An examination of linear and nonlinear causal relationships between commodity prices and U.S. inflation.
Mahadevan, Renuka ; Suardi, Sandy
I. INTRODUCTION
There is considerable literature on the use of commodity price
information (particularly oil prices) in predicting economic activity
and inflation. This, in large part, reflects the view that higher
commodity prices, such as oil prices, in particular, tend to be followed
by inflation and recessions (Barksy and Kilian 2002, 2004; Hamilton
2003). While much of the literature has focused on the oil
price-macroeconomy relationship, policymakers and economists, for a long
time, have been interested in the commodity-consumer price nexus, and
increasingly so with the rise in inflation targeting as part of an
objective of monetary policy. (1)
More recently, the inflation experience of the 2000s before the
global financial crisis was attributed to rising prices for globally
traded commodities. This event has generated a renewed interest on
inflationary consequences of commodity prices despite considerable
debate on the usefulness of commodity prices as a leading indicator for
inflation. (2)
This study empirically examines the relationship between changes in
commodity prices and inflation. Its contributions are fourfold. Firstly,
we investigate the performance of a variety of commodity price indices
as standalone indicators of inflation. We consider a myriad of Commodity
Research Bureau (CRB) grouping indices including, among others, metals,
raw industrials, textiles and fibers, livestock and products, and a
composite index comprising all grouping indices. One reason for this is
that by lumping together a diverse group of commodities, the indices
could obscure their components' predictive power. This would be the
case if some commodities were not good inflation predictors or if the
timing of the inflation signals varied among different kinds of
commodities. The results, by and large, indicate that the empirical link
between commodity prices and inflation has changed dramatically over
time, partly because of the changes in the extent to which movements in
commodity prices reflect idiosyncratic shocks, and partly due to the
change in macroeconomic fundamentals such as low inflation and lower
uncertainty about future inflation and output growth that prevailed
during the period of Great Moderation. Secondly, this article
demonstrates that there is significant evidence of nonlinear causality
between inflation and changes in metal and raw industrial price indices
during the Great Moderation, even though this nonlinear causal
relationship was absent prior to that period. As for other commodity
price changes and inflation the degree of linear causal relationship has
changed, albeit moderately, during the Great Moderation, supporting
earlier studies of Whitt (1988), Furlong (1989), and Blomberg and Harris
(1995). Thirdly, we show evidence that the nonlinear relationship is due
primarily to the rate of information flow which occurs during periods of
high volatility in these commodity prices in late 2000. This suggests
that the widely documented linear relationship is more complex than
initially thought. The existing research on the nature and sources of
causal relationship between commodity prices and inflation has, to date,
focused relatively on a linear causal relationship and has ignored the
possibility of nonlinear causal relationship. This is surprising and
indicates an important gap in this line of research given a priori that
economic theory does not predicate a linear functional form for the
relationship between changes in commodity prices and inflation.
Finally, on the methodology front, this study employs a robust
approach for testing the presence of nonlinear causal relationship
between commodity prices and inflation. One potential concern is that
the nonlinear causality inferred from the Baek and Brock (1992) test may
be affected by the presence of linear relations in the data. (3) For
this reason, we fitted a vector autoregression (VAR) model to the data
and apply the modified Baek and Brock test for Granger non-causality to
the resulting residuals. This approach, which differs from the standard
approach of applying the Baek and Brock (1992) test to the original
untreated observations, may lead to erroneous inferences because of
unaccounted estimation uncertainty. Specifically, there is a potential
difference of the null distribution when the test is applied to
residuals rather than to original observations (Randles 1984). This
difference arises because the parameter uncertainty is not reflected in
the test statistics of the standard Baek and Brock (1992) test. To
circumvent the problem of erroneous inference, we use a re-sampling
scheme that incorporates parameter estimation uncertainty. This is done
by using the test statistics of the modified Baek and Brock (1992) test
and the resampling procedure of Diks and DeGoede (2001) to yield
empirical p values of the nonlinear Granger causality tests. We also go
beyond identifying the presence of nonlinear causal relationship between
the series to determine the possible sources of the nonlinearities. To
this end, we filter the data by including differential reaction to
information flow as proxied by generalized autoregressive conditional
heteroskedasticity (GARCH) effects. We estimate a multivariate GARCH
model and obtain essentially the same results for linear causality as in
the VAR model. This implies that the evidence of linear causality
persists when we control for the GARCH nonlinearity. On the other hand,
diagnostic tests on the residuals of the multivariate GARCH model
indicate that the nonlinear causal relationships that previously
characterized the residuals of the VAR are substantially reduced and
eliminated. This implies that one possible source of nonlinear causal
relationship arises from the transmission of commodity price shocks
caused by the highly volatile changes in metal and raw industrial price
indices in late 2000.
The remainder of this article is structured as follows. Section II
outlines the methodology while Section III describes the data. Section
IV presents the results of cointegration tests, and both linear and
nonlinear Granger causality tests. It also provides a plausible
explanation for the observed nonlinear Granger causal relationship
between commodity price indices and inflation. Section V summarizes and
concludes.
II. EMPIRICAL METHODS
This section describes the methodologies used to undertake the
standard linear causality tests proposed by Granger (1969) and the
modified Baek and Brock (1992) method for testing nonlinear Granger
causality. However, prior to testing for the causal relationships
between the series, we undertake unit root tests and cointegration
tests, with and without structural breaks. Any evidence of cointegration
will need to be accounted in the VAR specification to determine whether
or not changes in commodity price indices and inflation are linearly
causally related. As the tests used to determine the presence of a unit
root and cointegration are standard in the literature, and for the sake
of brevity, we do not discuss the tests at great length here. (4)
Standard unit root tests involving the Augmented Dickey Fuller and
Phillips Perron tests are initially employed to determine the stationary
property of the series. Given that standard tests for unit root suffer
from low power in the presence of a neglected structural break (Perron
1989), we employ the Lumsdaine and Papell (LP) (1997) unit root tests
under the null of a unit root and the alternative hypothesis of a
stationary series with breakpoints. Notwithstanding that, Lee and
Strazicich (LS) (2003) demonstrate that there is significant size
distortion associated with the LP test statistics when a structural
break occurs under the null of a unit root. They develop a procedure
that allows for two breaks under both the null and alternative
hypotheses. The results suggest that there is evidence of a unit root in
all series with two break dates identified in all commodity indices. The
break dates determined by the LP and LS tests do not differ
significantly from each other; the first break revolves around the
commodity price shock of the early 1970s and the second break date
occurs in the late 1990s. The break dates for commodity price indices
are apparent when viewed from the series plotted in Figure 1. As for
consumer price index (CPI), the break dates are associated with the
peaks of inflation in 1974:07 and 1983:06 (see Figure 2).
