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o r K in g r a p e r s e rie s
Evidence on Structural Instability in
Macroeconomic Times Series Relations
J a m e s H . S to c k a n d M a rk W . W a ts o n
3
■)
W o rk in g P a p e rs S e rie s
M a c ro e c o n o m ic Is s u e s
R e s e a rc h D e p a rtm e n t
F e d e ra l R e s e rv e B a n k o f C h ic a g o
S e p te m b e r 1 9 9 4 (W P -9 4 -1 3)
1 iBR A S Y
OCT 0 6 1994
F DR L RS R E
E E A EE V
9ANK Of CHICAGO
FEDERAL RESERVE BANK
O F CHICAGO
Evidence on Structural Instability in Macroeconomic Time Series Relations
James H. Stock
Kennedy School of Government, Harvard University
and the NBER
and
Mark W. Watson
Northwestern University, Chicago Federal Reserve Bank
and the NBER
July 1994
*We are grateful to Neil Shephard for helpful discussions and participants at the 14th Annual
International Symposium on Forecasting in Stockholm, Sweden for useful comments. This
research was supported by National Science Foundation grant no. SES-91-22463.
Abstract
An experiment is performed to assess the prevalence of instability in univariate and bivariate
macroeconomic time series relations and to ascertain whether various adaptive forecasting
techniques successfully handle any such instability. Formal tests for instability and out-ofsample forecasts from sixteen different models are computed using a sample of 76 representative
U.S. monthly postwar macroeconomic time series, constituting 5700 bivariate forecasting
relations. The tests indicate widespread instability in univariate and bivariate autoregressive
models. However, adaptive forecasting models, in particular time varying parameter models,
have limited success in exploiting this instability to improve upon fixed-parameter or recursive
autoregressive forecasts.
Keywords: forecasting; time-varying parameters; structural stability; breakpoints; recursive
least squares
1 Introduction
.
Time series econometrics typically involves drawing inferences about the present or future
using historical data. In some cases these inferences are about how the economy operates or
how economic policy affects key variables. For example, much empirical work on monetary
economics currently rests on inferences drawn from so-called structural vector autoregressions
(VAR’s); Bemanke and Blinder (1992) and Christiano, Eichenbaum and Evans (1993) provide
two recent examples. In other cases these inferences are in the form of forecasts. Both
applications typically require that the model at hand be stable (that the future be like the past)
for such inferences to be valid. For example, giving advice to current policy makers based on a
structural VAR requires, among other things, that the historically estimated model remains
relevant today. Although studies occasionally include some analysis of stability, it is often
limited in scope, perhaps consisting of reestimating the model on a single subsample. The
importance of stability and the current lack of systematic evidence on it therefore leads us to
ask, how generic is instability in multivariate time series relations?
To answer this, we undertake a two-part experiment. The first part assesses the prevalence
of parameter instability in economic time series relations using a battery of recently developed
tests for instability. This is done using a sample of 76 monthly time series for the postwar U.S.
economy over the period 1959:1 - 1993:12 (420 observations), among which are 5700 distinct
(although not independent) bivariate forecasting relations. These series are chosen to provide
relations which are representative of those of interest to macroeconomists and macroeconomic
forecasters. This sample is then used to compute empirical distributions of various tests for
structural stability, including Nyblom’s (1989) test for parameter stability, CUSUM tests, and
break point tests such as the Quandt (1960) likelihood ratio statistic.
The second part of the experiment examines whether current state-of-the-art adaptive
forecasting models capture the instability found by the stability tests and thereby improve upon
more naive forecasts. This entails the empirical evaluation of different forecasting models
- 1-
which exhibit different degrees of adaptivity, ranging from no adaptivity (fixed parameter
models) through moderate adaptivity (recursive least squares, rolling regression, and random
walk coefficient time varying parameter (TVP) models with small coefficient evolution) to high
adaptivity (TVP models with large coefficient evolution). Although work on regression models
with stochastically time varying parameters (or "stochastic coefficients") dates to Cooley and
Prescott (1973a, 1973b, 1976), Rosenberg (1972, 1973), and Sarris (1973), and although TVP
models been applied to selected series, we know of no systematic evidence on whether these
techniques might be widely useful in economic forecasting applications. * Eight univariate
models are considered for each of the 76 series for a total of 608 univariate forecasting
equations, and eight bivariate models are considered for each of the 5700 bivariate forecasting
relations for a total of 45,600 bivariate forecasting equations. Models are compared using onemonth-ahead mean square errors (MSE’s). In the spirit of Makridakis et al. (1982) and Meese
and Geweke (1984), who applied univariate forecasting techniques to large numbers of series,
this part of this experiment yields a forecasting comparison suggesting which models typically
do best in macroeconomic applications. The results also provide an opportunity to assess model
robustness by identifying models which successfully guard against the most severe out-ofsample forecasting failures.
Looking ahead to the results, the tests indicate that instability is widespread. For example,
one version of the Nyblom (1989) test rejects stability (at the 10% level) in more than 70% of
the 5,700 bivariate relations. This instability is more prevalent in certain classes of series, such
as measures of aggregate output, than in others. However, our results also suggest that
forecasting models explicitly designed for time varying parameters (rolling regressions or
random walk TVP models) often fail to perform as well as traditional fixed coefficient or
recursive least squares models: in 57% of the 5700 pairs, fixed-coefficient or recursive least
squares forecasts have the lowest out-of-sample MSE among the sixteen competing models,
while in only 10% of the pairs do bivariate TVP models have the lowest out-of-sample MSE.
When they are best, the gains associated with the TVP forecasting models typically are small.
- 2 -
In a small fraction of the cases, the TVP and recursive least squares models perform well while
the fixed coefficient models perform quite poorly, and in this sense the TVP and recursive least
squares models are more robust than the fixed coefficient models. Overall, however, the TVP
models fail to exploit successfully the time variation found by the stability tests.
The outline of the paper is as follows. Section 2 describes the data set. The stability tests
are described in section 3, and section 4 summarizes the empirical testing results. The
forecasting models are described in section 5 and are evaluated empirically in section 6. Section
7 concludes.
2. The Data Set
Our objective in constructing the data set was to obtain a sample of economic time series
for the U.S. which is representative of the relations of primary concern to macroeconomists and
macroeconomic forecasters. While one could in principle draw series at random from a large
macroeconomic database, a simple random sample would oversample certain classes of heavily
represented series, such as industry-specific deflators, interest rates, or financial flows. Such a
sample would be representative of the monthly data which are produced but not of the
forecasting relations of interest to macroeconomists. Moreover, such a sample would omit
important forecasting variables which are constructed from the primary data, such as interest
rate spreads. In theory, stratification could eliminate the problem of oversampling certain
classes of series which are produced in detail, but would not address the issue that many of the
important forecasting series are constructed by researchers and thus are not contained in
standard databases. Moreover, mechanical simple or stratified sampling would produce many
series with definitional changes or other internal inconsistencies.
Our sample of series therefore was obtained by applying subjective judgment, using four
criteria as guidelines:
-3 -
1. The sample should include the main monthly economic aggregates and coincident
indicators. This resulted in the inclusion of series such as industrial production,
weekly hours, personal income and inventories.
2. The sample should include important leading economic indicators. This led us to
include series such as monetary quantity aggregates, interest rates, interest rate spreads,
stock prices, and consumer expectations.
3. The series should represent different broad classes of variables which can be expected
to have quite different time series properties.
4. The series should have consistent historical definitions or, when the definitions are
inconsistent (for example different base years for different segments of a real series) it
should be possible to adjust the series with a simple additive or multiplicative splice.
These criteria were used to select 76 monthly U.S. economic time series. Most of the raw data
were obtained from the CITIBASE data base, although many series were subsequently modified
(for example by creating interest rate spreads). The series can be grouped into eight categories:
output and sales; employment; new orders; inventories; prices; interest rates; money and
credit; and other miscellaneous series including exchange rates, government spending and taxes,
and miscellaneous leading indicators. The complete list of series and their mnemonics are given
in the appendix.
The sample runs from 1959:1 to 1993:12. The starting date was chosen because this is the
earliest date for which many of the series, in particular the monetary aggregates, are available.
Four series (the series on government finance) start in 1967:6. The statistics in question were
computed using the longest possible sample for which all relevant data were available.
-4-
Each series was screened to detect breaks and outliers due to changes in definitions or
reporting practice. Most series were also transformed to induce approximate stationarity by
taking either first differences or first differences of logarithms. For consistency, the
stationarity transformation was in general applied to entire classes of series rather than on a
case-by-case basis. For example, production, employment, prices, and money were all
transformed using first differences of logarithms, and interest rates were transformed by first
differencing. Some series which did not fit naturally into a broader category were analyzed on
a case-by-case basis using visual inspection, a-priori reasoning, and unit root test statistics, and
then transformed accordingly. The transformation for each series is listed in the appendix. It
should be emphasized that many of the procedures are only slightly affected by the use of first
differences vs. levels. In particular, the forecasting models in section 5 produce similar shortrun forecasts using levels or first differences (this would not be the case if many of the series
were cointegrated, but there are neither theoretical nor empirical reasons to suspect widespread
cointegration among these series).
3. Description of Stability Tests
The empirical analysis uses variants of three classes of tests for parameter stability: tests for
random (time-varying) coefficients; tests based on cumulative forecast errors (CUSUM tests);
and tests based on sequential Wald tests for a single break. For completeness, we briefly
summarize these tests here, although details are available in the original references. For
additional discussion of these tests see the review by Stock (1994) and for additional references
to stability tests more generally see Hackl and Westlund (1989, 1991).
The general model considered is,
(1)
yt =
+ “t ^ t - l + ^ t ^ xt-l + e
t
-5-
where a t(L) and j8t(L) are p-th order lag polynomials which in general are time varying and
where et is serially uncorrelated with mean zero and variance < Let k denote the total
?".
number of regressors. Each test has as its null hypothesis that the parameters are constant, that
is,
« t(L)=a(L) and j8t(L)=j8(L). The derivation of the null distributions of the test statistics
also assumes that the regressors are jointly second order stationary, along with additional
technical conditions. When the discussion below refers to univariate tests, it is understood that
the terms in xt_j in (1) are omitted.
A. Tests for time-varving parameters
The first set of tests for randomly time-varying coefficients are Nyblom’s (1989) locally
most powerful tests against the alternative that the coefficients follow a random walk, where the
random walk error is independent of et. Nyblom (1989) derived his statistic against the
alternative that all the coefficients are stochastic, and this requires some modification since we
also test subsets of coefficients. Rewrite the regression (1) as yt = 0t’zt + et in obvious
notation. Suppose under the alternative that q < k linear combinations of 0t follows a random
walk; that is, R0t = R0t_j + ijt, where R is a q x k matrix of constants which are either known
or can be consistently estimated under the null, and
the modified Nyblom statistic is L = T
9
is i.i.d. and uncorrelated with et. Then
j. i
n
r
£ t
4
-
St, where St = R ^ 5 _ jZ ses, where {es} are
the residuals from OLS estimation of (1), and where ^ = (RT"* £ ^ _ i Z tzt’R’)o^, where
The test is evaluated for three different sets of coefficients:
(2a)
L ^ : test nv «t(L), 0t(L);
test nv 0t(L);
(2b)
(2c)
L
test /zt, j3t, where /3t(L)=/Stb(L) and b(l) is normalized so b(l) = l.
-6-
For Lgjj, R is the k x k identity matrix; for
Rzt = (1, xt_ j,...,x t_p)’; and for
q,
R is the matrix of ones and zeros such that
^ y R is the matrix such that Rzt = (1, S(L)xt_j), where
6(L)=$(L)/$(1) where $(L) is the OLS estimator of /3(L) under the null. The
statistic tests for stochastic evolution of the cumulative effect of xt on the forecast of yt.
Heteroskedasticity-robust variants of the Nyblom statistics were also computed by replacing
P with V = RT~* X ^ = i e^ztzt’R’ (Hansen (1990)). The heteroskedasticity-robust versions
of the tests in (2a), (2b) and (2c) are respectively denoted by iJjj, I_£
and L ^ g ^ .
Also computed were two variants of the Breusch-Pagan (1979) LM test for random
coefficients, for which the alternative hypothesis is that the coefficients are i.i.d. draws from a
distribution with constant mean and finite variance. The two statistics are,
(3a)
2
2
2
2
BPjji = TR from the regression of e^ onto (1, yj_j,...,yj_p,Xj_j,...,Xj_p);
(3b)
BP^j = TR^ from the regression of e^ onto (1, x^_j,...,Xj_p);
(3c)
2
BP^g(i) = TR^ from the regression of e^ onto (1, (6(L)xt_j)^).
B, Tests based on cumulative forecast errors
One of the tests based on cumulative forecast errors is the maximal OLS CUSUM statistic
proposed by Ploberger and Kramer (1992), which is similar to Brown, Durbin and Evans’ (1975)
CUSUM statistic except that the Ploberger-Kramer (1992) statistic is computed using OLS
rather than recursive residuals. Let f-p(5) =
£ ^ ] e s, where [•] is the greatest lesser
integer function. The Ploberger-Kramer (1992) maximal CUSUM statistic is,
(4)
PKsup = suPS E [0,1] I
I•
A related statistic is the mean square of f p
(5)
PKmsq = J
-7-
2
The PKSU and PKmSq statistics respectively have limiting representations as the supremum and
p
the integral of the square of a one-dimensional Brownian bridge.
