This paper employs the unrestricted extended constant conditional correlation GARCH specification proposed in Conrad and Karanasos (2008) to examine the intertemporal relationship between the uncertainties of inflation and output growth in the US. We find that inflation uncertainty effects output variability positively, while output variability has a negative effect on inflation uncertainty.Bivariate GARCH process, negative volatility feedback, inflation uncertainty, output variability

Christian Conrad

Menelaos Karanasos

English

Research Papers in Economics

 
 
 
 
University of Heidelberg   
 
 
 
Department of Economics  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Discussion Paper Series    No. 475 
Modeling Volatility Spillovers between the 
Variabilities of US Inflation and Output:  
the UECCC GARCH Model¤ 
Christian Conrad and
Menelaos Karanasos
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
September 2008 Modeling Volatility Spillovers between
the Variabilities of US In°ation and Output:
the UECCC GARCH Model¤
Christian Conrady y and Menelaos Karanasosz z
yUniversity of Heidelberg, Heidelberg, Germany
zBrunel University, West London, UK
September 2008
Abstract
This paper employs the unrestricted extended constant conditional correlation
GARCH speci¯cation proposed in Conrad and Karanasos (2008) to examine the
intertemporal relationship between the uncertainties of in°ation and output growth
in the US. We ¯nd that in°ation uncertainty e®ects output variability positively,
while output variability has a negative e®ect on in°ation uncertainty.
Keywords: Bivariate GARCH process, negative volatility feedback, in°ation
uncertainty, output variability.
JEL Classi¯cation: C32, C51, E31.
¤We would like to thank James Davidson, Marika Karanassou and Ruey Tsay for their valuable
suggestions. We have also bene¯ted from the comments received from participants at the Seminars held
at Brunel University, University of Zurich, and University of Exeter.
yCorresponding author: Christian Conrad, University of Heidelberg, Alfred-Weber-Institute, Graben-
gasse 14, 69117 Heidelberg, Germany; email: christian.conrad@awi.uni-heidelberg.de, tel:
+49(0)6221543173, fax: +49(0)6221543578.
zMenelaos Karanasos, Economics and Finance, Brunel University, Uxbridge, West London, UB3 3PH,
UK; email: menelaos.karanasos@brunel.ac.uk, tel: +44(0)1895265284, fax: +44(0)1895269770.
11 Introduction
During the last decade researchers have employed various bivariate GARCH-in-mean mod-
els to investigate the relationship between the uncertainties of in°ation and output growth
and/or to examine their impact on the levels of in°ation and growth (see, for example,
Grier et al., 2004).1
The two most commonly used speci¯cations are the diagonal constant conditional
correlation (DCCC) model (see, for example Grier and Perry, 2000, Fountas et al., 2006,
and Fountas and Karanasos, 2007) and the BEKK representation (see, for example, Lee,
1999, and Grier and Grier, 2006). However, these two speci¯cations are characterized
by rather restrictive assumptions regarding potential volatility spillovers. At the one
extreme, the former assumes that there is no link between the two uncertainties, whereas,
near the other extreme, the latter only allows for a positive variance relationship.
In sharp contrast, several economic theories predict either a positive or a negative
association between the variabilities of in°ation and growth (for more details and a review
of the literature, see, Karanasos and Kim, 2005). Obviously, the extent to which there is
an interaction of either sign between the two variances is an issue that cannot be resolved
on merely theoretical grounds. These considerations reinforce a widespread awareness of
the need for more empirical evidence, but also make clear that a good empirical framework
is lacking.
In this paper we employ the unrestricted extended CCC (UECCC) GARCH model to
examine how the US nominal and real uncertainties are interrelated. This speci¯cation,
de¯ned in Conrad and Karanasos (2008), allows for feedback e®ects between the two
volatilities that can be of either sign, i.e. positive or negative.2 More speci¯cally, Conrad
and Karanasos (2008) derive necessary and su±cient conditions which ensure the positive
de¯niteness of the conditional covariance matrix even in the case of negative volatility
feedback. While negative values of the GARCH coe±cients have commonly been thought
