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Working Paper Series
Forecasts of Inflation From Var Models
WP 94-08
This paper can be downloaded without charge from:
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Roy H. Webb
Federal Reserve Bank of Richmond
Working Paper 94-8
FORECASTS OF INFLATION FROM VAR MODELS
Roy H. Webb*
Federal
Research Department
Reserve Bank of Richmond
July 1994
This is a preprint of an article published in the Journal of Forecasting , vol. 14, May 1995, pp. 267-85.
*The author
is grateful
to James Lothian, Stephen McNees, Yash Mehra, Radha
referees for helpful comments.
The views and
opinions expressed
in this paper are solely those of the author and should
be attributed
to any other person or the Federal Reserve Bank of Richmond.
Murthy, and two anonymous
not
FORECASTS
OF INFLATION
FROM
VAR MODELS
ABSTRACT
Why are forecasts of inflation from VAR models
forecasts
of
performance,
real
variables?
This
paper
Accounting
actions,
two
changes
of
that
relatively
a VAR model
poor
fitted to U.S.
Statistical work by other authors has found
that coefficients in such price equations
monetary
documents
and finds that the price equation
postwar data is poorly specified.
so much worse than their
in
may
not be constant.
monetary
policy
for those two shift5 yields significantly
more
Based on specific
regimes
are
proposed.
accurate forecasts and
lessens the evidence of misspecification.
Key words:
regimes.
Inflation forecasts, vector autoregressive
models, monetary policy
FORECASTS
OF INFLATION
Forecasts of real macroeconomic
FROM
VAR
MODELS
statistics for the United State5 have
been successfully produced by vector autoregressive
(VAR) models.
McNees
(1986), for example, compared a series of published forecasts produced in the
early 1980s by Robert Litterman's VAR model with forecasts made by conventional macroeconomic
forecasters.
McNees’s
comparisons showed Litterman's
forecasts to be more accurate than six competitors for real GNP growth and the
unemployment
ahead.
rate for forecasts varying in length from one to eight quarters
In addition, Webb (1991) compared simulated forecasts of business
cycle turning points generated by a VAR model adjusted by individually set lag
lengths (AVAR) with actual forecasts from three large forecasting services.
That study found little difference for one-quarter-ahead
superior performance
by the AVAR for four-quarter-ahead
forecasts but
forecasts.
VAR forecasts of inflation have been less successful relative to other
forecasts.
McNees's data shows that the average error of the series of
Litterman's
inflation forecasts was typically the worst of several that were
compared, with his errors often double those of the best forecaster.
Zarno-
wit2 and Braun (1991) also simulated VAR forecasts from a model with restrictions similar to Litterman's, which is often referred to as a BVAR model.
In
comparison with actual forecasts included in the National Bureau of Economic
Research-American
Statistical Association survey, they found "The BVAR
forecasts of [real GNP] perform relatively well . . . . but the BVAR forecasts of
... [the GNP implicit price deflator) are apparently much weaker."
fOreCaSt
Simulated
of inflation from models that are presented below are often less
accurate than simply predicting no change.
This paper attempts to diagnose the source of inaccurate inflation
forecasts and to suggest a direction for improvement.
performance of several models is first documented.
evidence of misspecification
from two models.
The relatively poor
In the next section
is presented for representative
Possible sources of misspecification
price equations
are next examined;
2
evidence is presented that the price equations' coefficient5
to two monetary policy
regime
changes.
are unstable due
A possible remedy is evaluated by
means of simulated forecasts.
RELATIVELY
POOR PERFORMANCE OF VAR INFLATION FORECASTS
VAR MODELS
A VAR model uses historical data to predict future values.
unrestricted
X=
VAR (WAR)
p + p(L)X+
where
X
is a
terms,
f3(L)
a
vector
kxl
An
model can be written
e
(1)
vector of variables,
kxl
u
is a
vector of constant
kxl
is a polynomial of degree m in the lag operator
of
error
terms.
(L), and
e
is
In practice, a k-variable model is often
simply k separate equations for which the coefficients can be estimated by
ordinary
least squares.
pt = PLp+ gg
P&L-,
The price equation in the WAR
considered below is
+ et
(2)
where p is the inflation rate, t indexes time, v indexes the particular
variable,
j indexes the lag number, and
pI, and the
fl,,@ are coefficients.
Note that even for a relatively small VAR model such as the one used in
this paper, with the number of variables k=5 and the lag length m=6, the price
equation contains 31 coefficients to be estimated.
It is often believed
desirable to limit the number of estimated coefficients.
As Doan put it,
"Forecasts made using unrestricted vector autoregressions
often suffer from
the overparameterization
forecast
errors.
n’
of the models ... [which] causes large out-of-sample
In addition, Hafet and Sheehan (1989) presented evidence
that led them to conclude that "relatively short-lagged
be
more
[VAR] models
accurate, on average, than longer-lagged specifications."
typically,
performance
[tend] to
More
Engle and Yoo (1987) have simply asserted that "The forecasting
of unrestricted
VARs has not been particularly good . ..."
3
In order to limit the number of estimated parameters, this paper's
strategy is to reduce many lag lengths in an adjusted VAP (AVAP) model.
For
each equation, set two lag lengths -- one for the dependent variable and
another for all independent variables -- in order to optimize
criterion.
Criterion
some
statistical
The particular criterion used in this paper is the Schwarz
(SC), which typically gives a parsimonious specification.'
It is
defined as
SC = T In a2 + Nln
T
(3)
where T is the number of observations, N is the number of estimated coefficients, and
a2
is the estimated residual variance.
By minimizing
the
Schwarz Criterion one is trading off the lower residual variance from adding
an additional coefficient against a penalty term that rises with the number of
estimated coefficients.
The strategy for selecting lag lengths in this paper
is to (1) specify a maximum lag length; (2) assume that all independent
variables
in each equation have the same lag length; and (3) compute the
Schwarz value for all possible lag length combinations.
lengths for several variants of
the Appendix.
a
basic
five
variable
The resulting lag
model are presented in
Of special interest is the price equation for the AVAP model,
which is
PC=Pp+g bP% +c, g b,J”,t-j + Et
(4)
where the number of lags m, of the dependent variable is three and the number
of lags q of each independent variable is one.
Forecast error statistics from representative VAR models are presented
in Tables 1 and 2; a detailed account of the models' construction
used is given in the Appendix.
five U.S.
time
series:
base, the manufacturing
bill
rate.
real
and the data
Unless otherwise noted, each model contains
GNP, the GNP implicit price deflator, the monetary
capacity utilization rate, and the go-day Treasury-
Two models illustrate the basic result of relatively poor infla-
4
tion forecasts.
