Asymmetric volatility of real GDP: some evidence
from Canada, Japan, the United Kingdom
and the United States
Kin-Yip Ho, Albert K.C. Tsui*
Department of Economics, National University of Singapore, 10 Kent Ridge Crescent,
Singapore 119260, Singapore
Received 6 December 2001; received in revised form 21 January 2003; accepted 11 February 2003
Abstract
The recent empirical investigation of conditional volatility in real GDP growth rates of Japan, the
United Kingdom, and the United States by Hamori [Jpn. World Econ. 12 (2000) 143] finds no
evidence of asymmetry. This paper re-visits the issue of asymmetric volatility using a similar
approach with some modifications. We find statistically significant evidence of asymmetric volatility
in the real growth rates of the United States and Canada. As such, it may be premature to conclude
that business cycle indicators generally do not exhibit volatility asymmetry.
# 2003 Elsevier B.V. All rights reserved.
JEL classification: E32; E37
Keywords: EGARCH model; Real growth rates; Asymmetric volatility
1. Introduction
In a previous issue of Japan and the World Economy, Hamori (2000) analyses the
conditional volatility of the real growth rates of Japan, the United Kingdom and the United
States. Among others, he applies the exponential generalised autoregressive (AR) conditional heteroskedasticity (EGARCH) model to the real GDP growth rates of these three
OECD countries and finds no evidence of volatility asymmetry.
We re-visit the issue of asymmetric volatility using enlarged data sets and more general
time series filters for the conditional mean equation. We find evidence of asymmetric
volatility in the real GDP growth rates of the United States and Canada. However, like
Japan and the World Economy
15 (2003) 437–445
*Corresponding author. Tel.: þ65-6874-3952; fax: þ65-6775-2646.
E-mail address: [email protected] (A.K.C. Tsui).
0922-1425/$ – see front matter # 2003 Elsevier B.V. All rights reserved.
doi:10.1016/S0922-1425(03)00026-4
those findings of Hamori (2000), we do not detect asymmetric GARCH effects in the
growth rates of Japan and the United Kingdom. These mixed empirical results imply that
asymmetric conditional volatility is possible in real GDP growth rates.
The rest of the paper is divided as follows. In Section 2 we provide a brief exposition of
the econometric methodology. Section 3 discusses the data sets used in this study. Section 4
reports the empirical results and the battery of diagnostic tests employed to evaluate model
adequacy. Finally, Section 5 concludes.
2. Methodology
Following Nelson (1991), Hamori (2000) applies the EGARCH (1, 1) model to the real
GDP growth rates of Japan, the United Kingdom and the United States, assuming the
conditional normal distribution. More specifically, the conditional mean and variance
equations employed by Hamori (2000) are reproduced as follows:
mean equation : yt ¼ p0 þX
k
i¼1
yt#i þ et; k ¼ 2; 4 (1)
variance equation : log s2
t ¼ o þ a
et#1
st#1
!
!
!
!
!
!
!
!
þ g
et#1
st#1
þ b log s2
t#1 (2)
where yt is the quarterly real growth rate (in percentage) compounded on discrete time
basis:
yt ¼ Yt # Yt#1
Yt#1
$ 100 (3)
where Yt is the seasonally adjusted quarterly real GDP. Essentially, Hamori’s methodology
involves the following steps. First, obtain yt and confirm the stationarity of the series
using the Augmented Dickey–Fuller (ADF) test. Next, estimate the mean and variance
equations using the maximum likelihood estimation procedure available in EViews
version 3.1 (Quantitative Micro Software). The lag order of the autoregressive process
in the mean Eq. (1) is set to either 2 periods (half a year) and/or 4 periods (1 year)
to examine the robustness of the empirical findings. Third, estimate the standard errors
of the model parameters using the heteroskedastic-consistent variance–covariance
matrix advocated by Bollerslev and Wooldridge (1992). Finally, look for evidence of
asymmetric conditional volatility by assessing the estimated value of g. If it differs
significantly from 0, then asymmetric volatility is regarded as present in the growth rate
of real GDP.