Having established the presence of breaks in all series, it is
pertinent that the next stage of analysis on cointegration accommodates
these breaks. We perform the Engle and Granger (1987) test without a
structural break followed by the Gregory and Hansen (1996) test which
accommodates a single break. The test of Kejriwal and Perron (2010) is
further employed to determine the number of breaks in a cointegration
framework when it is believed that more breaks could exist. This test
has better power than the Gregory and Hansen (1996) test which assumes
only a single break in the cointegration relationship. However, for all
three tests we fail to find any evidence of cointegration in the
commodity price indices and the CPI. The Kejriwal and Perron (2010) test
also points to a single break date with the break date identified around
the 1980s coinciding with that identified by the Gregory and Hansen
test. (5)
A. Linear Granger Causality
Suppose two variables are changing over time, [X.sub.t] and
[Y.sub.t]. Linear Granger causality determines whether past values of
[X.sub.t] have significant linear predictive power for current values of
[Y.sub.t] given past values of [Y.sub.t]. If the coefficients associated
with past values of [X.sub.t] are statistically significant, then
[X.sub.t] is said to linearly Granger cause [Y.sub.t]. Bidirectional
causality is said to exist with Granger causality runs in both
directions. In the absence of cointegration, the test for linear Granger
causality between commodity prices and inflation involves estimating the
following equations in a VAR framework:
(1) [DELTA][X.sub.t] = [I.summation over (i=1)]
[[alpha].sub.i][DELTA][X.sub.t-i] + [I.summation over (i=1)]
[[beta].sub.i][DELTA][Y.sub.t-i] + [[epsilon].sub.1,t]
(2) [DELTA][Y.sub.t] = [I.summation over (i=1)]
[[delta].sub.i][DELTA][Y.sub.t-i] + [I.summation over
(i=1)][[theta].sub.i][DELTA][X.sub.t-i] + [[epsilon].sub.2,t].
[DELTA][X.sub.t] and [DELTA][Y.sub.t] are the first difference of
commodity prices and inflation on day t, respectively; the parameters
[alpha], [beta], [delta], and [theta] are to be estimated.
[[epsilon].sub.1,t] and [[epsilon].sub.2,t] are zero-mean error terms
with constant variance-covariance matrix. The optimal lag length I is
determined using the Akaike information criterion (AIC).
Linear causal relationships are inferred from Equations (1a) and
(1b). To test for linear Granger noncausality at specific lags, we
examine the joint statistical significance of the [[beta].sub.i] and
[[theta].sub.i] coefficient estimates for all i. For example, the null
hypothesis of [DELTA][Y.sub.t] (say, inflation) does not Granger cause
[DELTA][X.sub.t] (change in commodity prices) we can test that
[[beta].sub.i] = 0 jointly for all i.
[FIGURE 1 OMITTED]
[FIGURE 2 OMITTED]
B. Nonlinear Granger Causality
Baek and Brock (1992) developed the following nonparametric test to
detect nonlinear causal relationships from the residuals of linear
Granger causality models. Given that their procedure is a nonparametric
test, it does not provide any information about the functional form or
the sign of the detected causal relationship, hence it should be used as
a diagnostic tool that encompasses a modeling process, rather than a
modeling device. Nonetheless, the test is an important development in
detecting nonlinear Granger causality.
Consider the same two series as described in the subsection II.A,
[DELTA][X.sub.t] and [DELTA][Y.sub.t]. Let the m-length lead vector of
[DELTA][X.sub.t] be denoted by [DELTA][X.sup.m.sub.t], and let L and S
be the lengths of the lag vectors [DELTA][X.sup.L.sub.t-L] and
[DELTA][Y.sup.S.sub.t-S], respectively. For given values of m, L, and S
[greater than or equal to] 1 and an arbitrarily small constant d > 0,
[DELTA][[bar.Y].sub.t] does not strictly nonlinearly Granger cause
[DELTA][X.sub.t] if
(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where Pr(*) is probability, [parallel]*[parallel] is the maximum
norm, and s, t = max(L, S) + 1, ..., T - m + 1.
The left-hand side of Equation (2) is the conditional probability
that two arbitary m-length lead vectors of [DELTA][X.sub.t] are within a
distance d of each other, given that two corresponding L-length lag
vectors of [DELTA][X.sub.t] and two S-length vectors of
[DELTA][Y.sub.t], respectively, are within a distance d of each other.
The right-hand side of Equation (2) is the probability that the two
m-length lead vectors of [DELTA][X.sub.t] are within a distance d of
each other, conditional only on their corresponding L-length lag vectors
being within distance d of each other. The intuition is that if
commodity price changes do not nonlinearly Granger cause inflation, then
the probability of the distance between two conformable vectors of
inflation being less than d will be the same whether the probability is
conditioned on the past inflation and commodity price changes or only on
the own past inflation.
The test in Equation (2) can be expressed in terms of the ratios of
joint and conditioning probabilities associated with each part of the
test as follows:
(4) (CI(m + L, S, d)/CI(L, S, d)) = (CI(m + L, d)/CI(L, d))
whereby the joint probabilities are defined as
(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
The condition in Equation (3) can be tested using the
correlation-integral estimators of the joint probabilities in Equation
(4). The correlation-integral is a measure of "closeness" of
realizations of a possibly multivariate random variable in two different
times and is estimated as a proportion of the number of observations
that are within the distance d of each other to the total number of
observations. Assuming that [DELTA][X.sub.t] and [DELTA][Y.sub.t] are
strictly stationary and meet the required mixing conditions as specified
in Denker and Keller (1983), under the null hypothesis that
[DELTA][Y.sub.t] does not strictly Granger cause [DELTA][X.sub.t], the
test statistic T is asymptotically normally distributed. In other words,
(6) T = [(CI(m + L, S, d)/CI(L, S, d)) - (CI(m + L, d)/CI(L, d)] ~
N(0, 1/[square root of n][[sigma].sup.2](m, L, S, d))
where n = T + 1 - m - max(L,S) and [[sigma].sup.2](*) is the
asymptotic variance of the modified Baek and Brock test statistic. (6)
The test statistic in Equation (5) is applied to the two estimated
residual series from the VAR model in Equations (1a) and (1b),
[[epsilon].sub.1,t] and [[epsilon].sub.2,t], respectively. Hence, if the
null hypothesis of Granger noncausality is rejected, the detected causal
relationship between commodity price changes and inflation must
necessarily be nonlinear.
One potential drawback of the application of the Baek and Brock
test to estimated residuals is an unaccounted estimation uncertainty
that may give rise to erroneous inference. Accordingly, we test the
robustness of our nonlinear findings to this problem by using the
re-sampling procedure of Diks and DeGoede (2001) to obtain empirical p
values. Diks and DeGoede perform several experiments to determine the
best randomization procedure to obtain empirical p values. They
demonstrate that the best finite sample properties of the tests are best
obtained when only the causing series were bootstrapped. Here, the
causing series are the residuals. Accordingly, we adopt their
methodology and employ the Stationarity boostrap of Politis and Romano
(1994) to preserve potential serial dependence in the causing series.
The resampling scheme which is robust with respect to parameter
uncertainty is implemented by undertaking the following steps.
