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Chaim Ehrman, received his Ph.D. in Marketing from the Wharton School, Univer- sity of Pennsylvania, 1984. He is an Assistant Professor at Loyola University, and this ... a consultant to corporations and as an expert witness to law firms and to ...
INFORMATION SEARCH FOR FOREIGN DIRECT INVESTMENT USING TWO-STAGE COUNTRY SELECTION PROCEDURES: A NEW PROCEDURE? Chaim Meyer Ehrman * The University of Illinois at Chicago and Morris Hamburg * University of Pennsylvania Abstract. This paper reports the development and testing of a normative model for determining how firms should select the countries to be used in the information search for foreign direct investment. After a subset of countries is selected in the first stage of the decision process, a final selection process chooses the country with the best score within the subset: A "percentile method of subset selection" for singling out clusters of countries for information search and for identifying the best of the subset is presented that performs better than maximum country rankings ("top means") and maximum uncertainty ("top variance") techniques of subset selection. As an illustration, the percentile selection method is applied empirically. The research reported in this paper focuses on the development and testing of a normative model on how firms should specify the countries for which to carry out the information search for foreign direct investment (F.D.I.) The term "foreign direct investment" is used herein as the investment in a manufacturing facility in a host country to produce a given *

Chaim Ehrman,receivedhis Ph.D. in Marketingfrom the WhartonSchool, University of Pennsylvania,1984. He is an AssistantProfessorat Loyola University,and this paper was written while he was an AssistantProfessor at the University of Illinois at Chicago.His currentresearchinterestsareInternationalMarketing,ConsumerBehavior, and subset Selection Proceduresfor BusinessApplications. ** MorrisHamburgis Professor of Statistics and OperationsResearchat the Wharton School. His publications include researchmonographs,books, and numerousarticles in professionaljournals. He has been the directorof researchstudies and has servedas a consultant to corporationsand as an expert witness to law firms and to government agencies.This article stemmed from work begun by Chaim Ehrmanin a doctoral dissertationsupervisedby MorrisHamburg. t The authors acknowledge The Center for International Business Studies at The WhartonSchool of the University of Pennsylvaniafor financial support. The first author acknowledgesProfessorAbba Krieger,Universityof Pennsylvaniaand Dr. Harold R. Shire for guidance,encouragement,and for constructivecomments. Date Received:April 1985; Revised:Augustand September1985; Accepted:October1985.

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commodity with the intention of selling the finished commodity to the markets of the host country as well as other markets. We exclude foreign investment for extractive purposes, such as mining and oil drilling, from our definition of F.D.I. The term "foreign investment" ordinarily includes investment for extractive purposes, such as mining and oil drilling. The focus of our research is to propose new screening methodologies for country selection in foreign investment endeavors. When an investment is geared toward extraction of raw materials, the screening problem may be trivial. A country can be considered a viable candidate for extraction investment if two prerequisites are met. First, the country must have deposits in sufficient quantities to make extraction economically feasible. Second, the government of the host country must be willing to allow foreign companies to extract these resources at a profit. Because the number of countries that meet these two prerequisites is not large, a two-stage screening process may be unnecessary. Therefore, we restrict our focus to foreign direct investment for manufacturing and production, and not to extractive activities. Many countries are. suitable candidates for production and manufacturing, and for these activities country selection is a significant problem. The field of foreign direct investment is becoming increasingly important for both American firms and foreign firms. Many forces that encourage F.D.I. have been documented in the literature. Economic factors include comparative advantages, such as access to factors of production (land, labor, and capital) at lower cost or with greater efficiency than in the firm's country of origin (Caves, 1971). Marketing motives include expansion of market share at the host country (Dymsza, 1966). This expansion may include the opportunity for maintaining close proximity to the market, thereby minimizing the distance between markets and production facilities. Closeness to the market enables the firm to capitalize on future opportunities. Foreign direct investment may also be defensive in nature, safeguarding the firm's current (export driven) market share in the host country (Johanson and Vahlne, 1966). For example, F.D.I. may thwart a competitive advantage of local producers. In addition, F.D.I insulates the firm's exports from protective tariffs as well as from local suspicion and reluctance to purchase imports that threaten jobs of the host country's labor force (Bilkey, 1973). Financial factors, such as interest rates, inflation rates and currency stability and accounting factors, such as governmental tax incentives, are also significant, both at the host country and the country of origin. Finally, there are numerous political considerations. Many firms may have to abandon their foreign investments due to political conflicts with the government of the host country. Texaco and Mobil, for example, had to abandon their operations in Libya because of the political climate. Other political factors include degree of stability of the host country's government, and type of government, for example, repressive, autocratic, or democratic. Political factors represent a constraint, unless the government

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of the host country asks the government of the country of origin to encourage its firms to direct their F.D.I efforts toward the host country. Clearly, many factors encourage and motivate firms to engage in foreign direct investment. A key component in the decision to go ahead with F.D.I. is information on important variables regarding market potential, costs and revenues to be earned in the host country. This information is more reliable if it is collected at the host country by a team of investigators (Aharoni, 1966). Business International, Country Assessment Service (1981) (henceforth abbreviated as BI/CAS) explains why many firms want on-site investigation: A top-level managementteam is sent to a country in which a businessopportunity has been identified. In addition to examining the prospect in question, meetings are usually held with government officials, chambers of commerce, trade associations,etc. This quick review of the country environmenthelps the managementteam decide whether the climate is favorable or unfavorablefor its business. The advantageof this method is that it allows the key decision makers within a corporation to arrive at firsthandjudgments of the business environmentandto relatethese observationsto their particularstrategies(p. 12).

At the same time, this investigation merely compounds the cost-of-information problem. To quote Raymond Vernon (1966): "..

. we cannot af-

ford to disregardthe fact that information comes at a cost; and that entrepreneurs are not readily disposed to pay the price of investigating overseas markets of unknown dimensions and unknown promise." As indicated earlier, this paper develops and tests a normative model for deciding which countries should be chosen for the information search for F.D.I. We assume that the cost of information acquisition is a fixed constraint and allows information search only for a fixed number of countries. Initially we cite descriptive models used as a country rating device by firms pursuing F.D.I. These descriptive models identify the significant variables that firms use to rate potential host countries for F.D.I. After identifying these variables, we present and test a normative model for information acquisition.

