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A Series of Occasional Papers in Draft Form Prepared by Members'o
A LINEAR MODEL OF THE LONG-RUN
NEUTRALITY OF MONEY
Thomas A. Gittings
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A Linear Model of the Long-Run Neutrality of Money
by
Thomas A. Gittings
Department of Research
Federal Reserve Bank of Chicago
The views expressed herein are solely those of the author and do not
necessarily represent the views of the Federal Reserve Bank of Chicago
or the Federal Reserve System. The material contained is of a prelimi
nary nature, is circulated to stimulate discussion, and is not to be
quoted without permission.
A LINEAR MODEL OF THE LONG-RUN NEUTRALITY OF MONEY
by
Thomas A. Gittings
Federal Reserve Bank of Chicago
One of the oldest verbal theories of economics is the quantity theory
of money.
Over the last two hundred years this general theory has been
presented in a variety of forms.
The common assumption of these alternative
approaches is that a change in the quantity of money causes a proportional
change in the level of prices and does not affect the level of real output
in the long run.
The theoretical arguments for this theory have been
developed most clearly by Irving Fisher (2) and Milton Friedman (4).
An extension of the quantity theory of money assumes that a change in
the rate of growth of money
causes, in the long run, an equal change in
the rate of inflation and does not have any permanent effect on real output
or employment.
In recent times this additional argument has been developed
in the extensive literature on the lack of a permanent trade-off between the
rate of inflation and the rate of unemployment.
The pioneering work in this
area was done by Milton Friedman (3) and Edmund Phelps (5).
The purpose of this paper is to translate these economic assumptions into
mathematical constraints that then are imposed on a small macroeconomic model.
Mathematically this model consists of linear ordinary difference equations,
where one of the external forces is assumed to be a weighted average of a
money variable.
The first part of this paper reviews the basic assumptions
of the quantity theory of money.
The derivation of the constraints and
the corresponding transformation of the data are presented in the
appendixes.
The second part of the paper examines some of the problems
that arise in estimating and comparing these constrained models.
-2-
In its simplest form the quantity theory of money states that, other
things being equal, doubling the quantity of money will cause of doubling of
prices.
This theory is an assumption about how an economic system works
in our physical universe.
logic.
It is not merely a simple if-then statement of
As Alexander Del Mar (1) was careful to specify over a hundred years
ago, this latter interpretation does ,fviolence to Nature, whose movements
are performed only in time; an element, of which logic has usually taken but
little account."
Del Mar went on to state "whilst the volume of money might
be increased or diminished instantly, the resulting movement of prices would
only occur after an interval of time."
Irving Fisher (2) was very careful to make this distinction between the
short-run and long-run effects of a change in the quantity of money.
In his
words:
We have emphasized the fact that the strictly proportional
effect on prices of an increase in M [the quantity of money]
is only the normal or ultimate effect after transition periods
are over. The proposition that prices vary with money holds
true only in comparing two imaginary periods for each of
which prices are stationary or are moving alike upward or
downward and at the same rate.
He characterizes the dynamic relationship between money and the general
level of prices with the following analogy:
The peculiar effects during transition periods are
analogous to the peculiar effects in starting or stopping
a train of cars. Normally the caboose keeps exact pace
with the locomotive, but when the train is starting or
stopping this relationship is modified by the gradual
transmission of effects through the intervening cars. Any
special shock to one car is similarly transmitted to
all the others and to the locomotive.
)
-3-
Although Fisher saw that a "sudden" change in the quantity of money
initially would affect the volume of real output or trade, he considered this
effect to be a temporary one.
In terms of long run or "ultimate" effects, he
assumed that "An inflation of the currency cannot increase the product of farms
or factories, nor the speed of freight trains or ships.
The stream of business
depends on natural resources and technical conditions, not on the quantity of
money."
Under a system of fiat money, like we have today, Fisher thought that a
change in the quantity of money would not "appreciably affect the quantity of
goods sold for money."
He concluded that
. . .the issue of paper money may affect the paper and printing
trades, the employment of bank and government clerks, etc. In
fact, there is no end to the minute changes in the Q Ts [measures
of real output] which the changes mentioned, and others might
bring about. But from a practical or statistical point of view
they amount to nothing, for they could not add to nor subtract
one tenth of 1 per cent from the general aggregate of trade.
Notice that the quantity theory of money focuses on the hypothesized effects
of just one of a myriad of factors that determine the levels of prices
and real outputs.
On this matter Fisher wrote
The importance and reality (sic) of this proposition are
not dimished in the least by the fact that these othere causes
do not historically remain quiescent and allow the effect on the
p fs [prices] of an increase in M [money] to be seen alone. The
effects of M are blended with the effects of changes in the other
factors in the equation of exchange just as the effects of gravity
upon a falling body are blended with the effects of the resistance
of the atmosphere.
In order to translate the preceeding arguments into a mathematical
model, it is necessary to articulate some of the assumptions that are being
-4-
made.
This theory hypothesizes a dynamic relationship between a measure of
the quantity of money (M) and measures of a price index (P) and the rate of
producing real output or the level of real transactions (Q).
We can define
a measure of nominal output or transactions (Y) as the product of the price
index and the measure of real output.
Algebraically this definition can be
written as
1)
Y = PQ
Let y, p, q, and m represent the logarithms of Y, P. Q, and M, respectively.
Equation 1 can be expressed in the following linear form
2)
y = P + q,
where each of these variables are assumed to be functions of m, time (t),
and whatever other variables one wishes to included in a model.
Given the accounting identity between y, p, and q, there are three possible
ways to formulate a model.
We can specify directly the dynamic linkage
between m and y and p, between m and y and q, or between m and
p and q,
and then use the accounting identity (equation 2) to determine the corresponding
values of q, p, or y, respectively.
These three alternatives correspond to
the three versions of the dynamic model that are estimated in this paper.
