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Working Paper Series
A Federal Funds Rate Equation
WP 95-03
This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/
Yash P. Mehra
Federal Reserve Bank of Richmond
This is a preprint of an article which appeared in Economic Inquiry, vol. XXXV, No. 3, July 1997.
WORKING PAPER 95-3
A FEDERAL FUNDS RATE
EQUATION
Yash P. Mehra*
Research Department
Federal Reserve Bank of Richmond
May 1995
Abstract
This paper presents evidence that indicates that U.S. interest rate policy
during most of the 1980s can be described by a reaction function in which the
federal funds rate rises if real GDP rises above trend GDP, if actual
inflation accelerates, or if the long-term bond rate rises. Money growth when
included in the reaction function is significant, indicating that money also
influenced policy. The results presented here however indicate that in recent
years the fed has discounted the leading indicator properties of money.
In
contrast, the bond rate has been a key determinant of the funds rate during
the period I979 to 1992.
*Vice President and Economist.
The views expressed in this paper are those of
the author and do not necessarily represent those of the Federal Reserve Bank
of Richmond or the Federal Reserve System. The author thanks Tim Cook, Marvin
Goodfriend, Robert Hetzel, and Bennett McCallum for comments.
This paper estimates an equation that explains the behavior of the
federal funds rate during the period 1979 to 1992.
This funds rate equation
has two parts: a long-run part and a short-run part.
The long-run part, which
assumes that the funds rate moves with the inflation rate and that the real
federal funds rate is mean stationary, determines the long-run, equilibrium
component of the funds rate.
In the short run, however, the funds rate
differs from its long-run equilibrium value.
The short-run part has the
feature that the federal funds rate rises if real GDP is above trend GDP, if
inflation rises, or if the long-term bond rate rises.
The funds rate equation estimated here incorporates some salient
features of monetary policy in Taylor (1992) and Goodfriend
(1993).
The
policy rule in Taylor (1992) has the property that the funds rate rises if
real income rises above trend income, if inflation increases above an assumed
target of 2 percent, or if the long-run equilibrium funds rate rises.'
Taylor
shows that this policy rule, though not estimated from the data, is consistent
with the actual path of the funds rate during the period 198741 to 199243.
Goodfriend
(1993) has argued that in order to establish and maintain
credibility the Fed has during the period 1979 to 1992 reacted to the
information in the bond rate about long term, expected inflation.
Goodfriend,
however, does not estimate any policy reaction function.'
'The particular policy rule studied there is FR, = 2 + p +, .5 (y - y*) +
.5 (p-2), where FR is the federal funds rate, y is real GDP; y is trend GDP;
and p is the inflation rate. The term (2tp) captures the long-run equilibrium
component of the federal funds rate, which equals the assumed, equilibrium
real rate (2 percent) plus the inflation rate. The rule assumes that the Fed
has a short-run inflation target of 2 percent and real output target equal to
the trend rate. The funds rate rises one-for-one with inflation and responds
equally to positive discrepancies between the actual inflation rate and the
inflation target and between actual real GDP and trend GDP. Thus, if the Fed
achieves its real output and inflation objectives, then the proper funds rate
is given by its long-run equilibrium component (2+p).
- 2 Money growth is not included in the policy rule examined in Taylor
(1992), nor does it receive much prominence in Goodfriend
(1993).
In
contrast, the reaction functions recently reported in McNees (1992) indicate
that the funds rate has reacted to money growth during the 1970s and the
1980s.
The funds rate equation here is estimated with and without
money.
Money growth when included in the equation is however highly
including
The results indicate that over the longer sample period, 1979 to
significant.
1992, the funds rate equation with money predicts better the funds rate.
Furthermore, money appears to be a significant determinant
of the funds rate
during the 197Os, when direct measures of inflation and/or real output are not
significant
in the funds rate equation.
These results indicate that money
influenced policy during the 1970s and the 1980s.
However, over the recent
shorter sample period 1987 to 1992 considered in Taylor (1992),
the funds rate
equation without money is quite consistent with the actual path of the funds
rate.
This result indicates that the Fed may have discounted
in recent years
the leading indicator properties of money.
The bond rate is found to be a key determinant
during the sample period 1979 to
1992.
of the funds rate
The funds rate equation without the
bond rate significantly
underpredicts
indicate that movements
in the funds rate accounted for by the bond rate are
significant
the funds rate.
The results here
during periods when inflation scares occurred
The plan of this paper is as follows.
(Goodfriend 1993).
Section 2 discusses
the
premises that underlie the federal funds rate equation estimated here.
also discusses
the estimation methodology.
Section 3 presents empirical
results, and Section 4 contains conclusions and summary observations.
It
-3-
2.
The Model and the Method
2.1
A Discussion of the Determinants of the Federal Funds Rate
This paper assumes that the Fed targeted the federal funds rate
directly or indirectly during the period 195443 to 199244.'
This section
discusses the factors that might have influenced the Fed in setting the funds
rate.
The federal funds rate equation studied here has two parts: a longrun part and a short-run part.
determinants
Equation (1) specifies the long-run economic
of the funds rate.
FR; = rr: t pi
(1)
where FR is the federal funds rate; rr* is the economy's underlying
equilibrium
real rate; and p* is the long-term expected inflation rate.
The
real rate rr* can be viewed as the rate which equates the flows of desired
saving and investment in the economy.
Wicksell.
It is the real natural rate of
The term rr: t pi in (1) thus measures the nominal natural rate.
'This assumption is only approximately correct. During the sample period
examined here the Fed has not always focused on the federal funds rate and
when it did it has used varying monetary policy operating procedures to manage
it. Thus, the Fed focused on free reserves and short-term money market rates
(including the funds rate) during most of the 1950s and 196Os, used 'direct'
funds rate targeting during most of the 197Os, and 'indirectly' managed the
funds rate using the nonborrowed reserves procedure during 197944 to 198243
and the borrowed-reserve procedure during 198244 to 199294 (Cook and Hahn
1989; Wallich 1984; and Thornton 1988). Goodfriend (1993) has argued that the
period from 197944 to 198243 should be viewed as the one of 'aggressive'
federal funds rate targeting rather than one of nonborrowed reserve targeting.
The'reason is that during this period only one-third of funds rate changes
resulted from automatic adjustments of non-borrowed reserve targets. The
remainder of funds rate changes during this period resulted rather from
'judgmental' actions of the Fed (Cook 1989).
-4
-
Equation 1 says that the nominal federal funds rate depends upon the nominal
This relationship which holds in the long run assumes that the
natural rate.
Fed lets the federal funds rate move with the real rate plus the long-term
inflation rate expected by the public.
Failure to hold this equality in the
long run results in monetary accelerations
(decelerations)
and inflation
(deflation).
In the short run, however, the funds rate can differ from the longrun equilibrium
value (the nominal natural rate) determined
number of reasons.
unobservable
Both the real rate and long-term expected inflation are
variables.
The Fed has to track them in the short run, which it
may do so by focusing on the behavior of observables
real growth, inflation etc.
objectives
in (1) for a
pertaining
such as actual money,
More importantly, the Fed may have some short-run
to real growth and inflation.
For all these reasons the
actual funds rate differs from the nominal natural rate in the short run.3
How will then one specify a short-run federal funds rate equation?
Following Taylor
(1992), this paper investigates a funds rate equation that
focuses directly on real output and inflation as in (2).