The need for efficient information search across countries is not restricted to decisions where F.D.I. is the mode of entry. Information search is necessary for other modes of entry, such as foreign sales office, foreign branch office, joint venture and assembly manufacturing. We address F.D.I because it usually represents the mode of entry with the highest commitment of resources in the host country. Clearly, country selection methodology for information search using F.D.I. is equally applicable to country selection for information search using other modes of entry. We also note that the methodology introduced here is focused on the country selection problem. Many firms enter F.D.I. without a country selection problem. For instance, a firm may be approached by a foreign government to set up a manufacturing facility with extremely favorable conditions. Here, the decision is country-specific: go/no go, and a heuristic to solve country selection for information search may be unnecessary.

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The first section presents a brief review of the related literature. The second section spells out the problem in greater detail. Section three presents the methodology and analysis and the fourth section gives a summary of simulation studies. REVIEWOF THE LITERATURE

In the field of foreign direct investment, there is a keen sense of awareness by line managers of the need for precise information on demographics and key variables for the host country. As noted earlier, the precision of these data is enhanced by on-the-spot investigation teams that collect information for the specific firm. Cost limits the number of countries to be investigated. The criteria for selecting countries for information search represent an important problem. Limiting the scope of on-site information acquisition and identifying "key" countries that are prime candidates for F.D.I. are joint objectives. Several techniques for identifying prime candidate countries have been suggested. Stobaugh (1969a) operationalizes an "imitation lag"' measure for the host country, using a product/country matrix. The combined score by country and by product for imitation lag identifies the leading candidates for foreign direct investment in the appropriate sequence. Another approach examines common decision factors among countries and groups of countries into relatively homogeneous markets. Terpstra (1967) uses cluster analysis for five levels of economic and societal factors to group countries. Sethi (1971) identifies four clusters based on the Euclidean distance of the countries from cluster centroids. These clusters group countries by common values along various dimensions. The clustering of countries can represent a cost-minimizing approach to country investigation for F.D.I. The information search for F.D.I. would focus only on clusters as opposed to countries, limiting thereby the need for information search across all countries of the world. Another technique limits the search for F.D.I. initially to those countries that are similar to the firm's country of origin in terms of culture, language and methods of doing business. The distance along this dimension has been termed "psychic distance" (Beckermann, 1956). In a comprehensive review of F.D.I. literature, Bilkey (1978) highlights this similarity between the host country and country of origin for the firm. His findings show F.D.I. to be motivated by the need to protect the export trade of the host country. Since export trade develops along cultural, language and transactional similarities between the country of origin and the host country, it is appropriate that F.D.I. follow the same pattern. Davidson (1980) postulates an "experience effect." As firms become more experienced in F.D.I., they are willing to invest in countries characterized by greater psychic distance. Davidson corroborates this hypothesis empirically by analyzing F.D.I. using a sample of 1,000 products for 57 "Fortune 500" U.S. firms over a 30-year period. This idea that firms become more global

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in their outlook has also been postulated by Thorelli (1966), Perlmutter (1969) and applied by Wind (1973). Aharoni (1966) has a different opinion. He points out that firms do not look at the entire world when seeking potential location sites for F.D.I. In fact, according to various empirical studies the entire decision-making process that is used in connection with F.D.I. does not follow a rational decision model (Vernon, 1966). Many firms analyze each individual opportunity as it comes. There is practically no analysis of alternative opportunities nor is there a formal limiting of the search process. The search effort is conducted with the objective of gathering information for a proposed project in a specific country for a specific endeavor. Under this scenario, there is no need to develop algorithms to narrow the search across all possible candidates, because other countries do not enter the decision set! Again, these comments are empirically based and do not attempt to suggest a normative approach to the problem of limiting the information search for F.D.I. Stobaugh (1969b) stresses the need for rational decision processes in F.D.I. He applies decision theory using "expected value" (EV) as a means for appraisingreturns on investment resulting from F.D.I. For each investment, the decision maker assigns probabilities to relevant events and projects the revenues/losses that will occur for these events. The expected value of a F.D.I. is the sum of the product of the probabilities of the events and the payoffs associated with these events. These values are summed over the life expectancy of the particular investment. The rational decision process approach enables the decision maker to evaluate and select countries for F.D.I. He may select, for example, the country with the highest EV score. Dymsza (1973) illustrates the use of a weighted linear compensatory model when evaluating countries for F.D.I. Weights are assigned to various predictor variables, and each country in the decision set is rated on these given dimensions. The "best" country is the one with the most points. We have presented a brief overview of methods that firms can use to limit the number of countries qualifying for F.D.I. The aforementioned techniques of establishing ratings for countries have basic premises: countries can be scored for F.D.I. and countries with highest ratings are the most likely ones for the attainment of a firm's goals in F.D.I. We are concerned with the method by which firms should restrict the information search process using a rational decision-making approach. ELABORATIONOF THEPROBLEM

In the previous section, we cited several distinct approaches for narrowing the scores of potential countries suited for F.D.I. These approaches require information on "key" or crucial variables on all countries to be considered as possible location sites for F.D.I. The initial data for this narrowingprocess are not collected in host countries by teams of investigators. The investigative procedure for the host country mentioned earlier refers to the

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"semifinalists," the countries that are prime candidates to be the single "finalist," assuming the firm has the resources to undertake only one foreign direct investment project. The firm interested in F.D.I. evaluates countries based on predictor variables for which there are only imprecise data. The shortcomings using imprecise data include the following. A prime country for F.D.I. initially may score poorly based on imprecise data that the firm has at its disposal. Therefore, this country may be rejected from further consideration. Conversely, a country ill-suited for F.D.I. may initially be a high scorer based on imprecise data and be kept for further costly information collection and analyses. The countries that are to be considered as serious contenders for F.D.I. can be viewed as members of a "subset," a small set of countries from the larger set of all possible countries available for F.D.I. The problem of deciding which countries should be further investigated can be generalized to one of subset selection. A firm interested in F.D.I. may limit the number of countries targeted for additional search and investigation by using data on key variables. These select few countries are defined as members of the subset. The selection of the subset is the first stage of the decision process. The final selection process will choose the country with the best score within the subset. Those countries that are outside the subset are excluded from final selection. In this type of procedure the objective of the decision maker is to select the "best subset" of countries, that is, the subset in which the country with the highest score on average, will outperform or exceed the countries with highest scores in any other subsets. Performance of a subset may be measured by the maximum score of countries within the subset. The decision rule at the second stage is to select the country with the maximum score in the subset once the additional investigative effort has been conducted. It is important to note that few firms actually use the specific narrowing down procedure known as subset selection. Sequential investigation is an alternative approach, in which the firm directs its search effort on the one prime candidate. After a final score has been determined, the decision must be made as to whether the investigation should continue or be terminated. There is no initial determination of a specific subset and members of that set. A case can be made to use subset selection for obtaining information for F.D.I. purposes. There are advantages to subset selection over sequential selection. As stated earlier, information search for F.D.I. usually entails a team of investigators visiting the countries in question. The team's objective is to increase the precision of various estimates for F.D.I. These estimates may pertain to the country's productivity, cost of resources, market size for the given commodity, and so forth. A main component of the cost of conducting field studies is the transportation cost of sending a team of qualified investigators abroad. The costs of time and transportation incurred when gathering information can be minimized by selecting, prior to the search, a subset of countries for information search. Sequential