of as resulting either from sampling error or model misspeci¯cation, they show that this
is not necessarily the case. Interestingly, negative volatility spillovers may be in line with
economic theory.
This is the ¯rst paper to apply this °exible bivariate formulation to investigating the
relation between the variabilities of US in°ation and output. We ¯nd strong evidence
supporting the Logue and Sweeney (1981) theory that in°ation uncertainty has a positive
impact on the volatility of growth. In sharp contrast, real variability a®ects nominal
uncertainty negatively as predicted by, among others, Fuhrer (1997). Clearly, this negative
e®ect could not have been detected by applying the restrictive DCCC or BEKK GARCH
speci¯cations.
1We will use the terms variance, variability, uncertainty and volatility interchangeably in the remainder
of the text.
2The speci¯cation is termed `unrestricted extended' because it can be viewed as an unrestricted ver-
sion of the extended CCC (ECCC) speci¯cation of Jeantheau (1998) which allows for positive volatility
feedback only.
22 The Bivariate GARCH Model
We use a bivariate model to simultaneously estimate the conditional means, variances,
and covariances of in°ation and output growth. Let Yt = (¼t yt)0 represent the 2 £ 1
vector with the in°ation rate and real output growth. The symbols ¯ and ^ denote the
Hadamard product and the elementwise exponentiation respectively. Further, Ft¡1 =
¾(Yt¡1;Yt¡2;:::) is the ¯ltration generated by the information available up through time
t ¡ 1.
De¯ne the residual vector as "t = ("¼;t "y;t)0 = zt ¯ h
^1=2
t , where the stochastic vector
zt = (z¼;t zy;t)0 is independent and identically distributed (i:i:d:) with mean zero, ¯nite
second moments, and 2 £ 2 correlation matrix R = [½ij]i;j=¼;y with diagonal elements
equal to one and o®-diagonal elements absolutely less than one. ht = (h¼;t hy;t)0 denotes
a vector of Ft¡1 measurable conditional variances.
We estimate the following bivariate AR(p)-GARCH(1;1)-in-mean model
Yt = ¡0 +
p X
l=1
¡lYt¡l +
s X
r=0
¢rht¡r + "t; (1)
where I is the 2 £ 2 identity matrix, ¡0 = [°i]i=¼;y, ¡l = [°
(l)
ij ]i;j=¼;y and ¢r = [±
(r)
ij ]i;j=¼;y.
We assume that the roots of
¯
¯I ¡
Pp
l=1 ¡lLl¯
¯ lie outside the unit circle. Note that our
speci¯cation allows the conditional variances to e®ect the level variables with some time
delay r.
Following Conrad and Karanasos (2008), we impose the UECCC GARCH(1,1) struc-
ture on the conditional variances:
ht = ! + A"
^2
t¡1 + Bht¡1; (2)
where ! = [!i]i=¼;y, A = [aij]i;j=¼;y and B = [bij]i;j=¼;y.3
Finally, we assume that the above model is minimal in the sense of Jeantheau (1998,
De¯nition 3.3) and invertible (see Assumption 2 in Conrad and Karanasos, 2008). The
invertibility condition implies that the inverse roots of jI ¡ BLj, denoted by Á1 and Á2,
lie inside the unit circle.
Conrad and Karanasos (2008) show that the following four conditions are necessary
and su±cient for ht > 0 for all t: (i) (1¡byy)!¼ +b¼y!y > 0 and (1¡b¼¼)!y +by¼!¼ > 0,
(ii) Á1 is real and Á1 > jÁ2j, (iii) A ¸ 0 and (iv) [B¡max(Á2;0)I]A > 0. These constraints
do not place any a priori restrictions on the signs of the coe±cients in the B matrix. In
particular, this implies that potential negative volatility spillovers are allowed.4
3 Economic Theory
Various economic theories make predictions about the link between the volatilities of in°a-
tion and output on the one hand and the e®ects of these uncertainties on the levels of the
3This speci¯cation nests the DCCC model when A and B are diagonal matrices and Jeantheau's
(1998) ECCC model when aij ¸ 0 and bij ¸ 0.
4Nakatani and TerÄ asvirta (2008) derive a similar set of conditions for the more restricted case that
the two diagonal elements of the GARCH matrix are positive.
3Table 1: Economic Hypothesis.
Panel A: volatility feedback + ¡
b¼y Devereux (1989) Fuhrer (1997)
by¼ Logue-Sweeney (1981) Fuhrer (1997)
Panel B: in-mean e®ects + ¡
±
(r)
¼¼ Cukierman and Meltzer (1986) Holland (1995)
±
(r)
¼y Devereux (1989) Taylor (1979)
±
(r)
y¼ Dotsey and Sarte (2000) Friedman (1977)
±
(r)
yy Black (1987) Pindyck (1991)
Note: Standard business cycle models imply that there is no link between the level of output and
output variability and, hence, imply ±
(r)
y¼ = 0.
respective variables on the other hand. For example, according to Fuhrer (1997) there ex-
ists a trade-o® between nominal uncertainty and real variability. However, in Devereux's
(1989) model higher output variability is associated with more in°ation uncertainty. Sim-