1. Estimate a parametric model and obtain the fitted values of the
conditional mean and estimated residuals.
2. Resample the residuals following the null hypothesis. Let N
denote the length of the series and [P.sub.S] is the stationary
bootstrap switching probability. We start a new bootstrapped sequence
from a random position in the initial series selected from the uniform
distribution between 1 and N. The next element in the bootstrapped
sequence corresponds to the next element in the initial series with
probability 1 - [P.sub.S]. We randomly select an element from the
initial sequence with probability [P.sub.S] and put it as the next
element in the bootstrapped sequence. The procedure continues until we
obtain a bootstrapped sequence of length N. To ensure stationarity of
the bootstrapped sequence, we connect the beginning and the end of the
initial sequence.
3. Create artificial data series using the fitted values and the
re-sampled residuals.
4. Re-estimate the model using the artificial data and obtain a new
series of residuals.
5. Compute test statistics [T.sub.i] for the artificial residuals.
The empirical distribution of the test statistics under the null
can be obtained by repeating the bootstrap B-times and calculating test
statistic [T.sub.i] for each bootstrap i = 1, ..., B. The p value of the
test is obtained by comparing the test statistics computed from the
initial data [T.sub.0] with the test statistics under the null [T.sub.i]
:
p = [B.summation over (i=0)]I([T.sub.0] [less than or equal to]
[T.sub.i])/B + 1
where I([T.sub.o] [less than or equal to] [T.sub.i]) denotes an
indicator dummy which takes the value 1 when the event in the brackets
is true and 0 otherwise. The test rejects the null hypothesis in the
direction of nonlinear Granger causality whenever [T.sub.0] is large. B
is set to 5000 and the bootstrap switching probability [P.sub.S] is set
to 0.05. The results of the bootstrapped empirical p values are reported
in Table 3.
III. DATA
The empirical investigation covers the period from January 1957 to
December 2007. The data consist of monthly CRB grouping and composite
indices and CPI, all taken from the CRB (http://www.crbtrader.com). We
use a number of commodity price indices because we are interested to
know whether certain group indices provide greater predictive power of
inflation compared to others. The CRB indices comprise composite spot
index, liverstock sub-index, fats and oils sub-index, foodstuffs
sub-index, raw industrials sub-index, textiles sub-index, and metals
sub-index. The composition of each of the six grouping indices (or
sub-indices) are described in Table 1. The CRB spot index is a measure
of the collective movement in the prices of 22 basic commodities from
these six commodity groups.
One benefit of using indices of commodity groups rather than
individual commodity prices is that idiosyncratic factors impacting on
individual commodity markets should have far less influence at the level
of a multi-commodity, broadly based index. Annual percentage change in
the indices is computed using the index based on the formula
[([CRB.sub.t]/[CRB.sub.t-12]) - 1] x 100%. Likewise annual percentage
change in CPI (or inflation) is computed in the same way. Figure 1
provides plots of the logarithmic commodity indices and CPI, while
Figure 2 provides plots of changes in the commodity price indices and
inflation.
From Figure 1 it can be seen that all the commodity price indices
exhibit greater volatility than the CPI. It is apparent that a
significant shift in the level of commodity price indices occurred
around the early 1970s with a sudden and sharp increase in the
individual grouping and composite commodity price indices. This sharp
increase in commodity price indices can also be seen in Figure 2 as
evidenced by the sharp spike in changes in commodity prices by as much
as 100% in the case of foodstuffs price index. Another apparent increase
in commodity price indices occurred in the late 1990s or early 2000 when
there was a sudden peak in price indices, particularly for metals,
foodstuffs and the composite spot index.
To better visualize the extent of volatility in commodity price
indices, we turn to Figure 2 where it can be seen that among the
different sub-indices, the change in metal sub-index is the largest,
rising by as much as 80% in 2007. Foodstuffs index and oils and fats
index have also risen by as much as 40% and 60%, respectively toward the
end of the sample period. Seasonally adjusted inflation in the United
States reaches its highest level of around 15% in the late 1970s but
drops dramatically in the early 1980s. By the mid-1980s, inflation has
dropped dramatically to below 2.5% before stabilizing below 5% from 1985
to 2007.
IV. EMPIRICAL RESULTS
A. Linear Granger Causality
Results of the linear Granger causality tests are reported in Table
2. For the whole sample spanning the period 1957:1-2007:12, there is
overwhelming evidence of unidirectional Granger causality running from
changes in commodity price indices to inflation with the exception of
raw industries and metal indices which exhibit evidence of bidirectional
causality at the 5% significance level. Our results for the CRB
composite index are comparable with those of Bhar and Hamori (2008) who
also document evidence of causal relationship from CRB futures prices to
CPI, although they employ the Toda and Yamamoto (1995) approach of
estimating a level VAR which accommodates variables of unknown
integration or cointegration order.
To better understand the causal relationship between commodity
price indices and inflation over time, we split the sample into two
sub-samples; one prior to the Great Moderation (1957:1-1984:12) and the
other during the Great Moderation (1985:1-2007:12). We then estimate the
VAR model on these two subsamples and perform the linear Granger
causality tests. (7) The results are interestingly different. In the
sample period prior to the Great Moderation, there is evidence of
bidirectional Granger causality between changes in CRB composite price
index, raw industrials index, metals index, and inflation. However, in
the period during the Great Moderation, there is no evidence that the
composite index provides any predictive power on inflation, although all
the individual group of commodity price indices continue to showcase,
albeit with lower degree of predictive power, causal relationship from
changes in commodity price indices to inflation. Taken together, there
is evidence to suggest that the predictive power of many of the
commodity price indices has weakened during the Great Moderation period.
It is not certain what could have caused this empirical observation
given that there could be many factors that could have altered the
relationship between the movements in commodity prices and inflation. It
could be that the relatively lower level of inflation and greater
certainty about the future level of inflation during the Great
Moderation could have caused overall prices to drift away from the price
changes in various non-oil commodities, thus leading to a weaker causal
relationship between them. Our results seem to concur with the findings
of Blomberg and Harris (1995) who demonstrate that there is diminished
signalling power of commodities since the mid-1980s. They argue that
this could be due to commodities playing a smaller role in U.S.
production relative to earlier periods, and to some extent, the absence
of major food and oil price shocks for the period being investigated
(ending in mid-1990s) could mean that commodity price fluctuations may
not have been big enough to be passed through to consumer prices.
Although our sample extends their sample by more than a decade and
includes periods of greater commodity price volatility, particularly
subsequent to late 1990s, the evidence remains robust and demonstrates
weaker causal relationships between commodity price indices and
inflation.