FR, - FR,-,= d, t d, (FR,*_, - F&e,) + d, ( (Y - Y*)/Y*),
t d, Ap; + et ; d,, d,, d, ' 0
where
FR: = rri t pr
3Hetzel (1994) discusses
some of these issues in detail.
(2)
-5-
where y is real GDP; y* is trend or potential GDP; FR* is the long-run
equilibrium funds rate and et is a random disturbance term.
The funds rate
equation given in (2) makes some key assumptions about Fed behavior.
It
assumes that in the short run the Fed targets real GDP and change in
The Fed raises the funds rate if real GDP rises above trend GDP,
inflation.
or if the long-term expected inflation accelerates.
The parameters d, and d,
measure the vigor with which monetary policy "leans against the winds": the
larger are d, and d,, the more vigorously the Fed moves the funds rate in
response to deviations of output from trend and accelerations
in long-term
inflation.
While the Fed is free to pursue its short-run objectives,
its short-
run behavior is assumed to be constrained by the long-run relationship
postulated
in (1).
Thus, equation (2) also assumes that the Fed raises the
funds rate if the actual funds rate is below its long-run equilibrium value.
The long-run equilibrium funds rate equals the nominal natural rate.
The
parameter d, measures the vigor with which the Fed keeps the actual level of
the funds rate in line with the nominal natural rate.4
If this parameter is
unity, then the funds rate equation given in (2) can be expressed as in (3).
FRt=
d, t FR,*_,+ d, ((Y - Y*)/Y*) + d, (P: - P:,) + ct
(3)
4The funds rate equation (2) is specified in first differences rather
than levels of the funds rate. Moreover, the Fed is assumed to target changes
in the inflation rate. These features permit drift in the rate of inflation
and hence in the nominal natural rate. The inclusion of the term d,
( FR,*, - FR,-,) in (2) however ensures that the nominal funds rate converges to
the nominal natural rate in the long run.
-6-
As cdn be seen, the actual funds rate differs from the nominal natural rate
If the Fed
only to the extent that the Fed pursues some short-run objectives.
achieves such objectives
(y = y*, pf = constant), the proper funds rate is the
nominal natural rate, which equals the real natural rate plus the long-term
inflation rate expected by the public.
2.2
Data, Definition of Variables, and Empirical Specifications
Funds Rate Equation
of the
The empirical work uses quarterly data over 195443 to 199244.
The long-
run part of the funds rate equation estimated here is given in (4).
FRt =atbp,tU,
(4)
where FR is the actual, nominal
federal funds rate (average for the quarter);
p is the actual, annualized quarterly inflation rate measured by the behavior
of the implicit'GDP deflator;
specification
to inflation.
The
(4) thus assumes that actual inflation is a good proxy for the
long-term, expected
stationary.
and U is a stationary random disturbance.
inflation and that the random disturbance
term is
The parameter b measures the long-run response of the funds rate
If this parameter
is unity, then the real federal funds rate
(FR, - pt = a t U,) is mean stationary.
If the Fed has on average kept the
real federal funds rate in line with the natural real rate, then the real
federal funds rate may be a good proxy for the real natural rate.
assumptions,
the long-run equilibrium nominal funds rate equation
proxies the nominal natural rate) may be expressed as follows
FR; = pt t 6
Under these
(which
-7-
where iiiis the mean real federal funds rate.
The short-run funds rate equation given in (3) requires proxies for
the equilibrium funds rate (FR*), the long-term expected inflation rate (p*),
and trend GDP (y*).
The empirical work here calculates the equilibrium funds.
rate (FR*) from equation (4).
As indicated above, actual inflation (p) is
used as a proxy for long-term expected inflation (p*).
The variable (y -
y*)/y* is measured as lny - lny', where lny is the natural logarithm of real
GDP; and lny* is the value predicted by the regression of lny on a constant
and linear trend.
This reflects the assumption made here that the long-run
secular component of real GDP can be approximated by a linear trend.
I
however also examine results using potential GDP as proxy for the secular
component.5
Goodfriend
(1993) has convincingly argued that in order to establish
and maintain credibility the.Fed has reacted to the information the long-term
bond rate has had about long-term, expected inflation.
The empirical work
here captures this reaction by including in equation (3) an additional
variable measured as the ten-year bond rate (RlO) minus the actual inflation
rate (p).
This variable provides information about long-term expected
inflation (plus perhaps about real rate) that is not in the actual inflation
rate.
Hence, the short-run funds rate equation estimated is of the form (5).
'The empirical work uses the data--real GDP and the implicit GDP
deflator--that reflect latest revisions, assuming that data revisions are
unlikely to alter the long-run secular nature of the series. The interest
rate data are averages for the quarter. All the data are from Citibank's data
base, except the series on potential GDP which is from the Board of Governors.
-8-
AFR, = d, t d, (FR,L_,- FR,-,I + d, (1nyt-, - Jw$T, )
+ d, Apt-, + d, (RlO, - P,) + ;: d,, AFR,,
s=l
where all variables are as defined before.
+
(5)
[2t
Lagged values of changes in the
funds rate included in (5) capture short-run dynamics of the funds rate
behavior.
If the funds rate moves rapidly in response to the short- and long-
run economic variables discussed above, then changes in the funds rate are
likely to be serially uncorrelated.
to known information,
Furthermore, the Fed is assumed to react
so that only lagged values of inflation and real output
are included in the funds rate equation, except for the long-term bond rate
that enters contemporaneously.
2.3
Estimation
issues: The Long-run Federal Funds Rate Equation
If the time series FR, and pt are nonstationary
in Engle and Granger
consistently
(1987),
estimated
then the long-run equation
by ordinary least squares.
but cointegrated
as
(4) can be
The coefficient
b that
appears on the inflation rate in this equation captures the long-run response
of the funds rate to inflation.
Tests of the hypothesis that b = 1 in (4) can
be carried out by estimating Stock and Watson's
(1993)
dynamic OLS regressions
of the form
FRt =
a t b pt t
s
C, Apt-, t E,
s=-n
(6)
- 9 -
where all variables are as defined before.
Equation (6) includes, in addition
to current inflation, past, current and future values of changes in the
inflation rate.
In order to determine whether the series FR, and pt have unit roots
or whether they are mean stationary, unit root tests are performed by
estimating the Augmented Dickey-Fuller
(1979) regression of the form
k
xt
= a t p X,., t
I:
s=l
es AX,-, t nt
(7)
where X, is the pertinent variable; nt is the random disturbance term; and k
is the number of lagged first-differences
uncorrelated.
of X, necessary to make nt serially
If p=l, X, has a unit root and is thus nonstationary
The null hypothesis p=l is tested using the t-statistic.
in levels.
The lag length k
used in tests is chosen using the procedure given in Hall (1990), as advocated
by Campbell and Perron (1991).
Recently, some authors including Dejong et al. (1992)
that Dickey-Fuller
tests have low power in distinguishing
and mean stationarity.
The long-run relationship
have shown
between unit roots
(4) is therefore estimated
under the alternative that the series FR, and pt may be mean stationary.
that case, the long-run relationship
FR,
nl
In
is estimated as (8).6
n2
= a +X b,, ptms +C b,, FR,, + U,
s=o
s-l
(8)
61f the series are non-stationary, then the long-run relationships among
the series can be estimated without completely specifying short-run dynamics.
