# Evidence for Nonlinear Asymmetric Causality in  US Inflation, Metal, and Stock Returns

Authors: D. Hristu-Varsakelis; C. Kyrtsou
Journal: Discrete Dynamics in Nature and Society
 (2008)

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## Abstract

The purpose of this paper is to propose a version of causality testing that focuses on
how the sign of the returns affects the causality results. We replace the traditional VAR
specification used in the Granger causality test by a discrete-time bivariate noisy Mackey
glass model. Our test reveals interesting and previously unexplored relationships in US
economic series, including inflation, metal, and stock returns.


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## Body

## 1. Introduction

Since the concept of Granger causality was first
proposed [1], there
has been a growing body of research devoted on the subject of causal
relationships between economic variables. Among the central goals of that
research are (i) the elucidation of the connections that exist between
different components of an economic system, with all its complexity and (ii)
the use of causality links (where detected) in designing efficient economic
policies. Granger's pioneering work on linear causality has been extended and
adapted to take into account nonlinear dependencies in the quantities studied,
as in [2, 3] and more recently [4, 5], among others.The contribution of this paper is to propose a
nonlinear version of Granger causality test, which conditions on samples of
the causing series being positive or negative. This “asymmetric” causality test,
which builds on [6],
is a way to “sharpen” the usual version of causality testing in which the
series being tested are considered in their entirety, all at once. Asymmetric
causality testing can reveal interesting information about the inherent
dynamics of the processes studied. For example, detection of a causality
relationship does not give information on whether a positive or negative shock
has significant predictive value. Conversely, lack of an apparent causality relationship
does not preclude the existence of causality when we condition for certain
features, such as a positive or negative sign.The remainder of this paper is organized as follows.
In  Section2, we describe a model to be used for studying causality
relationships, together with a Granger-type asymmetric causality test. We
replace the linear VAR specification used by Granger by a discrete-time, noisy
Mackey-Glass model, in order to detect nonlinear “links” between the
variables studied. The choice of model is not unique and was guided by the
recently investigated ability of the model to reproduce properties of real
economic data [7–9]. The deterministic Mackey-Glass equation has also
been used in the neural network literature [10, 11].
Section 3 discusses the application of our test in
real economic series which include data on US inflation, as well as metal and
stock returns.


## 2. A Test for Nonlinear Asymmetric Causality

The causality test proposed in this paper extends the
test for nonlinear Granger causality introduced in [6]. In its symmetric version,
it assumes an underlying process with a special type of nonlinear structure,
known as the bivariate noisy Mackey-Glass (hereafter “M-G”) model [7]. The model is as
follows:  (2.1)Xt=α11Xt−τ11+Xt−τ1c1−δ11Xt−1+α12Yt−τ21+Yt−τ2c2−δ12Yt−1+ϵtϵt∼N(0,1),Yt=α21Xt−τ11+Xt−τ1c1−δ21Xt−1−α22Yt−τ21+Yt−τ2c2−δ22Yt−1+utut∼N(0,1), where t=τ,…,N, τ=max⁡(τ1,τ2) and X0,…,Xτ−1,Y0,…,Yτ−1 are given. The αij,
and δij are parameters to be estimated, τi are integer delays, and ci are constants. The model (2.1) can produce
various types of dependencies by adjusting the parameters ci and τi.
The principal advantage of (2.1) over simple VAR alternatives is that the M-G
terms are able to capture more complex dependent dynamics in a time series. As
discussed in [7, 12], the presence of statistically significant M-G terms
in a pair of series suggests nonlinear feedback as the generating mechanism of
the interdependences between X and Y.The M-G-based causality test attempts to detect
whether past samples of a variableY have a significant nonlinear effect (of the
type Yt−τ2/(1+Yt−τ2c2)) on the current value of another variable X.
Operationally, the test is similar to the linear Granger causality test, except
that the models fitted to the series are M-G processes. One begins by
estimating the parameters of an M-G model that best fits the given series,
using ordinary least squares. To test for causality from Y to X,
a second M-G model is estimated, under the constraint α12=0.
The latter equation is our null
hypothesis. Let ϵ^t,⁢θ^t be the residuals produced by the unconstrained
and constrained best-fit M-G models, respectively. We compute and compare the
sums of squared residuals Sc=∑t=1Nθ^t and Su=∑t=1Nϵ^t.
Let nfree=4 be the number of free parameters in our M-G
model and nrestr=1 be the number of parameters set to zero when
estimating the constrained model. If the test statistic(2.2)SF=(Sc−Su)/nrestrSu/(N−nfree−1)⁢∼Fnrestr,N−nfree−1is greater than a specified
value, then we reject the null hypothesis that Y causes X
(recall that SF follows an F distribution under the null hypothesis.)
The p-value for the test is computed from(2.3)p=1−Fnrestr,N−nfree−1cdf(SF,nrestr,N−nfree−1),where Fa,bcdf is the cumulative distribution function for
the Fa,b distribution. The test requires prior
selection of the parameters of the M-G process, namely, τ1,⁢τ2,⁢c1,⁢c2.
The best delays τ1,⁢τ2,
are to be chosen on the basis of likelihood ratio tests and the Schwarz
criterion. We will refer to the above test as “symmetric” to distinguish it
from its specialized counterpart described next.Regardless of the causality relationships identified
in the pairX and Y,
we would like to know whether those relationships hold when conditioning for
positive or negative returns. For example, one may want to know whether
positive returns observed in one series give predictive information that is
more significant than that given by negative returns; alternatively, there may be
settings in which causality exists only in a subset of the observations, (e.g.,
for positive returns). To investigate such relationships, we consider an
“asymmetric” causality test that conditions on the causing series being
nonnegative or nonpositive. Briefly, the asymmetric test is based on the M-G
test described above, where the best M-G model is calculated using only those
observations that satisfy the appropriate condition on the returns. To make
matters concrete, suppose that in model (2.1) we would like to test whether
nonnegative returns in the series Y cause the series X.
An observation (Xi,Yi) is included for regression only if Y(t−τ2)≥0.
The same restricted set of observations is used to compute the model
corresponding to the null hypothesis (in this case α12=0). The procedure is then repeated, with the
order of the series reversed (to investigate whether positive returns in X cause Y) and again with the subset of observations
that correspond to nonpositive returns.In this work, we have chosen to condition on the sign
of the causing series because of its practical relevance. Of course,
nonpositivity (resp., nonnegativity) is by no means the only type of
conditioning possible, so that in addition to varying the choice of dynamics
fitted to the series, one can test for causality given various other events
(e.g., start/end of the week, price movement thresholds, etc.).


