Monthly Labor Review July 2003 3

Consumer Gasoline Prices

Consumer gasoline prices:

an empirical investigation

A structural vector autoregression model

indicates that price changes for consumer gasoline

have been driven by changes in supply

rather than changes in demand

Jonathan Weinhagen

Jonathan Weinhagen

is an economist in the

Office of Prices and

Living Conditions,

Bureau of Labor

Statistics. The views

expressed in this

article are those of

the author and do

not necessarily reflect

those of the Bureau of

Labor Statistics or the

Department of Labor.

A

ccording to the BLS Consumer Expenditure Survey, the average consumer spent

approximately $1,300 on gasoline and

motor oil in 2000, an increase of 22.4 percent over

the 1999 figure. Over the same period, the average

price of gasoline increased 36.3 percent,1 indicating

that price changes within the gasoline market can

substantially affect consumers’ expenses. Conventional reasoning suggests that the high level of

volatility for gasoline prices is the result of supply

forces, as the price of crude petroleum changes rapidly

due to production decisions of the Organization of

Petroleum Exporting Countries (OPEC) nations.

However, shifts in demand also can cause variations

in gasoline prices. The purpose of this article is to

examine the nature of price changes for consumer

gasoline, using econometric techniques as well as

historical evidence.

The second section of the article analyzes the

impact of crude-oil supply shocks on prices at

various stages of gasoline production by visually

examining those price changes for crude oil,

producer gasoline, and consumer gasoline which

occurred subsequent to interruptions in the supply

of crude petroleum. The major supply shocks

considered are the Yom Kippur War, the Iranian

Revolution, the Iran-Iraq War, the Persian Gulf

War, and a 1999 OPEC production cut.

The article’s third section constructs a structural

simultaneous-equations model of the market for

consumer gasoline to determine the effects of

changes in supply and demand on the price of

gasoline. The model developed is a five-variable

structural vector autoregression constructed from

the Producer Price Indexes (PPI’s) for crude

petroleum and gasoline, the Consumer Price

Index (CPI) for gasoline, the quantity of gasoline

consumed domestically, and the industrial

production index. The final section of the article

presents its conclusion.

Historical evidence

The impact of supply shocks on prices at various

stages of processing within the gasoline market

can be analyzed by visually examining historical

price movements for crude petroleum, producer

gasoline, and consumer gasoline. The actions of

the OPEC cartel enable petroleum-based supply

shocks to be easily identified and their effects

on prices throughout the gasoline market to be

examined. The analysis begins with a historical

overview of OPEC.

OPEC’s history. OPEC was established in

September 1960 at the Baghdad Conference.

Initially, the cartel included Iran, Iraq, Kuwait,

Saudi Arabia, and Venezuela. By the end of 1971,

Qatar, Indonesia, Libya, the United Arab Emirates,

and Nigeria had joined the organization. From

OPEC’s inception until the early 1970s, the cartel

was unable to exert any significant control over

crude-petroleum prices. Prices for crude petroleum remained relatively stable in nominal

terms at around $3.00 per barrel from 1958 to 1970

and fell in real terms over the same period.2

During the 1970s, OPEC’s ability to influence

crude-petroleum prices increased substantially

due to rising demand for petroleum products 3

and the strength the organization gained from
Consumer Gasoline Prices

4 Monthly Labor Review July 2003

the addition of new members. OPEC’s increasing power in the

petroleum market becomes apparent from the effects its supply

decisions have had on petroleum prices since the 1970s.

OPEC supply shocks. Chart 1 displays the PPI’s for crude

petroleum and gasoline and the CPI for gasoline. For a simplified

comparison, the three indexes were rebased to March 1973 = 100.

The first major interruption in OPEC’s petroleum supply resulted

from an oil embargo launched in connection with the Yom Kippur

War. As a result, U.S. imports of crude petroleum fell by

approximately 30 percent while the embargo was in place.4 The

drastic reduction in the supply of crude petroleum caused

domestic prices to rise 67 percent from October 1973 to the end

of 1974. Over the same period, domestic prices for wholesale and

consumer gasoline increased 67 and 32 percent, respectively,

indicating a strong pass-through relationship between crudepetroleum prices and gasoline prices. (See chart 1.)