The model labeled AVAB is identical to one described in Webb
(19851, except that the lag lengths are set by the simpler procedure described
above.
The model labeled BVAB uses Bayesian "priors" that have been advocated
by Litterman
(1984) to effectively
limit the model's number of parameters that
need to be estimated.
FORECASTING
PROCEDURE
A rolling regression procedure was used to simulate forecasts to compare
with data actually observed through 1990 Q4.
Coefficients
in each model were
first estimated based on actual data from 1952 Q2 to 1977 Ql; one-quarterahead forecasts were made for each variable, which were then used to prepare
forecasts for the next quarter, and
50
forth up to eight quarters ahead.3
Each model's coefficients were then reestimated based on data from 1952 Q2 to
1977 Q2, and forecasts were again produced for each variable up to eight
quarters ahead.
accordingly
The procedure was repeated through 1990 Q3.
100 observations
quarter-ahead
There were
on which the first estimate was based, 55 one-
forecasts, and 48 eight-quarter-ahead
forecasts.
Each forecast
was compared with actual values and error statistics were calculated.
error is simply the actual value minus the predicted value.
An
The square root
of the mean squared error (RMSE) was then calculated for all available
forecasts.
Exhibit 1 {place here} contains BMSE values for the levels of the
interest rate and percentage changes of real GNP and the implicit deflator.
For the latter two variables the columns labeled "Single Quarter
were derived from percentage
Forecasts”
changes from the preceding quarter stated as a
compound annual rate; the column labeled "Cumulative Forecasts" was derived
from the percentage
change over four quarters.
The forecast horizon is the
number of quarters beyond the latest data that were used to estimate the
coefficients
for the model.
Thus for a one-quarter-ahead
forecast for 1977
Q2, only data referring to 1977 Ql and previous quarters would be used; that
5
same data would also produce a two-quarter-ahead
forecast for 1977 Q3, and so
forth.
RMSEs are useful for giving a quick view of model accuracy.
Another way
to view the same data is presented with Theil U statistics in Exhibit 2.
{Place here} Those values simply represent the ratio of the RMSE described
above to the RMSE of a forecast of no change from the previous period.
with RMSEs,
smaller
U values represent
more
As
accurate forecasts; in addition,
values greater than 1.0 indicate a forecast of little value under most loss
functions.
BASIC
RESULTS
The U Statistics in Exhibit 2 confirm that the inflation forecasts from
the AVAR model are relatively inaccurate.
While forecast5 of the T-bill rate
are similar or slightly worse than a no-change forecast, real GNP forecasts
are substantially better.
worse.
Inflation forecasts, however, are substantially
The BVAR model embodies a substantially different strategy for
reducing a VAR model's parameterization.
While its inflation forecasts are
more accurate than the AVAP model, its single-quarter inflation forecasts are
either worse than or not significantly better than a no-change forecast at
each horizon.4
change forecast.
Its cumulative inflation forecast is much better than a noThe BVAR is less accurate than the AVAR, however, for GNP
and interest rate forecasts.5
A more formal comparison of model and no-change forecasts can
be
made by
testing the hypothesis that a series of forecast errors from a model is not
significantly different from the series of errors that would result from a nochange forecast.
Diebold and Mariano
(1991) have proposed such a test that
has several desirable properties; perhaps
valid even if the two
series
most
important for this paper, it is
of forecast errors are contemporaneously
lated or if either series has significant autocorrelation.
corre-
The DM test
statistic is calculated from autocovariances of the difference in squared
errors6 from two forecasts, and is asymptotically normally distributed.
6
For
each model listed in Exhibits 1 and 2, the inflation forecasts at
each horizon were compared with no-change forecasts.
DM statistics were
calculated and are presented in Exhibit 3 {place here}; negative values
indicate that the average squared error from the model was larger than the
squared error of a no-change ,forecast.
For the AVAE model, inflation fore-
casts at each horizon were significantly worse than the no-change forecasts.
For the BVAE, forecasts one quarter ahead were also significantly worse than
the no-change
forecast, but were not significantly different at two, three,
and four quarter horizons. The four-quarter cumulative forecast was significantly more accurate than the no-change comparison.
There are many possible explanations
few can be examined in a single article.
for the poor performance;
Any small macroeconomic
only a
model omits
variables that could be important, and there is certainly a long list of
omitted variables that might affect prices.
While examination of potential
additions will be left for future research, poor choices may have been made in
choosing the five variables included in the model.
To examine that possibili-
ty, two additional models were used to produce simulated fOreCastS.
labeled AVAB-M2,
The rationale
substitutes the M2 monetary aggregate for the monetary base.
is that many researchers have documented a relationship
M2 and prices.'
One,
Another,
between
labeled AVAR-CPI, substitutes the Consumer Price
Index for the GNP implicit price deflator, on the grounds that movements
in
the deflator reflect not only price changes but also changes in the relative
quantities of goods produced.
lengthy intervals and is a
ClOBer
In contrast, the CPI has fixed weights for
approximation to the type of price index
most analysts would prefer.
As indicated in Exhibit 1, substituting M2 for the monetary base failed
to
improve inflation forecasts and also produced less accurate forecasts for
other variables.
significantly
At each forecasting horizon the inflation forecasts were
worse than the no-change comparison.
The results from substituting the CPI for the implicit deflator are
slightly better.
The U statistic for the inflation term fell below unity at
7
the shorter horizons in the table, as well as for the four-quarter
forecast.
cumulative
The GNP and interest rate forecasts were slightly worse than in the
original model.
A final variation is to simulate forecasts from a six-lag WAR
as a
check on the conventional wisdom that substantially reducing the number of
estimated coefficients will normally improve the accuracy of VAR forecasts.
The results for prices were surprisingly accurate (to the author, at least);
these results should not have been too surprising, however, in light of
Lupoletti and Webb (1986), who found simulated WAR
inflation forecasts at all
but a one-quarter horizon to be competitive with those from major forecasting
services.
Attempting to isolate the reason for the WAR
a hybrid model was simulated.
rates are highly persistent,
model's greater accuracy,
The intuition here is that since inflation
long lags might be especially useful.
The AVAR
specification was adopted for every equation except the price equation, and
the Unrestricted
six-lag version was used for prices.
roughly between the AVAR and WAR
in forecast accuracy.
was then simulated, using unrestricted
the monetary base.
The results were
Another hybrid model
lags in the equations for prices and
The results are shown in the table, labeled A/WAR.
This
model produced the most accurate longer-term inflation forecasts, with little
penalty for other variables.
