Zia Wadud
email: zia.wadud@imperial.ac.uk

Robert B Noland

Daniel J Graham
email: d.j.graham@imperial.ac.ukr.noland@imperial.ac.uk

A cointegration analysis of gasoline demand in the United States

Keywords: gasoline demand, elasticities, cointegration gasoline demand, elasticities, cointegration

Time-series estimation of gasoline demand elasticities often does not take into account the possibility of non-stationarity in the underlying data, which may render the parameter estimates spurious. Studies have shown that the time trending variables used to explain gasoline demand could be difference stationary and therefore may require cointegration analysis to assess the relationship among the trending variables. In this work we use the cointegration technique to derive long run and short run demand elasticities of non-commercial gasoline consumption using time-series data for the USA from 1949 to 2004. We also attempt to incorporate the presence of a structural break in the data generation process of the time trending variables. Our results show that the consumption of gasoline and lifetime income have a long term stable relationship after the second oil shock of 1978. Prior to the first oil shock of 1973, no such long run relationship could be established through cointegration.

Introduction and motivation

The demand for personal transport fuel has been increasing in the world for the past half a century and currently remains one of the principal sources of aggregate oil demand. The management of gasoline demand is a crucial policy issue in many economies, especially in developed countries, due to concerns about energy security, the environment and global climate change, and the effect of high prices on different societal groups. There are many estimates of gasoline demand that have been derived for different empirical contexts, using different determinants of demand interacting in various ways. The analytical techniques used to derive elasticity estimates also vary widely. 1 Apart from a few disaggregate cross sectional and panel data models, most gasoline demand models employ time-series data. A specific issue with the use of time-series data that has received a great deal of attention over the past two decades concerns the stationarity properties of the data. If two time dependent variables follow a common trend that causes them to move in the same direction, it is possible to find a good correlation between them despite not having any 'true' association. This potential feature of time-series data can give rise to spurious regression [START_REF] Granger | Spurious Regressions in Econometrics[END_REF] and the parameter estimates obtained through these regressions are then unreliable. 2 A time-series is non-stationary when its mean and/or variance is changing over time. Two types of non-stationary series are possible: trend stationary and difference stationary. Trend stationary series have a deterministic time trend around which the series is stationary. The time detrended data is therefore stationary. On the other hand, if the data series are to be differenced 1 [START_REF] Dahl | Analysing gasoline demand elasticities: A survey[END_REF], [START_REF] Sterner | Modelling transport fuel demand[END_REF], [START_REF] Dahl | Demand for transport fuels: A survey of demand elasticities and their components[END_REF], [START_REF] Goodwin | Elasticities of road traffic and fuel consumption with respect to price and income: A review[END_REF], de [START_REF] De Jong | Recent evidence on car cost and time elasticities of travel demand in Europe[END_REF], and [START_REF] Graham | The demand for automobile fuel: A survey of elasticities[END_REF]Glaister (2002, 2004) -provided a survey of existing literature on fuel demand elasticities. 2 One such spurious regression constructed by Hendry (1980) could explain that cumulative rainfall is a better indicator of price inflation than money stock in the UK! 3 to obtain a stationary series, then this is known as a difference stationary series. If differencing the level data once results in a stationary series, then the original non-stationary series is integrated to the order 1, or described as an I(1) series. Similarly, if the non-stationary series is differenced twice to produce stationarity, then the series is I(2). Although the regression of difference stationary series in levels may be spurious, a regression on the differenced series, if stationary, is valid.

A simple level regression of an I(1) series with another I(1) series will also normally be a spurious regression, however in some cases, there may exist a linear combination of different I(1) series which are stationary. This means that there is a long term relationship between those variables, which tends to bind them together. If such a combination does exist, then the variables are known to be cointegrated and their long run relationship in levels is a valid one [START_REF] Granger | Time series analysis of error correction models[END_REF]. [START_REF] Engle | Co-integration and error correction: Representation, estimation, and testing[END_REF] prove that the cointegrated series have an Error Correction Mechanism (ECM) representation, which allows inference on the short run behaviour. An excellent description of the cointegration method appears in [START_REF] Hendry | Explaining cointegration analysis: Part I[END_REF].

Since many of the related macroeconomic data series used in gasoline demand modelling tend to be difference stationary, the use of conventional econometric methods may be inappropriate.

Therefore, it is necessary to address the non-stationarity of the underlying data series and assess any cointegration among them. Recent studies do report a cointegrating long-run relationship between gasoline consumption and other macroeconomic variables. [START_REF] Bentzen | An empirical analysis of gasoline demand in Denmark using cointegration techniques[END_REF] first estimated the cointegrating relationship between gasoline demand, vehicle stock and price for Denmark, where vehicle stock was used as a proxy for income. Eltoni and Al-Mutairi (1995), [START_REF] Ramanathan | Short-and long-run elasticities of gasoline demand in India: An empirical analysis using cointegration techniques[END_REF], [START_REF] Ramanathan | An empirical analysis of gasoline demand in the Sultanate of Oman using Cointegration Techniques[END_REF], and Cheung and [START_REF] Banerjee | Exploring equilibrium relationships in econometrics through static models: some Monte Carlo evidence[END_REF] have also shown via Monte Carlo studies that the OLS estimation of long run parameters may have substantial small sample bias. With this background, we opt to use a dynamic OLS formulation to assess whether cointegration between gasoline consumption, income and gasoline price is present in US data, and to estimate corresponding elasticity parameters.

The Model and Methodology

Following the previously mentioned studies, we choose the constant elasticity functional form for gasoline demand for our econometric analysis:

(1) Where g t is gasoline consumption per capita, p t the real price of gasoline and y t real income per capita, all at time t. Taking the logarithm of both sides gives the familiar log-linear form:

G t = 0 + Y Y t + P P t + $ t (2)
Where, G t , Y t and P t refers to respective logarithms ln(g t ), ln(y t ) and ln(p t ), $ t is the residual.

We assume that G, Y and P are generally all non-stationary and I(1) series, hence the parameter estimates will not be spurious only if the variables have at least one stable long run relationship and their combination is stationary for that specific set of parameters. The economic behavioural theory behind the ECM is that there exists a long run desired relationship between the three variables in discussion. At equilibrium the residual $ t is zero.

However, the consumption at any time 't' generally is not at equilibrium ($ t & 0), and economic agents respond to the disequilibrium of the previous time period by adjusting their consumption toward the equilibrium target in the next period. At the same time, consumers also respond to changes in exogenous stimuli, and for gasoline consumption, carry through the 'inertia' of previous consumption. Thus the adjustment in consumption, formed as the ECM, is:

(3) where the lags m, n and s are chosen such that ' t is white noise, and $ t-1 is the disequilibrium from the previous time period. As long as Y, P and G are I(1), and they cointegrate, all the variables in this equation are stationary and therefore inference on the parameter estimates would be valid. Since the economic behavioural theory suggests that the agent responds to the disequilibrium from the desired long run consumption, the estimated parameter * res should be statistically significant if a valid long run relationship exists. Engle-Granger's two step procedure thus involves solving the static Eq. ( 2) by OLS to obtain the long run cointegrating parameters and the residual $ t ; the lag of $ t is then used as an explanatory variable in Eq. (3).