## 3. Results and Discussion

We applied our
test to two pairs of series. The first pair included US consumer price (CPI)
and metal price (MET) indices. The consumer price index data are from the BLS
CPI—all urban consumers CUUROOOOAA0 index, reported in Free Lunch. BLS
producer price index WPU10 for metals and metal products from the BLS website.
In the second pair, the metal price index was replaced by the Dow-Jones stock
price index (PSDJ). The Dow Jones Stock Prices
Index includes the NBER series #11009 representing 20 stocks until 1928,
and 30 stocks thereafter, ending on 12/1968. Beginning in 01/1969, it is
updated from the Dow Jones Index for Industrials: 30 Industries (index
1920 = 100, monthly end)—reported in Free Lunch.
The series contained monthly observations from 01/1960 to 07/2002 (511 obs.)
and were not seasonally adjusted. That metal price index was selected because
it is one of the commodity price series having a significant impact on
manufacturing activities, and hence inflation. Prior to causality testing, we
performed the Dickey-Fuller test [13, 14] to identify unit roots in all series. For each series
CPI, MET, and PSDJ, the test showed that a unit root was present, therefore the
log-differenced version of the series was used, giving us DLCPI, DLMET, and
DLPSDJ, respectively.


### 3.1. Series DLCPI, DLMET

Table1 shows
the results of our symmetric causality test on the series CPI and MET. The test
identified bidirectional causality, confirming the nonlinear feedback mechanism
proposed in [12, 15].
To find out
whether specifying the direction of series movement has a significant effect on
the causality relationships identified, we performed the asymmetric causality
test on the same series. The results are shown in Tables 2
and 3. The exponent
“p” (resp., “n”) indicates that only the positive (resp., negative) values
of a series were included. The only relationship that differs qualitatively
from what is identified in the symmetric test is the lack of M-G feedback
between DLCIPp and DLMET. Of course, we cannot exclude other possible forms of
nonlinear causality, however, based on the M-G structure adopted here, the
causality from DLCPI to DLMET is mainly due to the predictive information
contained in the negative DLCPI movements on the DLMET series. We also note
that, based on Table 1 only, one could assert that DLCPI causes DLMET. That
statement would be in doubt, however, in the event that CPI movement was
positive, as Tables 2 and
3 show. The fact that positive MET returns strongly
affect inflation suggests that one should consider more seriously the impact of
asymmetries in commodity price behavior. As highlighted in [15], the implications of such
relationships can be of profound interest if we take into account the fact that
traditional US policy relies heavily on monetary policy to control inflation
effects.Table 1
Test for nonlinear causality (symmetric case).
IfP>.05,
then at 5% we accept the null hypothesis that A does not cause B.
The parameters for the M-G model were τ1=3, τ2=1,
and c1=c2=2.
Relation(A→B)F-statisticProbabilityDLCPI→ DLMET4.57450.0329DLMET→ DLCPI3.92020.0483Table 2
Test for nonlinear causality (asymmetric case
for positive returns). IfP>.05,
then at 5% we accept the null hypothesis that A does not cause B.
The parameters for the M-G model were τ1=3, τ2=1,
and c1=c2=2.
Relation(A→B)F-statisticProbabilityDLCPIp→ DLMET2.63890.1049DLMETp→ DLCPI18.98061.6E-5Table 3
Test for nonlinear causality (asymmetric case
for negative returns). IfP>.05,
then at 5% we accept the null hypothesis that A does not cause B.
The parameters for the M-G model were τ1=3, τ2=1,
and c1=c2=2.
Relation(A→B)F-statisticProbabilityDLCPIn→ DLMET5.95870.0150DLMETn→ DLCPI4.14060.0424


