National borders and international trade
The influence of the value of trading partner countries, increasing the distance between them on the level of trade. Transport costs, tariffs as constraints of trade between countries. The modelling of trade flows and the paradox of the "border puzzle".
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Content
 Introduction: model and concept
 Chapter 1. The model specification and its evolution
 1.1 The distance variable
 Chapter 2. My model
 2.1 Methodology
 3. The Data
 4. Results of Estimation
 Conclusions
 References
 Appendix
Introduction: model and concept
This model was first introduced by Tinbergen (1962) and Linneman (1966) as an empirical specification. It is based on the Newtonian theory that the physical points attracts to each with the force which is proportional to their masses and inversely proportional to the distance (squared) between them. In other words, applied to the framework of trade modeling, it can be stated that the trade flow between two countries is equal to the product of their GDPs (`masses') divided by the distance between them.
T_{ij} = A Ч Y_{i} Ч Y_{j}/D_{ij} (1)
Equation (1) is therefore known as a gravity model of world trade after the law of gravity. In fact, economists tried to estimate the equation in more general form, raising all the variables, not only distance, to the power of some positive constants, namely
T_{ij} = A Ч Y_{i}^{a}Ч Y_{j}^{b}/D^{c}_{ij}_{ }(2)
andthe estimates have found that (1) is a good approximation of the reality. (for example, Kepaptsoglou, Karlaftis, Tsamboulas, 2010)
Broad intuition of this result is the following. First of all, large economies tend to spend larger amounts on imports since they can afford it with their higher incomes. They also produce more and thus attract larger shares of other countries' spending as their exports. Thus, other things being equal, the larger either of trading partner is, the higher is the level of trade between them. But of course on practice there are lots of factors limiting international trade flows and countries tend to spend more at home. For instance, such huge trading partners as USA and European Union each attracts only about 2 percent of the other's spending, as said by Krugman (2012). [10]
So, such factors are the socalled impediments or impedance to trade. Examples of them are transporting costs and distance, tariffs and other barriers, borders, cultural dissimilarity, etc. It is useful, since it allows explaining the differences or “anomalies” in the volume of trade. For instance, if considering Figure 1, it can be inferred that Netherlands, Belgium and Ireland all trade considerably more with the USA than other European countries of similar size. Most probable explanation for the first two is the geography and transportation costs. Indeed, both the Netherlands and Belgium have such geographic position that they traditionally have been the “point of entry” to many other European countries.
Figure 1. The Size of European Economies and the Value of Their Trade with the United States Source: U.S. Department of Commerce, European Commission.
Indeed, both countries are situated near the mouth of river Rhine which is one of the most important river routes in Europe. This then opens the trade route to the United States through the Atlantic Ocean. Measured by the tonnage, Rotterdam in the Netherlands is the largest port in North Europe and Antwerp in Belgium is the second largest. This consequently shows the key role played by the transportation costs in the determining the volume of international trade. In case of Ireland, such volume of trade is explained not solely by the location but additionally by cultural affinity. Ireland has the same language as the United States and there are lots of Irish immigrants there. Basically, Ireland is not a typical case due to its policy of being a host country for many multinational corporations, and so there are several factors affecting the volume of international trade in this case.
So, it is obvious that more closely the countries are the higher will be the volume of trade. Typical estimates provide that “a 1 percent increase in the distance between trading partners is associated with a fall of 0.7 to 1 percent in the trade between them” (Krugman, Obsfeld, 2012). This fall is though partly associated with increased transportation costs and partly by diminishing in personal contact between countries. In addition to being even neighbors, the countries can be members of free trade agreements, which eliminate tariffs and other institutional and economic barriers to trade. There are lots of studies that use gravity model to assess the impact of such agreements (for instance, Baier and Bergsrtand, 2007). As stated in Krugman (2012), if a free trade agreement is effective, then the volume of trade would be much higher than the model predicts using the countries' GDPs and bilateral distance. FTAs are widely used tools for enhancing international trade. Examples of them include NAFTA, FTAA, ASEAN FTA, etc.