The main result of this section is that the VAR models generated price
forecasts that, for most horizons, were significantly worse than simply
predicting no change.
In order to isolate potential problems, the price
equation from the AVAR and A/WAR
models is examined more closely below.
SPECIFICATION TESTS FOR PRICE EQUATIONS
The price equations from two VAR models will be examined more closely in
this section.
Equation
(5) is from the AVAR model with shorter lag lengths,
and the other is equation
reprinted below.
(6) with longer lag lengths.
Both equations are
8
Pt = b + k$
v-1
01
Pj,GG,,-j + et
Residuals
from both equations were first tested for serial correlation.
The specific test' is a Lagrange multiplier test that takes into account the
presence of lagged values of the dependent variable and little prior knowledge
of the exact form of serial correlation.
The procedure is to first estimate
the price equation and its residuals and then add
residuals to the equation before reestimating
it.
m
lagged values of the
An F test on the signifi-
cance of the lagged values is asymptotically valid for a null hypothesis of no
serial correlation
and an alternative hypothesis of either
AR(m)
or
MA(m)
errors.
As the first line in Exhibit 4 {place here) indicates, the null hypothesis of no serial correlation
For equation
is rejected for equation
(6) with unrestricted
when the alternative
(5) with short lags.
lags, the null hypothesis
is not rejected
is first order correlation, an unsurprising
six lagged values included in the equation.
fifth order autocorrelation,
With an alternative hypothesis of
however, the null hypothesis can be rejected.
The next test, for autoregressive
conditional hetereoskedasticity
(ARCH) I is also a two-step procedure; the first step
price equation and its residuals.
is
also to estimate the
The second step is to regress the squared
residual on a constant term and m lags of the squared residual.
observations
times
the R2
statistic
is chi-squared with
under the null hypothesis of no ARCH.
equation
(5) with
m=l,
log-level price
was accepted.
series,
m
The number of
degrees of freedom
The null hypothesis is rejected for
and is also rejected for equation
A final test is whether it is appropriate
in first differences.
result with
to
estimate
(6) with
m=5.
the price equation
A standard Dickey-Fuller test was used to examine the
and the null hypothesis of the presence of a unit root
Similar test results for the series of first differences were
9
ambiguous, however.
shown in Exhibit 4, either acceptance or rejection of
As
the null hypothesis could occur, depending on the lag length of the differenced inflation rates.g
Since there is no conclusive reason a priori to
prefer one lag length to another, the choice of working with first or second
differences of the price data is a close call.
Since overdifferencing
risks
losing information that could be useful in forecasting, the basic models in
this paper use the first difference of the implicit deflator.
FORECAST PERFORMANCE WITH MODIFIED MODELS
STATIONARITY
AND POSSIBLE
COINTEGRATION
The VAR models discussed above contained three variables in differences:
the price level, the monetary base, and real GNP.
Formal tests for stationar-
ity have already been discussed for the price series.
Dickey-Fuller
also failed to reject the null hypothesis of nonstationarity
tests
for the monetary
base, real GNP, and the interest rate, but did reject the null when data were
differenced one time.
The capacity utilization rate, which is stationary by
construction, was not tested.
ARE THE VARIABLES COINTEGRATED?
It is also possible that the
equations in differences may be misspecified
of those variables that is stationary.
the variables
if there is a linear combination
It is especially important to check
in this model since (1) other authors, for example Mehra (1991),
have found that closely related series are cointegrated, and (2) theory
suggests that the variables could be related by the quantity equation with
interest-sensitive
velocity and a money-multiplier
monetary base and the money supply.
relation between the
If cointegration were to be found, then
the use of an error correcting model might well improve forecast accuracy.
A cointegrating
equation was therefore used to estimate the parameters
of such a linear combination.
found to be nonstationary,
A cointegrating
If the residuals from that equation are not
the hypothesis of cointegration
can be accepted."
equation for the four nonstationary variables was therefore
estimated, and the results are displayed in Exhibit 4.
An augmented Dickey-
10
Fuller test was used to test the null hypothesis that the residuals were
nonstationary,
which would imply that the series are not cointegrated.
While
the test result could possibly be affected by the lag length m, the value
shown in the table is the one that maximizes the
?
statistic, and is also
the shortest lag length for which the Ljung-Box Q statistic does not indicate
significant
serial correlation.
The failure to reject the null hypothesis is
consistent with no cointegration.
SHOULD_THE?
At this point it is useful
to question the economic significance of the finding that the interest rate is
nonstationary.
It is well known that the Dickey-Fuller
test is not powerful
against a highly persistent alternative, such as a root of 0.98.
And there is
a reason that one would expect the interest rate series to be highly persistent.
Goodfriend
(1991) discusses theoretical and empirical work on central
bank smoothing of nominal interest rates -- that is, daily changes in reserve
supply that keep the federal funds rate within a narrow band that is only
changed infrequently.
Such smoothing could impart a high degree of persis-
tence to even a quarterly average of a daily rate, as is used in this paper.
It may therefore be appropriate to treat the interest rate as stationary, as was done in the models presented earlier.
A regression with the
interest rate in levels would not be spurious, and differencing
valuable information.
could lose
But if that assumption were wrong and the interest rate
were actually a random walk, the regression in levels would be spurious and
differencing
would be appropriate.
In order to empirically examine which specification
lag lengths were reset
in the A/WAR
is more appropriate,
model with the interest rates in differ-
ences; series of forecasts were then generated.
The somewhat ambiguous
results are shown in Exhibit 5 for the model labeled A/WAF+l?D.
The interest
rate forecasts from the latter model were slightly more accurate, but forecasts of GNP growth and inflation were less accurate.
Those results are
consistent with the notion that the information 1055 from differencing may be
large enough to justify entering the interest rate in level form.
11
ANOTHER POSSIBILITY OF COINTEGRATION:
Admitting the possibility
the interest rate and inflation series are nonstationary,
that
could there be a
stable "real" rate, that is, the nominal rate minus the quarterly inflation
rate?ll
To test this possibility,
it is not necessary to estimate a cointe-
grating equation since the coefficients are known to be one and minus one.
real rate series was therefore constructed; a Dickey-Fuller
A
test, presented in
Exhibit 4, rejects the null hypothesis of the presence of a unit root.
Given the stationary combination of two arguably nonstationary
series,
the next step is to construct a vector error correcting version of the A/WAR
model (VECM).
The strategy used here was to difference the nominal rate and
inflation series, and add the first lag of the constructed real rate series to
each equation.