The second crude-petroleum supply shock took place at the

time of the Iranian revolution, in conjunction with the Iran-Iraq

War. The shock began as a result of panic in the world oil market

caused by the revolution. The situation worsened when Iran

prohibited oil exports to U.S. firms after the U.S. administration

froze Iranian assets in the United States.5 The war between Iran

and Iraq exacerbated the crisis, and Iran’s oil production declined

3.9 million barrels a day from 1978 to 1981. Furthermore, the war

caused other Persian Gulf countries to reduce their oil

production. By 1981, OPEC’s oil production fell 7 million barrels

per day, decreasing world oil production by 11.6 percent from its

1978 average.6 From November 1978 to October 1981, the price

of crude petroleum rose 172 percent. Increasing prices were

passed forward through the chain of production, with prices for

wholesale and consumer gasoline rising 150 and 103 percent, respectively, over the same period. (See chart 1.)

The third crude-oil interruption occurred in 1990 as

tensions between Iraq and Kuwait rose. On July 17, 1990,

Iraq accused Kuwait of overproducing oil and of stealing oil

from the Iraqi Rumaila oil fields. Iraq invaded Kuwait on

August 2, 1990, and the ensuing Gulf War resulted in a

reduction of about 4.3 million barrels of oil per day from Iraq

and Kuwait. This decrease in the oil supply caused world

production to decline by approximately 7.2 percent from its

average 1989 level. However, non-OPEC countries in Central

America, Western Europe, and the Far East, as well as the

United States, supplemented OPEC production to offset some

of the losses.7 Chart 1 shows that crude-petroleum prices

rose 155 percent between July 1990 and October 1990. Over

Chart 1. Historical price movements and supply shocks, 1973–2001

Percent Percent

NOTE: All year labels are for January.

Iran–Iraq War

Iranian Revolution

Persian Gulf War

1973 1975 1977 1979 1981 1983 1985 1987 1989 1991 1993 1995 1997 1999 2001

0

100

200

300

400

500

600

700

800

0

100

200

300

400

500

600

700

800

PPI gasoline CPI gasoline Crude oil

OPEC production cuts

Yom Kippur War
 Monthly Labor Review July 2003 5

the same period, domestic prices of gasoline at the wholesale

and consumer levels increased by 45 percent and 26 percent,

respectively, to reflect skyrocketing input costs.

The fourth significant crude-petroleum supply shock took

place in 1999, after OPEC reduced its production of oil by 1.7

million barrels per day, representing a 2.5-percent decline in world

oil production.8 In addition, U.S oil production decreased

approximately 6 percent from 1998 to 1999.9 Crude-petroleum

prices soared 277 percent from February 1999 to November 2000

in response to drastically reduced supplies. Gasoline prices at

the wholesale and consumer levels rose 114 and 55 percent,

respectively, due to increasing crude-petroleum input prices.

An examination of historical price movements for crude

petroleum, wholesale gasoline, and consumer gasoline indicates

that the production decisions of OPEC nations have considerable effects on prices within the gasoline market at all stages

of production. The historical price trends also suggest that price

volatility resulting from supply shocks diminishes at progressively more advanced stages of processing. In three out of

four instances, supply shocks increased prices for crude

petroleum more than they did wholesale gasoline prices, and in

all four instances crude-petroleum prices rose more than

consumer gasoline prices.

Model of gasoline price movements

To examine the source of variations in consumer gasoline prices

more rigorously, a five-variable structural vector autoregression

model of supply and demand within the gasoline market is

presented. Vector autoregressions are an econometric tool used

to study systems of interrelated time series in which all variables

in a system are expressed as a linear function of the lagged

values of every variable in the system. 10 A structural vector

autoregression model is developed by imposing theoretically

plausible contemporaneous restrictions on the error terms of

the unrestricted vector autoregression.

Unrestricted vector autoregression. A five-variable

unrestricted vector autoregression model was constructed

with 1974–2001 monthly data of the PPI’s for crude petroleum

and gasoline, the CPI for gasoline, the quantity of domestically consumed gasoline, and the industrial production index.

The PPI’s for crude petroleum and gasoline were included in

the model as supply variables, because both are major inputs

into the production of consumer gasoline. To account for

shifts in demand, the industrial production index, a major

determinant of gasoline demand, was included in the model.