Estimation of Eq. ( 3) by OLS provides an estimate of the short run parameters.

t t res s i i t Gi n i i t Pi m i i t Yi t G P Y G + + + + + = = = = 1 0 1 0 0 0 F o r P e e r R e v i e w 7
Although the regression of various I(1) variables is valid when their combination is I(0), as in Eq. ( 2), an OLS estimation of I(1) variables may result in substantial residual autocorrelation.

Thus the t-statistics on the long run coefficients do not have the usual t-distribution, even asymptotically [START_REF] Patterson | An Introduction to Applied Econometrics: A Time series Approach[END_REF]. This makes the inference on the long run parameters from OLS estimation of Eq. ( 2) inappropriate. In addition, as previously mentioned, [START_REF] Banerjee | Exploring equilibrium relationships in econometrics through static models: some Monte Carlo evidence[END_REF] showed that the OLS estimator can be substantially biased in small samples and suggest using a dynamic model to get long run parameters. Such a dynamic formulation is an autoregressive distributed lag model:

t s i i t Gi n i i t Pi m i i t Yi t G P Y G + + + + = = = = 0 1 0 0 0 (4)
If the dynamics are correctly specified, the residual ' t is serially uncorrelated in Eq.( 4) and the parameter estimates have valid t-statistics. Another approach that combines the two stages of the Engle-Granger process into one is the single equation error correction model. Substituting the value of $ t-1 from Eq. (2) in Eq. ( 3) and considering a simplified case of i=0,

t t P t Y t res s i i t Gi n i i t Pi m i i t Yi t P Y G G P Y G + + + + + = = = = ) ( 1 1 1 0 1 0 0 1 (5)
where * 1 = * 0 -* res 0 . Eq. ( 5), however, is not linear in parameters (because of the multiplication of * res with 's) and has to be estimated by the non linear least squares (NLLS) method. If G, Y and P are cointegrated, then (G t-1 -Y Y t-1 -P P t-1 ) is stationary and the residual ' t is a white noise process, as long as the dynamics are correctly specified. The nonlinear estimation of Eq.( 5) is a one step process and gives the long run ( ) and short run (*) elasticities directly. As ' t is white noise, the usual inference procedure based on t-statistics is Prior to estimating the model, however, each of the variables is tested for its difference stationarity properties. Difference stationarity of a time-series is detected through the presence of unit roots in the data generation process of the variable.4 Among the 'unit-root' tests, the ADF (Dickey and Fuller 1979) and PP [START_REF] Phillips | Testing for a unit root in time series regression[END_REF] tests are the most popular.

Previous studies in the gasoline demand literature utilising cointegration methods and cited above have opted for the ADF test. However, the ADF and PP tests both tend to under reject the presence of unit roots and have been superceded by alternative tests [START_REF] Maddala | Unit roots, Cointegration and Structural Change[END_REF]; one such test is a modified version of the ADF test known as the DF-GLS test, where the data are first GLS detrended as per [START_REF] Elliott | Efficient tests for an autoregressive unit root[END_REF]. We use this test to detect the presence of a unit root in the data generation process of each of the variables.

Still, all unit root tests have a significant drawback in that they tend to accept the presence of a unit root, when the data is actually trend stationary, but with a break in the trend [START_REF] Perron | The great crash, the oil price shock, and the unit root hypothesis[END_REF]).

Since our data consists of the time period from 1949 to 2004, there is at least one major incident, the 1973 oil shock, that could cause a break in the underlying data generation process for the variables. Also we expect another break due to the second round of oil price hikes in 1978. As the time-series between 1974 and 1978 is very small we cannot statistically test for any difference for these years. We assume this brief period as a transitory phase when the data generation process may have changed, and choose to divide the sample into two, pre-1974 and post-1978 and test for the presence of unit roots in different variables for both samples. In addition to a break in the stationarity properties and thus the data generation process of any of the variables, there could also be a structural change between the cointegrating relationship among the variables, i.e. the parameters corresponding to the cointegration relationship may have changed between 1974 and 1978. The stability of the cointegration parameters across these two periods can be tested using [START_REF] Chow | Test of equality between sets of coefficients in two linear regressions[END_REF] or [START_REF] Quandt | Tests of hypothesis that a linear regression system obeys two separate regimes[END_REF] type tests, which have been extended to I(1) processes by [START_REF] Hansen | Test for parameter instability in regressions with I(1) processes[END_REF]. We also assess the possibility of a cointegrating relationship for the whole sample. It is also possible that there would be a structural change in the cointegrating relationship (provided there is one) between these three variables.

Data

Results

Variables were initially checked for their stationarity properties using the DF-GLS test. This test, however, is sensitive to the choice of lag length. There are different methods to choose the optimum lag length, such as the Schwarz Information Criteria, the Ng-Perron sequential t [START_REF] Ng | Unit root tests in ARMA models with data-dependent methods for the selection of the truncation lag[END_REF], and the Modified Akaike Information Criteria [START_REF] Ng | Lag length selection and the construction of unit root tests with good size and power[END_REF]. Often the three tests report contradictory lag lengths for the same variable. We therefore employed all three tests but chose the lag length supported by the majority of them. All three variables have been found to be non-stationary at level, for the entire sample, as well as the two subsamples (Table 2), as indicated by the test statistics being less than the critical value. 5 The test statistics for the DF-GLS tests on the first differences, exceed the critical value for the full sample and the post-1978 sample. These are stationary at first differences, whereas the variables from the pre-1974 sub-sample are still non stationary (Table 2). Thus the variables, when considered in the full sample and in the post-1978 sub-sample are I(1), but this cannot be said about the pre-1974 sub-sample. This is an indication that there could have been a break in the data generation process during the 1974 and 1978 oil price escalation. 3. For the post-1978 sub-sample the residuals from the OLS estimation of Eq. ( 7) are stationary confirming a cointegrating relationship between the three variables. For the full sample case, the residuals still show a unit root (the absolute value of the test statistic 0.824 is smaller than the critical value of 2.245), thus no cointegration among these variables can be found. This is not surprising as the full sample contains the pre-1974 sub-sample and cointegration among the variables for this subset could not be assessed because of inconclusive unit root test results. The results suggest that it is not possible to estimate a long term stable relation between gasoline consumption, income and price prior to 1974.

The error correction model (Table 4) gives the short run response to price and income change for the post-1978 sub-sample. The residuals of the ECM fulfill the conditions of a normally distributed, white noise process (the test statistic does not reject the null of normality, and the Q-test statistic for white noise does not reject the null of a white noise process). The long run income elasticity (0.565) is higher than the short run elasticity of 0.473. 6 The price response in the long run (-0.102) is also larger but not very different from the short run response (-0.065).

The coefficient of the error correction term is 0.778, meaning around 78% of the disequilibrium in the previous period is adjusted in the current time period. This adjustment parameter is statistically significant, another verification that a cointegrating relationship exists between the variables. 5. We selected the dynamic model on the basis of the highest adjusted R 2 and passing of the misspecification tests such as residual autocorrelation, normality, ARCH and white noise tests. The resulting model contains three lags of gasoline consumption, but no lags of income or price. The parameter estimates of Y and P in this model give the short run elasticities directly (0.457 and -0.091 for income and price respectively). Corresponding long run estimates are calculated to be 0.593 and -0.118 respectively (Table 7). A DF-GLS test on the residuals involving only long run parameters finds that a cointegrating relationship exists between the three variables. It is important to note that if a unit root had been found in the residuals with the long run parameters, then the dynamic model estimates would be spurious parameters. In the dynamic model, cointegration is thus tested after the model is estimated.