C. Tests based on sequential Wald statistics
The third set of tests statistics consists of functionals of the sequence of Wald test statistics,
F t (5), which test the null hypothesis that the parameters are constant against the alternative that
they have a single break at a fraction 8 through the sample. The break date is treated as
unknown a-priori, so that the tests involve computing the sequence F^(t/T) for t= tQ ,...,tj, and
then computing a functional of this sequence. Three such functionals are considered. The
Quandt (1960) likelihood ratio (QLR) statistic, in Wald form, is given by
(6)
QLR = suP5e(50,5i)FT ^ -
The mean Wald statistic (Hansen (1992), Andrews and Ploberger (1992)) is
(7)
MW = f ^ F T(5)d5.
The Andrews-Ploberger (1992) Wald statistic is the exponential average,
(8)
APW = ln{ J |jexp(lAFT(5))dS}.
These statistics have asymptotic representations as functionals of a k-dimensional Brownian
bridge; see Andrews (1993) and Andrews and Ploberger (1992) for the details. The tests are
implemented with 15% symmetric trimming (5q =1-5 j =.15).
-8-
4. Stability Tests: E m pirical Evidence
A. Univariate Tests
The values of the univariate stability test statistics, along with summary statistics on the
fraction of rejections, are given for all 76 series in table 1. The final column contains the
regression F statistic testing the hypothesis that the transformed series follows an AR(0). The
first panel reports summary measures of rejections for each series. The second panel reports
each of the individual test statistics. For all regressions, p=6.
The answer to the question of whether there is evidence of widespread instability in these
univariate autoregressions evidently depends on which stability test one uses. On the one hand,
50% of the series reject at the 5% level using the QLR statistic, and similar results obtain for
the APW statistic. There are also many, if fewer, rejections using the MW statistic. These
results provide evidence of one-time shifts in the parameters of the univariate autogressions.
While the Breusch-Pagan (1979) test often rejects, this test also has power against
heteroskedasticity so it is not clear whether this indicates heteroskedasticity or time variation in
the parameters. The Nyblom test has a lower rejection rate (20% at the 5% level); when an
adjustment is made for heteroskedasticity, the rejection rate drops to the level of the test.
However, because the random walk time variation introduces heteroskedasticity, the
heteroskedasticity-robust Lr statistics seems likely to have lower power than the L statistics, so
the drop in significance from the L to Lr statistics need not be interpreted as evidence against
time variation in the data but rather simply as evidence that the adjusted test has lower power.
The rejection rates for the PK CUSUM statistics are low, suggesting that shifts in the intercept
are not a major feature in these data.
The instability is more heavily concentrated in certain classes of series than others. For
example, the QLR statistic rejects at the 5% level for all interest rate and inflation series. In
contrast, other than the Breusch-Pagan test which could be detecting heteroskedasticity, none of
the tests rejects for business failures, the government finance series, and several of the orders
and inventories series.
-9-
B. Bivariate Tests
There are 5700 bivariate forecasting relations among our 76 series so rather than present all
of the test statistics we present various graphical and tabular summaries. Summary rejection
rates of the bivariate tests for parameter stability are presented in table 2. The final column
reports the Granger causality Wald statistic testing the hypothesis that /3(L)=0 in (1). For all
regressions p= 6, and all tests have level 10%.
The results for these 5700 bivariate relations are summarized in panel A of table 2. The
main feature is the evidence of widespread instability in these relations, although this instability
is only detected by a subset of the tests. The
q
statistic rejects in over 70% of the cases,
and its heteroskedasticity-robust variant rejects in almost 60% of the cases. The QLR and APW
statistics also reject in approximately 60% of the cases. A large fraction (58%) of cases also
have significant Granger causality statistics, a result which is perhaps surprising since no apriori economic reasoning was used to select which variables should be used to forecast any
particular dependent variable. As in the univariate results, the CUSUM-based tests have low
rejection rates, which suggests that the instability does not arise from breaks or drift in the
direction of the mean regressors. As the final two rows of panel A indicate, there is only
slightly more instability among statistically significant predictive relationships (based on the
Granger causality test) than among insignificant relationships.
These results can be used to examine stability in relations involving those variables which
commonly appear in structural VAR modeling. Industrial production, real personal income,
manufacturing employment, the CPI, the PPI, the 90-day Treasury bill rate, and the commercial
paper-Treasury bill spread each reject stability in at least 93% of their 75 respective bivariate
relations based on either the
relations based on the
^ or QLR statistics, and M l rejects in 77% of its bivariate
^ statistic, when these series are used as dependent variables (panel
B). When these series appear as predictor variables (panel C), for each the QLR rejects in at
least 59% of the 75 pairs. For five of the seven price series, the QLR statistic rejects stability
- 10-
in each of the 75 bivariate forecasting relations in which inflation is a dependent variable.
When any of these five price series is instead used as a predictor, the QLR statistic again rejects
in more than half the cases. If anything, it appears that instability in bivariate relations
involving these key series is even more prevalent than on average across all 5700 relations.
These marginal distributions provide one window on the extent of instability in these 5700
relations. However, it is possible that some of this instability is in relations which would be of
little interest from a forecasting perspective because they have low overall predictive content.
Exploring this possibility requires examining the joint distribution of the instability and
Granger-causality test statistics. This is done graphically in figures 1-4, which are scatterplots
of selected stability test statistics against the Granger causality test statistic. These figures
confirm that the stability and Granger causality test statistics are only weakly correlated. In a
sense, each forecasting relation can be thought of as having a temporal average level of
predictive content, and deviations from that predictive relation over time are largely
uncorrelated with the average predictive content.
Figure 5 summarizes the estimated break dates ([TS], where S maximizes F^(5)) for the
bivariate relations for the which the corresponding QLR statistics are significant at the 5 %
level. Instability is concentrated around 1974-75, 1980-81, and at the endpoints (5q and
in
(6)).
Figure 6 is a scatterplot of the QLR vs. the APW statistics. Although these are rather
different functionals of the sequential Wald statistics, respectively the maximum and an
exponential average, the statistics are clearly highly correlated and give quite similar inferences
in these data. Evidently little is lost by considering only one or the other of these statistics.
The large number of rejections when the Granger causality statistic is insignificant presents
an intriguing opportunity. This is most easily seen for the QLR statistic. If there is in fact no
predictive content in a bivariate relation in any subsample, then the fraction of QLR rejections
will tend towards the level of the test. In contrast, a rejection by the QLR test but not by the
Granger causality test suggests that the bivariate relation has predictive content in at least one
- 11 -
continuous subsample, even though on average over the full sample it does not (ignoring type I
errors). For example, oil prices (pw561) was a useful predictor for only 23% of the other
series, but 61 % of the corresponding QLR tests rejected. This raises the possibility that models
which adapt to changing relations might find new, albeit transitory, forecasting relations to
exploit.
5. D escription of Forecasting M odels
The second stage in this investigation is an examination of the performance of sixteen fixed
and adaptive forecasting models. Eight of the forecasting models are univariate while eight are
bivariate. Throughout, a (pseudo) in-sample estimation period is used for preliminary
estimation of the parameters and a (pseudo) out-of-sample period is used for forecasting.
The eight univariate models consist of a fixed-parameter autoregression, two autoregressions
estimated by rolling regression, one autoregression estimated by recursive least squares, and four
random walk TVP models. The eight multivariate models are a fixed-parameter bivariate
model, two bivariate models estimated by rolling regression, one model estimated by recursive
least squares, and four bivariate models with random-walk time TVP. All models are of the
form (1), with the coefficients fixed or time-varying as appropriate. The bivariate models will
be referred to as vector autoregressions (VAR’s), although because only one-step ahead
forecasts are considered only the single equation (1) of the (yt, xt) VAR needs to be estimated.
The specification of the TVP models is conventional and assumes the coefficients follow a
random walk. Let 81 = (/xt, a j t,..., «pt, j8jt,..., /3pt) (where
application). Then,
(9)
- 12-
are omitted in the univariate
9
where 1^ is the k x k identity matrix, so that X is the ratio of the variance of the parameter
disturbance
to the variance of the regression error et. The parameters of the TVP models
are O , < ", and X. For each TVP model the value of 0q is set to its OLS estimate over the inq P
sample period, and <" is estimated using the in-sample data. (The out-of-sample forecasts and
?
their relative performances are insensitive to choice of initial conditions because of the long insample period.) One-step ahead forecasts are then produced using period-by-period updating
with the Kalman filter. We consider four TVP models that differ only in their choice of X. So
that a single value of X could be applied to series measured in different units, all series were
first rescaled by dividing through by their in-sample standard deviation.
The eight univariate models are:
(10a)
AR:
AR(6); n, a(L) estimated by OLS, then fixed at in-sample values for outof-sample forecasts
(10b)
RRA1:
AR(6) estimated using rolling regression with 120 observations
(10c)
RRA2:
AR(6) estimated using rolling regression with 240 observations
(10d)
RLSA:
AR(6) estimated by recursive least squares
(lOe)
ATVP1: AR(6) estimated by TVP with X = .005
(100
ATVP2: AR(6) estimated by TVP with X = .010
(10g)
ATVP3: AR(6) estimated by TVP with X = .020
(10h)
ATVP4: AR(6) estimated by TVP with X = .030.
The eight bivariate models are:
(Ha)
VAR:
VAR(6); ix, cc(L), /3(L) estimated by OLS, then fixed at in-sample values
for out-of-sample forecasts
(11b)
RRV1:
VAR(6) estimated using rolling regression with 120 observations
(He)
RRV2:
VAR(6) estimated using rolling regression with 240 observations
- 13-
(lid )
RLSV:
(He)
VTVP1: VAR(6) estimated by TVP with X = .005
(Ilf)
VTVP2: VAR(6) estimated by TVP with X = .010
d ig )
VTVP3: VAR(6) estimated by TVP with X = .020
(llh )
VTVP4: VAR(6) estimated by TVP with X = .030.
VAR(6) estimated by recursive least squares
Two related models are univariate and bivariate TVP models with X estimated over the insample period. These models are not examined because this estimation is currently
computationally impractical for the large number of forecasting relations under consideration.
For the forecast comparisions, the in-sample period ends in 1978:12. This cutoff date was
chosen so that the models are tested in the turbulent economic conditions of the late 1970’s and
early 1980’s. For series ending in 1993:12, 180 observations remain for the out-of-sample
comparison.
6. Forecasting Model Comparison: Empirical Results
Comparing the various models using these data entails examining 608 univariate forecasting
systems (76 variables, eight models each) and 45,600 bivariate forecasting systems (5700
bivariate forecasting relations, eight models each). All comparisons are made using out-ofsample one-month-ahead forecast MSE’s, although in principle other loss functions could be
used. The term "best model" will be used to refer to the model which minimizes this out-ofsample forecast MSE, relative to some comparison group. One objective of this comparison is
to see which models do best most frequently. However, because of the instability found in
section 4, another objective is to ascertain which if any of the models protect the forecaster
from making extreme forecast errors resulting from parameter instability.
The question of which model performs best out-of-sample most frequently is examined in
table 3. For each bivariate relation, MSE’s from the eight bivariate and eight univariate models
- 14-
were computed; for the purposes of this tabulation, the model producing the lowest out-ofsample MSE among these sixteen was then deemed the "best" model for that (yt, xt) pair. Two
sets of tabulations are presented. Panel B presents the fraction of times the column model is
best among the 75 bivariate relations, broken down by forecasted variable. For example, for
industrial production, in 16% of the 75 bivariate pairs, RRV2 produces the smallest out-ofsample MSE; in 61% of these pairs, ATVP1 outperforms not just the other seven univariate
models but also the eight bivariate models. Panel C presents analogous results, broken down by
forecasted variable, except that for each forecasted variable the comparison is only among the
top ten of the 75 pairs, as measured by the Bayes Information Criterion (BIC) for the in-sample
OLS estimation of (1) with fixed parameters. Thus, among forecasts of IP based on the ten
variables with the lowest in-sample BIC’s, in one case (10%) RRV2 has the lowest out-ofsample MSE, but in 7 cases ATVP1 outperforms the other univariate and the eight bivariate
models. The first two rows of panel A respectively summarize the results of panels B and C,
where the fractions are computed over all the forecasted variables. The final row of panel A
presents results for bivariate relations with significant time variation, as measured by
significance (at the 10% level) of the
^ statistic evaluated for the in-sample period.
Several conclusions are evident from table 3. Overall, there is no clearly dominant model;
no model performs best in more than 17% of the 5700 pairs. However, there is strong evidence
that the adaptive models (the rolling, recursive and TVP models) outperform the two fixedparameter models. Among the set of models with predictors based on the top 10 BIC’s, 73% of
the best-performing models are adaptive. However, the extreme TVP models (with X=.02 and
X=.03) are rarely the top performers out of sample. Interestingly, comparing the final row of
panel A with the first two rows indicates that the adaptive models perform similarly whether or
not in-sample instability is detected. Consistent with the stability test evidence, the results in
panels B and C show that different variables tend to be forecast best by different models. For
example, exchange rates (exnwt2) have no rejections using the univariate stability tests but a
moderately high fraction of rejections using the bivariate Nyblom tests, and in 88% of the 75
- 15-
pairs the best forecasting model is the fixed-coefficients AR. In contrast, real personal income
(gmyxp8) has widespread rejections by univariate and bivariate stability tests, and in 97% of the
cases the best forecasting models are adaptive, primarily the univariate or bivariate recursive
least squares models.
