TRANSMISSION OF PRODUCER PRICES THROUGH STAGES OF PROCESSING

Tae-Hwy Lee, University of California, Riverside and Stuart Scott, U. S. Bureau of Labor Statistics

Stuart Scott, 2 Massachusetts Ave., NE, Room 4915, Washington, DC 20212

KEY WORDS: Cointegration, VECM, inflation, CPI

1. Introduction

At the time of the 1973-74 oil price shock, the

featured index for the U.S. Bureau of Labor Statistics

(BLS) Producer Price Index (PPI) program was “All

Commodities”. For some time, this index was

dominated by oil prices as their effects spread to

refined petroleum producers and other producers

experiencing higher energy prices. Critics complained

that the single summary index gave a very limited

picture of what was happening in prices. In 1978, BLS

shifted its publication emphasis to a stage of processing

(SOP) system. As explained by Gaddie and Zoller

(1988),

The basic idea of a stage of process system is

that the economy can be subdivided into distinct

economic segments which can be arranged

sequentially so that the outputs of earlier

segments become inputs to subsequent ones, up

through final demand.... To the extent that

such a sequential system of processing stages

can be defined, it is possible to trace the

transmission of price change through the

economy and to develop information on both

the timing and magnitude of price passthroughs to final demand.

Blanchard (1987), Clark (1995), Baillie (1989),

Boughton & Branson (1991), and many others have

looked for evidence of price transmission among stages

or for evidence that producer price increases presage

consumer price increases. Both Blanchard and Clark

find some explanatory power from the stages, but Clark

and others find the relationships to be weak, especially

in the sense of foretelling consumer price changes.

Baillie and Boughton & Branson find that there is no

discernible long-run relationship between commodity

or producer prices and the CPI (the two time series are

not cointegrated), yet Boughton & Branson find a weak

short-run relationship in which commodity prices help

predict future CPI inflation, while Baillie does not.

This study, still in progress, is part of a BLS effort

to examine the usefulness of SOP’s, including

comparisons of alternative partitions of covered

industries. The study employs multiple time series

methods, and benefits from greater data availability for

some of the indexes than some of the previous studies.

The focus of the present paper is on price transmission

among stages of producer prices and on transmission to

consumer prices. Our results show that consumer

prices are strongly related to Finished Goods prices,

and that meaningful relationships exist between

processing stages to be defined and discussed below.

2. Stages of Processing

The initial SOP system (Popkin, 1974) is based on

allocating products or commodities to three stages,

Crude, Intermediate, and Finished, based on their

degree of fabrication and end use. These will be

denoted CSOP, since they are commodity-based. The

Finished Goods index, representing goods nearest final

consumption, is usually emphasized in press releases.

Crude and Intermediate indexes may be viewed as

possible indicators of future movements in Finished

Goods.

A second SOP system (Gaddie & Zoller, 1988),

denoted ISOP, with data available from June 1985,

dovetails with an improved, industry-based sample

design, introduced gradually over the 1978-1986

period. In statistical terms, the redesign represents

probability sampling of products made by individual

industries under the Standard Industrial Classification

(SIC) system. In conceptual terms, the ISOP’s

represent an interindustry flow model for the economy.

Using the Bureau of Economic Analysis Input/Output

(I/O) tables, transaction flows between producing and

consuming industries can be estimated. Four stages,

Crude, Primary, Semifinished, and Finished Goods

producers, are derived as weighted averages of

component SIC indexes. In addition, “input” indexes,

input to one of the above stages or to Final Demand,

can be estimated. These again use I/O table data and

are based on the assumption that input products to

consuming industries come proportionally from

industries making these products.

Tables 1 and 2 show ISOP industry composition

and transaction flow. Overall, the Crude stage

represents about 10% of covered transactions, and the

other stages roughly 30% each. The CSOP distribution

for the three stages Crude, Intermediate, and Finished

is roughly 10-50-40. Since these stages are formed by

putting together commodities, wherever made, other

statistics like those in Tables 1 and 2 are not available

for the CSOP. Following many analysts, we emphasize

“core” SOP’s, that is, indexes which exclude Food and

Energy sectors, each representing about 15% of the

total (Table 1A). These components are obviously

important, but their volatility may mask other
relationships in the data. For core ISOP’s, from Table

1B, Primary is reduced to about 20%, and the last two

stages increase somewhat.