We would like to re-visit the issue of volatility asymmetry using a similar approach
with some modifications. First, we compute the quarterly growth rates on a continuously
compounding basis using the first difference of the natural logarithm of the real GDP
figures:
rt ¼ log Yt
Yt#1
” # $ 100 (4)
438 K.-Y. Ho, A.K.C. Tsui / Japan and the World Economy 15 (2003) 437–445
where rt is the real GDP growth rate (in percentage) and Yt is the seasonally adjusted
quarterly real GDP series. Second, like Hamori (2000), we use the standard Augmented
Dickey–Fuller test to check the stationarity of the real GDP growth series. Third, we apply
a more general Box and Jenkins ARMA (p, q )-type time series filter for the conditional
mean equation. The lag orders (p, q ) of the autoregressive (AR) and moving average (MA)
parts are selected by the Schwarz Bayesian information criterion (SBIC). SBIC is preferred
to the Akaike information criterion (AIC) because it is asymptotically consistent and
corrects the over-fitting nature of the AIC. Following Nelson (1991), we specify the
ARMA (p, q )–EGARCH (1, 1) model as below:
mean equation : rt ¼ p0 þX
p
i¼1
pirt#1 þ et þX
q
j¼1
yjet#j (5)
variance equation : log s2
t ¼ o þ a
et#1
st#1
!
!
!
!
!
!
!
! #
ffiffiffi
2
p
! r
þ g
et#1
st#1
þ b log s2
t#1 (6)
The effect of asymmetric volatility is captured by g when it takes values significantly
different from zero. Particularly when g < 0, it implies that negative shocks generate
higher volatility than positive shocks of the same magnitude, and vice versa. We apply the
model to the growth rates of Canada, Japan, the United Kingdom and the Unites States. All
parameters of the conditional mean and variance equations (5) and (6) are estimated using
Bollerslev and Wooldridge’s (1992) quasi-maximum likelihood estimation (QMLE)
technique, assuming normally distributed errors. Finally, we report a battery of diagnostic
tests to ascertain the adequacy of the selected model. These tests include the standard runs
test, Ljung–Box Q-statistic (see Ljung and Box, 1978), the Mcleod–Li test (see McLeod
and Li, 1983), the BDS test (see Brock et al., 1996), and the ARCH LM test (see Engle,
1982), respectively.
3. Data
Our data sets start from 1961:Q1 to 1997:Q4, yielding a total of 148 quarterly
observations for each of the four OECD countries. They are extracted from the OECD
website OECD Data Online: Quarterly National Accounts. Appendix A reports details of
the data sources by country. In addition, the validity of data sets used in this paper is verified
against various issues of OECD’s monthly bulletin entitled Main Economic Indicators. In
order to ensure the consistency of our data sets, we have, wherever feasible, compared our
real GDP figures with those reported by the International Monetary Fund in International
Financial Statistics CD-ROM. We choose to start from 1961 because the quarterly real
GDP figures for Canada are available only from that year onwards. Figs. 1–4 display the
plots of growth rates of the four OECD countries.
Panel A of Table 1 provides a summary of the descriptive statistics of the data. It can be
seen that except for the United States, all the growth rates are positively skewed. In
addition, all the series except for Canada are highly leptokurtic. The mean and the standard
deviation of the growth rates for Japan are the highest among the four OECD countries.
K.-Y. Ho, A.K.C. Tsui / Japan and the World Economy 15 (2003) 437–445 439
This is consistent with Hamori (2000). In Panel B, the Jarque–Bera test statistics for Japan,
the UK and the US are highly significant at the 5 percent level, suggesting that the growth
rates are not normally distributed. It may be due to the fat-tailed nature of the distribution.
Particularly in the case of the United Kingdom, the kurtosis is about two times that of a
standard normal distribution. Turning to Panel C, the BDS test statistics unequivocally reject
the hypothesis that all the real GDP growth rate series are independently and identically
distributed (IID) at the 1 percent significance level. It seems that the departure from IID
could be ascribed to the presence of conditional heteroskedasticity in all the series.
4. Results and discussions
Table 2 tabulates the ADF test results for all the real GDP growth rate series. It can be
observed that all ADF test statistics are statistically significant at the 1 percent level,
thereby indicating that all the series are stationary. In addition, the Ljung–Box Q-statistics
and Breusch–Godfrey test statistics are insignificant at the 5 percent level. This implies that
the residuals obtained from the ADF test equations are approximately white noise.
Figs. 1–4. Quarterly real GDP growth rates (1961:Q2 to 1997:Q4).