Table 4 summarizes pairwise comparisons of the sixteen models over all 5700 bivariate
relations. The ATVP3 and ATVP4 models typically have MSE’s worse than the other
univariate models; the VTVP3 and VTVP4 models also typically perform worse than the other
bivariate models. Among the univariate models, only RLSA and ATVP1 outperform the simple
AR in more than 50% of the cases, and RLSA in turn outperforms ATVP1 in 63% of the cases.
Among the bivariate models, RLSV outperforms all others at least 64% of the time. The RLSA
and ATVP1 models typically outperform all bivariate models. While the univariate models
often outperform the bivariate models, this is perhaps not surprising since a-priori reasoning
would lead one to suspect that many of the 5700 pairs would have forecasting links which are
weak at best.
Table 4 and the test results from section 4 provide additional evidence of instability. While
58% of the 5700 Granger causality statistics reject at the 10% level, only 38% of the recursive
VAR forecasts (the best-performing bivariate forecast) outperform the recursive AR model out
of sample. Presumably some of these Granger causality rejections are Type I errors, but with
75% power against the "true" VAR’s and a 10% level test these 58% rejections correspond to
74% of the VAR’s having nonzero coefficients on the predictor variable. The much lower
fraction of pairs for which bivariate techniques improve performance out of sample thus is
another indication of instability in the bivariate relations.
It is useful to go beyond these assessments of which model typically performs best to
quantify the extent to which the various models reduce the possibility of extremely poor
performance. Table 5 presents the empirical quantiles of the MSE’s of the various models. To
make results comparable across series, the MSE’s are relative to the MSE for the recursive least
squares AR forecast (RLSA). Panel A shows the distribution of these relative MSE’s for the
- 16-
univariate forecasting models. The median values all exceed 1.00, consistent with the finding
in Table 4 that RLSA has lower MSE than the other forecasts more than 50% of the time. The
results also suggest that RLSA is the most robust univariate forecasting model, in the sense that
its worst performance is significantly better than the worst performance of the other models.
For example, the minimum relative MSE for the fixed coefficent AR model is .959 while its
maximum relative MSE is 1.158. Thus, at its best, the AR forecast outperforms the RLSA
forecast by 4.1 %, while at its worst, the AR forecast underperforms the RLSA forecast by
13.6% (=1-(1.158)"*). Panel B presents quantiles for the 5700 bivariate forecasts. Robustness
at the a ’th quantile can be determined by comparing the relative MSE at a to the inverse of the
relative MSE at the (l-a )’th quantile. For example, the relative MSE of VTVP1 at <*=.001 is
.600, so that in . 1% of the cases, VTVP1 outperforms RLSA by at least 40%. At <*=.999, the
relative MSE of VTVP1 is 1.146, so that in .1% of the cases, RLSA outperforms VTVP1 by
more than 12.7% (=1-(1.146)~*). In this sense, in the .1% extremes, VTVP1 produces better
forecasts than RLSA. The table indicates that at the . 1% quantile, all of the bivariate models
dominate RSLA, while at c*=.5% and 1%, only bivariate models with small time variation (RLSV
and VTVP1) dominate RLSA. Similar results obtain at c*=1% for the best 10 BIC-selected
models shown in panel C.
7. Conclusions
Some caveats are warranted. The stability tests are all evaluated using asymptotic critical
values and some finite-sample size distortions are to be expected. However, Monte Carlo
results for sequential Wald tests in Diebold and Chen (1992) suggest that these distortions are
only moderate and could plausibly account for only a small fraction of the empirical rejections.
Also, the quantitative results of the forecasting comparison depend on the choice of out-ofsample period.
- 17 -
With these caveats in mind, these results suggest some general observations about the two
time series applications laid out in the introduction, structural VAR modeling and forecasting.
One finding relevant to VAR modeling is that relations involving key macroeconomic variables
such as industrial production, personal income, employment, prices, and interest rates are, if
anything, more likely to be unstable than the average of these 5700 bivariate relations. While
most VAR modeling involves more than two variables, a stable multivariate VAR implies a
stable VAR for any subset of variables, so these tests can be seen as testing an implication of
the hypothesis of stability of larger models. While it is possible for some (but not all) equations
in a multivariate VAR to be stable despite instability of the bivariate VAR formed from two of
the variables, impulse responses involve all equations in the VAR so instability in the bivaraite
relations implies instability in at least some multivariate impulse responses. One practical lesson
which this emphasizes is the importance of performing systematic stability analysis as part of a
structural VAR modeling exercise.
Given the widespread evidence of structural instability, the comparison of the forecasting
models yielded some surprising results. While the fixed-parameter models occasionally worked
very poorly, models with only small degrees of adaptivity performed well. In particular the
univariate and bivariate recursive least squares models typically were either best or nearly best.
In 38% of the cases, the recursive VAR outperformed the recursive AR out of sample; in 99%
of the 5700 bivariate relations, the recursive VAR model produced an out-of-sample MSE
which was at most 7.8% higher than the recursive AR model. Moreover, in 99% of the
bivariate relations, the recursive VAR model produced an out-of-sample MSE which was at
most 8.7% higher than the best-performing model for each bivariate relation (including the
univariate models) and in 50% of the cases its MSE was within 1.4% of the best model. Of
course, if the parameters are in fact constant, then the recursive estimator will be efficient
relative to the fixed parameter models. One striking result is that the more adaptive models
such as rolling regression or the TVP models typically failed to improve upon recursive least
squares, and indeed did not even guard against extreme failures as well as did recursive least
- 18 -
squares. This negative finding suggests that the class of models considered here, which are the
models most widely studied in adaptive forecasting, are largely unsuccessful in modeling and
exploiting the instability we found in typical macroeconomic applications.
- 19 -
Appendix: Definitions of Series
The entries for each series are the series mnemonic, the transformation code, and the definition
of the series. For series obtained from CITIBASE, the CITIBASE mnemonic has been used.
The transformation codes are: 0 = first difference, 1 =log first difference, 2 = level.
A. Output and Sales
ip 1 index of industrial production
ipxmca 2 capacity util rate: manufacturing,total(% of capacity,sa)(frb)
gmpy 1 personal income: total (bil$,saar)
gmyxp8 1 personal income (real) less transfers
rtql 1 retail trade: total (mil.87$)(s.a.)
gmcq 1 personal consumption expenditure:total (bill. 1987$)
ipcd 1 industrial production: durable consumer gds (1987=100,sa)
ced87m 1 personal consumption expenditures:durable goods,87$
xci 1 stock-watson index of coincident indicators
mt82 1 manuf. and trade sales
B. Employment
lpmhuadj 1 total employee hours (adjusted)
lphrm 2 avg. weekly hrs. of production wkrs.: manufacturing (sa)
lhell 1 index of help-wanted adv.
lhnaps 1 persons at work: part time eas-slack wk,nonag (thous,sa)
luinc 2 avg wkly initial claims,state unemploy.ins. ,exc p.rico(thous;sa)
lhu5 1 unemploy.by duration: persons unempl.less than 5 wks (thous.,sa)
lhur 0 unemployment rate: all workers, 16 years & over (%,sa)
lhelx 2 employment: ratio; help-wanted ads: no. unemployed elf
C. New Orders
hsbp 2 housing authorized: index of new priv housing units (1967=100;sa)
mdu82 1 mfg unfilled orders: durable goods industries, 82$
mpcon8 1 contracts & orders for plant & equipment in 82$(bil$,sa)! 2
mocm82 1 mfg new orders: consumer goods & material,82$(bil$,sa)! 2
mdo82 1 mfg new orders: durable goods industries,82$(bil$,sa)! 2
ivpac 2 vendor performance: % of co’s reporting slower deliveries(%,nsa)
pmi 2 purchasing managers’ index (sa)
pmno 2 napm new orders index (percent)
D. Inventories
invmt87 1 manufacturing & trade inventories:total,87$(bil$,sa)
invrd 1 inventories, retail (sa)
invwd 1 inventories, wholesale (sa)
ivmld8 1 mfg inventories: materials & supplies, all mfg indus 87$(sa)
ivm2d8 1 mfg inventories: work in process, all mfg indus 87$(sa)
ivm3d8 1 mfg inventories: finished goods, all mfg industries 87$(sa)
ivmtd 1 manufacturing & trade inventories: total
ivmld 1 mfg inventories: materials & supplies, all mfg indus (mil$,sa)
-2 0 -
ivm2d 1 mfg inventories: work in process, all mfg indus (mil$,sa)
ivm3d 1 mfg inventories: finished goods, all mfg industries (mil$,sa)
invrd8 1 inventories, retail 87$ (sa)
invwd8 1 inventories, wholesale 87$ (sa)
E. Prices
gmdc 1 pce,impl pr defl:pce (1987=100)
punew 1 cpi-u: all items (82-84 = 100,sa)
pw 1 producer price index: all commodities (82=100,nsa)
pw561 1 producer price index: crude petroleum (82=100,nsa)
pw561r 1 pw561/punew
jocci 1 dept, of commerce commodity price index
joccir 1 jocci/punew
F. Interest Rates
fyff 0 interest rate: federal funds (effective) (% per annum,nsa)
fygm3 0 interest rate: u.s.treasury bills,sec mkt,3-mo.(% per ann,nsa)
fygm6 0 interest rate: u.s.treasury bills,sec mkt,6-mo.(% per ann,nsa)
fygtl 0 interest rate: u.s.treasury const maturities,l-yr.(% per ann,nsa)
fybaac 0 bond yield: moody’s baa corporate (% per annum)
fygtlO 0 interest rate: u.s.treasury const maturities, 10-yr.(% per ann,nsa)
cp6_gm6 2 yield on 6 month commercial paper - fygm6
gl0_gl 2 fytlO - fygtl
gl0_ff 2 fygtlO - fyff
baa_gl0 2 fybaac - fygtlO
G. Money and Credit
fcbcuc 2 change in bus and consumer credit outstand.(percent,saar)(bcdl 11)
fcbcucy 2 fcbcuc-annual percentage growth in GMPY
delinqcr 0 delinq. rate, total install, credit
cci30m 0 consumer instal.loans: delinquency rate,30 days & over, (%,sa)
fmld82 1 money stock: m-1 in 1982$ (bil$,sa)(bcd 105)
fm2d82 1 money stock: m-2 in 1982$(bil$,sa)(bcd 106)
fmbase 1 monetary base, adj for reserve req chgs(frb of st.louis)(bil$,sa)
fml 1 money stock: ml(curr,trav.cks,dem dep,other ck’able dep)(bil$,sa)
fm2 1 money stock:m2(ml+o’nite rps,euro$,g/p&b/d mmmfs&sav&sm time dep(bil$,
fm3 1 money stock: m3(m2+lg time dep,term rp’s&inst only mmmfs)(bil$,sa)
fmbaser 1 monetary base: fmbase/punew
H. Other Variables
exnwt2 1 Trade weighted average nominal exhange rate
fspcom 1 s&p’s common stock price index: composite (1941-43 = 10)
fspcomr 1 fspcom/punew
fail 1 business failures: current liabilities (mil$,nsa)
failr 1 fail/punew
gfosa 1 federal government outlays seasonally adjusted
gfrsa 1 federal government receipts seasonally adjusted
gfor 1 Real federal government outlays, gfosa/punew
gfrr 1 Real federal government receipts, gfrsa/punew
hhsntn 2 u. of mich. index of consumer expectations(bcd-83)
-21
-
Footnotes
1. Applications of adaptive forecasting include Baudin, Nadeau, and Westlund (1984), Guyton,
Zhang and Foutz (1986), Engle, Brown and Stem (1988), Sessions and Chatteijee (1989),
Schneider (1991), Young, Ng, Lane and Parker (1991), Zellner, Hong and Min (1991), Edlund
and Sdgaard (1993), Min and Zellner (1993), and the time-varying VAR’s developed in Doan,
Litterman and Sims (1984), Highfield (1986), and Sims (1982, 1993). Surveys of TVP models
are provided by Chow (1984), Nichols and Pagan (1985), Engle and Watson (1987), and Harvey
(1989).
