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A P P E N D I X D Interpreting the Estimates in the Presence of Missing Data This appendix explains and proves four results cited in the argument in the body of Chapter 6. Result 6.1 Because all notation is defined at the relevant points of the text, I do not repeat definitions here. Equation (6.8) of the text stated a relationship between factor endowments and the structure of trade: Equation (B.2) of Appendix B (see also Chapter 3) gives a relationship between GNP and resource endowments, which when converted to per capita variables is the following: (D.2) Let us suppose that endowments is the factor "sophistication," which cannot be measured across countries. (Of course, any other interpretation of this factor could be used in the following derivation.) However, GNP can be measured. Therefore, solving out endowment S from the above equations: (D.3) This equation has a structure identical to that of equation (6.10) in Chapter 6. Hence, model HG is derived. Because an analogous derivation is possible for model PG, there is no need to provide the details here. 250 Appendix D Result 6.2 In discussing possible biases resulting from the omission of reporting countries from the mirror trade statistics, I stated that, for the group of Western countries used in estimating trade relationships, the average predicted endowments would be equal to the average of those countries' actual endowments (see Chapter 6). Here, I prove this statement. The proof uses the properties of the econometric techniques developed in Chapter 5. The reader is referred to that chapter for definitions of the notation. Let us suppose that we have applied the estimating method to a sample of Western countries and CPEs. One obtains: B = YZ'(ZZ')' (D.4) In exactly the same way that the unknown endowments for the CPEs were predicted, one could obtain predicted values for the endowments of the Western countries. The predictions for all countries can be written in the following form: Z, = Z21 Z2: Then the predictions are derived using equation (5.7) of Chapter 5: Z2 = (B2'W-1 B2)->B2'W~] (Y - B1Z1) . (D.5) Now equation (D.4) is just the matrix representation of a set of ordinary least squares regressions. In such regressions, the average of the actual values of the dependent variable (i.e., the trade measures) is equal to the average of the predicted values of that variable, when those predictions are obtained using the actual values of the independent variables and the estimated coefficients . This means that if J is a vector of l's of appropriate size, then: YJ = [Y1 Y2]J = Bx B2 Z|| Z|2 Z2i Z22> • (D.6) Note that the estimated value of Z22 is used in the above equation since, because the actual values are unknown, these estimated values are the values used in the regressions to obtain B. Using the dot notation to rewrite (D.6), one obtains: YJ = [B1Z11B1Z12]J + [B2Z21 B2Z22]J Hence, (Y - BxZ, )J = [B2Z21 B2Z22]J Appendix A 251 Substituting in (D.5), one obtains: Hence, or (where J, is a vector of l's, whose dimension is equal to the number of Western countries used to estimate the trade equations). Thus, on average, the techniques cannot produce biased predictions for countries with resource endowments lying in the range of the Western countries' endowments. For such countries the R2 's in Table 6.1 give the correct sense of how accurate the predictions are. Result 6.3 This and the following result establish reasonable possibilities for the types of biases that can be expected when the mirror statistics of one country omit some trade that is not omitted from the mirror statistics of the countries used to estimate the trade equations. The first case is rather trivial. Suppose the group of countries whose trade reports are missing from the mirror statistics of this one country, say j, are a representative sample of / s trading partners. Then, it is reasonable to assume the following: a ^ = X'^ and = M"n where an a superscript denotes the actual value of / s trade variables, an o superscript represents the observed value of the trade variables, and Then it follows that otjWl = W"r Hence, the observed structure of the trade variable is: Because the a / s cancel, the observed value of the dependent variable in the trade equation, wu, is equal to the actual value of that variable. There is no effect...


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