The basic form of each of these models consists of an ordinary difference
equation, where the time step should be relatively "small11 with respect to the
speed of adjustment of the economy.
Within the context of this mathematical
framework, the current value of an economic variable is assumed to be a
function of the lagged values of this variable plus a summation of other
forces, one
of which is a weighted average of the quantity of money.
-5-
Furthermore, these difference equations are assumed to be linear and have
constant coefficients.
Each version of the models consists of the accounting
equation plus two of the following three equations that need to be estimated:
3)
N
y(t) = £
ciyCrOyCt-i) +
i=i
V
4)
p(t) =
bp (j)m(t-j) + 0p
j=°
N
q(t) =
b (j)m(t-j) + 0
y
y
J=°
M
ap (i)p(t-i) + £
i=l
5)
M
£
M
aq (i)q(t-i) +
i=l
bq (j)m(t-j) + 0q
j=o
where 0__, 0D and 0a can represent an intercept term plus whatever other
y
r
H
variables one wishes to include.
Notice that each of these difference equations
is of the same order (N) and includes the same number (M) of lagged values of
money.
Since this model is linear the incremental effect of change in the quantity
of money is independent of the initial conditions of the model and of the
effects of other variables that might be added to the model.
Furthermore, the
effects of an equal increase or decrease in the quantity of money are exactly
equal in absolute magnitude, although they have opposite signs.
Because of
these inherent restrictions, the model is presented with the caveat that it is
intended to predict the effects of relatively "small" changes in the quantity
of money or its rate of growth.
Given the functional form of this model, the next step is to specify the
mathematical constraints that correspond to the assumptions about the long-run
-6-
neutrality of money.
Recall that the quantity theory assumes that a change
in the quantity of money causes a proportional change in the price index and
level of nominal output, in the long run, and that a change in the rate of
growth of money causes an equal change in the long-run rates of inflation and
growth of nominal output.
In order for these two conditions to exist, the
coefficients on equations 3 and 4 must satisfy the following constraints:
N
M
6) E a(i>+ E
7)
i=l
j=o
N
M
£
+E
i=l
j=o
b (j ) = 1
b(j) = 0
where a(i) represents ay (i) or a^(i) and b(j) represents b^(i) or bp (j).
See Appendix A for the derivation of these constraints.
It should be pointed out that the second constraint (equation 7) differs
from one that Dean Taylor (6) used in estimating Friedman’s dynamic model of
nominal output.
This particular model is a second order difference equation
(N=2) with one lag of the money variable (M=l).
b(0) = 1 + (A1 e
-A^e
His constraint is
Ai)
where A^ and A2 are the roots of the characteristic polynomial of this
difference equations.
The fact that this constraint is mathematically incorrect
can be shown by checking the derivation of equation 7 or by solving Taylor’s
model with his estimated coefficients.
In his model a change in the quantity
of money does not cause a proportional change in the level of nominal output,
in the long run.
-7-
When the equation for the logarithm of real output (equation 5) is
estimated, the following constraints will impose the assumptions of the
quantity theory of money:
M
N
8)
£ a (i) +
i=l
£
b (j) = 0
j=°
M
N
9)
£
laq (i) +
£
Jbq(j) = 0
i=i
These constraints correspond to the assumptions that neither a change in the
quantity of money nor a change in the rate of growth of money has a permanent
effect of the level of real output.
In order to reduce the number of variables that have to be estimated, the
lag weights in each of the equations of the model are assumed to be generated
by a third-degree polynomial, e.g.
a(i) = cxq + a^i + o^i2 + a3i3
b(j)
=
B0 +
Bij
+
S2j2 +
B 3j 3
where a(i) represents a^(i), a^(i), or a^(i) and b(j) represents by(j) or b^Cj).
The lag weights for the money variable in the equation for real output are
assumed to be on a fourth-degree polynomial.
Furthermore, the end-points are
constrained to equal zero or
a(N+l) = 0
b(M+l) = 0.
The above constraints and the assumptions about the long-run neutrality of
money can then be used to determine the corresponding transformation of the
-8-
data.
After the data has been transformed the model can be estimated by
any appropriate statistical technique.
See Appendixes B and C for the
derivation of these transformations.
Given the preceeding method of imposing the assumptions about the
long-run neutrality of money, the next step is to estimate a version of
the model.
This linear transfer equations model is general enough so that
it can be fitted to any time series of a monetary aggregate, a price index,
a measure of real economic activity, and the corresponding measure of
nominal economic activity.
The difference equations can be estimated
with monthly or quarterly data, provided that the chosen time step is
relatively small with respect to the speed of adjustment of the economy.
This latter proviso is an inherent restriction of the mathematics of
difference equations.
Notice the enormous number of combinations that can be tried for any
given economy or country.
Possible measures of nominal economic activity
include retail sales, personal income, gross national product (GNP),
and personal consumption expenditures.
For a measure of real economic
activity, one can try using industrial production, man-hours in private
nonagricultural industry, or a deflated series of nominal economic activity.
Some of the available price indices that could be estimated include the
consumers1 price index, wholesale price index, a wage rate series, and
a price deflator for any nominal income or output series.
In terms of a
measure of monetary aggregate, one can try a series for the monetary base
or a money supply series such as M-l, M-1+, M-2, M-3, . . .
For each set
of economic time series, it is necessary to specify a range in historical
time for the purpose of estimation.
-9-
In order to provide a specific example of some of the problems that
arise in estimating this model, I shall concentrate in the remainder of
this paper on the dynamic linkage between M-l and GNP.
Because GNP data
is available only on a quarterly basis, the time step of the model is a
quarter of a year.
The sample period for each model is from the first
quarter of 1959 through the fourth quarter of 1976.
Beginning in the first
quarter of 1977, an eight quarter dynamic simulation of each model is run
so as to provide measures of how well the models fit outside their period
of estimation.