However, it should be still kept in mind that the FTAs do not make national borders being irrelevant. There is still much more trade between the regions of the same country than between the different countries with the similar distance. This paradox is known as the socalled “border puzzle”. It was first discovered by John McCallum (1995) who has found that the border between Canada and USA in 1988 led to the fact that the trade between Canadian provinces was 2,200 percent (22 times higher) than the trade between U.S. states and Canadian provinces. It was a surprising result that contradicted the economists' prior beliefs.One of the possible explanations to this is the difference in national currencies.
Overall, gravity model is a valuable tool used for analyzing not only general trade flows, but also trade of specific goods (for instance Pelletiere and Reinert (2004)), as well as effects on trade of free trade agreements, currency unions, and common markets (Baier and Bergsrtand (2007), Arghyrou (2000)). Other researchers used it in analyzing effects of trade policy implications, such as foreign direct investment (Gopinath and Echeverria (2004)), domino effects (Sapir (2001)), regulatory quality and export performance (Ivanov and Kirkpatrick (2007)), trade union rights and democracy effects (Kucera and Sarna (2006)), etc.
So far, there was a brief overview of the concept behind the gravity model and some issues involved in trade flows modeling that are in the scope of research.
Chapter 1. The model specification and its evolution
Research community recently has paid a wide interest towards gravity model specification which generally is as follows.The dependent variable is trade flow that is usually expressed as bilateral trade flow or exports. It is explained by the set of explanatory variables that can be roughly divided into two groups:
(1) Factors indicating the size of countries involved in trade, and thus demand for imports and supply of exports, and
(2) Factors indicating the impedance imposed on a trade between countries. (Kepaptsoglou, Karlaftis, Tsamboulas, 2010)
Commonly used proxies for a country's economic and market size (and consequently for demand and supply) are income level, GDP, GDP per capita, population or sometimes area size. For instance, GDP per capita indicates the purchasing power of importing and exporting countries as stated by Sohn (2005). Besides, it should be kept in mind that two countries with considerably different populations may have similar GDPs but totally different economic development. Alternatively, some specifications incorporated size similarity, firstly motivated by Helpman (1987), which is a function of the involved countries' GDPs. Overall there is a strong positive empirical relationship between the size of a country's economy and the levels of its imports and exports.
Impedance factors include variables that additionally affect trade, usually in negative manner. The main resistance elements are transportation costs. These include actual freight transportation costs, tariffs, quality of infrastructure, etc Sohn (2005). Typically, those are approximated as the distance between trading partners, usually taken between the countries' economic centers. It is the great circle distance and is calculated by the longitudes and latitudes of centers. Nitsch (2000), however, proposed a more detailed approach to calculating intracountries distances that expresses distance as a function of country size. A deep analysis of transportation costs is suggested by MartinezZarzoso and SuarezBurguet (2005). They explicitly have estimated a transportation costs equation and then included it in the gravity model. For the variables they used “distance, weighttovalueratio, volume of imports, the case of landlocked countries and common language and infrastructure indices”. They reached the conclusion that high distance and poor infrastructure notably increase transportation costs.
The simplest version of equation is that estimated, for example, by John McCallum (1995):
ln x_{ij} = k + ln y_{i} + ln y_{j} + lnd_{ij} + д_{ij} + е_{ij}_{ }(3)
For his work, McCallum took the data on USCanadian trade in 1988, so that it was a bilateral model (only two countries are included). Here, the two variables are used for the impedance factors: bilateral distance and a dummy variable д_{ij} that illustrates the boarder effect. Correspondingly, dummy д_{ij} was equal to one for interprovincial trade and equal to zero if trade was between a province and a state. Besides, in this particular case of analyzing these two countries, the inclusion of other variable that states for example for cultural affinity or language difference is irrelevant, since USA and Canada are very similar in terms of culture, language and institutions. The data incorporated 10 Canadian provinces and 30 states which accounted for almost 90% of trade between US and Canada. The data was collected from Statistics Canada which included both interprovincial trade flows and flows between each province and state in USA. But, the data on interstate trade flows within the USA was then missing.
The data consisted of exports and imports for each pair of 10 Canadian provinces and of exports and imports between each province and each state out of 50. However, author has decided to limit the scope of calculations and took only 30 states which were defined as 20 largest states with respect to population and all boarder states, in fact remaining 10. As was already mentioned above, those 30 states accounted for 90 per cent of all trade between Canada and US. Finally he got the number of 683 observations, but still it was a crosssectional analysis using OLS method.
