Joakim Westerlund
email: joakim.westerlund@nek.lu.se.

Fredrik Wilhelmsson

Estimating the gravity model without gravity using panel data

Keywords: JEL Classification: F10, F15, C15, C23 Gravity model of trade, Poisson regression model, Panel data, Monte Carlo simulation

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Estimating the gravity model without gravity using panel data

Introduction

The gravity model of trade has been widely used to estimate the impact of various policy issues, including preferential trade agreements, currency unions, and border effects. The model has a long tradition in social sciences where it has been used to model, for example, migration. In economics, the model has become very popular due to its success in explaining trade The gravity model has traditionally been estimated using cross-sectional data. However, this has been shown to generate biased results since heterogeneity among the countries is typically not controlled for in an appropriate way, see [START_REF] Cheng | Controlling heterogeneity in gravity models of trade and integration[END_REF], and [START_REF] Cheng | Estimating the staged effects of regional economic integration on trade volumes[END_REF]. To mitigate this problem, researchers have turned towards panel data, which have the advantage that they permit more general types of heterogeneity. For example, consider estimating the impact of currency unions on trade while controlling for country-pair propensity to trade. For a single cross-section, these controls can only depend on observed country-pair attributes such as common language, and estimates can thus be biased if there is additionally an unobserved component to the propensity to trade. With panel data, such unobserved heterogeneity can be readily controlled for by means of a country-pair fixed effects model, which is more general than both the pooled cross-sectional and country specific fixed effects panel data models.

The single most popular approach to estimating the gravity model using panel data is to first make it linear by taking logarithms and then to estimate the resulting log-linear model by the fixed effects least squares (LS). However, although simple to implement, this approach is problematic because the log-linearized model is not defined for observations with zero trade. Moreover, even though the proportion of observations with zero trade may vary somewhat depending on, among other things, the size of the sample, it is usually quite significant, suggesting that the proper handling of these zeros is potentially very important.

Another problem is that the LS estimator of the log-linearized model may be both biased and inefficient in the presence of heteroskedasticity. Two of the most common approaches to handle the presence of zero trade are to either simply discarding the zeros from the sample, or to add a constant factor to each observation on the dependent variable. The first strategy is correct as long as the zeros are randomly distributed. However, if the zeros are not random, as is usually the case, then this induces a selection bias. This problem is often ignored in applied work, but could be handled by using sample selection correction. In a recent contribution, [START_REF] Helpman | Estimating trade flows: trading partners and trading volumes[END_REF] propose a theoretical model rationalizing the zero trade flows and suggest estimating the gravity equation with a correction for the probability of countries to trade. To estimate the model they apply a two-step estimation technique similar to sample selection models. However, in order to implement the new estimator, the researcher needs to find a suitable exclusion restriction for identification of the second stage equation, which can be quite difficult. The problem with bias and inefficiency in the presence of heteroskedasticity has been largely ignored by applied researchers.

In this paper, we explore and extend upon an idea first pointed out by [START_REF] Wooldridge | Econometric analysis of cross section and panel data[END_REF], namely that the fixed effects panel poisson maximum likelihood (ML) estimator can be applied also to continuous variables. We therefore propose estimating the gravity model directly from its non-linear form by using the poisson ML estimator. Since this removes the need to linearize the model by taking logarithms, the problem with zero trade disappears. A similar approach has recently been proposed by Silva and Tenreyro (2006), who also use the poisson ML estimator. However, they use cross-sectional data, and focus mainly on the issue of heteroskedasticity. Our approach is more general in the sense that it permits one to get rid of the problems of zero trade and heteroskedasticity while simultaneously taking care of the bias caused by country specific heterogeneity, which cannot be accomplished when using cross-sectional data.

Our simulation results suggest that the new estimation method is superior to the conventional approach of applying LS to the log-linearized model. In particular, it is shown that the conventional approach is likely to result in severe bias and misleading inference even if the fraction of observations with zero trade is very small. On the other hand, the poisson ML estimator generally performs very well with only small bias and size distortion. Therefore, since the poisson ML estimator is becoming increasingly available using standard statistical software packages, these results suggest that it should be a valuable tool for econometric analysis of the gravity model. As an empirical illustration, we consider the trade effects of the 1995 European Union (EU) enlargement.

The remainder of this paper is organized as follows. Section 2 briefly outlines the gravity model and the problems of zero trade. Section 3 then presents the Monte Carlo simulations, while Section 4 contains the application. Section 5 concludes.

The problem of zero gravity

Let M ijt denote the bilateral trade between countries i = 1, ..., n and j = 1, ..., n with i = j at time t = 1, ..., T , as measured by the imports of country i from country j. For convenience, the total number of observations per time period, which is given by n(n -1), is henceforth denoted by N .1 A common empirical formulation of the gravity model for bilateral trade includes the GDP levels of the two countries, Y it and Y jt say, as well as D ijt , a dummy variable representing for example some contiguity, common language or free-trade agreement effect. This formulation of the gravity equation can be written algebraically as

λ ijt = E(M ijt |Y it , Y jt , D ijt ) = exp(γD ijt )Y β 1 it Y β 2 jt . ( 1
)
Because only a very limited amount of heterogeneity between the country pairs is allowed in the parametrization of the regression function, conventional cross-section estimates of the gravity model are generally biased. With panel data, on the other hand, we can easily permit for such heterogeneity by means of N country-pair specific effects, denoted α ij . These effects may be different depending on the direction of trade and enters (1) multiplicatively in the following fashion

E(M ijt |Y it , Y jt , D ijt , α ij ) = exp(α ij + γD ijt )Y β 1 it Y β 2 jt = exp(α ij )λ ijt .
This implicitly defines the following regression

M ijt = exp(α ij )λ ijt + e ijt ,
which can be written equivalently as

M ijt = exp(α ij )λ ijt v ijt , (2)
where e ijt is a mean zero disturbance that is independent of the regressors, and where

v ijt = 1 + e ijt / exp(α ij )λ ijt is a heteroskedastic disturbance term with E(v ijt |Y it , Y jt , D ijt , α ij ) = 1.
Moreover, since α ij will generally be correlated with the explanatory variables, random effects estimation of (2) will be inconsistent. To circumvent this, it is common to treat α ij as fixed.

Suppose for a moment that M ijt is strictly positive. One of the most common approaches to estimate the regression in (2) is to first make it linear by taking logarithms, which yields

ln(M ijt ) = α ij + ln(λ ijt ) + ln(v ijt ) = α ij + γD ijt + β 1 ln(Y it ) + β 2 ln(Y jt ) + ln(v ijt ). ( 3
)
Since the model is now linear, it is readily estimable using LS. However, this is only possible as long as M ijt is nonzero, which is not always the case. Indeed, a common feature of trade data is that the bilateral trade can sometimes be zero. Although this poses no problem when estimating the gravity model based on its multiplicative form in (2), as the logarithm is defined only for positive outcomes, the log-linear regression in (3) is no longer admissible.

A common solution to this problem is to drop all observations with zero trade, and then to estimate (3) based on the resulting truncated sample. However, although this approach certainly eliminates the zeros, it simultaneously induces a bias to the LS estimator, which is why truncating the sample should be avoided as a matter of practice.

