Miscellaneous Topics - Time series, Business Mathematics and Statistics | SSC CGL Tier 2 - Study Material, Online Tests, Previous Year PDF Download

Seasonal variation indices are calculated as below :
Seasonal variation index =
 

Meaning of “Normal” in Business Statistics
Business is often said to be “above normal” or “below normal”. When so used the term “normal” is generally recognized to mean a level of activity which is characterized by the presence of basic trend and seasonal variation. This implies that the influence of business cycles and erratic fluctuations on the level of activity is assumed to be insignificant. Therefore, the product of trend value for any period when adjusted by the seasonal index for that period gives us an estimate of the normal activity during that period.
 

Measuring Cycle as the residual
Business cyclical variations are measured either as the difference between the observed value and the “normal”. Whatever remains after elimination of secular trend and seasonal variations from the time series, is said to be composed of cyclical variations and Irregular movements.
 

Second degree Parabola
The simplest form of the non-linear trend is the second degree parabola. It is used to find long term trend. We use the following equation for finding second degree trend –
Yc = a + bX + cX²
To know the value of a, b and c we use the following three normal equations –
∑Y = Na + b∑X + c∑X²
∑XY = a∑X + b∑X² + c∑X³
∑X2Y = a∑X2 + b∑X³ + c∑X⁴
A second degree trend equation is apporpriate for the secular trend component of a time series when the data do not fall in a straight line.

Illustration: Fit a parabola (Yc = a + bX + cX²) from the following
Years   1   2  3   4   5   6  7
Values 35 38 40 42 36 39 45

– 84c = – 4
c = 4/84 = 0.05
By substituting the value of c in equation (i) we get the value of a
7a + 28 × 4/48 = 275
7a = 275 – 1.33
a = 273.67/7 = 39.09

We may get the value of b with the help of equation (ii)
28b = 28
b = 1
The required equation would be:
Yc = 39.09 + 1X + 0.05 X²
= 39.09 + X + 0.05 X²
With the help of above equation we can estimate the value for year 8 where x = 4
Yc = 39.09 + 4 + 0.05 (4)²
= 39.09 + 4 + 0.8 = 43.89
 

Exponential Trend
The equation for exponential trend is of the form: y = abx
Taking log of both sides we get log y = log a + x log b
To get the value of a and b we have normal equation
∑logy = Nlog a + logb ∑X
∑(x. log y) = log a∑x + log b∑X²
When we slove these equations we get –
log a = and log b =

Illustration : The production of certain raw material by a company in lakh tons for the years 1996 to 2002 are given below:
Year :          1996 1997 1998 1999 2000 2001 2002
Production : 32      47    65    92    132  190   275
Estimate Production figure for the year 2003 using an equation of the form y = ab1 where x = years and y = production

Solution :

log y = 1.9704 + .154 x
for 2003, x would be 4 and log y will be
log y = 1.9704 + .154(4) = 2.5864
y = AL 2.5864 = 385.9
Thus estimated production for 2003 would be 385.9 lakh tons.