Whereas earlier literature, such as Tinbergen (1962) or Linneman (1966) analyzed the trade flows between countries to obtain themultinational effect on trade patterns, McCallum (1995) have used both intra and international trade flows to figure out the effect of state on the trade patterns. The result was quite surprising and, as also mentioned above, showed the intranational trade to be 22to1 compared to international trade, accounted for the boarder effect (McCallum 1995). It was contrary to previous studies, such as Frankel (1993) who gained much smaller figures for regional effects on trade patterns; and more similar to the result of Krugman (1991) on difference between the economic integration of American states and European Union and impact on trade.
Also, 1988 was the last year for the author for which the data was available, so it was taken. But also this was the year when the Free Trade Agreement (FTA) between the USA and Canada was reached. Then there is an argument that such extreme results may change dramatically since thus year due to the effect of FTA. Author addresses this question by brief discussion of evolution of trade and tariffs during the period from 1950 to 1993, but does not give a final answer.
The sharpest criticism of the model was referred to the lack of theoretical foundation. For example, Bergstrand (1985) reported that, “despite the model's consistently high statistical explanatory power, its use for predictive purposes has been inhibited owing to an absence of strong theoretical foundations”. Thus, after the mentioned above early works concerning this model, Anderson (1979) was the first who attempted to derive the economic justification of the model. After him though, there were several alternative theoretical foundations. Anderson for instance (1979) tried to introduce the micro foundations, in particular the assumption that each nation produces the unique commodity which is an imperfect substitute for other nations' commodities. However this assumption could be considered as ad hoc at that time. Thus was a period of 1970s and 1980s when the gravity model became unpopular among economists and fell into disrepute. For instance, Deardorff (1984) stated that the theoretical heritage of the model is somewhat “dubious”.
The next attempt to introduce theoretic foundation came with Bergstrand in 1985 and this time was based on the classical trade theory, namely there was made a connection between the bilateral trade and factor endowments. However, he got a complicated price indices included into his equation which did not manage to calculate. After several years, Bergstrand (1989, 1990) repeated his attempt using the model developed by Krugman and Helpman (1985) which combined old and new theory and instead of using complicated price indices justified by his theory, he approximated them with the existing ones. At that time, as stated by Baldwin and Taglioni (2006), the gravity model “went from having too few theoretical foundations to having too many”.
Sometimes, to impedance factors the factor of remoteness is added. It also indicates the geographic position of countries in another manner. Originally introduced by Deardorff (1998) , remoteness is defined as “the GDP weighted negative of distance between countries”. The estimated regression then becomes:
ln x_{ij} = a + ln y_{i} + ln y_{j} + ln d_{ij} + lnREM_{i} + lnREM_{j} + д_{ij} + е_{ij }(4)
where remoteness of region `i' equals to:
REM_{i} = (5)
This variable was introduced to reflect the distance of a region i on average from all other regions except j. These variables were quite commonly used in the gravity literature, but again they were not enough proven by the economic theory. As shown by Anderson and Wincoop (2003), adding the remoteness variable adds little explanatory power based on the adjusted Rsquared.
Another recent wellknown attempt to provide micro foundations to the model is Anderson and Van Wincoop (2003) which was similar to Anderson (1979). Their contribution to the theory was, as they stated, mainly methodological which made it possible to conduct a comparative statics analysis. Anderson and van Wincoop suggestedestimating a slightly different equation and the innovation is concerned mostly with formalization of those impedance factors to trade.They considered the endogenous trade costs and institutional barriers to trade. The factor of “multilateral resistance” was implemented that reflected the average barrier of each of two trading partners to trade with all other partners. It was based on the trade theory that states that the costs of bilateral trade between two countries are affected by the trading costs of each country with the rest of the world. They also, as McCallum (1995) analyzed the boarder effect on trade and claimed that they have solved the “Boarder puzzle”.
They used the parameter у that refers to the constant elasticity of substitution (CES) of all goods on which the foundation of Anderson (1979) was based. However, Anderson and van Wincoop (2003) suggested a manipulation of CES expenditure system to simplify the model.They tried to provide micro foundations to the model. They first solved the utility maximization problem of consumers subject to their budget constraint to obtain the nominal demand for each region. Then they have implemented the market clearance condition to solve for the prices and substitute them into the demand function. Besides, this implementation of market clearance constraints was the crucial point that allowed estimating the model which made it operationally useful. After further substitutions the final equation becomes of the following form:
x_{ij} = (6)
subject to:
P_{j}^{1у} = У_{i} P_{i}^{у1 }и_{i}t_{ij}^{1у } , for any j (7)
This expression significantly simplifies the equation derived by Anderson (1979) in his previous work. It is also important to notice that the assumption of trade barriers' symmetry was made to simplify the derivation. The multilateral resistance is then expressed by price indices P_{i} since they are dependent on all the bilateral resistances. Following, the index will rise if barrier to trade with all trading partners rises. Then it becomes easy to show that the increase in multilateral resistance of country j if it is an importer increases its trade with country i. In other words, given a particular bilateral barrier countries i and j, the higher barrier between j and the rest of the world reduces relative prices of goods from i and thus increases imports. Similarly, the conclusion could be made if country i is an exporter. Thus, the key proposition that can be made from theoretical foundation of Anderson and van Wincoop (2003) is that tarde between countries is affected by the relative barriers to trade.
After taking logarithms, the following equation was estimated:
ln x_{ij} = k + ln y_{i} + ln y_{j} + (1  у)с ln d_{ij} + (1  у) lnb_{ij}  (1  у) ln P_{i}  (1  у) lnP_{j} + е_{ij}_{ }(8)
As can be seen the first two terms are identical to McCallum equation (three with constant) while others are different. Additionallyto the distance, the price indices of trading countries are added as well as a barrier b_{ij}whichstate for the impedance factors. Price indices represent multilateral resistance while trade cost factor t_{ij} stays for bilateral resistance, where
t_{ij}= b_{ij}d_{ij}^{p}.
Here also the problem arises of different ways of computing price levels across countries and thus the failure of PPP (Purchasing Power Parity).The presence of nontraded goods also adds some inaccuracy.
In their work, Anderson and van Wincoop (2003) used data on USA and Canada, the same as McCallum (1995) and on other 20 industrialized countries. The trade flows were combined from four data sets. The first is trade between Canadian provinces and the second is stateprovince international trade (both from the Statistics of Canada following McCallum). The third is trade between states, the one which was missing in McCallum (1995), taken from Commodity Flow Survey by the U.S. Census. And the last is international trade between other 20 industrialized countries taken from IMF. Since those sets were quite different, the adjustments were required to make them more comparable.
As a method of estimation the Nonlinear least squares was used. In other works, however, such as Baier and Bergstrand (2009) an alternative technique was used, namely the fixed effect. It is a more simply way of treating the multilateral resistance.
Moreover to these studies, there were attempts to consider as an impedance factor the exchange rate between trading partners, since its volatility was expected to negatively affect the volume of trade flows. Examples are Rose (2000), Arghyrou (2000), Egger (2002), where the first explicitly considered in his paper the monetary union and the effect of common currencies on the level of trade flows.
Many other impedance factors, that are expected to promote or, vice versa, restrain trade, are included as dummy variables. Most commonly used factors are contiguity of two countries, common language, and landlocked country. Participation in free trade agreements or customs and monetary unions are also used as dummies. Even if the use of dummies has once been criticized for the fact that they may include the effects of other not relevant factors (Polak (1996)), they compose a most typical approach for analyzing such kind of impedance factors. Similarly, the effects of having the same currency, being parts of the same nationalities, or being colonies in the past or in present are also caught through dummy variables.
1.1 The distance variable
One variable which is of a particular interest is the distance between the trading countries. The question of how to calculate distances is crucial for obtaining correct estimates for trade impedance factors. Though, it is a quite nontrivial question how to define the distance between two countries.
The common approach in gravity literature is to calculate distance between countries as the distance between countries' “centers”. In the earlier works the simplest case to define the center was presented, specifically just the distance between the capital cities. In some countries, where the capitals were just nominal, the more appropriate way was to take the largest economic center instead. Occasionally, just a centrally located large city was taken. If countries are small and economic activity is concentrated, it is not a big problem. But for large countries, such as Russia or US, which are geographically or economically dispersed, there is a great case for concern.
