This paper investigates comovements between the United States and Australia. Our nonlinear model allows the dynamic response to shocks to differ if countries are in recession. Generalised Impulse Response Functions highlight a significant asymmetric response to positive and negative shocks.Current Depth of Recession Models, Generalised Impulse Response Functions, Asymmetries

Kalvinder Shields

Nilss Olekalns

Olan T. Henry

English

Research Papers in Economics

Non-linear Co-Movements in Output Growth: Evidence from
the United States and Australia.
by
´ Olan T. Henry∗ , Nilss Olekalns and Kalvinder Shields
The University of Melbourne
Abstract
This paper investigates comovements between the United States and Australia. Our non-
linear model allows the dynamic response to shocks to diﬀer if countries are in recession.
Generalised Impulse Response Functions highlight a signiﬁcant asymmetric response to
positive and negative shocks.
Keywords: Current Depth of Recession Models, Generalised Impulse Response Functions.
Asymmetries
J.E.L. Reference Numbers: E0, N1
∗Corresponding author: Department of Economics, The University of Melbourne, Victoria 3010, Aus-
tralia. Tel: + 613 8344-5312, Fax: +613 8344-6899. Email: oth@unimelb.edu.au
11I n t r o d u c t i o n
This paper examines the linkage between the rate of economic growth in the United States
and the growth rate in Australia. The tendency for countries’ outputs to co-move posi-
tively with the output of the United States is well known. Backus, Kehoe and Kydland
(1992), for example, document correlation coeﬃcients for a number of countries’ respective
outputs with that of the United States; with only one exception, these correlation coeﬃ-
cients are positive. Similar ﬁndings have been reported by other researchers (for example
Gregory, Head and Raynauld 1997; and Ambler, Cardia and Zimmermann 1999).
This previous literature has derived measures of co-movement on the assumption that
the relationship between United States’ growth and growth in other countries is linear.
An exception is the paper by Henry and Summers (2000) who document the existence of
a threshold non-linearity in the relationship between Australian and US economic growth.
In their paper, the size and sign of shocks inﬂuence the nature of the dynamic adjustments
that follow.
Our approach allows for non-linearity in the comovement between the United States
and Australia. In particular, the dynamic response to a shock may diﬀer if one, or
both, economies are in recession. Beaudry and Koop (1993), in the context of a single
country’s economic growth path, refer to this as a “bounce-back” eﬀect, where the rate
of growth accelerates in the aftermath of a recent recession. To date, the question of
whether recessions impact on the international co-movement of output growth has not
been investigated.
2 The Modelling Framework
Our starting point is the standard, linear, Vector Autoregressive Moving Average (VARMA)
representation of output growth. This is given by,
Θ(L)∆yt = µ + Φ(L)εt (2.1)
where ∆yt is a vector of output growths (assumed to be stationary) so that ∆yt =( ∆yUS,t
∆yAUS,t)0, Θ(L)a n dΦ(L)a r e2×2 matrix polynomials in the lag operator L,w i t hﬁxed
2 × 2c o e ﬃcient matrices Θk and Φk,w i t hΘ(0) = Φ and (0) = I2. µ =( µUS,µ AUS)
0 is
a vector of constants to capture any drift in growth and εt =( εUS,t,ε AUS,t)
0 is a vector
of i.i.d. error terms with a 2 × 2 variance-covariance matrix Σ = σij,i , j= US,AUS.
Assuming that stability and invertibility conditions hold, such a representation is general
enough to directly accommodate the interdependencies in the determination of ∆yUS,t
and ∆yAUS,t through the lag ﬁlters Θ(L)a n dΦ(L) and indirectly through the covariance
[1]matrix Σ. Therefore the modelling framework is potentially able to incorporate the eﬀects
of shocks which inﬂuence both output growth in the US and output growth in Australia
directly, and the feedbacks from output growth in the US to output growth in Australia,
and vice versa. We introduce non-linearity by allowing for regime changes are governed by