Each of the equations that are estimated directly in the models is
fitted by minimizing the sum of squared errors with respect to the
coefficients of the equation.
The data for each model are first transformed
into rates of growth by taking the first difference of the logarithms of
the data.
Next, these data have been transformed, according to the procedure
that has been developed in this paper, so as to build in the long-run
assumptions about the neutrality of money.
Finally, an intercept term
and some additional variables have been added to each equation and the
least-squares estimates have been calculated.
The money variable is the rate of growth of M-l between two successive
quarters.
The quarterly data for M-l is equal to the average of the three
months' seasonally adjusted data of each quarter.
The nominal output
variable is the rate of growth of GNP minus Federal Government purchases
of goods and services.
Federal purchases have been subtracted since one
of the additional variables in each equation is a weighted average of
its rates of growth.
deflator.
For each of these quarterly series, seasonally adjusted data
have been used.
The price variable is the rate of change of the GNP
-10-
The lag weights on the weighted average of the rates of growth of
Federal Government purchases are generated by a third-degree polynomial.
In order to simplify the model, the number of lagged values of Federal
purchases is assumed to be equal to M, the number of lagged values of
the money aggregate.
The end-point is constrained to equal zero so that
only three coefficients need to be estimated for this fiscal policy variable.
Two dummy variables have been included in each equation to provide
estimates of the effects of wage and price controls during the Nixon
administration and the effects of the quadraupling of crude oil prices in
1973 by OPEC.
Those dummy variables are third-degree time polynomials with
an end-point constraint.
For example, the dummy variables for wage-price
controls are generated by the following equation:
d„n (t
wr
Wp
+
t)
= 6n + 6 it + 62T2 + <53t 3, x=0,l___ _ NTWP
d (t
+ NTWP) « 0
w p N wp
where twp is the first quarter with controls and NTWP is the number of
quarters these dummy variables are applied.
The first quarter for the
wage-price dummy variables is 1971-3, and the first quarter for the OPEC
dummy variables is 1973-4.
Each of these dummy variables requires that
three additional coefficients be estimated in each equation.
Even after selecting the economic time series to be included, the
sample period, and the method of estimation, there is a large number of
possible models that can be tried.
of the differences equations?
For example, what should be the order
How many lagged values of the monetary
-11-
aggregate and Federal purchases should be used?
of the dummy variables be applied?
How long should each
In terms of the parameters of the
models, these issues are concerned about the appropriate values of N, M,
NTWP, and NTOIL.
NTOIL is the number of quarters the OPEC dummy variables
are applied.
In order to reduce the number of possible models, I restricted the
first series of regressions to three values for each of these four parameters.
The corresponding set of 81, or 3^, parameter combinations are summarized
in the following table:
N
M
NTWP
NTOIL
3
8
8
8
4
12
10
10
5
16
12
12
For each of these parameter combinations, three equations were estimated
directly —
one for the rate of growth of nominal GNP minus Federal
Government purchases, the second for the rate of growth of the GNP
deflector, and the third for the rate of growth of real GNP minus real
Federal purchases.
These three estimated equations and the accounting identity for the
rate of growth of nominal output determine the three versions of this
model.
Version I of the model uses the estimated equations for the rates
of growth of nominal output and the price index and determines the
corresponding values of the rate of growth of real output from the accounting
identity.
Version II of the model uses the estimated equations for the
rates of growth of nominal and real output and determines the corresponding
-12-
rate of growth of the deflator from the accounting identity.
Version III
of the model uses the estimated equations for the rates of growth of the
price deflator and real output and determines the corresponding rate of
growth of nominal output from the accounting identity.
In total there were
243, or 81 times 3, equations and models estimated in the first set of
regressions.
The next problem that logically arises is how to evaluate each of these
alternative models and parameter combinations.
My heuristic approach for
selecting the "best" model developed along the following line.
For each
parameter combination I calculated two sets of summary statistics.
The
first set of statistics includes measures of fit within the sample period.
The second set of statistics includes measures of fit within the forecast
period.
In the first set of summary statistics I include fifteen numbers.
Six
of these numbers are the coefficients of determination and the root mean
squared errors of the three equations that have been estimated directly.
These statistics are labeled R
2
(Dy), R
RMSE (Dp), RMSE (Dq), respectively.
2
(Dp), R
2
(Dq) and RMSE (Dy),
Each of the three versions of this
model uses two of the directly estimated equations and the accounting identity
to calculate estimates of the "residual" variable.
Therefore, there are
also a coefficient of determination and a root mean squared error for each
of the three possible residual variables.
These six numbers are labeled
R2 (Dye), R2 (Dpe), R2 (Dqe) and RMSE (Dye), RMSE (Dpe), RMSE (Dqe),
respectively.
-13-
The next summary statistics that are calculated are just the sums of
the three root mean squared errors for each version of the models.
These
numbers are labeled RMSE (I), RMSE (II), and RMSE (III), and are defined by
the following equations:
RMSE (I) = RMSE (Dy) + RMSE (Dp) + RMSE (Dqe)
RMSE (II) = RMSE (Dy) + RMSE (Dpe) + RMSE (Dq)
RMSE (III) = RMSE (Dye) + RMSE (Dp) + RMSE (Dq) .
These statistics provide a single measure of how well any model estimates
the rates of growth of nominal output, the price deflator, and real output,
within the sample period.
While there are fifteen summary statistics in this
first set for any parameter combinations, only seven of these numbers are
related to any particular version of the model.