However, that is not the case if the series are stationary (Wickens and
Breusch 1988).
- 10 The coefficient that measures the iong-run response of the funds rate to
inflation can be calculated as (z
b,,.(l - ii b,,)) .
this long-run coefficient equals'zity
impliz
The restriction that
that slope coefficients
sum to
unity ( "c'b,, + "c'b,, = 1 in (8)).
s=l
S=O
The empirical work on policy reaction functions summarized
(1990)
indicates that the long-run relationship postulated
stable during the sample period 195443 to 199244.
in Khoury
in (4) may not be
The reason is that the
relative weight the Fed assigned to the inflation objective may have varied
over time.
The power of the conventional test for cointegration
Engle and Granger (1987)
falls sharply when the cointegrating
subject to a structural break.
the one proposed
cointegration
given in
relationship
Hence, the test for cointegration
in Gregory and Hansen (1992).
is
used here is
This test examines
under the possibility that the cointegration
regression
(4) is
subject to a one-time regime shift of unknown timing.
The structural change considered here is of the form (9).
FRt = a, t a2 D,, + b, Pt + b, Dt, Pt + u,
(9)
where Dt, is a dummy variable that is zero if t 5 TT and unity otherwise.
unknown parameter 7 c(O,l)
T is the sample size.
The
denotes the relative timing of the change point and
In the new parameterization
intercept and slope coefficients
a, and b, represent
in the cointegrating
regression before the
regime shift and a2 and b, denote changes in them.
The test for cointegration
given in Gregory and Hansen (1992)
examines the presence of a unit root in the residuals of equation
(9) for all
- 11 The test uses residuals (6) from (9) and is implemented
possible breakpoints.
by running an Augmented Dickey-Fuller
Ait7 = p, it7 t
regression of the form
k es A?Q~-~
s=l
(10)
and then computing a t-statistic for the hypothesis p, = 0.
hypothesis in this test is that of non-cointegration.
The null
The test rejects the
null hypothesis if the largest (absolute) of t-statistics exceeds the critical
value (given in Gregory and Hansen (1992)).
The test also generates the date
of the break suggested by the data.
3.
3.1
Estimation Results
The Long-run Federal Funds Rate Equation
Table 1 presents unit root tests for determining whether the series
FR,, pt, and FR, - pt have a unit root, or are mean stationary.
The t-
statistic for the hypothesis p = 1 in (7) is small for FR, and pt, but large
for FR, - pt.
These results indicate that the series FR, and pt have a unit
root, whereas the series FR, - pt does not.
The latter result implies that
the series FR, and pt are cointegrated as in Engle and Granger (1987).7
7The conventional Engle-Granger test for cointegration examines whether
the residuals in (4) have a unit root or not. The t-statistic for the
hypothesis p = 0 in an Augmented Dickey-Fuller regression (with one lag) is
3.23 (The 5 percent critical value taken from Table 3 in Engle and Yoo 1987 is
3.17).
This result indicates that the series FR, and pt are cointegrated.
- 12 -
Panel A in Table 2 presents the dynamic OLS estimates of the
long-run funds rate equation (4).
x: is a Chi square statistic that tests the
hypothesis that the coefficient that appears on pt in (4) is unity.
This
statistic is small, indicating that the funds rate does adjust one-for-one
with actual inflation in the long run.
Panel B in Table 2 reports an estimate of the long-term coefficient
on pt in (8) under the alternative assumption that the series FR, and pt are
mean stationary.
either.
This estimated coefficient
is not different from unity
These results together then suggest that the long-run, equilibrium
federal funds rate equation is of the form
FR; = iii+ Pt
where iiiis the mean real federal funds rate.
3.2
Stability of the Long-run Federal Funds Rate Equation
As indicated before, the long-run funds rate equation reported in
Table 2 may not be stable during the sample period 195443 to 199244.
Though
the unit root test results discussed above suggest that the real federal funds
rate does not have a unit root and thus the series FR, and pt are
cointegrated,
I nevertheless
re-examine this issue using the test of
cointegration
proposed
cointegration
under the possibility that the cointegrating
in Gregory and Hansen (1992).
This test examines
relationship may be
subject to a one-time regime shift of unknown timing.
Chart 1 graphs the relevant t-statistic
the Augmented
Dickey-Fuller
regression
seen, this Chart has a well-defined
for the hypothesis p, = 0 in
(with k=2) of the form (10).
As can be
minimum and at this minimum the absolute
- 13 -
value of the t-statistic
is large.
(The t-value is 4.88, which exceeds the 10
percent critical value 4.68 given in Table lA, Gregory and Hansen (1992).)
This result indicates that the series FR, and pt are cointegrated
and that
this cointegrating relationship may have shifted once during the sample period
195443 to 199244.
The date of the break suggested by the test is 198043.
This date is
very close to the date 197944, when the Fed changed its monetary policy
operating procedures.
I therefore examine the nature of the shift in the
cointegrating relationship
197944.
(4), assuming that the data of the break instead is
Table 3 presents the dynamic versions of the cointegrating
with slope and intercept dummies.
regression
As can be seen, the long-run coefficient
that appears on inflation is close to unity and the slope dummy is generally
small and not statistically significant.
This result indicates that the
federal funds rate has adjusted one-for-one with actual inflation during preand post-1979 periods.8t9
However, the intercept shift dummy is large and
‘1 also tested for the presence of cointegration between inflation and
the federal funds rate using the test for cointegration proposed by Johansen
and Juselius (1990).
The test procedure consists of estimating a VAR model
that includes levels as well as differences of variables.
The matrix of
coefficients that appear on levels of these time series contain information
about the long-run properties of the model.
The VAR model (with lag length set at 4) estimated here included also a
dummy defined to be unity over 197944 to 199244 and zero otherwise.
The trace
test statistic has a value of 18.9 (the 5,percent critical value is 17.8) and
the maximum eigen value test statistic a value of 15.0 (the 5 percent critical
value is 14.6). These test results are consistent with the presence of
cointegration between the funds rate and inflation. The cointegrating
regression generated by this procedure is
w
= 1.1 pt
The coefficient
that appears on pt is not different from unity.
- 14 -
statistically
significant.
This may be because the real natural rate had
increased or because the Fed may have been reacting differently
economic factors than it did before this subperiod.
to certain
These issues are examined
in the next section where the short-run funds rate equation is estimated.
3.3
A Short-run Funds Rate Equation
The empirical results presented in the previous section suggest that
the short-run funds rate equation
(5) may not have stable parameters during
pre- and post-1979 sample periods.
The short-run equation is, therefore,
estimated
Since the bond rate enters
including slope dummies.
'Alternatively, if FR, and pt are stationary, then stability of the longrun slope coefficient on pt can be examined by estimating a regression of the
form
nl
F{
= a +X
S=O
b,, pt% + z
b,, FRt, + D, + :
A,, (D-p),, +sT, 62,
(D-&s
j=a
s-l
(a)
where D is a dummy variable that is unity uver 197944 to 199244 and zero
n2
otherwise.
The long-run slope parameter b is (g
S=O
7993 period and
("c' b,, + z
a,,)/(l-(%
b,,/l - C b2s) for the pres=l
n2
b,s + Z a,,)) for the period
thereafter.
