Monthly Labor Review November 2002 3

Price Transmission

I

n 1978, the Bureau of Labor Statistics began

emphasizing the stage-of-processing (SOP)

system as the key structure used in analyzing

the behavior of producer prices. This system

allocates commodities among three categories:

crude goods, intermediate goods, and finished

goods. Crude goods are defined as unprocessed

commodities that are not sold directly to the

consumer. Intermediate goods are either commodities that have been processed, but that still

require further processing, or nondurable, physically complete goods purchased by business firms

as inputs for their operations. Finished goods are

commodities that are ready for sale to the finaldemand user, either an individual consumer or a

business firm.1 The SOP model can be extended to

encompass consumer prices by including the

Consumer Price Index (CPI) as the fourth “stage of

processing.”

According to the SOP system, commodities at

earlier stages of processing can be considered

inputs to commodities at later stages of processing.

Economic theory predicts that price changes may

be transmitted forward through the stages of

processing. The study presented in this article (1)

uses econometric techniques to determine the

causal directions of price changes by means of the

SOP system and (2) examines the stability over time

of the causal relationships found.

Several authors have investigated the causal

relationship between commodity prices and

Jonathan Weinhagen

is an economist in the

Division of Producer

Price Indexes, Bureau

of Labor Statistics. The

opinions expressed

here are the author’s

and do not

necessarily reflect

those of the Bureau of

Labor Statistics.

An empirical examination of causal price relationships

from a stage-of-processing approach reveals that

changes in the indexes for crude- and intermediate-goods

prices often preceded changes in the CPI during the 1970s

and 1980s; since the early 1990s, however,

the relationship has become more tenuous

Jonathan Weinhagen

An empirical analysis of price

transmission by stage of processing

consumer inflation. S. Brock Blomberg and Ethan

S. Harris examined the relationship between the

core CPI, on the one hand, and the Commodity

Research Bureau spot index, the Journal of

Commerce index, the crude-goods PPI, the

National Association of Purchasing Managers

price index, and the Federal Reserve Bank of

Philadelphia’s prices-paid index, on the other. 2

The authors discovered that the commodity

indexes had a statistically significant positive

effect on core CPI inflation from 1970 to 1986, but

that from 1987 to 1994 all of the commodity

indexes, except for the Journal of Commerce

index, had a negative effect on core CPI inflation.

Fred Furlong and Robert Ingenito analyzed the

relationship between CPI inflation and the

Commodity Research Bureau’s indexes for all

commodities and for raw materials.3 The authors

showed that both price indexes, excluding oil

prices, were strong indicators of CPI inflation in

the 1970s and early 1980s, but from the mid-1980s

to the mid-1990s, the indexes performed poorly

as CPI inflation indicators. Todd Clark studied

the relationship between PPI SOP models and the

CPI by building models that forecast the CPI

using producer price indexes.4 He found that

using information about the PPI improved

forecasts of the CPI for the overall period from

1977 to 1994 and for the subperiod from 1977 to

1980. However, for the subperiods from 1986 to

1989 and 1991 to 1994, including the PPI in
Price Transmission

4 Monthly Labor Review November 2002

forecasts of the CPI detracted from the forecasts, suggesting a

breakdown in the relationship between producer price indexes

and the CPI. C. Alan Garner discovered that the explanatory

power of the price of gold, the Commodity Research Bureau

index of commodities futures prices, the Journal of Commerce

index, the Center for International Business Cycle Research index,

the Paine Weber index, and the past CPI to explain movements in

the current CPI decreased significantly since 1983.5 Tae-Hwy

Lee and Stuart Scott used vector error correction models to

examine price transmission within the SOP system and found

significant forward price transmission from 1985 to 1996.6 The

common finding in the majority of these studies was that the

power of commodity prices to predict CPI inflation has

diminished since the 1980s.