-22-
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- 2 5 -
Table 1
Univariate Tests for Stability
A. Summary:
Percent rejections over all series
T e s t -----
Test Size
Lall
Lall
PKsup
PKmsq
10%
5%
1%
28.9
19.7
9.2
10.5
5.3
0.0
21.1
10.5
0.0
11.8
6.6
0.0
BPall
QLR
72.4
65.8
55.3
53.9
50.0
30.3
MW
APW
34.2
25.0
14.5
52.6
42.1
27.6
F
98.7
97.4
96.1
B. Results for Individual Series
Series
Sample
PK
sup
Lall
PK
msq
BP
all
QLR
MW
APW
F
A. Output and Sales
ip
59:2 93:12
1.72*
1.06
0.83
0.19
18.61***
21.94**
8.74
6.70
ipxmca
59:1 93:12
1
.8 8 *
1.17
0.81
0.13
27.73***
19.87*
9.30
7.05*
14.22***
2450.44***
gmpy
59:2 93:12
3.58***
1.14
1.23*
0.45**
145.24***
56.81***
20.98***
25.51***
gmyxp8
59:2 93:12
5.46***
1.54
1 .0 2
0.25
133.62***
70.15***
30.76***
32.03***
rtql
59:2 93:12
0.98
0.72
0.60
0.04
43.37***
16.66
gmcq
59:2 93:12
1.83*
1.58
1.23*
0.37*
47.51***
24.47**
ipcd
59:2 93:12
1.50
1.09
0 .6 6
0.09
87.19***
23.61**
ced87m
59:2 93:10
3.02***
1.73*
0.54
0.05
55.30***
46.60***
xci
59:3 93:12
1 .6 8
1.20
0.87
0 .2 1
23.47***
18.33
9.63
5.90
20.08***
mt82
59:2 93:12
1.32
1.17
0 .6 8
0.07
10.85*
13.93
8 .00
4.95
2.06*
6.70
12.17**
9.77
20.44***
3.04***
3.95***
5.20
4.68***
8.50**
2.63**
7.90**
18.47***
1.76
5.37***
B . Employment
lpmhuadj 59:2 93:12
1.48
1.26
0.56
0.07
15.84**
17.53
9.20
5.92
6.27***
lphrm
59:1 93:12
0.93
1.33
1.30*
0.24
27.65***
20.92*
5.61
5.10
317.38***
lhell
59:2 93:12
1.85*
1.70*
0.44
0.03
13.72**
30.11***
lhnaps
59:2 93:12
1.26
1.08
0.56
0.05
32.22***
11.96
luinc
59:1 93:12
1.41
1.17
1.04
0.24
56.08***
22.91**
lhu5
59:2 93:12
0.99
0.96
0.90
0.16
10.89*
12.36
lhur
59:2 93:12
1.25
1.10
0.78
0.04
18.23***
15.85
8 .2 2
5.13
lhelx
59:1 93:12
1 .1 2
0.76
0.89
0.17
62.00***
25.58**
8.36
8
21.81***
11.30*
11.52***
8.29
4.60
10.07
7.77*
5.64
3.68
.2 0 **
18.43***
3.07***
1 2 2 0
.6 6 ***
14.76***
9.68***
6275.90***
C. Hew Orders
hsbp
59:1 93:12
1.45
1.13
0.62
0.06
mdu82
59:2 93:12
2.23**
1.44
0.79
0.17
mpcon8
59:2 93: 9
0.58
0.42
0.58
0.05
24.72***
mocm82
59:2 93: 9
1.03
0 .8 6
0.69
0.08
11.31*
mdo82
59:2 93: 9
1.47
1.39
0.69
0 .1 0
ivpac
59:1 93:12
1.31
1.11
0.89
0 .1 2
pmi
59:1 93:12
0.98
0.90
0.87
0.16
7.72
11.49
5.99
3.74
504.26***
prano
59:1 93:12
1 .0 1
0.85
0.65
0.08
4.76
12.32
6.25
4.15
225.84***
3.54
21.95**
10.96*
7.57*
24.38**
14.08**
9.83**
1005.56***
55.09***
9.66
4.14
2.59
17.85***
15.50
7.13
4.99
3.15***
3.61
28.64***
9.95
9.50**
15.17**
22.27**
8.84
7.73*
8.26***
690.03***
D. Inventories
invmt87
59:2 93: 9
1 .1 2
0.98
0.92
0.24
4.90
12.60
6.26
4.09
invrd
59:1 93: 9
1.33
1.24
0.87
0.24
9.04
20.34*
7.62
5.87
invwd
59:1 93: 9
1.82*
1.32
ivmld8
59:2 93: 9
3.46***
2
ivm2 d8
59:2 93: 9
0.97
ivm3d8
59:2 93: 9
1.55
11.41*
.8 8 ***
4.47***
1 .1 2
0.23
25.33***
2 2
1.13
0.23
38.93***
38.58***
1.01
0.97
0.27
6 .0 0
13.66
5.41
4.35
25.43***
1.52
1.33**
0.59**
4.35
19.44
9.89
6.44
7.67***
.2 0 **
.1 2 **
2 0
19.54***
8.28**
14.39***
12.63***
19.83***
T a b le
Series
Sample
Lall
Lall
PK
sup
1 ,
PK
msq
c o n tin u e d
BPall
i
i
59:1 93: 9
0.98
0.91
1.03
0.23
15.26**
12.44
ivrald
59:1 93: 9
3.70***
2.08**
0.94
0 .2 0
60.32***
36.31***
ivmtd
m
QLR
6 .2 0
21.71***
APW
3.66
14.87***
F
71.58***
55.00***
13.28
5.06
3.84
51.62***
1 0 .1 1
19.43
9.47
6.30
24.58***
0.06
14.87**
15.40
8.32
5.69
4.92***
0.16
23.71***
19.54
10.37*
7.50*
3.30***
ivm2 d
59:1 93: 9
0.99
0.99
1.01
0 .2 1
ivm3d
59:1 93: 9
1.45
1.37
1.48**
0.37*
invrd8
59:2 93: 9
1.38
1.22
0 .6 8
invwd8
59:2 93: 9
1.77*
1.23
1.00
7.18
E. Prices
gmdc
59:2 93:12
1.40
1.05
0.77
0.18
45.15***
14.54***
18.49***
42.31***
punew
59:2 93:12
2.44***
1.42
0.73
0.16
18.06***
44.09***
2 0
.6 6 ***
18.10***
73.50***
0 .66
0.95
0.23
113.95***
93.39***
19.78***
41.02***
17.65***
0.71
1.27*
0.28
54.89***
56.30***
13.08**
24.13***
11.06***
pw
59:2 93:12
2.35**
pw561
59:2 93:12
2
.2 2 **
4.28
pw561r
59:2 93:12
1.91**
0.64
1.15
0.23
47.17***
49.08***
10.61*
20.37***
10.24***
jocci
59:2 93:11
1.09
0.92
0.61
0.08
22.54***
25.88**
7.23
9.22**
21.14***
joccir
59:2 93:11
0.98
0.92
0.82
0.07
16.85***
22.50**
6.45
7.61*
19.25***
F. Interest Rates
fyff
59:2 93:12
1.15
0.49
1.18
0.14
32.68***
39.82***
15.13***
15.05***
fygm3
59:2 93:12
1.15
0.54
1.41**
0 .2 1
71.15***
34.74***
9.70
12.45***
19.94***
fygm6
59:2 93:12
1.32
0.57
1.36**
0 .2 1
73.86***
32.45***
10.50*
11.49***
19.75***
fygtl
59:2 93:12
1.05
0.50
1.36**
0 .2 0
87.81***
30.78***
9.03
10.53***
23.86***
27.23***
11.53**
fybaac
59:2 93:12
0.76
0.56
1.42**
0.27
74.80***
27.13***
6.25
8.80**
fygtlO
59:2 93:12
0.83
0.50
1.41**
0.24
46.68***
25.64**
6.90
8.16**
cp6 _gm6
59:1 93:12
0.65
0.39
0.76
0.07
145.60***
34.04***
5.25
11.38***
17.91***
218.52***
glO_gl
59:1 93:12
1.18
1.16
1.27*
0.28
26.07***
glO_ff
59:1 93:12
1.45
0.45
0.65
0.06
114.79***
49.81***
11.56**
19.43***
216.13***
baa__glO
59:1 93:12
1.51
1.59
1.14
0.37*
13.04**
26.78***
12.54**
9.64**
936.64***
22.48**
9.37
7.23*
921.55***
G. Money and Credit
fcbcuc
59:1 92:11
0.95
0.80
1.05
0.25
fcbcucy
59:1 92:11
1.61
1.65
1.10
0.52**
1.52
1.45
0.98
0.16
1.19
1.03
0.99
0.17
delinqcr 59:2 93:
cci30m
6
59:2 93: 9
fmld82
59:2 93: 9
1.93**
1.65
fm2d82
59:2 93: 9
2.08**
2
fmbase
59:2 93:12
1.79*
fml
59:2 93:12
15.11
5.70
4.65
149.90***
17.79
8.69
6.14
15.71***
6.89
12.49
8.97
4.74
5.27
19.09
7.52
6.13
5.33***
6.4 7 ***
6 .2 1
29.15***
1.23*
0.27
26.96***
12.79**
9.46**
23.05***
0.97
0.24
1.59
26.43**
14.98***
10.08***
50.49***
1.87*
1.31*
0.58**
6 .1 1
19.19
10.60*
1.42
1.33
1.18
0.62**
15.38
8.51
.0 0 **
24.52***
39.42***
7.18*
15.69***
4.94
21.47***
fm2
59:2 93:12
1.06
1.07
1.32**
0.37*
3.25
18.13
fm3
59:2 93:12
2.44***
1.80*
1.11
0.33
3.98
24.97**
15.56***
10.17***
141.76***
frabaser 59:2 93:12
H. Other Variables
1.91**
1.53
1.30*
0.23
3.83
29.64***
13.12**
12.54***
19.72***
exnwt2
59:2 93:12
0.90
0.67
0.99
0 .1 1
15.07**
16.86
6.63
5.46
9.33***
fspcomr
59:2 93:12
1.14
1.04
0.99
0.16
18.60***
14.72
7.42
5.45
7.49***
6.91
5.68
73.20***
fspcom
59:2 93:12
1.15
1.07
0.96
0.18
15.32**
14.99
7.36
5.47
6.93***
fail
59:2 93:12
1.25
1.06
0.63
0.06
11.06*
16.56
7.92
5.91
36.23***
36.49***
failr
59:2 93:12
1.25
1.05
0.62
0.05
11.53*
16.56
7.91
5.93
gfosa
67:8 93:10
1.0 1
0.85
0.91
0 .2 2
43.70***
11.07
6.43
3.78
4.29***
gfrsa
67:8 93:10
1.00
1.12
0.89
0.18
10.07
5.85
3.34
11.94***
gfor
67:8 93:10
0.81
0 .6 8
0.49
0.04
10.21
5.07
3.27
4.88***
gfrr
67:8 93:10
1.04
1.20
0.55
0.06
10.72
5.82
3.31
11.46***
hhsntn
59:1 93:12
1.95**
2.03**
0.84
0.15
4.16
33.18***
5.30
41.74***
47.34***
20.23***
21.00***
1077.94***
*
**
^''
kkk
Notes: Tests are significant at the :
10% .
1% levels. All tests
5%, and
were performed for AR(6) models including a constant term. See the appendix for
series definitions and the text for descriptions of the tests.
Table 2
Bivariate Tests for Stability
Percent of Tests Significant at 10% Level
A. Summary of All Regressions
Test Statistic
PK
‘L .
Lall
sup
PK
msq
BP
all
BP*
BP
/HI)
QLR
MW
APW
GC
Combined
23.3
70.2
18.8
10.7
59.6
18.8
18.8
15.3
66.3
2 2 .6
23.8
60.2
37.9
59.3
GC significant
25.5
74.1
2 1 .0
13.0
62.9
2 1 .0
19.7
16.3
6 6 .6
27.6
30.3
61.8
42.0
60.5
100.0
GC insignif.