The non-linear single step estimation (Table 6) gives a similar long run relationship with income and price as the dynamic model. This is expected as both the models have lags of the variables incorporated, the principal difference being the different choice of the length of lags.

The significant coefficient of G t-1 -Y Y t-1 -P P t-1 indicates that there is a cointegrating relationship. The DF-GLS test of residuals involving the long run parameters again rejects the presence of a unit root, confirming cointegration. Of all three models, the dynamic model reports the smallest standard errors for the parameter estimates, and also the highest adjusted R 2 . A comparison of the elasticity estimates (Table 7)

show that the long run response for the dynamic model is slightly higher than that from the static Engle-Granger models, but statistically are not very different. Since the residuals of the static cointegration regression do not show any autocorrelation through the Breusch-Godfrey LM test, the long run parameters from the static model have a valid inference for this dataset. The short run price response is slightly higher for the dynamic model, but still statistically not different from the static model.

Conclusion

In this paper we have assessed the cointegration between gasoline consumption, gasoline price and income in US data using an annual time-series from 1949 to 2004. The results indicate that no stable and meaningful long run relationship exists between these three variables for the whole period. Dividing the sample into two subsets, pre-1974 and post-1978, similar tests indicate that cointegration exists between the three variables for the post-1978 sub-sample, but not for the pre-1974 sub-sample in which unit root tests could not detect the order of integration for two of the variables and therefore the long run association between the variables could not be estimated. This suggests there may have been a structural break in the data between 1974 and 1978, coinciding with the oil supply and price shocks, as well as during introduction of fuel economy standards in the US.

Due to the inference and bias issues associated with the popular Engle-Granger method of estimation, we employed single stage non linear least squares and dynamic models to test the sensitivity of the estimates for the post-1978 sub-sample. The models do not report a statistically significant divergence in parameter estimates, although some elasticities vary slightly. Among all three specifications, the dynamic model reports the lowest standard error for the parameter estimates. Even if the time-series models do not take into account the cointegrating relationships explicitly, the dynamic models give robust estimates of long and short run elasticities, provided there is underlying cointegration between the variables. Since gasoline consumption, income and price tend to cointegrate, at least during the post oil-shock F o r P e e r R e v i e w 14 period, previous studies employing dynamic models with these variables would tend to give valid elasticity estimates despite not testing for the cointegrating relationship explicitly.

Our results using the cointegration model for the post-1978 period give a lower elasticity of both short-run and long-run demand than most other studies. This is typical of most cointegration studies. Examples can be found in studies in Australia [START_REF] Samimi | Road transport energy demand in Australia: A cointegration approach[END_REF], Denmark [START_REF] Bentzen | An empirical analysis of gasoline demand in Denmark using cointegration techniques[END_REF], and France [START_REF] Nadaud | The demand for car gasoline in France: a long run econometric perspective[END_REF]). The consensus gasoline price elasticity is typically assumed to range from -0.25 in the short-run to -0.64 in the long-run [START_REF] Goodwin | Elasticities of road traffic and fuel consumption with respect to price and income: A review[END_REF].

The much lower elasticity estimates from this and most other cointegration studies provides a lower bound of estimates and suggest a need for further investigation of these effects. The level of aggregation used here of annual data, could mask many of the individual level effects found in other studies that estimate larger elasticities. However, this study provides an interesting perspective on the nationwide impacts of gasoline price changes.

  estimated parameters. All three models are estimated here to test the sensitivity of the parameter estimates with respect to model specification.

National

  Income and Product Account (NIPA, Bureau of Economic Analysis 2005) data on personal expenditure is used. The NIPA database contains records for per capita disposable personal income and total gasoline expenditure in the USA from 1929 to the present. Usability of the long time-series data is however limited by the availability of the gasoline price data, which is available only from 1949 from the US Energy Information Administration (2005). Total gasoline expenditure is converted to per capita consumption by dividing the consumption expenditure by mid-year total population, also available in the NIPA database. Gasoline consumption expenditure is then converted to gallons of gasoline consumed per capita by dividing it by nominal gasoline price for the respective years. All income and price data for the analysis are then normalized to 2003 US dollars using the consumer price index (CPI). The time-series data for logarithms of per capita income, gasoline consumption and gasoline price are depicted in Fig. 1, and the disruption to gasoline consumption and gasoline price in 1974 and 1978-79 are clearly visible. The slope of gasoline consumption also differs between the pre-1974 and post-1978 periods. In the income series, although there is no sudden drop or and 1978, the slope does differ between the pre-1974 and post-1978 periods. The data generation process, therefore, could have changed during the 1974-1978 period.

  of the unit root test for the pre-1974 sub-sample does not allow us to test for cointegration in this sample. Static estimation results involving the variables using the Engle-Granger two step process for the full sample and the post-1978 sample are presented in Table

6

  Long run elasticities are calculated from =2/(1-2 G ) of the dynamic model are presented in Table

Figure 1 .

 1 Figure 1. Gasoline consumption, gasoline price (real) and income per capita (real) from 1949-2004

  -run relationship can be determined from the parameters of Eq. (2). Once the long run relationship is established, the short run elasticities are determined from the ECM.

	then the long
											F o r
											P
	g	t	=	y	t	Y p	t	P	+	t	e
											e r
											R e v
											i e
											w
											If cointegration exists at
	that combination, $ t becomes stationary too. Thus a test to examine if the three variables
	indeed cointegrate is to test whether $ t is stationary. If the residual $ t in Eq. (2) is stationary,
											6

Table 1 .

 1 Estimates from cointegrating regression for gasoline demand

	Income elasticity	Price elasticity

Page 18 of 22 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript F o r P e e r R e v i e w 19Table 2 .

 2 DF-GLS test for the presence of unit root in level and differenced variables (H 0 : unit root exists, series non-stationary)

	Sample		Full sample		Post-1978 sub-sample	Pre-1974 sub-sample
	Variable	No of lags	DF-GLS	5% DF-GLS	No of lags	DF-GLS	5% DF-GLS	No of lags	DF-GLS	5% DF-GLS
			test	critical value		test	critical value		test	critical value
	G	1	-0.633	-3.195	1	-3.438	-3.498	2	-1.550	-3.349
	Y	1	-1.130	-3.195	7	-0.830	-3.014	1	-1.309	-3.509
	P	1	-2.498	-3.195	1	-1.202	-3.498	1	-2.020	-3.509
	QG	1	-4.751	-3.202	1	-5.140	-3.509	1	-2.858	-3.517
	QY	1	-3.844	-3.202	6	-4.344	-3.114	8	-1.875	-3.702
	QP	1	-5.195	-3.202	1	-6.156	-3.509	1	-2.802	-3.517
	Q 2 G							1	-4.517	-3.521
	Q 2 Y							2	-1.550	-3.332
	Q 2 P							2	-1.970	-3.332
	N		56			26			25
	Remarks		All 3 variables I(1)		All 3 variables I(1)	G is I(2), others inconclusive

Page 19 of 22 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

	Submitted Manuscript
	F o r P e e r R e v i e w
	20

Table 3 .