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selection requires choosing a single country for information search instead of a subset of countries. The decision on continuing the information search to another country is made only after the results of the information search on the first country have been tabulated and analyzed. If the total number of countries to be searched is large, sequential selection becomes a less attractive alternative than subset selection. The incremental costs of sending investigators to foreign countries once a few countries have been investigated becomes prohibitive and may even negate the benefits of the entire search process. High costs of transportation per country search can be the main reason for preferring subset selection to sequential selection. The F.D.I. literature mentions two rules for subset selection. One rule selects countries based on the maximum scores of potential countries evaluated for F.D.I. Another approach focuses on those countries that have high measures of uncertainty prior to site investigation, provided that initial rating scores are sufficiently high. These measures can be calculated by analyzing the evaluative scores of countries prior to the on-site search and by determining the respective mean and variance scores. In this section we apply the percentile method of subset selection to the field of F.D.I. We postulate the following situation. Each country has an associated crude summary estimate of performance potential. We assume that this multivariate estimate can be characterized by a single number using the weighted linear compensatory model described earlier. Prior to an extensive information search, the initial estimate is based on imprecise information and secondary data. Our crude estimate prior to subset selection is equivalent to a single observation of a random variable that can be referred to as a "score." The probability density function of the random variable "score" is characterized by a mean and a variance. We will use a conventional shorthand by stating that each country has a mean score and a variance score. The subset selection process mentioned earlier entails ranking countries in the set by (1) highest mean scores and by (2) highest variances provided the mean scores are "high scoring," i.e., above a control. A shortcoming of the "highest mean scores" is that we ignore measures of uncertainty or dispersion around the mean scores. The "highest variance" uses mean scores as a categorical variable (0-1) and ignores usage of mean scores as interval data. The percentile method incorporates both highest mean scores and variances for each item in the set. Country rankings are based on probabilities of scoring above a control. Initially we define measurement issues, the rating of countries and variable definition. Subsequently, we explain the methodology for subset selection using percentiles. Finally, we note the superiority of percentiles for subset selection compared to other methods and we apply the selection technique empirically.

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Rating of Countries

The countries selected for information search obviously depend on the scoring method used. Many firms do not use a scoring method for a country's desirability of investment. As stated earlier, many firms merely evaluate opportunities for F.D.I. as they are presented. For example, foreign governments, anxious for employment opportunities for their populations may actually solicit firms to invest in their countries. In that situation, a firm will evaluate a country's investment desirability only when the company is approached by the foreign government. This approach obviates the need for information search across a large number of countries before committing the firm to F.D.I. Using a normative approach, we assume that there are many countries with good opportunities for success in F.D.I. whose governments are not actively soliciting firms for investment. It is reasonable to assume that countries with favorable conditions for F.D.I. will not have the need to solicit firms for investment. In a normative approach, we want to evaluate as many "good" countries as possible to determine the "best" candidate for investment.-

At least the following three classifications of variables can be used to rate opportunities for F.D.I.: country-related, firm-related and product-related factors. Country-related variables are discussed below. Firm-related variables can be used to assess the commitment the firm has toward investing abroad. This commitment includes managerial orientation toward overseas markets (Cavusgil and Nevin, 1981a), availability of funds and resources, and high-quality personnel who are willing to relocate abroad.2 Product-related variablesinclude the commodity's position in the product life cycle, the strength of primary demand for the product, strength of the export markets, market size and market share. In our normative approach for F.D.I., we specify country-related variables. Researchers have surveyed executives on the identification of important variables for F.D.I. and the respective weights for these variables. By treating these variables as additive we are able to use a weighted linear compensatory model. There are numerous methods for rating countries. Davidson (1980) rates countries by cultural similarities to the home country. Douglas et al. (1973) focus on managerial orientation as a key ingredient toward country evaluation. We assume the weighted linear compensatory model merely as an example of country rating. We use it to select countries for information collection and analysis. Our method for country selection at the information gathering phase can easily be applied to other mechanisms for country rating. DEFINITION OF VARIABLES

Country evaluation3 for F.D.I. can be classified according to political issues, business or market-relatedissues and economic or monetary-related

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issues. For purposes of working through an illustration of subset selection for information gathering on a country's attractiveness for F.D.I., we will use BI/CAS (1981) weighting scales and variable definitions. An alternative rating scale has been introduced by Schollhammer (1973). He interviewed many executives involved in F.D.I., and has thereby identified mean and variance measures of executives' ratings for weights assigned to variables used to rate countries for F.D.I. BI/CAS simply identifies key variables and their respective country rating scores. We utilize data from Business International Country Assessment Service for our methodology. Clearly, Business International's classification of variables is not sacrosanct. There are many alternatives to its ratings and classifications. However, we utilize BI/CAS because it identifies variables and provides data for a ten-year period across 57 countries. The decision maker may use our subset selection methodology for any other variable identification paradigm. The variable definitions are as follows. There are three major categories: political, commercial and monetary. There are ten political variables, ten commercialization variables and ten monetary variables. Table 1 identifies each variable. Each variable has several possible discrete values as indicated in Table 1. The maximum score of the sum of all country scores for variables in each major category is 100. Methodologyfor Subset Selection

As noted earlier, the literature of country selection for foreign direct investment identifies two techniques for subset selection: highest mean scores; highest variance scores with mean above control. In the highest mean score technique, the best scoring items are chosen for this subset. In the highest variance technique, the mean scores are taken into account as a constraint: any country with a low mean score is omitted regardless of the variance score. Countries with mean scores above a cutoff value are ranked by variances and those countries with the highest variances are selected. The percentile technique may be explained as follows. Every country's score can be converted into a percentile, that is, the probability of scoring below a given value. For example, assume that country "A" has a mean of 10 and a variance of 4, and country "B" has a mean score of 9 and variance of 9. What is the percentile of each country at 12? Assuming normality, the probability that X1, the score for country "A" is below 12 is the probability that a standard normal variate scores below +1 ((1210)/v/4) which is .8413 or the 84th percentile. The probability that X2, the score for country "B" is below 12 is also the probability that a standard normal variables scores below +1 ((12 - 9)/1/9). Although the means and variances of these countries are quite different, the percentile for both countries at 12 are identical. Figure 1 identifies percentiles for different cutoff values.