In the first set of regressions, where I tried 81 different parameter
combinations, these summary statistics fell within the following ranges:
.543
R2 (Dy) >, .375
.608 _< RMSE (Dy) _< .702
.423 > R2 (Dye) >. .269
.683 _< RMSE (Dye) _< .760
.857 £ R2 (Dp) >_ .799
.764 1 R2 (Dpe) >. .513
.244 _< RMSE (Dp) _< .290
.314 _< RMSE (Dpe) _< .452
2
.577 _> R
2
(Dq) >_ .388
.695 _< RMSE (Dq) _< .816
1.501 j< RMSE (I) _< 1.774
1.631 _< RMSE (II) _< 1.900*
1.648 _< RMSE (III) _< 1.856
.656 >_ R
(Dqe) >. .436
.629 _< RMSE (Dqe) _< .783
-14-
Notice that the best fits as measured by the size of the root mean squared
errors, are provided by the inflation equation when it is estimated
directly.
This property is robust in the sense that the highest root mean
squared error of this equation is lower than the smallest root mean squared
error of any other equation.
By examining the summary statistics for each of the individual
regressions, it is possible to spot several other patterns.
For every
parameter combination in this set of regressions, the root mean squared error
for the rate of growth of nominal output is smaller when this variable is
estimated directly instead of being treated as the residual variable.
Furthermore, the root mean squared error for the rate of growth of real
output is at least as small when this variable is treated as the residual
variable instead of being estimated directly.
These patterns can be
summarized by the following inequalities:
RMSE (Dy) < RMSE (Dye)
RMSE (Dp)
< RMSE (Dpe)
RMSE (Dqe) _< RMSE (Dq)
when N = (3, 4, 5}, M = (8 , 12, 16}, NTWP = {8 , 10, 12}, and NTOIL = {8 , 10, 12}.
A direct implication of these findings is that the first version of the model
always provides the smallest sum of the root mean squared errors.
Like any
empirical findings, these results are dependent upon the economic data that
are used and the particular model that is estimated.
After selecting the first version of the model, where Dy and Dp are
estimated directly, I compared the sums of the root mean squared errors for
-15-
different values of NTWP and NTOIL.
In every regression this sum was
lowest when NTWP was equal to 10, instead of 8 or 12.
In 78 of the 81
regressions the sum of the root mean squared errors was lowest when
NTOIL was equal to 10, instead of 8 or 12.
In the three other regressions,
the model had a slightly better fit when NTOIL was equal to 12.
Given
these patterns, I concluded that ten quarters, or two and a half years,
was an appropriate length of time to apply the dummy variables for Nixon’s
wage and price controls and for the formation of the OPEC cartel.
By deciding to set the value of NTWP and NTOIL to equal 10, I had
narrowed down my search to nine parameter combinations in the first set
of regressions.
The sums of the root mean squared errors of these nine
regressions are presented in the following table:
TABLE 1:
RMSE (I) IN SAMPLE PERIOD
N/M
8
12
16______
3
1.605
1.521
1.524
4
1.619
1.508
1.504
5
1.572
1.506
1.501
Judging only by this criterion, the best fitting model within the sample
period would be the one with fifth-order difference equations that are
functions of a 16 quarter, or 4 year, average of the rates of growth of
M-l and Federal Government purchases.
-16-
In order to compare these models under alternative criteria, I next
examined the set of summary statistics for the forecast period.
This set
includes the three root mean squared errors in the forecast period and the
sum of these errors.
These statistics, for the first version of the model,
are RMSE (Df ), RMSE (Dp), RMSE (Dq), and RMSE (I), respectively.
The sums
of the root mean squared errors of these nine forecasts are presented in
the following table.
TABLE 2:
N/M
RMSE (I) IN FORECAST PERIOD
8 __,
12
16
3
2.423
2.492
2.465
4
2.415
2.487
2.453
5
2.471
2.542
2.514
Notice that, among these nine models, some of the parameter combinations
that provide the lowest sums of the root mean squared errors within the
sample period have the highest sums in the forecast period.
In fact, the
correlation coefficient between these two sets of root mean squared errors
is negative (-.78)!
Another set of summary statistics includes the accumulative errors
over the eight quarter forecast.
For example, between the first quarter of
1977 and the fourth quarter of 1978, the GNP deflator increased by 13.860
percent.
Over this time period, a model that predicts this deflator would
have increased by 14.988 percent would have an accumulative eight quarter
error of -1.128 percentage points.
Table 3 and 4 display the accumulative
forecast errors for the rates of growth of nominal output and the price
deflator during the two year dynamic simulation.
-17-
TABLE 3:
ACCUMULATIVE ERROR FOR Dy
N/M
8
3
2.844
3.626
4.131
4
2.774
3.617
4.046
5
3.134
3.750
4.299
TABLE 4:
12
16
ACCUMULATIVE ERROR FOR Dp
N/M
8
12
16
3
-.334
-1.128
-.811
4
-.366
-1.035
-.632
5
-.261
-1.148
-.716
By using the accounting identify, the accumulative error for the rate of
growth of real output can be determined by subtracting the accumulative
error for inflation from the accumulative error for the rates of growth
of nominal output.
An examination of the coefficients of the GNP deflator regressions
reveals an interesting pattern.
are unstable.
Five of these estimated difference equations
In these models an increase in the rate of growth of money
causes a temporary increase in the rate of inflation.
After a relatively
long period of time, the rate of inflation begins to decrease.
Eventually
these models enter a period of self-sustaining hyperdeflation, following
the initial increase in the rate of growth of money.
Since the equations
for the rate of growth of nominal output are stable, this version of the
model predicts that real output eventually will be growing at an ever
increasing rate!
Needless to say, these unstable models are inconsistent
with the assumptions about the long-run neutrality of money.
-18-
The "mechanics11 of this peculiar instability problem can be seen easily
by examining the sums of the coefficients for the lagged values of inflation.
Table 5 displays these sums for the nine regressions.
TABLE 5:
N/M
8
SUM OF ap(i)Ts, i = 1.2,..., N
12
16
3
1.088
.976
.858
4
1.077
1.053
.928
5
1.054
1.101
.967
With one trivial exception, the coefficients for the lagged values of
inflation in these regressions are always positive numbers that are less
than one.
