# 10.5 Calculating volume aggregates with the chain Fisher formula for investment in inventories

10.69 Like other expenditure-based gross domestic product aggregates, real investment in inventories is estimated by means of the chain Fisher formula. However, this formula cannot be used directly for investment in inventories aggregates because the series can take positive or negative values. To avoid the problems created by sign changes, the estimates are produced using inventory levels, which always take a positive value. Essentially, the indirect approach to estimating real investment in inventories involves estimating real inventory levels at the start and end of the period using the Fisher formula and subtracting one from the other.

## Calculation formulas

10.70 The Fisher formula used to compute non-chained volume indexes^{1} is

**Equation 10.1**

10.71 In practice, this formula is not useful because the quantities
would have to be observed directly. Consequently, the formula is transformed
to include only current dollars (C_{t} and C_{t−1})
and prices (P_{t} and P_{t−1}), two kinds of data
that are available to statisticians. Equation 10.1 becomes

**Equation 10.2**

10.72 Equation 10.2 produces a non-chained volume index. To obtain estimates
in chained dollars, the non-chained indexes must be added together from
a particular starting point (CFV_{0}). For us, the starting point
is 2002, the reference period for which the real and nominal estimates
of the aggregates are equal. The formula used is:

**Equation 10.3**

*CFV _{t} = CFV_{0} × FV_{1} ×
FV_{2} ... × FV_{t}*

10.73 For example, to calculate real personal expenditure on consumer goods and services in millions of chained (2002) dollars, the current value and the price index for each of the 130 categories of goods and services that make up the aggregate for all periods since 2002 are used.

### Using the formula for investment in inventories

10.74 In the case of investment in inventories, the non-chained Fisher
index (FV_{t}) cannot be used because of sign changes. It is
impossible to take the square root of a negative number. The solution
is to compute two "real" estimates using the chain Fisher
formula for each period (t), one for the inventory level at the end (*e*)
of period *t* (e.g., March 31, 2005) and the other for the inventory
level at the beginning (*b*) of period *t* (e.g., January
1, 2005). As a result, investment in inventories for period *t* will
be written as:

**Equation 10.4**

*CFV _{t} = CFV_{e} − CFV_{b}*

10.75 As in the case of any aggregate, the data needed to calculate
the end component (CFV_{e}) and the start component (CFV_{b})
are current dollar series and component prices series.

## Components of the calculation of investment in non-farm inventories

10.76 The current dollar series for each basic component are evaluated in Steps 9 and 10 (see paragraphs 10.49 and 10.50) of the 10-step method (see Table 10.4). The end-of-period inventory value is estimated by multiplying end-of-period inventory quantities by the average price for the quarter. By the same logic, the start-of-period inventory value is computed by multiplying start-of-period inventory quantities by the same average price for the quarter.

10.77 The price series corresponding to the current-dollar series are simply the revaluation price indexes (Step 5 of the 10-step method). For a description of these prices, see paragraphs 10.29 to 10.37. The prices are used for both the start-of-period estimates and the end-of-period estimates.

## Components of the calculation of investment in farm inventories

10.78 The calculation of real investment in farm inventories is also based on start-of-period and end-of-period inventory levels expressed in terms of the average prices of goods sold during the estimation period. The average prices are the same as those used in estimating investment in inventories at current prices. Inventory levels in current dollars are calculated in two steps: first, the inventory level is estimated in constant dollars, and then inventory value is computed at current prices.

10.79 Real end-of-period inventory levels (KBVe) are estimated for each
of the 17
basic farm components (Table 10.5).
To estimate the inventories at constant prices, a starting point is needed;
the fourth quarter of 2001 was chosen. By performing the calculation
for the periods before and after the fourth quarter of 2001, we can obtain
the inventories at constant prices. This calculation involves adding
business investment in inventories at constant prices for the period
(i.e., the value of physical change at constant prices (KVPC)) to inventories
for the previous period (KBV_{e−1}). This is shown in Equation
10.5 below.

**Equation 10.5**

*KBV _{e} = KBV_{e−1} + KVPC*

10.80 The end-of-quarter inventories at current prices (RBVe) are obtained
by multiplying the end-of-quarter inventories at constant prices (KBVe)
by the average price^{2} of the current
period (REVt) for each of the 17 basic components. This calculation is
shown in Equation 10.6, (Step 9 of the 10-step method used for non-farm
inventories).

**Equation 10.6**

*RBV _{e} = KBV_{e} × REV_{t}*

10.81 As in non-farm inventories, the start-of-quarter inventories at current prices (RBVb) are obtained by subtracting the value of physical change in inventories at current prices (VPC) from the end-of-period inventory value (RBVe) given by Equation 10.6. This calculation is shown in Equation 10.7, which is identical to Step 10 of the 10-step method used for non-farm inventories. This method of calculating the start-of-period inventory value is equivalent to multiplying the start-of-period inventory values in constant dollars by the average price of inventories during the period (see paragraph 10.50).

**Equation 10.7**

*RBV _{b} = RBV_{e} − VPC*

## Investment in inventories, level of aggregation

10.82 Real business investment in non-farm inventories is published for 14 aggregates while real business investment in farm inventories is published for four aggregates (Table 10.6). The estimates for these aggregates are produced using the chain Fisher formula.

10.83 It is important to note that the 38 basic components of business investment in non-farm inventories and the 17 basic components of business investment in farm inventories are included in the calculation of real GDP. Also included are start-of-quarter and end-of-quarter inventories.

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## Notes

1. Unchained indexes means that the index represents the growth relative to the preceding period. In other words, it is as if the preceding period value was always "1" or "100".

2. As noted, the same prices are used to revalue the level of farm inventories as are used in the investment in farm inventories at current prices. However, the seasonally adjusted version is used in the form of a price index since real inventory levels are estimated on a seasonally adjusted basis and in constant dollars rather than on a non-seasonally adjusted basis and in quantity terms, as is the case for investment in inventories.

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