The next section of this article visually examines historical

movements of various PPIs and the CPI in order to study the link

between producer and consumer prices. The aim is to confirm or

disconfirm previous authors’ findings that a change in the nature

of the inflationary relationships within the SOP system occurred

in the late 1980s. The section that follows estimates unrestricted

vector autoregression (VAR) models to further investigate the

causal price transmission relationships between the PPI SOP

indexes and the CPI. The final section constructs impulse

response functions and variance decompositions from the VAR

models to show how price changes at various stages of

processing are transmitted throughout the SOP system.

Historical movements of the SOP indexes

This section visually examines historical trends of SOP indexes

in order to determine whether movements of PPIs lead

movements of the CPI. Toward that end, instances are sought

in which changes in PPIs appear to precede changes in the

CPI. The examination also attempts to discern whether the

causal relationship between producer and consumer prices has

changed over time. Chart 1 presents the 12-month percent

changes in the PPIs for crude, intermediate, and finished goods

from 1975 to 2001 and in the CPI for commodities for those same

years. All indexes exclude food and energy.

The top panel of the chart compares price movements in

the PPI for crude materials with changes in the CPI. In some

instances, price changes in the index for crude materials

appear to precede changes in the CPI. In October 1977, prices

for crude materials began accelerating, a trend that continued

through March 1979. The CPI went through a similar period

of acceleration; however, it did not begin until March 1978, 5

months after the acceleration in prices for crude goods began,

and it persisted 8 months longer. In July 1985, the index for

crude goods began a long period of acceleration that lasted

through May 1988. This acceleration preceded a run-up in

consumer prices that began in May 1986 and continued

through February 1989. In contrast, since the early 1990s,

crude-goods prices have exhibited three noteworthy periods

of acceleration that were not followed by significant run-ups

in the CPI. These periods of faster increase in the prices of

basic industrial materials began in September 1991, July 1996,

and January 1999, respectively.

The middle panel of the chart presents historical price

movements in the PPI for intermediate materials and in the CPI.

Prior to the early 1990s, there were several periods of acceleration

and deceleration in the prices of intermediate materials that

foreshadowed similar movements in the CPI. A rise in the rate of

increase for intermediate-materials prices beginning in March of

1976 was followed by a period of acceleration in the CPI that

started in May 1976. An extended period of deceleration in

intermediate-goods prices began in March 1980 and lasted

through January 1983. This slower rate of increase in the prices

of intermediate goods preceded a similar deceleration in the CPI

that began in December 1980. Periods of slowing price increases

for intermediate goods beginning at the end of 1988 and the

early part of 1991 were followed by similar decelerations in the

CPI. However, three significant periods of price acceleration in

intermediate goods since the early 1990s cannot be traced to

similar movements in the CPI.

The bottom panel of the chart compares historical movements

of the PPI for finished goods with changes in the CPI. An

examination of these historical price changes reveals that,

since 1975, movements of the PPI for finished goods and

changes in the CPI have been similar. In most instances, the

two time series’ movements actually coincide with each other,

so a visual examination does not reveal any distinct causal

trend between them.

A visual comparison of the PPIs for crude and intermediate

materials with the CPI indicates that the link between them may

have weakened since early 1990. However, such a simple visual

inspection may not reveal the intricate causal relationships

between the SOP indexes. To examine the issue further, more

rigorous empirical testing is necessary.

Unrestricted vector autoregression model

This section uses VAR models to examine the nature and

consistency of causal price transmission relationships

among price indexes at various stages of processing. VAR

modeling involves estimating a system of equations in which

each variable is expressed as a linear combination of lagged

values of itself and all other variables in the system. 7

VARs were estimated from the PPIs for crude materials,

intermediate goods, and finished goods and from the CPI for

commodities, using monthly data from January 1974 through

December 2001. Core price indexes, which exclude food and

energy, were chosen because the excessive volatility of food

and energy prices can distort underlying trends between SOP

indexes. All data were seasonally adjusted and converted to
Monthly Labor Review November 2002 5