B. Nonlinear Granger Causality
Prior to testing for nonlinear Granger causality, we first
determine whether the data are characterized by nonlinearities. We
employ the BDS test due to Brock, Dechert, and Scheinkman (1987, revised
in 1996) which is by far the most widely adopted test for nonlinear
structure. The null hypothesis for the BDS test is that the data are
independently and identically distributed (i.i.d), and any departure
from i.i.d should lead to rejection of this null in favor of an
unspecified alternative. Hence the test can be considered a broad
portmanteau test which has been shown to have reasonable power against a
variety of nonlinear data generating processes (see Brock, Hseih, and
LeBaron 1991 for an extensive Monte Carlo study). (8) Results of the BDS
test, which are not reported here for brevity but are available from the
authors upon request, reveal that all the majority estimates of the BDS
statistics are statistically significant, indicating significant
nonlinearities in the univariate series for changes in the various
commodity price indices and inflation.
The nonlinear causality test is conducted using the residuals
obtained from the VAR model, from which any linear predictive
relationship has already been removed. To implement the Baek and Brock
test, values for the lead length, m, the lag lengths Lr1 and Lr2, and
the distance measure d must be selected. Following the results in
Hiemstra and Jones (1994), we set the lead length at m = 1 and set Lr1 =
Lr2 for all cases. We also use common lag lengths of one to five lags
and a common distance measure of d = 1.5[sigma], where [sigma] refers to
the standard deviation of the time series. For robustness check, we also
performed the analysis for d = 0.5[sigma] and 1.0[sigma] and there were
no qualitative differences in the results. (9) The discussion of the
empirical results focuses on empirical p values for the modified Baek
and Brock test given that they are computed using the re-sampling
procedure. The empirical p values are more reliable because they account
for estimation uncertainty in the residuals of the VAR model used in the
modified Baek and Brock test.
It can be seen in Table 3 for the entire sample period, the null
hypothesis of no nonlinear Granger causality from changes in commodity
price indices to inflation is strongly rejected at the 5% significance
level. Equally, we do not find any evidence in support of nonlinear
Granger causality from inflation to changes in commodity prices. It is
apparent from this test that the predictive power of commodity prices
tends to operate linearly on inflation. However, when we perform the
test on the residuals of the VAR models in sub-samples, an interesting
result emerges. While the period prior to the Great Moderation continues
to show a lack of evidence of nonlinear Granger causality (in either
direction) between commodity prices and inflation, the Great Moderation
period clearly and distinctly demonstrates overwhelming evidence of
unidirectional nonlinear leadlag relationships from changes in raw
industrials index and metals index to inflation. The null hypothesis of
no nonlinear Granger causality from commodity price changes to inflation
is strongly rejected at the 5% significance level for indices associated
with raw industrials and metals. These results are in sharp contrast to
those reported in previous studies that report linear predictability
from changes in commodity prices to inflation.
C. Nonlinear Shocks Transmission Between Commodity Price Indices
and Inflation
To determine the source of nonlinear Granger causality that
operates from changes in commodity price index (i.e., for metal and raw
industrials indices) to inflation during the Great Moderation, we model
the nonlinear Granger causality in volatility using the Baba, Engle,
Kraft, and Kroner (BEKK) model of Engle and Kroner (1995). It has been
argued by Ross (1989) and Andersen (1996) that the volatility of a time
series can measure the rate of information flow. A cursory look at the
plot of the change in commodity price index for metals and raw
industrials in Figure 2 suggests that the significant rise in volatility
for these two price indices in late 2000 could potentially explain the
transmission of shocks from changes in commodity price indices to
inflation. In addition, fitting the BEKK model also enables us to
account for conditional heteroskedasticity that is observed in the data
and allow us to apply the test for nonlinear Granger causality on the
resulting standardized residuals of the model. (10)
The BEKK model is specified by assuming that the joint distribution
of changes in commodity price index and inflation conditional on their
past observations is multivariate normal with conditional mean
[[mu].sub.t] and conditional variance [H.sub.t] such that
[Y.sub.t]|[[OMEGA].sub.t-1] ~ N([[mu].sub.t], [H.sub.t]). The mean
equation is specified by the VAR model as in Equations (1a) and (1b).
The variance equation, [H.sub.t], in matrix form is given as follows
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where [C.sub.0] is a (2 x 2) lower triangular matrix; A is a (2 x
2) matrix with the diagonal elements capturing the impact of unexpected
shocks of past inflation and commodity price changes on the current
conditional volatility of inflation and changes in commodity prices, and
the off-diagonal elements measure the corresponding cross effects. Both
the diagonal and off-diagonal elements determine the conditional
variance; B is a (2 x 2) matrix with the diagonal elements measuring the
impact of past conditional volatility of inflation and changes in
commodity price shocks on current conditional volatilities, whereas the
off-diagonal elements measure the corresponding cross effects; D is a (2
x 2) matrix that measures the impact of both positive inflation shocks
and changes in commodity price shocks on current conditional
volatilities such that [[zeta].sub.t-1] = max([[epsilon].sub.t], 0)
where [[epsilon].sup./.sub.t] = ([[epsilon].sub.1t [[epsilon].sub.2t]]).
The BEKK specification ensures that [H.sub.t] is positive definite. The
model is estimated using a maximum likelihood procedure and the results
of the simultaneous estimation of the VAR-asymmetric BEKK model for the
causality in means tests are reported in Table 4.
Consistent with our earlier results, we find that there is
significant evidence of linear Granger causality from changes in
commodity prices to inflation for both metals and raw industrials (see
Panel B of Table 4). The null hypothesis that the coefficients
associated with the five lagged changes in commodity prices are jointly
equal to zero is rejected at the 5% level of significance in both cases.
The magnitudes of the coefficient estimates in the VAR-asymmetric BEKK
model are slightly different and the standard errors (not reported) are
generally smaller than in the VAR model. (11) This is because of the
improved efficiency of the simultaneous procedure and accounting for
heteroskedasticity in variance.
The estimated coefficients of matrices [C.sub.0], A, B, and D are
reported in Panel A of Table 4. As noted earlier, off-diagonal elements
of matrix A capture the impacts of shocks arising from commodity price
changes on inflation conditional volatility and vice versa.
Specifically, the off-diagonal element [a.sub.12] ([a.sub.21]) captures
the impact of changes in commodity price shock (inflation) on current
conditional volatility of inflation (changes in commodity prices). |t
can be seen for both metals and raw industrials indices the
[[??].sub.12] is statistically and economically significant while the
[[??].sub.21] is not statistically significant. This implies that only
shocks emanating from commodity price changes are being transmitted to
the conditional volatility of inflation. There is also evidence that the
lagged conditional volatility of raw industrial price changes impact on
current volatility of inflation (i.e., coefficient estimate of
[b.sub.12] is statistically significant at the 5% level). As for the
coefficients of the D matrix, both [[??].sub.11] and [[??].sub.12] are
statistically significant for raw industrials price index. The former
suggests that an unanticipated positive inflation shock engenders
greater current conditional volatility of inflation than a negative
shock of equal magnitude. (12) The latter suggests that an unanticipated
positive raw industrials price shock has an economically ([[??].sub.12]
= 0.82) and statistically significant effect on current conditional
volatility of inflation. In the case of metals price index, only
[[??].sub.11] and [[??].sub.22] are statistically significant suggesting
that both positive shocks on inflation and changes in metals price index
exert a greater impact on their own conditional volatilities compared to
negative shocks of equal magnitude. There is, however, no evidence of
cross asymmetric effects arising from either positive inflation shocks
or positive metals price shocks. These evidences point to the importance
of not only the magnitude of commodity price shocks but also its sign as
a possible channel of information transmission that gives rise to
nonlinear Granger causality.
