Trend Series Analysis II

(a) Simple Average Method: Under this method, the average (

) of all the monthly or quarterly values for each year are found out. Each monthly or quarterly value (y_i) is divided by the corresponding average and the results are expressed as percentage:

Then the mean index or seasonal index (S_i) is calculated for each month or quarter. If the mean of all seasonal indices is not equal to 100, then they will be adjusted.

Take data from Example 1, and calculate the four seasonal indices by the ‘simple-average method’.

Now the above observed values are converted to indices using the following formula:

Year/Quarter	I	II	III	IV	Total
2003	50.52%	82.35%	148.79%	118.34%
2004	88.55	103.23	113.06	95.16
2005	119.78	71.83	99.00	109.39
Season Index (S_i)	86.28%	85.80%	120.28%	107.63%	400.00

(b) Link Relative Method: Under this method, the data for each month or quarter are expressed in percentage, known as ‘Link Relatives’. An appropriate average of the link relatives is taken, usually a median is taken. Convert these averages into a series of chain indices. The chain indices are adjusted for the fraction of the effect of the trend. The adjusted chain indices are further reduced to the same level as the first month or quarter.

Take the data from Example 1, and calculate seasonal indices by using ‘link relative method’.

The observed values are converted into price relatives or link relatives using the following formula:

and then the link relatives are converted into chain indices (chaining process) using the following formula:

Year	Quarter				Total
Year	I	II	III	IV	Total
2003	–	163.01%	180.67%	79.53%
2004	107.02%	116.58	109.53	84.17
2005	111.36	59.97	137.82	110.50
Median (link relative)	109.19	116.58	137.82	84.17
Chain Index	100	116.58	160.67	135.24	512.49
Adj. C. I.	78.05%^*	90.99%	125.40%	105.56%	400.00

^* Adjusted chain index for QI: 100 ÷ 512.49 × 400 = 78.05, and so for other quarters.

(c) Ratio to Moving Average Method: A 12-month or 4-quarter moving average centred is computed. The observed values are divided by the corresponding centred moving average and the results are expressed in percentage:

The monthly or quarterly averages of these percentages are found out. The adjusted values are the indices of the seasonal variations.

Take data from Example 1 and calculate seasonal indices using ‘ratio to moving average method’.

Quarter		y	4-Quarter Moving Total	8-Quarter Moving Total	4-Quarter Moving Average (Centred)	Ratio to Moving Average
						I	II	III	IV
2003	I	219				–
	II	357					–
			1734
	III	645		3798	475			135.8
			2064
	IV	513		4411	551				93.1
			2347
2004	I	549		4750	594	92.4
			2403
	II	640		4883	610		104.9
			2480
	III	701		5068	634			110.6
			2588
	IV	590		4930	616				95.8
			2342
2005	I	657		4526	566	116.1
			2184
	II	394		4378	547		72.0
			2194
	III	543						–

	IV	600							–

Mean Seasonal Index (total = 410.5)						104.3	88.5	123.2	94.5
Adjusted Seasonal Index (S_i) (total = 400)						101.6%^*	86.2%	120.1%	92.1%

^* Adjusted seasonal index for QI: 104.3 ÷ 410.5 × 400, and so on for other three quarters.

(d) Ratio to Trend Method: An average for each year is found out and a straight line is fitted by least squares method. The trend values for each month or quarter are calculated on the assumption that the data correspond to the middle of the month or quarter. Each original value is divided by the corresponding calculated trend values and expressed in percentage. A mean of these percentages are calculated for each month or quarter. The adjusted values are the indices of seasonal variation.

Take data from Example 1 and calculate seasonal indices using ‘ratio to trend method’.

Quarters		y	x		x		x²
2003	I	219	–11					455	48.13%
	II	357	–9					469	76.12
				433.5	–8	–3468	64
	III	645	–7					484	133.26

	IV	513	–5					498	103.01

2004	I	549	–3					512	107.23

	II	640	–1					527	121.44
				620	0	0	0
	III	701	1					541	129.57

	IV	590	3					556	106.12

2005	I	657	5					570	115.26

	II	394	7					584	67.47
				548.5	8	4388	64
	III	543	9					599	90.65

	IV	600	11					613	97.88
			0	1602	0	920	128

Year/Quarter	I	II	III	IV	Total
2003	48.13	76.12	133.26	103.01
2004	107.23	121.44	129.57	106.12
2005	115.26	67.47	90.65	97.88
Seasonal Index (S_i)	90.21	88.34	117.83	102.34	398.72
Adj. S.I.	90.50%^*	88.62%	118.21%	102.67%	400.00

The cyclical and random components of a time series are first isolated from the time series using the multiplicative model:

The Random component R_i will now be separated from the time series by using the smoothing technique, moving average. These moving averages show the indices of cyclical variation.

Quarters		y	^*	**S_i^****		C_i × Ri = ×100	3-Quarter Moving Average *(C_i)*
2003	I	219	455	86.28	392.57	55.79%	–
	II	357	469	85.80	402.40	88.72	85.10
	III	645	484	120.28	582.16	110.79	98.41
	IV	513	498	107.63	536.00	95.71	110.26
2004	I	549	512	86.28	441.75	124.28	120.51
	II	640	527	85.80	452.17	141.54	124.52
	III	701	541	120.28	650.71	107.73	115.95
	IV	590	556	107.63	598.42	98.59	113.30
2005	I	657	570	86.28	491.80	133.59	103.60
	II	394	584	85.80	501.07	78.63	95.86
	III	543	599	120.28	720.48	75.37	81.74
	IV	600	613	107.63	657.77	91.22	–

Year/Quarter	y				Mean
Year/Quarter	I	II	III	IV	Mean
2003	219	357	645	513	433.5
2004	549	640	701	590	620
2005	657	394	543	600	548.5