Equa&
(a) \:s estimatid'by IV '-dver 195444 to 199244. With
nl = 0, n2 = 8, slope coefficients sum to .97 over 5692 to 197943 and 1.22
over 5642 to 1992Q4. These results indicate that the long-run slope
coefficient on p is not different from unity over the subperiods 195642 to
197943 and 1956Qi to 199294.
It should however be pointed out that slope dummies are generally
significant, whereas the intercept shift dummy is not. These results suggest
the presence of different short-run dynamics during pre- and post- 1979Q4
periods.
- 15 -
contemporaneously
in (5), the equation is also estimated using the
instrumental variables procedure.
Table 4 presents ordinary least squares (OLS) and instrumental
variables
(IV) estimates of the short-run funds rate equation (5) over 195544 to 1992Q4.l’
Since OLS estimates are similar to IV estimates, the discussion hereafter focuses on
OLS estimates.
[All standard errors have been corrected for the possible
hetroscedascity
of the regression error.]
The coefficients that appear on various
economic variables are strikingly different over pre- and post-1979 sample periods.
In particular, the estimates reported there indicate that for the sample period
195544 to 197943 the funds rate equation is
AFR, = -.07
(1.5)
(FR - FR*),-,t .04 (lny - lny*),-,
(2.3)
t .57 AFR,-, - .38 AFR,-,
(4.1)
(2.6)
where parentheses contain t-values (absolute).
For the period 197944 to
, 199244 the funds rate equation is
AFR, = -.30
(FR - FR*),-,t .24 (lny - lny*),-,
(2.2)
+ .33 tq+
(3.0)
(2.0)
t .20
(3.0)
(RlO - p), + .09 AFT+-,
(2.6)
- .38 AFR,m*
V-6)
"Lag lengths on various economic variables in the funds rate equation
were selected on the basis of experimentation.
In OLS regressions only the
bond rate enters contemporaneously.
In IV regressions the instruments chosen
are just the lagged values of the economic variables included in the reaction
functions.
Thus, the instruments used in the reaction func;tion (without
money) are*a constant, one-period lagged values of (FR - FR )t,
and (RlO - p),, two-period lagged values of Ap,, and two lagged
(lny - lny )
values of AFtd,. The instruments for interactive-dummy variables enter
similarly.
- 16 -
Thus, during the samp le period 197944 to 199244 the funds rate moved strong lY
in response to the discrepancy
equilibrium
between the actual funds rate and its long-run
value, cyclical expansions in real GDP, accelerations
in actual
inflation, and the long-term bond rate.
The responses of the funds rate to above mentioned economic
variables are either weak or non-existent during the pre-1979 sample period.
In particular,
the funds rate equation presented above indicates that during
the pre-1979 sample period the funds rate has responded weakly to the
discrepancy
between the actual funds rate and its long-run equilibrium
and responded not at all to accelerations
coefficient
equation
value
The
in actual inflation.
that appears on the cyclical expansion variable in the funds rate
is small, though statistically
significant.
One possible explanation
of these results is that the Fed may have focused during this subperiod on
some other indirect measures of real growth and/or inflation.
McNees
(1992)
has in fact presented evidence that indicates that the Fed paid considerable
attention to money growth during the sample period 1970 to 1992.
robustness,
the funds rate equation here is also estimated
Following McNees
To test
including money.
(1992), money is defined by M2 over 198244 to 199244 and by
Ml over the period before, and slope coefficients
on money growth are assumed
to be different during the subperiods 195544 to 197943, 197944 to 198243, and
1982Q4 to 1992Q4.”
"The financial innovations and deregulation of the financial industry
that occurred during the early part of the 1980s changed the character of Ml
demand (Hetzel and Mehra 1989), leading the Fed to de-emphasize Ml in 198244.
The Fed however continued setting annual targets for other monetary and credit
aggregates including M2. Hence, money is measured by Ml over 195443 to 198243
and by M2 thereafter, necessitating the use of different.slope coefficients on
money during these subperiods.
- 17 -
The funds rate equation estimated including money is also presented
in Table 4.
As can be seen, money growth is highly significant.
Including
money in the reaction function reduces somewhat the magnitudes of coefficients
that appear on inflation and real output including the bond rate.
Nevertheless, direct measures of inflation and real output remain significant
in the reaction function that spans 197944 to 199244.
3.4
Examining the Predictive Ability of the Federal Funds Rate Equation over
1979 to 1992
This section examines whether funds rate equations reported in Table
4 are consistent with the actual path of the federal funds rate during the
period 197941 to 1992Q4.12
The equations given in Table 4 are re-estimated by
OLS over 1955Q4 to 198644 and then dynamically simulated over 1979Ql to
1992Q4.13
Predicted values of the funds rate generated using the funds rate
equation without money are reported in column (2) of Table 5 and those
generated using the one with money are in column (5).
these values, predicted as well as actual.
Charts 2 and 3 graph
As can be seen, the funds rate
equation with money tracks the actual path of the funds rate somewhat better
than does the one without money.
Both the mean error and the root mean
squared error decline when money is included in the funds rate equation
(see
12The federal funds rate equation reported here is less successful in
tracking the actual behavior of the funds rate during the pre-1979 period.
'3Simulations are partly within- and partly out-of-sample.
The out-ofsample period 198741 to 1992Q4 chosen here is the one studied b;,z;;lor (1992)
and happens to span most of Greenspan's term as Fed Chairman.
simulations thus implicitly assume that reaction functions display stable
parameters over the period 197944 to 1992Q4 that spans Volcker's and
Greenspan's terms as Fed Chairman.
- 18 The reason is that the funds rate equation with money explains the
Table 5).
actual path of the funds rate during the early part of the 1980s much better
than does the one without money.
reduces substantially
Including money in the funds rate equation
the size of the prediction error that occurs over the
subperiod 1979 to 1982 (compare columns (2) and (5), Table 5).
In order to evaluate further the role of money in the funds rate
equation, Table 5 presents dynamic simulations of the funds rate over the
shorter sample period 1987Ql to 1992Q4 examined by Taylor (1992).
Predicted
values given in column (3) are from the funds rate equation without money and
those in column (4) from the one with money.14
values, actual and predicted.
Charts 4 and 5 graph these
As can be seen, during this period the funds
rate equation without money tracks better the actual path of the funds rate
than does the one with money.
Both the mean error and the root mean squared
error rise when money is included in the funds rate equation.
One explanation
have discounted
measured
by M2.
of the results presented above is that the Fed may
in recent years the leading indicator properties of money as
The evidence reported in Carlson and Sharron (1991) and Mehra
(1992) indicates that the relationship between M2 demand and its traditional
determinants
recent years.
(like income, prices and interest rates) has deteriorated
in
Hence, the reaction function that focuses directly on prices
14Predicted values use OLS regressions estimated over rolling horizons
and are the dynamic, one-year ahead sample forecasts conditional on actual
values of other economic variables.
The forecasts are generated as follows.
The reaction functions are initially estimated over 195544 to 198644 and then
dynamically simulated over 198741 to 198744.
The end of the estimation period
is then advanced four quarters, reaction functions re-estimated and forecasts
prepared as above. This process is repeated until the end of the estimation
period reaches 1991Q4.
- 19 and real output (including the bond rate) can describe actual policy in recent
years much better than the one that also includes money, a finding that is
similar in spirit to the one in Taylor (1992).
Chart 6 highlights the role of the bond rate in predicting the
behavior of the funds rate during the period 1979 to 1992.