Panel B further reports the diagnostic tests for the adequacy of
the VAR-asymmetric BEKK model. The null of no GARCH effects is clearly
rejected for both cases thus confirming the importance of modeling the
volatility of the time series and validating the role played by
volatility in capturing the rate of information flow. The null
hypothesis of no cross effects (i.e., the null of diagonal conditional
variance) is also rejected at conventional levels of significance
implying that it is important to account for spillover effects arising
from shocks in commodity price changes to inflation volatility. Finally,
the null of symmetric conditional variance (i.e., the absence of
asymmetric volatility effects) is rejected at 5% levels of significance.
This result suggests that one should account for the sign of the shock
in measuring the rate of information flow that is transmitted from
shocks of both inflation and commodity price changes. The serial
correlation of the standardized residuals and squared standardized
residuals show no evidence of serial correlation up to 5th order
implying that the model is adequate in characterizing the dynamics of
the data.
Finally, Panel C shows the results of the nonlinear Granger
causality test when applied to residuals of the BEKK model. The test is
used as a diagnostic device to establish whether the BEKK model is
capable of fully capturing the Granger causal relationship. The
rejection of the test would mean either that there is some Granger
causality left beyond the second moment, or that the BEKK specification
is not adequate in fully reflecting the true relationship. Here, we
apply the test based on re-sampling procedure described above to account
for any biases in the test arising from estimation uncertainty. Note
that Step 1 of the re-sampling procedure now includes the conditional
variance-covariance equation from the BEKK model. When compared with the
results of the test applied to the residuals of the VAR model in Table 3
for both metals and raw industrials indices, we notice in all cases the
p value of the test statistic increases quite dramatically and leads to
failure in rejecting the null hypotheses. This implies that there is no
further evidence of nonlinear Granger causality but more importantly,
the BEKK model fully captures the nonlinear Granger causality detected
from the residuals of the VAR model.
V. CONCLUSION
This article presents a robust analysis of the causal relationships
between inflation and commodity price changes by focusing on a sample
period which includes the recent run-up of commodity prices that have
been the source of relatively high rates of inflation in late 2000. This
is the first study to examine the dynamic nonlinear linkages between
inflation and a wide array of commodity price indices. It first
establishes the existence of a long-run cointegrating relationship
between commodity price indices and CPI in the presence of structural
breaks. Given that the data span a long period covering periods of
extreme commodity price shocks and that inflation dynamics has
experienced significant changes in the U.S. economy during the Great
Moderation, we perform robust analysis on the cointegrating relationship
by accommodating for regime changes. The results indicate that
irrespective of the presence of structural breaks, there is no evidence
of cointegration between any of the commodity price indices and the CPI.
There is, however, strong evidence of a unidirectional linear
causal relationship from changes in commodity prices to inflation. The
evidence is pervasive for all commodity price indices considered. Be
that as it may, the degree of linear causal relationship is stronger in
the period prior to the Great Moderation. For instance, we find that
there is evidence of bidirectional causal linkages between inflation and
changes in metals and raw industrials indices before the Great
Moderation, but there are no causal effects from inflation to these
commodity price indices during the Great Moderation. These results could
well be explained by the weakening in association between inflation and
commodity price changes during the period of low inflation and inflation
uncertainty, and be attributed to the sharp decline in the commodity
composition of U.S. output.
An important finding not previously documented in the literature is
the evidence of nonlinear Granger causality from changes in metals and
raw industrials price indices to inflation. Existing studies have, in
general, only tested for a linear relationship. This empirical
observation of nonlinear Granger causality, interestingly, only occurred
during the Great Moderation. The results are proven to be robust having
filtered the data for linear causal relationships and having applied a
novel and robust approach to testing nonlinear Granger causality by
accounting for estimation uncertainty. We identify one potential source
of nonlinear Granger causality emanating from nonlinear transmission of
shocks of commodity price changes to inflation. Specifically, the
multivariate GARCH model is adequate in characterizing the rate of
information flow between commodity price indices and inflation beyond
the first moment. There is, in fact, overwhelming evidence that the
magnitude and sign of the shocks, and their interaction with the
conditional volatilities of inflation and commodity price changes are
important in explaining the transmission of information which gives rise
to this nonlinear Granger causality.
Given the peak in volatility of metals and raw industrial price
indices in late 2000, and despite the fall in importance of metals and
raw industrials in U.S. production, there is evidence that the extent of
Granger causality to inflation is not restricted to the linear model.
Unanticipated commodity price shocks, particularly metals and raw
industrials, contain useful information for predicting inflation, based
on results of in-sample statistics. These results highlight the
importance for future research to consider alternative nonlinear
approach in identifying the transmission process of information
contained in commodity prices to inflation. Moreover, there are
questions that remain unanswered and are worthy of further
investigation. That is, (a) whether commodity price changes are helpful
in predicting inflation out-of-sample; (b) whether other types of
nonlinearities could also capture the change in the relationship during
the Great Moderation; and (c) whether changes in both in-sample and
out-of-sample predictive abilities could have changed over time in a
smooth manner and not in a fashion captured by a single structural
break. (13)
REFERENCES
Andersen, T. G. "Return Volatility and Trading Volume: An
Information Flow Interpretation of Stochastic Volatility." Journal
of Finance, 51, 1996, 169-204.
Baek, E., and W. Brock. "A General Test for Nonlinear Granger
Causality: Bivariate Model." Working Paper, Iowa State University,
1992.
Bai, J., and P. Perron. "Estimating and Testing Linear Models
with Multiple Structural Changes." Econometrica, 66, 1998, 47-78.
--. "Computation and Analysis of Multiple Structural Change
Models." Journal of Applied Econometrics, 18, 2003, 1-22.
Barsky, R. B., and L. Kilian. "Do We Really Know That Oil
Caused the Great Stagflation? A Monetary Alternative," in NBER
Macroeconomics Annual 2001, edited by B. S. Bernanke and K. Rogoff.
Cambridge, MA: MIT Press, 2002, 137-83.
--. "Oil and the Macroeconomy Since the 1970's."
Journal of Economic Perspectives, 18(4), 2004, 115-34.
Bernanke, B. S., M. Gertler, and M. Watson. "Systematic
Monetary Policy and the Effects of Oil Price Shocks." Brookings
Papers on Economic Activity, 28, 1997, 91-157.
Bhar, R., and S. Hamori. "Information Content of Commodity
Futures Prices for Monetary Policy." Economic Modelling, 25, 2008,
274-83.