2 0 .2
64.8
15.6
7.4
55.0
15.6
17.6
13.8
65.9
15.7
14.7
57.9
32.2
57.6
0 .0
B. Percent rejections, listed by variable being forecasted
Test Statistic
Series
r
Lall
\ . 0
V/3U)
PK
Lall
sup
PK
msq
BP
all
%
BP
QLR
MW
APW
GC
66.7
0(1)
A. Output and Sales
ip
ipxmca
49.3
93.3
17.3
5.3
54.7
17.3
13.3
25.3
22.7
25.3
6 8 .0
52.0
62.7
50.7
94.7
21.3
8 .0
61.3
21.3
1.3
5.3
1 0 0.0
16.0
18.7
60.0
48.0
62.7
54.7
22.7
62.7
93.3
1 0 0 .0
18.7
12.0
100.0
100.0
100.0
100.0
100.0
100.0
gmpy
1 0 0 .0
100.0
22.7
14.7
gmyxp8
98.7
69.3
29.3
1 0 0 .0
1 0 0.0
25.3
22.7
97.3
25.3
16.0
40.0
1 0 0 .0
18.7
16.0
rtql
0 .0
2 0 .0
13.3
0 .0
37.3
13.3
4.0
1.3
100 .0
12.0
14.7
gmcq
46.7
6 8 .0
24.0
28.0
89.3
24.0
77.3
81.3
1 0 0 .0
10.7
9.3
81.3
70.7
77.3
57.3
ipcd
17.3
80.0
17.3
14.7
82.7
17.3
13.3
16.0
100 .0
24.0
32.0
86.7
52.0
86.7
78.7
17.3
29.3
ced87ra
93.3
xci
37.3
90.7
8 .0
66.7
mt82
100.0
98.7
17.3
5.3
68.0
12.0
9.3
2.7
68.0
12.0
9.3
30.7
9.3
30.7
56.0
60.0
2.7
4.0
1 0 0 .0
14.7
16.0
18.7
26.7
100 .0
32.0
32.0
52.0
49.3
54.7
64.0
5.3
5.3
41.3
21.3
33.3
1 2.0
10.7
14.7
70.7
1 0 0.0
1 0 0.0
100.0
72.0
B . Employment
Ipmhuadj
24.0
97.3
18.7
22.7
94.7
18.7
1.3
2.7
44.0
21.3
22.7
32.0
38.7
34.7
73.3
lphrm
13.3
52.0
22.7
29.3
60.0
22.7
56.0
34.7
98.7
17.3
34.7
38.7
16.0
32.0
78.7
lhel
53.3
98.7
10.7
42.7
97.3
10.7
2.7
2.7
45.3
2 0 .0
26.7
98.7
72.0
97.3
74.7
lhnaps
16.0
80.0
13.3
10.7
84.0
13.3
0 .0
1.3
1 0 0 .0
28.0
30.7
42.7
45.3
46.7
77.3
100 .0
luinc
1.3
80.0
14.7
1.3
46.7
14.7
9.3
10.7
lhu5
2.7
62.7
17.3
1.3
65.3
17.3
13.3
16.0
lhur
10.7
76.0
10.7
5.3
62.7
10.7
8 .0
5.3
4.0
70.7
18.7
6.7
52.0
18.7
0 .0
1.3
8 8 .0
1 2.0
4.0
66.7
12.0
5.3
2.7
94.7
16.0
10.7
85.3
0 .0
0 .0
13.3
24.0
lhelx
34.7
45.3
58.7
30.7
58.7
74.7
28.0
18.7
22.7
40.0
6.7
33.3
73.3
94.7
37.3
52.0
34.7
28.0
37.3
72.0
13.3
13.3
92.0
24.0
84.0
56.0
22.7
45.3
24.0
37.3
48.0
22.7
64.0
86.7
84.0
40.0
1 0 0 .0
C. New Orders
hsbp
mdu82
5.3
65.3
90.7
6.7
6.7
mpcon8
0 .0
13.3
10.7
0 .0
14.7
10.7
4.0
mocm82
0 .0
24.0
13.3
0 .0
24.0
13.3
10.7
mdo82
2 0 .0
54.7
14.7
14.7
ivpac
13.3
80.0
24.0
9.3
2.7
12.0
8 .0
22.7
1.3
1.3
1.3
65.3
33.3
17.3
22.7
8 .0
9.3
9.3
78.7
10 0 .0
68.0
14.7
6.7
9.3
13.3
22.7
36.0
90.7
41.3
78.7
77.3
72.0
24.0
5.3
0 .0
62.7
17.3
10.7
77.3
28.0
74.7
42.7
2 0 .0
77.3
29.3
84.0
pcni
1.3
49.3
1 2.0
4.0
30.7
12.0
17.3
16.0
34.7
33.3
25.3
21.3
6.7
pmno
8 .0
65.3
16.0
4.0
53.3
16.0
16.0
2 0 .0
14.7
25.3
22.7
25.3
10.7
17.3
D. Inventories
invmt87
1 2 .0
81.3
8 .0
4.0
64.0
8 .0
2.7
6.7
22.7
24.0
37.3
16.0
33.3
68 .0
invrd
21.3
69.3
21.3
10.7
73.3
21.3
24.0
30.7
1 2 .0
5.3
4.0
66.7
28.0
64.0
66.7
invwd
2 0 .0
97.3
13.3
2.7
57.3
13.3
17.3
6.7
1 0 0 .0
28.0
32.0
65.3
44.0
62.7
58.7
ivmld8
94.7
97.3
8 .0
38.7
97.3
8 .0
22.7
6.7
10 0 .0
22.7
14.7
97.3
94.7
94.7
69.3
ivm2 d8
0 .0
46.7
6.7
0 .0
64.0
6.7
8 .0
6.7
6.7
6.7
4.0
12.0
8 .0
12.0
52.0
ivm3d8
17.3
96.0
16.0
14.7
97.3
16.0
1.3
8 .0
56.0
42.7
45.3
61.3
60.0
6 8 .0
10.7
58.4
T a b le
2 ,
c o n tin u e d
Test Statistic
Series
Lall
Lm ,0 ( D
V^U)
PK
PK
sup
msq
ivmtd
13.3
49.3
16.0
0.0
28.0
16.0
8 .0
10.7
ivmld
94.7
97.3
14.7
45.3
97.3
14.7
8 .0
8 .0
BP
all
64.0
100.0
QLR
BP
%
0
MW
APW
(1 )
GC
36.0
41.3
33.3
18.7
33.3
72.0
2 0 .0
14.7
96.0
97.3
94.7
66.7
ivm2 d
2.7
40.0
5.3
0.0
56.0
5.3
2.7
4.0
4.0
5.3
1.3
13.3
5.3
10.7
60.0
ivm3d
10.7
76.0
18.7
9.3
73.3
18.7
58.7
25.3
16.0
10.7
13.3
46.7
30.7
42.7
78.7
invrd8
1 2 .0
6 8 .0
10.7
5.3
77.3
10.7
2.7
1.3
46.7
8 .0
8 .0
40.0
24.0
45.3
69.3
invwd8
10.7
94.7
10.7
2.7
66.7
10.7
1.3
1.3
98.7
14.7
17.3
49.3
29.3
52.0
46.7
10.7
52.0
9.3
5.3
41.3
9.3
1.3
0.0
6.7
E. Prices
grodc
25.3
29.3
100.0
100.0
100.0
36.0
17.3
100.0
64.0
100.0
32.0
100.0
34.7
100.0
26.7
32.0
68.0
2 0.0
20.0
12.0
8 .0
52.0
8 .0
42.7
52.0
42.7
5.3
100.0
12.0
50.7
6.7
36.0
50.7
4.0
4.0
100.0
12.0
17.3
0.0
97.3
13.3
16.0
93.3
17.3
92.0
41.3
82.7
17.3
14.7
76.0
12.0
66.7
58.7
62.7
100.0
2 0 .0
pw561
50.7
1 0 0 .0
pw561r
38.7
50.7
16.0
1.3
53.3
16.0
0.0
52.0
4.0
100.0
100.0
100.0
0.0
pw
14.7
14.7
0.0
56.0
14.7
4.0
1.3
69.3
92.0
1.3
24.0
joccir
100.0
10.7
100.0
4.0
100.0
24.0
86.7
jocci
100.0
22.7
8 .0
82.7
punew
98.7
5.3
18.7
F. Interest Rates
fyff
10.7
76.0
42.7
2.7
36.0
42.7
32.0
5.3
100.0
41.3
48.0
fygm3
10.7
77.3
46.7
2.7
38.7
46.7
77.3
5.3
100.0
49.3
48.0
98.7
41.3
97.3
70.7
fygm6
12.0
89.3
46.7
2.7
36.0
46.7
72.0
5.3
100.0
42.7
45.3
94.7
45.3
94.7
66.7
4.0
100.0
41.3
42.7
100.0
41.3
37.3
100.0
94.7
60.0
8 .0
89.3
46.7
89.3
10.7
82.7
48.0
10.7
97.3
40.0
22.7
62.7
62.7
6 8 .0
42.7
1.3
29.3
42.7
73.3
32.0
30.7
2.7
30.7
30.7
77.3
fygtlO
41.3
41.3
0.0
24.0
41.3
81.3
6.7
100.0
32.0
36.0
cp6 _gm6
1.3
28.0
22.7
1.3
10.7
22.7
4.0
1.3
100.0
49.3
46.7
100.0
glO_gl
0.0
41.3
26.7
1.3
16.0
26.7
42.7
6.7
100.0
38.7
41.3
6 8 .0
100.0
100.0
9.3
66.7
26.7
0.0
10.7
26.7
0.0
0.0
baa_glO
4.0
89.3
33.3
10.7
82.7
33.3
13.3
45.3
44.0
10.7
21.3
2 0 .0
10.7
8 .0
82.7
2.7
0.0
4.0
65.3
22.7
9.3
0.0
0.0
glO_ff
100.0
93.3
fygtl
fybaac
12.0
53.3
100.0
38.7
38.7
49.3
37.3
42.7
78.7
13.3
16.0
25.3
16.0
14.7
40.0
100.0
60.0
49.3
77.3
62.7
33.3
12.0
30.7
62.7
34.7
21.3
34.7
60.0
9.3
32.0
33.3
30.7
60.0
G . Money and Credit
fcbcuc
18.7
delinqcr
10.7
5.3
10.7
26.7
84.0
5.3
fcbcucy
10.7
12.0
78.7
10.7
56.0
18.7
100.0
100.0
cci30m
6.7
70.7
22.7
10.7
61.3
22.7
5.3
fmld82
49.3
92.0
29.3
29.3
80.0
29.3
33.3
8 .0
100.0
5.3
2.7
100.0
8 .0
9.3
10.7
64.0
14.7
60.0
69.3
34.7
18.7
92.0
80.0
93.3
78.7
fm2d82
66.7
30.7
62.7
30.7
13.3
26.7
14.7
29.3
26.7
93.3
93.3
93.3
70.7
fmbase
28.0
92.0
12.0
18.7
86.7
12.0
54.7
81.3
21.3
28.0
33.3
26.7
44.0
37.3
54.7
fral
16.0
77.3
24.0
14.7
58.7
24.0
24.0
78.7
32.0
25.3
42.7
25.3
36.0
56.0
fm2
9.3
54.7
25.3
16.0
62.7
25.3
72.0
64.0
18.7
13.3
45.3
26.7
42.7
69.3
£ro3
fmbaser
89.3
100.0
14.7
37.3
57.3
13.3
2 0 .0
2 0 .0
100.0
44.0
100.0
100.0
14.7
14.7
4.0
44.0
17.3
22.7
14.7
84.0
93.3
42.7
54.7
13.3
45.3
13.3
16.0
25.3
24.0
89.3
73.3
96.0
68 .0
2 0.0
2.7
0.0
0.0
0.0
0.0
0.0
21.3
100.0
100.0
H . Other Variables
exnwt2
2.7
45.3
4.0
33.3
fspcomr
0.0
0.0
56.0
2.7
1.3
58.7
2.7
4.0
52.0
2.7
1.3
48.0
2.7
1.3
fspcom
fail
2.7
6 8 .0
9.3
4.0
58.7
9.3
failr
2.7
6 8 .0
10.7
4.0
57.3
10.7
gfosa
2.7
2 0 .0
17.3
2.7
33.3
17.3
0.0
0.0
0.0
gfrsa
0.0
24.0
4.0
0.0
61.3
4.0
2.7
2.7
gfor
1.3
13.3
10.7
1.3
21.3
10.7
2.7
0.0
69.3
2.7
0.0
0.0
0.0
0.0
30.7
49.3
74.7
30.7
2.7
9.3
gfrr
hhsntn
0.0
41.3
36.0
100.0
1.3
64.0
28.0
24.0
34.7
10.7
33.3
82.7
21.3
2 0 .0
16.0
8 .0
16.0
48.0
56.0
22.7
14.7
16.0
5.3
16.0
46.7
42.7
26.7
2 0 .0
38.7
4.0
37.3
16.0
45.3
29.3
2 0 .0
38.7
4.0
38.7
14.7
100.0
26.7
100.0
25.3
100.0
8 .0
8 .0
4.0
9.3
46.7
30.7
36.0
2.7
0.0
4.0
45.3
12.0
5.3
10.7
9.3
2.7
8 .0
42.7
29.3
34.7
1.3
0.0
4.0
48.0
28.0
29.3
100.0
97.3
100.0
54.7
T a b le
2 ,
c o n tin u e d
C. Percent rejections, listed by variable used as predictor
---------Series
all
L
L
M D
.0C
all
Test Statistic ----
L1 flLr fl/1xPK
H,P
n,p(l)
sup
PK
msq
BP
all
BP.