440 K.-Y. Ho, A.K.C. Tsui / Japan and the World Economy 15 (2003) 437–445
Panel A of Table 3 displays the estimation results for various ARMA (p, q )–EGARCH
(1, 1) models. Based on the Schwarz Bayesian information criterion, the ARMA (1, 2) time
series filter is adequate for the growth rates of Canada, AR (3) filter for Japan, a constant
filter for the United Kingdom, and the ARMA (1, 1) filter for the United States,
respectively. As can be observed, Canada and the US exhibit significant asymmetric
volatility in the GDP growth rates. For Canada, the estimated value of the coefficient
Table 1
Summary statistics of real GDP growth rates
Canada Japan United Kingdom United States
Panel A: moments, minimum, and maximum
Mean 0.9229 1.2588 0.5921 0.8514
Median 0.9428 1.2105 0.5636 0.8264
Maximum 3.3417 5.5411 4.8376 3.7804
Minimum #1.3932 #3.4964 #2.5291 #2.0598
Standard deviation 0.9535 1.1690 1.0650 0.8985
Skewness 0.1144 0.0983 0.5442 #0.2325
Kurtosis 2.9190 5.3758 6.0328 3.9827
Observations 148 148 148 148
Panel B: Jarque–Bera test for normality
Jarque–Bera 0.3610 34.8094** 63.5937** 7.2390*
Panel C: BDS testa
e ¼ 3, l ¼ 1.0 4.9228** 5.2653** 3.5429** 4.9574**
e ¼ 5, l ¼ 1.0 6.7901** 6.7899** 5.6128** 6.7644**
e ¼ 3, l ¼ 1.5 5.1040** 3.4007** 2.7287** 4.2637**
e ¼ 5, l ¼ 1.5 5.9200** 4.6110** 4.3066** 5.1861**
a For the BDS test, e represents the embedding dimension whereas l represents the distance between pairs of
consecutive observations, measured as a multiple of the standard deviation of the series. Under the IID
assumption, the BDS test statistic is asymptotically distributed as standard normal. * Significance at 5 percent level.
** Significance at 1 percent level.
Table 2
Augmented Dickey–Fuller test results
Country ADF model Test statistic Diagnostic checking
Q-statistica BG testb
Canada Case 3 (7) #4.4443** 16.1940 11.2688
Japan Case 3 (5) #4.4641** 20.1010 14.2776
United Kingdom Case 3 (11) #4.0914** 19.2020 13.7437
United States Case 3 (14) #4.0471** 7.3695 13.8824
Note: Case 3 refers to the equation with both the intercept and the deterministic time trend. Figures in
parenthesis are the number of lagged difference terms. a The Ljung–Box Q-statistic with 20 lags.
b The Breusch–Godfrey test with a lag order of 8. ** Significance at 1 percent level.
K.-Y. Ho, A.K.C. Tsui / Japan and the World Economy 15 (2003) 437–445 441
Table 3
Estimation results of ARMA (p, q )–EGARCH (1, 1) models
Canada Japan United Kingdom United States
Panel A: parameters of mean and variance equations
Model ARMA (1, 2) AR (3) Constant ARMA (1, 1)
p0 0.0197 (0.0907) 0.4145 (0.2415) 0.7247 (0.0614) 0.2067 (0.1182)
p1 0.9677 (0.0095) 0.1190 (0.0482) – 0.7491 (0.1000)
p2 – 0.3165 (0.0872) – –
p3 – 0.2045 (0.0814) – –
y1 #0.6562 (0.0793) – – #0.5052 (0.1288)
y2 #0.3184 (0.0736) –––
o #0.6653 (0.1152) 0.4395 (0.3409) #1.1057 (0.1500) #0.5553 (0.1071)
a 0.3601 (0.1136) #0.3921 (0.2933) 0.6978 (0.2067) 0.3273 (0.1165)
g #0.2482 (0.0648) 0.1941 (0.1534) #0.2384 (0.1720) #0.1788 (0.0815)
b 0.8488 (0.0690) #0.4147 (0.5247) 0.8346 (0.0913) 0.9164 (0.0417)
Panel B: Schwarz Bayesian criteriona
SBC 4.8556 5.0445 5.1591 4.7426
Panel C: summary statistics of standardized residuals and diagnostic tests
Mean #0.0486 0.0009 #0.0834 0.0033
Median #0.0856 0.0273 #0.1556 #0.0436
Maximum 1.9295 3.0984 3.6967 3.3205
Minimum #2.2044 #3.6054 #2.9322 #3.2092
Standard deviation 1.0022 1.0040 0.9990 1.0026
Skewness #0.0464 #0.1864 0.5799 0.0974
Kurtosis 2.2478 4.5176 5.3361 3.4353
Jarque–Bera test 3.4940 14.6535** 41.6672** 1.3836
Q-statistic (20)b 23.0610 28.4300 32.3410* 26.0140
McLeod–Li test (20) 20.9130 14.6250 22.8820 22.3780
ARCH LM (4) 4.9734 1.9888 0.5064 2.7726
Panel D: BDS testc
e ¼ 3, l ¼ 1.5 #1.1889 0.7812 #0.6390 #0.9916
e ¼ 5, l ¼ 1.5 0.4254 1.0339 #0.3262 #0.6151
e ¼ 3, l ¼ 1.0 #1.7970 0.9293 #0.5039 #0.5100
e ¼ 5, l ¼ 1.0 #0.3342 1.1123 0.5575 #0.2309
Panel E: runs testd
R1 #0.3122 #0.6668 #1.3520 0.5578
R2 0.6155 #2.2518* 1.1096 1.1317
R3 0.4375 #0.2247 #0.0166 1.6569
a The Schwarz Bayesian criterion is calculated based on the following formula given in Judge et al.