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TABLE1 BusinessInternationalCountryAssessmentService,1971-1979 Possible Rating Scores Political, 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Legal, Social Factors Political stability Probability of Nationalization Restrictions on Capital Movements Desire for Foreign Investment Limits on Foreign Ownership Limits on Expansion of Foreign-Owned Firms Government Intervention in Business Likelihood of Internal Disorder and Vandalism Delays in Getting Approval Cultural Interaction

(15, 10, 8,5,2) (15, 12, 9,6, 3) (15, 12, 9,6,3) (10, 8,6,4,2) (10,8,6,4,2) (8, 5, 3) (8, 6, 4, 2) (8, 6, 4, 2) (6, 4, 2) (5,3,1)

Commercial Factors 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

Present Market Size as Indicated by GNP Annual Average Real GNP Growth-Past Five Years Annual Average Real GNP Growth-Next Five Years Present Market Sophistication as Indicated by Income Restrictions on Foreign Trade-Next Three Years Availability of Local Capital-Next Three Years Availability of Labor-Next Five Years Stability of Labor-Next Five Years Corporate Tax Level-Next Five Years Quality of Infrastructure-Next Five Years

Monetary/Financial Factors 21. Annual Inflation-Past Three Years 22. Annual Inflation-Next Three Years 23. Number of Devaluations-Past 10 Years 24. Percentage of Devaluation-Past 10 Years 25. Currency Forecast-Next Three Years 26. Overall Balance of Payments-Next Three Years 27. External Debt Position-Next Three Years 28. Reserves/Imports Ratio-Past 12 Months 29. Reserves/Imports Ratio-Next 12 Months 30. Convertibility in Foreign Currencies-Next Three Years

(12,9,6, 3, 1) (6,4,2, 1) (8, 6, 4, 2) (12,9,6,3,1) (12,9,6,3) (12,9,6,3) (12,9,6,3) (12,9,6,3) (8, 4, 2) (6, 4, 2) (8, 6, 4, 2) (16, 10,5,2) (5,4,3, 1) (5,4,3,2, 1) (16, 12, 8, 4) (8, 5, 2) (12,6,2) (6, 4, 2) (8,5,2) (16, 12,8,4)

The percentile represents the chances of scoring below the cutoff value. We see that at cutoff values over 12, the percentiles for country B are lower and hence, country B is a better choice than country A. At cutoff values less than 12, country A is a better choice. Using percentiles for subset selection, we first compute percentiles for all countries at the given cutoff value and rank countries by percentiles. Second, we select the countries with the lowest percentiles. The percentile technique that we have developed initially requires a stipulated minimum performance level, a cutoff value. There is no obvious way to select an "appropriate" cutoff value. An alternative to arbitrarily setting a cutoff value is to fix the percentile value, p*, for the best scoring item in the subset. We describe the use of percentiles to identify a cutoff value. A cutoff value, referred to as a "quantile," is determined by an assigned percentile

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Figure1 Percentilesfor Countries Cutoff Value 8 10 12 14 15 16

Country A. Mean: 10 Variance: 4 16th 50th 84th 98th 99.4th 99.9th

Country B. Mean: 9 Variance: 9 37th 63rd 84th 95th 98th 99th

for the subset maximum, that is, the probability that the best in the subset will score at most the quantile value. Initially, we select a cutoff value and compute percentiles for each member of the set, and select the items with the lowest percentiles. We compute the percentile of the subset maximum as the product of the individual percentiles of items in the subset.4 We have identified the desired quantile or cutoff value if the percentile for the subset maximum is close enough to the assigned percentile. The actual algorithm for the percentile technique for subset selection is described in Ehrman (1984). There are several objective functions that may be used to evaluate performance of subset selection techniques. One objective function is to maximize the expected value of the subset maximum, maximize E[Max Yi], t ieSt where t represents techniques for subset selection, and St is the subset generated by technique t, t = 1, 2, 3. An alternative objective function is to maximize the chances that the subset maximum will perform well, i.e., will perform above a given control c. Mathematically, Maximize Pr(Max Yi t

ieSt

> c). If c is known, Minimize F (c). If p* is given, Maximize c, where Max Yi t ie St

c = F-1 (p*). Mathematically it can be shown that the percentile techniMax Yi ieSt

que is optimal to maximize the p*th quantile for the best of the subset. A detailed outline of the technique is given in Exhibit A. Independence

A key assumption is independence among alternatives. Of course, there are actually dependencies among countries. For example, if the economy in Germany is experiencing a boom period, the economies of France, Switzerland, Luxembourg and Belgium are likely to be affected. Conversely, a depression in Germany will bring downward economic pressures in other countries. The purpose of subset selection in F.D.I. is to focus the information search on a select group of countries. If their economic, political and

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commercial variables are highly related and interdependent, the search for information on one country provides information on the dependent countries as well. There will be unnecessary search if similar countries are in the subset. One approach to solve the redundancy in information search among the countries selected in the subset is to utilize a different definition of individual "items." Instead of using "countries" as the competing "items," we identify clusters of homogeneous countries as the competing "items." The approach omitting "countries" as a unit of measurement has been well documented in the literature of F.D.I. For example, "global firms" or "geocentric firms" (Thorelli, 1966, Perlmutter, 1969, Wind et al., 1973) omit countries as separate entities. The clustering of homogeneous countries is synonymous with the indifference zone and preference zone approach in subset selection as explained by Gupta and Panchapakesan(1979). The heuristic that we use to group homogeneous countries together is cluster analysis. The clustering algorithm groups countries that are related by relevant variables, so that the economic events that take place in one cluster, for instance, may not necessarily have any economic effect in countries of another cluster.5 The clustering is devised to maximize the variance between clusters and minimize the variance within clusters. We identify an item to be a cluster6 of countries rather than one country. We note that the results of clustering are data specific. An important shortcoming in cluster analysis is that we have no allowablenmarginof uncertainty for data used. The assumption of point estimates for every variable without any interval estimation can pose a problem, especially when data are imprecise. The focus of this paper is on new superior methodology. Sensitivity analysis can be used by the decision maker to determine whether data imprecision affects the selection process. For example, after the initial results of cluster analysis have been documented (as in Table 2), the analyst can identify upper and lower bounds for each variable to generate interval estimates. The clustering routine can be used first on the upper bounds and then on lower bounds. If the clustering results for upper and lower bounds are identical, then data imprecision does not affect the results of the clustering algorithm. TABLE2 The Results of the Howard Harris (1967) Clustering Procedure