Bloomberg, B., and E. Harris. "The Commodity-Consumer Price
Connection: Fact or Fable?" FRBNY Economic Policy Review, October
1995, 21-38.
Brock, W. A., W. D. Dechert, and J. A. Scheinkman. "A Test for
Independence Based on the Correlation Dimension." Mimeo, Department
of Economics, University of Wisconsin at Madison, and University of
Houston, 1987.
Brock, W. A., D. A. Hsieh, and B. LeBaron. Nonlinear Dynamics,
Chaos, and Instability: Statistical Theory and Economic Evidence.
Reading, MA: MIT Press, 1991.
Brock. W. A., W. D. Dechert, J. A. Scheinkman, and B. LeBaron.
"A Test for Independence Based on the Correlation Dimension."
Econometric Reviews, 15, 1996, 197-235.
Clarida, R., J. Gali, and M. Gertler. "Monetary Policy Rules
and Macroeconomic Stability: Evidence and Some Theory." Quarterly
Journal of Economics, 115, 2000, 147-80.
Denker, M., and G. Keller. "On U-Statistics and von-Mises
Statistics for Weakly Dependent Processes." Zeitschrift fur
Wahrscheinlichkeitstheorie und Verwandte Gebiete, 64, 1983, 505-22.
Diks, C., and J. DeGoede. "A General Nonparametric Bootstrap
Test for Granger Causality," in Global Analysis of Dynamical
Systems, edited by H. W. Broer, W. Krauskopf, and G. Vegter. Bristol:
Institute of Physics Publishing, 2001.
Edelstein. P. "Commodity Prices, Inflation Forecasts and
Monetary Policy." Mimeo, Department of Economics, University of
Michigan, Ann Arbor, 2007.
Engle, R. F., and C. W. J. Granger. "Cointegration and Error
Correction Representation: Estimation and Testing." Econometrica,
55, 1987, 251-76.
Engle, R. F., and F. Kroner. "Multivariate Simultaneous
Generalised ARCH." Econometric Theory, 11, 1995, 122-50.
Francis, B., M. Mougoue and V. Panchenko. "Is There a
Symmetric Nonlinear Causal Relationship between Large and Small
Firms?" Journal of Empirical Finance, 17, 2010, 23-28.
Furlong, F. "Commodity Prices as a Guide for Monetary
Policy." Federal Reserve Bank of San Francisco Economic Review, 1,
1989, 21-38.
Gospodinov, N., and S. Ng. Forthcoming. "Commodity Prices,
Convenience Yields and Inflation." Review of Economics Statistics,
2012.
Granger, C. W. J. "Investigating Causal Relations by
Econometric Models and Cross-Spectral Methods." Econometrica, 37,
1969, 424-38.
Gregory, A. W., and B. E. Hansen. "Residual-Based Tests for
Cointegration in Models with Regime Shifts." Journal of
Econometrics, 70, 1996, 99-126.
Hamilton, J. D. "What Is an Oil Shock?" Journal of
Econometrics, 113, 2003, 363-98.
Hiemstra, C., and J. D. Jones. "Testing for Linear and
Nonlinear Granger Causality in the Stock PriceVolume Relation."
Journal of Finance, 49, 1994, 1639-64.
Kejriwal, M., and P. Perron. "Testing for Multiple Structural
Changes in Cointegrated Regression Models." Journal of Business and
Economic Statistics, 28, 2010, 503-22.
Kilian, L. "A Comparison of the Effects of Exogenous Oil
Supply Shocks on Output and Inflation in the G7 Countries." Journal
of the Eurapean Economic Association, 6, 2008a, 78-121.
--. "Exogenous Oil Supply Shocks: How Big Are They and How
Much Do They Matter for the U.S. Economy?" Review of Economics and
Statistics, 90, 2008b, 216-40.
Leduc, S., and K. Sill. "A Quantitative Analysis of Oil-Price
Shocks, Systematic Monetary Policy, and Economic Downturns."
Journal of Monetary Economics, 51, 2004, 781-808.
Lee, J., and M. C. Strazicich. "Minimum Lagrange Multiplier
Unit Root Test with Two Structural Breaks." Review of Economics and
Statistics, 85, 2003, 1082-89.
Lumsdaine, R., and D. Papell. "Multiple Trend Breaks and the
Unit-Root Hypothesis." Review of Economics and Statistics, 79,
1997, 212-18.
Perron, P. "The Great Crash, the Oil Price Shock, and the Unit
Root Hypothesis." Econometrica, 57, 1989, 1361-401.
--. "Dealing with Structural Breaks," in Palgrave
Handbook of Econometrics, Vol. 1: Econometric Theory, edited by K.
Patterson and T. C. Mills. New York: Palgrave Macmillan, 2006, 278-352.
Politis, D. N., and J. P. Romano. "The stationary
bootstrap", Journal of the American Statistical Association, 89,
1994, 1303-13.
Randles, R. H. "On Tests Applied to Residuals." Journal
of the American Statistical Association, 79, 1984, 349-54.
Ross, S. "Information and Volatility: The No-Arbitrage
Martingale Approach to Timing and Resolution Irrelevancy." Journal
of Finance, 44, 1989, 1-18.
Sanso, A., V. Arrado, and J. L. Carrion. "Testing for Change
in the Unconditional Variance of Financial Time Series." Revista de
Economia Financiera, 4, 2004, 32-53.
Toda, H. Y., and T. Yamamoto. "Statistical Inference in Vector
Autoregression with Possibly Integrated Processes." Journal of
Econometrics, 66, 1995, 225-50.
Whitt, J. A., Jr. "Commodity Prices and Monetary Policy."
Federal Reserve Bank of Atlanta Working Paper No. 88-8, 1988.
SUPPORTING INFORMATION
Additional Supporting Information may be found in the online
version of this article:
Table S1, Unit Root Tests Results with and without Structural
Breaks.
Table S2. Cointegration Tests Results with and without Structural
Breaks.
(1.) See Barsky and Kilian (2002, 2004), Bernanke, Gertler, and
Watson (1997), Clarida, Gall, and Gertler (2000), Leduc and Sill (2004),
and Kilian (2008a, 2008b) for interpretations of the evidence on the
effects of oil shocks.
(2.) Recent studies point to the usefulness of commodity prices as
leading indicators of inflation. Gospodinov and Ng (2012) find that
convenience yields constructed as principal components from commodity
price futures have economically and statistically important predictive
power for inflation. Edelstein (2007) show that commodity prices contain
predictive information not contained in the leading principal components
of a broad set of macroeconomic and financial variables. They also yield
inflation forecasts that improve the fit of a forward-looking Taylor
rule.
(3.) It should be highlighted that nonlinearities could also
potentially affect the inferences about linear causality. Thus, our
empirical approach provides a comprehensive and robust assessment of the
presence of both linear and nonlinear causal relationships between
commodity price changes and inflation following Francis, Mougoue, and
Panhchenko (2010).
(4.) Details of the various tests for stationarity and
cointegration are discussed in the Working Paper.