Aapt = IQ, +
kg
v=l
The resulting price equation is
Pj,A,t-j
+ y==t-1
+
‘lt
(7)
=l
where p is the log of the implicit deflator, rr is the error correcting
term,
that is the nominal rate minus the inflation rate, and the variables in X are
altered to include the first differences of the interest rate and inflation
rate.
Lag lengths were reset to minimize the Schwarz Criterion.
were then generated, with summary statistics given in Exhibit 5.
tion values indicate that the VECM
A/WAR
model.
Forecasts
The infla-
forecasts are much worse than the standard
It appears that the information lost from differencing the
inflation series was substantial, while the value of including the level of
the real rate was not large.
One final check of possible cointegration was made using the multivariate test proposed by Watson
(1992).
In this test the null hypothesis is the
existence of one cointegrating vector, the real interest rate; the alternative
is the existence of an additional cointegrating vector.
\
The test is a
likelihood ratio, and in this case involves comparing the largest eigenvalue
of a matrix computed from the appropriate VECM with critical values that were
12
computed from the statistics' asymptotic chi-squared distributions.
In this
case, the largest eigenvalue was 9.7, well below the critical value of 20.4
that would indicate rejection of the null hypothesis at the 10% level.
This
failure to reject the null is consistent with the results of the univariate
tests above, and does not indicate that further analysis of possible cointegration would prove useful.
In short, reconsidering
cointegration
the stationarity of the variables and possible
did not indicate a direction for improving the accuracy of
inflation forecasts from the VAB models examined here.
POSSIBLE
REGIME
CHANGES
It is common to think of nonstationary behavior of an economic variable
by analogy to a random walk with frequent small, permanent disturbances.
Indeed, widely used statistical tests for nonstationarity
are derived under
the assumption of such a random walk as the null hypothesis.
nonstationarity
are also possible, however.
Other types of
An alternative that is pursued in
this section involves large, infrequent shocks.
Several authors have found
that U.S. postwar inflation data appear to have been generated by a process
with one or more discrete shifts.
Evans and Wachtel
(1993) proposed a two-state Markov switching process
for the CPI, and found that it explained
some of the puzzles that are raised
by the ex post bias often found in series of inflationary anticipations.
Boschen and Talbot
(1991) found evidence of unstable coefficients
in regres-
sions of inflation on several variables, notably including the growth of the
monetary base, growth of real GNP, and the differenced T-bill rate.
Balke and
Fomby (1991) studied the GNP deflator from 1870 to 1988 and found four "level
shifts" to the inflation rate; the two in postwar data were in 1968 and in
1983.12
The last finding provides statistical evidence that something important
that affects inflation changed around the mid 1960s and early 1980s.
An
obvious possibility
I find
is that something about monetary policy changed.
it plausible that (1) the President and Congress chose to tolerate higher
13
inflation during the Viet Nam buildup for a variety of reasons, including
raising revenues via 'bracket creep' and the personal income tax;13 (2) as
long as this consensus held, monetary policy had an inflationary bias;14 (3)
this consensus was destroyed by the early 1980s;15 and (4) monetary policy
then included low inflation as an important goal.
It is therefore possible to
view monetary policy as making a discrete move toward more inflation at some
point in the middle of the 19605, with the monetary effects on prices becoming
apparent by 1968; a discrete move toward disinflation occurred somewhat later
and the inflation effects became apparent by 1983.
DATING REGIME CHANGES:
moves?
What were the exact dates of these discrete
The strategy here will be to look for specific actions that do not fit
the pattern of the Fed's usual behavior.
It is possible to loosely character-
ize monetary policy since the Federal Reserve - Treasury Accord in 1951 as
"leaning against the wind.'
That is, the Fed's day-to-day actions involved
open market operations that varied the quantity of bank reserves in'order to
keep the federal funds rate within a narrow band.
The funds rate band, in
turn, was set in order to respond to pressing economic conditions.
This often
involved raising the funds rate when inflation was rising and unemployment
relatively
low, and lowering the funds rate when inflation was low and real
growth weak.
slow
In addition, there were times during which unusually rapid or
money growth would bolster the case for changing the funds rate target.
The Fed departed from that strategy in the 1960s; the background account
in this paragraph
substantially
is taken from Kettl (1986).
In 1965 President Johnson
increased military action in Viet Nam, proposed domestic social
legislation requiring increased spending, but resisted increasing nominal
interest rates.
In June 1965 the Fed's chairman William McChesney Martin
learned that the Viet Nam buildup was larger than the administration
publicly admitting.
was
In October Martin wrote to the President arguing for an
immediate increase in interest rates.
On December 5 the Fed raised the
discount rate by 50 basis points, and 'Johnson was furious.
advisers believed that the decision was precipitous
. ...
He and his
The President saw
14
The President
the discount rate increase as a personally vindictive act."16
and the Chairman met at the President's ranch in Texas later that month.
In 1966 real growth was a rapid 6.0 percent, the unemployment
rate was a
very low 3.7 percent, and the inflation rate (the change in the annual average
CPI) rose from 1.6 in 1965 to 2.9 percent.
A consistent application of
leaning against the wind would have required increasing the funds rate until
inflation was clearly checked.
Although interest rates rose during the first
part of the year, in the fourth quarter "Monetary policy promptly moved to
relax the degree of reserve restraint.""
The inflation rate was 3.1 percent
in 1967, 4.2 percent in 1968, and 5.5 percent in 1969.
departure
Here is a clear
from the previous leaning against the wind strategy, namely lowering
interest rates, beginning in the fourth quarter of 1966, while inflation was
rising and unemployment
low.
The fourth quarter of 1966 was therefore the end of a low inflation
period.
enemy
By the end of the 1970s inflation was being described as "public
number
one."
That sentiment did not translate immediately into presi-
dential support for a substantial change in monetary policy.
In the lead
paragraph of his 1980 Economic Report to Congress, President Carter asserted
that higher oil prices were the
major
reason for rising inflation in 1979. In
that twelve page statement, there are only two sentences that mention monetary
policy.
The most relevant is, "Monetary policy will have to continue
in support of the same anti-inflationary
goals.111E [Emphasis mine]
firmly
In
January 1981 the political party controlling the Presidency and the Senate
changed.
The discussion of monetary policy also changed substantially.
example, in the first annual report of the Reagan Administration's
Economic Advisers
essentially
(1982), one encounters the following phrases:
Council of
"Inflation is
a monetary phenomenon." p.75; "The appropriate policy for reducing
the inflation rate is a decrease in the rate of money growth." p.76; "The
Administration
reduction
For
expects that the Federal Reserve will achieve an orderly
in the trend of money growth to a noninflationary
[Emphasis mine].
rate." p.64
15
Monetary growth was high and
example,
Ml
variable
in the 1970s and also in 1980; for
grew at a 16 percent annual rate between May and November of 1980.
