Trend Origin Shifting

For the simplicity and ease of computation trends are usually fitted to annual data with the middle of the series as urging. At many times it may be necessary to change the origin of the trend equation to some other point in the series. For e.g., the annual trend values must be changed to monthly or quarterly values if we wish to study the seasonal or cyclical patterns.

The shifting of the trend origin is a simple process. For an arithmetic straight line we have to find out new y intercept the value of a & of b remains unchanged as the slope of the trend line is the same by the expression.

Y_t = a + B (X + k)

Where k is the number of time units shifted. If the origin is shifted for d in time k is positive if shifted backward in time k is negative.

Illustration:

You are given the trend equation
Y = 110+ 2X
Shifts the origin to 2009,

Solution:
We are required to shift the origin to 2009 4 years forward here k = 4 the required equation can be obtained as follows:

Y₁ = a + b (X + k)
= 110 +2 (X + 4) = 110 + 2 X + 8 = 118 + 2 X

You are given the trend equation
Y = 210 - 1.5 X
Shift the origin to 2004.

Solution:

Changing origin form 2009 to 2004 means going back by 5 years. Using the formula

Y₁ = a + b (X + k)

= 210 - 1.5 (X - 5) = 210 - 1.5 X + 7.5 = 217.5 - 1.5 X

The formula explained above can be expanded to cover parabolic trend equations.

Illustration:

You are given the following equation

Y = 126.55 + 1804(X + 0.5) + 1.786 (X+ 0.5)²

= 126.55 + 18.04 x + 9.02 + 1.786 (X² + x + 0.25)

= 126.55 + 18.04 X + 9.02 + 1.786 X² + 1.786 X + 0.4465

= 136.0165 + 19.826 X + 1.786 X²

It must be noted that in the parabolic trend equation C is constant and hence its value would remain unchanged.