Membersof Cluster1, the missingdata group, are not shown CLUSTER 2 Argentina (71, 72, 73) Egypt (71, 72, 73) Pakistan (71, 72, 73) Turkey (78) CLUSTER 3 Israel (74, 75, 76, 77, 78, 79) Chile (76, 77, 78, 79) Uruguay (77, 78, 79)

Peru (75, 76) India (73, 74) Portugal (75) Yugoslavia (76, 77, 78, 79) Argentina (77, 78, 79) Mexico (77) Ireland (75, 76)

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TABLE 2 (Continued) CLUSTER 4 Kuwait (76, 77, 78, 79) CLUSTER 5 Iran (71,72, 73) Ivory Coast (73) Nigeria (71,72, 73) Thailand (71,72, 75) Italy (71) CLUSTER 6 U.S. (71, 72, 73, 74, 75, 76, 77, 78, 79) Australia (79) CLUSTER 7 Peru (71, 72, 73, 74) Egypt (74) India (71, 72, 75, 76, 77, 78, 79) CLUSTER 8 Iran (74, 75, 76, 77) Mexico (78, 79) Nigeria (74, 75, 76, 77) Malaysia (73, 74, 75, 76, 77, 78, 79) Italy (72) CLUSTER 9 Australia (74, 76, 77, 78) Denmark (74, 75, 76, 77, 78, 79) Norway (78, 79) CLUSTER 10 Argentina (72, 73, 75) Peru (77, 78, 79) Ghana (76, 77, 79) Zaire (75, 78, 79) Pakistan (74, 78, 79) South Korea (75) Portugal (76, 78, 79) CLUSTER 11 Australia (75) Singapore (75) Belgium (74, 75, 76) Luxembourg (74, 76) Norway (73, 74, 75, 76, 77) Switzerland (74, 75) CLUSTER 12 Brazil (74-79) Egypt (77) CLUSTER 13 Japan (74) Italy (73-79) United Kingdom (71-79) CLUSTER 14 Iran (78, 79) Ecuador (76, 77) Nigeria (78) Philippines (74, 76, 77, 78, 79) CLUSTER 15 Argentina (76) Chile (74, 75) Uruguay (76) Ghana (75, 78) Sudan (77) Indonesia (71, 72, 73) South Korea (71, 72, 73, 74) Portugal (77) CLUSTER 16 Israel (71,72, 73) Venezuela (73) South Africa (71, 72, 73)

Saudi Arabia (74, 75, 76, 77, 78, 79) Venezuela (71, 72) Kenya (73) Taiwan (71,72, 73, 76, 77) Greece (71,72, 73) Portugal (71, 73) Canada (71,72, 73, 75, 76, 78, 79) Belgium (77, 78, 79) Algeria (76, 77, 78, 79) Nigeria (79) Turkey (71,72, 73) Indonesia (78, 79) Ecuador (74, 78, 79) Venezuela (74) Japan (71, 72) Singapore (73) Taiwan (78, 79) New Zealand (74, 76, 77, 78, 79) Finland (74, 75, 76, 77, 78, 79) Sweden (73, 74, 77, 78, 79) Colombia (71) Egypt (78) South Africa (77, 78, 79) Bangladesh (75, 76, 77, 78, 79) Philippines (71,72) Greece (75) Turkey (79) New Zealand (75) Austria (74, 75) Ireland (77, 78, 79) Netherlands (72, 73, 74, 76, 77) Sweden (75, 76) Colombia (74, 75, 76, 77, 78, 79) Ecuador(75) Turkey (74, 75, 76) France (71-79) Spain (75, 76, 77, 78, 79) Argentina (74) Kenya (75, 76) Pakistan (75, 76, 77) Thailand (74, 75, 76, 77, 78, 79) Brazil (72) Colombia (72, 73) Egypt (75, 76, 79) South Africa (76) Zaire (76, 77) Philippines (73) Greece (76, 77, 78, 79) Turkey (77) Mexico (71,72, 73) Ivory Coast (75) Australia (72)

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TABLE2 (Continued) New Zealand (71, 72, 73) Denmark (71, 72, 73) Norway (71,72) CLUSTER 17 Australia (71) Singapore (76, 77, 78, 79) Belgium (71,72, 73) Netherlands (71) CLUSTER 18 Mexico (74, 75, 76) Algeria (75) Australia (73) Portugal (72) CLUSTER 19 Canada (74) Germany (71-79) CLUSTER 20 Argentina (71) Ivory Coast (74) South Africa (74, 75) Zaire (74) Philippines (75) Taiwyan(74, 75) Portugal (74)

Austria (71,72, 73) Finland (71,72, 73) Sweden (71,72) Hong Kong (77, 78, 79) Austria (76, 77, 78, 79) Luxembourg (73, 75, 77, 78, 79) Switzerland (71, 72, 73, 76, 77, 78, 79) Venezuela (75, 76, 77, 78, 79) Ivory Coast (76, 77, 78, 79) Singapore (74) Japan (73, 75, 76, 77, 78, 79) Netherlands (75, 78, 79) Brazil (71, 73) Kenya (74, 77, 78, 79) Sudan (78, 79) Indonesia (74, 75, 76, 77) South Korea (76, 77, 78, 79) Greece (74) Spain (71,72, 73, 74)

Each year is treated as a separate data point for each country. The clusters are assumed to be uncorrelated. The assignment of each country and respective years ('71-'79) to a cluster is shown in Table 3. We now comment on some characteristics of a few clusters to motivate the use of cluster analysis in the context of F.D.I. In order to identify cluster characteristics, we identify performance of cluster members using centroids for each variable. Group centroids for all thirty variables are computed for each cluster. The group centroids for variables 1, 2, 5, 8, 9, 10, 15, 16, 18, 20, 23, 24, 25, and 29 are very close to the maximum possible value for cluster 17. Therefore, cluster 17 contains the high growth opportunity countries in political, commercial and monetary values. In Table 2, we find that from 1976-1979, cluster 17 includes Singapore, Hong Kong, Austria, Luxembourg and Switzerland. Although the group centroids for cluster 19 also scored high on most of these variables, variable 19, projected corporate tax level for the next five years, shows a very low score for cluster 19 and a much higher score for cluster 17. Members of cluster 19 include Japan, West Germany, and the Netherlands. The best scorer on variable 19 is cluster 4, consisting of Kuwait and Saudi Arabia. In the context of F.D.I., a corporation may be faced with a tradeoff of higher projected taxes in one country versus projected restriction on capital movements (variable 3), low per capita income (variable 14), and projected problems in external debt position of another country (variable 27). Some firms may be willing to forego higher potential returns on investment if there are restrictions on capital movements and external debt problems in the host country. Another example of identification of characteristics of clusters through scores of group centroids can be shown with cluster 2, in which group

