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: ecsatsui@nus.edu.sg (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>

References

Bollerslev, T., Wooldridge, J.M., 1992. Quasi-maximum likelihood estimation and inference in dynamic models

with time varying covariances. Econometric Reviews 11, 143–172.

Brock, W., Dechert, D., Sheinkman, J., LeBaron, B., 1996. A test for independence based on the correlation

dimension. Econometric Reviews 15, 197–235.

Engle, R.F., 1982. Autoregressive conditional heteroskedasticity with estimates of the variance of United

Kingdom inflation. Econometrica 50, 987–1007.

Hamori, S., 2000. Volatility of real GDP: some evidence from the United States, the United Kingdom and Japan.

Japan and the World Economy 12, 143–152.

Jarque, C.M., Bera, A.K., 1987. A test for normality of observations and regression residuals. International

Statistical Review 55, 163–172.

Judge, G.G., et al., 1988. Introduction to the Theory and Practice of Econometrics, second ed. Wiley, Singapore.

Ljung, G., Box, G.E.P., 1978. On a measure of lack of fit in time series models. Biometrika 65, 297–303.

McLeod, A.I., Li, W.K., 1983. Diagnostic checking ARMA time series models using squared-residual

autocorrelations. Journal of Time Series Analysis 4, 269–273.

Nelson, D.B., 1991. Conditional heteroskedasticity in asset returns: a new approach. Econometrica 59, 347–370.

Tse, Y.K., Tsui, A.K.C., 1997. Conditional volatility in foreign exchanges rates: evidence from the Malaysian

ringgit and Singapore dollar. Pacific-Basin Finance Journal 5, 345–356.

K.-Y. Ho, A.K.C. Tsui / Japan and the World Economy 15 (2003) 437–445 445

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