BP
/3(1) QLR
v
MW
APW
GC
A. Output and Sales
ip
33.3
89.3
34.7
16.0
78.7
34.7
16.0
12.0
70.7
22.7
17.3
72.0
49.3
69.3
76.0
ipxmca
40.0
85.3
18.7
25.3
81.3
18.7
16.0
21.3
73.3
49.3
44.0
6 8 .0
56.0
69.3
82.7
gmpy
28.0
80.0
14.7
17.3
85.3
14.7
9.3
12.0
58.7
4.0
10.7
57.3
38.7
56.0
61.3
grayxp8
38.7
72.0
13.3
28.0
69.3
13.3
16.0
8 .0
52.0
2.7
1.3
64.0
49.3
66.7
52.0
rtql
14.7
6 8 .0
2 0 .0
4.0
53.3
2 0 .0
2 0 .0
10.7
62.7
12.0
25.3
58.7
25.3
60.0
66.7
gmcq
22.7
72.0
17.3
6.7
61.3
17.3
2 0 .0
13.3
65.3
16.0
21.3
56.0
32.0
56.0
72.0
ipcd
22.7
73.3
30.7
8 .0
62.7
30.7
14.7
8 .0
78.7
25.3
30.7
61.3
30.7
56.0
45.3
ced87m
2 0 .0
69.3
2 0 .0
5.3
54.7
2 0 .0
17.3
9.3
62.7
13.3
24.0
53.3
30.7
53.3
49.3
xci
40.0
90.7
29.3
16.0
82.7
29.3
2 0 .0
12.0
76.0
25.3
13.3
73.3
48.0
6 8 .0
88 .0
mt82
37.3
8 8 .0
36.0
16.0
70.7
36.0
16.0
13.3
66.7
26.7
34.7
6 8 .0
46.7
69.3
81.3
B . Employment
lpmhuadj
29.3
80.0
14.7
16.0
65.3
14.7
22.7
16.0
64.0
4.0
10.7
6 8 .0
42.7
66.7
72.0
lphrm
21.3
73.3
26.7
12.0
62.7
26.7
29.3
29.3
70.7
40.0
41.3
73.3
46.7
73.3
62.7
lhel
32.0
78.7
24.0
16.0
65.3
24.0
22.7
18.7
62.7
21.3
38.7
62.7
42.7
65.3
92.0
lhnaps
29.3
78.7
45.3
13.3
74.7
45.3
17.3
14.7
61.3
10.7
30.7
6 8 .0
42.7
6 8 .0
66.7
luinc
2 0 .0
73.3
13.3
6.7
60.0
13.3
6.7
8 .0
70.7
54.7
42.7
69.3
40.0
70.7
8 8.0
lhu5
24.0
82.7
34.7
77.3
34.7
21.3
10.7
65.3
5.3
18.7
58.7
38.7
58.7
49.3
lhur
24.0
82.7
24.0
14.7
72.0
24.0
32.0
24.0
69.3
16.0
29.3
66.7
42.7
65.3
81.3
lhelx
33.3
80.0
53.3
22.7
82.7
53.3
16.0
24.0
6 8 .0
33.3
2 0 .0
80.0
65.3
80.0
85.3
12.0
C. New Orders
hsbp
13.3
60.0
9.3
5.3
44.0
9.3
25.3
13.3
66.7
33.3
25.3
56.0
30.7
56.0
85.3
mdu82
21.3
84.0
29.3
10.7
74.7
29.3
18.7
16.0
62.7
13.3
13.3
49.3
33.3
48.0
57.3
mpcon8
17.3
6 8 .0
10.7
0 .0
56.0
10.7
2 0 .0
16.0
60.0
1.3
13.3
56.0
26.7
49.3
53.3
mocm82
30.7
77.3
29.3
17.3
70.7
29.3
13.3
10.7
65.3
16.0
29.3
48.0
36.0
46.7
77.3
mdo82
26.7
78.7
28.0
10.7
65.3
28.0
13.3
10.7
69.3
18.7
21.3
60.0
37.3
53.3
80.0
ivpac
12.0
70.7
13.3
4.0
54.7
13.3
17.3
12.0
65.3
21.3
2 0 .0
61.3
34.7
58.7
77.3
13.3
9.3
18.7
13.3
70.7
42.7
29.3
64.0
40.0
62.7
85.3
16.0
72.0
41.3
41.3
6 8 .0
41.3
66.7
81.3
prai
30.7
84.0
13.3
14.7
pnxno
36.0
77.3
16.0
16.0
64.0
65.3
16.0
D. Inventories
invmt87
22.7
74.7
14.7
8 .0
58.7
14.7
16.0
13.3
65.3
5.3
9.3
54.7
28.0
53.3
32.0
invrd
26.7
6 8 .0
10.7
12.0
62.7
10.7
26.7
10.7
66.7
10.7
10.7
57.3
28.0
54.7
45.3
38.7
invwd
22.7
69.3
5.3
1 2.0
54.7
5.3
6.7
8 .0
66.7
26.7
30.7
53.3
30.7
54.7
ivmld8
26.7
70.7
16.0
10.7
62.7
16.0
14.7
17.3
70.7
28.0
30.7
66.7
32.0
65.3
42.7
ivm2 d8
2 0 .0
73.3
16.0
12.0
65.3
16.0
8 .0
14.7
6 8 .0
8 .0
6.7
52.0
21.3
50.7
38.7
ivm3d8
28.0
78.7
13.3
1 2.0
73.3
13.3
34.7
37.3
73.3
25.3
21.3
6 8 .0
45.3
72.0
48.0
ivmtd
26.7
74.7
5.3
17.3
62.7
5.3
14.7
17.3
64.0
25.3
28.0
62.7
36.0
58.7
6 8 .0
ivmld
30.7
74.7
14.7
14.7
69.3
14.7
8 .0
14.7
73.3
37.3
36.0
62.7
40.0
61.3
66.7
ivm2 d
22.7
74.7
14.7
16.0
66.7
14.7
10.7
18.7
6 8 .0
22.7
25.3
49.3
24.0
45.3
65.3
32.0
66.7
48.0
66.7
56.0
12 .0
60.0
24.0
60.0
38.7
13.3
50.7
28.0
48.0
12.0
ivm3d
29.3
85.3
9.3
21.3
70.7
9.3
29.3
24.0
70.7
37.3
invrd8
22.7
72.0
17.3
10.7
65.3
17.3
24.0
13.3
61.3
9.3
invwd8
18.7
66.7
17.3
9.3
53.3
17.3
13.3
8 .0
6 8 .0
2 0 .0
T a b le
2 ,
c o n tin u e d
Test Statistic
Series
Lall
Lnf
.i
1
)
r
r
Lall
*V.0U>
PK
sup
PK
msq
BP
all
%
BP
0
(1 )
QLR
MW
APW
GC
E. Prices
gmdc
28.0
78.7
12 .0
22.7
74.7
12.0
29.3
25.3
74.7
46.7
42.7
50.7
36.0
50.7
50.7
punew
2 0 .0
69.3
10.7
13.3
60.0
10.7
26.7
26.7
70.7
46.7
37.3
62.7
40.0
61.3
70.7
pw
18.7
52.0
0.0
9.3
45.3
17.3
pw561
8 .0
38.7
0.0
18.7
21.3
64.0
24.0
28.0
61.3
33.3
60.0
64.0
1.3
28.0
17.3
17.3
10.7
62.7
5.3
9.3
61.3
41.3
61.3
22.7
9.3
45.3
17.3
0.0
29.3
17.3
18.7
10.7
62.7
5.3
9.3
52.0
40.0
52.0
20 .0
jocci
24.0
66.7
21.3
9.3
60.0
21.3
17.3
14.7
61.3
17.3
18.7
54.7
34.7
56.0
73.3
joccir
22.7
66.7
18.7
5.3
57.3
18.7
16.0
13.3
72.0
32.0
38.7
54.7
34.7
54.7
80.0
pw561r
F. Interest Rates
fyff
16.0
41.3
4.0
1.3
2 0 .0
4.0
13.3
8 .0
54.7
17.3
29.3
69.3
37.3
66.7
70.7
fygm3
14.7
42.7
1.3
1.3
21.3
1.3
10.7
8 .0
61.3
25.3
30.7
77.3
44.0
74.7
58.7
fygm6
17.3
53.3
1.3
2.7
26.7
1.3
13.3
8 .0
60.0
24.0
38.7
76.0
40.0
76.0
68.0
fygtl
16.0
50.7
2.7
1.3
29.3
2.7
12.0
8 .0
65.3
25.3
34.7
73.3
41.3
77.3
77.3
fybaac
13.3
56.0
14.7
2.7
38.7
14.7
24.0
21.3
70.7
29.3
30.7
65.3
45.3
64.0
76.0
fygtlO
13.3
54.7
6.7
2.7
33.3
6.7
16.0
12.0
76.0
29.3
33.3
69.3
42.7
61.3
78.7
cp6 _gm6
25.3
70.7
17.3
8 .0
44.0
17.3
22.7
17.3
66.7
38.7
36.0
6 8 .0
46.7
70.7
76.0
glO_gl
30.7
73.3
34.7
13.3
56.0
34.7
25.3
36.0
65.3
26.7
24.0
6 8 .0
57.3
72.0
89.3
glO_ff
12.0
52.0
8 .0
1.3
25.3
8.0
18.7
17.3
69.3
42.7
37.3
56.0
28.0
53.3
68 .0
baa_glO
22.7
72.0
8 .0
8 .0
60.0
8.0
17.3
2 0 .0
81.3
61.3
34.7
73.3
53.3
74.7
81.3
G . Money and Credit
fcbcuc
21.3
66.7
21.3
12.0
50.7
21.3
6.7
6.7
64.0
18.7
17.3
58.7
30.7
54.7
60.0
fcbcucy
25.3
65.3
26.7
13.3
65.3
26.7
9.3
5.3
64.0
9.3
9.3
62.7
32.0
57.3
21.3
delinqcr
16.0
61.3
22.7
6.7
50.7
22.7
17.3
12.0
58.7
13.3
21.3
46.7
25.3
48.0
26.7
1.3
53.3
22.7
2 0.0
13.3
58.7
16.0
25.3
54.7
37.3
56.0
36.0
65.3
cci30m
13.3
66.7
22.7
fmld82
36.0
81.3
46.7
28.0
76.0
46.7
30.7
24.0
70.7
33.3
36.0
66.7
50.7
65.3
fm2d82
22.7
70.7
9.3
9.3
57.3
9.3
24.0
2 0 .0
64.0
2.7
17.3
49.3
30.7
50.7
65.3
fmbase
26.7
80.0
33.3
16.0
72.0
33.3
29.3
26.7
72.0
21.3
13.3
56.0
46.7
60.0
28.0
fml
26.7
78.7
29.3
12.0
69.3
29.3
38.7
32.0
6 8.0
25.3
32.0
61.3
53.3
61.3
53.3
fm2
18.7
72.0
1.3
12.0
69.3
1.3
9.3
10.7
61.3
1.3
13.3
58.7
45.3
60.0
68.0
fm3
30.7
82.7
16.0
17.3
76.0
16.0
5.3
5.3
68.0
22.7
21.3
6 8 .0
54.7
6 8 .0
42.7
fmbaser
26.7
77.3
37.3
18.7
72.0
37.3
28.0
24.0
69.3
32.0
26.7
57.3
41.3
60.0
56.0
H. Other Variables
exnwt2
2 0.0
72.0
24.0
2.7
58.7
24.0
24.0
10.7
66.7
25.3
2 0 .0
58.7
38.7
57.3
29.3
fspcomr
28.0
76.0
18.7
9.3
65.3
18.7
24.0
10.7
62.7
22.7
24.0
50.7
37.3
49.3
66.7
fspcom
26.7
76.0
17.3
9.3
64.0
17.3
22.7
13.3
64.0
2 0 .0
25.3
50.7
38.7
50.7
65.3
fail
18.7
62.7
18.7
4.0
62.7
18.7
18.7
12.0
65.3
14.7
9.3
49.3
28.0
44.0
2 0 .0
failr
18.7
64.0
18.7
2.7
61.3
18.7
18.7
12.0
65.3
14.7
9.3
49.3
26.7
45.3
18.7
gfosa
18.7
70.7
2 0 .0
9.3
6 8 .0
20 .0
24.0
16.0
60.0
26.7
13.3
42.7
33.3
45.3
45.3
gfrsa
16.0
42.7
14.7
9.3
53.3
14.7
2 0 .0
12.0
60.0
4.0
8 .0
46.7
21.3
40.0
18.7
gfor
2 0 .0
73.3
29.3
10.7
69.3
29.3
25.3
17.3
57.3
18.7
12.0
44.0
32.0
46.7
41.3
gfrr
16.0
45.3
14.7
10.7
50.7
14.7
21.3
14.7
60.0
4.0
9.3
44.0
21.3
41.3
16.0
hhsntn
14.7
62.7
13.3
5.3
42.7
13.3
21.3
12.0
72.0
50.7
29.3
58.7
29.3
56.0
49.3
Notes: All statistics are based on regression (1) with 6 lags. See the
appendix for series definitions and the text for descriptions of the tests.
Table 3
Best Out-of-Sample Forecasting Models:
Percentage of cases in which model was best out-of-sample
A.
AR
KRAI
RRA2
RLSA
ATVP1
9
ATVP2
Summary
ATVP3
ATVP4
VAR
RRV1
RRV2
RLSV
VTVP1
VTVP2
VTVP3
VTVP4
All Models
17
4
8
15
4
4
0
13
1
5
12
6
3
1
0
10 BIC Sel.
15
2
6
14
10
3
3
0
12
2
8
12
8
6
1
0
Stab, rej.