Jan 1975 Jan 1977 Jan 1979 Jan 1981 Jan 1983 Jan 1985 Jan 1987 Jan 1989 Jan 1991 Jan 1993 Jan 1995 Jan 1997 Jan 1999 Jan 2001

–5

0

5

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20

25

30

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14

Intermediate goods

CPI

Jan 1975 Jan 1977 Jan 1979 Jan 1981 Jan 1983 Jan 1985 Jan 1987 Jan 1989 Jan 1991 Jan 1993 Jan 1995 Jan 1997 Jan 1999 Jan 2001

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Finished goods CPI

Chart 1. Historical movements of various price indexes, percent change, January 1975–

 December 2001

Jan 1975 Jan 1977 Jan 1979 Jan 1981 Jan 1983 Jan 1985 Jan 1987 Jan 1989 Jan 1991 Jan 1993 Jan 1995 Jan 1997 Jan 1999 Jan 2001

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Crude goods CPI

Crude goods and the CPI

Crude-goods index

NOTE: All plots exclude prices for food and energy. CPI is CPI for commodities.

CPI

Intermediate-goods index CPI

Intermediate goods and the CPI

Finished-goods index CPI

Finished goods and the CPI
Price Transmission

6 Monthly Labor Review November 2002

annualized percentage growth form by multiplying the change

in the logged data by 1,200.

A time series is stationary if the mean, variance, and

covariances of the series are not dependent on time. The

construction of a VAR with nonstationary data is problematic,

because the tests used to estimate the significance of the

coefficients of the VAR will not be valid. To test for stationarity,

the Dickey-Fuller test was applied; in this one-tailed test, the

null hypothesis is that the time series is not stationary. A large

negative test statistic rejects the null hypothesis and implies

that the time series is stationary.8 Dickey-Fuller tests performed

on the SOP indexes indicated that, when the data are converted

to annualized percentage growth form, all of the time series

involved are stationary.

The Akaike and Schwarz information criteria were used to

determine the optimal number of lags to include in the VARs.9

The Schwarz criterion suggested that estimating the VARs by

using one lag was optimal, whereas the Akaike criterion

indicated that four lags were best. Residual tests implied that

estimating the model with one lag would result in significant

levels of autocorrelation; therefore, the VARs were constructed by using four lags of all variables.

In order to investigate the stability of the causal price

transmission relationships, two separate VARs were estimated,

using data from 1974 to 1989 and from 1990 to 2001. The 1989–90

breakpoint was chosen because the visual examinations

conducted in the previous section suggested that the causal

relationships between PPIs and the CPI weakened around 1990.

Furthermore, a couple of earlier studies found that in the late

1980s there was a breakdown in the usefulness of commodity

prices as predictors of consumer inflation.10 These studies were

conducted with the use of data that began in either 1959 (Clark)

or the 1970s (Blomberg and Harris), ran through the mid-1990s,

and focused on price transmission from early stages of

processing to the CPI. This article expands the earlier investigations by including data through 2001 and examining

relationships among all of the SOP indexes.

Table 1 shows the results of VAR-model tests of the joint

statistical significance of the lagged values of the SOP indexes

in predicting the PPIs for crude, intermediate, and finished

goods, as well as in predicting the CPI. Wald tests were used

to test the null hypothesis that the joint significance of the

explanatory variables is zero. Wald tests are based on

measuring the extent to which the unrestricted estimates fail

to satisfy the restrictions of the null hypothesis.11 A small pvalue of the Wald statistic rejects the null hypothesis of no

feedback from lagged SOPs to the dependent variable, and a

large p-value of the Wald statistic implies that the null

hypothesis is not rejected.

Table 1 also shows the R-squared values of the equations

within the VAR systems. R-squared values measure the

amount of variation of the dependent variables explained by the

regression equation. An R-squared close to unity indicates a

good overall fit of the regression, while an R-squared of zero

suggests a failure of the regression equation to explain variations

of the dependent variable any better than could be explained by

the sample mean of the dependent variable.12

The results presented in table 1 indicate that, during both

subperiods, price movements within each stage of production

can be explained by price changes occurring at earlier stages.