A natural alternative approach in situations such as this, when the model cannot be loglinearized, is to estimate it from its multiplicative form directly. In so doing, note that the fixed effects conditional mean can be written as

λ ijt = exp(α ij + γD ijt + β 1 ln(Y it ) + β 2 ln(Y jt )), (4)
which is known as the exponential regression function. This regression follows naturally from the multiplicative form of (1) and ensures that λ ijt is nonnegative, which is very convenient as trade cannot be negative. Thus, the conventional additive regression in ( 3) is likely to be unsatisfactory here as it cannot ensure the nonnegativity of trade.

The estimation of (4) has been studied by [START_REF] Hausman | Econometric models for count data with an application to patents-R&D relationship[END_REF], who consider the special case when the data are measured in nonnegative integers. They propose using a version of the conventional poisson ML estimator, which is modified to account for the fixed effects. In so doing, the authors eliminate the fixed effects by conditioning on T t=1 M ijt , a sufficient statistic for α ij , which in our case yields the following log-likelihood function

ln(L) = n i =j T t=1 Γ(M ijt + 1) - n i =j T t=1 M ijt ln T s=1 λ ijs λ ijt ,
where Γ is the gamma function. As noted by the authors, given that the regression in ( 4) is correctly specified, consistency of the resulting fixed effects poisson ML slope estimator follows directly by standard ML theory, see for example [START_REF] Gourieroux | Pseudo maximum likelihood methods: theory[END_REF]. estimator that replaces the fixed effects by the pre-sample mean of the dependent variable, see for example [START_REF] Blundell | Individual effects and dynamics in count data models[END_REF] for a detailed discussion. 3Having estimated the slopes, an estimate of the fixed effects can be obtained by simply replacing λ ijt in T t=1 M ijt / T t=1 λ ijt by its ML estimate. Note that this gives an estimate of exp(α ij ), not of α ij , which is unidentified in the fixed effects formulation of the model.

In order to identify α ij , a random effects assumption is needed. But such assumptions are generally not satisfied in practice, and so we only consider the fixed effects specification.

Although the poisson ML estimator is consistent, valid inference requires the correct specification of both the conditional mean and variance, which necessitates that

λ ijt = var(M ijt |Y it , Y jt , D ijt ).
(5)

However, note that the validity of ( 4) and ( 5) does not require the data to be poisson distributed. In fact, M ijt does not have to be an integer at all. This suggests that we can use the fixed effects poisson ML to estimate the gravity model. Since this estimator does not require M ijt to be nonzero, it is expected to produce better results than LS in panels where some trade flows are zero. Moreover, if it is consistency that we are interested in, then (5) does not have to hold either, so the data do not have to be equidispersed. In the next section, we elaborate on this point.

Monte Carlo study

In this section, we investigate the small-sample properties of the LS and ML estimators in the presence of zero observations through Monte Carlo simulations. The data generating process used for this purpose is given by where α ij = γ = β = 1 for simplicity. Since Y ijt is usually positive in applied work, we set Y ijt ∼ U (0, 1). Moreover, if we let τ ij ∼ U (0, 1) denote the location of the break, then the dummy variable D ijt , representing for example a preferential trade agreement, is such that

M ijt = exp(α ij + γD ijt + βY ijt )v ijt , (6)
D ijt = 1 if t > τ ij T and zero otherwise.
The disturbance v ijt is key in this data generating process. In particular, it is assumed that v ijt is a log-normally distributed variable with mean one and variance σ 2 ij . We have two variance cases. In the case 1,

σ 2 ij = 1, which implies that var(M ijt |Y ijt , D ijt ) = exp(α ij + γD ijt + βY ijt ) 2 , while in case 2, σ 2 ij = 1/ exp(α ij + γD ijt + βY ijt ) so that var(M ijt |Y ijt , D ijt ) = exp(α ij + γD ijt + βY ijt ).
Thus, we expect the LS estimator to perform relatively well in case 1, while we expect the poisson ML estimator to perform relatively well in case 2, as condition ( 5) is now satisfied.4

In both cases, we generate data by drawing 1, 000 panels, each consisting of N observations on each of the T time series.

The results are organized according to the two cases described above. In each case, we want to examine the effect of zero observations in the data. Both the LS and poisson ML estimators are considered. 5 The former is implemented using both truncated data and ln(M ijt + 1) as dependent variable. However, note that since M ijt > 0 in this data generating process, the log-linear model is no longer inadmissible. Hence, to be able to study the effect of truncating the sample we use a positive truncation threshold parameter, which is such that the fraction of truncated observations is exactly δ. For brevity, we only report the mean bias and the size of a nominal 5% level t-test of the null hypothesis that the parameter of interest is equal to its true value versus the alternative that it is not.6

Besides the LS and poisson ML estimators, we also experimented with the negative binomial ML estimator of [START_REF] Hausman | Econometric models for count data with an application to patents-R&D relationship[END_REF], which relaxes condition (5). But since the performance was so unsatisfactory, the results are not included here but are available from the corresponding author upon request. The panel version of the quasi-ML estimator discussed in [START_REF] Gourieroux | Pseudo maximum likelihood methods: theory[END_REF] also performed very poorly, and was therefore removed. 7 Another possibility is to treat the zeros as a sample selection issue, and to estimate the model using an estimator that eliminates the selectivity bias. We tried the [START_REF] Kyriazidou | Estimation of a panel data sample selection model[END_REF] estimator, which is a popular two-step procedure to difference out both the bias and fixed effects. However, as with the negative binomial and quasi-ML estimators, the results from this estimator were very poor, and were therefore removed. 8

The results reported in Table 1 for the LS and poisson ML estimators can be summarized as follows. First, as expected, LS estimation with ln(M ijt + 1) as the dependent variable generally produces very poor results. In particular, it is seen that the estimators of γ and β both suffer from substantial downwards bias, which do not show any tendency to vanish as the sample size increases. Moreover, the results of the size of the t-tests suggest that inference based on this estimation method is likely to be highly deceptive. In fact, with this method, we always end up rejecting the null hypothesis. Thus, based on these results, we recommend not using LS estimation based on ln(M ijt + 1).

Second, the results on the truncated LS estimator are mixed. At one end of the scale, we have case 1 when there is no truncation, in which the performance, both in terms of bias and size accuracy, is very good. At the other end, we have the case when δ > 0, in which Table 1 shows that the performance is poor, and that the problems with bias and size distortion are highly potent, even for a truncation as small as 10%. Apparently, the truncation makes the LS estimator both downwards biased and unfit for inference. Thus, from an empirical point of view, it seems highly unlikely that the truncated LS is able to deliver any meaningful results at all.

In addition to the problems associated with truncating the data, Table 1 points to another important shortcoming with the truncated LS estimator. In particular, it seems as that the heteroskedasticity in case 2 induces both severe size distortions as well as a sizeable bias that persists even in large panels.

7 The quasi-ML estimator only requires that the conditional mean in (4) is correctly specified, and does not make use of (5), see for example [START_REF] Gourieroux | Pseudo maximum likelihood methods: theory[END_REF] and [START_REF] Wooldridge | Econometric analysis of cross section and panel data[END_REF].