13
5
8
14
10
4
5
0
10
1
6
12
8
4
1
0
B. Best models among all bivariate pairs, by variable being forecasted
AR
RRA1
RRA2
RLSA
ATVP1
ATVP2
ATVP3
ATVP4
VAR
RRV1
RRV2
RLSV
VTVP1
VTVP2
VTVP3
VTVP4
A. Output and Sales
ip
ipxmca
gmpy
gmyxp8
0
0
0
0
61
0
0
0
1
0
16
7
0
0
0
61
0
0
0
0
1
1
3
15
11
20
1
0
3
0
0
0
0
4
1
44
1
0
0
0
76
0
0
0
0
0
0
0
0
0
0
47
0
0
0
0
3
5
9
0
0
7
1
0
rtql
71
0
0
0
0
0
0
0
20
0
0
8
1
0
0
0
gracq
0
0
73
0
0
0
0
0
0
0
21
1
1
3
0
0
17
16
9
0
0
0
1
0
0
0
ipcd
0
0
0
0
49
0
0
0
3
0
5
ced87m
0
0
56
0
0
0
0
0
0
0
43
xci
0
0
0
68
0
0
0
0
0
0
9
12
7
4
0
0
mt82
0
0
0
65
0
0
0
0
11
0
0
20
4
0
0
0
B . Employment
lpmhuadj
0
61
0
0
0
0
0
0
0
39
0
0
0
0
0
0
0
21
lhel
0
40
0
0
0
0
0
0
lhnaps
0
0
0
0
51
0
0
0
0
0
51
0
0
0
0
0
12
52
0
0
0
0
0
0
0
12
lphrm
luinc
lhu5
13
5
1
15
4
0
0
0
7
15
15
4
0
0
0
3
0
16
37
4
0
0
0
3
1
7
3
0
0
19
17
0
0
1
9
20
12
24
1
0
0
5
0
0
lhur
0
0
0
59
0
0
0
0
4
1
0
16
19
1
0
0
lhelx
0
0
0
69
0
0
0
0
0
0
1
16
11
3
0
0
hsbp
0
0
67
0
0
0
0
0
0
0
5
20
mdu82
0
0
79
0
0
0
0
0
0
0
mpcon8
60
0
0
0
0
0
0
0
20
0
4
12
mocm82
51
0
0
0
0
0
0
0
28
0
0
16
C. New Orders
20
0
5
1
1
0
0
1
0
0
3
1
0
0
5
0
0
0
mdo82
0
0
0
0
48
0
0
0
4
7
0
25
15
1
0
0
ivpac
0
0
0
87
0
0
0
0
1
1
1
5
4
0
0
0
pmi
pmno
0
0
0
65
0
0
0
0
9
0
0
21
3
1
0
0
55
0
0
0
0
0
0
0
17
0
4
19
4
1
0
0
D. Inventories
invmt87
0
0
0
45
0
0
0
0
invrd
0
0
0
47
0
0
0
0
invwd
5
20
1
11
23
13
1
0
0
0
1
23
8
1
0
0
0
0
0
72
0
0
0
0
3
0
0
24
1
0
0
0
ivmld8
0
63
0
0
0
0
0
0
0
4
9
3
0
15
7
0
ivm2 d8
56
0
0
0
0
0
0
0
17
0
3
16
8
0
0
0
ivm3d8
0
0
0
0
41
0
0
0
0
3
5
0
0
13
12
25
T a b le
AR
RRA1
RRA2
RLSA
ATVP1
ATVP2
3 ,
ATVP3
c o n tin u e d
ATVP4
VAR
RRV1
RRV2
12
RLSV
3
VTVP1
VTVP2
VTVP3
VTVP4
0
3
4
0
9
16
4
0
0
0
5
27
25
7
4
0
0
16
0
16
13
7
0
0
0
0
3
0
0
8
13
1
0
0
0
0
16
0
0
1
3
4
1
1
85
0
0
0
3
0
0
3
9
0
ivrald
0
68
0
0
0
0
0
0
0
11
ivm2 d
0
0
0
44
0
0
0
0
27
0
ivm3d
0
0
0
0
32
0
0
0
0
invrd8
0
0
0
48
0
0
0
0
invwd8
0
0
0
0
75
0
0
0
0
0
0
0
73
0
E. Prices
gmdc
punew
0
0
0
0
0
pw
0
0
0
0
0
0
93
0
0
0
0
0
1
0
5
0
0
0
0
0
0
75
0
0
0
0
0
0
0
0
pw561
25
0
0
pw561r
36
0
0
0
0
0
0
0
64
0
0
0
0
0
0
jocci
65
0
0
0
0
0
0
0
15
0
0
16
4
0
0
0
0
0
0
64
0
0
0
0
11
0
0
23
3
0
0
0
joccir
F. Interest Rates
fyff
33
0
0
0
0
0
0
0
57
0
3
7
0
0
0
0
fygm3
49
0
0
0
0
0
0
0
48
0
0
1
0
1
0
0
fygm6
43
0
0
0
0
0
0
0
39
0
1
16
0
1
0
0
fygtl
47
0
0
0
0
0
0
0
39
0
1
12
1
0
0
0
fybaac
59
0
0
0
0
0
0
0
41
0
0
0
0
0
0
0
0
fygtlO
51
0
0
0
0
0
0
0
47
0
0
3
0
0
0
cp6 _gm6
91
0
0
0
0
0
0
0
9
0
0
0
0
0
0
0
glO_gl
63
0
0
0
0
0
0
0
37
0
0
0
0
0
0
0
glO_ff
0
0
44
0
0
0
0
0
36
0
8
9
3
0
0
0
baa_glO
0
65
0
0
0
0
0
0
17
1
12
3
0
1
0
0
0
G. Money and Credit
fcbcuc
0
0
0
55
0
0
0
0
12
0
0
23
0
0
fcbcucy
0
0
0
0
0
56
0
0
0
0
0
3
29
11
0
1
delinqcr
0
0
0
52
0
0
0
0
0
1
0
44
3
0
0
0
cci30m
33
0
0
0
0
0
0
0
37
0
0
19
11
0
0
0
fmld82
0
0
0
0
40
0
0
0
1
0
5
28
21
4
0
0
8
17
7
0
19
0
0
0
fm2d82
0
0
fmbase
0
0
11
0
0
0
63
0
0
0
1
1
0
,
0
53
0
0
0
0
0
5
23
3
fml
61
0
0
0
0
0
0
0
21
0
0
11
7
0
0
0
fm2
0
0
0
0
0
49
0
0
16
0
0
1
5
28
0
0
fm3
0
0
0
0
0
0
93
0
0
0
0
0
1
fmbaser
0
0
0
0
0
57
0
0
0
0
3
5
13
3
21
3
0
0
0
H. Other Variables
exnwt2
fspcomr
88
0
0
0
0
0
0
0
11
0
1
0
0
0
0
0
0
0
0
0
79
0
0
0
0
0
3
15
4
0
0
0
fspcom
0
0
0
0
84
0
0
0
0
0
1
13
1
0
0
0
fail
0
0
0
0
73
0
0
0
15
0
0
8
4
0
0
0
0
failr
75
0
0
0
0
0
0
0
16
0
0
7
3
0
0
gfosa
0
0
0
68
0
0
0
0
0
3
0
29
0
0
0
0
gfrsa
0
0
87
0
0
0
0
0
0
0
7
7
0
0
0
0
gfor
0
0
0
76
0
0
0
0
0
1
1
gfrr
0
0
84
0
0
0
0
0
0
0
11
hhsntn
0
0
55
0
0
0
0
0
0
7
19
21
4
20
0
0
0
0
1
0
0
0
0
0
0
0
T a b le
C.
3 ,
c o n tin u e d
Best models among those bivariate pairs with the 10-lowest in-sample BIC,
by variable being forecasted
AR
KRAI
RRA2
RLSA
ATVP1
ATVP2
ATVP3
ATVPA
VAR
RRV1
RRV2
RLSV
VTVP1
VTVP2
VTVP3
VTVPA
A. Output and Sales
ip
ipxmca
gmpy
gmyxp8
rtql
0
0
0
0
70
0
0
0
0
0
10
0
0
20
0
0
0
0
0
70
0
0
0
0
0
0
0
0
30
0
0
0
60
0
0
0
0
0
0
0
A0
0
0
0
0
0
0
0
0
0
0
20
0
0
0
0
20
0
0
60
0
0
0
0
AO
0
0
0
0
0
0
0
20
0
0
A0
0
0
0
0
0
gmcq
0
0
20
0
0
0
0
0
0
0
70
ipcd
0
0
0
0
80
0
0
0
0
0
10
ced87m
0
0
10
0
0
0
0
0
0
0
90
0
0
10
0
10
0
0
0
0
0
0
0
0
0
xci
0
0
0
60
0
0
0
0
0
0
0
10
20
10
0
0
mt82
0
0
0
60
0
0
0
0
20
0
0
20
0
0
0
0
0
B . Employment
lpmhuadj
0
20
0
0
0
0
0
0
0
30
0
0
50
0
0
lphrm
0
0
0
0
0
0
0
0
0
0
10
20
A0
30
0
0
lhel
0
30
0
0
0
0
0
0
0
0
0
10
50
10
0
0
30
lhnaps
0
0
0
0
A0
0
0
0
0
0
0
0
10
luinc
0
0
50
0
0
0
0
0
10
0
30
0
0
10
20
0
0
0
lhu5
AO
0
0
0
0
0
0
0
0
0
0
10
30
20
0
0
lhur
0
0
0
20
0
0
0
0
0
10
0
50
20
0
0
0
lhelx
0
0
0
60
0
0
0
0
0
0
10
10
20
0
0
0
C. New Orders
hsbp
0
0
80
0
0
0
0
0
0
0
0
10
0
10
0
0
mdu82
0
0
70
0
0
0
0
0
0
0
30
0
0
0
0
0
mpcon8
30
0
0
0
0
0
0
0
30
0
10
30
0
0
0
0
mocm82
AO
0
0
0
0
0
0
0
A0
0
0
20
0
0
0
0
0
mdo82
0
0
0
0
30
0
0
0
0
20
0
30
20
0
0
ivpac
0
0
0
100
0
0
0
0
0
0
0
0
0
0
0
0
pcni
0
0
0
50
0
0
0
0
10
0
0
30
0
10
0
0
AO
0
0
0
0
0
0
0
0
0
10
30
10
10
0
0
0
porno
D. Inventories
invmt87
0
0
0
20
0
0
0
0
0
10
20
20
20
10
0
invrd
0
0
0
50
0
0
0
0
0
0
0
20
20
10
0
0
invwd
0
0
0
80
0
0
0
0
0
0
0
20
0
0
0
0
ivmld8
0
20
0
0
0
0
0
0
0
10
20
0
0
A0
10
0
ivm2 d8
90
0
0
0
0
0
0
0
0
0
0
10
0
0
0
0
10
0
0
10
10
0
0
0
10
20
10
0
0
0
0
ivm3d8
0
0
0
0
70
0
0
0
0
ivmtd
0
0
0
20
0
0
0
0
A0
ivmld
0
20
0
0
0
0
0
0
0
20
30
0
0
20
10
ivm2 d
0
0
0
80
0
0
0
0
0
0
0
10
10
0
0
0
ivm3d
0
0
0
0
0
0
0
0
0
10
60
20
0
10
0
0
0
0
0
0
0
0
50
20
10
0
0
0
90
0
0
0
0
0
0
0
10
0
0
0
invrd8
0
0
0
20
invwd8
0
0
0
0
E. Prices
gmdc
0
0
0
0
0
90
0
0
0
0
0
0
0
10
0
0
punew
0
0
0
0
0
0
90
0
0
0
0
0
0
0
10
0
pw
0
0
0
0
0
0
80
0
0
0
0
0
10
0
10
0
pw561
20
0
0
0
0
0
0
0
80
0
0
0
0
0
0
0
pw561r
30
0
0
0
0
0
0
0
70
0
0
0
0
0
0
0
jocci
90
0
0
0
0
0
0
0
0
0
0
10
0
0
0
0
joccir
0
0
0
70
0
0
0
0
0
0
0
20
10
0
0
0
T a b le
AR
RRAl
RRA2
RLSA
ATVPl
ATVP2
3 ,
ATVP3
c o n tin u e d
ATVP4
VAR
RRV1
RRV2
RLSV
VTVP1
VTVP2
VTVP3
VTVP4
F. Interest Rates
fyff
0
0
0
0
0
0
0
0
90
0
0
10
0
0
0
0
fygm3
30
0
0
0
0
0
0
0
60
0
0
10
0
0
0
0
fygm6
30
0
0
0
0
0
0
0
50
0
0
20
0
0
0
0
fygtl
50
0
0
0
0
0
0
0
30
0
0
10
10
0
0
0
fybaac
50
0
0
0
0
0
0
0
50
0
0
0
0
0
0
0
fygtlO
40
0
0
0
0
0
0
0
50
0
0
10
0
0
0
0
cp6 _gm6
gl0 _gl
100
80
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
20
0
0
0
0
0
0
0
gl0 _ff
0
0
0
0
0
0
0
0
40
0
50
0
10
0
0
0
baa_gl0
0
50
0
0
0
0
0
0
40
0
10
0
0
0
0
0
G. Money and Credit
fcbcuc
0
0
0
30
0
0
0
0
20
0
0
20
30
0
0
0
fcbcucy
0
0
0
0
0
0
0
0
0
0
0
10
50
40
0
0
delinqcr
0
0
0
80
0
0
0
0
0
0
0
20
0
0
0
0
cci30m
20
0
0
0
0
0
0
0
40
0
0
20
20
0
0
0
fmld82
0
0
0
0
40
0
0
0
0
0
0
50
10
0
0
fm2d82
0
0
0
0
0
50
0
0
0
0
0
0
0
fmbase
0
0
0
0
60
0
0
0
0
0
0
30
10
fml
50
0
0
0
0
0
0
0
40
0
0
10
0
fm2
0
0
0
0
0
40
0
0
0
0
0
0
20
0
40
0
10
0
0
0
0
0
0
40
0
0
fm3
0
0
0
0
0
0
90
0
0
0
0
0
0
10
0
0
fmbaser
0
0
0
0
0
20
0
0
0
0
20
30
10
20
0
0
H. Other Variables
exnwt2
100
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
fspcomr
0
0
0
0
100
0
0
0
0
0
0
0
0
0
0
0
100
fspcom
0
0
0
0
fail
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
90
0
0
0
0
0
0
10
0
0
0
0
failr
90
0
0
0
0
0
0
0
0
0
0
10
0
0
0
0
gfosa
0
0
0
70
0
0
0
0
0
0
0
30
0
0
0
0
gfrsa
0
0
70
0
0
0
0
0
0
0
0
30
0
0
0
0
gfor
0
0
0
80
0
0
0
0
0
0
0
20
0
0
0
0
gfrr
0
0
70
0
0
0
0
0
0
0
10
10
10
0
0
0
hhsntn
0
0
60
0
0
0
0
0
0
0
20
20
0
0
0
0
Notes: See the appendix for series descriptions. All models included six lags plus
a constant. The first row of panel A shows results for all 76x76 models. Entries in
the second row are the corresponding fraction, except that the set of bivariate
relations is restricted from 75 to 10 for each forecasted variable, where the 10
predictors are chosen to be those with the lowest in-sample BIC for the forecasted
variable at hand. Entries in the third row are for the set of bivariate relations
restricted to be those for which the L n statistic is significant at the 10% level,
when calculated through 1978:12. Panel*a shows detailed results for all model for
each variable, and panel C shows detailed results for the 10 best fitting (insample) models for each variable. The in-sample period was from the later of 1959:1
or the first data for which data are available, to 1978:12, and the out-of-sample
period is from 1979:1 through the earlier of the final date for the series or
1993:12.