Highlighting this result is the finding that, in the regression

equations for intermediate goods, finished goods, and the CPI

from both subperiods, the coefficients of the lagged explanatory

variables of previous processing stages are jointly significant.

For example, during both subperiods, the lagged values of crude

and intermediate goods are jointly significant in the finishedgoods equation.

The Wald tests also suggest that in most cases price changes

within the SOP model cannot be explained by price movements

at later stages of processing. Only in the finished-goods

equation from 1990–2001, in which the lagged values of the CPI

are jointly significant, is an explanatory variable (or a

combination of explanatory variables) at a stage of processing

after the dependent variable statistically significant.

Results of the VAR estimation do not imply that commodity

prices have lost their power as predictors of consumer inflation.

The R-squared values, however, indicate a decrease in the

usefulness of commodity prices in that regard: R-squared fell

from 0.59 in the 1974–89 CPI equation to 0.30 in the 1990–2001

CPI equation. The R-squared values in the finished-goods

equations behaved similarly. In addition, in the CPI equations,

the chi-square statistics testing the joint significance of crude,

intermediate, and finished goods suggest that the coefficients

are statistically significant in both periods, but that the level of

significance declined in the 1990–2001 period. Similarly, in the

finished-goods equations, the combined lagged values of crude

and intermediate goods are statistically significant in both

estimation periods; however, the level of significance decreased

from 1990 to 2001.

Year-ahead, out-of-sample forecasts also were constructed,

using 8-year blocks of data to estimate the VARs. The 8-year

blocks were rolled forward on a monthly basis. In contrast to the

previous R-squared values and Wald tests, the average absolute

error of the forecasts from 1982 to 1988 was higher than from

1990 to 2001, indicating greater predictive power in the second

period. However, the out-of-sample tests were somewhat limited,

because 8 years of data were necessary to estimate the VAR

models. Accordingly, because the time series began in 1974, no

forecasts could be constructed prior to 1982.

Impulse response, variance decomposition

VAR coefficients are difficult to interpret, due to the multivariate

nature of the models. Accordingly, impulse response functions
Monthly Labor Review November 2002 7

and variance decompositions were developed to overcome this

difficulty. Impulse response functions measure the effect of a

one-standard-deviation innovation of a variable on current and

future values of the variables in a system of equations. Variance

decompositions show the percentage of forecast variance (that

is, the percentage of variance in the forecast) in one variable of

the VAR that is explained by innovations of all variables within

the VAR.

13 This section constructs impulse response functions

and variance decompositions that examine the causal

relationships between the SOP indexes and the CPI.

Innovations within a VAR are generally not contemporaneously independent of each other: a random innovation to

one variable often occurs simultaneously with innovations to

other variables in the system. To overcome this problem,

innovations can be orthogonalized by a Cholesky decomposition in which the covariance matrix of the resulting

residuals is lower triangular. Therefore, an innovation to one

variable within a VAR affects only variables ordered after that

variable, and the VAR is given a causal interpretation.14

In order to calculate the impulse response functions and

variance decompositions, the innovations were orthogonalized

by a Cholesky decomposition following the ordering crude

goods, intermediate goods, finished goods, CPI. This ordering

was chosen because economic theory predicts that price

changes transmit forward through the stages of processing as

input costs of producers change. The unrestricted VARs

estimated in the previous section suggest that this has been the

case historically. Chart 2 presents cumulated impulse response

functions of one-standard-deviation innovations from 1974 to

1989 and from 1990 to 2001. In addition, standard-error bands of

two standard deviations representing the statistical significance

of the responses are shown. The standard-error bands were

calculated with the software package EVIEWS 4.0.

The impulse response functions in chart 2 indicate that

during both subperiods price changes at various stages of

processing were transmitted forward through the SOP system.

While some causal price relationships appear to have weakened

since the 1990s (as previous analysis has suggested), others

have remained stable or even have strengthened.