All data are seasonally adjusted and were transformed into

percentage growth form by taking the first differences of the

natural logarithms of the data. Converting data to percentage

growth form usually induces stationarity, indicating that the

mean, variance, and covariance of the time series are independent

of time. Estimation of vector autoregressions with nonstationary

data is problematic, because tests used to estimate the significance

of the regressions’ coefficients will not be valid.11 Accordingly,

to test for stationarity, the augmented Dickey-Fuller test was

applied to the variables in percentage growth form; this is a onetailed test of the null hypothesis that the time series is not

stationary. A large negative test statistic rejects the null

hypothesis and implies that the time series is stationary.12 As

the following tabulation shows, the tests suggested that, at the

significance level of p = 0.01, all five time series were stationary

when they were expressed in percentage growth form:

Augmented

Dickey-Fuller

Variable statistic

Crude petroleum ........................................................... –8.87

PPI for gasoline .............................................................. –5.02

CPI for gasoline .............................................................. –5.22

Quantity of gasoline ..................................................... –4.71

Industrial production .................................................... –5.97

The Akaike, Schwarz, and Hannan-Quinn information criteria

were implemented to compare the performance of the vector

autoregression model with various lag length specifications. The

Schwarz and Hannan-Quinn criteria indicated that a vector

autoregression whose equations have two lags is optimal, while

the Akaike criterion suggested a three-lag regression. The twolag specification suggested by the Schwarz and Hannan-Quinn

criterion was chosen, and the unrestricted vector autoregression

was estimated by using ordinary least squares.

Structural vector autoregression. Innovations within a vector

autoregression are generally contemporaneously correlated with

each other: a random innovation to one variable often occurs

simultaneously with innovations to other variables in the

system.13 To recover the contemporaneous relationships among

the vector autoregression’s innovations, allowing for economically meaningful conclusions, it is necessary to orthogonalize

the residuals from the unrestricted vector autoregression. The

conventional method of orthogonalization is based on the

Cholesky decomposition, which assumes that the residuals have

a recursive structure.14 However, this approach is often not

supported by economic theory and leads to a series of orthogonal shocks that have no particular meaning. Alternatively, the

structural impulses can be obtained by imposing theoretically

plausible restrictions on the vector autoregression’s residuals.15

The latter of these two approaches is taken in this article.

The estimated variance-covariance matrix of the unrestricted vector autoregression’s residuals contains n(n + 1)/2

distinct elements. Recovering the structural disturbances

requires the estimation of an n × n matrix of parameters.

Therefore, n

2 – n(n + 1)/2 = n(n – 1)/2 additional restrictions

are required to recover the structural disturbances. These
Consumer Gasoline Prices

6 Monthly Labor Review July 2003

additional restrictions can be obtained by letting the coefficients of the structural parameters vanish. Consequently, in

the case of the five-variable vector autoregression model that

was constructed, identification of the structural disturbances

requires at least 10 restrictions.

The following structural specification of the contemporaneous interactions among the vector autoregression’s

innovations was estimated:

(1) PCP = ß1QIP + u1;

(2) PPG = ß2PCP + u2;

(3) PCG = ß3PCP + ß4PPG + u3;

(4) PCG = –ß5QCG + ß6QIP + u4;

(5) QIP = u5.

In the preceding equations, PCP, PPG, PCG, QCG, and QIP refer

to the innovations in, respectively, the PPI for crude petroleum,

the PPI for gasoline, the CPI for gasoline, the quantity of gasoline

consumed, and the quantity of industrial production, as

estimated by the unrestricted vector autoregression. The u’s are

uncorrelated error terms. For symmetry purposes, the fourth

equation is normalized to the price of consumer gasoline.

In all of the equations, all of the coefficients are positive, with

the exception of the coefficient of QCG . The structure of the

contemporaneous relationships in the system is derived by

assuming a horizontal supply curve and a downward -sloping

demand curve in the market for consumer gasoline. Under this

framework, shifts in the supply curve affect price and quantity,

whereas shifts in the demand curve change only quantity. The

error terms u1, u2, and u3 represent supply shocks, u4 is a demand

shock, and u5 is a simultaneous shock to supply and demand.

Given the assumptions about the slopes of the demand and supply

curves, the following relationships hold: (1) crude-petroleum prices

vary as a result of innovations in industrial production, reflecting

shifts in demand due to changes in the level of production; (2)

producer gasoline prices are affected by innovations to crude

petroleum, which is a major material input to the production of

producer gasoline; (3) consumer gasoline prices vary with

innovations to crude petroleum and producer gasoline, both of

which are inputs to the production of consumer gasoline; (4)

consumer gasoline prices are affected by innovations to both

the quantities of gasoline consumed and industrial production;

and (5) the quantity of industrial production is exogenous to the

system and is not affected by innovations to any variables.