(5.) Results for the unit root and cointegration tests are
available in Tables S1 and S2 (Supporting Information).
(6.) For a complete and detailed derivation of the variance see the
appendix in Hiemstra and Jones (1994).
(7.) We also performed the cointegration tests with and without
structural breaks for the two sub-samples. The results indicate that
there is absence of a cointegrating relationship in any of the
sub-samples, much like the results for the overall sample.
(8.) For a formal description of the BDS test refer to Brock et al.
(1996).
(9.) Results for these robustness analyses can be obtained upon
request from the authors.
(10.) Prior to model specification, we performed the Bai and Perron
(1998, 2003) test to determine the presence of breaks in the conditional
mean. In addition, we used the test of Sanso. Arrado, and Carrion (2004)
to also detect breaks in the variance of the commodity price indices,
both metals and industrials, and inflation. The identified break dates
for both mean and variance tall outside of the Great Moderation period,
thus the BEKK model does not have to incorporate these breaks.
(11.) Results for the VAR estimates are available from the authors
upon request. We do not report it here for the sake of brevity. Instead,
we only report the results for linear Granger causality.
(12.) Expansion of the last term of Equation (7) yields
[[[[??].sub.11] max([[epsilon].sub.1t], 0) + [[??].sub.12]
max([[epsilon].sub.2t], 0)].sup.2] which implies a positive inflation
shock has an impact of [[??].sub.11] on the its own conditional variance
while a negative inflation shock has no impact.
(13.) We thank the referee for suggesting qualifications for the
implications of results on the usefulness of commodity price changes in
predicting inflation.
RENUKA MAHADEVAN and SANDY SUARDI *
* The authors gratefully acknowledge helpful comments from referees
but retain responsibility for any remaining errors.
Mahadevan: School of Economics, University of Queensland, Colin
Clark Building, QLD 4072, Australia. Phone +617 3365 6595, Fax +617 3365
7299, E-mail r.mahadevan@uq.edu.au
Suardi: School of Economics, La Trobe University, Donald Whitehead
Building, Room 307, VIC 3086, Australia. Phone +613 9479 2318, Fax +613
9479 1654, E-mail s.suardi@latrobe.edu.au
doi: 10.1111/j.1465-7295.2012.00503.x
TABLE I
Composition of Commodity Resource Bureau (CRB) Grouping Indices
Grouping Index Composition
CRB Metals Copper scrap, lead scrap, steel scrap,
tin, and zinc
CRB Textiles and Fibres Burlap, cotton, print cloth, and wool
tops
CRB Livestock and Products Hides, hogs, lard, steers. and tallow
CRB Fats and Oils Butter, cottonseed oil, lard, and tallow
CRB Raw Industrials Hides, tallow, copper scrap, lead scrap,
steel scrap, zinc, tin, burlap,
cotton, print cloth, wool tops, rosin,
and rubber
CRB Foodstuffs Hogs, steers, lard, butter, soybean oil,
cocoa, corn, Kansas City wheat,
Minneapolis wheat, and sugar
CRB Spot This is made up of 22 commodities
from two major subdivisions (Raw
Industrials, and Foodstuffs) and four
smaller groups (Metals, Textiles and
Fibres, Livestock and Products, and
Fats and Oils). Note that the
groupings are nonxmutually exclusive.
Note: For relative weights and other details concerning these series,
refer to the Commodity Research Bureau.
TABLE 2
Test Results for Linear Granger Causality
Between Change in Commodity Price Index
and Inflation
Ho: [DELTA]CP not Ho: INF not
[right arrow] INF [right arrow]
[DELTA]CP
Panel A: Whole Sample (1957:1-2007:12)
Composite index 12.9001 [0.0000] 1.3906 [0.2258]
Livestocks 3.6528 [0.0029] 1.3079 [0.25881
Fats 2.2776 [0.0455] 2.0288 [0.0728]
Food 4.3033 [0.0007] 0.9819 [0.4279]
Raw industries 12.7217 [0.0000] 2.5485 [0.0269]
Textiles 6.4812 [0.0000] 2.0813 [0.06611
Metals 10.2014 [0.0000] 2.6210 [0.0234]
Panel 13: Sample Prior to the Great Moderation
(1957:1-1984:12)
Composite index 11.9201 [0.00001 3.3558 [0.0057]
Livestocks 4.3909 [0.0007] 0.5876 [0.7095]
Fats 3.1298 [0.0089] 1.2361 [0.2918]
Food 5.1965 [0.0001] 1.7460 [0.1237]
Raw industries 12.3627 [0.0000] 4.7872 [0.0003]
Textiles 5.7025 [0.0000] 1.1605 [0.3284]
Metals 8.3715 [0.0000] 4.6415 [0.0004]
Panel C: Sample During the Great Moderation
(1985:1-2007:12)
Composite index 1.5719 [0.1683] 1.1240 [0.3479]
Livestocks 2.5568 [0.0279] 0.5044 [0.7728]
Fats 3.2646 [0.0071] 0.3007 [0.9122]
Food 2.3900 [0.0383] 0.7198 [0.6090]
Raw industries 2.2156 [0.0531] 1.3245 [0.2539]
Textiles 2.3120 [0.0444] 1.4812 [0.1957]
Metals 4.8890 [0.00021 1.3568 [0.2409]
Note: Figures in columns 2 and 3 are F-test statistic for
the respective null hypothesis. Figures reported in [ ] are
p values. A VAR lag length of five is estimated for all cases
based on the AIC.