 3 Engle-Granger static estimation of long run behaviour

	Full sample	Post-1978 sub-sample

Page 20 of 22 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

	Submitted Manuscript
	F o r
	P
	e
	e r
	R e v
	i e
	w
	21

Table 4 .

 4 ECM based on Engle-Granger static long run estimates for the post-1978 sub-sample

	Dependent variable	QG t
	Explanatory variables	Coefficient	Standard error	t-statistic
	QY t	0.473	0.109	4.35
	QP t	-0.065	0.018	-3.68
	QG t-1	0.441	0.093	4.74
	QG t-2	-0.040	0.077	-0.51
	S t-1	-0.778	0.162	-4.79
	Constant	-0.004	0.002	-1.54
	Test diagnostics
	Adjusted R 2	0.843
	Breusch-Godfrey LM test of residual	1.435	(p=0.231)
	autocorrelation
	Shapiro Wilk W test for residual normality	1.207	(p=0.114)
	Engle's LM test for ARCH	2.420	(p=0.120)
	Portmanteau Q test for white noise	11.019	(p=0.274)
	N	25

Table 5 .

 5 OLS estimation of dynamic model for post-1978 sub-sample

	Dependent variable	G t
	Explanatory variables	Coefficient	Standard	t-statistic
			error
	Y t	0.457	0.053	8.61
	P t	-0.091	0.012	-7.34
	G t-1	0.720	0.112	6.45
	G t-2	-0.566	0.123	-4.62
	G t-3	0.076	0.066	1.15
	Constant	0.406	0.231	1.76
	Test diagnostics
	Adjusted R 2	0.991
	Breusch-Godfrey LM test of residual	0.118	(p=0.731)
	autocorrelation
	Shapiro Wilk W test for residual normality	0.084	(p=0.466)
	Engle's LM test for ARCH	1.025	(p=0.311)
	Portmanteau Q test for white noise	10.712	(p=0.296)
	DF-GLS test for unit root for G -T Y Y -T P P	-3.761 a
	N	25
	a 5% critical DF-GLS value -3.498

Table 6 .

 6 Single step NLLS estimation of the ECM for post-1978 sub-sample

	Dependent variable	QG t
	Explanatory variables	Coefficient	Standard error	t-statistic
	QY t	0.520	0.102	5.07
	QP t	-0.085	0.018	-4.70
	QG t-1	0.556	0.109	5.12
	QY t-1	-0.177	0.121	-1.46
	QG t-2	-0.066	0.081	-0.81
	G t-1 -T Y Y t-1 -T P P t-1	-0.782	0.151	-5.19
	Y	0.592	0.035	16.89
	P	-0.116	0.020	-5.77
	Test diagnostics
	Adjusted R 2	0.880
	Shapiro Wilk W test for residual normality	-0.699	(p=0.758)
	Portmanteau Q test for white noise	6.444	(p=0.695)
	DF-GLS test for unit root test for G -T Y Y -	-3.839 a
	T P P
	N	25
	a 5% critical DF-GLS value -3.498

Table 7 .

 7 Comparison of elasticity estimates for post-1978 sub-sample from various models (standard

	error in brackets)
	Parameter estimates		Estimation procedure
		Engle-Granger	Dynamic OLS	NLLS
	Income elasticity, long run	0.565	0.593	0.592
		(0.037)	(0.025)	(0.035)
	Price elasticity, long run	-0.102	-0.118	0.116
		(0.019)	(0.015)	(0.020)
	Income elasticity, short run	0.473	0.457	0.520
		(0.109)	(0.053)	(0.102)
	Price elasticity, short run	-0.065	-0.091	0.085
		(0.018)	(0.012)	(0.018)
	Speed of adjustment	-0.778	-0.771	-0.782
		(0.162)	(0.075)	(0.151)

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

For example, a simple unit root test on G with a single lag involves testing for , = 1 in the equationG t = -+ ,G t-1 + (.t) + ' t

Phillips P.C.B., and Loretan, M., 1991, Estimating long-run economic equilibria, Review of Economic Studies, Vol. 58, pp. 407-436.


Bruno Ori

Geoff Pugh

Prof Geoff Pugh

The effects of exchange rate variability on international trade: a Meta-Regression Analysis

Keywords: Meta-regression analysis, Exchange rate variability, International trade, Hedging, Trade promotion

The trade effects of exchange rate variability have been an issue in international economics for the past 30 years. The contribution of this paper is to apply metaregression analysis (MRA) to the empirical literature. On average, exchange rate variability exerts a negative effect on international trade. Yet MRA confirms the view that this result is highly conditional, by identifying factors that help to explain why estimated trade effects vary from significantly negative to significantly positive. MRA evidence on the pronounced heterogeneity of the empirical findings may be instructive for policy: first, by establishing that average trade effects are not sufficiently robust to generalise across countries; and, second, by suggesting the importance of hedging opportunities -hence of financial development -for trade promotion. For the practice of MRA, we make a case for checking the robustness of results with respect to estimation technique, model specification and sample.

Introduction

Since the onset of generalized floating, there has been extensive theoretical and empirical investigation into the effects of exchange rate variability on international trade. This issue has also been prominent in policy debate. There is a consensus that exchange rate movements cannot be anticipated and, hence, create uncertainty in international trade. However, the literature gives no such clear guidance on the trade effects of exchange rate variability and uncertainty. [START_REF] Gros | Exchange Rate Variability and the Foreign Trade in the Presence of Adjustment Costs[END_REF], [START_REF] Dhanani | The management of strategic exchange risk: evidence from corporate practices[END_REF]Groves (2001), De Grauwe (1988) and [START_REF] Dellas | Real Exchange Rate Volatility and International Trade: A Reexamination of the Theory[END_REF] develop models in which exchange rate variability may have either a positive or a negative impact on trade.

Unfortunately, the ambiguous implications of the theoretical literature are not resolved by the empirical literature. The conclusions from the 58 studies analysed below are presented in Table 1. In each case, the recording of negative, no statistically significant effects, positive or not conclusive (studies reporting a combination of the previous three categories) reflects authors' own interpretations of their results. The largest category, 33 studies, concludes that exchange rate variability exerts an adverse effect on trade. 1 The other 25 studies reach conclusions suggesting that this is not the case. Indeed, six studies report findings that suggest the precise opposite. This range of published results corresponds to the range of possibilities allowed by theory and suggests that the results reported in this literature are unlikely to be driven by publication bias. The theoretical ambiguity in the relationship between exchange rate variability and trade, together with the corresponding non-conclusive nature of the empirical evidence, are likely to reduce the probability that journal editors and authors have systematically favoured studies and results biased in one or other direction or even towards higher levels of statistical significance irrespective of sign.

This paper uses meta-regression analysis (MRA) to make two contributions to the literature on the trade effects of exchange rate variability: to help explain the wide variation of results -ranging from significantly positive to significantly negative effects -in the empirical literature; and to suggest new lines of enquiry. Because of the pronounced heterogeneity in this literature, we focus on the direction and significance of estimated trade effects. Correspondingly, we do not conclude with a representative estimate of the trade effect, as this would be misleading for most particular contexts of concern to policy makers. 1 Although some studies concluding that the trade effect is negative nonetheless contain some positive results; for example, Stokman, 1995, reports two positive effects and one zero effect among otherwise consistently negative effects. The work is structured as follows. Section 2 explains how the data was collected and the choice of effect size. Section 3 explains the MRA of the trade effects of exchange rate variability. Section 4 reports and interprets the results. Section 5 concludes.

