Spatial Notes Kelejian

Spatial Notes Kelejian - Spatial Models in Econometrics...

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Spatial Models in Econometrics: Section 13 1 1 Single Equation Models 1.1 An over view of basic elements Space is important: Some Illustrations (a) Gas tax issues (b) Police expenditures (c) Infrastructure productivity (d) Cities and budgets (e) regulation issues (f) exchage market contagion (h) volitility of GDP (i) Spatial spill-overs relating between governments relating to the quality measures Consider a cross sectional framework: i = 1 .... , N Concept of Neighbor: Neighboring units are units that interact in a meaningful way. This interaction could relate to spill-overs, externalities, copy cat policies, geographic proximity issues, industrial structure, similarity of markets, sharing of infrastructure, welfare bene fi ts, banking regulations, tax issues, re-election issues, etc. Example of Geographic Neighbors : For the i th unit, N denotes a ”close” neighbor, NN a neighbor that is less close etc. 1 These notes are mostly based on work I have done with Ingmar Prucha, and parts of them were taken from his notes. 1
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NN NN NN NN NN NN N N N NN NN N i N NN NN N N N NN NN NN NN NN NN The pattern described by N is called a Queen; the pattern described by N and NN is called a double Queen. A weighting matrix: A matrix that select neighbors, and indicates how important each neighbor is. For example, suppose we have N observations on the dependent variable Y 0 = ( Y 1 , ..., Y N ) . Suppose the neighbors corre- sponding to the i th observation ( i th cross sectional unit) are units 1, 2, and 3. Then the i th row of the weighting matrix, W N × N will have non-zero elements: w i 1 , w i 2 , and w i 3 . If P N j =1 w ij = 1 for all i, the weighting matrix is said to be row normalized. Because a unit is not viewed as its own neighbor w ii = 0 , i = 1 , ..., n. Example of use: Let W i. be i th row of W and let X i be a scalar . Then a model such as Y i = b 0 + b 1 X i + b 2 W i. X + ε i X 0 = ( X 1 , ..., X N ) suggest that Y i depends upon X i (a within unit e ff ect), as well as P N j =1 w ij X j , which is a weighted sum of the regressor in neighboring units. Typically, W is speci fi ed to be row normalized so that Y i depends on X i and a weighted average of this regressor corresponding to neighboring units. Clearly, the simplest weighted average is the uniform: e.g. if Y i has 5 neighbors, then the non-zero weights in the i th row of W are all 1/5. Other weighting schemes will be considered below. A Further Elaboration and Some Speci fi cations of w ij Consider again the above relation in scalar terms Y i = b 0 + b 1 X i + b 2 Σ n j =1 w ij X j + ε i Y i = b 0 + b 1 X i + b 2 ¯ X i + ε i ; ¯ X i = Σ n j =1 w ij X j 2
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In this form one can clearly see that w ij relates to the e ff ect that X j has on Y i . That is, the dependent variable depends on the “within unit value” of X , and a weighted sum (which could be a weighted average) of the values of X corresponding to neighboring units.
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