In 1981, however, "growth in Ml-B adjusted for shifts into NOW accounts was
about 2 l/4 percent -- 1 l/4 percentage points below the lower end of its
targeted range.""
That behavior of the main intermediate target of Fed
policy, shift-adjusted Ml-B, was substantially different from what had been
seen in the past decade.
Turning to specific actions, in the first quarter of 1981 the unemployment rate was 7.4 percent, compared to an average 7.0 percent in 1980 and 5.8
percent in 1979.
The trend in real activity was difficult to interpret in
light of the monetary volatility in the last year as well as the imposition
and removal of credit controls in 1980.
shift-adjusted
During the second quarter of 1981
Ml-B was at or below the lower bound of the target range that
the Fed had announced in February.
Yet the federal funds rate rose by five
full percentage points from April 1 to July 8.20 Here is an extraordinary
departure from previous behavior; the second quarter of 1981
is
therefore the
last of the inflationary period.21
ECONOMETRIC
RESULTS:
Based on the evidence discussed above, three
periods are studied: an early low inflation period from 1952 Q2 to 1966 44; a
middle inflationary period from 1967 Ql to 1981 42; and a disinflationary
period from 1981 Q3 to 1990 44.
Separate regressions were run for each
subperiod and the entire sample; the results are displayed in Exhibit 6.
Perhaps most striking is the complete failure of the restricted equation
for the early period, with essentially no explanatory power from its regressors.
Regressions
for the middle and late periods appear to work somewhat
better, at least as indicated by the usual summary statistics.
For the whole
period the sum of the coefficients on nominal variables, that is lagged
prices, the interest rate, and the monetary base, is approximately unity.
the middle period, however, it
is
In
1.26 while in the late period it is 0.80.
The sum of coefficients on lagged prices rises substantially
in the two later
periods; the strong persistence of inflation is not evident in the early
16
period.
The coefficient on the monetary base is significantly different from
zero only in the middle period.
These statistical results are consistent with the story of monetary
regime changes.22
regimes.
There are several options when forecasting under changing
Perhaps the simplest is to use a few dummy variables to allow some
coefficients to abruptly shift.
O-l dummies were constructed
Dl, respectively.
Based on the results summarized in Exhibit 6,
for the middle and late periods, labeled Dm and
In addition, the product of Dm and the monetary base is
labeled Db, and the product of the implicit deflator and sum of Dm and Dl is
labeled Dp.
equation
Dm, Dl, and one lagged value each of Db and Dp were added to
(5), with the results also displayed in Exhibit 6. {place here) As
might be expected, the apparent fit of the equation improved and the four
dummy variables were each significant at conventional
Diagnostic
levels.
tests were again repeated and are displayed in Exhibit 6.
The F test for serial correlation was repeated on the price equation for the
entire sample.
In contrast to the results presented in Exhibit 4, the null
hypothesis of no serial correlation is not rejected at conventional
levels.
The test statistic for ARCH is also presented, and the hypothesis of no ARCH
is again rejected at the one percent level.
statistics
were
constructed
Finally, Dickey-Fuller
test
for the inflation rate over the three intervals.
In the early and middle periods the null hypothesis of a unit root is rejected
at the one percent level.
For the late period, the null is rejected at either
the five or ten percent level, again depending on a lag length used in the
test.
Adding the dummy variables to the price equation of the AVAB model
substantially
improved its forecasting performance,
The modified AVAB model has by far the
most
as shown in Exhibit 5.
accurate inflation forecasts at
each horizon of any of the models considered in the paper, with the GNP
forecasts remaining as accurate as any of the models considered.
same dummy variables to the A/WAR
Adding the
model slightly improved its inflation
17
forecasts, although the AVAR-D model remained significantly
more
accurate.
Neither model forecasts the interest rate well.
To put these figures in perspective, consider the root-mean-square
errors of the ASA-NBER inflation forecasts from 1968 to 1990, taken from
Zarnowitz and Braun, Table 5.
forecasts for the same period.
The AVAR-D model was used to produce simulated
One-quarter-ahead
forecasts from the VAR model
were less accurate, with an RMSE of 1.48 percent versus 1.00 percent.
Four-
quarter cumulative forecasts from the VAR model were much more accurate, with
an RMSE of 0.88 percent versus 1.92 percent.
VAR's performance
If taken at face value, the
is competitive with actual forecasters of inflation over
that interval. The major reason not to take the results at face value is the
amount of experimentation
that was used to construct the AVAR-D model.
In
addition, the model had the benefit of using revisions of GNP and implicit
deflator data that were unavailable to the real-time forecasters.
CONCLUSION
Several small VAR models have been examined in this paper, with particular emphasis on inflation forecasts and the inflation equation of the models.
Two models are of particular interest.
stricted lags in two equations.
One has unfashionably
long, unre-
The other uses dummy variables to represent
the political decision made in the 1960s to allow higher rates of inflation
and the subsequent decision made in the early 1980s to lower the inflation
rate.
The latter model produced the most accurate inflation forecasts of the
models examined while its GNP forecasts retained their accuracy.
That result
suggests a resolution to the puzzle of relatively inaccurate inflation
forecasts from VAR models; two monetary policy regime changes led to inaccurate inflation forecasts from models with constant coefficients.
Moreover,
either adding dummy variables or dividing the entire sample into three
subperiods also removed serial correlation from the residuals of the price
equation.
18
These results may be of interest to several groups.
To anyone inter-
ested in price behavior, the improved performance of the price equation after
an adjustment for temporal instability points out a danger of using constant
coefficients
over most of the postwar period.
the finding that long, unrestricted
To users of small macro models,
lags led to an equation that worked well
relative to a highly restricted equation
suggests that long, unrestricted
lengths should not be rejected out of hand.
Students of monetary policy may
find the dating of regime changes of interest.
nonstationarity
lag
The finding that possible
of the inflation rate over the postwar period is not apparent
in the three subperiods of consistent monetary policy may be of interest to
empirical macroeconomists
contexts.
concerned about potential nonstationarity
in other
Finally, if one wishes to forecast inflation, using the admittedly
ad hoc dummy variables may be preferable to ignoring the monetary regime
changes altogether.
Several directions
for future research are also evident.
First, do the
results that are presented above overstate the probable accuracy of current
inflation forecasts, due to too much experimentation?
Of course, new data
over time will reveal whether the post-sample performance of these models is
in line with those results.