Therefore, whenever their sum is greater than one, the model is
unstable.
Recall that one of the constraints that is imposed upon these
models is that the sum of the coefficients for lagged values of inflation
plus the sum of the coefficients for the money variables are equal to
one (equation 6).
Whenever the model is unstable because the sum of the
ap(i) coefficients is greater than one, the sum of the bp(j) coefficients
must be negative.
This is why these unstable models predict an eventual
hyperdeflation following an increase in the rate of growth of money.
Even when the inflation equations are stable, the sums of the
coefficients for the lagged values of inflation are close to one.
models display high degrees of "inflationary momentum".
These
By this expression,
economists mean that an economy takes a relatively long period of time
to adjust to a one period disturbance in the rate of inflation.
One of
the perplexing questions raised by these regressions is why does the
inflation equation predict a relatively slow speed of adjustment when the
-19-
directly estimated equations for the rates of growth of nominal and real
output predict a relatively fast speed of adjustment.
In order to see how the three versions of this model fit for
alternative values of M, I ran a second set of regressions and calculated
some of the dynamic impact multipliers.
In this set of regression I estimated
third-order difference equations and varied M between 10 and 16.
By
setting N equal to 3, the end-point and third-degree polynominal constraints
on the coefficients of lagged values of the dependent variables are
nonbinding.
Figure 1 plots some of the summary statistics for these
regressions.
The three graphs in the first column plot the root mean squared errors
within the sample period.
the horizontal axes.
The different values of M are arranged along
There are two distinct patterns in these graphs.
Each version of the model fits about the same within the sample period
regardless of the value of M.
In other words, the plots of the root
mean squared errors are very flat.
The second pattern is that the first
version of the model consistently provides the best fit within the sample
period.
Recall that the relevant root mean squared errors for this version
of the model are RMSA (Dy), RMSE (Dp), and RMSE (Dqe).
The three graphs in the middle column plot the root mean squared
errors in the forecast period.
Notice that these errors always are higher
in the forecast period than they are in the sample period.
Furthermore,
none of the three versions of the model consistently provides the best fit
within the forecast period.
(
RM SE (Dye) • • •
RM SE (D y) --------
1.0
-
.8
-
RMSE (Dve) • • •
RM SE (D y) --------
£ 6 (Dye) • • •
2 6 (Dy) -------
.6
-J— «— i— i— *— i— i—
10 11 12 13 14 15
--1--1--1--1--1 1— M
m
16
RM SE (Dpe) • • •
RM SE (Dp) -------
10 11 12 13 14 15 16
10 11 12 13 14 15 16
2 6 (Dm )* • •
RM SE (Dpe) • • .
RM SE ( b p ) --------
2 e ( D p j --------
..1
1 I___ 1 i____ *- i
10 11 12 13 14 15 16
- i ----- 1----- L _J----- 1----- 1----- 1 _ M
10 11 12 13 14 15 16
RM SE (Dqe) • • •
RM SE (Dq) -------
-I--1-1— « «-- i— J— M
RM SE (Dqe) • • •
RM SE (Dq) --------
S e (D q e ) *
_
• •
£e (Dq} ---
.8 -
-l
1-1— I 1-- 1--»— M
10 11 12 13 14 15 16
_J
1----- 1------1-----I----- 1
1—
10 11 12 13 14 15 16
Figure 1: Summary Statistics
M
- J ------1------1----- 1------1------1------ 1__ M
10 11 12 13 14 15 16
-20-
The graphs in the third column plot the accumulative errors for each
variable in the two year forecast period.
Notice that most of the accumulative
errors for the GNP deflator estimates are less than one percentage point.
These errors are very small when compared with the 13.86 percent increase in
the GNP deflator during 1977 and 1978.
On the other hand, the accumulative
errors for the real GNP estimates are substantial.
For the different
models they range between 3 and 5 percentage points, compared with a 9.85
percent increase in real GNP between the first quarter of 1977 and the fourth
quarter of 1978.
After weighing how well these models fit within the sample and the
forecast periods, I subjectively selected 14 to be the value of M.
This
model uses weighted averages of the current rates of growth of money and
of Federal government purchases and their lagged values for the previous
three and a half years.
The dynamic impact multipliers for a one percentage
point change in the rate of growth of money are plotted in Figure 2.
The
three graphs correspond to the three versions of the model where N, M,
NTWP, and NTOIL are equal to 3, 14, 10, and 10, respectively.
Given these parameter combinations, the equation that is estimated
directly for the rate of growth of nominal income predicts that Dy will
increase rapidly, overshoot, and then converge within five percent of
its long-run value by eight quarters.
The inflation equation that is
estimated directly predicts that Dp will pass through its new equilibrium
value in two years, overshoot by about fifteen percent and then very
gradually converge onto its long-run value.
Therefore, the implicit
prediction of the first version of the model is that real output will
increase for two years and then gradually return to its equilibrium
Dy
Figure 2: Dynam ic Impact Multipliers for a Change
in the Rate of Growth of Money
-21-
value.
By summing the first eight values of Dqe, one obtains 0.886.
This
is the maximum percentage increase in output that temporarily is caused by
a one percentage point increase in the rate of growth of money.
However, when the rate of growth of real output is estimated directly,
the regression predicts the Dq will be positive for only six quarters or
one and a half years.
According to this regression, the maximum percentage
increase in real output is only 0.518 following a one percentage increase
in the rate of growth of money.
In the second version of the model, where
inflation is the residual variable, the model predicts that Dpe will
increase by its equilibrium value in six quarters, overshoot, and then
converge to within five percent of its equilibrium value by four years.
The coefficients that are estimated directly for this particular set
of data and combination of parameters are listed in Table 6.
Except for
the intercept terms, these are some of the coefficients of the various
polynomial generating functions of the model.
are listed in parentheses.