8 We used the T = 2 version of the [START_REF] Kyriazidou | Estimation of a panel data sample selection model[END_REF] estimator, which is relatively easy to compute, but preliminary results suggest that the poor performance extends also to the case when T > 2. Also, for this experiment, the data generating process was adapted so as to fit the sample selection setting of [START_REF] Kyriazidou | Estimation of a panel data sample selection model[END_REF]. Although this may appear somewhat counterintuitive at first, as pointed out by Silva and Tenreyro (2006), it is actually a direct consequence of the well-known Jensen inequality.

To appreciate this, consider the data generating process in ( 6) where E(v ijt |Y ijt , D ijt ) = 1. The LS estimator of the parameters in the log-linear model (3) are consistent only if E(ln(v ijt )|Y ijt , D ijt ) = 0. However, although ln(E(v ijt |Y ijt , D ijt )) = 0, by the Jensen equality,

E(ln(v ijt )|Y ijt , D ijt ) = 0. Indeed, since E(v ijt |Y ijt , D ijt ) 2 =
1 in our case, by using the properties of the log-normal distribution, we have that

E(ln(v ijt )|Y ijt , D ijt ) = ln 1 1 + σ 2 ij ,
which is not equal to zero unless of course σ 2 ij is zero too. As a result, the LS estimator in (3) will generally be biased.

Third, except possibly for case 1 when there is no truncation, the results show that the poisson ML consistently outperforms the other estimators in terms of bias. In fact, by looking at Table 1, it would appear as that the bias is practically nonexisting even for as small panels as T = 10 and N = 500, which correspond approximately to 10 time series observations for 23 countries. We also see that the size is very close to the nominal 5% level in case 2 but that it is distorted in case 1, which is partly expected since condition (5) is not satisfied in this case.

One possibility to get rid of the distorted standard errors of the ML estimator is to use the bootstrap. This approach has become very popular in applied work, and it will therefore be used in this paper. The particular algorithm used is taken from [START_REF] Cameron | Regression analysis of count data[END_REF], who make a very simple proposal, in which the dependent and independent variables are resampled in pairs. 9 Some simulations of the resulting bootstrapped t-statistic based on 100 bootstrap replications are reported in Table 2. As expected, we see that the size of the bootstrapped test generally lies much closer to the 5% level than the size of the asymptotic test. Also, the t-statistics appear to be well centered around zero.

In summary, we find that the poisson ML show smaller bias than the two LS estimators considered and, at the same time, maintain relatively good size properties in small samples.

Since the poisson ML with bootstrapped standard errors is now readily available through existing software packages such as STATA, it should be considered a feasible alternative to estimation by LS. 9 Another possibility is to use the wild bootstrap, see [START_REF] Cameron | Regression analysis of count data[END_REF] for a discussion. The estimated gravity equation can be written as

M ijt = exp(α ij + µ t + γ 1 D it + γ 2 D jt + γ 3 D ijt )Y β 1 it Y β 2 jt N β 3 it N β 4 jt v ijt , ( 7
)
or equivalently in its log-linear form

ln(M ijt ) = α ij + µ t + γ 1 D it + γ 2 D jt + γ 3 D ijt + β 1 ln(Y it ) + β 2 ln(Y jt ) + β 3 ln(N it ) + β 4 ln(N jt ) + ln(v ijt ), (8)
where M ijt denotes the nominal imports of country i from country j, Y it and Y jt denote the real GDP of the two countries, and N it and N jt denote their population. The fixed effects α ij capture all types of unobserved country-pair specific heterogeneity that is constant over time, while the time effects µ t capture all forms of time-varying heterogeneity that is shared among the country pairs.

The dummy variables D it , D jt and D ijt are key in this model. The variable D it equals one if country i is a member of the EU at time t while country j belongs to the rest of the world. The second dummy variable D jt equals one if country j is a member of the EU while i belongs to the rest of the world. Similarly, D ijt equals one if both i and j are members of the EU at time t. In other words, the three dummy variables take the value one for EU imports from the rest of the world, EU exports to the rest of the world and intra-EU trade, respectively.

The rest of the world is defined as all countries in the sample that are not members of the EU at any given time in the sample. This enables us to identify the effect of the enlargement on the trade of new EU members as opposed to the effect of changes in the size of the rest of the world. To appreciate this, note that if the rest of the world also included new members, the dummy variable D it would capture not only the import effect on the new members but also the effect of the change in the composition of the rest of the world, as the imports from the new members to the old ones would no longer be classified as imports from the rest of the world. A similar argument applies to the construction of D jt .

A consequence of this definition of the rest of the world is that, since fixed effects absorb all heterogeneity that is constant over time, the trade effect for countries that have been members of the EU for the whole sample period cannot be identified. Thus, the dummy variables capture only the effect on countries that have changed their EU status at least one time. That is, the dummy variables capture the effect of the Austrian, Finnish and Swedish accession to the EU. Specifically, γ 1 measures the trade diversion or changes in EU imports from the rest of the world. Similarly, γ 2 measures the effect on EU exports to the rest of the world, sometimes called export diversion. Finally, γ 3 measures trade creation, resulting from the increased intra-EU trade following the enlargement.

Economic integration should increase trade between countries integrating. Thus, we expect the trade creation, as measured by γ 3 , to be positive. This effect can be separated into pure trade creation, or increased trade due to lower prices on imports from the other countries in the EU, and trade diversion, which implies a shift in imports from more efficient producers in the rest of the world to less efficient producers within the EU. A negative sign on γ 1 would thus indicate trade diversion. Similarly, export diversion occurs if exports to the rest of the world decreases as a result of the integration process, but exports could also increase. The expected sign of γ 2 is therefore ambiguous.

The empirical results are contained in Table 3. It is seen that the enlargement of the EU induced significant trade diversion but no trade creation. This absence of trade creation is, however, not surprising since the new members were part of a free trade area with the EU prior to the membership. When joining the EU, the new members implemented the Common External Tariff, which changed the tariffs on their imports from the rest of the world. Note that the trade diversion effect is rather large in comparison to the trade creation effect. Although counterintuitive at first, one should keep in mind that several countries with preferential access to the EU market, such as those that joined the EU in 2004, have been excluded from our sample, so trade might have been diverted away from suppliers on the world market to suppliers with preferential access to the EU market. Moreover, taken as a fraction of total trade, the diversion effect is probably quite small since the estimation results only capture the effect on imports to Austria, Sweden and Finland and not changes in the total imports of the EU.

F

Even though the number of zeros is comparatively small in our sample, only 10%, when comparing the results obtained from the various estimators, we see that the difference can be substantial. In particular, for the GDP and population variables, the poisson ML estimates are typically larger than their LS counterparts. This finding is well in line with the Monte Carlo evidence suggesting that both LS estimators are downwards biased. Moreover, while the truncated LS estimator indicates that changes in GDP of importing countries does not effect imports, the ML estimator gives a more plausible estimate close to unity.