As a starting point for partitioning industries into

stages, an I/O table, a matrix like Table 2, but with

roughly 500 detailed industries, shows transaction

flows between producing and consuming industries.

As indicated in the SOP definition, the aim is to order

these industries so that for a given row, representing a

producing industry, most of the output is to subsequent

industries, i.e., to industries to the right of the matrix

diagonal. Companies, however, make such a variety of

products, and their products are consumed by such a

variety of industries, that no ordering produces a purely

upper triangular matrix. Gaddie & Zoller’s efforts to

maximize “forward flow” and limit “internal flow”

(consumption within the stage where produced) and

“backflow” (consumption by previous stages) are rather

successful. The flow summary of Table 2 shows that

the ISOP achieves a forward flow exceeding 80%,

while backflow and internal flow are 6% and 11%,

respectively. A shortcoming, however, is that 30% of

transactions skip one or more stages. For example,

roughly 30-40% of output of each of the first three

stages goes to Final Demand. Thus, for instance,

output from Primary differs considerably from input to

Semifinished. This has led to the construction of the

input indexes. Since these indexes are indirectly

constructed, using simplifying assumptions, they are

not as firmly based as the output indexes. Currently,

BLS is examining additional industry partitions.

Briefly describing the flows among stages in the

ISOP in Table 2, output from Crude is about onequarter each to Crude, Primary, and Final Demand,

with the rest divided between Semifinished and

Finished. From Primary, about 20% each are

consumed by Finished, Semifinished, and the

combination of Crude and Primary, with the

remainder, close to 40%, to Final Demand. Nearly

80% of Semifinished goes to Finished or to Final

Demand, and 90% of Finished goes to Final Demand.

Caution in interpreting Table 2 numbers is advised,

since it is based on I/O data for all industries, while, as

seen in Table 1, the current ISOP industry coverage is

quite limited. (In recent years, many indexes in the

service-producing industries have been added, but they

are not yet in the existing ISOP’s).

Success in analyzing price transmission among

stages and to consumer prices has been limited. For

example, Clark (1995) compares forecast performance

of the Core CPI (All Items, Less Food and Energy) and

the Core Goods CPI (omitting services) in VAR

models with and without the three CSOP’s over the

period 1977-1994. For the entire period, there are

modest gains from including the PPI indexes, but for

certain subperiods, forecast error is larger. Clark

points out that a shortcoming of his results are the use

of full SOP’s for the PPI data and core indexes for the

CPI, for data availability reasons.

Blanchard (1987) argues for nominal rigidity in

both wages and prices. That is, nominal wages and

prices respond slowly to forces acting on them,

including each other. The rigidity on the prices side is

perhaps less accepted by economists in general.

Individual price chain equations with CSOP’s show

fairly rapid response of wages to price shocks and vice

versa. These two results are consistent with a

cumulation hypothesis that short lags in price

transmission at detailed levels become relatively long

lags at the aggregate level. He forms four regression

equations with response variables personal

consumption deflator, PPI Finished Nonfood Consumer

Goods, PPI Finished Food, and PPI Intermediate

Nonfood Goods. These equations include an input

price index and a wage variable as explanatory

variables, and both input prices and wages have

significant long-run effects.

Mattey (1990), who makes an extra effort to

extend the ISOP data available at the time, has the only

analytic study we have seen which explicitly compares

CSOP and ISOP. Starting from a Cobb-Douglas

formulation, he obtains a model of output prices as a

function of input prices and labor and capital costs.

Pointing out large flow differences between output

from one stage and input to the next stage, due to skips

and leakage, in the ISOP equations he uses the ISOP

input indexes for input prices, an advantage with the

ISOP. Modeling diagnostics and forecast performance

are similar for the two sets of equations. However,

Mattey prefers the ISOP results, since the regression

coefficients are more in accord with economic theory.