Table 4
Comparsion of out-of-sample forecasts
among all 5700 bivariate bivariate combinations
Percentage of times that row forecast MSE is less than column forecast MSE
Model
AR
RRA1
RRA2
RLSA
AR
ATVP1
ATVP2
ATVP3
ATVP4
—
66
55
36
45
53
79
87
RRA1
34
—
26
12
20
30
72
RRA2
45
74
—
20
34
62
RLSA
64
88
80
—
63
ATVP1
55
80
66
37
ATVP2
47
70
38
18
ATVP3
21
28
16
11
ATVP4
13
11
VAR
35
RRV1
VAR
RRVl
RRV2
RLSV
VTVPl
VTVP2
65
79
60
50
54
63
79
88
89
53
80
48
36
41
58
82
91
84
93
61
85
65
51
57
70
85
93
82
89
93
69
88
72
62
68
76
88
94
—
86
93
93
66
86
67
56
63
77
89
95
14
—
93
97
59
78
55
45
48
67
88
94
VTVP3
VTVP4
7
7
—
100
44
60
33
27
27
36
73
90
0
—
34
42
20
16
15
19
46
77
45
68
80
20
61
87
7
7
7
3
47
39
31
34
41
56
66
—
65
42
26
33
21
20
15
12
14
22
40
58
35
—
9
7
6
RRV2
40
52
35
28
33
45
67
80
58
91
—
22
37
63
89
96
RLSV
50
64
49
38
44
55
73
84
74
93
78
—
64
78
91
96
VTVP1
46
59
43
32
37
52
73
85
67
94
63
36
—
86
95
98
VTVP2
37
42
30
24
23
33
64
81
55
80
37
22
14
—
98
99
VTVP3
21
18
15
12
11
12
27
54
32
39
11
9
5
2
—
100
VTVP4
12
9
7
6
10
23
20
13
4
2
1
0
—
Notes:
See the notes to table 3
5
6
4
Table 5
Selected Quantiles of Distributions of Mean Square Forecast Errors
Relative to MSE for the AR Recursive Least Squares (RLSA) Forecast
A. Univariate Forecasts
----------- Percentile -----------Model
Min
0.050
0.250
0.500
0.750
0.950
Max
1.158
AR
0.959
0.972
0.998
1.003
1.016
1.032
RLA1
0.967
0.987
1.006
1.017
1.049
1.076
RLA2
0.977
0.991
1.004
1.022
1.010
1.023
1.076
1.002
1.004
1.025
1.035
1.007
1.014
1.089
1.107
ATVP1
0.988
0.992
ATVP2
0.976
0.987
1.001
1.000
1.002
ATVP3
0.969
0.984
1.014
1.024
1.039
1.233
1.269
ATVP4
0.973
0.989
1.029
1.040
1.063
1.368
1.413
B. Bivariate Forecasts (All)
------------------------------- Percentile -------------------------------------Model
Min
0.001
0.005
0.010
0.050
0.250
0.500
0.750
0.950
0.990
0.995
0.999
Max
2.278
VAR
0.512
0.607
0.914
0.931
0.960
0.994
1.018
1.048
1.145
1.276
1.338
1.570
RLV1
0.493
0.612
0.920
0.934
0.979
1.018
1.037
1.057
1.093
1.131
1.145
1.200
1.392
RLV2
0.491
0.602
0.906
0.922
0.964
0.998
1.013
1.029
1.059
1.093
1.105
1.159
1.268
RLSV
0.486
0.598
0.903
0.919
0.961
0.992
1.006
1.019
1.047
1.078
1.094
1.152
1.263
VTVP1
0.491
0.600
0.903
0.921
0.961
0.995
1.009
1.024
1.050
1.076
1.096
1.146
1.274
VTVP2
0.499
0.602
0.905
0.923
0.964
1.001
1.020
1.038
1.094
1.123
1.133
1.183
1.320
VTVP3
0.517
0.615
0.917
0.935
0.978
1.020
1.048
1.074
1.220
1.280
1.289
1.329
1.454
VTVP4
0.534
0.639
0.934
0.951
0.996
1.043
1.077
1.114
1.331
1.428
1.446
1.485
1.574
C. Best 10 bivariate models as selected using in-sample BIC
---------------------------- Percentile ------------------------Model
Min
0.005
0.010
0.050
0.250
0.500
0.750
0.950
0.990
0.995
Max
VAR
0.512
0 .6 8 6
0.901
0.940
0.980
1.017
1.066
1.195
1.519
1.947
2.278
RLV1
0.493
0.712
0.889
0.936
1.003
1.031
1.052
1 .101
1.175
1.240
1.392
RLV2
0.491
0 .6 8 8
0.875
0.928
0.981
1.009
1.031
1.071
1.157
1.193
1.268
RLSV
0.486
0.689
0.870
0.922
0.977
1.002
1.024
1.061
1.146
1.194
1.263
VTVP1
0.491
0.701
0.873
0.922
0.978
1.008
1.027
1.059
1.130
1.149
1.274
VTVP2
0.499
0.724
0.884
0.926
0.983
1.018
1.042
1.096
1.157
1.201
1.320
VTVP3
0.517
0.772
0.904
0.939
1.001
1.043
1.077
1 .222
1.306
1.368
1.454
VTVP4
0.534
0.809
0.926
0.958
1.022
1.070
1.112
1.331
1.437
1.517
1.574
Note: Mean square forecast errors are relative to mean square error of the
recursive least squares AR forecast. See the note to Table 3 for a description
of the models.
G r a n g e r —C a u s a l i t y S t a t i s t i c
Figure 1
Scatterplot of Nyblom (1989)
s a i t c vs. Granger causality F s a i t c
ttsi
-ttsi
S o lid lin e s d e n o te 10% c r it ic a l v a lu e s
G r a n g e r —C a u s a l i t y S t a t i s t i c
0 .2
0.4
0 .6
0 .8
1 .0
1.2
1.4
1.6
1.8
2.0
PKsup
Figure 2
Scatterplot of Ploberger-Kramer (1992) P K sup s a i t c vs. Granger causality F s a i t c
ttsi
-ttsi
S o lid lin e s d e n o te 10% c r it ic a l v a lu e s
o
ro
CM
G r a n g e r —C a u s a l i t y S t a t i s t i c
IN
CM
00
m
CM
a>
co
m
o
0
40
80
120
160
200
QLR
Figure 3
Scatterplot of Quandt (1960) Q L R s a i t c vs. Granger causality F-s a i t c
ttsi
ttsi
S o lid lin e s d e n o te 10% c r it ic a l v a lu e s
240
o
n
rCJ
N
G r a n g e r —C a u s a l i t y S t a t i s t i c
CJ
N
(N
C
O
m
V
—
<
N
<>
7
to
to
o
0
10
20
30
40
50
60
70
80
MW
Figure 4
Scatterplot o f MW statistic vs. Granger causality F-statistic
S o lid lin e s d e n o te 10% c r it ic a l v a lu e s
90
100
C
0 .08
0 .06
0 .0 4
0.0 2
0.00
Frequency
Relative
1964
1968
1972
1976
1980
1984
Year
Figure 5
Histogram of break-dates from Q L R s a i t c
ttsi
1988
APW
o
QLR
Figure 6
Scatterplot of Andrews-Ploberger (1992) APW statistic vs. QLR statistic
Working Paper Series
A s ries of research studies on regional economic issues relating to the Seventh Federal
e
Reserve D s r c ,and on financial and economic t p c .
itit
ois
REGIONAL ECONOMIC ISSUES
Estimating Monthly Regional Value Added by Combining Regional Input
With National Production Data
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P h ilip R. Israilevich and Kenneth N . Kuttner
Local Impact of Foreign Trade Zone
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D a v id D . Weiss
Trends and Prospects for Rural Manufacturing
WP-92-12
W illiam A . Testa
State and Local Government Spending--The Balance
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WP-92-14
Richard H . Mattoon
Forecasting with Regional Input-Output Tables
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P.R . Israilevich, R . M ahidhara, and G J D . Hew ings
A Primer on Global Auto Markets
WP-93-1
P au l D . Ballew and Robert H . Schnorbus
Industry Approaches t Environmental Policy
o
in the Great Lakes Region
W P-93-8
D a v id R . Allardice, Richard H . Mattoon and William A . Testa
The Midwest Stock Price Index-Leading Indicator
of Regional Economic Activity
W P-93-9
W illiam A . Strauss
Lean Manufacturing and the Decision to Vertically Integrate
Some Empirical Evidence From the U.S. Automobile Industry
WP-94-1
Thomas H . K lie r
Domestic Consumption Patterns and the Midwest Economy
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Robert Schnorbus and P au l Ballew
1
W orking p aper series continued
ISSUES IN FINANCIAL REGULATION
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Capital Adequacy and the Growth of U.S. Banks
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H erbert Baer and John M cElravey
Bank Contagion: Theory and Evidence
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George G. Kaufman
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James T. Moser
Preferred Sources of Market Discipline: Depositors v .
s
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D ouglas D . Evan o ff
An Investigation of Returns Conditional
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James T. M oser and Jaclcy C. So
The Effect of Capital on Portfolio Risk a Life Insurance Companies
t
WP-92-29
E lijah Brew er I II , Thomas H . Mondschean, and P hilip E . Strahan
A Framework forEstimating the Value and
Inte e tRate Risk of Retail Bank Deposits
rs
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D a vid E . Hutchison, George G . Pennacchi
Capital Shocks and Bank Growth-1973 t 1991
o
WP-92-31
H erbert L . B aer and John N . M cElravey
The Impact of S&L Failures and Regulatory Changes
on the C D Market 1987-1991
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E lijah Brew er and Thomas H . Mondschean
Junk Bond Holdings, Premium Tax Offsets, and Risk
Exposure a Life Insurance Companies
t
WP-93-3
E lijah Brewer I I I and Thomas H . Mondschean
2
W alk in g p aper series continued
Stock Margins and the Conditional Probability ofPrice Reversals
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P a u l Kofman and James T. M oser
I There Lif(f)e After DTB?
s
Competitive Aspects of Cross Listed Futures
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P a u l Kofman, Ton y Bouwman and James T. M oser
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H erbert L . Baer, Virginia G . France and James T. M oser
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ntttos
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Hesna Genay
Origins of the Modem Exchange Clearinghouse: A History ofEarly
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James T. M oser
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E lija h Brew er I I I , Bernadette A . Minton and James T. M oser
Small Business Investment Companies:
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E lija h B rew er I I I and Hesna Genay
MACROECONOMIC ISSUES
An Examination of Change i Energy Dependence and Efficiency
n
in the Six Largest Energy Using Countries-1970-1988
WP-92-2
Ja c k L . H ervey
Does the Federal Reserve Affect Asset Prices?
WP-92-3
Vefa Tarhan
Investment and Market Imperfections i the U.S. Manufacturing Sector
n
WP-92-4
Paula R. Worthington
Business Cycle Durations and Postwar Stabilization of the U.S. Economy
WP-92-6
M ark W. Watson
3
W orking p aper series continued
A Procedure forPredicting Recessions with Leading I d
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W P -9 2 -7
Jantes H . Stock and M ark W. Watson
Production and Inventory Control a the General Motors Corporation
t
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WP-92-10
A n il K . Kashyap and D a vid W. W ilcox
Liquidity Effects, Monetary Policy and the Business Cycle
W
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Lawrence J. Christiano and M artin Eichenbaum
Monetary Policy and External Finance: Interpreting the
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neet
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Kenneth N . Kuttner
Testing Long Run Neutrality
WP-92-18
Robert G. K ing and M ark W. Watson
A Policymaker’ Guide to Indicators of Economic Activity
s
Charles Evans, Steven Strongin, and Francesca Eugeni
WP-92-19
Barriers t Trade and Union Wage Dynamics
o
WP-92-22
Ellen R. Rissman
Wage Growth and Sectoral S i t : Phillips Curve Redux
hfs
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Ellen R. Rissman
Excess Volatility and The Smoothing of I t r s Rates:
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Steven Strongin
Market Structure, Technology and the Cyclicality of Output
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Bruce Petersen and Steven Strongin
The Identification of Monetary Policy Disturbances:
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Steven Strongin
Earnings Losses and Displaced Workers
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Louis S. Jacobson ,
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4
W orking pap er series continued
Some Empirical Evidence of the Effects on Monetary Policy
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M artin Eichenbaum and Charles Evans
An Unobserved-Components Model of
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Kenneth N . Kuttner
Investment, Cash How, and Sunk Costs
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Lessons from the Japanese Main Bank System
for Financial System Reform in Poland
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Takeo H oshi, A n il Kashyap, and G a ry Loveman
Credit Conditions and the Cyclical Behavior of Inventories
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A n il K . Kashyap, Owen A . Lam ont and Jerem y C. Stein
Labor Productivity During the Great Depression
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M ichael D . B ordo and Charles L . Evans
Monetary Policy Shocks and Productivity Measures
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Consumer Confidence and Economic Ructuations
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John G . Matsusaka and A rg ia M . Sbordone
Vector Autoregressions and Cointegration
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M ark I . Watson
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M ichael T. K . H orvath and M ark W. Watson
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Jeffrey R. Campbell
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Cyclical Productivity i a Model of Labor Hoarding
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Algorithms for Solving Dynamic Models with Occasionally Binding Constraints
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ntblt n
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James H . Stock and M ark W. Watson
6
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