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TABLE 3 Cluster Identification For Each Country, by Year, Using 20 Clusters Clusters for countries are presented using the Howard Harris (1967) clustering technique. The first column identifies the country, and the other columns are the cluster identification for each country. The mode is the cluster with the highest frequency. A tie is resolved by assigning the country to the most recent cluster, giving greater prominence to most recent data. Countries have been clustered into 2, 3, 4, . ., 19 and 20 clusters. Due to space considerations, we only present results using 20 clusters. The numbers in parenthesis next to each country are for identification purposes only. YEAR 1974 1975 1976 1977 1978 1979 MODE 1972 1973 1971 MIDDLE EAST Iran (501) Israel (502) Kuwait (503) Saudi Arabia (504) SOUTH AMERICA Argentina (401) Brazil (402) Chile (403) Colombia (404) Ecuador(405) LATIN AMERICA Mexico (406) SOUTH AMERICA Peru (407) Uruguay (408) Venezuela (409) NORTH AMERICA U.S.A. (101) Canada (102) AFRICA Algeria (601) Egypt(602) Ghana(603) Ivory Coast (604) Kenya (605) Nigeria (606) South Africa (607) Sudan (608) Zaire (609) ASIA Australia (301) Bangladesh (302) Hong Kong (303) India (304) Indonesia (305) Japan (306) Malaysia (307) New Zealand (308) Pakistan (309) Philippines (310) Singapore (311) South Korea (312)

5 16 1 1

5 16 1 1

5 16 1 1

8 3 1 4

8 3 1 4

8 3 4 4

8 3 4 4

14 3 4 4

14 3 4 4

8 3 1 4

20 20 2 10 1

10 15 2 15 1

10 20 2 15 1

14 12 15 12 8

10 12 15 12 12

15 12 3 12 14

3 12 3 12 14

3 12 3 12 8

3 12 3 12 8

3 12 3 12 8

16

16

16

18

18

18

3

8

8

18

7 1 5

7 1 5

7 1 16

7 1 8

2 1 18

2 15 18

10 3 18

10 3 18

10 3 18

7 1 18

6 6

6 6

6 6

6 19

6 6

6 6

6 6

6 6

6 6

6 6

1 2 1 1 1 5 16 1 1

1 2 1 1 1

1 2 1 5 5

18 15 15 16 14 8 20 1 10

7 15 10 18 14 8 15 1 15

7 12 10 18 20 8 10 15 15

7 10 15 18 20 14 10 20 10

7 15 10 18 20 7 10 20 10

7 15 10 18 20 8 10 1 10

11 10 1 7 20 19 8 11 14 20 11 10

9 10 1 7 20 19 8 9 14 14 17 20

9 10 17 7 20 19 8 9 14 14 17 20

9 10 17 7 7 19 8 9 10 14 17 20

6 10 17 7 7 19 8 9 10 14 17 20

9 10 1 7 20 19 8 9 10 14 17 20

17 1 1 7 15 8 1 16 2 10 1 15

16 1 1

16 1 1

1 7 1 20 20 8 20 1 20

16 1 1 7 15 8 1 16 2 10 1 15

18 1 1 2 15 19 8 16 2 15 8 15

9 1 1 2 20 13 8 9 10 14 18 15

5

5

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TABLE3 (Continued)

Taiwan (313) Thailand (314) EUROPE Austria (201) Belgium (202) Denmark (203) Finland (204) France (205) West Germany (206) Greece (207) Ireland (208) Italy (209) Luxembourg (210) Netherlands (211) Norway (212) Portugal (213) Spain (214) Sweden (215) Switzerland (216) Turkey (217) Utd. Kingdom (218) Yugoslavia (219)

YEAR 1975 1976

1971

1972

1973

1974

5 5

5 5

5 5

20 14

20 14

16 17 16 16 13 19 5 1 5 1 17 16 5 20 16 17 7 13 1

16 17 16 16 13 19 5 1 8 1 11 16 18 20 16 17 7 13 1

16 17 16 16 13 19 5 1 13 17 11 11 5 20 9 17 7 13 1

11 11 9 9 13 19 20 1 13 11 11 11 20 20 9 11 12 13 1

11 11 9 9 13 19 10 3 13 17 19 11 2 13 11 11 12 13 1

1977

1978

5 14

5 14

8 14

8 14

5 14

17 11 9 9 13 19 15 3 13 11 11 11 10 13 11 17 12 13 2

17 6 9 9 13 19 15 11 13 17 11 11 15 13 9 17 15 13 2

17 6 9 9 13 19 15 11 13 17 19 9 10 13 9 17 2 13 2

17 6 9 9 13 19 15 11 13 17 19 9 10 13 9 17 10 13 2

17 6 9 9 13 19 15 1 13 17 11 11 10 13 9 17 12 13 1

1979 MODE

centroids for practically all variables score near the minimum except for variable 17, availability of labor for the next five years. Consider a firm producing a product that requires labor-intensive inputs for production. The planning horizon may be for a short time, perhaps one year. If the host country has a labor force that is ready and willing to work, then the country is a desirable location. American Can is said to be willing to erect a canning facility for any country with a dependable labor force that will guarantee one year's purchase of cans.7 When the planning horizon is so short, the presence of a few key variables may be sufficient for F.D.I. A final important issue affecting subset selection is the assignment of countries to more than one cluster. Clearly, the evaluation of countries for F.D.I. is not over a given year but over a number of years, such as five or ten. Therefore, it is intuitively appropriate to cluster countries by year. However, the same country experiences change and may be in different clusters in different years. For example, Switzerland is in cluster 17 in seven of nine years, U.S.A. is in cluster 6 in nine of nine years. On the other hand, Egypt is in cluster 2 (1971-1973), cluster 7 (1974), cluster 15 (1975, 1976, 1979), cluster 12 (1977) and cluster 10 (1978). Where should Egypt and other countries be assigned? This issue of grouping items that are dispersed among clusters is non-trivial and has evidently not been resolved in the literature. A possible solution is to assign a country to the cluster that country appeared in most frequently. Therefore, Egypt will be assigned to cluster 15. Of course, there are many other ways of resolving this problem. For