The upper panel in
this Chart graphs the funds rate predicted with and without the bond rate in
the funds rate equation."
there.
Actual values
of the funds rate are also charted
The lower panel graphs changes in the funds rate that are predicted by
the bond rate against changes in the bond rate.
observations.
This Chart suggests two
First, the bond rate is quantitatively
the funds rate over 1979 to 1992.
important in predicting
The funds rate equation without the bond
rate seriously underpredicts the level of the funds rate (see the upper
panel).
Second, movements
bonds rate are significant
in the funds rate accounted for by movements
in 1981,
coincide with what Goodfriend
3.5
(1993)
1983-1984,
and 1987.
in the
These periods
calls periods of inflation scare.
Additional Results
The short-run federal funds rate equations reported in Table 4 here
are estimated over the period that includes the late-1950s and 1960s.
During
most of the 1950s and 1960s the Fed's attention was focused more on free
"These predictions, which use the funds rate equation without money
reported in Table 4, were generated as follows. The funds rate predicted
including the bond rate is given by the dynamic simulations of the funds rate
equation in which the bond rate takes the historical values over the
simulation period 197944 to 199244 (see Table 5). The funds rate predicted
without the bond rate is then given by the dynamic simulations in which the
bond rate is held fixed at the 197944 value during the simulation period. The
differences between these two sets of simulations give predictions of the
funds rate that are due to the bond rate.
- 20 reserves and money market rates in general (Poole 1971) than on the federal
funds rate.
To test robustness, the funds rate equation is also estimated
excluding the 1950s and 1960s.
Furthermore, potential GDP is alternatively
used as proxy for the long-run secular GDP, which the Fed is assumed to use as
a target.
Table 6 presents funds rate equations estimated over 197044 to
199244.
The long-run part of the funds rate equation is still measured as the
inflation rate plus the mean real funds rate, the latter now approximated
its sample mean over 197041 to 1992Q4?
highly significant
by
As can be seen, money growth is
in these reaction funct i ons.
inflation and real GDP rema in significant.
However, direct measures of
The estimated funds rate equation
still indicates that during the sample peri od 197944 to 1992Q4 the funds rate
responded strongly to cyclical expansions
in real GDP, accelerations
in actual
inflation, and increases in the long-term bond rate and th,e long-run
equilibrium
funds rate.
to the alternative
regressions
Furthermore, the results are also robust with respect
proxy used for the secular component of real GDP (see
in Table 6).
The funds rate equations reported above indicate that the Fed has
reacted to accelerations
1992.
in actual inflation during the subperiod 1979 to
This behavior is tantamount
to inflation targeting
in which the short-
term inflation target at any time is the previous period's inflation rate.
now examine a version in which the Fed's short-term inflation targets are
assumed to be viewed differently.
During the sample period 1979 to 1992
actual inflation declined by almost 6 percentage points from 8,3 percent in
161 get similar results if the sample mean over 195443 to 199244 is
instead used.
I
- 21 -
1979 to 2.7 percent in 1992.
However, most of this deceleration
occurred during two subperiods 1979 to 1982 and 1990 to 1992.
in inflation
Inflation
declined by about 4 percentage points from 8.3 percent in 1979 to 4.3 percent
in 1982, then hovered around 4.0 percent until 1990, declining thereafter to
2.7 percent in 1992.
I assume that the decline in inflation observed during
this subperiod was due to Fed policy and that the Fed behaved as if it had
short-term inflation targets.
Hence, I assume short-term inflation targets
that successively decline over time, so that they roughly match the temporal
pattern and the overall reduction of inflation rates during this subperiod.
In the particular scenario assumed here, the inflation target variable takes
values 8.3 in 1979, 7.3 in 1980, 6.3 in 1981, 5.3
4.0 in 1985-1990,
3.5 in 1991,
and 3.0 in 1992.
in 1982, 4.3
in 1983 -1984,
Table 7 presents funds rate
equations estimated with this new measure of the inflation target.
rate equation is estimated with and without including money growth.
The funds
As can be
seen, all variables appear with theoretically correct signs and are
statistically
significant.
The dynamic within-sample
simulations graphed in
Chart 7 indicate that this reaction function is consistent with the actual
path of the federal funds rate during the period 1979 to 1992.17
The funds rate equations reported in some other recent studies
(Khoury 1990, McNees 1992) use forecasts of inflation and real GDP (growth).
These studies thus assume that the Fed raises the funds rate if predicted real
17Alternatively, one could use as target values the midpoint of inflation
predictions made by FOMC members at their July meetings each year. These
predictions are made public by the Chairman as part of his Humphrey-Hawkins
testimony to Congress.
In this scenario, the inflation target variable takes
values 10.2 in 1979, 9.5 in 1980, 8.2 in 1981, 5.4 in 1982, 4.6 in 1983, 3.9
in 1984, 3.8 in 1985, 2.2 in 1986, 3.6 in 1987, 3.4 in 1988, 4.6 in 1989, 4.2
in 1990, 3.2 in 1991 and 3.0 in 1992.
The funds rate equation estimated using
this measure of inflation target gave qualitatively similar results.
- 22 -
GDP rises, or if predicted inflation increases.
The predicted values used in
these reaction function studies usually come from the forecasts presented at
the Federal Open Market Committee meetings and the private forecasts prepared
by some prominent commercial forecasting services.
In contrast, the funds
rate equation here includes actual, lagged values of real GDP and inflation.
The results here however do not rule out the Fed behavior assumed in these
other studies.
If the Fed and private forecasters use,past inflation and real
GDP in predicting
future inflation and real GDP, then the funds rate will be
correlated with past inflation and real GDP.18
3.6
A Counterfactual
Simulation
The short-run,
federal funds rate equations estimated here use the
latest revised data, rather the data the FOMC actually observed at the time.
This raises the question whether the short-run reaction functions are robust
'18This can be easily seen as follows.
function is of the form
Assume that the Fed's reaction
AFR, = a, + a, (P: - pP_,) + a, (lny, - lny:)",
where pe is predicted inflation; and (lny, - lny:)e is predicted real GDP gap.
Assume i hat these variables are determined as follows.
f-4= d, + d, Pi-, + e,,
= f,+ f, (1ny+,- W,T-,
1 + c2t
(lw, - lny:)
If the Fed and private forecasters use these equations to generate their
forecasts, then the funds rate equation that include only know information can
be expressed as follows.
AFR, = a, + a, f, + a, d, (P,-, - ptm2) + a2 f, (lny,-, - lnyl-')
Thus, the funds rate will be correlated with lagged inflation and real GDP.
- 23 -
to revisions in the data used.
Rather than re-estimate the equations using the
preliminary available data I examine this robustness issue in a different way.
I begin with the assumption that the Fed behavior since 1979 can be
described by a reaction function of the form
AFR, = 25
(1v,_, - In&?,)
+ .4(P,-, - Ti,.,)
t .2 (RN,,, - pt-,) + .4 (FR,+_,- FR,.,)
(11)
where
FR; = iii
+ Pt
The reaction function (11)
embodies the key properties of the short-run
federal funds rate equations estimated here.
economic variables in (11)
conduct a counterfactual
come from those reported in Table 4.
simulation of (11)
actual but model-generated
The coefficients that appear on
over 1979 to 1992,
values of real GDP and inflation.
I then
using not
The results of
this exercise provide a somewhat different evidence on the issue whether the
reaction function estimated here is consistent with actual policy during this
subperiod.