Data and effect size

We used the EconLit data base (period ending March 2003) to identify as far as possible all econometric investigations of the trade effect of exchange rate variability that have been published in refereed economics journals. 2 As is the norm in MRA, we gathered close to but not necessarily the complete population of studies [START_REF] Rose | A Meta-Analysis of the Effect of Common Currencies on International Trade[END_REF]. EconLit search is a common approach to minimising the influence of poorly designed and/or executed studies [START_REF] Stanley | Wheat from Chaff: Meta -Analysis as Quantitative Literature Review[END_REF]. However, this approach on its own was not sufficient to identify the population of relevant papers. On the one hand, key word(s) search may fail to identify important papers that include estimates of the trade effects of exchange rate variability but do so only as a subsidiary theme (e.g., [START_REF] Rose | One Money, One Market: Estimating the Effect of Common Currencies on Trade[END_REF]. Other papers may be overlooked because the key search words are insufficiently comprehensive and/or authors use terminology that differs from the mainstream of the literature. On the other hand, many papers thus identified may not be relevant (e.g., some will be purely theoretical studies) or report no usable effect size. In practice, therefore, we implemented a more flexible strategy. At first, we used our own knowledge of the literature and existing narrative literature reviews to identify the most cited papers. Next, systematic EconLit search added further papers.

In addition, still further papers were brought to our attention during the normal process of informal and formal review of this study. Finally we expanded our (1984), [START_REF] Akhtar | Exchange Rate Uncertainty and International Trade: Some Conceptual Issue and New Estimates for Germany and the United States[END_REF] and De [START_REF] De Grauwe | Long-run Exchange Rate Variability and International Trade[END_REF], because these were frequently cited in subsequent studies. Altogether, we identified 58 papers, most of which report multiple results. Accordingly, our 58 studies generated 835 observations. For comparison, Table 2 displays the number of studies and corresponding observations together with goodness of fit measures from three respected MRAs in economics.

Table 2: Number of studies and observations in examples of MRA

A summary measure (effect size) has to be chosen:

1. to combine and compare effect sizes among studies, obtain their mean value, and test their differences for statistical significance; and 2. as the dependent variable of the MRA.

We follow [START_REF] Stanley | Meta -Regression Analysis: A Quantitative Method of Literature Surveys[END_REF] recommendation that in economics the t-value of the regression coefficient is the natural effect size. From each result (regression) reported in each study, the t-value of the estimated coefficient measuring the trade effect of exchange rate variability was chosen as the effect size. 3 This exchange rate variability effect size (ERVES) is independent of the units in which variables in different studies are measured and, given the large sample, under the null of no genuine effect approximates the standard normal distribution (Stanley, 2005), which makes it suitable for the statistical analysis outlined in the following section.

3 Some studies employ more than one measure of exchange rate variability (e.g., by including both current and lagged values). An appendix detailing how the effect size was selected in each such case is available on request. It is excluded here for reasons of length.

Meta-analysis of the ERVES

835 ERVES were pooled from the 58 collected studies; 52 studies contain more then one estimation of the trade effect of exchange rate variability. The mean ERVES value is -1.31 with standard deviation of 2.93,4 which by common standards in meta analysis can be characterised as close to a medium (0.5 ) effect size [START_REF] Stanley | New Wine in Old Bottles: A Meta Analysis of Ricardian Equivalence[END_REF].

The null hypothesis -that the mean ERVES is zero -was rejected at the one percent level (t = -12.96; p=0.000). This statistically significant negative mean effect size suggests a negative relationship between exchange rate variability and trade. Yet, because the ERVES are t-values, the mean ERVES suggests that in the typical regression the coefficient on exchange rate variability falls short of conventionally accepted levels of statistical significance. Moreover, this negative effect is not uniform across the literature. The observed ERVES ranges from -22.00 to 14.77, which suggests considerable variation around the mean. However, if the differences among observed ERVES are random sampling effects, then under the null the standard deviation of the ERVES distribution should be one ( ERVES = 1); otherwise, in the presence of systematic variation from the mean, the standard deviation exceeds one ( ERVES > 1). The null was rejected ( 2 = 2,441; p=0.000). This result supports the alternative hypothesis that the variations of the observed ERVES around their mean are the product of systematic differences in the design of the primary studies. MRA is a method to analyse the specification characteristics that determine differences among the observed ERVES. Hence, in the following section, we discuss the specification of our meta regression model.

Independent variables

The key to explaining variation among observed ERVES is selection of appropriate moderator variables. This selection was guided both by our interpretation of the studies that provide the data for our MRA and by suggestions from the two most recently published narrative literature reviews [START_REF] Mckenzie | The Impact of Exchange Rate Volatility on International Trade Flows[END_REF][START_REF] Pugh | Exchange Rate Variability, International Trade and the Single Currency Debate: a survey[END_REF].

Moderator variables are constructed as dummy variables (i.e., one for studies with a particular characteristic; otherwise zero). First, we explain those that are needed to account for different definitions of both the dependent variable (trade flows) and the independent variable of interest (exchange rate variability). Some researchers argue that analysis of aggregate trade flows is misleading [START_REF] Mckenzie | The Impact of Exchange Rate Volatility on International Trade Flows[END_REF] and, instead, use bilateral trade flows. However, because of near perfect multicollinearity between the moderator variable for bilateral trade flows and the moderator variable for bilateral exchange rates (BILATERAL), we use the latter to capture the effect on the ERVES of both of these study characteristics. A few studies examine the impact of exchange rate variability on sectoral trade flows.

Hence, we construct a moderator variable for sectoral trade flows (SECTALT) with aggregate trade flows as the benchmark. Researchers also have to make a choice between the effects of exchange rate variability on export supply and the effects on import demand. Because of differences in the currency of invoicing, levels of risk aversion and elasticities of export supply and import demand, the impact of exchange rate variability is likely to vary. Hence, we construct a moderator variable for import demand (IMPORT), with export supply as the benchmark. The definition of the independent variable of interest is also contested. There are differences in the literature over both the appropriate exchange rate measure and the appropriate measure of exchange rate variability. The choice between nominal and real exchange rates is related to the choice of high or low frequency exchange rate variations. Over short periods, all prices are more or less known except the nominal exchange rate. However, as the planning horizon of traders is lengthened, the relevant exchange rate becomes that between domestic cost of production and foreign sale prices converted into domestic currency (IMF 1984). Hence, we construct a moderator variable to test the impact of researchers' choice of real exchange rate series (REALER) on the ERVES, with nominal exchange rate data as the benchmark. [START_REF] Pugh | Exchange Rate Variability, International Trade and the Single Currency Debate: a survey[END_REF] distinguish between studies focussing on high-frequency variability and those focussing on low-frequency variability. This issue is important, because of the different time horizons of business contracts, and correspondingly different hedging possibilities. Since low-frequency exchange rate movements are less subject to hedging [START_REF] Bodnar | Exchange rate exposure and market value[END_REF][START_REF] Cooper | Towards a Common Currency[END_REF], any mitigating effect will be correspondingly reduced. Hence, we constructed moderator variables to test the impact of researchers' choice of daily, weekly, monthly and annual frequency of exchange rate variability on the ERVES (DAILYER, WEEKLYER, MONTHER and ANNUALER), with the most used frequency (quarter-to-quarter variations) as the benchmark. Studies also differed over the choice of measure to proxy exchange rate uncertainty. The most common measure, the standard deviation of either exchange rate changes or percentage changes, is used as the benchmark. However, we identified 13 alternative measures in the literature (MERV 1-13; see Appendix A for definitions) and so constructed moderator variables to analyse the effect of each of these on the ERVES.