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example, points can be assigned for each year with the more recent years receiving the highest points. The cluster with the highest number of points will be the cluster to which the country will be assigned. Alternatively, rating scores can be averagedover ten years, and clustering is based on average country scores. For purposes of illustration, we treat each year as a separate data item and we assume that the weights are the same for each year. Hence, the cluster with highest number of points corresponds to the modal cluster as described above. In the event of ties, a country will be assigned to the most recent cluster. Table 3 identifies the cluster membership for countries over the period 1971-1979 using this approach for 20 clusters. We may calculate mean and average variance scores for each cluster. We assume a naive model of equal importance8 among political, commercial and monetary variables for every year. We can compute an average score for each year by summing points for all variables and dividing by 3 (there are 3 categories, and each category has 100 points distributed among all sub-categories). We then compute the variance across years for each country. A high variance indicates large swings in the performance among variables, and underscores a "high uncertainty" measure, i.e., the need for further search. The mean and variance scores of all members of a cluster are averaged to generate a cluster-wide mean and variance score. We apply subset selection techniques across clusters, in which clusters are selected for information search. We can evaluate different subset selection techniques for clusters of countries by analyzing the "best" term in each subset. Subset Selection for F.D.I.

Since an item refers to a cluster of countries, let Xk = score for cluster

k, k = 1, . . . , 20, ua = variance for cluster k, k = 1, . .,

20, in order to

evaluate the performance of different subset selection techniques. To this end, we consider Wijp = score for country j in year i on variable p;

i= 1,...,9,

and p=1,...,30.

j=1,...,57

The first ten variables correspond to the political variables, the next ten variables to the commercial variables and the last ten variables to the monetary variables. The maximum score for the sum of all political variables is 100. Similarly, the highest possible score for the sum of all commercial variables is 100, and the same is true for the sum of all monetary variables. We assume that all variabes are weighted equally to obtain a 30 score for each country for each year, i.e., Wij

= 2 p= 1

Wijp/3, Wij < 100.

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110

Means and standard deviations are obtained for each country by giving equal weights to each year, i.e., 9

Wj =

i=1

9

= Wij/9 and oj

2 i=l

(Wij

-

Wj)2/8)

Finally, the mean and standard deviation are calculated for each cluster by averaging over all countries in the cluster. Note that only 17 clusters were used. When cluster assignments were computed by mode, three clusters were empty. Formally, and Xk = 2 Wj/nk ; k = 1, .. .17, jeS ok = ( oj) /nk ;k = 1, . . ., 17, where S and nk denote the subset of jeS clusters and number of countries in cluster k. There are other ways of defining Xk and ok in terms of Wijp and the cluster membership. Alternative measures might affect the items (i.e., clusters) in the subset, but they would not affect the structure of the subset selection techniques. The basic assumption of weighting year, country and variables equally in the computation of Xk and ok is, therefore, meant only for the purpose of simplifying the exposition. A percentile p* is subjectively chosen. The problem is to determine the subset S of clusters so as to maximize c, where Pr(max Yk < c) = p*. We keS assume that Yk, the post-information searchscore, is the criterion for selection and Yk - N(Xk, oU). We select members of subsets using highest means, highest variances or percentiles. In Figure 2, the subsets of clusters selected by different techniques are enumerated. We assume a subset size of 4, and apply the subset selection algorithm described earlier for the percentile technique. We assume an assigned percentile of 50. Figure 2 Subsets for F.D.I. Information Search

Cluster number:

Top Mean Score Technique

Top Variance Technique (with Mean > 65)

19 17 4 6

19 5 11 9

Percentile Technique 19 17 11 6

The objective function, maximize c, the quantile for the subset maximum, can be used to evaluate the performance of the three subset selection techniques. The 50th quantile for the percentile technique is 82.34; for the top variance technique it is 80.84, and for top mean score technique it is 82.23. We see that percentiles generate a higher cutoff value or quantile for the subset maximum. This improvement, although seemingly insignificant

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in terms of rating points, can be worth thousands of dollars. The magnitude of improvement offered by percentile subset selection, however, is datadependent. In addition, the required "closeness of fit" between the computed percentile, P(c) and the stipulated percentile, p*, i.e., e, can determine the magnitude of improvement between percentiles and other subset selection techniques. If we allow for a "loose fit" (IP(c) - p* I< e, with e arbitrarily large), the qualities for the top mean and percentile selection techniques may be very close if not identical. A worthwhile goal for a decision maker is to maximize the pth quantile for the subset maximum. The percentile technique is optimal for this objective. The quantile for the subset maximum always will be maximized when using percentiles for subset selection. Although, under certain conditions, other techniques may select a subset in which the "best" scorer has the same pth quantile, the percentile technique will always do as well if not better. Proof for optimality is shown in Exhibit B (page 115). An alternative to maximizing quantiles as an objective function to evaluate performance of subset selection is to maximize the expected value (EV) of the subset maximum. A rationale for this objective function is that oftentimes top managers evaluate productivity by average performance and average rates of return. However, this calculation is very cumbersome assuming a normal distribution. A simulation study with many replications can be used as a means of approximating EV. In an example reported here, we applied a simulation study with 1,000 replications. We found that the subset maximum for the top scorer technique averaged 82. 730; the subset maximum for the top variance technique averaged 81.36 7; and the subset maximum for the percentile technique averaged82.807. Although the percentile technique performs only slightly better than top scores and top variance techniques for subset selection in terms of this criterion, we still notice that other selection techniques do not surpass the expected value of the best of the subset selected by percentiles. The magnitude of improvement of percentile subset selection is enhanced when a high-scoring item with a high variance just misses the "highest mean" subset, but is included in the "percentile" technique. For example, consider a set consisting of clusters #14, 20, 3, 15, 12 in Table 4. If we assume a cutoff value of 54.8, there is convergence among all techniques for subset selection of size 2: we choose items 14 and 3. However, if we make a very minor change in the mean score for cluster 20, from 54.730 to 54.890, a change of +. 16, we find that the "highest mean" subset selects items # 14 and 20, and the "percentile" method selects items 14 and 3. The magnitude of improvement can be estimated using simulation with 5,000 replications. The results are as follows: Members of the Subset: Average of Subset Maximum:

Highest Mean {14, 20 } 58.252

Percentile {14, 3 } 59.356

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TABLE4 AverageMeanand StandardDeviationper Cluster,Using20 Clusters Cluster # 2__2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Mean

S.D.