It should also be mentioned that the LS estimates of the GDP and population parameters appear to be rather unstable, and to a large extent dependent on the time period used, which is probably due to the fact that these variables seem to be quite highly correlated. On the other hand, the corresponding LS estimates of the effects of trade liberalization appear to be very robust, and show almost no variation between time periods. Similarly, all ML estimates seem vary robust to changes in the time period.

For the dummy variables, the differences are less marked. In particular, although the sign and significance of the estimates do not differ much, the magnitude of the estimates varies quite substantially. The LS estimator indicates that the trade diversion is twice as large as implied by the ML estimator and, while the LS estimate of the trade creation effect is slightly negative, it is positive for the ML estimator.

In summary, the results presented in this section highlight the importance of using appropriate estimation techniques to be able to draw correct inference.

Conclusions

The gravity model has become a standard tool for evaluating policies affecting trade and it is widely used to assess the effects of preferential trade agreements and currency unions or to calculate trade potential, among other things. It is well known that the gravity model should be estimated by panel data to mitigate the bias due to failure to fully control for country heterogeneity. A very popular way to accomplish this is to first linearize the model by taking logarithms and then to apply the conventional fixed effects LS estimator.

In this paper, we argue that this approach is likely to be very misleading with severely biased estimates and t-statistics. be zero in the log-linearized model, all zeros must either be discarded or replaced by some arbitrary positive value, which induces a sample selection bias. Secondly, the heteroskedasticity inherent in the log-linear formulation of the gravity model can render the LS estimates both biased and inefficient. By contrast, being based on the gravity model in its original non-linear form, the fixed effects poisson ML estimator does not suffer from these weaknesses and is therefore expected to yield more accurate results.

Our assertion is verified by means of Monte Carlo simulations and illustrated via an application to the 1995 EU enlargement. The simulations show that the performance of the log-linear approach is likely to be so poor that it may not even be meaningful to interpret the results. On the other hand, the poisson ML estimator performs well with only a very small bias and good size accuracy in most cases. Still, in some data generating processes, the results show that the estimated standard errors can be downward biased. To alleviate this, we suggest using bootstrapped standard errors. The empirical application points to a significant difference between the estimators with respect to both the main explanatory variables and the trade effects of the 1995 EU enlargement, thus underlining the importance of using the proper estimation technique.

To conclude, we recommend not estimating the gravity model from its log-linear form.

Instead, we propose estimating the model directly from its non-linear form using the fixed effects poisson ML estimator with bootstrapped standard error. Our proposal provide researchers with a simple framework for analyzing the gravity model while at the same time avoiding potential bias due to zero trade. This, together with the fact that the poisson ML estimator can now be implemented using many standard statistical software packages such as

  . Some critique for the lack of theoretical underpinnings has emerged but much progress has been made and now the gravity model rests on a solid theoretical foundation. Instead, the focus has shifted towards the estimation techniques used.

2

 2 [START_REF] Hausman | Econometric models for count data with an application to patents-R&D relationship[END_REF] poisson conditional ML estimator is the same as the poisson ML estimator with individual specific constants, which in turn is equivalent to the moment estimator in a model where the fixed effects are replaced by 1 , the ratio of within group means. Alternative estimators of the fixed effects poisson model include the quasi-differenced generalized method of moments estimator and the pre-sample mean

4

  An application to the 1995 EU enlargementWe have shown that log-linear LS estimation of the gravity model yields biased results. In this section, we demonstrate these findings by estimating the trade effects of the adhesion of Austria, Finland and Sweden to the EU in 1995. The sample that we use for this purpose cover the period 1992 to 2002 and consists of import data for EU and other developed countries from all trade partners except oil exporting countries and formerly planned economies in Central and Eastern Europe, as defined in Direction of TradeStatistics (International Monetary Fund, 2005). The GDP and population data comes from World Development Indicators(World Bank, 2005).

  The value δ refers to the fraction of truncated observations, γ ls and β ls refer to the LS estimates, and γ ml and β ml refer to the poisson ML estimates. Case 1 refers to the data generating process with σ 2 ij = 1, while case 2 refers to the data generating process with var(Mijt |Y ijt , D ijt ) = E(M ijt |Y ijt , D ijt ).The reported bias results refer to the mean bias times 100. aThe LS estimator is based on truncating the sample.

Page 12 of 17 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

  There are two reasons for this. Firstly, since trade cannot

	Submitted Manuscript
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Table 1 :

 1 Simulated bias and tests size for the ML and LS estimators.

	STATA, makes our proposal definitely seem worthwhile.

Table 2 :

 2 Simulation results for the bootstrapped ML t-test.Notes: The values t( γ ml ) and t( β ml ) refer to the conventional asymptotic ML t-tests, while t * ( γ ml ) and t * ( β ml ) refer to thir bootstrapped counterparts. See Table1for an explanation of the remaining features of the table.

				Case 1				Case 2
	N	T	t( γ Size at the 5% level
	500 10	23.8	33.4	9.2	10.0	5.8	4.0	10.2	7.4
	1000 10	25.2	31.4	10.4	9.6	4.6	5.8	10.0	7.2
	500 20	28.4	33.2	7.6	9.6	5.0	6.6	7.8	10.0
	1000 20	26.4	39.4	8.4	10.6	6.4	4.8	8.2	6.6
						Mean
	500 10	0.0	-0.1	0.0	-0.1	0.1	0.0	0.1	0.1
	1000 10	0.1	0.0	0.0	0.0	0.0	0.0	0.0	0.0
	500 20	0.0	-0.1	0.0	0.0	0.1	0.0	0.1	0.0
	1000 20	0.1	0.0	0.1	0.0	0.0	-0.1	0.0	-0.1
				Standard deviation
	500 10	1.7	2.1	1.2	1.2	1.0	1.0	1.2	1.1
	1000 10	1.8	1.9	1.2	1.2	1.0	1.0	1.2	1.1
	500 20	1.8	2.1	1.1	1.2	1.0	1.1	1.1	1.1
	1000 20	1.8	2.2	1.1	1.2	1.0	1.0	1.1	1.1

ml ) t( β ml ) t * ( γ ml ) t * ( β ml ) t( γ ml ) t( β ml ) t * ( γ ml ) t * ( β ml ) 15 Page 15 of 17 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

Table 3 :

 3 Empirical estimation results. The numbers within the parantheses are the robust LS standard errors or the bootstrapped poisson ML standard errors. The superscripts ( * * * ), ( * * ) and ( * ) denote significance at the 1%, 5% and 10% levels, respectively.

	Estimator	LS	LS	poisson ML
	Dependent variable	ln(M ijt )	ln(M ijt + 1)	M ijt
	β 1	-0.091	0.229 * * *	0.931 * * *
		(0.191)	(0.062)	(0.173)
	β 2	1.438 * * *	0.820 * * *	1.483 * * *
		(0.084)	(0.039)	(0.110)
	β 3	4.055 * * *	1.765 * * *	2.471 * * *
		(0.612)	(0.267)	(0.629)
	β 4	-1.275 * * *	-0.979 * * *	-0.580
		(0.190)	(0.074)	(0.357)
	γ 1	-0.403 * * *	-0.211 * * *	-0.232 * * *
		(0.046)	(0.016)	(0.074)
	γ 2	0.000	0.102 * * *	0.041
		(0.032)	(0.023)	(0.047)
	γ 3	-0.002	0.033 *	0.035
		(0.025)	(0.018)	(0.034)
	No. of country-pairs	2719	2748	2719
	No. of observations	32487	35600	35256
	Notes:

16 Page 16 of 17 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

Note that since each country is both an exporter and an importer in a bilateral trade relation, each country pair is observed twice. The number of observations is therefore twice the number of country pairs.