observable variables. This is achieved by augmenting (2.1) with a measure of the “current
depth of recession” (CDR). The CDR term can be written as,
CDRi,t =m a x{yi,t−s}
t
s=0 − yi,t. (2.2)
where i indexes the country. The CDR is deﬁn e da st h eg a pb e t w e e nt h ec u r r e n tl e v e l
of output and its historical maximum. That is, CDR will take non-zero values when
output dips below its trend value due to a negative shock. The asymmetry implied by
the CDR term is reﬂected in the “bounce-back” eﬀect, the tendency for output growth to
recover relatively strongly following a recent recession. Hence, the CDR approach treats
the historical maximum level of output as an attractor which inﬂuences the dynamics
of output growth when output falls below its previous peak. Beaudry and Koop (1993)
hypothesise that there is a non-linearity in this “peak reversion”; the further output falls
from its peak, the greater is the pressure that builds up for output to return to its historical
maximum. As a result, the speed at which output recovers varies according to the severity
of the recession.
Adding CDR terms to (2.1) implies that the conditional expectation of future output
is inﬂuenced by whether the current level of output is above, below or at its historical
maximum. Further, these eﬀects can impact across countries so that, for example, US
output growth being above, below or at its historical maximum, may directly or indirectly,
aﬀect Australian output growth (and vice versa).
With the introduction of the CDR term, equation (2.1) becomes
Θ(L)∆yt = µ + Ξ(L − 1)CDRt + Φ(L)εt (2.3)
where CDRt =( CDRUS,t,CDR AUS,t)
0 and Ξ is a 2 × 2 matrix of lag polynomials with
ﬁxed 2×2c o e ﬃcient matrices Ξk. This parameterisation for ∆yt nests the VARMA model
(2.1) while capturing the possibility of asymmetric responses to positive and negative
shocks to growth in both economies.
Other approaches to capture regime switching behaviour, such as Markov, or Thresh-
old switching models, have been employed in the literature. Our approach has a major
advantage over such models. Tests of the null of linearity based on (2.3) are not subject
to unidentiﬁed parameters under the null (the Davies 1987 problem) and, consequently,
asymptotic inference is valid.
[2]To investigate the eﬀect of shocks on the future values of output growth in either
the linear VARMA model in (2.1), and the non-linear VARMA-CDR model in (2.3), we
use Generalised Impulse Response Functions (GIRFs) introduced by Koop et al (1996).
Whereas the VARMA speciﬁcation easily admits analytical solutions for the impulse re-
sponses, simulation methods must be used to trace the impact of a shock to the elements
of the state vector, or to the entire system, for the non-linear VARMA-CDR case.
If Yt is a random vector, the GIRF for a speciﬁcs h o c kυt and history ωt−1 is deﬁned
as,
GIRFY(n,υt,ω t−1)=E [Yt+n|υt,ω t−1] − E [Yt+n|ωt−1], (2.4)
for n =0 ,1,2,.... Hence, the GIRF is conditioned on υt and ωt−1 and constructs the
response by averaging out future shocks given the past and present. Given this, a natural
reference point for the impulse response function is the conditional expectation of Yt+n
given only the history ωt−1, and, in this benchmark response, the current shock is also
averaged out. Assuming that υt and ωt−1 are realisations of the random variables Vt
and Ωt−1 that generate realisations of {Yt}, then, following Koop et al (1996), the GIRF
deﬁned in (2.4) can be considered to be a realisation of a random variable given by,
GIRFY(n,Vt,Ωt−1)=E [Yt+n|Vt,Ωt−1] − E [Yt+n|Ωt−1]. (2.5)
The computation of the GIRF for the linear VARMA model in equation (2.1) is relatively
straightforward, where the distribution of the GIRF is given by GIRFY(n,Vt,Ωt−1) ∼
N(0,ΨnΣΨ0
n) for the case where Vt ∼ N(0,Σ). Ψn are 2 × 2 matrices of coeﬃcients
assumed to be square-summable, where
Ψ(L)=Θ(L)
−1 Φ(L)=
∞ X
n=0
ΨnL
n, (2.6)
Ψ0 = I2, and the (i,j)th element of Ψ(L) is denoted by lag polynomials ψi,j(L). The GIRF
is characterised by the variance-covariance matrix ΨnΣΨ0
