each year.

Hence, seasonality is present quarter wise.

To calculate trend, we eliminate seasonality.

By using moving averages, we have 4- quarter moving average (as that will remove the effect of seasonality from years) and we will be able to obtain trend.

Then, we take centerd 4-quarter moving average (by taking average of 2 consecutive obs in 4-moving average)

Then, we assume it is multiplicative series, so we take ratio of original series to trend to obtain seasonal component or detrended series.

Table:

Period	Sales	Trend - 4MA	Centered 4MA	Detrended
Mar-01	26.84
Jun-01	26.95	28.33
Sep-01	32.21	28.33	28.33	1.14
Dec-01	27.30	29.05	28.69	0.95
Mar-02	26.87	29.08	29.07	0.92
Jun-02	29.82	29.58	29.33	1.02
Sep-02	32.34	30.38	29.98	1.08
Dec-02	29.28	31.02	30.70	0.95
Mar-03	30.07	31.79	31.41	0.96
Jun-03	32.40	32.51	32.15	1.01
Sep-03	35.41	32.95	32.73	1.08
Dec-03	32.15	33.62	33.29	0.97
Mar-04	31.84	34.22	33.92	0.94
Jun-04	35.08	34.55	34.38	1.02
Sep-04	37.80	35.05	34.80	1.09
Dec-04	33.46	35.46	35.25	0.95
Mar-05	33.85	35.53	35.50	0.95
Jun-05	36.73	35.41	35.47	1.04
Sep-05	38.09	35.18	35.30	1.08
Dec-05	32.98	34.90	35.04	0.94
Mar-06	32.93	35.32	35.11	0.94
Jun-06	35.61	35.91	35.61	1.00
Sep-06	39.76	36.83	36.37	1.09
Dec-06	35.32	37.91	37.37	0.95
Mar-07	36.61	38.64	38.28	0.96
Jun-07	39.95	39.30	38.97	1.03
Sep-07	42.69	39.84	39.57	1.08
Dec-07	37.94	40.24	40.04	0.95
Mar-08	38.76	41.05	40.65	0.95
Jun-08	41.57	42.13	41.59	1.00
Sep-08	45.94	43.38	42.75	1.07
Dec-08	42.25	45.06	44.22	0.96
Mar-09	43.75	47.18	46.12	0.95
Jun-09	48.29	48.65	47.91	1.01
Sep-09	54.43	50.49	49.57	1.10
Dec-09	48.11	51.75	51.12	0.94
Mar-10	51.11	53.07	52.41	0.98
Jun-10	53.33	53.99	53.53	1.00
Sep-10	59.71	54.21	54.10	1.10
Dec-10	51.81	54.31	54.26	0.95
Mar-11	52.00	53.56	53.93	0.96
Jun-11	53.71	52.95	53.25	1.01
Sep-11	56.72	52.27	52.61	1.08
Dec-11	49.36	52.50	52.39	0.94
Mar-12	49.30	52.25	52.38	0.94
Jun-12	54.63	52.52	52.38	1.04
Sep-12	55.71	53.19	52.85	1.05
Dec-12	50.43	53.78	53.48	0.94
Mar-13	51.99	54.50	54.14	0.96
Jun-13	56.97	55.15	54.82	1.04
Sep-13	58.61	56.59	55.87	1.05
Dec-13	53.02	57.31	56.95	0.93
Mar-14	57.75	57.86	57.59	1.00
Jun-14	59.87	58.68	58.27	1.03
Sep-14	60.80	58.66	58.67	1.04
Dec-14	56.28	59.65	59.15	0.95
Mar-15	57.67	61.10	60.37	0.96
Jun-15	63.86	62.64	61.87	1.03
Sep-15	66.57	64.36	63.50	1.05
Dec-15	62.44	66.33	65.34	0.96
Mar-16	64.57	68.26	67.30	0.96
Jun-16	71.73	69.62	68.94	1.04
Sep-16	74.31
Dec-16	67.86

Plot for original Series:

Plot for only trend (deseasonalised series) - Plotting colume (Centered 4MA):

Detrended plot for seasonality (plotting Detrended):

Short Explanation:

As, the data is given Quarter wise and seasonality is a component within year or of the duration 0-1 years. Hence, removing it by taking moving averages gave us the true picture of Trend over the years (as it can be seen in 2nd plot) and also, after removing trend from the series in column Detrended, its plot (3rd one) shows the seasonality pattern without trend. It can be seen that sales go up for month of June and go down for the month of September & December confirming our criteria for taking period of seasonality as 4.

Both the graphs 2nd and 3rd combine to make the 1st one.

2nd graph clearly shows only trend (no seasonality) as it is a smooth line moving upwards meaning that Sales are increasing over the years.

Please rate my answer and comment for doubt. It took a lot of effort, thanks.