As long as (4) holds the poisson estimator works, see for example[START_REF] Wooldridge | Econometric analysis of cross section and panel data[END_REF] Winkelmann (2008). In fact, neither (4) nor the maximization of the log-likelihood function require that the dependent variable is a count. It could be a binary variable or, as in our case, a nonnegative continuous variable. This property of the estimator has been used bySilva and Tenreyro (2006). The interpretation of the estimated coefficients is similar to the interpretation of the coefficients in the log-linear model. That is, the estimated coefficient reflects the elasticity of the dependent variable with respect to the relevant independent variable. In the case of an dummy variable, the estimated coefficient provides a reasonable approximation for small estimated values, seeWinkelmann (2008) for a more elaborative discussion.

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

Another possibility is to use the zero inflated poisson (ZIP) model. But so far it seems that the estimation of this model with fixed effects has not yet been analyzed in the literature. In fact,[START_REF] Winkelman | Econometric analysis of count data[END_REF] points out that the properties of the fixed effects poisson ML estimator does not carry over to the ZIP model, and that the estimation of this model is still an open issue.

Other values of σ 2 ij produced very similar results and are thus not reported.

The poisson ML estimator is implemented using the GAUSS optimization library OPTMUM. We use the BFGS gradient algorithm with numerical derivatives. The standard errors of the estimated parameters are computed based on the conventional Hessian method, which generally worked best in the simulations. The truncated LS is used to start up the estimation.

We also simulated the power of the t-tests. However, since the size of the LS based tests turned out to be heavily distorted, with rejection frequencies close to 100% in most experiments, power is not very interesting, and the results are therefore not reported.

Westerlund gratefully acknowledges financial support from the Jan Wallander and Tom Hedelius Foundation, research grant number W2006-0068:1.


Peter Gripaios
email: p.gripaios@plymouth.ac.uk

Paul Bishop

Steven Brand

A Lottery within a Lottery? An Examination of the Distribution of Lottery Funds in England

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

The UK National Lottery has been in existence since 1994. Since its inception, a proportion of the revenue from ticket sales has been channelled to "good causes" through some thirteen award bodies in the constituent parts of the UK. Each award body decides on how it allocates funds to projects but, though free from day to day Government control, it has to work within a regulatory framework of Policy and Financial Directions laid down by the Department of Culture Media and Sport (DCMS) in England and the relevant statutory authorities elsewhere (National Audit Office 2000). By June 2006 over £17bn had been raised for the "good causes" (DCMS, Lottery Grants Database).

The National lottery is effectively a tax as some of the proceeds go directly to the government. Moreover half of the value of ticket sales is hypothecated to good causes, at least some of which would otherwise have been financed by more formal taxation. [START_REF] Feehan | Distribution of UK National Lottery grants across local authority areas[END_REF] point out that it is an exceptionally high tax at around 80% of consumers' net expenditure. They also note that there is little justification for such a high tax in terms of negative externalities (as in the case of petrol) or grounds of individual and public health (as in the case of tobacco). Indeed, it might be argued that the Lottery imposes a negative externality by inducing addiction to gambling. Farrell et al (1999), for example, report that the existence of roll-overs gives a boost to ticket sales, which lasts for some time. There are also welfare implications arising from the fact that the Lottery does not seem to be a very efficient means of raising funds (Farrell et al, 1999;[START_REF] Forrest | Elasticity of demand for UK National Lottery tickets[END_REF]Walker, 1998) Indeed, Farrell et al (1999) argue that allocating a higher proportion of sales revenue to prizes would generate a sufficiently large increase in sales that awards to 'good causes' could be increased A further problem is that, though the tax is voluntary as no-one is forced to play, survey data suggest that it falls most heavily on the poorer sections of the community, with lower class groups more likely to play and spend more than other groups (Sproston, 2003). This conclusion is confirmed by an analysis of earlier data by Farrell and Walker (1999) which concludes that the income elasticity of demand is positive but inelastic. This may reflect expenditure switching with reduced spending on necessities (Smith, 2007) by the relatively poor. On the other hand, richer income groups potentially gain from lower progressive taxes associated with the use of lottery funds as an alternative funding source for good causes. They may also gain from the fact that those good causes (e.g. the Arts) feature more strongly in their utility function than lower income consumers. It is also worth noting that such welfare issues are of wider concern than just for the UK as many countries use lottery funding for areas such as arts and sport [START_REF] Feehan | Distribution of UK National Lottery grants across local authority areas[END_REF]. Studies of these issues in other countries include [START_REF] Kearney | State lotteries and consumer behaviour[END_REF], [START_REF] Scott | Probability of purchase, amount of purchase and the demographic incidence of the Lottery tax[END_REF], Stanahan and Borg (1998), Walker (1998) and Worthington (2001).

One welfare issue which has received relatively little attention is the regional and sub-regional distribution of lottery expenditure. This is surprising as there has been considerable research and debate on the general spatial distribution of state spending in the UK. Recent contributions include, for example, Mackay (2001), [START_REF] Gripaios | Regional spending: a comment on Mackay[END_REF], [START_REF] Heald | The regional dimension of public expenditure in England[END_REF], McClean and McMilllan (2003), HM Treasury (2003) and [START_REF] Gripaios | Government output and expenditure in UK regions and sub-regions: an analysis of the new experimental accounts data[END_REF]. These studies have revealed wide variations across regions both in terms of "identifiable" public expenditure, for the benefit of a particular population, and "non-identifiable" spending such as that on defence, which is seen, in principle, as benefiting citizens of the UK in total.

As far as the distribution of Lottery grants is concerned, there may have been some in-house studies given the hints in the consultation process prior to the reform of the National Lotteries Act in 2006 that, in the early years: "There were concerns that too much lottery money was going on buildings, and not enough on people, that it was easier for bigger well-established organisations to get access to funding than smaller groups and that some groups and areas were missing out entirely" (DCMS, 2002, p4). Some changes were made as early as 1998 to address these concerns but the DCMS seems to have remained worried that not enough was continuing to get through to deprived communities and locations.

The only independent research on the topic of the spatial distribution of UK lottery funds appears to be the recent study by [START_REF] Feehan | Distribution of UK National Lottery grants across local authority areas[END_REF]. This paper covers some of the same ground as that of [START_REF] Feehan | Distribution of UK National Lottery grants across local authority areas[END_REF] but also extends it. In particular it utilises different explanatory variables and examines directly the impact of levels of deprivation on Lottery grant funding. Further research is justified both by the important welfare issues outlined above and the paucity of work on the regional and local distribution of this aspect of government expenditure relative to mainstream spending. It is also timely in view of the fact that Lottery Funds are to be a major source of funding required infrastructure for the London Olympics.