Thus, there were attempts to deal with this problem. For instance, Head and Mayer (2002) decided to go further in obtaining country level distances in a consistent way. They argued that the mismeasurement of distance in the existing literature may lead to the fact that boarder effects are also mismeasured which is a problem of “systematic overstatement” or “illusory” boarder and adjacency effects. They tried to calculate interstate bilateral distances in more complicated way using the intercity data based on a geographic distribution of population inside each state. The idea was to calculate weighted distance between two states based on bilateral distances between largest cities of these two states. In Head and Mayer (2002) they just took two cases, namely the arithmetic mean and the harmonic mean of intercity distances as special cases. They then argue that they have managed to shrink the boarder effect puzzle, though not eliminate it.
Chapter 2. My model
This work focuses particularly on the distance variable and corresponding issues involved. In it I am analyzing which is the best way of measuring the economic distance between trading partners. To do it I have suggested my own approach before which made a comparative analysis of existing ways of measurements to make a starting point. My work is in a way similar to Head and Mayer (2002), but I have generalized the approach which will be described later.
I will start with pointing out the specification of the model I used and describe the variables. I have chosen the standard gravity model since there is a huge range of contradicting alternative specifications, described in previous chapter, though without any one generally acknowledged among others. It can be represented by the following equation, similar to (1):
x_{ij} = в_{0} Ч Y_{i} Ч Y_{j}/D_{ij, }_{ }(9)
where
Y_{i}=pop_{i}*gdpcap_{i. }_{ }(10)
The trade flows x_{ij} is bilateral, meaning that the one observation involves two countries. Another point is that the flow between country i and country j is not the same as vice versa between country j and country i. Next question is which attributes of exporter and importer to include. Due to the structure of data, the values of simultaneously population and GDP per capita for one country do not change which give the basis for perfect multicollinearity. Therefore the one of two should be used or GDP alone, which are close substitutes.
For the estimation, then, the general approach is to take logs of both sides to get an equation that is linear in parameters and set an error term that represents the remaining variation in trade flows. I got the following equation:
lnx_{ij} = lnв_{0} + в_{1 }ln gdpcap_o + в_{2 }ln gdpcap_d + в_{3 }ln d_{ij} + в_{4 }contig + в_{5 }comlang_off + в_{6 }comcurr + е_{ij}_{, }(11)
where:
gdpcap_o  is the GDP per capita of country 1 (country of origin), and
gdpcap_d  is correspondingly the GDPper capita of country 2 (country of destination)
d_{ij } is the bilateral distance between trading countries (while estimating I am using four different ways of distance measurements, which would be discussed later)
contig, comlang_off, comcurr  are dummy variables representing correspondingly contiguity of two countries, common official language and common currency.
There is a huge range of dummies included into specification. I have chosen to use most commonly appropriate ones. One should pay attention to the dummy indicating contiguity (or adjacency). Many studies include it but since it is rarely of interest they include it without explanation. One argues that it might be related to freight costs, since adjacent countries are directly connected via roads or railways; or to political costs which are beard every time while crossing the border. However, neither argument is really justified. Those about freight costs suggest that the different modes of transports should be instead interacted with the distance and the latter states that the number of crossings should be calculated. Thus, later I will show that contiguity tend to have a positive effect on trade since the distance between neighbors is systematically lower that average distance and often lower than center to center distance as well.
Also the assumptions of identical preferences and perfect competition on goods markets are made.
Now I switch to describing the main part of my work that suggests a method how to calculate economic distance between two countries in the most accurate way. The basic idea is similar to Head and Mayer (2002), namely to obtain the intercountry distances based on geographic distribution of population inside each country. Indeed, the tracks to the final destinations of international trade flows are not as simple as may seem and may change their paths many times within the country of importer. The final point of destination mainly depends on the consumption at that point of the imported goods and thus on the population density inside the country. The same inference could be made about production and the exporter. Thus, the population is a good proxy for economic activity.
Then, if considering international trade more deeply it is the trade between the largest cities of two countries, which are the most economically active. So, to obtain the accurate measurement of distance between the two countries one should take the average of all the distances between trading cities in each country. The intercity distances are then being weighted by the share of the city in the overall country's population. Formally it could be written as follows, as generalized mean of citylevel data:
d_{ij} = [У_{k belong to i}(pop_{k}/pop_{i})У_{l belong to j}(pop_{l}/pop_{j}) d^{и}_{kl}]^{1/и}^{ }(12)
The parameter и could be viewed as a sensitivity of trade flows to bilateral distance between cities and is of a particular interest. It is an important issue which value it should take.
So far, the researchers have used no more than two variants of calculations, namely the arithmetic mean and harmonic mean, as for the distance measures; but there were no analysis on which one is strictly preferable. Also, in previous works the researchers concentrated more on solving the boarder puzzle (or finding the boarder effect) and did not pay much attention to the choice of distance measurement.
Therefore the impact of my work is, first that I did the precise comparison of different ways of distance calculations and second, introduced my own way of distance calculating by searching for an optimal parameter theta (и). The method how I did it and the estimation results are provided below.
2.1 Methodology
I have used several different methods, namely OLS, Random effects method, and Between effects method. I started my analysis by running several regressions in order to choose the best measurement of existing distances. The results of it are reported in the Table 2. I estimated the relationship between trade flows, GDP of both the country of origin and the country of destination, and distance variables which were described earlier. For distance variable there were four different ways of measures and I was testing the same regression specification with different measurements. They are distance between capitals (distcap), between largest cities (dist), arithmetic mean of intrecity distances (distw) and harmonic mean of intrecity distances (distwces). Since the specification did not change the Rsquared was the best criterion of comparison. The results will be described later.
Next, for estimating the parameter theta I have introduced the special procedure, but before describing it I will first explain how I have obtained the figures. For calculating intercities distances I have used the Haversine formula presented below:
Figure 2. The illustration of the distance between any two points on Earth.
For simplicity I made an assumption of equality of the Earth radius for all the cities, though it has some variation in it.
Now I will describe my procedure of theta estimation. I took the 12 cities in each country. After obtaining city level distances I have calculated the weight of each city according to its population. Then using the range of theta I have calculated 150 different distances for each particular pair of cities and then for countries. Then I made a cycle procedure of 150 linear regressions and compared them with respect to goodness of fit. Since the specification was the same for all regressions I again used the criterion of Rsquared. Namely, I have chosen among these regressions the one with maximum Rsquared and found which value of theta corresponds to this particular regression. But before the estimation of cycle of my regressions, I first need to choose the range for the parameter и. This I made based on the conclusion of panel data estimation.
3. The Data
partner trade flow tariff
The data used in this paper I have collected from different resources. The main source from which the economic data was taken, is the CEPII web site, which is a French independent institute of research in the sphere of international economics. The data base of geographical coordinates of cities, more precisely latitudes and longitudes, was taken from WORLD CITY LOCATIONS DATABASE prepared and made publicly available by H.U. Bahar. The population data of large cities were gathered by hand The populations of cities were taken, instead of agglomerations, to make it more standardized since not in all countries the data on agglomerations are available and they are of different sizes in statistics across the world. As populations are used to calculate relative weights, this point is not crucial. ^{}, since there is no universal database of citylevel population across different countries. And finally, the intercity distances were calculated by my own using latitudes and longitudes and using the formula described in the section Model. All these sources are available at the end of the paper in the Appendix.
In my work I have used the sample of 24 largest countries in terms of their territory all other the world. As it was stated earlier, the problem of distance measurement is most severe for large countries, such as Russia or US, which are geographically or economically dispersed. Since my analysis is devoted to measuring distance, the criterion of large territory was chosen to be the leading so that the variation in intracountry distances would be the biggest and most representative. Besides, in Head and Myer (2002) they used data only on United States and some members of European Union. The list of countries is given below (in the Appendix the corresponding ISO codes in three characters for each country are given):
Table 1. List of countries in the sample.
1 
Russia 
13 
Bolivia 