Researchers are also divided over the choice between bilateral and effective exchange F o r P e e r R e v i e w 10 rates. Hence, a moderator variable for bilateral exchange rates was constructed (BILATERAL), with effective exchange rates as the benchmark. The grounds for different choices between bilateral and effective exchange rates are similar to [START_REF] Cushman | Has Exchange Risk Depressed International Trade? The Impact of Third Country Exchange Risk[END_REF] case for modelling third-country effects. This third-country effect suggests that overall decrease in trade occasioned by increased exchange rate variability will be lower than is likely to be suggested by studies of purely bilateral trade flows, because traders substitute markets with low exchange rate variability for markets with higher variability. Hence, a moderator variable is included for all models that include third-country effects (THIRDCOUN).

We construct moderator variables not only to model different definitions of the dependent and independent variable of interest but also to account for other differences in datasets and model specification. Many studies have used data from within floating exchange rate periods only or from within fixed periods only. The reason is to preclude possible specification bias associated with structural changes in the relationship between exchange rate variability and trade (Pugh and[START_REF] Pugh | Will the single currency promote intra-European trade?[END_REF]Arize, 1997a). Hence, moderator variables were constructed for studies using only fixed (FIXPER) or floating (FLOPER) periods, with studies using both periods as the benchmark.

The type of country can also influence the trade effects of exchange rate variability. In particular, there are reasons to expect stronger effects on developing economy trade: these include underdeveloped or nonexistent forward markets; and different trade structures, with typically greater dependence on primary products. Hence, we construct moderator variables both for studies focussing solely on trade among In addition, we construct a moderator variable for studies that focus exclusively on US trade flows (US). Possible differences between the impact of exchange rate variability on US trade and the trade of other countries might arise from the ability of

US traders to invoice in USD.

There is likewise no consensus over the choice of model. Most studies have employed a conventional utility maximisation approach to analyse the trade effects of exchange rate variability. However, since [START_REF] Abrams | International Trade Flows Under Flexible Exchange Rates[END_REF] some researchers have argued that a gravity model provides a better explanation of international trade flows; and, hence, that the impact of exchange rate variability on trade should be examined within the gravity framework. Other researchers have specified time series models to estimate conventional models: at first, with lagged independent variables; subsequently, error correction modelling; and, finally, cointegration analysis in the context of error correction modelling. Accordingly, we construct moderator variables to test the influence on the ERVES of a gravity framework (GRAVITY), lagged independent variables (LAGTEST), error-correction modelling (ERRORCOR) and cointegration analysis (LRCOINT), with those studies otherwise estimating conventional utility maximisation models as the benchmark. We also construct moderator variables to investigate the effect on the ERVES of cross-section (CROSS) or panel (POOLED) strategies, with time-series estimation as the benchmark. [START_REF] Akhtar | Exchange Rate Uncertainty and International Trade: Some Conceptual Issue and New Estimates for Germany and the United States[END_REF] observed that data on trade flows usually exhibits seasonal patterns. Hence, a moderator variable was constructed for all models using seasonally (2000) were on average estimated with unusually high levels of statistical significance. Accordingly, we constructed a moderator variable for regressions from Rose's study (ROSE). Finally, moderator variables were included for all studies that control for structural breaks (DOCKSTR -including dock strikes, oil shocks, changes in monetary regime and wars) and to control for the possibility that the trade effects of exchange rate variability may change through time (T, which is a continuous variable defined as the mean year of the estimation period).

We also include in the MRA the square root of the degrees of freedom (SqRt_df) from each regression to test for the existence of an authentic empirical effect in the literature rather than mere reflection of publication bias (Stanley 2005). In the present study, theory permits the trade effect of exchange rate variability to be either positive or negative, and neither alternative is excluded by the empirical evidence. If either negative or positive effects dominate, then two conditions are necessary to confirm the existence and the direction of an authentic empirical effect: that a statistically significant relationship between the effect size and the square root of the degrees of freedom exists; and that the relationship has the same sign as the estimated average effect size. In addition, in the presence of the square root of the degrees of freedom, the intercept term may be interpreted as a test for publication bias and, if statistically significant, its estimated coefficient measures the direction and strength of publication bias (Stanley 2005). Hence, non significance constitutes non rejection of the null of no publication bias.

Model specification

Our benchmark MRA model (following [START_REF] Stanley | Meta -Regression Analysis: A Quantitative Method of Literature Surveys[END_REF]) is specified in Equation ( 1):

(1)

j jk k j j u Z DF Int ERVES + + + =
where • j = 1, …, 835 indexes the regressions in the literature,

• k = 1, …, 37 indexes the moderator variables discussed above,

• ERVES j is the exchange rate variability effect size -i.e., the reported t-value for the coefficient on exchange rate variability in the j th regression,

• Int is the intercept term,

• DF j is the degrees of freedom in the j th regression,

• the coefficient is to be estimated and measures the relationship between the square root of DF j and the effect size,

• Z jk are k moderator variables, which reflect the main data and specification characteristics of the j th regression,

• k are k coefficients to be estimated, each of which measures the effect of a moderator variable on the effect size,

• and u j is the usual regression residual.

Results

We check the robustness of our results with respect to estimation technique by reporting not only OLS but also weighted least squares (WLS) and cluster-robust

estimates. There are two potential problems with OLS estimation. First, there is wide variation in the number of results reported by each study: while the mean number of results per study is 14.15, the range is from one to 54. Weighting studies reporting results [START_REF] Jarrell | A Meta-Analysis of the Union-Nonunion Wage Gap[END_REF]. Accordingly, to adjust for disparity in the number of reported results, we weight each result from a particular study by the inverse of the number of results reported in that study, so that each study is equally weighted, and estimate the MRA model by Weighted Least Squares (WLS). The second problem is that reported results in MRA are not sampled independently, but are sampled in groups (most studies report a group of results). Accordingly, we add cluster-robust estimates to both our OLS and our WLS results. 5 For these two reasons, in Table 3 we report four sets of results: OLS; weighted least squares (WLS), cluster-robust linear regression; and cluster-robust WLS. In all cases, t-statistics and p-values reflect robust standard errors.

We also check the robustness of our results with respect to model specification.