It is, of course, important to note that it has never been the intention of the government to ensure an equal distribution of lottery funds in per capita terms. Indeed, it has been recognised that specific towns and cities might be the natural locations for some of the activities (e.g. Arts, Heritage) funded by the lottery regardless of wider welfare issues. However, since there was unequal provision of facilities prior to the lottery, for historical reasons, it is a perfectly valid use of lottery funding to try to reduce these cultural inequalities.

Moreover, spatial disparities in spending inevitably have important economic implications and an understanding of the nature of such disparities is of interest from both an academic and policy perspective. This article examines the spatial distribution of lottery funding both in total and in terms of the six good causes identified in the amended National Lottery Act of 1998, namely arts, sport, heritage, charitable causes, projects to mark the year 2000 and the beginning of the new millennium and, projects on or connected with health, education or the environment (National Audit Office 2000). The analysis begins by assessing the regional and local distribution of awards in broad terms before developing a model to try and explain the spatial pattern of funding. This is then tested using published data and the main findings are outlined. Finally, the article discusses the wider implications of the study.

Lottery Funding in the UK

There are wide variations in lottery funding per head (accumulated through to June 2006) at the broad regional level, ranging from a high of £575 in London to a low of just £163 in the adjacent East of England Government Office region (Table 1). In addition, the North East, Scotland, Northern Ireland and Wales also seem to do well from the lottery, partly replicating the findings for Government expenditure in total (Mackay (2001), [START_REF] Gripaios | Regional spending: a comment on Mackay[END_REF], [START_REF] Heald | The regional dimension of public expenditure in England[END_REF], McClean and McMilllan (2003), HM Treasury (2003) and [START_REF] Gripaios | Government output and expenditure in UK regions and sub-regions: an analysis of the new experimental accounts data[END_REF]). Whether such a distribution is "fair" is an arguable proposition depending on the extent to which one believes that lottery funding should go on iconic projects, wherever they are most properly located, or, whether it should be skewed towards regions of greater need. A further dimension is whether some notion of fairness should take into account the regional distribution of lottery ticket sales. Table 2 shows that there are considerable variations in participation rates and household expenditure across regions. Thus, some 63% of households in the North East played the lottery in the period 2001/2-2003/4, whereas only 39% of London households did likewise. The data suggest that London and Northern Ireland do particularly well from Lottery grant expenditure relative to ticket sales.

There is no breakdown of ticket sales below regional level, but figures on the distribution of lottery expenditure are published for all local authority jurisdictions in the UK. Table 3 shows the relevant figures for the ten highest and lowest lottery expenditure areas in England, the focus of the work in this article. The differences are considerable, ranging from £29 per head in Hart to £4,407 per head in the City of London. Even given the broad regional figures discussed above, it is perhaps a little surprising that London boroughs occupy eight of the top ten places. It might be thought that London would feature particularly strongly in specific categories of expenditure such as Heritage and the Arts, given the importance of these sectors in the capital both at the present time and historically. Table 4 demonstrates that this is the case with London boroughs occupying the first four places for Arts and the first three for Heritage. However, they are also strongly represented for other categories of expenditure that might be expected to be more evenly distributed. Indeed, London boroughs occupy the first eight places for Charitable expenditure, and the first three for spending on Health, Education and the Environment. In every category, there are at least 3 London boroughs in the top ten. There are also differences in the extent of spatial variation for the different categories of funding. In England, the smallest variation in expenditure is for the two categories Health, Education and Environment and Sports while the greatest is for Millennium and Heritage (Table 5).

Explaining the spatial distribution of funding

Given the large variations in Lottery funding across spatial areas both in total and for all specific award categories, it is of some interest to try to identify the factors that might explain the observed pattern. Consequently, a model was developed with the dependent variable defined as the log of lottery funding per head of population from January 1995 through to June 2006 (LLOTTPH).

More disaggregated models were also estimated for arts (LARTS) Charitable Expenditure (LCHAR), Heritage (LHER), Millennium funding (LMIL), Sport (LSPORT) and Health, Education and the Environment (LHEE). The source of the lottery award expenditure is the DCMS Lottery Grants Database and the area of study was the 354 LAs in England. Of course, there are limitations in using this dependent variable as a measure of local benefits as there may be extensive spill-over benefits across neighbouring LAs. These may be particularly extensive in London and the other large metropolitan areas which contain multiple LAs. However, they are also likely between adjacent urban and rural areas where the city limits are tightly defined so as to exclude economic hinterlands.

As far as explanatory variables are concerned, an important point to emerge from the discussion and evidence presented in this paper is that some London boroughs do extremely well from lottery funding. There are a number of reasons that might explain this. One is that London, as the capital city, has always been the main location of Arts, Heritage, Sports and other facilities such as teaching hospitals and major universities that could qualify for awards to upgrade or extend existing provision. Moreover, given that London is a major domestic and international tourist destination, it is likely to be a sensible location for completely new projects, on the grounds that any such projects might be expected to be more viable there than elsewhere in the UK and might add to the general attractiveness of the UK as a tourist destination.

Although the unique features of London are important, one might expect the "London effect" to operate particularly in inner London and specific London boroughs. For example, the City of London might be a special case because of its low population, while Westminster is unique because it is the location of many facilities visited by tourists and Londoners alike. Similarly, Greenwich may attract a high level of lottery funding per head because it has the somewhat ill-fated Dome within its boundaries, whilst Kensington and Chelsea is the location for some key museums and educational establishments. Of course, such location-specific factors may be important outside London. A major northern city such as Manchester may attract funding for similar The impact of the rurality or urban nature of areas is, therefore, a complex issue and two alternative variables are included to try to account for this. The first is population density (LPOPDENS -the log of population per hectare 2002), which may be a good indicator of scale effects in the provision of new services and, therefore of project viability. The second is a measure of the degree of rurality -the proportion of employment in agriculture and forestry in 2001 (AGRI). There are, of course, important limitations to this variable as it may not account for all self-employment. The above arguments suggest that these variables could potentially take a positive or negative sign. The source of the data is Regional Trends for population and the Census of Population for employment.

Given Government priorities and advice to award boards (see, for example, National Audit Office, 2000, Appendix 2), a further potential explanatory variable is the level of deprivation in specific local authority areas. A high level of deprivation might be expected to lead to more favourable treatment, ceteris paribus. The variable used to measure deprivation is the Index of Deprivation residents in an area with qualification levels 4 and 5 (QUAL). Our assumption is that persons involved in writing bids in a professional capacity and/or with better qualifications will be better able to bring their skills to bear on making the case for support by local sports clubs, amateur dramatics and the like.

Interestingly, there could be a different explanation. [START_REF] Feehan | Distribution of UK National Lottery grants across local authority areas[END_REF] also use a similar QUAL variable arguing that it is correlated with income and so can be used as a proxy for income. They argue that more affluent areas will benefit disproportionately from lottery funding as the type of activities which the lottery supports are attractive to high income groups and organisations with high income members will find it easier to generate requisite match funding.