(1988): log(residual sum of squares) þ (number of parameters $ log(number of observations))/(number of
observations).
b The Q-statistic refers to the Ljung–Box Q-statistic with 20 degrees of freedom. c For the BDS test, e represents the embedding dimension whereas l represents the distance between pairs of
consecutive observations, measured as a multiple of the standard deviation of the series. The BDS test statistic is
asymptotically distributed as standard normal, under the IID assumption.
d Ri for i ¼ 1–3 denote the runs tests of the series Rt, |Rt| and Rt
2
, respectively. Under the null hypothesis that
successive observations are independent, the test statistic is asymptotically normally distributed. * Significance at the 5 percent level.
** Significance at the 1 percent level.
442 K.-Y. Ho, A.K.C. Tsui / Japan and the World Economy 15 (2003) 437–445
capturing the asymmetric effect (g) is #0.2482, which is significant at the 1 percent level;
whereas for the US, the estimated value of g is #0.1788, which is significant at the 5
percent level. Both cases indicate that negative shocks to the GDP growth rates induce
greater volatility than positive shocks of the same magnitude. We note in passing that our
estimation results are reasonably robust to the period chosen, the lag lengths of the ARMA
time series filter for the conditional mean equation, and the method of computing the real
GDP growth rates.
Panels C–E present the summary statistics of the standardized residuals of the fitted
models and the corresponding diagnostic checks. It can be seen that the computed
standardized residuals of Canada and the US from the fitted models pass all the diagnostic
tests. More specifically, the Ljung–Box Q-statistic, McLeod–Li test, the ARCH LM test
and the BDS test are all insignificant at the 5 percent level, thereby implying that the series
have been effectively filtered. Adequacy of the models is further corroborated by the runs
test and the Jarque–Bera test (see Jarque and Bera, 1987). Both tests demonstrate that the
null hypothesis of normality cannot be rejected. In addition, it can be observed that the real
GDP growth rates of both Canada and the US display rather high persistence to volatility
shocks. For example, the estimated values of b are 0.8488 and 0.9164 for Canada and the
US, respectively, and they are statistically significant at the 1 percent level. In the case of
the UK, the estimated value of g is #0.2384, which is insignificant even at the 10 percent
level. The standardized residuals do not pass the Jarque–Bera test and the Q-test. In fact,
they are highly leptokurtic, indicating a significant departure from normality. As for Japan,
the estimated value of g is 0.1941, and it is insignificant at the 10 percent level. These
findings are basically similar to those reported by Hamori (2000).
5. Concluding remarks
We have examined the issue of asymmetric volatility in the real GDP growth rates of
Canada, Japan, the United Kingdom and the United States. Based on the sample data sets
from 1961:Q1 to 1997:Q4, we find significant evidence of conditional volatility asymmetry
in the growth rates of Canada and the United States. As such, it is premature to conclude
that business cycle indicators generally do not exhibit volatility asymmetry.
What causes the asymmetric volatility in GDP growth rates is still unclear to researchers.