2 
Canada 
14 
Venesuella 

3 
China 
15 
Pakistan 

4 
United States 
16 
Turkey 

5 
Brazil 
17 
Chile 

6 
India 
18 
France 

7 
Argentina 
19 
Ukraine 

8 
Kazakhstan 
20 
Spain 

9 
Mexico 
21 
Sweden 

10 
Indonesia 
22 
Japan 

11 
Peru 
23 
Germany 

12 
Columbia 
24 
Finland 

The reason why the whole world is taken and not a particular region is so as to the results would be more universal and representative. But, as can be seen, there is no Arabic and African countries since for simplicity they were excluded from the sample, because in those regions there may arise the problems of data collection or it standardization.
Also, when the countries are far away from each other, the difference in distance measures is becoming less important. Intuition is the following. When the countries are far enough all the points of one country are relatively far from all the points of another country making different ways of calculations being irrelevant. To make a definite criterion, I made an assumption that 7000 km is quite far for the trade and thus excluded from the sample the pairs of countries that are more than 7000 km away from each other.
It should be noted, that data on trade flows is bilateral, meaning that the one observation involves two countries. Also, it should be specified that the flow between country and country j is not the same as vice versa between country j and country.
As I mentioned my analysis is divided in two major parts. I start from estimating the panel including 220 pairs of countries in the period of 2001 to 2006. The results are reported in Table 2. I sorted 24 largest countries and their corresponding trading partners from the above mentioned 24. Further I limited the sample on the basis of the distance: I excluded pairs with trade flows greater than 7 000 km. As it later could be seen, the best fitted models are associated with the variable distwces, where theta is equal to 1.
In order to compute distances in my way, for the purpose of simplicity of calculations I reduced the sample. In each country I need to take a representative number of largest cities, I took 12. Then, among 24 countries, I took not all possible pairs of partners, with 6 largest trading partners, which have resulted in the number of 57 observations. Note that, due to the structure of the matrix, there were exact multicollinearity between variables pop_o and gdpcap_o and I had to exclude the variable of population. Since I finally included 6 variables, such a sample size was enough.
4. Results of Estimation
In this section I turn to the regression results. As I mentioned earlier I started my analysis by running several regressions using OLS, RE and Between effects methods in order to choose the best measure among existing distances. The results are shown in the table 2. The measures used are distance between capitals (distcap), between largest cities (dist), arithmetic mean of intrecity distances (distw) and harmonic mean of intrecity distances (distwces). I will repeat, that since the specification did not change the Rsquared is the best criterion of comparison.
Note that the highest Rsquared is while using distwces  the harmonic mean of city level data in all three methods.
Table 2. Panel data estimation. Coefficients standard errors are reported in parentheses. * significant at 10% 

Method used 
OLS 
Random Effects 
Between Effects 

Dependent var: flow 
Coef. Std. 
Coef. Std. 
Coef. Std. 
Coef. Std. 
Coef. Std. 
Coef. Std. 
Coef. Std. 
Coef. Std. 
Coef. Std. 
Coef. Std. 
Coef. Std. 
Coef. Std. 

gdp_o 
0.0061149 (0.000316) 
0.0060808 (0.0003139) 
0.006 (0.00032) 
0.0060626 (0.0003123) 
0.0062 (0.0004) 
0.006211 (0.000413) 
0.006208 (0.000414) 
0.006206 (0.000412) 
0.006072 (0.000772) 
0.006106 (0.000776) 
0.006061 (0.000775) 
0.006053 (0.000768) 

gdp_d 
0.0079233 (0.000316) 
0.0078893 (0.0003139) 
0.008 (0.00032) 
0.007871 (0.0003123) 
0.0088 (0.0004) 
0.00877 (0.000413) 
0.00877 (0.000414) 
0.008764 (0.000412) 
0.007866 (0.000772) 
0.007901 (0.000776) 
0.007856 (0.000775) 
0.007848 (0.000768) 

contig 
10974.4 (1666.957) 
10854.76 (1648.861) 
10761 (1670.99) 
9598.605 (1676.421) 
10595 (4040) 
10491.93 (3998.979) 
10379.68 (4052.395) 
9223.623 (4065.818) 
10866.42 (4015.785) 
10986.62 (4060.653) 
10773.39 (4070.224) 
9610.652 (4082.261) 

comlang_off 
6515.676 (1563.007) 
6605.553 (1555.176) 
6898 (1559.96) 
6407.485 (1549.4) 
7146.1 (3748.2) 
7279.945 (3727.388) 
7591.013 (3738.027) 
7099.589 (3713.365) 
6583.879* (3789.026) 
6495.365* (3808.707) 
6876.017* (3801.208) 
6385.249* (3774.362) 

distw 
3.706688 (0.382217) 
3.8152 (0.9238) 
3.77451 (0.889381) 

dist 
3.777481 (0.3651599) 
3.8701 (0.88444) 
3.70319 (0.931148) 

distcap 
3.724 (0.37901) 
3.82222 (0.91799) 
3.72057 (0.923229) 

distwces 
4.083658 (0.3743325) 
4.17101 (0.907552) 
4.08085 (0.911549) 

cons 
4593.505 (1795.197) 
4312.481 (1686.46) 
3893 (1716.27) 
5876.706 (1743.775) 
4016.5 (4329.6) 
3597.996 (4045.011) 
3177.603 (4120.584) 
5118.852 (4182.792) 
4335.444 (4108.791) 
4612.094 (4373.721) 
3916.341 (4181.839) 
5901.054 (4247.75) 