ilarly, models with a stable in°ation-unemployment trade-o® imply a positive relationship
between nominal and real uncertainty (see, Logue and Sweeney, 1981). We refrain from
discussing the various economic theories in detail and instead present in Table 1, Panel
A, a summary of the signs implied by the respective theories about volatility interactions
for the parameters of our empirical model. Table 1, Panel B, includes the economic theo-
ries regarding the e®ects of the uncertainties on the level variables. A detailed survey of
the theories is provided, e.g., in Karanasos and Kim (2005) and Fountas and Karanasos
(2007).
4 Empirical Results
Monthly data, obtained from Datastream, are used to provide a reasonable number of
observations. The in°ation and output growth series are calculated as the monthly dif-
ference in the natural log of the Consumer Price Index and Industrial Production Index,
respectively. The data range from 1960:01 to 2007:12 and, hence, comprise 576 usable
observations. Applying various unit root tests to both series, we came to the conclusion
that in°ation as well as output growth can be treated as stationary variables.
Within the bivariate UECCC GARCH-in-mean framework we will analyze the dy-
namic adjustments of the conditional variances of US in°ation and output growth, as well
as the implications of these dynamics for the direction of causality between the two un-
certainties. Parameter estimates were obtained by quasi maximum likelihood estimation
(QMLE). To check for the robustness of our estimates we used a range of starting values
and, hence, ensured that the estimation procedure converged to a global maximum. The
best model was chosen on the basis of Likelihood Ratio (LR) tests and three alternative
information criteria. For reasons of brevity, we refrain from presenting the estimation
results for the autoregressive parameters; instead, in Table 2 we concentrate on the main
4parameters of interest.
As we are most interested in the potential spillover e®ects between the two volatilities,
we ¯rst discuss the implications of equation (5) in Table 2.5 First, since by¼ is positive and
signi¯cant there is strong evidence that nominal uncertainty has a positive impact on real
volatility, as predicted by Logue and Sweeney (1981). Second, the negative and signi¯cant
value of b¼y indicates that output variability a®ects the uncertainty of in°ation negatively
and provides support for the Fuhrer (1997) theory.6 Most importantly, although one o®-
diagonal GARCH coe±cient is negative the necessary and su±cient conditions for the
positive de¯niteness of the conditional covariance matrix are satis¯ed.
Table 2: Parameter Estimates for the AR-UECCC-GARCH(1;1)-in-mean model.
¼t = ::: + 0:130
(0:043)
h¼;t¡1 ¡ 0:006
(0:004)
hy;t + "¼;t (3)
yt = ::: ¡ 0:137
(0:081)
h¼;t¡1 + 0:017
(0:006)
hy;t¡3 + "y;t (4)
ht =
0
@
0:3816
(0:119)
5:3901
(3:098)
1
A +
0
@
0:0684
(0:013)
0:0064
(0:003)
¡ 0:2945
(0:051)
1
A"
^2
t¡1 +
0
@
0:9145
(0:028)
¡0:0077
(0:004)
1:0250
(0:345)
0:5259
(0:058)
1
Aht¡1; (5)
with ½ = ¡0:0396 (0:0449).
Notes: This table reports parameter estimates of the bivariate AR-UECCC-GARCH(1;1)-in-
mean model for the US in°ation (¼t) and output (yt) data. h¼;t and hy;t denote the conditional
variances of in°ation and output, respectively. The numbers in parenthesis are robust standard
errors.
Finally, the parameter estimates in equation (3) in Table 2 show that higher nominal
uncertainty leads to higher in°ation rates as suggested by Cukierman and Meltzer (1986),
while increasing output volatility appears to lower the average in°ation rate (the so-called
Taylor e®ect). The results in equation (4) support the Friedman (1977) hypothesis that
increasing in°ation uncertainty has a negative e®ect on output growth.7 Moreover, higher
real variability appears to increase output growth as predicted by Black (1987). Note that
in three out of the four cases the e®ects from the uncertainties to the levels arise with
some time delay (insigni¯cant contemporaneous parameters are not presented), which is
5At ¯rst sight, the di®erence in the size of the estimated b¼y and by¼ coe±cients is surprising. However,
these di®erences can be explained by the fact that the variance in output is much higher than in in°ation.
In particular, the descriptive analysis of the two series shows that the variance of output is about ¯ve
times higher than that of in°ation.
6Both results are in line with the ¯ndings in Karanasos and Kim (2005) for the US. However, in