The corresponding t-statistics
The extremely low t-statistics for 33 in the
equations for Dy and Dp and for (3^ in the equation of Dq indicate that
lower order polynomials can be used to generate the lag weights of the
monetary aggregates.
In none of these equations is a weighted average of
Federal government purchases of goods and services statistically
significant at any conventional level.
On the other hand, the dummy
variables, especially the ones for the quadrupling of crude oil prices
by OPEC, are statistically significant in the sample period.
These
patterns are found in all of the equations in the second set of regressions.
TABLE 6 : CONSTRAINED ESTIMATES OF THE LONG-RUN NEUTRALITY OF MONEY
N=3, M=14, NTWP=10, NT0IL=10, (t-statistics in parentheses)
Dependent
Variable
Intercept
a0
Coefficients of the Polynomial Constraints
Federal Government Purchases
Lagged Dependent Variables
Monetary Aggregate
a2
Y0
12
60
63
1.2537
-0.4231
(3.60)
(-1.50)
-0.0555
1.7193
(-0.46)
(1.26)
0.9656
0.4608
(1.81)
(0.14)
0.6342
-0.0008
-0.0138
-0.0036
0.0004
(1.74)
(-0.003)
(-0 .D2)
(-0 .002)
(0.0003!
-0.1628
0.1220
-0.000008
0.0256
-0.0151
0.0021
(-1-22)
(1.36)
(-0.00008)
(0.09)
(-0 .02)
(0.004)
0.5626
0.3372
&,=3xl0~6
-0.0296
0.0075
-0.0008
(0.13)
(2.41)
(0 .00002)
(-0.04)
(0.004)
(-0 .001)
-0.0236
Dy
Dp
(-0.08)
-0.9966
Dq
(-0.13)
Coefficients of the Polynomial Dummy Variables
Wage-Price Controls
OPEC Cartel
Dependent
Variable
Dy
R
2
60
60
«2
^3
-1.5294
-0.1323
0.0059
0.4249
0.4247
-0.0029
(-2.03)
(-0.91)
(0.64)
(0.61)
(2.98)
(3.27)
-0.0855
-0.0067
0.2483
-0.1516
0.0010
(-1:45)
(-1.78)
(0 .86)
(-2.55)
(2.71)
-1.1428
-0.2863
0.0156
-0.1730
0.4885
-0.0311
(-1 .11)
(-1.70)
(-0 .21)
(2.95)
(-3.02)
.837
(-1.73)
Dq
53
.518
-0.5160
Dp
62
.537
(1.45)
-22-
In conclusion, let us review the path followed in this paper.
The
main purpose of this paper is to translate the assumptions about the
long-run neutrality of money into mathematical constraints that then
can be imposed when estimating a small macroeconomic model.
For a
clear verbal presentation of this theory, I used Irving Fisher’s
description of the quantity theory of money.
Next, I specified that
the mathematical form of the models to be estimated would consist of
linear ordinary difference equations.
For this type of model, the
assumptions about the long-run neutrality of money correspond to certain
linear restrictions on the coefficients of each equation.
In the
appendixes, these restrictions were derived and the appropriate trans
formations of the data were presented.
The second part of this paper uses M-l and GNP quarterly data to
estimate three versions of the model under a variety of parameter
combinations.
The different parameters tried include the order of the
difference equations, the number of lagged values of the monetary
aggregate, and the length of time to apply dummy variables for Nixon’s
wage-price controls and the quadrupling of crude oil prices by OPEC.
In order to compare these alternative models after they have been
estimated, I calculated two sets of summary statistics.
These two sets
include measures of fit within the sample period and within a two-year
forecast period.
This latter set of statistics was used in selecting
the ’’best” model, since many parameter combinations provide about the
same degree of fit within the sample period.
On the other hand, some
of the estimated models that did not fit relatively well within the
sample period did predict relatively well within the forecast period.
-23-
Furthermore, some of the better fitting models within both of these
periods contain an unstable inflation equation, where an increase in the
rate of growth of money ultimately causes a self-generating hypderdeflation.
After subjectively weighing these summary statistics and stability
conditions, I chose
a single set of parameters and calculated the dynamic
impact multipliers associated with a change in the rate of money.
The major contribution of this paper is that it provides economists
with a simple method of imposing the assumptions about the long-run
neutrality of money within a system of linear difference equations.
Furthermore, these models can be estimated with a large variety of
economic time series for money, prices, nominal output, and real output.
Additional explanatory variables may be added, and these alternative
models may be estimated and compared by any method one chooses to use.
The final test for any of these models is how well do they predict the
future vis-a-vis other existing models.
REFERENCES
Del Mar, Alexander. The Science of Money, third edition.
New York: Cambridge Encyclopedia, 1899.
Fisher, Irving. The Purchasing Power of Money, revised edition.
New York: Macmillan, 1916.
Friedman, Milton. "The Role of Monetary Policy," American
Economic Review, Vol. LVIII No. 1 (March 1968), pp. 1-17.
Friedman, Milton. "A Theoretical Framework for Monetary Analysis,"
Journal of Political Economy, Vol. 78 No. 2 (March/April 1970),
pp. 193-237.
Phelps, Edmund S. "Phillips Curves, Expectations of Inflation and
Optimal Unemployment Over Time," Economica, Vol. XXXIV
(August 1967), pp. 254-81.
Taylor, Dean. "Friedman’s Dynamic Models: Empirical Tests,"
Journal of Monetary Economics, Vol. 2 (1976), pp. 531-38.
APPENDIX A
CONSTRAINTS FOR A NOMINAL SERIES
Assume the following linear transfer function that relates the logarithm
of a nominal output or price series y(t) to its lagged values and to the
current and lagged values of the logarithm of a monetary aggregate m(t):
1)
N
M
y(t) =■ Z a(i)y(t-i) + Z
i=l
j=o
b(j)m(t-j).