McCallum
[The macromodel used is the Keynesian modelI
(1988) and Judd and Motley (1992),
employed recently by
and simulations assume the economy
"The Keynesian model, estimated over 195941 to 199244, consists of four
equations (Judd and Motley 1992).
The first is the real aggregate demand
equation in which the growth rate of real GDP is a function of the lagged
growth rate of real M2, real government spending, and its own lagged value.
The second is the aggregate supply equation in which the current inflation
rate depends upon past inflation and the gap between actual and trend GDP.
The third equation defines trend GDP, lny , as the fitted values of a log
linear time trend of real GDP. The fourth equation is the real M2 demand
equation in which the growth rate of real M2 is a function of current and
lagged growth rates of real GDP, short-term interest rates (measured here by
the funds rate) and own lagged values.
In addition, lagged levels of these
variables also appear in the real money demand equation, because M2 is found
cointegrated with real GDP and interest rates.
- 24 was hit by the same set of shocks that actually occurred during this
The variable j!jin (11)
subperiod.]
rates or followed the disinflation
was either measured by lagged inflation
path assumed in the previous section.
Table 8 presents values of the funds rate, simulated and actual.
graphs these values.
Chart 8
As can be seen, simulated paths of the funds rate
generated by (11) are fairly close to actual paths.
These results confirm
that short-run reaction functions estimated here capture the key determinants
of the funds rate during the period 1979 to 1992.
4.
Concluding Observations
This paper finds that the federal funds rate and the inflation rate
are cointegrated
during the sample period 195443 to 199244.
The results
indicate that the funds rate adjusts one-for-one with the actual inflation
rate in the long run.
substantially
Furthermore,
In the short run, however, the funds rate differs
from the value given by this cointegrating
relationship.
in the short-run the funds rate has responded to some direct and
and indirect measures of inflation and real GDP, the two final goal variables
the Fed cares about.
These short-run responses however have not been stable over time.
In particular,
the evidence reported here indicates that the actual behavior
of the funds rate during most of the 1980s is well predicted by a reaction
function in which the funds rate rises if real GDP is above trend GDP, if
actual inflation accelerates,
or if the long-term bond rate and the
equilibrium
Many of these short-run responses are missing,
funds rate rise.
or found to be weak during the pre-1979 period.
- 25 -
Money growth when included in the short-run funds rate equation is
generally significant,
indicating that in the 1970s and the 1980s the funds
rate has reacted to the information in money about inflation and/or real
growth.
The evidence reported here however indicates that in recent years the
Fed may have discounted the leading indicator properties of the empirical M2
measure of money.
- 26 -
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Campbell, J.Y. and P. Perron.
Macroeconomists Should Know About Unit
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in O.J. Blanchard and S.
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pp. 141-200.
"The Demand for M2, Opportunity Cost,
Carlson, John B. and Sharon F. Parrott.
Federal Reserve Bank of Cleveland Economic
and Financial Change."
Review, vol. 27, 42, 1991, pp. 2-11.
Federal
Cook, Timothy. "Determinants of the Federal funds Rate: 1979-1982”,
Reserve Bank of Richmond Economic Review, vol. 75, January/February
1989, pp. 3-19.
"The Effect of Changes in the Federal Funds
Cook, Timothy and Thomas Hahn.
Rate Target on Market Interest Rates in the 1970s.’
Journal of Monetary
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Integration vs. Trend Stationarity in Time Series." Econometrica
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Engle, Robert F. and Byung Saun Yoo. "Forecasting and Testing in
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Journal of Econometrics, vol. 35, May 1987,
pp. 143-59.
Engle, Robert F. and C.W. Granger.
'Cointegration and Error-Correction:
Representation, Estimation and Testing."
Econometrica, vol. 55,
March 1987, pp. 251-76.
Goodfriend,
Marvin.
1979h992. I’
Winter 1993.
"Interest Rate Policy and the Inflation Scare Problem
Federal Reserve Bank of Richmond Economic Quarterly,
'Residual-Based Tests for
Gregor y, Allan N. and Bruce E. Hansen.
Cointegration Models with Regime Shifts." Mimeo. November 1992.
Hall, A. "Testing for a Unit Root in Time Series with Pretest Data Based
Model Selection."
Manuscript, North Carolina State University, 1990.
Hetzel , Robert L. "Why the Price Level Wanders Aimlessly":
Bank of Richmond, 1994, Mimeo.
Federal Reserve
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Khoury, Salwa S. "The Federal Reserve Reaction Function: A Specification
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"Maximum Likelihood Estimation and Inference on
Johansen, S. and K. Juselius.
Cointegration--With Applications to the Demand for Money." Oxford
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Judd, John P. and Brian Motley.
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Federal Reserve Bank of San Francisco Economic Review,
Number 3, 1992, pp. 3-22.
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in Carnegie-Rochester Conferences Series in Public Policy, 1988,
pp. 173-204.
McNees, Stephen K. "A Forward Looking Monetary Policy Reaction Function:
Continuity and Change." New Ensland Economic Review. November/December
1992.
Mehra, Yash P. "Has M2 Demand Become Unstable?"
Richmond Economic Review, September/October
Federal Reserve Bank of
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Poole, William.
"Rules-of-Thumb for Guiding Monetary Policy." in Open Market
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Thornton, Daniel.
"The Borrowed-Reserves Operating Procedure: Theory and
Evidence."' Federal Reserve Bank of St. Louis Economic Review, Vol. 10,
January/February 1988, pp. 30-54.
Wallich, Henry C. "Recent Techniques of Monetary Policy."
Federal Reserve
Bank of Kansas City Economic Review, Vol. 69, May 1984, 21-30.
Wickens and Breusch.
"Dynamic Specification, The Long-Run and the Estimation
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of Transformed Regression Models."
1988, pp. 189-205.
Table 1
Unit Root Test Results; 195443 - 199244
*t
P
t-statistic
for
Lag
p=l
x2(l)
x2(4)
Q(36)
n
FRt
.95
-1.97
7
.35
.97
33.6
pt
.89
-2.09
2
.47
.55
24.2
FR, - pt
.85
-2.62*
Notes:
3
.06
.04
18.3
FR is the federal funds rate; and pt is the inflation rate measured
by the behavior of the implicit GDP deflator.
The t-statistic and p
above are from the Augmented Dickey-Fuller regression of the form
*t = Q t p Ztvl t
i
e, AZ,-,,
where Z is the pertinent series.
The
s=I
number of lagged first differences (n) included in these regressions
are chosen using the procedure given in Hall (1990).
X2(l) and
x2(4) are Lagrange Multiplier tests for lst- and fourth-order serial
correlation in the residuals of the Augmented Dickey-Fuller
regression.
Q(36) is the Ljung-Box Q-statistic, which tests for the
presence of higher order serial correlation.
significant
at the 10 percent level.
Table 2
Long-run Federal Funds Rate Equation
A.
FR, and pt Nonstationary; Dynamic OLS Regressions;
195443 - 199244
Leads and
Lags
(-4,
4)
1.
FR, = 1.6 t 1.1 pt ; MA(lO); x:(l)
= .08
(-8,
8)
2.
FR, = 2.3 t
= .02
B.
FR, and pt Stationary;
(1.3)
(1.6)
(3.7)
.96 pt ; MA(l0);
(3.3)
x:(l)
IV Estimates;
195544 - 199244
3.