Also, despite the experimentation
a basic stability in the model structure.
there has been
Since 1984 the author has published
results from VAR models with the same variables, with the same starting date
for regressions,
and with the same choices for differencing.23
Changes have
been limited to different lag lengths and the addition of the dummy variables
for the two shifts in monetary policy.
This paper views monetary policy changes as being responsible
persistent
fied.
shifts in the inflation process that several authors have identi-
Are there other candidates such as energy
plausible?
for large,
prices that are equally
And if the monetary policy explanation
suggested dates of regime change robust?
is accepted, are the
One approach would be to estimate a
structural relation between instruments and goals of monetary policy and
19
examine its stability in the neighborhood of the dates of the suggested
changes.
Residuals
from the best performing price equation still displayed ARCH,
and that specification
failed to explain inflation during the early subperiod.
It is therefore apparent that this model, like many other small
macro
models,
might benefit from additional variables that help predict inflation rates.
The challenge will be to narrow the field, given the large number of variables
that analysts have used with apparent success in price equations in the past.
20
APPENDIX:
TEE DATA AND MODELS
The following data series, with
used in this paper:
Citibase
mnemonic in parentheses,
were
P, the implicit price deflator for gross national product
(GD); Y, real gross national product (GNP82); M, the monetary base as estimated by the Federal Reserve Bank of St. Louis (FMBASE); CU, the capacity
utilization
rate in manufacturing
three month treasury bills (FYGM3);
urban consumers
(IPXMCA); R, the secondary market yield on
CPI, the consumer price index for all
(PUNEW), and the M2 monetary aggregate after 1959 (FM2).
All
except the interest rate were seasonally adjusted by the agency compiling the
data.
Natural logarithms were taken of each data series except the interest
rate and the capacity utilization rate.
maintained
by Citibase on July 19, 1991.
Robert Hetzel; see Hetzel
The table
collection
below
Data were the latest revisions
Pre-1959
M2
data were obtained from
(1989) for further information on its construction.
describes each model in the paper.
of equations that are individually described.
DEP VAR contains the dependent variable for each eguation.
Each model is a
The column labeled
The column labeled
DIFF? contains a Y for differenced series, N for levels, and 2 for a series
differenced
twice.
The number of lagged terms for the dependent variable and
the common lag length for all the independent variables are given in the next
two columns.
as described
The final column notes if an equation contained dummy variables
in the final section, or RR, the difference between the nominal
interest rate and the inflation rate; a constant term was also included in
each equation.
21
Model Descriptions
MODEL
DEP VAR
DIFF?
# OWN LAGS
# LAGS OF IND VARS
A/WAR
P
Y
t4
C
R
A/WAR-D
P
Y
6
6
1
1
M
6
2
6
6
C
R
A/WAR-RD
P
Y
M
C
R
AVAR
P
Y
M
C
R
AVAR-CPI
OTHER TERMS
Y
Y
Y
N
N
4. 4 d, %
CPI
Y
M
C
R
AVAR-D
P
Y
d, d, d, 4
M
C
R
AVAR-M2
P
Y
M2
C
R
BVAR"
P
Y
M
C
R
VECM
P
Y
M
C
R
RR
RR
RR
RR
RR
22
EXHIBIT 1
Forecast Error Statistics from Several VAR Models
Root Mean Square Errors
Model
Variable
Single Quarter Forecasts
Horizon: IQ
4Q
x2
Cumulative Forecasts
As2
AVAR
IPD
GNP
RTB
1.57
3.40
1.20
1.92
3.07
1.95
2.62
3.50
2.44
1.84
1.58
IPD
GNP
RTB
1.58
3.81
1.24
1.59
3.75
2.03
1.79
4.21
2.69
1.11
1.92
IPD
GNP
RTB
1.86
3.75
1.26
2.32
3.41
2.05
3.17
3.84
2.60
2.30
2.15
CPI
GNP
RTB
1.91
3.40
1.24
2.56
3.24
2.04
3.22
3.58
2.60
2.16
1.80
IPD
GNP
RTB
1.69
3.40
1.20
1.64
3.09
1.97
1.73
3.60
2.50
1.11
1.60
BVAR
AVAR-M2
AVAR-CPI
A/WAR
Note : Model construction and data sources are described in detail in the
Appendix.
Variable abbreviations'include the quarterly average level of RTB,
the go-day Treasury-bill rate, and annualized rates of change from the
previous quarter of GNP, real gross national product; IPD, the GNP implicit
price deflator; and CPI, the consumer price index. The numerical entries are
root-mean-squared error statistics described in the text, covering synthetic
forecasts from 1977 Q2 to 1990 43.
23
EXHIBIT 2
Forecast Error Statistics from Several VAR Models
Theil U Statistics
Model
Variable
Single Quarter Forecasts
Horizon: u
22
Cumulative Forecasts
AC2
AVAR
IPD
GNP
RTB
1.05
0.75
1.00
1.22
0.63
1.08
1.44
0.66
1.04
1.40
0.47
IPD
GNP
RTB
1.06
0.84
1.03
1.01
0.77
1.12
0.98
0.80
1.15
0.85
0.58
IPD
GNP
RTB
1.25
0.83
1.05
1.46
0.70
1.13
1.74
0.73
1.11
1.74
0.65
CPI
GNP
RTB
0.85
0.75
1.03
0.92
0.66
1.13
1.02
0.68
1.11
0.96
0.54
IPD
GNP
RTB
1.13
0.75
1.00
1.04
0.63
1.09
0.95
0.68
1.06
0.84
0.48
BVAR
AVAR-M2
AVAR-CPI
A/WAR
Note: Model construction and data sources are described in detail in the
Appendix.
Variable abbreviations include the quarterly average level of RTB,
the go-day Treasury-bill rate, and annualized rates of change from the
previous quarter of GNP, real gross national product; IPD, the GNP implicit
price deflator; and CPI, the consumer price index. The numerical entries are
Theil U statistics described in the text, covering synthetic forecasts from
1977 Q2 to 1990 43.
24
EXHIBIT 3
VAR Inflation Forecasts Compared to No-Change Forecasts
Model
Variable
Single Quarter Forecasts
Horizon: u
2sL
!?Q
Cumulative Forecasts
!a
AVAR
U Statistic
DM Statistic
Significance,
%
1.05
-3.7
eo.1
1.22
-11.7
CO.1
1.44
-11.6
co.1
1.40
-5.2
co.1
%
1.06
-3.2
0.1
1.01
-0.2
>lO
0.98
0.63
>lO
0.85
2.4
1.7
%
1.25
-12.8
co.1
1.46
-18.0
<O.l
1.74
-10.0
co.1
1.74
-4.9
CO.1
%
0.85
14.5
co.1
0.92
7.3
<O.l
1.02
-1.0
>lO
0.96
0.9
>lO
1.13
-6.1
co.1
1.04
-1.2
>lO
0.95
1.6
>lO
0.84
2.2
2.9
BVAR
U Statistic
DM Statistic
Significance,
AVAR-M2
U Statistic
DM Statistic
Significance,
AVAR-CPI
U Statistic
DM Statistic
Significance,
A/WAR
U Statistic
DM Statistic
Significance,
%
:
Model construction and data sources are described in detail in the
Appendix.