Match funding may certainly be important to demonstrate commitment, to reduce the risk to award bodies and to otherwise "oil the wheels". It is not easy to get direct data on this, even in the case of the public sector, for there are many funds operated by bodies such as the European Union, National Government Departments, Local Authorities and various quangos. Moreover, access to some sources may be dependent on some of the factors hypothesised as determining lottery grant funding. This paper includes a limited attempt to examine this issue directly by assessing whether designation as an Objective One European region affects lottery funding. Thus, a dummy variable (OBONE -eligible authorities coded 1) is used to designate the local authorities eligible for such funding in Merseyside, South Yorkshire and Cornwall. Clearly, given the points by Feehan and Forrest (2007), the match funding explanation may be also partly picked up by other regressors.

Results

The model was tested for the 354 Unitary and District Local Authorities in England using OLS multiple regression. Various formulations were tried but there were clear instances of multicollinearity between two sets of variables. These were AGRI and LPOPDENS which were both used to test for rurality/ scale effects (r=-0.82) and QUAL and PROF (r=0.89) which were used as a proxy for the quality of bids. As a result, it was decided to use one of each pair in a variety of combinations. In practice, it made little difference to the results and so we do not present all variations for either the total model or for individual good causes. In no case was OBONE significant, suggesting that it is either an inadequate measure of availability of match funding or that this is not important in the distribution of grants. Hence it is not included in the reported equations. It also made little difference which inner London dummy was used, so only the equations utilising IL2 are reported.

The results for lottery funding in total are presented in Table 6. The three variations of the basic equation that are reported fit reasonably well for crosssection data, with their explanatory power ranging from 47% to 56%.

Equation (1) passes normality, reset and heteroscedasticity tests, whereas version (2) fails the normality test. There was some evidence of heteroscedasticity in equation ( 3) and the presented results for this equation All coefficients are significant in equation ( 1). Deprivation, the London dummy and QUAL all have the expected positive impact on funding. Of course, the significance of QUAL may either reflect its role as a measure of professional bid writing capability (as we hypothesise), or of some combination of income and availability of match funding as suggested by [START_REF] Feehan | Distribution of UK National Lottery grants across local authority areas[END_REF].

The coefficient on LPOPDENS, for which there was no clear prediction, is negative. This suggests that far from being at a disadvantage, rural areas do relatively well from lottery funding once other factors such as deprivation and the London effect are allowed for. This might also be the conclusion drawn from the alternative equation ( 3), where AGRI is positive and significant. Details of the results for the individual lottery funds are presented in Table 7.

In the reported equations, LPOPDENS is used as the measure of rurality and QUAL as the measure of local professional capabilities. All equations bar Arts and Sports had evidence of heteroscedasticity and hence the standard errors for these equations were corrected using the jacknife method. The explanatory power of the equations differs considerably with the best results being for Charitable expenditure and Arts. For Arts, Heritage funding and Charitable expenditure all variables have the same signs as that for lottery funding in total though IL2 is not significant for Arts or Charitable expenditure.

It should be noted that only the Arts and Millennium equations passed normality tests. This may reflect the fact that large sums of money have gone to a small number of key projects in specific locations. As argued above, this is likely to become even more the case as lottery funds are diverted to finance the London Olympics in 2012.

Millennium funding was directed at special projects in a restricted set of locations and might be expected to follow different rules. This is confirmed as it has one of the weakest fitting equations and only DEPRIV, LPOPDENS and QUAL are significant. Sports spending might be expected to be much more widely distributed and to replicate the pattern of lottery funding in total. In fact, once again only DEPRIV, LPOPDENS and QUAL are significant. Turning finally to Health, Education and Environment spending, the statistically significant coefficients are those on QUAL and DEPRIV.

Conclusions

The welfare aspects of the National Lottery are both interesting and complex.

Relevant issues include the general question as to whether the promotion of lotteries is something which the government should be encouraging given the issue of gambling addiction and the consequences of that for individuals and health service provision. In addition, it is debatable as to whether a National Lottery is the most efficient method of raising money for good causes, whether the effective tax rate is appropriate and whether it is regressive. The previous research outlined above suggests that there are problems regarding all of these issues. In this paper, the main focus has been on a specific welfare issue -whether the regional and local distribution of grant expenditure is "fair". One important aspect of this whether the pattern of expenditure reflects the pattern of ticket sales, a topic which can only be examined at broad regional level. Our conclusion here is that there are clear anomalies. For example, although the North East and East regions have the joint highest average household ticket expenditure per head on the lottery, they are almost at opposite ends of the league table in terms of lottery grants per head. If the East of England does very badly in these terms, then the adjacent region, London, does especially well. Such disparities also apply to Lottery grant expenditure per se, for which London heads the league table followed by the North East, Scotland and Wales. Apart from the North East, this seems to replicate the distribution of overall "identifiable" Government expenditure (McKay 2002) for which Scotland, Wales and London are the top three recipients in mainland Britain. So should we be concerned at these inequalities? It is certainly not surprising that London heads the list -a fact which can be justified on a range of grounds. One is that London, for historical reasons, is the location of many cultural, sporting, heritage, education and health facilities that good cause lottery funding was designed to help. A second is that there are economies of scale in concentrating expenditure in the capital. A third is that improving and concentrating facilities there is the best way of attracting international tourists and their spending. A fourth is that London is the most accessible location for the citizenry of the UK. These are all valid arguments, such that the real question is the extent to which high levels of per capita spending in London can be justified. Is it, for example, justifiable that lottery expenditure per head in the capital is 3.5 times that in the East of England and 3.3 times that in the South East, even allowing for the fact that the residents of (some parts of) the latter two regions have easier access to London facilities than those of more peripheral regions?

There are, of course other welfare issues associated with lottery grant expenditure. One is that, though all UK citizens have access to major projects in London and other large urban centres, they do not have equal access. It is, for example, much less costly in both actual and psychic terms for a resident of London to visit the Millennium Dome or Wembley Stadium or Twickenham than a resident of the South West and that will affect the number of visits. On the other hand, South West residents benefit more from the extensive Lottery Funding given to the Eden Project in Cornwall because it is far easier for them to get there than it is for other UK residents. Of course, there would be winners and losers even if lottery expenditure per head was the same by region because the type and range of facilities supported by the Lottery varies over space. London horticulturalists, for example, may derive little benefit from better sports facilities there or from facilities such as the Eden Project located more than 250 miles away. This paper has demonstrated that, at local authority level, disparities are even more marked than at the broad regional level, both for Lottery expenditure per head in total and for all types of Lottery award. On the face of it, many small, relatively rural authorities seem to do poorly while some in large urban areas and especially in Inner London and specific provincial regional centres do very well. However, the empirical results suggest that whilst levels of deprivation are an important predictor of funding per head, perhaps surprisingly, rural areas do not seem to lose out once this and other factors have been taken into account. The qualifications of residents also seems to make a difference to funding, possibly through influencing the quality of bids and/or, as [START_REF] Feehan | Distribution of UK National Lottery grants across local authority areas[END_REF] suggest, because qualifications are a proxy for high income and high income areas may be more capable than poor in delivering match funding.