To illustrate how shocks to demand and supply affect

consumer gasoline market equilibrium, chart 2 shows the effects

Chart 2. Effects of an industrial production shock on the market for consumer gazoline

Q0 Q1 Q

S1

S0

P

P1

P0

D1

D0

S = supply; D = demand; P = price; Q = quantity 

Effects of an industrial production shock on the market for consumer gasoline
 Monthly Labor Review July 2003 7

of a shock to u5 (industrial production) on the equilibrium

price and quantity of consumer gasoline. A shock to industrial

production causes individuals to desire more gasoline and

shifts the demand curve from D0 to D1. In turn, the supply

curve shifts from S0 to S1, reflecting increased production

costs as input prices are driven up by the change in industrial

production. The effect on price is positive and results from

the shifting supply curve. The effect on quantity is ambiguous and depends on the relative size of shifts in the

demand and supply curves. In the chart, it is clearly seen that

the positive effect on quantity resulting from the shift in

demand outweighs the negative effect on quantity from the

shift in supply.

The results of the estimation of the structural coefficients

are as follows, where (1) indicates significance at the level of p

= 0.1, (2) indicates significance at the level of p = 0.05, and (3)

indicates significance at the level of p = 0.0001:

PCP = 0.36QIP + u1;

PPG = 0.38PCP(3) + u2;

PCG = 0.014PCP + 0.42PPG(3) + u3;

PCG = –12.5QCG(2) + 3.13QIP(1) + u4;

QIP = u5.

The signs of the estimated coefficients are as anticipated.

Innovations to crude petroleum are positively affected by

shocks to industrial production. Innovations to producer

gasoline prices are positively correlated with crude-petroleum

innovations. Shocks to the CPI for gasoline are positively

affected by innovations to crude petroleum and to the PPI for

gasoline. Innovations to the CPI for gasoline are negatively

correlated with shocks to the quantity of consumer gasoline

and are positively correlated with industrial production

shocks.

The system of structural disturbances is overidentified,

because estimation required only 10 restrictions, whereas 14

were provided. The overidentification of the system allowed

the likelihood ratio (LR) test for overidentification to be

applied. The LR test is a test of the validity of the system’s

restrictions, where the null hypothesis is that the identifying

restrictions are valid. A p-value of 0.01 or 0.05 is required to

reject the null hypothesis. The test’s chi-square statistic and

p-value were 4.24 and 0.37, respectively. Therefore, the null

hypothesis was not rejected, and the restrictions were found

to be valid.

Accumulated impulse response functions were constructed

from the vector autoregression’s coefficients with the use of the

orthogonalized set of residuals. Impulse response functions

measure the effect of a one-standard-deviation innovation of a

variable on current and future values of the other variables in a

system of equations.16 Standard error bands demonstrating the

statistical significance of the impulse response functions also

were constructed, using analytical methods. The impulse

response function is statistically significant when both standard

error bands either are above zero or are below zero on the y-axis.

 Chart 3 presents the accumulated impulse response

functions. The first row of the chart indicates that, on the one

hand, the PPI for crude petroleum is not significantly affected

by unanticipated changes in the PPI for gasoline, the CPI for

gasoline, or the quantity of consumer gasoline. On the other

hand, shocks to the quantity of industrial pro-duction result

in marginally significant changes in crude-petroleum prices.

The second row suggests that innovations to crudepetroleum prices strongly affect producer gasoline prices and

that unanticipated changes in the CPI for gasoline and the

quantity of industrial production produce only marginal changes

in the PPI for gasoline. By contrast, shocks to the quantity of

gasoline consumed do not affect producer gasoline prices. The

third row indicates that innovations in crude-petroleum prices

and producer gasoline prices produce highly significant changes

in the CPI for gasoline and that shocks to the quantity of

industrial production affect consumer gasoline prices only

marginally. Conversely, unanticipated changes in the quantity

of gasoline consumed do not affect the CPI for gasoline. The

fourth row of the chart shows that price shocks to crude

petroleum, producer gasoline, and consumer gasoline tend to

reduce the quantity of gasoline consumed, whereas innovations

to the quantity of industrial production in-crease the quantity of

gasoline consumed. The last row suggests that none of the

variables in the system significantly affect the quantity of

industrial production.