n, for n =0 ,1,2....1. If we scale
the GIRF by a unit shock, (deﬁned by one standard error (
√
σii)), then the eﬀect of a
unit shock to the ith equation in the model on the jth variable at time horizon n is given
by the jth element of ΨnΣe0
i/
√
σii, or equivalently, as ejΨnΣe0
i/
√
σii.e i (ej)i sa1× 2
selection vector, with unity in the ith (jt h )e l e m e n ta n dz e r oi nt h eo t h e re l e m e n ta n dΣ
is the 2×2 variance-covariance matrix of εt with individual elements σij,i ,j= US,AUS.
The computation of GIRFs for non-linear models is less straightforward since analytical
expressions for the conditional expectations in the GIRFs are not usually obtainable.
1See also Lee and Pesaran (1993) who propose a scaled version of this measure, namely, ‘persistence
proﬁles’, and applied by Pesaran and Shin (1995) and Lee and Shields (2000).
[3]Monte Carlo methods of stochastic simulation need to be used to compute the conditional
expectations, see Granger and Ter¨ asvirta (1993, Ch. 8), and Koop et al (1996) for detailed
descriptions of the various methods that can be used.
One advantage of the VARMA-CDR model is that there is no assumption of symmetric
output response to positive and negative shocks. GIRFs can be used to measure the extent
to which negative shocks are more persistent than positive shocks as well as to assess the
potential diversity in the dynamics in the eﬀects of positive and negative shocks on output.
Let GIRFY(n,V
+
t ,Ωt−1) denote the GIRF from conditioning on the set of all possible
positive shocks, where V
+
t = {υt|υt > 0} and GIRFY(n,−V
+
t ,Ωt−1)b et h eG I R Ff r o m
conditioning on the set of all possible negative shocks. These GIRFs will be referred to
‘asymmetric’ GIRFs. Following van Dijk et al (2000), the distribution of the random
asymmetry measure given by,
ASYY(n,V
+
t ,Ωt−1)=GIRFY(n,V
+
t ,Ωt−1)+GIRFY(n,−V
+
t ,Ωt−1), (2.7)
will be insigniﬁcantly diﬀerent from zero if positive and negative shocks have the same
eﬀect, i.e. a positive unit shock will have exactly the same eﬀect as a negative unit shock.
The distribution of this measure can provide an indication of the asymmetric eﬀects of
positive and negative shocks.
3 Modelling US and Australian Output Growth
We employ data on real GDP for the USA and Australia, sampled at a quarterly frequency
over the period 1959 III to 2001 I. Given the non-stationary nature of the GDP series
and our focus on the transmission of growth shocks we transform the data into diﬀerence
stationary growth rates as:
∆yi,t =l o g( Yi,t/Yi,t−1)( 3 . 8 )
where ∆ represents the ﬁrst diﬀerence operator and Yi,t represents the level of output in
period t for i = US, AUS. Figure 1 presents time series plots of the data. It is not possible
to reject the null of a unit root in the logarithm of GDP for either country at the 5% level
(ADFAUS = −1.9829,A D F US = −3.1359, critical value = −3.4381). However the re-
spective growth rates are clearly stationary (ADFAUS = −11.7027,A D F US = −6.0528).2
Table 1 displays the parameter estimates of the unrestricted VARMA and the VARMA-
CDR models. On the basis of the Swartz (1978) information criterion a two lag VAR was
chosen for both models. The models pass the usual diagnostic tests for serial correlation
2Using the Johansen approach there was no evidence of cointegration between the logarithms of GDP.
The results are available upon request.
[4]and heteroscedasticity3. As the VAR-CDR nests the VAR we perform a LR test for the
joint exclusion of the CDR terms. The results of the LR test (LR =1 1 .5438, p − value =0.0211)
suggest that the VAR-CDR provides a superior characterisation of the data to the linear
VAR.
O nt h eb a s i so fas y s t e mb a s e dl i k e l i h o o dr a t i ot e s tw ew e r ea b l et oe x c l u d et h e
Australian CDR variable (LR =3 .7553, p−value 0.1530) but unable to exclude the corre-
sponding US variable (LR =7 .9201 , p−value = 0.0191). We were further able to exclude
lags of Australian growth from the US equation of the VAR-CDR model.