3. Modeling Price Transmission

In this section we describe how restricted vector

autoregressions can be employed to examine the

sources of inflation and its transmission (direction,

speed, and magnitude). We show how the

cointegration restriction can be used to identify a VAR

system with common stochastic trends, subject to

permanent and transitory changes in inflation rates,

and how we may investigate the system’s responses to

the permanent shocks, i.e., to innovations to the

stochastic common trends.

We begin with some assumptions. Let

X x x t = t pt ( ⋅⋅⋅ )′

1

 be a PPI SOP system with p stages.

For example, in the ISOP output index system, p = 4 ,

and the elements of Xt are PPI indexes for Crude,
Primary, Semifinished, and Finished processors. Let

Xt be I(1) and cointegrated with cointegrating rank r

(that is, there exists a p × r matrix b of rank r (< p)

such that b ′Xt is I(0)). Then the system can be

generated from the common trend representation

Xt = X0 + mt + Aht + Xt

~

, (1)

where X0 is the initial value at t = 0 , m is p ×1 , A

is p × ( p − r) , ht is ( p − r) × 1 I(1) common

stochastic trends, and ~Xt consists of p ×1 I(0)

transitory components. Apart from the initial values

and the deterministic trend, Xt can thus be

decomposed into X Ah t

P

= tand Xt X

T

= t

~

. The

elements of Xt can be explained in terms of a smaller

number ( p − r) of I(1) variables, ht, which are thus

called common factors.

Because an SOP system is constructed based on

the principle of maximizing forward flow and

minimizing backward flow, it will be assumed that the

price indexes in the later stages include the permanent

components of the previous stages ( xi

P

+1

 contains

x i p i

P

, = 1,2, , − 1), but not vice versa. No such

assumption is imposed for the transitory component

Xt

T

. As seen in Table 2, backflow represents only

5.8% of the total shipments in ISOP. Reflecting this,

we make an assumption on the structure of the matrix

A:

Assumption 1. The p × ( p − r) factor loading matrix

A can be written ~AΘ , where Θ is a

( p − r) × ( p − r) lower triangular matrix.

Then the permanent component is

X Ah A h Af

t

P

= t = t ≡ t

~ ~

Θ ,

where ht and f

t

 are vectors of size ( p − r) .

Let hit (i = 1, , p − r) be the innovation to each

stochastic trend hit , i.e., hit = hi,t−1 + hit . Define

ht = (ht ht )

1 2

' ' ' such that ht

1

 consists of hit

(i = 1, , p − r) and ht

2

 is r × 1. ht

1

 may be called the

permanent shock because it determines Xt

P

, while ht

2

may be called the transitory shock. If it is true that

supply shocks are persistent and demand shocks are

transitory, if cost-push inflation may be persistent in

the long run while the demand-pull inflation is only

temporary, then ht

1

 can be referred to as the supply

shock and ht

2

 as the demand shock.

Next, we make an assumption on the matrix 

~

A .

Suppose for now that E Xt(Δ = 0) , and that the p × r

matrix of r cointegrating vectors is b = (− f I r )' ,

where f is the r × ( p − r) submatrix of unknown

parameters to be estimated and I

r

 is the identify

matrix of dimension r . We assume there is no

additional restriction on f . The triangular

representation for Xt is

ΔX u t t

1 1

= , X X u t t t

2 1 2

= f +

where Xt is partitioned as (Xt Xt )

1 2

' ' ' , Xt

1

 is

( p − r) ×1 and Xt

2

 is r ×1, and ut = (ut ut )

1 2

' ' ' is a

stationary stochastic process with full rank spectral

density matrix. This representation has been used by

Phillips (1991), Campbell (1987), and Stock & Watson

(1993).

Assumption 2. f

t Xt

P

=

1

, where

Xt (X X )

P

t

P

t

P

=

1 2

' ' ' , the partition conformable with

Xt = (Xt Xt )

1 2

' ' ' .