Variance decompositions were also constructed from the

model. Variance decompositions show the percentage of

forecast variance in one variable of the vector autoregression

caused by innovations in the other variables.17 The variance

decompositions obtained from the analysis are presented in

table 1.

The variance decomposition of the CPI for gasoline implies

that the majority of the forecast error variance in consumer

gasoline prices results from price shocks to production inputs.

Innovations in crude petroleum and in the PPI for gasoline

account for 73.74 percent of the forecast errors in the CPI for

gasoline (40.14 percent from crude petroleum and 33.6 percent

from producer gasoline). Conversely, shocks to the quantities of

gasoline and industrial production account for only 0.11 and

1.82 percent, respectively, of the CPI’s forecast error variance.

THIS ARTICLE HAS PRESENTED BOTH HISTORICAL AND

EMPIRICAL EVIDENCE in examining the source of price variations

within the gasoline market. The main finding of the article is that

price changes for consumer gasoline have histori-cally been

driven by changes in supply as opposed to demand.

The initial approach taken was to identify historical supply

shocks within the crude-petroleum market and examine how
Consumer Gasoline Prices

8 Monthly Labor Review July 2003

Plus or minus two 

standard deviations

Chart 3. Accumulated impulse response functions

Response of

PPI for crude 

petroleum

Shock to PPI for crude petroleum Shock to PPI for gasoline

1 2 3 4 5 6 7 8 9 10 11 12

-0.020

0.000

0.020

0.040

0.060

0.080

0.100

0.120

1 2 3 4 5 6 7 8 9 10 11 12

-0.020

0.000

0.020

0.040

0.060

0.080

0.100

0.120

1 2 3 4 5 6 7 8 9 10 11 12

-0.010

0.000

0.010

0.020

0.030

0.040

0.050

0.060

0.070

1 2 3 4 5 6 7 8 9 10 11 12

-0.010

0.000

0.010

0.020

0.030

0.040

0.050

0.060

0.070

1 2 3 4 5 6 7 8 9 10 11 12

-0.010

0.000

0.010

0.020

0.030

0.040

1 2 3 4 5 6 7 8 9 10 11 12

-0.010

0.000

0.010

0.020

0.030

0.040

1 2 3 4 5 6 7 8 9 10 11 12

-0.010

-0.005

0.000

0.005

0.010

0.015

0.020

1 2 3 4 5 6 7 8 9 10 11 12

-0.010

-0.005

0.000

0.005

0.010

0.015

0.020

1 2 3 4 5 6 7 8 9 10 11 12

-0.004

0.000

0.004

0.008

0.012

0.016

0.020

1 2 3 4 5 6 7 8 9 10 11 12

-0.004

0.000

0.004

0.008

0.012

0.016

0.020

Response of

PPI for 

gasoline

Response of

CPI for 

gasoline

Response of 

quantity of 

gasoline

Response of 

Industrial 

production

Percent 

change

Percent 

change

Period Period

Response function
 Monthly Labor Review July 2003 9

Chart 3. Continued—Accumulated impulse response functions

Shock to quantity of gasoline Shock to industrial production

1 2 3 4 5 6 7 8 9 10 11 12

-0.020

0.000

0.020

0.040

0.060

0.080

0.100

0.120

1 2 3 4 5 6 7 8 9 10 11 12

-0.020

0.000

0.020

0.040

0.060

0.080

0.100

0.120

1 2 3 4 5 6 7 8 9 10 11 12

-0.010

0.000

0.010

0.020

0.030

0.040

0.050

0.060

0.070

1 2 3 4 5 6 7 8 9 10 11 12

-0.010

0.000

0.010

0.020

0.030

0.040

0.050

0.060

0.070

1 2 3 4 5 6 7 8 9 10 11 12

-0.010

0.000

0.010

0.020

0.030

0.040

1 2 3 4 5 6 7 8 9 10 11 12

-0.010

0.000

0.010

0.020

0.030

0.040

1 2 3 4 5 6 7 8 9 10 11 12

-0.010

-0.005

0.000

0.005

0.010

0.015

0.020

1 2 3 4 5 6 7 8 9 10 11 12

-0.010

-0.005

0.000

0.005

0.010

0.015

0.020

1 2 3 4 5 6 7 8 9 10 11 12

-0.004

0.000

0.004

0.008

0.012

0.016

0.020

1 2 3 4 5 6 7 8 9 10 11 12

-0.004

0.000

0.004

0.008

0.012

0.016

0.020

1 2 3 4 5 6 7 8 9 10 11 12

-0.020

0.000

0.020

0.040

0.060

0.080

0.100

0.120

1 2 3 4 5 6 7 8 9 10 11 12

-0.010

0.000

0.010

0.020

0.030

0.040

0.050

0.060

0.070

1 2 3 4 5 6 7 8 9 10 11 12

-0.010

0.000

0.010

0.020

0.030

0.040

1 2 3 4 5 6 7 8 9 10 11 12

-0.010

-0.005

0.000

0.005

0.010

0.015

0.020