Among others, the heterogeneous expectations of market participants may be one of the
plausible explanations. As noted by Tse and Tsui (1997) in explaining the existence of
asymmetric conditional volatility of the Malaysian ringgit in the foreign exchange market,
the persistence and spill-over of exchange rate volatility could be caused by traders in the
foreign exchange market who may have heterogeneous expectations about the market
movements. A similar argument may be applied to the case of real GDP shocks. When
economic agents perceive negative GDP shocks, they may be inclined to curtail private
consumption and investment, which leads to a further contraction in the real GDP. The
uncertainty associated with deflationary shocks will be greater if economic agents have
heterogeneous beliefs about the future outlook of the economy. This greater sense of
uncertainty about the future may induce (risk-averse) economic agents to be even more
cautious about their consumption and investment decisions. On the other hand, when
K.-Y. Ho, A.K.C. Tsui / Japan and the World Economy 15 (2003) 437–445 443
economic agents perceive expansionary shocks, their desire to increase consumption and
investment expenditure is constrained by the potential productive capacity of the economy
in the short-run. As such, the supply-side constraints may partially account for the asymmetric volatility of the real GDP growth series in well-developed countries like Canada and
the United States.
What are the possible policy implications for asymmetric effects? A significant
conditional volatility asymmetry in the real GDP growth rates may further vindicate
the government’s role in stabilising the macroeconomic environment during recessions.
This is because the adverse impact of negative shocks would hopefully be mitigated, on
condition that the countercyclical measures are effective. In addition, the need for stronger
international policy co-ordination would be more imperative for countries experiencing
negative growth shocks. The reason is that the negative economic disturbances arising
from one country would affect another country fairly quickly through the international
transmission of business cycles. This in turn might generate an adverse impact on the future
volatilites of the real growth rates if countries do not co-operate speedily to ameliorate
the impact of such negative shocks. As such, further research should be conducted to
investigate the co-movements of conditional volatilities in real GDP growth rates across
countries.
Acknowledgements
We would like to thank the editor of this Journal, Ryuzo Sato and the anonymous
referees for their helpful comments and suggestions. Albert Tsui acknowledges the support
by the National University of Singapore academic research grant: R-122-000-059-112.
Appendix A. Data sources
This appendix provides details pertaining to the sources of our data sets, which are based
on the latest figures available at the time of writing. The data sets are available upon request
from the authors. All (seasonally adjusted) quarterly GDP figures are expressed in constant
prices at a chosen base year, which may differ among the economies. Readers interested in
the methodology employed in calculating the figures may consult OECD’s publication
entitled Main Economic Indicators—Sources and Definitions 1997.
The principal sources of our data sets are:
OECD. OECD Data Online: Quarterly National Accounts
<http://oecdnt.ingenta.com/oecd/selected_table.asp?TableId¼570&lang¼eng>
OECD. Main Economic Indicators. Various issues. Paris: OECD
US Bureau of Economic Analysis, Department of Commerce website
<http://www.bea.doc.gov/bea/dn/st-tabs.htm>
Canada
Variable: (seasonally adjusted) GDP at 1992 prices in millions of C$ (expressed at
annual rates)
444 K.-Y. Ho, A.K.C. Tsui / Japan and the World Economy 15 (2003) 437–445
Series identifier (if any): CAN.VNBARSA.1992.S1 — Gross Domestic Product
Source: OECD Data Online: Quarterly National Accounts
<http://oecdnt.ingenta.com/oecd/selected_table.asp?TableId¼570&lang¼eng>
Japan
Variable: (seasonally adjusted) GDP at 1990 prices in millions of Yen (expressed at
annual rates)
Series identifier (if any): JPN.VNBARSA.1990.S2 — Gross Domestic Product
Source: OECD Data Online: Quarterly National Accounts
<http://oecdnt.ingenta.com/oecd/selected_table.asp?TableId¼570&lang¼eng>
United Kingdom
Variable: (seasonally adjusted) GDP at 1995 prices in millions of £ (expressed at
quarterly rates)
Series identifier (if any): GBR.VNBQRSA.1995.S1 — Gross Domestic Product
Source: OECD Data Online: Quarterly National Accounts
<http://oecdnt.ingenta.com/oecd/selected_table.asp?TableId¼570&lang¼eng>
United States
Variable: (seasonally adjusted) GDP at 1996 prices in billions of US$ (expressed at
annual rates)
Series identifier (if any): nil
Source: US Bureau of Economic Analysis, Department of Commerce website
<http://www.bea.doc.gov/bea/dn/st-tabs.htm>
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