From 1974 to 1989, price changes at the producer level were

transmitted forward to the CPI. During this period, shocks to

crude goods, intermediate goods, and finished goods produced

statistically significant responses from the CPI. (The impulse

response function is statistically significant when both standarderror bands are above or below zero on the y-axis.) For example,

a one-standard-deviation shock to crude goods (approximately

VAR test results

Null hypothesis

Chi-square Probability Chi-square Probability

 Dependent variable: crude

Intermediate = 0 .................................................................................. 3.0820 0.5442 4.2260 0.3763

Finished = 0........................................................................................ 5.2486 .2627 5.9309 .2044

CPI = 0 ............................................................................................... 3.5224 .4745 4.9347 .2941

Intermediate/finished/CPI = 0................................................................ 17.7327 .1241 13.4744 .3355

R 2..................................................................................................... .25 ... .38 ...

 Dependent variable: intermediate

Crude = 0 ........................................................................................... 21.3029 .0003 10.4637 .0333

Finished = 0........................................................................................ .7899 .9398 4.5748 .3338

CPI = 0 ............................................................................................... 3.0544 .5488 1.5539 .8170

Finished/CPI = 0 ................................................................................. 3.6722 .8854 5.6183 .6899

R 2..................................................................................................... .69 ... .61 ...

 Dependent variable: finished

Crude = 0 ........................................................................................... 7.9022 .0952 8.1216 .0872

Intermediate = 0 .................................................................................. 40.7242 .0000 14.7711 .0052

Crude/intermediate = 0 ......................................................................... 44.3922 .0000 15.6977 .0469

CPI = 0 ............................................................................................... 7.5823 .1081 10.0163 .0402

R 2..................................................................................................... .59 ... .19 ...

 Dependent variable: CPI

Crude = 0 ........................................................................................... 6.1217 .1902 13.1299 .0107

Intermediate = 0 .................................................................................. 5.7063 .2222 1.8955 .7550

Finished = 0........................................................................................ 11.6461 .0202 23.4214 .0001

Crude/intermediate/finished = 0............................................................. 49.8345 .0000 37.5086 .0002

R 2..................................................................................................... .59 ... .30 ...

NOTE: Probability is the p-value of the Wald statistic.

Table 1.

1974–89 1990–2001
Price Transmission

8 Monthly Labor Review November 2002

Chart 2. Accumulated responses to a one-standard-deviation shock, 1974–89 and 1990–2001

1 6 11 16 21 26 31 36 -30

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1974–89 Response function Plus or minus two 

standard deviations

Crude response to crude shock Crude response to intermediate shock Crude response to finished shock Crude response to CPI shock

1 6 11 16 21 26 31 36 -30

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Intermediate response to crude shock Intermediate response to intermediate shock Intermediate response to CPI shock

1 6 11 16 21 26 31 36 -10

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Intermediate response to finished shock

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Finished response to crude shock Finished response to intermediate shock Finished response to finished shock Finished response to CPI shock

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CPI response to crude shock CPI response to intermediate shock CPI response to CPI shock

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CPI response to finished shock

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NOTE: For each plot, horizontal axis is number of months and vertical axis is percent change.

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Monthly Labor Review November 2002 9

Chart 2. Continued—Accumulated responses to a one-standard-deviation shock, 1974–89 and 

 1990–2001

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1990–2001 Response function Plus or minus two 

standard deviations

Crude response to crude shock Crude response to intermediate shock Crude response to CPI shock

1 6 11 16 21 26 31 36

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Crude response to finished shock

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Intermediate response to crude shock Intermediate response to intermediate shock Intermediate response to CPI shock

1 6 11 16 21 26 31 36 -8

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Intermediate response to finished shock

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Finished response to crude shock Finished response to intermediate shock Finished response to CPI shock

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Finished response to finished shock

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CPI response to crude shock CPI response to intermediate shock CPI response to CPI shock

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CPI response to finished shock

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NOTE: For each plot, horizontal axis is number of months and vertical axis is percent change.
Price Transmission

10 Monthly Labor Review November 2002

22.1 percent) resulted in a 4.4-percent increase in the CPI.