54.889 77.333 65.889 77.037 48.896 64.383 66.800 48.155 75.334 51.000 64.333 55.667 52.000

8.8065 1.0328 4.3716 2.3489 3.7751 5.5140 3.3847 5.7369 3.4969 3.5636 3.5352 5.5319 5.2691

Israel, Argentina, Chile Saudi Arabia Taiwan U.S.A., Canada, Belgium Peru, Algeria, India Iran, Ecuador, Nigeria, Malaysia Australia, New Zealand, Denmark, Finland, Sweden Ghana, South Africa, Zaire, Bangladesh, Pakistan, Portugal Netherlands, Norway Brazil, Colombia, Turkey France, Italy, Spain, United Kingdom Philippines, Thailand Egypt, Greece

79.675 64.290 80.500 54.730

3.3833 2.5869 4.4130 3.9398

Singapore, Austria, Luxembourg, Switzerland Mexico, Venezuela, Ivory Coast Japan, Germany Kenya, Indonesia, South Korea

We note that the magnitude of' improvement of "percentile" subset selection compared to "highest means" subset selection increases as the variance of an item omitted from "highest means" subset also increases. CONCLUDING REMARKS The objective of this research is to highlight a subset selection technique for foreign direct investment that performs better than other selection techniques. The goal of the selection is to single out clusters of countries for information search and to identify the best of the subset. The analysis presented herein has been directed primarily to subset selection issues. The concept of clustering countries is not new to the field of foreign direct investment. For example, clustering has been applied by Sethi (1972) using marketing decision variables. The percentile technique has been shown to perform better than the top mean and top variance methods for subset selection. The magnitude of improvement using percentiles should not be viewed as insignificant because of a relatively small increment in rating points. A "small" improvement in quantiles for a stipulated percentile may be translated into very significant increases in dollar earnings. Since the total value of F.D.I. for a firm can involve millions of dollars, a small improvement of the dollar value return generated by the subset maximum can be a very significant issue for the foreign investor. We have demonstrated that the percentile technique is optimal for the goal that the best of the subset performs well. Future research endeavors include applying percentiles to actual dollar

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investments and illustrating the superiority of percentiles over other selection techniques using quantiles in terms of actual dollar earnings. NOTES 1. "Imitation lag" is defined as the number of years it will take domestic firms from the host country to imitate the advanced technology of the investing firms, and to compete with the foreign firm. This measure is country-specific and product-specific. The underlying assumption is that the investing firm will lose its comparative advantage once domestic firms can effectively compete using the same technology. 2. Bilkey (1978) and Cavusgil & Nevin (1981b) present a view of the literature on firm-related variables associated with F.D.I. 3. We generally assume that the host country is evaluated on the variables mentioned here. However, each F.D.I. per country per industry is to be evaluated individually. Country-of-origin biases as well as biases for certain product categories will generate different ratings for different investors and for different product groups. 4. It can be shown that the probability that the subset maximum will score at most at a given cutoff value is equal to the product of the probabilities that items in the subset will score at most this same cutoff value. 5. It may be noted that the intuitive argument that unrelated clusters should be members of the subset is consistent with the mathematical assumption of independence. 6. Cluster analysis is used to generate "uncorrelated" groups. A large number (20) of these clusters was selected for illustrative purposes only. The averages of within-group variances were computed for different numbers of clusters. However, cross-validation checks using other clustering techniques such as K-means clustering (BMDP), for example, were not performed. 7. Susan Douglas, oral communication, 1982. 8. The concept of "equal importance" is used only for illustrative purposes. Indeed, it is unlikely that monetary, political and commercial variables have equal importance for all products, for all industries. Clearly, decision makers will want to weight variables by importance, using perhaps ratio weights, so the sums of all weights equal one. The exact weight to assign is a function of priorities of the decision maker, and this may vary by country, by product category. Our objective is to focus on the new methodology; weights assigned will not alter the methodology introduced.

EXHIBIT A Steps for Subset Selection Using Percentiles

Step 1. Collect mean scores and standarddeviation scores for each country in the set. Determine k, the number of countries to consider for further investigation, which is defined as the subset size. FIXED QUANTILE Step 2. Decide on the minimally acceptable "score" for the best country in the subset. We define this value as c. Step 3. Compute, for each member of the set, the likelihood of scoring below this standard. Assuming a normal probability density function, a table of areas under the normal distribution can be used, with the transformation Pr(X < c) =Pr(Z < (c - Li)/ ), with Z representingstandard units in a normal distribution. Select k members with the lowest values to enter the subset.

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JOURNAL OF INTERNATIONALBUSINESSSTUDIES, SUMMER1986

Step 4. Multiply the respective probabilities for all subset members to derive the probability that the "best" item in the subset scores below c. (We assume independence.) Step 5. The "best" subset will have the lowest percentile scores. FIXED PERCENTILE Step 2. Decide on p*, a fixed percentile value, 50th for example. Step 3. Determine upper and lower bounds for the cutoff value by the following procedure: upper bound - maximum of {(zp ( 1/k))(oi) lower bound - maximum of {(zp )(ai)

+

+ui }; i

where k is the size of the subset, and zp and zp (1/k) are the z-values from the standard normal table corresponding to the pth and p( 1 /k)th percentiles, respectively. Step 4. For a first pass, set c, the cutoff value for the subset percentile, as the average between upper bound and lower bound (U.B. + L.B.)/2). Compute percentiles for each member of the set and select k with the lowest percentiles (using Steps 3, 4 for the fixed quantile method). Step 5. Determine if the product (from Step 4) is close enough (userspecified accuracy) to be the fixed percentile value decided on in Step 2. If yes, then terminate and choose the correct subset. The correct value of the cutoff is then the p* quantile of the best item in the subset. If not, update the cutoff value and repeat this process until the computed percentile is within designed range. Algorithmfor SubsetSelection How to select c, the cutoff value to compute the percentile for the subset maximum. Given: I items in a set of size n, p*, the stipulated value for subset maximum percentile, c, allowable error, and the subset, k. Define C as the upper bound for c and C as the lower bound. Assume that each Ith item is characterized by a mean X(I) and standard deviation a(I). LET c =(C + C)/2 (1) DEFINE

I) = I((c

- X(I)/O

(I)),

,[I] -= I in ascending order,

k

and compute P(c) =

4)[I] I= 1

If: Ip* - P(c)t > e and p* < P(c), set C = c, go to (1). If: |p* - P(c)l > e and p* > P(c), set C = c, go to (1). STOP. If: Ip* - P(c)l < e

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115

EXHIBITB Optimalityof PercentileTechniquesfor the ObjectiveFunction: Maximizethe Quantilefor the Subset Maximum,p* the stipulatedpercentile,

(1) Weshow the limits of c: c(LB)< c < c(UB) C(LB)=Max {Zp (ai)+ i }; which is optimal for i= 1, and cannot decrease in i > 1. C(UB) = Max {Zp ( 1/K)(ai) Pr(Yi


c(LB)