Taken in isolation the parameter estimates themselves are uninteresting, save for the
sign and size of the CDR terms in the US and AUS equations. The positive sign of
CDRUS,t−1 i nt h eU Se q u a t i o ni sc o n s i s t e n tw i t hab o u n c e - b a c ke ﬀect. In contrast the
negative sign of CDRUS,t−1 in the Australian equation is consistent with the widely held
view that “when the US sneezes, Australia catches the ’ﬂu”.
4 Generalised Impulse Responses
Figure 2 presents the cumulative generalised impulse response functions. We report GIRFs
from both the estimated VAR and VAR-CDR models for comparative purposes. We report
the GIRFs from the best ﬁtting VAR to guage the economic importance of allowing for
non-linearity.
The upper panel of Figure 2 highlights the dynamic time proﬁle for Australian output
of a shock that causes Australian growth to rise by one standard error on impact and
suggests that the implied dynamic response to a domestic shock to Australian output is
largely comparable across the models over the short and long-run, despite the presence of
the US CDR term in the Australian equation. This similarity occurs becuse the US CDR
term never turns on in response to an Australian shock.
The middle panel of Figure 2 plots the response of Australian growth to an external
shock, in this case a shock that causes US growth to rise by one standard error on im-
pact. The short-run dynamics of the impulse response diﬀer markedly across models; for
instance, the response function from the VAR model predicts that there will be a short
and medium term increase in the level of output. In comparison, the VAR-CDR model
predicts that there will be more volatility in the periods immediately after the shock (up
to 6 quarters). Unlike the VAR which predicts a uniformly positive reaction, the response
of the VAR-CDR becomes negative after roughly four quarters following the shock. As
the time horizon increases, the dynamics of the responses fall into line with each other.
3Results of the diagnostic tests are available upon request.
[5]However, the diﬀerence between the responses remains markedly distinct, indicating the
importance of the role of the US CDR term.
The impact of a one standard deviation domestic shock to US growth is displayed
in the bottom panel of Figure 2. Here the VAR predicts a more persistent response to
the shock than is implied by the VAR-CDR. This is consistent with a bounce-back eﬀect
following a negative shock.
Are the dynamic responses for the VAR-CDR model to positive and negative shocks
signiﬁcantly diﬀerent from one another? The linear VAR explicitly assumes that the sign
of a shock is of no importance. The asymmetry measures suggest the following. First,
there is a statistically signiﬁcant diﬀerence in the Australian response to positive and
negative domestic and overseas shocks. The asymmetry measure for domestic shocks is
small in magnitude but the measure itself, -0.0021, with a t-ratio of -15.2139, indicates
signiﬁcance at any level of conﬁdence. The Australian response to an overseas shock is
-0.00138, with a t-ratio of 1.6635 This is signiﬁcant at the 10% level. The asymmetry
measure for the US response to US shocks proved insigniﬁcant.
5 Concluding Comments
This paper compares the predicted responses from a linear and non-linear multivariate
model of Australian and US economic growth. The linear model is rejected on the basis
of a series of LR tests. Further exclusion restrictions suggest that Australian growth
responds to lagged US growth and a measure of the depth of the US recession. The
simulation experiments suggest that Australia responds signiﬁcantly diﬀerently to positive
and negative domestic and overseas shocks, although the magnitude of this response is
small in an economic sense.
6A c k n o w l e d g e m e n t s
We thank the University of Melbourne for funding this research. We are grateful to Peter
M. Summers, Don Harding, James D. Hamilton and delegates at the 2002 Australasian
Meeting of the Econometric Society for comments and suggestions on earlier drafts of this