Beginning in 1990, the power of commodity indexes as

predictors of CPI inflation appears to have declined: only

innovations in the finished-goods index affected the CPI

significantly since then. However, in comparison with the

earlier subperiod, shocks to finished goods resulted in

stronger and more statistically significant responses from

the CPI.

Price changes at the crude and intermediate stages of

processing were passed forward to finished goods from 1974

to 1989. Shocks to the indexes for crude and intermediate

goods caused statistically significant changes in the prices

of finished goods. Over the 1990–2001 subperiod, the power

of the crude- and intermediate-goods indexes to predict

movements in finished-goods prices decreased substantially.

Impulse response functions estimated during this period show

that changes in crude-goods prices did not significantly affect

the finished-goods index and that shocks to intermediate-goods

prices produced only marginally significant responses from the

finished-goods index.

Price changes in crude goods have consistently caused

changes in the index for intermediate goods. During both

subperiods, an unanticipated shock to crude-goods prices

can be traced forward to a statistically significant movement

in the index for intermediate goods.

Table 2 presents variance decompositions of the SOP

indexes after 36 months. Like the impulse response functions,

the variance decompositions were estimated from the two

subperiods 1974–89 and 1990–2001.

The variance decompositions in table 2 reinforce the

earlier finding in this article that the nature of the transmission

of inflation within the SOP system has changed over the past

three decades. The share of the CPI’s forecast error variance

that can be explained by commodity price shocks differed

substantially between the two subperiods. Innovations to

crude and intermediate goods accounted for a significantly

larger share of the CPI’s forecast error variance from 1974 to

1989 than from 1990 to 2001. However, the share of the CPI’s

forecast error variance explained by shocks to finished goods

has increased approximately 12 percent since 1990.

Variance decompositions at the earlier stages of processing

also differ between the two subperiods. Shocks to crude and

intermediate goods accounted for a considerably smaller share

of finished goods’ forecast error variance from 1990 to 2001 than

from 1974 to 1989. However, since 1990, the portion of intermediate goods’ forecast error variance that can be explained by

price changes in crude goods has increased by approximately

15 percent.

THIS ARTICLE HAS PRESENTED AN EMPIRICAL INVESTIGATION of the nature and consistency of causal price

relationships within the stage-of-processing (SOP) system. A

visual examination of price movements shows that, during

the 1970s and 1980s, changes in the indexes for the prices of

crude and intermediate goods often preceded changes in the

CPI. Since the early 1990s, however, price movements at early

stages of processing do not appear to have foreshadowed

movements in the CPI. Wald tests of VA R coefficients

estimated with data from 1974 to 1989 and from 1990 to 2001

indicated that, in the equations for the CPI, finished goods,

and intermediate goods, the combined lagged values of all

variables in earlier stages of processing to the dependent

variable were jointly significant. However, the levels of

significance, as well as the equations’ R-squared values,

declined in the 1990–2001 subperiod.

Confidence regions of impulse response functions estimated

from the VAR models’ coefficients illustrate that, from 1974 to

1989, price movements in crude, intermediate, and finished

goods were transmitted forward to the CPI. In comparison, from

1990 to 2001, only changes in the PPI for finished goods caused

movements in the CPI. Variance decompositions told a similar

story, with the share of the CPI’s forecast error variance explained

by shocks to crude and intermediate goods declining from

1990 to 2001. However, the share of the CPI’s forecast error

variance explained by finished goods increased over the same

period.