In any event, no version of the model picks up more than 60% of the variation in the dependent variable suggesting that other factors play an important role and/or that the variables included here are inadequate proxies for hypothesised determinants. There is certainly scope for further research work on, for example, the extent to which different local authorities put in bids and the extent to which they encourage, promote and improve the quality of bids from organisations in their area through, for example, the employment of dedicated and competent Lottery Officers. Another potentially important avenue of research would be the use of better measures of bid quality, which is likely to be difficult, and of the availability and role of match funding in securing Lottery grants. Finally, there is the issue of the extent to which lottery funding is "additional" in a specific locality. Central Government may, for example, make compensating reductions in other grants to Local Authorities that attract lottery spending. Equally, it may mean that the need for match One problem with examining welfare issues at the spatial level is the appropriate unit of analysis. Regions may be too large while, given extensive spillover benefits, local authority areas may be too small. Using functional city regions would be preferable but data problems preclude this at the present time. However, whatever the scale of the analysis, it seems likely that future studies will throw up even more marked disparities in the distribution of Lottery funds given that ministers have indicated that funds are to be diverted from other potential good causes to that of providing the necessary infrastructure for the 2012 Olympics. As these are to be held very largely in London, it will again be the capital which benefits. No doubt some of the money will find its way to the deprived residents of East London but they are not the only or necessarily the most deprived ones in the UK.

  This study examines the distribution of lottery grants across local authority areas (LAs) in England and Wales to examine the issue of regressivity in lottery taxation. Their model relates the level of awards per capita to various economic characteristics and dummies for London and other Metropolitan areas. They find a tendency for big cities, particularly London, to win a disproportionate share of Lottery grants (especially in the Arts). The percentage of graduates (positive impact) and proportion of the population in socio-economic groups 1 and 2 (negative impact) are other significant explanatory variables.

  and also for unique factors such as the need to rebuild the centre after the IRA bomb.Given the potential importance of the "London effect", three alternative dummy variables were utilised to approximate the effect, with the relevant London areas coded 1 in every case. The first (IL1) defined inner London to represent the City of London alone, the second (IL2) included the City of London, Westminster and Greenwich and the third (IL3) also added Islington and Camden.It is important to recognise that the London dummies only pick up the most obvious outliers in the dataset. More generally, it might be expected that large urban areas would do better than rural or smaller urban ones for the same sort of reasons that might explain the London effect, such as the location of main hospitals, arts facilities, universities and serving markets well beyond local authority boundaries. However, an alternative hypothesis, at least for some aspects of funding, would be that rural and small urban areas might do better from the Lottery because of a greater sense of community enabling them to develop and engender local support for projects. Indeed, such communities may have higher lottery funding per head because many have not had access to other sorts of funds (e.g. Single Regeneration Budget, City Challenge, Neighbourhood Renewal Fund, New Deal for Communities) available elsewhere. Finally, they might do better because of the very fact that, traditionally, they have not had the facilities of large urban areas, leaving open the possibility of remedying the deficit through lottery bids.

( 2004 )

 2004 Average Score (DEPRIV) available from the Department of Communities and Local Government. One important factor which may affect whether specific bids get funding is the quality of the bids. It is not possible to assess this directly. Instead, we use two proxy variables. The first is the proportion of managers and professionals in the local population in 2001 (PROF); the second is the percentage of

  corrected for heteroscedasticity using the jacknife methods. Given these specification issues, version (1) is the preferred variant.

  authorities to switch money from other potential expenditures, especially if good cause projects generate a large amount of local interest.

Table 1 : Lottery funding accumulated to June 2006

 1

	Area	Total	Lottery
			funding
			(per '000
			persons)

Table 2 : Participation in the National Lottery (2001/2-2003/4)

 2

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

Table 5 : Standard Deviation of Lottery Funding to June 2006, by category

 5

	Arts	107,614
	Charitable Expenditure	88,835
	Heritage	119,311
	Millennium	150,729
	Sports	49,945
	Health, Education, Environment	35,961
	Source: DCMS Lottery Grants Database.

Table 6 : Model estimates for total lottery funding per head

 6 Figures in brackets are t statistics; the standard errors (and hence t statistics) for equation (3) were corrected for heteroscedasticity using the jacknife method. *indicates significance at 10% level, ** significance at 5% level.

	Dependent	Llottph (1)	Llottph (2)	Llottph (3)
	Constant	10.9			10.3	9.97
	AGRI				.097
						(7.9)	**
	PROF				.053
						(8.6)**
	DEPRIV	.061			.065	.052
			(16.9)**	(14.3)**	(16.3)	**
	LPOPDENS	.-.17			-.15
			(-7.5)	**	(-5.8)	**
	QUAL	5.2			4.6
			(12.7)**	(9.9)	**
	IL2	1.7			2.0	1.7
			(5.6)	**	(6.2)**	(4.0)	**
	R	2 adj	0.56			0.47	0.55
	F		112.9 **	78.4**	107.2

**

Notes:

Table 7 : Model estimates for category of lottery funding per head

 7 Figures in brackets are t statistics; the standard errors (and hence t statistics) for Char, Heritage, Millen and Health were corrected for heteroscedasticity using the jacknife method *indicates significance at 10% level, ** significance at 5% level. Sproston, K (2003) Report on participation, expenditure and attitudes, National Lottery Commission, London. Smith S (2007) Lotteries as a source of revenue, Centre for Market and Public Organisation, University of Bristol. Stanahan H and Borg M (1998) Horizontal equity implications of the lottery tax, National Tax Journal, 51, 71-78.

	Dependent	Arts		Char		Heritage	Millen	Sports	Health
	Constant	7.1		9.8		10.1	3.8		9.6	8.2
	QUAL	10.1		4.7		7.6	12.7		3.8	2.6
			(11.7)	**	(9.8)**	(7.6)**	(5.3)* *	(5.4)	**	(3.7)	**
	DEPRIV	..09		.06		.097	.14		.05	.06
			(11.9)** (16.4)**	(10.6)**	(8.1)**	(7.5)**	(9.6)**
	LPOPDENS	-.2 (-4.5)* *	-.2 (-9.6)	**	-.50 (-9.3)**	-.36 (-3.4)	**	-.17 (-4.2)	**	.04 (0.9)
	IL2	.6		.84		1.9	-.49		.27	.48
			(0.9)		(1.0)**	(5.3)**	(-.0.1)	(.51)	(0.9)
	R	2 adj	.41		.60		.35	.20		.16	.33
	F		60.45** 130.7 **	46.8 **	21.1 **	17.0 **	43.3 **
	Notes:

Page 29 of 29 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

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

Acknowledgement.

We acknowledge the helpful comments of an anonymous referee on an earlier version of this paper.

G38 -Government Policy and Regulation < G3 -Corporate Finance and Governance < G -Financial Economics, H59 -Other < H5 -National Government Expenditures and Related Policies < H -Public Economics, D63 -Equity, Justice, Inequality, etc. and Measurement < D6 -Welfare Economics < D -Microeconomics, R51 -Finance in Urban and Rural Economies < R5 -Regional Government Analysis < R -Urban, Rural, and Regional Economics