The assumption that a change in the quantity of money causes a proportional
change in the level of the nominal variable, in the long run, implies that
2)
N
M
y(t) + A = Z a(i)(y(t-i) + A) + Z b(j)(m(t-j) + A).
i=l
j=o
Subtract equation 1 from equation 2 and divide by A to obtain the first
constraint:
3)
N
M
1 = Z a(i) + Z b(j)
i=l
j=o
Without loss of generality we can normalize the long-run values of y(t)
and m(t) so that they equal one.
The assumption that a change in the rate of
growth of money (6) causes an equal change in the long-run rate of inflation
implies that the values of y(t-i) and m(t-j) are equal to 1-iS and l-j6,
respectively.
4)
In terms of our linear transfer function this means that
N
M
1 = Z a(i)(l-iS) + Z b(j)(1—j 6)
i=l
j=o
Subtract equation 2 from equation 4 and divide by 6 to obtain the
second constraint:
5)
N
M
0 = E ia(i) + Z
i=l
j=o
jb(j)
Notice that this constraint means that at least one of the coefficients of
the transfer function must be negative.
It is interesting to notice the different roles that these two
constraints play when the model is expressed as a rate of change equation.
Let D represent a difference operator that is defined such that
Dx(t)
= x(t) - x(t-l).
The transfer function in terms of rates of changes can be written as
N
M
Dy(t) = E a(i)Dy(t-i) + E b(j)Dm(t-j)
i=l
j=o
In this model the constraint that the sum of the coefficients equals one
(equation 3)
imposes the assumption that a change in the rate of growth
of money causes, in the long run, an equal change in the rate of
inflation.
The additional constraint that the weighted sum of
coefficients equals zero
(equation 5) means that a change in the quantity
of money will cause a proportional change in the level of the nominal
variable.
APPENDIX B
The dynamic relationship between the logarithms of a money variable
m(t) and a nominal variable y(t) is assumed to be represented by the
following linear transfer function, where the lag weights are generated
by third-degree polynomials.
1)
M
N
y(t) = Z a(i)y(t-i) + Z
i=l
2)
a(i) = aQ + a^i + a2i2 + a3i3
3)
b(j) - 60 + Bij + &zj 2 + e3j 3
b(j)m(t-j)
There are eight basic coefficients in this model - aQ, ai, a2> 013,
3q > $1, $2 » an<* ^3 “
Four of these coefficients can be determined by
constraining the end points —
a(N+l) and b(M4-l) —
to equal zero and by
making the two assumptions about the long-run neutrality of money (see
Appendix A).
These constraints are summarized by the following four
equations:
4)
a(N+l) = 0
5)
b(M+l) = 0
6)
N
M
E a(i) + E b(j) * 1
i=l
j=*o
7)
N
M
E ia(i) + E jb(j) = 0
i=l
j=o
These equations can be substituted into the model to determine a linear
transformation of the data that will impose the four constraints.
The
appendix will outline the transformation that can be used when a ^ 9 a 3,
So> and $3 are estimated directly.
The remaining coefficients can then
be determined from equations 2-6 .
Use equations 2-4 to evaluate the two end-point constraints and
rearrange to obtain the following expressions for a0 and 3i:
8)
9)
ao = -(N+l)a1 -(N+l)2a2 -(N+l)3a3
= -$0/(M+l) - (M+l)62 - (M+l)233
Next expand the two economic constraints by substituting in the polynomial
generating function to obtain
10}
Na0 + a 1Si + c ^E i 2 + a 3Z i 3 + ( Mf l) B0 + 6 Zj
+ g2 Z j 2 + @ E j 3 « 1
3
11)
a^Ei + c^Zi2 + a2Zi3 + a ^ i 1* + eQEj + g ^ j 2
+ g2Zj3 + ggEj1* = 0
The ranges of summation have been omitted with the understanding that the
terms with i's are summed from 1 to N and terms with j ’s are summed from
0 to M.
Use equations 8 and 9 to factor out ag and 61 in equations 10 and
11 and rearrange into matrix format.
Zi
-
N(N+l)
Zj2
Zi2 - (N+l)Zi
-
(M+l)Ej
e.
Zj3 - (M+l)Z j2
1
N(N+1)2-Zi2
N(N+1) 3-Zi3
Zj/(M+l)-(M+l)
(M+-1) 2Zj-Z j 3
0
(N+l)2Zi-Zi3
(N+l) 3Zi-Zil+
Zj2/(M+1) - Zj
(M+l)2Zj-Zj4
or
Au = Bv,
where u = (o^
B2) " and v = (1 a2 a 3 ^0 ^3^
The next several steps require a fair amount of tedius algebra that
can be described briefly as follows.
Recall (or look up in any mathematical
handbook) that the sums of powers os positive integers are given by the
following equations:
Zk - K(K+l)/2
Zk2 = K(K+1)(2K+1)/6
Zk3 = K2(K+l)2/4
Zk4 = K(K+1)(2K+1)(3K2+3K-l)/30,
where the summations are from 0 or 1 to K.
Use these equations to
evaluate the summation terms in matrix A and invert this matrix to get
-M(M+-1) 2 (M+2) /12
-M(M+l) (M+2)/6
-N(N+1)(N+2)/6
-N(N+1)/2
where
1A | = M(M+1)(M+2)N(N+1)(3M-2N-1)/72.
In order to use this transformation it is necessary that 3M-2N-1 does not
equal zero.
This special case can be solved by selecting a different set
of four coefficients that are to be estimated directly and by calculating
the corresponding transformation of the data.
