Statistical methods:  Chain Fisher Volume Index

Chain Fisher Volume Index

Statistics Canada: Chain Fisher Volume Index - Formulas

The Laspeyres volume index

The best know volume index is Laspeyres. For this index, weighting is done using prices from a predetermined base year:

(1)

or, the chained version:

(2)

Where:

LV is the Laspeyres volume index for the period t using the base period 0;
p are the price series;
q are the quantity series.

By using the identity C=pq (value equals price multiplied by quantity), formula (2) can be expressed in a more usable form (chained version):

(3)		(4)
hence:	(5)

The Paasche volume index

As well as referring to a point in time in the past, a volume index can be based on prices from the current period. This is known as a Paasche index:

(6)

and the chained version:

(7)

By performing the same substitutions that we did with the Laspeyres index, we get:

(8)		(9)
hence:	(10)

The Fisher volume index

The Fisher volume index is the geometric mean of the Laspeyres and Paasche indexes:

(11)   

The Fisher index carries the property of factorization, that is, the Fisher price index multiplied by the Fisher volume index is equal to the value index:

(12)   

The contribution to percentage change formula

The contribution to percentage change formula which is being used permits the calculation of the contribution of each series to the percentage change of an aggregate series. Unlike the contribution series resulting from ad hoc calculations on constant dollar series, the contributions generated from this formula are fully additive. More specifically, the contribution of a component i to the percentage growth of an aggregate ( ) at time t is written:

(13)   

or, in a more usable format:

(14)   

Where:

FV_t is the Fisher volume index for the period t and the reference period t-1;
C_t is the current dollar series for the period t;
p_t and p_t-1 are the prices for the period t and t-1 respectively.

The j index sums represent the most detailed components of the aggregates.