Rsquared 
0.4583 
0.4633 
0.459 
0.4678 
0.4592 
0.4641 
0.4602 
0.4686 
0.4653 
0.4604 
0.4613 
0.4698 

Now come the results of my procedure. As I already mentioned in methodology section, I took the 12 cities in each country. After obtaining city level distances I have calculated the weight of each city according to its population. Then using the range of theta I have calculated 150 different distances for each particular pair of cities and then for countries. Then I made a cycle procedure of 150 linear regressions and compared them with respect to goodness of fit. Since the specification was the same for all regressions I again used the criterion of Rsquared. Namely, I have chosen among these regressions the one with maximum Rsquared and found which value of theta corresponds to this particular regression.
Using Matlab I ran a cycle of 150 equations with theta parameter varying from 1.5 to 0. Since 1 was already the best fitted variable, I decided to run the model with theta varying in the range that includes 1 to check it. The optimal theta is equal to 1.23
Table 3. Results of “optimal” regression from the cycle
variable 
beta 
standard errors 
tstat 

const 
486,05 
17873,55 
0,02719 

contig 
23842,73 
12298,01 
1,93874 

comlang_off 
8432,38 
13821,98 
0,61007 

gdpcap_o 
1,91 
0,37 
5,07089 

gdpcap_d 
1,77 
0,36 
4,90704 

comleg 
25363,18 
16172,42 
1,56830 

Distance, Theta = 1.23 
12,42 
3,35 
3,7077 

The next graph represents the dynamics of Rsquared for 150 different regressions with respect to variation in theta paremeter. The maximum value of Rsquared is at и = 1.23. Coefficients tend to be significant except the dummy of common language. Note that while estimating the optimal regression, I did not include the dummy variable common currency since none of the countries in the sample had the common currency and it is useless to include the variable consisting of only zeros.
Figure 3. The dynamics of Rsquared with respect to parameter и.
At least, it is a local maximum. The inference whether it is a global maximum is in scope of further research.
Conclusions
I have compared the different ways of distance measurements and concluded that the changes in measurements have significantly improved the goodness of fit of the regression with the same specification. It is important to use the correct distance measure since it is closely connected with the bias of boarder effects.
The impact of my work is that it provides the first analytical comparison of different distance measurements. My results showed that the best way of distance measuring is taking the harmonic mean of citylevel bilateral distances. It is consistent with both parts of my empirical work. The cycle procedure has also obtained the result of theta that is close to 1.
Also an impact was to test distance measurements on international data, since in the earlier work, those calculations were introduced on statetostate trade flows.
Important conclusion is that variable contig (contiguity) tend to be significant and it has a positive effect on trade contradicting to previous results. Thus it could be inferred that the adjacency of countries leads to the decrease in distance and that the distance between neighbors is smaller systematically than the average one.
But still, this result needs to be repeated on even larger sample and future research will endeavor to perform the NLS procedure for obtaining theta.
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Appendix
http://www.cepii.fr/CEPII/en/bdd_modele/bdd.asp website of CEPII which is a French independent institute of research in the sphere of international economics.
askbahar.com/2010/07/02/worldcitylocationsdatabase/ (World Cities Locations)
Code for Matlab for optimal theta search.
clear all
close all
clc
alldata=importdata('C:\MATLAB\final.xls');
tf=alldata(:,1);
first_theta_column_number=7;
for i=first_theta_column_number:size(alldata,2)
x=[alldata(:,2:6) alldata(:,i)];
reg=regstats2(tf,x);
statRsq(ifirst_theta_column_number+1,1)=reg.rsquare;
end
[maxRsq, coord]=max(statRsq);
Optimal_Theta_Column=coord+first_theta_column_number1
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