### 3.2. Series DLCPI, DLPSDJ

Table4 shows
the results of our symmetric causality test on the series CPI and PSDJ. The
test did not identify M-G causality. When we conditioned on the causing series
being positive or negative, strong unidirectional M-G causality
from DLPSDJp to DLCPI and from DLCPIn to DLPSDJ was revealed
(Tables 5
and 6, resp.).Table 4
Test for nonlinear causality (symmetric case).
IfP>.05,
then at 5% we accept the null hypothesis that A does not cause B.
The M-G model parameters were τ1=τ2=1,
and c1=c2=2.
Relation(A→B)F-statisticProbabilityDLCPI→  DLPSDJ0.20140.6538DLPSDJ→  DLCPI0.30020.5840Table 5
Test for nonlinear causality (asymmetric case
for positive returns). IfP>.05,
then at 5% we accept the null hypothesis that A does not cause B.
The M-G model parameters were τ1=τ2=1,
and c1=c2=2.
Relation(A→B)F-statisticProbabilityDLCPIp→  DLPSDJ0.47800.4896DLPSDJp→  DLCPI19.75061.0853E-5Table 6
Test for nonlinear causality (asymmetric case
for negative returns). IfP>.05,
then at 5% we accept the null hypothesis that A does not cause B.
The M-G model parameters were  τ1=τ2=1,
and c1=c2=2.
Relation(A→B)F-statisticProbabilityDLCPIn→  DLPSDJ53.98458.1768E-13DLPSDJn→   DLCPI3.73900.0537Although the results of our test do not preclude the
existence of other models that “explain” the series interdependence, they do
confirm that the M-G model has significant predictive power when we take into
account positive values of PSDJ or negative values of DLCPI. Furthermore, the
coefficients in the best-fit M-G model indicate a positive relation between
DLPSDJp and DLCPI (meaning that the coefficients of both DLPSDJp terms in the DLCPI equation were positive).
This is in line with
the empirical findings in [16]
which identified a positive relationship between stock returns and inflation,
brought about by demand shocks (e.g., money supply fluctuations). The
coefficients of the DLCPIn terms in the DLPSDJ equation were both negative, indicating a negative
relation between DLCPIn and DLPSDJ. This agrees with the explanation advanced in [17] that inflation may
indirectly be responsible for negative fluctuations in stock returns via real
output movements, when supply shocks perturb the market.


## 3.1. Series DLCPI, DLMET

Table1 shows
the results of our symmetric causality test on the series CPI and MET. The test
identified bidirectional causality, confirming the nonlinear feedback mechanism
proposed in [12, 15].
To find out
whether specifying the direction of series movement has a significant effect on
the causality relationships identified, we performed the asymmetric causality
test on the same series. The results are shown in Tables 2
and 3. The exponent
“p” (resp., “n”) indicates that only the positive (resp., negative) values
of a series were included. The only relationship that differs qualitatively
from what is identified in the symmetric test is the lack of M-G feedback
between DLCIPp and DLMET. Of course, we cannot exclude other possible forms of
nonlinear causality, however, based on the M-G structure adopted here, the
causality from DLCPI to DLMET is mainly due to the predictive information
contained in the negative DLCPI movements on the DLMET series. We also note
that, based on Table 1 only, one could assert that DLCPI causes DLMET. That
statement would be in doubt, however, in the event that CPI movement was
positive, as Tables 2 and
3 show. The fact that positive MET returns strongly
affect inflation suggests that one should consider more seriously the impact of
asymmetries in commodity price behavior. As highlighted in [15], the implications of such
relationships can be of profound interest if we take into account the fact that
traditional US policy relies heavily on monetary policy to control inflation
effects.Table 1
Test for nonlinear causality (symmetric case).
IfP>.05,
then at 5% we accept the null hypothesis that A does not cause B.
The parameters for the M-G model were τ1=3, τ2=1,
and c1=c2=2.
Relation(A→B)F-statisticProbabilityDLCPI→ DLMET4.57450.0329DLMET→ DLCPI3.92020.0483Table 2
Test for nonlinear causality (asymmetric case
for positive returns). IfP>.05,
then at 5% we accept the null hypothesis that A does not cause B.
The parameters for the M-G model were τ1=3, τ2=1,
and c1=c2=2.
Relation(A→B)F-statisticProbabilityDLCPIp→ DLMET2.63890.1049DLMETp→ DLCPI18.98061.6E-5Table 3
Test for nonlinear causality (asymmetric case
for negative returns). IfP>.05,
then at 5% we accept the null hypothesis that A does not cause B.
The parameters for the M-G model were τ1=3, τ2=1,
and c1=c2=2.
Relation(A→B)F-statisticProbabilityDLCPIn→ DLMET5.95870.0150DLMETn→ DLCPI4.14060.0424