Variance decompositions

Crude Intermediate Finished CPI

Variance decompositions after 36 months, 1974–89:

 Crude ............................................................................................. 92.03 4.05 2.14 1.79

 Intermediate ................................................................................... 29.69 68.54 .56 1.21

 Finished ......................................................................................... 10.35 32.99 53.45 3.20

CPI ................................................................................................................................................... 12.06 24.43 8.05 55.46

Variance decompositions after 36 months, 1990–2001:

 Crude ............................................................................................. 92.01 2.24 2.51 3.23

 Intermediate ................................................................................... 44.87 50.29 2.24 2.60

 Finished ......................................................................................... 5.24 9.89 79.83 5.03

CPI ................................................................................................................................................... 8.02 4.58 20.06 67.34

Decomposition variable

Percentage of forecast errors due to—

Table 2.
Monthly Labor Review November 2002 11

At the earlier stages of processing, impulse response

functions show that innovations to crude- and intermediategoods prices caused changes in the finished-goods index

from 1974 to 1989. As with the CPI, movements in the prices

of crude goods were not passed forward to finished-goods

prices from 1990 to 2001. Price changes at the intermediate

stage of processing were passed forward from 1990 to 2001;

however, the relationship was relatively weaker during that

subperiod. Variance decompositions show a dramatic

decrease in the forecast error variance of finished-goods

prices explained by innovations to crude- and intermediategoods prices between the two subperiods. Impulse response

functions and variance decompositions illustrate that there

prices of intermediate goods during both periods. In fact, the

relationship was relatively stronger in the second subperiod.

Taken as a whole, the results from the various empirical

tests suggest that there was significant forward price

transmission in both subperiods examined. However, from

1990 to 2001, price transmission from the earlier stages of

processing—crude and intermediate goods—to the later

stages of processing—finished goods and the CPI—was

weaker relative to the earlier subperiod. By contrast, the

causal price relationships between the finished-goods index

and the CPI and between crude- and intermediate-goods

prices have strengthened since 1990.

Notes

1 Handbook of Methods (Bureau of Labor Statistics, 1997), esp.

Chapter 14, “Producer Price Indexes.”

2

 S. Brock Blomberg and Ethan S. Harris, “The CommodityConsumer Price Connection: Fact or Fable?” Federal Reserve Bank of

New York Economic Policy Review, October 1995, pp. 21–38.

3

 Fred Furlong and Robert Ingenito, “Commodity Prices and

Inflation,” Federal Reserve Bank of San Francisco Economic Review,

1996, no. 2, pp. 27–47.

4

 Todd Clark, “Do Producer Prices Lead Consumer Prices?” Federal

Reserve Bank of Kansas City Economic Review, third quarter, 1995,

pp. 26–39.

5

 C. Alan Garner, “How Useful Are the Leading Indicators of

Inflation?” Federal Reserve Bank of Kansas City Economic Review,

second quarter, 1995, pp. 5–18.

6

 Tae-Hwy Lee and Stuart Scott, “Investigating Inflation Transmission by Stages of Processing,” in Robert F. Engle and Halbert

White (eds.), Cointegration, Causality, and Forecasting: A Festschrift

in Honor of Clive W. J. Granger (Oxford, U.K., Oxford University

Press, 1999), pp. 283–300.

7

 William H. Greene, Econometric Analysis (Upper Saddle River,

NJ, Prentice Hall, 1997), esp. pp. 815–16.

8

 David A. Dickey and Wayne A. Fuller, “Distribution of the

Estimators for Autoregressive Time Series with a Unit Root,” Journal

of the American Statistical Association, vol. 74, 1979, pp. 427–31.

Also in John Dinardo and Jack Johnston, Econometric Methods (New

York, McGraw Hill, 1996), esp. pp. 224–25.

9

 Philip Hans Franses, Time Series Models for Business and

Economic Forecasting (Cambridge, U.K., Cambridge University Press,

1998).

10 See Clark, “Do Producer Prices Lead Consumer Prices?” and

Blomberg and Harris, “The Commodity-Consumer Price Connection.”

11 Greene, Econometric Analysis, p. 161.

12 Ibid., p. 253.

13 Dinardo and Johnston, Econometric Methods, pp. 298–301.

14 Ibid.

was a causal link between the prices of crude goods and the