Evaluate the summation terms in matrix B and then premultiply by A
to obtain the following equations:
12)
a l = c0 + c xa 2 + c2a 3 + c 3£Q + C/+63
13)
$2 = ^0 + ^ia2 + ^2a3 + d 3BQ + d^f^
where
c0 =
-6(M+1)
N(N+1)(3M-2N-1)
C 1 = ~M(4N+5)+3N2+5N+l
3M-2N-1
c
= (3N+4)(-15M(N+1)+12N2+17N+1)
10(3M-2N-1)
c3 =
(M4-2) (M+3)
N(N+l)(3M-2N-1)
ck = (M-1)M(M+1) (M+2) (M+3)
ION(N+l)(3M-2N-1)
d„ =
12 (N+2)_______
M(Mfl)(Mf2)(3M-2N-1)
d l = -(N-1)N(N+1)(N+2)
M(Mfl)(Mf2)(3M-2N-1)
d2 = -3(N-1)N(N+1)(N+2)(3N+4)
5M(M+L)(M+2)(3M-2N-1)
d3 =
6(M-N-2)
M(Mfl)(3M-2N-1)
du = 3(5N(M+l)-2(4M2+3M-2))
5 (3M-2N-1).
The accuracy of the preceding steps can be checked by first assuming
arbitrary values for N, M, a2 , a 3, 0Q, and 01.
By using equations 8 , 9,
12, and 13 it is then possible to determine the corresponding values of
ocq ,
oij, 0j, and 02 .
These coefficients’ values can then be plugged into
equations 10 and 11 to see if these equations sum to 1 and 0 , respectively.
Equations 8 , 9, 12, and 13 can be substituted into polynomial
constants to yield
14)
a(i) = <J>0 (i) + <f>^(i)a2 + <P2 (i)a 3 + <j>3(i)g0
15)
b(j) = Y0(j) + 'P1(j)o2 + V j ) a 3 + V ^ o
+ <j>i+(i)g3
+ Vj)e3
where
*0 (i) - -(N+l)c0 + c0i
<(>3(i) = -(N+l) (c^+N+1) + c xi + i2
4>2 (i) = -(N+l) (c?+(N+l)2) + c2i + i3
4>3 (i) = -(N+1)c 3 + c3i
♦^(D « -(N+l)Ctf + c k±
V0(j) = - (M+l)d0j + d0j2
^l(j) = -(M+l)d1j + d xj2
4'2 (j) = -(M+l)d2j •+ d2j2
^3(j) = l-((M+l)d3 + 1/(M+l))j + d 3j2
MJ)
= -(M+l)(d4+M+l)j + d^j2 + j 3
Equations 13 and 14 can then be substituted into the original transfer
function to determine the corresponding linear transformation of the data.
The constrained model is given by
z(t) = a2x 3(t) + a3x2(t) + gQx 3(t) + g ^ C t )
where
z(t) = y(t) - £<t>0 (i)y(t-i) -
(j )m(t-j )
x (t) = Z4> (i)y(t-i) + ZY, (j)m(t-j) ,
k
k
K
k = 1,2,3,4
After transforming the data, this model can be estimated by any standard
regression technique.
The estimated coefficients will determine how
quickly the nominal variable responds to a change in the quantity of
money.
By calculating the dynamic input multipliers of this model, it
is possible to check the accuracy of this transformation by seeing if the
long-run multipliers correspond to the two economic constraints.
APPENDIX C
CONSTRAINTS FOR A REAL SERIES
Assume that the current level of the logarithm of real output
q(t) is a function of its lagged values plus a weighted average of the
logarithms of a money variable m(t).
M
1) q(t) = f(q(t-i))+ E b(j)m(t-j)
j=0
The assumptions that the level of the money variable and its rate of
growth do not have a long-run impact of the level of real output imply
that
2)
Zb(j) - 0
3)
Zjb(j) - 0
where terms with j fs are summed from 0 to M.
When the lag coefficients
are generated by a polynomial, these two constraints can be used to
eliminate two of the coefficients that must be estimated.
Suppose the lag weights are determined by the following fourthdegree polynomial
4)
b(j) - e0+ Bjj + e2j2+ e3j 3 +
where the end point b(Mfl) is constrained to equal zero.
5)
b(Mfl) - 0
Equations 2, 3 and 5 can be used to factor out three of the coefficients.
Suppose g ,
into equation 4.
of equations:
and
are factored out by substituting these equations
In matrix format we can derive the following system
(M+l)2
(M+l)3
Ej
Ej2
Ej3
Ej2
Ej3
Ej 4
-1
h
*2
-(M+l)4
= -(M+l)
"Ej4
_~Zj
-Ej5
_ $3
I--O
ca
1___
M+l
or
Au = Bv
where u = (3^
3^
3^) ^and v = (3Q
3i+)."
The summation terms can be
evaluated by using the equations for sum of powers of positive integers
(see Appendix B).
The equation for the sum of fifth powers is
Ek5 = K2 (K+l)2 (2K2+2K-1)/12,
where k is summed from 0 to K.
Invert matrix A and then premultiply matrix B by A ^ to obtain the
following equations
CQ
II
*2
"
*2*0
*3
"
^3e 0
♦ ie0
+
b
+ v
2 b
+ VaBi*
where
<j>! - - (3M+1) (3M4-2) / ((M-l)M(MH) )
<J>2 = 6(3M+2)/((M-l)M(M+l))
<p3 = -10/( (M-1)M(M+1) )
4^ = - (M+l) (Mz+2M+2) /5
^2 = (6M2+12M+7)/5
4*3 = -2 (M+l)
These constraints can then be used to transform the money data into two
sets that can be used to estimate the model.
B0xi(t) + 34x2 (t) = Eb(j)m(t-j)
where
x l(t) - Zm(t-j)(l +
+ <j)2j2 + 4>3j 3 )
x2 (t) = Em(t-j)('t'1j + y2j2 + Y3j 3 + j^)
The same transformation can be used when the model is estimated using
the first difference or rates of growth of the real input and money
variables.
Notice that it is independent of the function that includes
the lagged values of the real output variable.
@FedFRASER