Synthetic forecasts inflation from 1977 42 to 1990 44 are compared
with no-change forecasts.
The U Statistic ie the ratio the root-mean-squared
The DM statistic
error of the model forecast to that of a no-change forecast.
tests the null hypothesis that the model and no-change forecast have equal
accuracy; negative values indicate that the sum of squared forecast errors is
larger for the model forecast.
Note
25
EXHIBIT 4
Specification Tests for Price Equations
Equation
(5)
pc = pP +
Serial correlation test:
ARCH test:
Equation
x2 (1)
F(1,145) = 3.93;
Serial correlation test:
ARCH test:
Dickey-Fuller
x2(5)
level = .049.
= 5.13; Significance level = .024.
pt = cr,+ ej?
GL-j
v-1 -1
(6)
Significance
+ et
F(5,114) = 2.48;
Significance
level = .036.
= 14.64; Significance level = .012.
test for a unit root
Z, = p + m P,A= t-j; P&-l + et
F-1
e,tp=11 = 0.80; 10% sig. level = -2.58;
Price Level
( Z, = P, ): m=2;
Differences
( Z, = AP, ): m=l;
9,(p=l) = -3.42; 5% sig. level = -2.89;
Differences
( Z, = AP, ): m=5;
$,(p=l) = -2.06.
"Real" Rate ( Z, = Rt- (pt-ptel) ): m=2 ;
Cointegrating
equation
Q,(p=l)
=
14.1.
P, = 1.29 + .83&l,- .15Y, + .016R, + I?,
Augmented Dickey-Fuller
test on residuals from cointegrating
AQ, = -XI?,-,
+ * PjAQt-j
F-1
t
(x=0)
= 3.12;
Note: This significance
10% level = 3.89
level is from Engle and Yoo (1987).
equation:
26
EXHIBIT 5
Forecast Error Statistics from Several Models
Theil U Statistics
Model
Variable
Single Quarter Forecasts
Horizon: lJJ
!T!Q
22
Cumulative Forecasts
22
A/WAR
IPD
GNP
RTB
1.13
0.75
1.00
0.95
0.68
1.06
1.06
0.65
1.03
0.84
0.48
IPD
GNP
RTB
1.19
0.81
0.98
1.02
0.75
0.98
0.98
0.61
1.03
0.94
0.69
IPD
GNP
RTB
1.30
0.80
0.95
1.55
0.73
1.08
2.09
0.61
1.20
1.49
0.71
IPD
GNP
RTB
1.16
0.75
1.00
0.69
0.68
1.05
0.65
0.65
1.02
0.65
0.48
IPD
GNP
RTB
1.05
0.75
1.00
1.44
0.66
1.04
1.54
0.64
1.00
1.40
0.47
IPD
GNP
RTB
0.83
0.75
1.00
0.67
0.66
1.04
0.56
0.64
1.00
0.58
0.47
A/WAR-RD
VBCM
A/WAR-D
AVAR
AVAR-D
Note: Model construction and data sources are described in detail in the
Appendix.
Variables include the quarterly average level of RTB, the go-day
Treasury-bill rate, and annualized rates of change from the previous quarter
of GNP, real gross national product; and IPD, the GNP implicit price deflator.
The numerical entries are Theil U statistics described in the text, covering
synthetic forecasts from 1977 Q2 to 1990 Q3.
27
EXHIBIT 6
Regression Results for Several Time Periods
1952 Q2 to 1966 Q4
fit= 0.28 - 0.08~~~, + O.OBp,-, + O.llp,-, + 0.07r,-, + O.O2c,-, - o.Olm,-, + O.OSy,-,
(0.76)
(-0.01)
(0.51)
(0.72)
(0.28)
(0.24)
(0.06) (-0.54)
2
= -0.08
8 =1.85
F(1,149) = 0.17
Q, = -7.90' (m=O)
x2 (1) = 3.48"'
1967 Ql to 1981 Q2
$, = -2.78 + 0.30&s, - o.o4p,-, + o.o4p,-, + 0.33r,,, + 0.02~~~, + 0.6Om,-, - 0.08~~~,
(-1.52)
(4.87)
(-0.54) (2.30)
(-0.28)
(0.27)
(0.26)
(2.56)
Rz = 0.57
a = 1.59
F(1,148) = 2.42
x2(1)=2.31
Q, = -3.64' (m=O)
1981 43 to 1990 44
8t = -8.87 + 0.21P,-, + O.O9p,-, + 0.20~~~, + 0.20r,-, + O.lOc,-, + O.lOm,-, - 0.15~~~~
(-0.21)
(0.53)
(1.02)
(-1.54) (1.16)
(1.15)
(1.07)
(1.68)
Ra = 0.51
6 = 1.06
9, = -2.63
9, = -2.63"'
(m=l)
(m=l)
F(1,28) = 0.45
9, = -2.95"
x2(1) = 0.46
(m-2)
1952 Q2 to 1990 44
rr,= -3.84 + 0.3Op,-, + 0.23~~~~ + 0.22p,+ + o.o05r,-, + o.o5c,-, + 0.17m,-, - 0.22~~~
(-0.59)
(-1.42) (6.38)
(2.89)
(2.71)
(0.07)
(2.95)
(1.54)
R1 = 0.59
8 = 1.75
F(1,145)
= 3.93"
x2(1) = 5.13.'
28
EXHIBIT 6
(Continued)
1952 Q2 to 1990 44 with Dummies
3, = -1.29 - 0.05pcw1 + 0.02~~-~ + O.O6p,-, + 0,2lr,-, + 0.04~~~, + O.O2m,-, - O.O2y,-,
(-0.51) (-0.45)
(0.29)
(0.30)
(-0.44)
(0.82)
(2.34)
.(1.16)
- 2.11d, - 1.32d, + o.38dD + 0.42d,
(-2.66)
(-2.00) (2.53)
(3.61)
R'"= 0.68
Significance
levels:
6 = 1.56
l%, *;
F(1,141)
5%, *;
= 0.25
X2(l) = 23.4'
lO%, -
Note:
For each equation t-statistics are in parentheses.
