Box: Basic Formulas for Calculating Chain-Type Quantity and Price Indexes

Annual indexes

This box shows the basic calculations used to prepare annual and quarterly chain-type quantity and price indexes. The formula used to calculate the annual change in real GDP and other components of output and expenditures is a Fisher index ($Q_t_^F^$) that uses weights for 2 adjacent years (years t-1 and t).

The formula for real GDP in year t relative to its value in year t-1 is

$Q_t_^F^ =\left\{\sum p_t-1_q_t_\sum p_t-1_q_t-1_\right\} × \left\{\sum p_t_q_t_\sum p_t_q_t-1_\right\},$

where the p's and q's represent prices and quantities of detailed components in the 2 years.

Because the first term in the Fisher formula is a Laspeyres quantity index ($Q_t_^L^$), or

$Q_t^L = \left\{\sum p_t-1_q_t_\sum p_t-1_q_t-1_\right\},$

and the second term is a Paasche quantity index ($Q_t_^P^$), or

$Q_t_^P^ = \left\{\sum p_t_q_t_\sum p_t_q_t-1_\right\},$

the Fisher formula can also be expressed for year t as the geometric mean of these indexes as follows:

$Q_t_^F^ =Q_t_^L^ × Q_t_^P^.$

The percent change in real GDP (or in a GDP component) from year t-1 to year t is calculated as

$100 \left(Q_t_^F^ - 1.0\right).$

Similarly, price indexes are calculated using the Fisher formula

$P_t_^F^ =\left\{\sum p_t_q_t-1_\sum p_t-1_q_t-1_\right\} × \left\{\sum p_t_q_t_\sum p_t-1_q_t_\right\},$

which is the geometric mean of a Laspeyres price index ($P_t_^L^$) and a Paasche price index ($P_t_^P^$), or

$P_t_^F^ =P_t_^L^ × P_t_^P^.$

The chain-type quantity index value for period t is

$I_t_^F^ = I_t-1_^F^ × Q_t_^F^,$

and the chain-type price index is calculated analogously. Chain-type real output and price indexes are presented with the base year (b) equal to 100; that is, $I_b_ = 100$. The current-dollar change from year t-1 to year t expressed as a ratio is equal to the product of the Fisher price and quantity indexes:/1/

$\left\{\sum p_t_q_t_\sum p_t-1_q_t-1_\right\} = P_t_^F^ × Q_t_^F^.$

Quarterly indexes

The same formulas are used to calculate the quarterly indexes for the most recent quarters, called the "tail" period; quarterly data are substituted for annual data. The tail period begins in the third quarter of the most recent complete year that is included in an annual or comprehensive NIPA revision, so the specific quarters covered change annually. Modified formulas are used to calculate the indexes for the other quarters, called the "historical" period. Quarterly quantity data are used for the quantity indexes, and quarterly price data are used for the price indexes, but the weights—prices for a quantity index and quantities for a price index—are annual data.

The weights that are used for the indexes in the historical period depend on the quarter being estimated. For each quarter, the weights for the closest 2 years are used: For the first and second quarters of a year, the weights are from that year and the preceding year; while for the third and fourth quarters, the weights are from that year and the next year.

All quarterly chain-type indexes for completed years that have been included in an annual or comprehensive revision are adjusted so that the quarterly indexes average to the corresponding annual index. When an additional year is completed between annual revisions, the annual index is computed as the average of the quarterly indexes, so no adjustment is required to make the quarterly and annual indexes consistent. For example, until the 1998 annual revision is released, the chain-type indexes for the year 1997 are derived by averaging the four quarterly indexes for 1997.

Chained-dollar estimates

The chained-dollar value ($CD_t^F$) is calculated by multiplying the index value by the base-period current-dollar value ($\sum p_b_q_b_$) and dividing by 100./2/ For period t,

$CD_t_^F^ = \sum p_b_q_b_ × I_t_^F^ / 100.$

Implicit price deflators

The implicit price deflator ($IPD_t_^F^$) for period t is calculated as the ratio of the current-dollar value to the corresponding chained-dollar value, multiplied by 100, as follows:

$IPD_t_^F^ = \left\{\sum p_t_q_t_CD_t_^F^\right\} × 100.$

Footnotes:

2. For exceptions to this procedure, see footnote 8 in the text.

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