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Price indices and dynamical 

expenditure shares

Preliminary Estimation of Chained CPI-U

Kate Eckerle

Daniell Toth, Joshua Klick, Jeffrey Wilson 

Bureau of Labor Statistics, OSMR, MSRC

FCSM

October 26th, 2023
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Outline

 Price indices + consumer substitution

Tornqvist Equation

BLS: two stage process

Timeliness 

Preliminary Chained CPI-U

 New idea: Forecast monthly shares

Vector timeseries model

With exogenous terms
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 A price index is a number 

 Should capture % change in cost of a set of items

 Inputs: 

-prices (what do people pay), before and now

-amounts (how much do people buy)

 How should you combine these into one number?

What is a price index?
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 One way: ratio of weighted averages of prices

How to weight by importance? Item quantities 

 Expenditure = Quantity x Price

total cost of a basket at today’s prices, vs total cost of same 

basket at past prices

 Vector notation (items go down rows)

Example
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Lowe Index

If fixed quantity vector:

 -is concurrent with time of past prices -> called Laspeyres

 -is defined at or over a time period non-overlapping with times of prices, called Lowe

*we’ll indicate this with a b subscript for “base period” (for us precedes t and t-1)
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...but, when item price goes up, people tend to buy less than 

before. 

They substitute relatively cheaper alternatives, to differing 

degrees for different items.

Consumer Substitution

more Same amount less more

Fixed quantity -> tend to overstate rise in cost of living when prices rise 
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Two Extremes

Fixed quantity

weighted arithmetic mean

Fixed Expenditure Share 

weighted geometric mean

“upper bound” “lower bound” 

Absolute price insensitivity Perfect price sensitivity

items
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Lowe + Geo Means connected by a continuous family of indices

Lloyd-Moulton 

One parameter: “Elasticity of Substitution”

Bridge Between Extremes
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Tornqvist Formula

geometric mean index but with dynamical quantity information

The idea is that this would capture the “true” amount of substitution, 

which is item and time dependent.

monthly expenditure 

shares
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BLS: two-stage process

 Inputs: Prices (establishment) + Expenditures (household) 

Market basket: 243 basic items (hierarchical)

Geography: 32 areas (Primary Sampling Units of CE) 

 1st stage: compute a price index for each item-area combination

 243 x 32 = 7,776 basic price indices

 2nd stage: aggregate to reflect desired broader group

 Broadest level for CPI-U: All items, U.S. city average 

most with Geo Means

item-areas:
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Aggregation

Type equation here.

Plug basic price indices and aggregation weights 

into ANOTHER price index formula.

months: 

item-areas:

Chained 

CPI

Regular

CPI

*both plutocratic, not democratic

capture upper-level substitution
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Hold-up: monthly shares, NOT prices

CE data comes with ~one year lag

While CPI-U final upon release, C-CPI-U issued as preliminary in month t.

 

receive ~one year late!

Timeliness problem
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2002 through 2014: 

-calculated using Geometric Means (downward bias!)

Since Jan 2015: 

-via constant elasticity of substitution model (i.e. based on LM)

Preliminary Estimate
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Originally based on work of Greenlees (2010)

sigma from Feenstra-Reinsdorf model:

annualized shares + prices relatives to obtain stable estimates (Y=year) 

Fixed value selected: 

Why .6?
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Surely an improvement, but 

assumes stability of this 

effective/net sigma...

Errors correlated 

Shocks

Great Recession

COVID economy

Seasonal effects

Can we do better?

Geo Means CES model 

COVID

Great

Recession
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Alternative: Forecast monthly shares

Cage + Wilson (2009): 

Forecast monthly budget shares -> plug into Tornqvist

- Univariate time series models (ARIMA)

- Worked well for highly seasonal

...but, this will capture budget share dynamics that can be predicted 

without knowing about prices, or what other shares are doing.

If one drops Feenstra-Reinsdorf, optimal sigma varies enormously year-to-year, and 

month-to-month. (Robert Cage + Joshua Klick)
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Our idea

Whole idea behind superlative indices is that the state of relative prices changes 

affects allocation of spending across items... 

Build an expenditure forecast model that incorporates item-area price indices 

(which recall are not lagged). 

1. Avoid constraints

-model item-area expenditures, not shares

2. Allow interactions among item-area expenditures

-univariate -> vector timeseries model

3. Allow couplings to item-area prices (external) 

-exogenous terms
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Vector ARMAX

model with inputs
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Contact Information

Kate Eckerle

Research Mathematical Statistician

MSRC, OSMR

eckerle.kate@bls.gov