1 2 3 4 5 6 7 8 9 10 11 12

-0.004

0.000

0.004

0.008

0.012

0.016

0.020

Period Period Period

Percent 

change

Percent 

change

Percent 

change

Shock to CPI for gasoline

Plus or minus two 

standard deviations Response function
Consumer Gasoline Prices

10 Monthly Labor Review July 2003

prices at various stages of processing responded to the shocks.

In all cases examined, interruptions in the supply of crude

petroleum resulted in significant increases in the prices of crude

petroleum, wholesale gasoline, and consumer gasoline.

To analyze pricing relationships within the gasoline market

more formally, a five-variable structural vector autoregression

model of the gasoline market was developed, using the PPI’s

for crude petroleum and gasoline, the CPI for gasoline, the

quantity of domestically consumed gasoline, and the inVariance decompositions after 12 months

Percentage of forecast errors due to—

PPI for crude PPI for CPI for Quantity of Industrial

 Petroleum gasoline gasoline gasoline production

PPI for crude petroleum ................................................................ 97.82 0.31 .26 0.16 1.46

PPI for gasoline ........................................................................... 43.08 52.57 2.42 .11 1.82

CPI for gasoline .......................................................................... 40.14 33.60 23.97 .40 1.88

Quantity of gasoline .................................................................... .49 1.28 .97 92.79 4.47

Industrial production ................................................................... .71 .32 .05 .56 98.36

Decomposition

variable

Table 1.

dustrial production index. Impulse response functions

constructed from the model’s coefficients imply that price

changes of inputs to consumer gasoline (crude petroleum

and PPI gasoline) significantly affect the CPI for gasoline, but

that changes in demand (industrial production) affect gasoline

prices only marginally. In addition, variance decompositions

indicated that the majority of the forecast variance in consumer

gasoline prices can be explained by price shocks to inputs, as

opposed to shocks to demand.

Notes

1

 The 36.3-percent figure represents the percent increase in the

annual average of the Consumer Price Index for gasoline from 1999

to 2000.

2

 James L. Williams, Energy Economics Newsletter, on the Internet

at http://www.wtrg.com/prices.htm.

3

Ibid.

4 Petroleum Chronology of Events 1970–2000 (U.S. Department

of Energy, 2002).

5

 Williams, Energy Economics Newsletter.

6 Petroleum Chronology of Events.

7

Ibid.

8 Oil Reserve Fact Sheet (U.S. Department of Energy, 2000).

9

 Eleni Xenofondos and William F. Snyders, “Rising producer prices

in 1999 dominated by energy goods,” Monthly Labor Review, August

2000, pp. 15–25.

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

NJ, Prentice Hall, 1997); see especially pp. 815–16.

11 Philip Hans Franses, Time Series Models for Business and

Economic Forecasting (Cambridge, U.K., and New York, Cambridge

University Press, 1998).

12 Jack Johnston and John DiNardo, Econometric Methods (New

York, McGraw-Hill, 1997); see especially pp. 224–25.

13 Ibid., p. 299.

14 Christopher A. Sims, “Macroeconomics and Reality,” Econometrica, January 1980, pp. 1–48.

15 Ben Bernanke, “Alternative Explanations of the Money-Income

Correlation,” in Karl Brunner and Allan Meltzer (eds.), Real Business

Cycles, Real Exchange Rates, and Actual Policies, Carnegie-Rochester

Conference Series on Public Policy, Autumn 1986, pp. 49–99

(Amsterdam, North-Holland, 1986); see also Christopher A. Sims,

“Are Forecasting Models Usable for Policy Analysis,” Federal Reserve

Bank of Minneapolis Quarterly Review, Winter 1986, pp. 2–16.

16 Johnston and DiNardo, Econometric Methods, pp. 299–300.

17 Ibid., p. 301.