FR, = .05 t
(.3)
.18 pt t 1.09 FR,_, - .43 FR,-,
(2.9)
(11.3)
(3.3)
t .30 FRtm3 - .10 FRtm4
(2.4)
(1.3)
Long-run Coefficient on pt = 1.04
DW = 1.9
Q(36) = 34.1
(23.6)
Notes:
Standard errors in Dynamic OLS
Parentheses contain t-values.
regressions have been corrected for the presence of moving-average
MA(l0) indicates the presence of tenth-order
serial correlation.
The order of
moving-average serial correlation in the residuals.
moving-average is chosen by examining autocorrelations of residuals
X: is the Chi-square statistic that tests the
at various lags.
hypothesis that the slope coefficient on pt is unity and is
distributed with one degree of freedom.
Regression (3) above is estimated by instrumental variables (IV).
The instruments used are a constant, one lagged pt, and four lagged
values of FR,. The long-run coefficient on pt is calculated as the
short-run coefficient on pt divided by one minus the sum of
coefficients that appear on lagged values of FR,.
Table 3
Stability Test Results; Long-run Federal Funds Rate Equation
Leads and
Lags
A.
Dvnamic OLS Rearessions
(-4, 4)
1.
FR, = 1.0 t .91 pt t 2.4 D, t .34 D,p,
(2.0) (6.7)
(2.9)
(2.1)
; MA(3)
(4%
6)
2.
FR, = 1.1 t .90 pt t 2.7 D, t .32 D,p,
(2.3) (6.5)
(2.3)
(2.3)
; MA(3)
(-8,
81
3.
FR, = 1.3 t .86 pt t 3.1 D, t .23 D,p,
(3.9)
(1.4)
(2.7)(6.4)
; MA(3)
B.
Dvnamic GLS Reqressions
(-4, 4)
4.
FR, = .6 t 1.1 pt t 2.9 D, t .05 D,p,
(.6) (6.1)
(2.9)
(-4)
t-6,
6)
5.
FR, = .51 t 1.1 pt t 3.2 D, t .03 D,p,
(05) (5.6)
(3.1)
w
; AR(l)
(-8,
f3>
6.
FR, = 1.1 t .96 pt t 3.5 D, t .02 D,p,
(1.1)(5.2)
(4
(3.4)
; AR(l)
Notes:
;
AR(l)
Parentheses contain t-values corrected for the presence of serial
correlation.
MA(3) indicates that residuals have third-order moving
The order of moving-average is chosen
average serial correlation.
by examining autocorrelations of residuals at various lags.
Regressions (4) through (6) are the dynamic GLS regressions
estimated assuming that the residuals follow a first-order
autoregressive process (AR(l)).
D is a zero-one dummy variable that takes values 1 over
1992Q4 and 0 otherwise.
1979Q4 to
Table 4
Short-run Federal Funds Rate Equations; 195544 - 199244
Variable: AFR,
Dependent
Independent
Variables
(4)
(1)
(2)
(3)
OLS
IV
OLS
IV
(FR- FR*It-,
-.07
(1.5)
-.07
(1.5)
-.05
(1.3)
-.05
(1.3)
Dt-,
(FR- FR*It-,
-.23
(2.2)
-.21
(1.7)
-.37
(4.9)
-.36
(4.0)
WY - lw*),-,
.04 (2.3)
.03 (2.3)
D,.,
WY - Iv*)t-,
.20 (2.0)
.19 (1.7)
.17 (3.3)
.16 (2.6)
D
.33 (3.0)
.34 (2.9)
.27 (3.4)
.27 (3.2)
.20 (3.0)
.18 (2.2)
.I2
.I1
.57 (4.1)
.57 (4.0)
.45 (3.0)
t-2
Apt-2
DtW
- P),
AFRt-I
q-2
Dtm,
q-,
(1.8)
(1.2)
.45 (3.0)
-.38
(2.6)
-.38
(2.5)
-.19
(1.8)
-.19
(1.7)
-.48
(2.6)
-.48
(2.0)
-.44
(2.4)
-.44
(2.4)
DIMlr 1
.03 (1.2)
.03 (1.2)
Dly-2
.07 (2.3)
.07 (2.3)
.28 (4.7)
.28 (4.8)
.13 (3.6)
.14 (3.3)
D2Ml
D3M2
t-1
t-1
-2
R
DW
Q(36)
Notes:
.37
.37
2.0
2.0
30.8
30.4
.54
.54
1.87
1.87
46.0
45.7
FR* is the long-run equilibrium value determined as FR: = 1.89 t pt;
lny is the natural logarithm of real GDP; lny* is the value
predicted from a regression of lny on constant and linear trend; p
is the inflation rate; RlO is the ten-year bond rate; and Ml and Mh,
respectively, are Ml (one-quarter annualized growth rate) and M2
(four-quarter growth rate) measures of money. Dl is a dummy
variable that is 1 over 195443 to 197943 and 0 otherwise; D2 is a
dummy that is 1 over 197944 to 198243 and 0 otherwise; and 03 is a
dummy that is I over 198244 to 199244 and 0 otherwise.
Parentheses
contain heteroscedastic-consistent t-values.
Table 5
Actual and Predicted Values of the Funds Rate
Reaction Function
Reaction Function
With Monev
Without Money
Year
Actual
Predicted
(1)
(3)
(Error1
Predicted
(Error\
Predicted
(Error)
Predicted
(Error1
(5)
(4)
(2)
11.0
(.l)
(1.4)
13.5
(-.l)
13.4
(2.9)
16.2
(.2)
11.6
( .6)
12.4
(-.2)
1979
11.1
10.4
1980
13.3
11.9
1981
16.4
1982
12.2
1983
9.1
8.3
(.7)
8.7
(.3)
1984
10.2
9.9
(.2)
9.9
(.3)
1985
8.1
9.3(-1.2)
8.6 (-.5)
1986
6.8
7.6 (-.8)
7.2 (-.4)
1987
6.6
7.0 (-.3)
7.3 (-.6)
7.0 (-.4)
7.2 (-.5)
1988
7.5
7.7 (-.2)
7.9 (-.3)
7.3
7.2
1989
9.2
8.1 (1.1)
8.2
(.9)
7.6 (1.6)
7.4 (1.8)
1990
8.1
8.1
(.O)
7.4
(.6)
7.7
6.8 (1.2)
1991
5.7
6.1 (-.4)
5.6
(.l)
6.1 (-.4)
5.2
(.5)
1992
3.5
3.6 (-.I)
3.5
(.O)
3.7 (-.2)
3.0
(.5)
.38
.21
.26
.73
.68
Mean Error
.Ol
RMSE
.49
Notes:
(.8)
1.07
(.3)
(.3)
(.3)
Predicted values given in columns (2) and (5) above are generated using the policy
reaction functions given in Table 4 that are re-estimated (by OLS) over 1955Q4 to
198644 and then dynamically simulated over 197941 to 199244.
Predicted values
given in columns (3) and (4) above are the dynamic, one-year ahead forecasts
generated using rolling regressions (see footnote 10 in the text).