letter. The usual disclaimer applies to any errors or omissions.
References
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cles: What are the Facts?” Center for Research on Economic Fluctuations and
[6]Employment,U n i v e r s i t ´ ed uQ u ´ ebec ` aM o n t r ´ eal, Working Paper No. 90.
Backus, D.K., P.J. Kehoe and F.E. Kydland. (1992). “International Real Business
Cycles”, Journal of Political Economy, 100(4), August, 745-775.
Beaudry, P. and G. Koop. (1993) “Do Recessions Permanently Change Output?”
Journal of Monetary Economics, 31, 149-63.
Davies, R.B. (1987) “Hypothesis Testing when a Nuisance Parameter is Present only
under the Alternative,” Biometrika, 74, 33-43.
Granger, C. and T. Ter¨ asvirta (1993), “Modelling Nonlinear Dynamic Relation-
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Gregory, A.W., A.C. Head, and J. Raynauld. (1997) “Measuring World Business
Cycles”, International Economic Review, 38(3), August, 677-701.
Koop, G., M.H Pesaran and S.M Potter. (1996), “Impulse Response Analysis in
Non-Linear Multivariate Models”, Journal of Econometrics, 74, 119-147.
Henry, ´ O.T. and N. Olekalns. (2002) “The Eﬀect of Recessions on the Relationship
Between Output Variability and Growth”. Southern Economic Journal,6 8 ( 3 ) ,6 8 3 -
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Henry, ´ O.T. and P.M. Summers (2000) “Australian Economic Growth: Nonlineari-
ties and International Inﬂuences”, The Economic Record, 76, 235, 355-365.
Lee, K.C. and M.H. Pesaran. (1993) “Persistence Proﬁles and Business Cycle Fluc-
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[7]Tables and Figures
Table 1: Parameter Estimates - VAR and VAR-CDR Models
VAR Model Restricted VAR-CDR Model
∆yUS,t ∆yAUS,t ∆yUS,t ∆yAUS,t
µ 0.0047 (0.0011) 0.0079 (0.0017) 0.0033 (0.0014) 0.0119 (0.0025)
∆yUS,t−1 0.2499 (0.0778) 0.1212 (0.1204) 0.3046 (0.0875) -0.0225 (0.1337)
∆yUS,t−2 0.1444 (0.0768) 0.4215 (0.1189) 0.2201 (0.0863) 0.2827 (0.1313)
∆yAUS,t−1 0.0857 (0.0479) -0.1089 (0.0742) * -0.1350 (0.0738)
∆yAUS,t−2 -0.0529 (0.0476) -0.2368 (0.0737) * -0.2594 (0.0730)
CDRUS,t−1 * * 0.2033 (0.1325) -0.4763 (0.2052)
CDRAUS,t−1 * * * *
LR (χ2(2)) * * 7.760 [0.95] 2.945 [0.91]
Notes: Standard Errors are in parentheses and p-values are in square brackets.
[8]AUS GDP Growth
1959 1962 1965 1968 1971 1974 1977 1980 1983 1986 1989 1992 1995 1998 2001
-0.04
-0.03
-0.02
-0.01
0.00
0.01
0.02
0.03
0.04
0.05
US GDP Growth
1959 1962 1965 1968 1971 1974 1977 1980 1983 1986 1989 1992 1995 1998 2001
-0.03
-0.02
-0.01
0.00
0.01
0.02
0.03
0.04
Figure 1: The growth rate data
[9]Australia: Cumulative Response to a Domestic Shock
0123456789 1 0 1 1 1 2 1 3 1 4
0.00
0.16
0.32
0.48
0.64
0.80
0.96
1.12
cdr
var
Australia: Cumulative Response to an External Shock
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
cdr
var
U.S.A:  Cumulative Response to a Domestic Shock
0123456789 1 0 1 1 1 2 1 3 1 4
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
cdr
var
Figure 2: The Generalised Impulse Responses.
[10]

Australian Economic Growth:

Cointegration and the Speed of Convergence to

Do Recessions Permanently Change Output?”

Expectations Formation and Business Cycle Fluctuations; An Empirical Analysis of Actual and Expected Output

Hypothesis Testing when a Nuisance Parameter is Present only under the Alternative,”

Impulse Response Analysis in Non-Linear Multivariate Models”,

International Business Cycles: What are the Facts?” Center for Research on Economic Fluctuations and [6]Employment,U n i v e r s i t ´ ed uQ u ´ ebec ` aM o n t r ´

Measuring World Business Cycles”,

Modelling Nonlinear Dynamic Relationships”.

Persistence Proﬁles and Business Cycle Fluctuations in a Disaggregated

The Eﬀect of Recessions on the Relationship Between Output Variability and Growth”.

Non-linear Co-Movements in Output Growth: Evidence from the United States and Australia

http://www.economics.unimelb.edu.au/downloads/wpapers-02/857.pdf

Non-linear Co-Movements in Output Growth: Evidence from the United States and Australia

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