## 3.2. Series DLCPI, DLPSDJ

Table4 shows
the results of our symmetric causality test on the series CPI and PSDJ. The
test did not identify M-G causality. When we conditioned on the causing series
being positive or negative, strong unidirectional M-G causality
from DLPSDJp to DLCPI and from DLCPIn to DLPSDJ was revealed
(Tables 5
and 6, resp.).Table 4
Test for nonlinear causality (symmetric case).
IfP>.05,
then at 5% we accept the null hypothesis that A does not cause B.
The M-G model parameters were τ1=τ2=1,
and c1=c2=2.
Relation(A→B)F-statisticProbabilityDLCPI→  DLPSDJ0.20140.6538DLPSDJ→  DLCPI0.30020.5840Table 5
Test for nonlinear causality (asymmetric case
for positive returns). IfP>.05,
then at 5% we accept the null hypothesis that A does not cause B.
The M-G model parameters were τ1=τ2=1,
and c1=c2=2.
Relation(A→B)F-statisticProbabilityDLCPIp→  DLPSDJ0.47800.4896DLPSDJp→  DLCPI19.75061.0853E-5Table 6
Test for nonlinear causality (asymmetric case
for negative returns). IfP>.05,
then at 5% we accept the null hypothesis that A does not cause B.
The M-G model parameters were  τ1=τ2=1,
and c1=c2=2.
Relation(A→B)F-statisticProbabilityDLCPIn→  DLPSDJ53.98458.1768E-13DLPSDJn→   DLCPI3.73900.0537Although the results of our test do not preclude the
existence of other models that “explain” the series interdependence, they do
confirm that the M-G model has significant predictive power when we take into
account positive values of PSDJ or negative values of DLCPI. Furthermore, the
coefficients in the best-fit M-G model indicate a positive relation between
DLPSDJp and DLCPI (meaning that the coefficients of both DLPSDJp terms in the DLCPI equation were positive).
This is in line with
the empirical findings in [16]
which identified a positive relationship between stock returns and inflation,
brought about by demand shocks (e.g., money supply fluctuations). The
coefficients of the DLCPIn terms in the DLPSDJ equation were both negative, indicating a negative
relation between DLCPIn and DLPSDJ. This agrees with the explanation advanced in [17] that inflation may
indirectly be responsible for negative fluctuations in stock returns via real
output movements, when supply shocks perturb the market.


## 4. Conclusions

This paper
proposed an asymmetric version of a nonlinear (Mackey-Glass) causality test
introduced in [6].
The test attempts to detect causality relationships in pairs of economic
series, conditioning on the sign (negative or positive) of returns of the
causing series. We have demonstrated the use of our test in data concerning US
inflation, metal, and stock prices. For the relationship between inflation and
metal returns series, we have found that the bidirectional causality identified
is mainly due to positive movements of metal prices and negative movements of
inflation. For inflation and stock returns, we found strong evidence of
unidirectional causality, from positive stock returns and negative inflation.
Our test confirms existing economics-based explanations of the relationships
between the series studied. The application of conditioning to reveal
“asymmetric” causality is useful for (i) refining the origins of detected
causality and (ii) detecting nonlinear feedback structures when no such
structure is apparent in the original series. In this work, we used the M-G
process to expose an interesting relationship between the three variables
studied, however, one could consider other (non-M-G) dynamics, as well as
asymmetric versions of other causality tests, such as [2, 3, 18]. Ongoing work includes an
investigation of our test via Monte Carlo experiments, in order to explore its
properties with samples which are significantly smaller than the ones used in
this work.



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*Source: 138547-2008-05-22.xml*