Under Assumption 2, the first ( p − r) rows of ~A

form an identity matrix. With p = 4 stages and r = 1 ,

~

A =

























1 0 0

0 1 0

0 0 1

f1 f2 f3

,

with b = (− f Ir )'= (− f − f − f )'

1 2 3 1 . Also,

x f i

it

P

= it , = 1,2,3, and x f

t

P

4 = t

f . Since f

1t = q11h1t

,

f

2t = q21h1t + q22h2t

, and f

3t = q31h1t +q32 h2t + q33h3t

,

we may examine the innovation propagation from the

earlier SOP’s to the later ones.

Finally, we make two assumptions on

Σh = E htht( ') in order to examine dynamic responses

of prices to shocks ht.

Assumption 3. The permanent shocks

hit (i = 1, , p − r) are uncorrelated.

Assumption 4. The permanent and transitory shocks

are uncorrelated.

Thus, Σ

Σ

h Σ

h

h

=













1

2

0

0

 is block diagonal

(Assumption 4). Σh

1

 is diagonal (Assumption 3), so
the common factors h i p r it ( = 1, , − ) are

uncorrelated random walks.

We can now present the vector error correction

model (VECM). Let the PPI stages

Xt = (x1tx pt)' be I(1) and cointegrated with

cointegration rank r . Then,

ΔXt = m + ΠXt − + Γ ΔXt− + +Γk ΔXt −k + te 1 1 1

, (2)

where Π and Γ1 to Γk are p × p , and et is a

p ×1 vector white noise. The long-run impact matrix

Π will have rank r , and can be expressed as Π = ab'

for suitable p × r matrices a and b . Xt is nonstationary, ΔXt

 is stationary, and the linear

combinations given by b' Xt are stationary.

We estimate the VECM in (2) following Johansen

(1991), and then transform it to a vector moving

average model

ΔXt = m + C B t( )e , (3)

where C(B) is a p × p matrix polynomial in the

backshift operator B . To identify the common factor

ht, we rewrite (3) as

ΔXt = m + Γ B ht( ) , (4)

where Γ(B) = C(B)Γ0 and h e t = t

−

Γ0

1

 for some

nonsingular p × p matrix Γ0. We choose Γ0 so that

(i) the ( p − r) -vector ht

1

 represents innovations to the

common stochastic trend with the partition

ht = (ht ht )

1 2

' ' ' , and (ii) Γ(1) = (A 0), with 0 being

a p × r null matrix as Γ(1) is of rank ( p − r) . Then,

using the well known expansion

Γ( ) Γ( ) ΔΓ ( )

* B = 1 + B , (4) becomes

Xt = X0 + mt + Aht + Γ B ht

*

( ) ,

which is the common trend model expressed in (1),

denoting the stationary component Γ

*

(B)ht by ~Xt.

The elements of A are the long-run multipliers of

the permanent shock ht

1

, that is, lim /

n→∞ Xt t−n

¶ ¶h1

because ht t n

n

= −

=

∞

∑ h

0

. The long-run multiplier of

the transitory shock ht

2

 is zero because Γ

*

(B) is

absolutely summable.

Under Assumption 1, the long-run multiplier A

can be written as ~AΘ so that the permanent

component is Xt Ah A h

P

= t = t

~

Θ . Noting that b' Xt

should be stationary if Xt is cointegrated,

b' b'

~

A = A = 0 . Thus, ~A is chosen from '

~

b A = 0

where b is an estimate of b . For example, with

p = 4 stages and r = 1, under Assumption 2 

~

A can

be estimated with last row f' = (− b1 − b2 − b3 ).

The first permanent shock potentially affects all of the

variables in Xt in the long run, the second potentially

affects the second variable and those lower in the

ordering, etc.

The approach of King et al (1991) for estimating

the long-run multiplier A is followed. The first

( p − r) columns of Γ(B) show how the series in ΔXt

respond to the permanent shocks ht

1

, and their

accumulated sums show how the series in Xt respond

to the permanent shocks. The response of

xit (i = 1, , p) to each permanent shock

hj,t−n occurring n periods ago is denoted by

i Rj n it j t n

( ) x / = ¶ ¶h , −. It may be noted that we do

not standardize a shock in order to make the ijth

elements of A the long-run multipliers of the

permanent shock hjt , i.e., limn→∞ i Rj(n) .