Accordingly, we estimated a fully specified model with all the variables discussed in Section 3.2 and then two successively more parsimonious versions. Our final parsimonious model is reported in Table 3; the full and an intermediate parsimonious model are available on request. Estimates from the fully specified model cannot be regarded as valid, because both the unweighted and the weighted models are misspecified with respect to functional form and display evidence of substantial collinearity. Accordingly, five variables were deleted: "LAGTEST", "MERV4" and "MERV9" to ensure adequate specification with respect to functional form; and 5 This procedure relaxes the assumption of independence between observations within the same group, requiring only that observations be independent between groups, and produces "correct" standard errors 'even if the observations are correlated' [START_REF] Statacorp | STATA User's Guide[END_REF]; see also Deaton, 1997, pp.73-78 and[START_REF] Baum | Instrumental variables and GMM: Estimation and Testing[END_REF]. The adjusted R 2 measures are within the range typically reported by MRA studies (see Table 2; also Stanley, 2005, pp.319 and 332). Across all estimation methods in the two parsimonious models, according to standard criteria, variance inflation analysis suggests that collinearity is not a problem; moreover, the Ramsey RESET test rejects the null of omitted variables or incorrect functional form at all conventional levels of significance. Standard diagnostic tests establish that the baseline OLS model is well specified as a statistical model with respect to normality and heteroskedasticity; however, following common practice in MRA, we report robust standard errors.

F
MRA requires additional diagnostic tests to distinguish between authentic empirical effects and the consequences of publication bias. MRA literature (Stanley, 2005) distinguishes between publication bias that is directional (Type 1) and publication bias that merely favours statistical significance regardless of sign (Type 2). We have already noted the a priori grounds on which Type 1 publication bias is less likely to be present in this study than in most MRAs; namely, with respect to the trade effects of 'conservative approach to the identification of an empirical effect'.

To test for both Type 1 publication bias and for an underlying 'genuine empirical effect, irrespective of publication bias' (Stanley, 2005, p.328; see also pp. 320-23 and 329-332), we embed the "Funnel asymmetry precision effect" test within each of our models reported in Table 3, as well as in the full and intermediate models not reported, by including among the independent variables the square root of the degrees of freedom (SqRt_df). In the full and parsimonious models alike, three of the four estimation methods (the exception in each case being simple OLS) result in intercept terms (_cons) not significantly different from zero at conventional levels, which suggests non rejection of the null of no publication bias. Moreover, in each case, the coefficient on the square root of the degrees of freedom is negative. In both parsimonious models the coefficients on the square root of the degrees of freedom (SqRt_df) are estimated as -0.02 in the two unweighted regressions and as -0.04 in the two weighted regressions; moreover, from these eight estimated coefficients, only the two from the cluster-robust least squares estimator fail to achieve statistical significance at conventional levels. Because the ERVES is the t-value on the exchange rate variability coefficient in each regression in our dataset, this result suggests that studies with larger samples on average are more likely to find a statistically significant negative relationship between exchange rate variability and trade. 6 This is consistent with a genuine empirical effect (Stanley, 2005, p.332).

Sampling theory predicts that, holding all other variables constant, a quadrupling of degrees of freedom doubles the effect size. For example, our weighted meta regression coefficient of -0.04 suggests that the t-statistic on the measure of exchange rate variability in a trade regression estimated with 100 degrees of freedom will be -0.4 (= -0.04 100 ); -0.8 with 400 degrees of freedom (= -0.04 400 ); and so on. At the unweighted mean degrees of freedom (1077.35; SD = 4151.76) the predicted tstatistic is -0.76; and at the weighted mean (430.44; SD = 2432.29) -0.91 (both calculated using the exact weighted and unweighted coefficients). However, considerable variation around such predicted values is caused by study characteristics modelled by the moderator variables, which are discussed below. 7

Type 2 publication bias is manifested as 'an excessive likelihood of reporting significant results' (Stanley, 2005, p.318). The corresponding test was implemented for both parsimonious models by substituting the absolute value for the actual value of our dependent variable (ERVES). In this case, a significant intercept term (_cons) would indicate Type 2 publication bias (Stanley, 2005, pp.325 and 332). (For reasons of space, these results are not reported in full; they are available on request.) In the first parsimonious model, only the intercept in the baseline least squares regression proved to be significant (p=0.03); the three remaining estimators yielded intercept 6 The 573 t-values estimated from 100 or fewer degrees of freedom have a mean of -0.75 (SD=2.02);

the 114 estimated from between 101and 500 degrees of freedom have a mean of -1.08 (SD=2.60); and the 148 estimated from more than 500 degrees of freedom have a mean of -3.66 (SD=4.56). 7 In bivariate regressions of ERVES on a constant and the square root of the degrees of freedom (SqRt_df), the R 2 measures are, respectively, 0.27 (unweighted) and 0.16 (weighted). In comparison, the R 2 measures reported in Following a suggestion in Stanley (2005) the standard errors were bootstrapped but without making any noteworthy difference to this pattern of results. Hence, the preponderance of the evidence does not indicate the presence of Type 2 publication bias.

Taken together, these tests suggest that the negative mean ERVES in the empirical literature (a t-value of -1.31), although small, reflects a genuine negative relationship between exchange rate variability and international trade rather than publication bias.

Given that both theory and empirical findings on the trade effects of exchange rate variability permit both negative and positive effects, these findings are consistent with the understanding in the meta-regression literature that 'publication bias will be less problematic whenever there are countervailing research propositions' (Stanley, 2005, p.335). We now turn to the estimated effects of the moderator variables. We restrict our discussion to those variables that display a consistently significant influence on the effect size across all specifications (while, because of statistical misspecification in the full model, giving preference to the parsimonious models) and all different approaches to estimation.

The consistently negative and, with one exception, significant coefficient on real exchange rate variability (REALER) supports the view that forward markets have a role in reducing exchange rate uncertainty. If a significant negative trade effect is more likely to be discovered by analysing real variability than by analysing nominal variability, this might be because real variability diverges from nominal variability [START_REF] Taylor | Exchange rate Modelling and Macro Fundamentals[END_REF] and thus cannot be hedged. This interpretation is supported by the predominantly negative (six from eight estimates in the parsimonious models) and significant (half of all estimates) effect of year-to-year variability (ANNUALER), which is much less subject to hedging than high-frequency exchange rate variability. In both parsimonious models, the consistently negative and predominantly significant coefficient on trade among less developed countries (LDC) also points to the importance of hedging. Underdeveloped or nonexistent forward markets for LDC currencies, together with capital movement constraints in the LDCs, reduce the possibility and increase the price of hedging, thereby causing higher exposure to exchange rate uncertainty and a correspondingly greater trade effect.

The consistently and significantly negative coefficients measuring the effect of gravity (GRAVITY), cross-section (CROSS), cointegration (LRCOINT) and, to a lesser extent, error-correction (ERRORCOR) modelling suggest that these are all much more likely to discover statistically significant negative trade effects and less likely to discover positive effects than other modelling strategies. However, while there is little distinction to be made between the implications of cross-section and time-series studies, both contrast with the uniformly insignificant effect of panel estimation (POOLED). Together, these results suggest that choice of modelling strategy accounts for some of the wide variation of results in this literature.

Moreover, the consistently and significantly negative results for dummy variables used to model structural breaks (DOCKSTR) in time series data demonstrate that these are important controls for estimating the trade effects of exchange rate variability. studies employing these definitions are more likely to detect a significant negative relationship. However, these measures are typically used in only a few studies and observations. 8 Together with the non-significant effects of the other seven alternative measures, these results suggest that alternative measures have so far added little to the conventional approach represented by the reference category.