Symbol definitions
are as follows: p is the inflation rate, t is a time subscript, r is the level
of the 3-month T-bill rate, c is the manufacturing capacity utilization rate,
m is the percentage change of the quarterly average monetary base, y is the
percentage change of real GNP, dm is a dummy variable that is unity from 1967
Ql to 1981 42 and zero otherwise, dl is a dummy variable that is unity from
1981 Q3 to 1990 44 and zero otherwise, dp is the product of p and dm+dl, and
db is the product of dm and m.
The
9,
statistic allows a Dickey-Fuller
test
The F statistic
of the hypothesis of a unit root in the dependent variable.
allows a test of the null hypothesis that serial correlation is zero. The
chi-squared statistic allows a test of the null hypothesis of no ARCH.
29
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(eds.), Leading Economic Indicators:
New Approaches
and Forecasting
Records.
New York: Cambridge University Press.
Yi, Chang and George Judge. 1988. Statistical Model Selection Criteria.
Economic Letters 28: 47-51.
Zarnowitz, Victor and Phillip Braun. 1991. Twenty-Two Years of the NBER-ASA
Quarterly Economic Outlook Surveys: Aspects and Comparisons of Forecasting Performance. Working Paper 3965, National Bureau of Economic
Research, Cambridge, Massachusetts.
-
31
ENDNOTES
1.
Doan
(1990). p. 8-16.
One
2. Why was the Schwarz Criterion used instead of some other statistic?
reason, which is only applicable for this paper, is simply to have a sharp
difference between the restricted and unrestricted models. More generally, Yi
and Judge (1988) studied the asymptotic performance of three widely used
statistics for model selection. Only the Schwarz Criterion excluded irrelevant
variables with a probability approaching one as the sample size increased.
3. These would be referred to as early quarter forecasts, since they assume
knowledge of the previous quarter's implicit deflator and GNP values, which are
now first released near the end of the first month of a quarter. In real time,
a forecaster would also have other valuable information by the time those values
are released, such as some values in the current quarter for interest rates and
the monetary base.
4. Forecasts from one to eight quarters ahead were examined, although only a few
horizons are shown in the table.
The discussion in the text refers to all
periods, whether or not included in the table.
5. These BVAR model results are based on a model in which real GNP, the implicit
price deflator, and the monetary base are differenced.
Many users of BVAEs,
however, prefer to use levels and trend terms. In other words, they prefer to
assume trend stationarity rather than difference stationarity. To see if these
results are sensitive to that assumption, the BVAR model was also simulated with
all variables in levels and with a time
trend in the equations. The results of
simulated forecasts were generally worse, especially for the inflation rate.
6. While this paper uses squared errors, other functions of the error series
would also be valid.
7.
A particularly
8.
For a discussion of this particular test see Godfrey
well-known example is Hallman, Porter, and Small
(1989).
(19881, section 4.4.
The shorter lag length was chosen to minimize the Schwarz Criterion.
A
longer lag was also examined to check the robustness of the test to the lag
length.
This was particularly important for prices since earlier work, Webb
(1988), had found that the results of Dickey-Fuller tests for unit roots in the
CPI series were also dependent on the lag length.
9.
10.
Engle and Granger
(1987) is the classic reference on the subject.
11.
For a proper ex post real rate, the 3-month nominal rate R would be the
value at the beginning of the quarter, the log price level P would be the value
at the end of the quarter, and the real rate would be R, - (P,-P,.,).The measures
for the nominal rate and the price level that are actually used in this paper are
32
quarterly averages.
would be preferable.
Of course, one could also argue that an ex ante real rate
12. It is interesting to note that they also applied their procedure to real GNP
growth over the same period and did not find any level shift.
13. See, for example,
and a nonindexed tax
inflation rate raised
falling real personal
Hetzel (19901, who emphasizes the interaction of inflation
code.
In 1974 alone he estimates
that.aneleven
percent
the growth in tax revenues to seventeen percent despite
income.
14. While the Federal Reserve has considerable independence in its day to day
operations, if it were to pursue goals outside a political consensus for very
long, legislation putting permanent restrictions on the Fed could be enacted.
Since officials would most likely not wish to see such legislation enacted, as
a practical matter one would not expect to see the Fed continue to pursue goals
that the President and Congress opposed.
15. The Economic Recovery Tax Act of 1981 included substantial indexing of the
personal income tax for inflation, thereby reducing the government's revenue
gains from inflation. According to Steurle (19911, "The major individual reform
instituted in 1981 was not the direct reduction in tax rates, but the establishment of indexing of tax brackets .... Eventually, indexing will dominate all
other provisions of the 1981 Act."
16.
Kettl, p. 104.
17.
Martin
(19671, p. 215.
The phrase "lowering the degree
restraint" can be translated as lowering the fed funds rate.
of
reserve
18. The other is "The October actions of the Federal Reserve Board to change the
techniques of monetary policy helpedmoderate inflationary expectations whichhad
been partly responsible for the pressure on the [foreign exchange value of the1
dollar."
19.
Volker
(1982) P. 129.
20.
For an account of specific Fed actions during this period see Cook (1989).
21. Some analysts prefer to date the monetary regime change as October 6, 1979,
the time of a Fed announcement of a major change in operating procedures.
Arguing against that date are (1) at the time, the President did not publicly
support a strong, disinflationary monetary policy, preferring instead to focus
on wage-price guidelines, energy conservation, and the risks of monetary policy
being too tight; (2) an important political incentive for inflation, the lack of
indexing of the personal income tax, was not removed until 1981; and (31 monetary
growth in 1980 was not consistent with a shift to a disinflationary monetary
policy.
22.
The statistical hypothesis of structural stability could of course be
formally tested with a standard Chow test. That test, however, is invalid when
the full-sample regression has residuals that display ARCH, as in this case. It
33
is however possible to construct a likelihood ratio test of the joint hypothesis
of constant coefficients and constant residual variance against the alternative
of changing coefficients and/or changing residual variance. The LR statistic in
this case has a chi-squared (18) distribution; the calculated LR value, 43.45,
indicates the null hypothesis is rejected at the 1% level.
23.
See Webb
(1984, 1985, 1991) and Lupoletti and Webb
(1986).
An important part of the BVAR specification procedure is to set "hyperzirameters" that could in principle be used to impose prior beliefs on the data.
Here these parameters were chosen to minimize the log-determinant of the
variance-covariancematrixconstructedfromone-quarter-aheadforecasterrors
for
the model.
The resulting choices are summarized in the RATS statement
SPECIFY(type=symmetric,decay=.5,tight=,6)
.9
(see Doan, ch. 8.8 for further details).
@FedFRASER