Since the VECM can be used for forecasting, we

also compute the fractions of the h -step ahead forecast

error variance of Δxi,t h(i , , p) + = 1 attributed to each

permanent shock hjt ( j = 1, , p − r) . The estimates

provide information about the relative importance of

permanent and transitory shocks in h -step ahead

forecasts of ΔXt.

4. Price Transmission among SOP’s

The CSOP’s have been generated back to 1947,

while the ISOP’s start in June, 1985. Thus, our

analyses are carried out for the 11-year span, 6/85-

5/96. All PPI and CPI series are core indexes,

excluding Food and Energy. In addition, the CPI

series excludes Used Cars. All series are monthly,

seasonally adjusted, and in logarithms. In our

empirical study the logarithms of all series are

characterized as I(1) processes based on the augmented

Dickey-Fuller (ADF) and Phillips-Perron tests. Some

wage and CPI series display upward trends and their

first differences show rather smoother series than PPI

series. In other studies where wages and CPI series are

used (e.g., Mehra, 1991), it is often found that the

inflation series, the first difference of log prices, are

I(1). However, it was not the case in our data,

especially in PPI series which show clear meanreverting in both ISOP and CSOP.
First, we examine results for the ISOP system.

Lag length k = 5 in the VECM is chosen using the

Akaike and Schwarz information criteria (AIC and

SIC), as well as the battery of residual diagnostics.

Table 3A reports the results of testing for

cointegration. The ISOP series X x x x x t = t t t t( )' 1 2 3 4

are cointegrated with three common stochastic trends.

In Table 3B, following Johansen (1991), we test

for weak exogeneity and for long-run exclusion. The

latter test is based on the hypothesis that a subset of the

variables in Xt do not enter the cointegration space,

which, if not rejected, implies that the variables in

question can be omitted from the long-run relations

between the ISOP’s. The hypothesis is H0: b 0 i = for

i=1 to 4. The results show that all of them are

significant. Similar tests have been performed on the

rows of a , corresponding to tests for weak exogeneity.

They can be formulated as H0:ai = 0 for each i=1 to

4, that the i

th component in Xt

 is not adjusting

toward the estimated long-run relations. If not

rejected, it implies that the variable in question itself

takes the role of a common trend in the system.

From Table 5A, it is observed that the first

permanent shock h1, the permanent shock to the

Crude stage, explains a significant portion of the

inflation fluctuations in all stages, the amount

declining through the stages, but increasing in the long

run (as the forecast horizon h increases). The second

permanent shock h2, which is another permanent

shock to Primary, accounts for substantial variations in

Primary and Semifinished in the short run. Its role

declines for the longer forecast horizons. The third

permanent shock h3that arrives at the third stage of

processing shows very significant inflation

transmission from Semifinished to Finished.

Turning to Figure 1, which graphs the dynamic

responses to the three shocks along with the two

standard deviation confidence bands computed by

Monte Carlo simulation using 300 replications (dashed

lines), we see significant long-run effects of the first

shock on Primary, the second shock on Semifinished,

and the third shock on Finished. None of the other

effects in the forward direction appears significant. In

the backward direction, some of the short-run effects

are substantial, implying that demand shocks matter in

the short run. However, their long-run effects are zero

by construction.

To sum up, ISOP exhibits inflation transmission

through the stages very strongly, with significant onestep forward flow. The multi-step forward

transmissions are insignificant. An alternative

bivariate model with Finished and Input to Finished

indicates cointegration. The latter index is intended to

reflect the appropriate contribution of the previous

stages to Finished.

Table 4, Table 5B, and Figure 2 present the results

for CSOP. Depending on the significance level

chosen, the Johansen tests suggest r=1 or 2. Here, we

proceed with r=1 and two common factors. The error

correction coefficients are all significant except a1.

Its non-significance indicates that Crude PPI does not

adjust toward the estimated long-run relations, that

Crude takes the role of a common trend in the system.

The impulse responses in Figure 2 show a strong, fairly

rapid response of Crude and Primary, but not Finished,

to the first permanent shock. Finished responds

rapidly to the second permanent shock. Similarly, the

variance decomposition statistics (Table 5B) show that

the first permanent shock strongly influences the first

two stages and the second permanent shock influences

stages 2 and 3. In general, the results for CSOP are

similar to those for ISOP, showing significant inflation

transmission through the three stages. The multi-step

forward transmission is insignificant.