F

The existence of an earlier Working Paper (available on request) using the same modelling strategy enables robustness also to be investigated with respect to the sample. The earlier study was based on a smaller sample of 40 papers (544 observations). Accordingly, to address the corresponding possibility of biases resulting from having in effect a very large sample rather than the complete population, we treat the databases of our earlier and present studies as different size samples. By comparing the results from our earlier and the present studies, we conclude that the evidence reported above on publication bias, the role of hedging and the influence of various modelling strategies is robust to a major enlargement of the 8 MERV11 is used in one study with 17 observations; MERV8, three and 11, respectively; MERV3, three and 24; MERV7, two and 37; and MERV13, one and six.

Conclusion

We This heterogeneity is consistent with Gagnon (1993, p.279) who calibrates a theoretical model of the trade effects of exchange rate variability and finds that not only is the negative effect of increasing variability on the level of trade small but also that the effect 'would not be statistically significant in the sample sizes typically available to researchers'. Lack of precision of estimates from small sample studies helps to explain the wide range of findings in the literature; indeed, 68.62 percent of the reported regressions analysed in this study are estimated from samples of 100 or fewer observations, which is the number that Gagnon uses as his benchmark.

Accordingly, our finding that the relationship between sample size (degrees of freedom) and effect size is negative and statistically significant is consistent both with [START_REF] Gagnon | Exchange rate variability and the level of international trade[END_REF] insights and with Stanley's (2005) criterion from the metaregression literature for an authentic empirical regularity.

Two recent studies have suggested that the negative trade effect of exchange rate variability is 'by no means a robust, universal finding' (Clark et al., 2004, p.6; see also [START_REF] Solakoglu | Exchange Rate Volatility and Real Exports: A Sensitivity Analysis[END_REF]. This MRA is able more conclusively to establish that the trade effects of exchange rate variability are highly conditional. This finding may be instructive for policy by providing evidence that the average trade effects suggested by this literature are not sufficiently robust to generalise across countries.

This MRA not only confirms that the relationship between exchange rate variability and trade is highly conditional, but also identifies factors that help to explain why estimated trade effects vary from significantly negative to significantly positive. One of these suggests policy implications and new lines of enquiry. with [START_REF] Brodsky | Fixed Versus Flexible Exchange Rates and the Measurement of Exchange Rate Instability[END_REF] who critically reviewed an earlier literature on the measurement of exchange rate variability and concluded that the standard deviation is the appropriate measure. So far at least, innovatory measurement of exchange rate variability does not appear to have yielded interesting new results on the determinants of international trade. Our results indicate that a statistically significant negative impact of exchange rate variability is more likely when it is beyond the range of forward markets. In particular, the substantially more negative effect associated with LDC trade connects our investigation with the large literature connecting financial development, trade and growth. Further investigation of the impact of hedging on trade may well be a fruitful source of ideas for trade promotion.

To check the robustness of our estimated influences on effect size and of our tests for publication bias and an authentic empirical effect, we make a case for applying different estimation techniques together with a standard testing down procedure. The presence in MRA databases of studies typically reporting varying numbers of interdependent results suggests the use of both weighted and cluster-robust estimators separate judgements about potential moderator variables rather than being derived from theory, specification may be more arbitrary than is the norm in econometric studies in economics. If so, then [START_REF] Leamer | Reporting the Fragility of Regression Estimates[END_REF] and [START_REF] Leamer | Sensitivity Analyses Would Help[END_REF] strictures concerning the robustness of regression estimates may apply with particular force to MRA. Although we make no attempt to enact Leamer's methods, it is in their spirit that we recommend robustness checking across a full model including all moderator variables and successively more parsimonious specifications. In addition, where it is difficult to identify the entire population either of relevant studies or of relevant results for MRA, we suggest that it may be instructive to check the robustness of meta-regression estimates with respect to different samples.

  papers published in refereed economics journals to include IMF

  DC) and for those focussing solely on less developed economy trade (LDC), with studies pooling data on both type of trade as the reference category.

  testing for seasonality, or including seasonal dummy variables (SESONADJ). The coefficients on exchange rate variability in Rose's regressions

  results more heavily than studies reporting fewer may distort MRA

  the effects of different proxy measures of exchange rate uncertainty are not.

  substantially influence estimates of the trade effects of exchange rate variability: in particular, gravity models, both cross sectional and modern time series approaches and the use of bilateral exchange rates and trade flows all make more likely the estimation of a significantly negative trade effect. Much of the research effort in this literature has been devoted to developing and testing proxy measures of exchange rate uncertainty as alternatives to the standard deviation of exchange rate changes. However, most of these alternative measures do not robustly influence the statistical significance of estimated trade effects in a way that differs from the conventional measure and none are widely used. These results are consistent

  estimation. Similarly, because MRA models are specified by

Table 1 : Econometric studies on the trade effects of exchange rate variability (1978-2003)

 1

	F o r
	P
	e
	e r
	R e v
	i e
	w

Table 3 : MRA results for the final (second) parsimonious model (835 observations from 58 studies)

 3

  Table 3 are, respectively, 0.48 and 0.35.

	terms with a uniform lack of statistical significance (p 0.17). In the second
	parsimonious model, the intercept terms were uniformly insignificant (p 0.14).
	F o r
	P
	e
	e r
	R e v
	i e
	w
	18

  applied MRA to the empirical literature on the trade effects of exchange rate variability. 58 papers published between 1978 and 2003 provide 835 usable estimates of our effect size, which is the t-value of the estimated coefficient measuring the trade effect of exchange rate variability.

	The theoretical literature does not yield an unambiguous prediction on how exchange F o r
	rate variability affects trade, while the corresponding empirical literature has yielded P
	the full range of possible results and thus has not generated consensus. MRA enables e
	overall assessment of precisely this kind of diverse and contradictory empirical literature. Simple "vote counting" (Table 1) and the negative mean effect size (-1.31) e r
	both suggest a negative relationship between exchange rate variability and trade. Moreover, MRA finds little evidence of publication bias together with mainly positive R e v evidence that this relationship is an authentic empirical effect. Yet the same evidence i e also suggests that this negative relationship is not robust: the vote count is not overwhelming; the average effect falls short of conventionally accepted levels of w
	statistical significance; and MRA identifies sources of variability and corresponding
	non-robustness. Accordingly, our main conclusion is that the empirical literature on
	exchange rate variability and trade reveals a modestly negative relationship with
	pronounced heterogeneity.
	22

Table 2 : Number of studies and observations in examples of MRA

 2

		Number of studies	Number of observations	R	2 (or range of R 2 )
	Card & Kruger (1995)	15	15	0.02 -0.10 *
	Görg & Strobl (2001)	21	25	0.05 -0.69
	Rose and Stanley (2005)	34	754	0.54 -0.68

The main combinations of key words used were "exchange rate variability", "exchange rate volatility", "exchange rate uncertainty/risk" and "trade effect".

The mean and standard deviation weighted to give each study equal influence on the estimates are, respectively, -1.21 and 2.55.

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

Acknowledgements

The authors acknowledge helpful suggestions from Tom Stanley and other participants at the International Colloquium on Meta-Analysis in Economics, Sønderborg (September 2007). In addition, the paper was greatly improved by guidance from our two anonymous referees. The authors alone are responsible for shortcomings.

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

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