Table 6
Short-run Federal Funds Rate Equations; 197044 - 199244
Dependent Variable: AFR,
Independent
Variables
lnv*: Trend GDP
lnv*:Potential GDP
IV
IV
OLS
.03 (.5)
.03 ( .5)
.03 ( .5)
.04 ( .6)
-.44 (6.0)
-.44 (6.0)
-.44 (5.9)
-.44 (5.9)
Uw - W*)t-,
.19 (3.6)
.19 (3.9)
.19 (3.2)
.18 (3.6)
D
.37 (4.8)
.37 (4.9)
.36 (4.8)
.36 (4.9)
.24 (3.4)
.23 (3.3)
.23 (3.3)
.23 (3.2)
(RIO - P),
.25 (2.7)
.23 (3.5)
.25 (2.7)
.24 (3.5)
DlMl
t-1
.ll
.ll
.lO (2.4)
.lO (2.2)
DIMlt 2
.17 (3.4)
.17 (3.6)
.17 (3.3)
.17 (3.4)
DZ”lt-,
.32 (6.6)
.32 (6.5)
.33 (6.9)
.33 (6.8)
D3MZt ,
.12 (4.1)
.12 (4.8)
.13 (4.9)
.14 (5.5)
-.20 (1.7)
-.20 (1.7)
(FR - FR*Jt-,
Dt-, (FR - FR*J,
D
t-2
Apt-2
t-3
Apt-3
AFR
t-1
ii2
DW
QF’)
Notes:
(2.7)
-.23 (1.9)
.67
-.23
.67
(2.5)
(1.9)
.66
OLS
.66
2.1
2.1
2.1
2.0
23.8
23.8
23.2
23.1
See notes in Table 4. The long-run federal funds rate equation used
above is 2.6 + pt, where 2.6 is the sample mean of the real funds
rate over 197041 to 199244.
Table 7
Short-run Federal Funds Rate Equations With Assumed
Inflation Targets Over 1979 to 1992;
Independent
Variables
Dependent
Iv
195544 - 199244
Variable: AFR,
Iv
VR - FR*lt-,
-.07 (1.8)
-0.5 (1.3)
Dt-, VR
-.45 (3.3)
-.53 (6.7)
(lw
- FR*)+,
- lny*)tm,
Dtm, (6
- lw*)t_,
.04 (2.5)
.26 (2.8)
.16 (2.6)
D t-2 (P - if)t-2
.41 (2.7)
.39 (3.4)
D t-, (RIO - P),
.33 (3.8)
.18 (2.3)
AFRt-l
.57 (4.3)
.45 (3.0)
-.40 (3.4)
-.20 (2.3)
-.48 (2.3)
-.42 (2.3)
AFRt-2
D
t-1
AFRt-l
DlMlt-1
.02
DIMlt 2
.08 (2.4)
D2Mlt ,
.24 (4.9)
D3MZt-,
.17 (4.5)
i*
.39
.55
DW
1.84
1.84
Q(36)
Notes:
36.8
(.8)
46.2
p, is the inflation target. All other variables are
defined as before (see notes in Table 4). it takes
values 8.3 in 1979, 7.3 in 1980, 6.3 in 1981, 5.3 in
1980, 4.3 in 1983-1984, 4.0 in 1985-1990, 3.5 in 1991,
and 3.0 in 1992.
Table 8
Actual and Simulated (Using a Policy Rule)
Values of the Funds Rate
Inflation Target (it):
Disinflation Path
Given in Table 7
Inflation Target (it):
Last Period Inflation
Rate
Year
Actual
Simulated
Error
Simulated
Error
12.4
-1.20
13.6
-.2
2.8
15.6
.8
11.5
1.7
13.0
-. 7
9.1
7.6
1.5
6.6
2.4
1984
10.2
9.9
.3
9.6
.6
1985
8.1
9.4
1986
6.8
7.1
-. 3
6.8
1987
6.6
6.6
.l
5.4
1.2
1988
7.6
8.2
-. 7
7.8
-. 2
1989
9.2
9.0
.2
9.7
-. 4
1990
8.1
8.3
-. 2
8.5
-. 4
1991
5.7
6.3
-. 6
6.4
-. 7
1992
3.5
3.8
-. 3
3.5
0
1979
11.2
12.1
1980
13.3
13.3
1981
16.4
13.6
1982
12.2
1983
Mean Error
RMSE
Notes:
-.9
.06
-1.3
10.2
-2.1
.O
.l
-. 1
1.00
1.08
Simulations use the policy rule (11) and the Keynesian model
summarized in footnote (12) of the text. Simulations begin in 1979.
Chart
Test
for
Cointegration
1
With Regime Shift
-2
Breakpoint
T-statistic
tests the null hypothesis
that rho=0
in the
Notes:
(12) given in the text for
Dickey-Fuller
regression
of the form
breakpoint
in the interval
(.15T, .85T), where T is the sample
10 percent critical
value.
at -4.66 represent
Augmented
a given
size.
Solid
line
Chart
The
Federal
2
Funds
Rate; Quarterly,
1979Ql to 1992Q4
20
16
12
4
a
-4
1979
Notes:
columns
1981
1980
Predicted
(2) and
1982
1983
1984
1985
1986
values are the quarterly
(5). Table 5.
1987
values
1988
of
1989
annual
1990
numbers
1991
1992
reported
c
in
The
Federal
Chart
3
Funds
Rate;
1979
-
Annual,
1992
20
16
12, __
Rodlcted
Without
MOnW
+J
c
:
eI -kl
a
t!
I --
Error
I-
,/-.
MOW
“\,
.L---c--\
--z-VA
(1 --
Error
-f1-
Without
1979
1980
Predicted
Notes:
Table 5.
\
-+
With Yonry
1981
1982
values
1983
1984
Xv---
1985
__-----
1986
_--- /
1987
1988
are the annual numbers reported
1989
1990
in. columns
1991
1992
(2) and
(5).
The
Chart 4
Funds Rate;
Federal
Quarterly,
10
Prmdietad
Without
Honay
6
6
4
2
Pror
With Itonoy
0
-2
1967
Notes:
columns
1988
Predicted
(3) and
1969
values
(4).
are
Table
5.
the
1990
quarterly
values
1991
of
annual
1992
numbers
reported
in
The
Federal
Chart
5
Funds
Rate;
1987
-
Annual,
1992
10
AmJDl
6
4
21-.
Error With Manly
C1 -t3ror
-C
Ylthout
wonay
cI-
Notes:
Table 5.
Predicted
1989
1966
1967
values
are
the
annual numbers
19YO
reported
lYY1
1YYd
in columns
(3) and
(4),
Chart
The
16
A
15
Role
of
6
the
Bond
Rate
ActuDl
Fund* RltD
12
9
6, --
3I --
C) .-
-2 i-
1979
Notes:
money.
1980
1961
1985 1986 1987 1966 1969 1990 1991 1992
are generated
using the funds rate equation without
15 in the text.
1982
values
Predicted
See footnote
1983
1984
Chart 7
Actual
and Predicted
Funds
Annual
Averages
Rate;
17
16 -15 -14 -13 -12 -11 -10 -9 -6 -7 -*
6 -5 -4 -3 -2 -.
Error
Without
Nmry
Error
With Money
1 -0
-1
--G---e-
1962 1983 1964
Predicted
values are the dynamic, within-sample
Notes:
using the regressions
reported
in Table 7.
I
1979
1980
1961
forecasts
generated
Chart
Actual
and
8
Simulated
Funds
Annual Averages
Rate;
6
1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992
Notes:
Simulated values are generated using the polic’v rule
(II) of the text.
@FedFRASER