5. Price Transmission to the CPI

We next examine the results of adding the CPI to

both ISOP and CSOP systems. Space limitations

prevent including the tables corresponding to Tables 3-

5. For the system consisting of the ISOP and the CPI,

cointegration is present, but the cointegrating rank is

still 1, and the CPI does not enter the cointegrating

relationship significantly. Overall, all permanent

shocks except the first have a sizable long-run impact

on the CPI, most rapidly for the fourth shock.

For the CSOP plus the CPI, the cointegration test

statistics are unclear. Given that the CSOP system

appears to have cointegrating rank either 1 or 2, it is

reasonable to examine results for these values. With

rank 2, an interesting picture emerges that one

cointegrating vector relates Finished and the CPI

alone. The other cointegrating vector can be

formulated among the three CSOP stages.

Confirmation for this comes from independent testing

for cointegration for the pair Finished and the CPI.

Figure 3 looks similar to Figure 2 for the three CSOP

stages. The CPI responds strongly to the second shock,

but not the first.

CSOP Finished relates more closely to the CPI

than ISOP Finished. The latter pair has slightly

inferior cointegration test statistics, and the entire

ISOP output system relates less strongly to the CPI.

A key criterion for usefulness of time series

models is forecast performance. Our evaluation of CPI
forecasts from the SOP’s proceeds as follows. The

VECM’s are estimated for ISOP plus CPI (with p=5,

k=5, and r=1), and CSOP plus CPI (with p=4, k=2,

and r=2) using the observations up to 3, 4, and 5 years

prior to 5/96, obtaining post-samples of size 36, 48,

and 60, respectively. Based on the estimated models,

one-step ahead forecasts are generated for those postsamples, and mean square error (MSE) and mean

absolute error (MAE) losses are calculated. The sign

test, the Wilcoxon signed-rank test, and tests of

Granger & Newbold, Meese & Rogoff, and Diebold &

Mariano are all computed. Diebold & Mariano (1995)

describes and discusses the performance of all these

tests. Since the CPI has grown fairly slowly and

steadily over most of the last decade, great differences

cannot be expected. Still, for all three post-samples,

CSOP forecasts have smaller loss, with some tests

achieving significance at the 5% level.

Thus, our results seem to favor the CSOP over the

ISOP system in terms of explaining the CPI. These

results, however, are based on ISOP output indexes

only. When we form a system with the series Input to

Final Demand added to the ISOP output indexes and

the CPI, forecast error loss is sometimes smaller for

CSOP, sometimes for ISOP, depending on the postsample and the loss criterion, with the differences not

significant whichever test statistic is used. Input to

Final Demand from the ISOP is closely related to the

CPI, similar to CSOP Finished.

6. Conclusions

Using the vector error correction models (VECM),

the sources of inflation and its transmission (direction,

speed, and magnitude) are examined. We show how

the cointegration restriction can be used to identify

common stochastic trends and how we may investigate

the system’s responses to the permanent shocks. Two

BLS stage of processing (SOP) systems are compared.

Forecasts of CPI inflation using the two SOP systems

are evaluated.

The results on the cointegrating relations, on the

response to shocks, and on the forecast error variance

decomposition all give support to Popkin’s original

notion of stages. Later stages respond to earlier stages,

and the CPI responds to Finished Goods. In the CSOP,

separating out Finished is beneficial, since it is less

volatile than Crude or Intermediate, and similarly for

the ISOP. By a small margin, CSOP outperforms the

four output ISOP indexes in post-sample forecasts of

the CPI.

Acknowledgments

We thank Jim Buszuwski, Brian Catron, Bill

Cleveland, Jack Galvin, Irwin Gerduk, Soon Paik, and

Roslyn Swick for helpful discussions, and Dimitri

Paliouras for research assistance. The work was

conducted while the first author visited BLS as an

ASA/NSF/BLS Fellow. The views expressed are those

of the authors and do not necessarily represent those of

BLS.

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