contrast to our one-step estimation approach, Karanasos and Kim (2005) employed an ine±cient two-
step strategy.
7Both cross e®ects, i.e., the coe±cients ±
(0)
¼y and ±
(1)
y¼ are signi¯cant at the 10% level.
5to be expected when working with monthly data.8
In order to check for the robustness of our results and to control for possible policy
changes we split our sample into the sub-periods 1960-1979 and 1980-2007. While our
conclusions regarding the link between the variabilities of in°ation and output remained
unchanged, it appears that the e®ects from nominal and real uncertainty on output growth
are stronger in the second sub-period.9
5 Conclusions
This is the ¯rst paper which employs the UECCC GARCH model to investigate the
in°ation-growth uncertainty link using US data. The main advantage of this new spec-
i¯cation is that it allows for volatility feedback of either sign, i.e., positive or negative.
Thus, we are able to test economic theories which suggest a trade-o® between the vari-
abilities of in°ation and output. Our results show that real volatility a®ects nominal
uncertainty negatively, as predicted by Fuhrer (1997). In sharp contrast, we ¯nd strong
evidence supporting the Logue and Sweeney (1981) theory that in°ation uncertainty has
a positive impact on the volatility of growth.
References
[1] Black, F., 1987, Business Cycles and Equilibrium. (Basil Blackwell, New York).
[2] Conrad, C. and M. Karanasos, 2008, Negative volatility spillovers in the unrestricted
ECCC-GARCH model. KOF Working Papers No 189.
[3] Cukierman, A. and A. Meltzer, 1986, A theory of ambiguity, credibility, and in°ation
under discretion and asymmetric information. Econometrica 54, 1099-1128.
[4] Devereux, M, 1989, A positive theory of in°ation and in°ation variance. Economic
Inquiry 27, 105-116.
[5] Dotsey, M. and P. Sarte, 2000, In°ation uncertainty and growth in a cash-in-advance
economy. Journal of Monetary Economics 45, 631-655.
[6] Fountas, S. and M. Karanasos, 2007, In°ation, output growth, and nominal and
real uncertainty: empirical evidence for the G7. Journal of International Money and
Finance 26, 229-250.
[7] Fountas, S., Karanasos, M. and J. Kim, 2006, In°ation uncertainty, output growth
uncertainty, and macroeconomic performance. Oxford Bulletin of Economics and
Statistics 68, 319-343.
8In the previous studies which employed GARCH-in-mean models the uncertainties were restricted to
a®ecting the levels contemporaneously, often resulting in insigni¯cant parameter estimates.
9For reasons of brevity, the estimation results for the sub-periods are not presented.
6[8] Friedman, M., 1977, Nobel lecture: in°ation and unemployment. Journal of Political
Economy 85, 451-472.
[9] Fuhrer, J., 1997, In°ation/output variance trade-o®s and optimal monetary policy.
Journal of Money, Credit, and Banking 29, 214{234.
[10] Grier, P. and K. B. Grier, 2006, On the real e®ects of in°ation and in°ation uncer-
tainty in Mexico. Journal of Development Economics 80, 478-500.
[11] Grier, K., Henry, O., Olekalns, N. and K. Shields, 2004, The asymmetric e®ects
of uncertainty on in°ation and output growth. Journal of Applied Econometrics 5,
551-565.
[12] Grier, K. and M. Perry, 2000, The e®ects of real and nominal uncertainty on in°ation
and output growth: some GARCH-M evidence. Journal of Applied Econometrics 15,
45-58.
[13] Holland, S., 1995, In°ation and uncertainty: tests for temporal ordering. Journal of
Money, Credit, and Banking 27, 827-837.
[14] Jeantheau, T., 1998, Strong consistency of estimators for multivariate ARCH models.
Econometric Theory 14, 70-86.
[15] Karanasos, M. and J. Kim, 2005, On the existence or absence of a variance relation-
ship: a study of macroeconomic uncertainty. WSEAS Transactions on Computers 4,
1475-1482.
[16] Lee, J., 1999, The in°ation and output variability trade-o®: evidence from a GARCH
model. Economics Letters 62, 63-67.
[17] Logue, D. and R. Sweeney, 1981, In°ation and real growth: some empirical results.
Journal of Money, Credit, and Banking 13, 497-501.
[18] Nakatani, T. and T. TerÄ asvirta, 2008, Positivity constraints on the conditional vari-
ances in the family of conditional correlation GARCH models. Finance Research
Letters 5, 88-95.
[19] Pindyck, R., 1991, Irreversibility, uncertainty, and investment. Journal of Economic
Literature 29, 1110-1148.
[20] Taylor, J., 1979, Estimation and control of a macroeconomic model with rational
expectations. Econometrica 47, 1267-1286.
7

