Measurement of Seasonal Variation by Ratio to Trend Method Lecture 07

 Measurement of Seasonal Variation by Ratio to Trend Method 

Lecture 07

The ratio to trend values approach is similar to the ratio to moving average method. The method for calculating tend values (TC) is the only distinction. The trend values are calculated by using the least squares method. The ratio-to-trend method, which isolates the seasonal effect, is used to determine the following ratios:

TCSI / TC * 100

The steps of the method are summarised as follows:

1. Use the least-squares approach to compute the trend values.

2. Take the trend value out. By dividing the original data values by the associated trend values and multiplying these ratios by 100 in a multiplicative model, the trend is removed. The results produced in this manner are trend-free.

3. For each year, group the percentage data values from Step (ii) according to the months or quarters that apply. 

De Trending

The process of eliminating the impacts of the trend component from time series data that have been seen. Detrending is mostly used to identify seasonal or cyclical patterns efficiently.

Assume multiplicative model; the observed time series is given by

Yt = TCSI

SI = TCSI / TC * 100

Assume additive model

Yt = T + C + S + I

SI = T + C + S + I - T - C

The detrended time series is also known as the stationary time series.

There are two common methods used to detrend the time series data:

Detrend by the Ratio-to-Trend method

In ratio to the trend method, the observed time series (Yt = TCSI) is divided by trend estimated values (Yt^ = TC). 

i.e., 

SI = Yt / Y^t * 100

Detrend by Differencing

In this method a new data set is developed by the difference between itself and previous observations.

∆Y^t = Y^t - Y^t-1

Practice Question

The data given below is the average annual sale (00, 000, 000) of the shop.

Year	1990	1991	1992	1993	1994	1995	1996	1997	1998	1999
sale	8	13	18	14	13	18	24	20	18	24
Year	2000	2001	2002	2003	2004	2005	2006	2007	2008
Sale	27	20	28	32	29	28	30	37	34

Assume both multiplicative and additive models and detrend the data by the least squares method.

Solution

Assume a multiplicative model and use the ratio to the trend line.

year	Yt		Yt x t	t^2	Y^t = 39.52 + 1.3174 t	SI = Yt / Y^t * 100
1990	8	-9	-72	81	27.6625	28.92002
1991	13	-8	-104	64	28.98	44.85852
1992	18	-7	-126	49	30.2975	59.41084
1993	14	-6	-84	36	31.615	44.28278
1994	13	-5	-65	25	32.9325	39.47468
1995	18	-4	-72	16	34.25	52.55474
1996	24	-3	-72	9	35.5675	67.47733
1997	20	-2	-40	4	36.885	54.22258
1998	18	-1	-18	1	38.2025	47.11734
1999	24	0	0	0	39.52	60.72874
2000	27	1	27	1	40.8375	66.1157
2001	20	2	40	4	42.155	47.44396
2002	28	3	84	9	43.4725	64.40853
2003	32	4	128	16	44.79	71.44452
2004	29	5	145	25	46.1075	62.89649
2005	28	6	168	36	47.425	59.04059
2006	30	7	210	49	48.7425	61.54793
2007	37	8	296	64	50.06	73.91131
2008	34	9	306	81	51.3775	66.17683
	435		751	570

Estimated trend line

Assume additive model and using differencing method

Year	Yt	SI = Y^t - Y^t-1
1990	8
1991	13	5
1992	18	5
1993	14	-4
1994	13	-1
1995	18	5
1996	24	6
1997	20	-4
1998	18	-2
1999	24	-6
2000	27	3
2001	20	-7
2002	28	8
2003	32	4
2004	29	-3
2005	28	-1
2006	30	2
2007	37	7
2008	34	-3

Practice Question

The profit of a grocery store is given below:

Year	Q 1	Q 2	Q 3	Q 4
2010	122	125	118	117
2011	119	114	114	109
2012	105	99	93	89
2013	86	80	83	84

 Compute seasonal indices by using the ratio-to-trend method.

Solution:

Year	Quarter	Y = TCSI	t	Y x t	t^2	Y^t = TC	SI
2010	1	122	-15	-1830	225	127.5736	95.63107
	2	125	-13	-1625	169	124.3721	100.5048
	3	118	-11	-1298	121	121.1706	97.38333
	4	117	-9	-1053	81	117.9692	99.17846
2011	1	119	-7	-833	49	114.7677	103.6877
	2	114	-5	-570	25	111.5662	102.1815
	3	114	-3	-342	9	108.3647	105.2003
	4	109	-1	-109	1	105.1632	103.6484
2012	1	105	1	105	1	101.9618	102.9798
	2	99	3	297	9	98.76028	100.2427
	3	93	5	465	25	95.5588	97.32228
	4	89	7	623	49	92.35732	96.36486
2013	1	86	9	774	81	89.15584	96.46031
	2	80	11	880	121	85.95436	93.07265
	3	83	13	1079	169	82.75288	100.2986
	4	84	15	1260	225	79.5514	105.5921
		1657		-2177	1360

Estimated Trend Line:

The average seasonal indices are:

Year	Q1	Q2	Q3	Q4
2010	95.63107	100.5048	97.38333	99.17846
2011	103.6877	102.1815	105.2003	103.6484
2012	102.9798	100.2427	97.32228	96.36486
2013	96.46031	93.07265	100.2986	105.5921
Total	398.7589	396.0017	400.2045	404.7838
Mean	99.68972	99.00043	100.0511	101.196	399.9372
SI	99.70537	99.01596	100.0668	101.2118

 The SI row is obtained by multiplying the mean row by CF:

Alternative way to compute seasonal indices:

Compute the average of trend values (TC):

	Q1	Q2	Q3	Q4
2010	127.5736	124.3721	121.1706	117.9692
2011	114.7677	111.5662	108.3647	105.1632
2012	101.9618	98.76028	95.5588	92.35732
2013	89.15584	85.95436	82.75288	79.5514
Total	433.4589	420.653	407.847	395.0411
Mean	108.3647	105.1632	101.9618	98.76028	414.25
Corrected Mean	104.637	101.5457	98.45432	95.36297	400

The corrected mean is obtained as:

Corrected Mean = Mean x CF

The seasonal indices for each year are obtained as:

SI for 2010:

116.5935

123.0973

119.8525

122.6891

 

SI for 2011:

113.7265

112.2648

115.7897

114.3001