TABLE 3
Test Results for Nonlinear Granger Causality Between Change
in Commodity Price Index and Inflation
Panel A: 1957:1-2007:12
Whole Sample
Ho: INF not
Ho: [DELTA] CP not [right arrow]
[L.sub.[DELTA]CP] = [right arrow] INF [DELTA]CP
[L.sub.INF] p value p value
Composite
1 0.08 0.10
2 0.18 0.13
3 0.16 0.20
4 0.09 0.25
5 0.14 0.28
Livestocks
1 0.15 0.13
2 0.23 0.14
3 0.14 0.15
4 0.12 0.18
5 0.13 0.14
Fats
1 0.15 0.32
2 0.16 0.18
3 0.21 0.30
4 0.17 0.42
5 0.25 0.19
Food
1 0.14 0.35
2 0.18 0.16
3 0.17 0.28
4 0.25 0.45
5 0.36 0.38
Raw industries
1 0.04 0.10
2 0.06 0.15
3 0.15 0.17
4 0.08 0.15
5 0.38 0.08
Textiles
1 0.05 0.08
2 0.12 0.16
3 0.18 0.21
4 0.11 0.28
5 0.19 0.25
Metals
1 0.05 0.15
2 0.06 0.13
3 0.05 0.16
4 0.08 0.18
5 0.42 0.21
Panel B: 1957:1-1984:12
Prior to Great Moderation
Ho: INF not
Ho: [DELTA] CP not [right arrow]
[L.sub.[DELTA]CP] = [right arrow] INF [DELTA]CP
[L.sub.INF] p value p value
Composite ACP
1 0.15 0.11
2 0.21 0.13
3 0.20 0.10
4 0.10 0.56
5 0.14 0.45
Livestocks
1 0.72 0.52
2 0.53 0.41
3 0.28 0.53
4 0.17 0.31
5 0.13 0.28
Fats
1 0.15 0.48
2 0.44 0.53
3 0.75 0.45
4 0.76 0.55
5 0.81 0.32
Food
1 0.25 0.25
2 0.14 0.32
3 0.23 0.30
4 0.16 0.35
5 0.11 0.15
Raw industries
1 0.51 0.55
2 0.48 0.48
3 0.56 0.53
4 0.44 0.51
5 0.60 0.45
Textiles
1 0.20 0.07
2 0.11 0.11
3 0.13 0.15
4 0.15 0.21
5 0.20 0.18
Metals
1 0.10 0.28
2 0.13 0.19
3 0.15 0.16
4 0.11 0.17
5 0.38 0.25
Panel C: 1985:1-2007:12
During Great Moderation
Ho: INF not
Ho: [DELTA] CP not [right arrow]
[L.sub.[DELTA]CP] = [right arrow] INF [DELTA]CP
[L.sub.INF] p value p value
Composite
1 0.25 0.30
2 0.34 0.75
3 0.45 0.89
4 0.30 0.53
5 0.38 0.25
Livestocks
1 0.53 0.60
2 0.10 0.32
3 0.22 0.55
4 0.37 0.53
5 0.15 0.40
Fats
1 0.78 0.73
2 0.29 0.45
3 0.27 0.58
4 0.25 0.65
5 0.20 0.47
Food
1 0.85 0.55
2 0.63 0.44
3 0.40 0.75
4 0.16 0.58
5 0.15 0.92
Raw industries
1 0.01 0.58
2 0.02 0.43
3 0.02 0.60
4 0.03 0.52
5 0.02 0.67
Textiles
1 0.14 0.83
2 0.12 0.77
3 0.15 0.55
4 0.18 0.40
5 0.21 0.43
Metals
1 0.01 0.32
2 0.02 0.21
3 0.01 0.09
4 0.01 0.10
5 0.01 0.18
Notes: This table reports parametric bootstrap p values
for the standard Back and Brock nonlinear Granger causality test.
The test is applied to the estimated VAR residuals. [L.sub.[DELTA]CP]
= [L.sub.INF] denotes the number of lags on the residuals series used
in the test. In all cases, the tests are applied to the unconditional
standardized residuals. The lead length, m, is set to unity and the
distance measure, d, is set to 1.5.
TABLE 4
Estimation Results of the VAR Asymmetric BEKK Model
Panel A C
Inflation
and metals [C.sub.11] 0.0005 (0.0002) **
[C.sub.21] 0.0153 (0.0074) **
[C.sub.22] 0.0003 (0.0000) **
Inflation
and raw [C.sub.11] 0.0003 (0.0001) **
industrials [C.sub.21] 0.0256 (0.0038) **
[C.sub.22] 0.0002 (0.0000) **
Panel A A
Inflation
and metals [a.sub.11] 0.2411 (0.0773) **
[a.sub.12] 0.4510 (0.0247) **
[a.sub.21] -0.0083 (0.0068)
[a.sub.22] 0.1711 (0.0826) *
Inflation
and raw [a.sub.11] -0.1886 (0.0692) **
industrials [a.sub.12] 0.2769 (0.0843) **
[a.sub.21] 0.0043 (0.0065)
[a.sub.22] -0.4284 (0.1038) **
Panel A B
Inflation
and metals [b.sub.11] 0.9328 (0.0330) **
[b.sub.12] 0.3849 (0.5308)
[b.sub.21] 0.0043 (0.0073)
[b.sub.22] -0.8674 (0.1048) **
Inflation
and raw [b.sub.11] 0.9438 (0.0221) **
industrials [b.sub.12] 0.0130 (0.0047) **
[b.sub.21] 0.7481 (0.8254)
[b.sub.22] -0.2479 (0.1012) **
Panel A D
Inflation
and metals [d.sub.11] 0.3764 (0.1445) **
[d.sub.12] 0.0849 (0.1445)
[d.sub.21] 0.0076 (0.0054)
[d.sub.22] -0.3787 (0.0752) **
Inflation
and raw [d.sub.11] 0.3707 (0.1488) **
industrials [d.sub.12] 0.8158 (0.1970) **
[d.sub.21] -0.0229 (0.0196)
[d.sub.22] -0.4521 (0.3358)
Panel B
Linear Granger Inflation and
Causality Inflation and Metals Raw Industrials
Ho: [DELTA]CP not 7.4262 [0.0000] 2.4684 [0.03041
[right arrow] INF
Ho: INF not [right 1.7594 [0.1218] 1.8250 [0.1083]
arrow] [DELTA]CP
Ho: no GARCH effects 4790.1468 [0.0000] 4284.4615 [0.0000]
Ho: no spillover 14.0880 [0.0000] 5.7841 [0.0000]
effects
Ho: no asymmetric 19.6404 [0.0000] 2.5371 [0.0405]
GARCH effects
Serial [[epsilon] [[epsilon]
correlation .sub.lt] .sub.lt]
[square root [square root
of [h.sub.1t] of [h.sub.2t]
Q(5) 1.994 [0.8499] 2.346 [0.7995]
[Q.sup.2](5) 1.527 [0.9099] 3.216 [0.6636]
Serial [[epsilon [[epsilon
correlation .sub.lt] .sub.lt]
[square root [square root
of [h.sub.1t] of [h.sub.2t]
Q(5) 1.691 10.8901] 2.370 [0.7960]
[Q.sup.2](5) 2.784 [0.7332] 2.830 [0.7263]
Panel C
Nonlinear Granger Inflation and Metals
Causality
[L.sub.[DELTA]CP] = Ho: Ho: INF not
[L.sub.INF] [DELTA]CP [right
not [right arrow]
arrow] INF [DELTA]CP
1 0.18 0.45
2 0.23 0.38
3 0.15 0.21
4 0.16 0.18
5 0.20 0.29
Inflation and
Nonlinear Granger Raw Industrials
Causality
[L.sub.[DELTA]CP] = Ho: Ho: INF not
[L.sub.INF] [DELTA]CP [right
not [right arrow]
arrow] INF [DELTA]CP
1 0.12 0.68
2 0.20 0.53
3 0.16 0.82
4 0.28 0.63
5 0.43 0.70
Notes: Figures in [] and () are p values and robust standard errors,
respectively. [[epsilon].sub.lt] [square root of [h.sub.1t] and
[[epsilon].sub.lt] [square root of [h.sub.2t] are the standardized
residuals of inflation and changes in commodity price index,
respectively. Coefficient estimates of matrices C, A, B, and D follow
the BEKK variance equation (13).
* and ** denote statistical significance at the 5% and 1% levels,
respectively.