A positive theory of in°ation and in°ation variance.

A theory of ambiguity, credibility, and in°ation under discretion and asymmetric information.

Business Cycles and Equilibrium.

Estimation and control of a macroeconomic model with rational expectations.

In°ation and real growth: some empirical results.

In°ation and uncertainty: tests for temporal ordering.

In°ation uncertainty and growth in a cash-in-advance economy.

In°ation uncertainty, output growth uncertainty, and macroeconomic performance.

In°ation, output growth, and nominal and real uncertainty: empirical evidence for the G7.

In°ation/output variance trade-o®s and optimal monetary policy.

Irreversibility, uncertainty, and investment.

Negative volatility spillovers in the unrestricted ECCC-GARCH model. KOF Working Papers No 189.

On the existence or absence of a variance relationship: a study of macroeconomic uncertainty.

On the real e®ects of in°ation and in°ation uncertainty in Mexico.

Positivity constraints on the conditional variances in the family of conditional correlation GARCH models.

Strong consistency of estimators for multivariate ARCH models.

The asymmetric e®ects of uncertainty on in°ation and output growth.

The e®ects of real and nominal uncertainty on in°ation and output growth: some GARCH-M evidence.

The in°ation and output variability trade-o®: evidence from a GARCH model.

Modeling Volatility Spillovers between the Variabilities of US Inflation and Output: the UECCC GARCH Model

http://www.uni-heidelberg.de/md/awi/forschung/dp475.pdf

Modeling Volatility Spillovers between the Variabilities of US Inflation and Output: the UECCC GARCH Model

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics