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Unformatted text preview: This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. This may not be resold, copied, or distributed without the prior consent of the publisher. 92 CHAPTER 16 SOLUTIONS TO PROBLEMS 16.1 (i) If α 1 = 0 then y 1 = β 1 z 1 + u 1 , and so the righthandside depends only on the exogenous variable z 1 and the error term u 1 . This then is the reduced form for y 1 . If α 1 = 0, the reduced form for y 1 is y 1 = β 2 z 2 + u 2 . (Note that having both α 1 and α 2 equal zero is not interesting as it implies the bizarre condition u 2 – u 1 = β 1 z 1 − β 2 z 2 .) If α 1 ≠ 0 and α 2 = 0, we can plug y 1 = β 2 z 2 + u 2 into the first equation and solve for y 2 : β 2 z 2 + u 2 = α 1 y 2 + β 1 z 1 + u 1 or α 1 y 2 = β 1 z 1 − β 2 z 2 + u 1 – u 2 . Dividing by α 1 (because α 1 ≠ 0) gives y 2 = ( β 1 / α 1 ) z 1 – ( β 2 / α 1 ) z 2 + ( u 1 – u 2 )/ α 1 ≡ π 21 z 1 + π 22 z 2 + v 2 , where π 21 = β 1 / α 1 , π 22 = − β 2 / α 1 , and v 2 = ( u 1 – u 2 )/ α 1 . Note that the reduced form for y 2 generally depends on z 1 and z 2 (as well as on u 1 and u 2 ). (ii) If we multiply the second structural equation by ( α 1 / α 2 ) and subtract it from the first structural equation, we obtain y 1 – ( α 1 / α 2 ) y 1 = α 1 y 2 − α 1 y 2 + β 1 z 1 – ( α 1 / α 2 ) β 2 z 2 + u 1 – ( α 1 / α 2 ) u 2 = β 1 z 1 – ( α 1 / α 2 ) β 2 z 2 + u 1 – ( α 1 / α 2 ) u 2 or [1 – ( α 1 / α 2 )] y 1 = β 1 z 1 – ( α 1 / α 2 ) β 2 z 2 + u 1 – ( α 1 / α 2 ) u 2 . Because α 1 ≠ α 2 , 1 – ( α 1 / α 2 ) ≠ 0, and so we can divide the equation by 1 – ( α 1 / α 2 ) to obtain the reduced form for y 1 : y 1 = π 11 z 1 + π 12 z 2 + v 1 , where π 11 = β 1 /[1 – ( α 1 / α 2 )], π 12 = − ( α 1 / α 2 ) β 2 /[1 – ( α 1 / α 2 )], and v 1 = [ u 1 – ( α 1 / α 2 ) u 2 ]/[1 – ( α 1 / α 2 )]. A reduced form does exist for y 2 , as can be seen by subtracting the second equation from the first: 0 = ( α 1 – α 2 ) y 2 + β 1 z 1 – β 2 z 2 + u 1 – u 2 ; because α 1 ≠ α 2 , we can rearrange and divide by α 1 − α 2 to obtain the reduced form. This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. This may not be resold, copied, or distributed without the prior consent of the publisher. 93 (iii) In supply and demand examples, α 1 ≠ α 2 is very reasonable. If the first equation is the supply function, we generally expect α 1 > 0, and if the second equation is the demand function, α 2 < 0. The reduced forms can exist even in cases where the supply function is not upward sloping and the demand function is not downward sloping, but we might question the usefulness of such models....
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This note was uploaded on 09/11/2011 for the course ECONOMICS eco375 taught by Professor Suzuki during the Spring '11 term at University of Toronto Toronto.
 Spring '11
 Suzuki
 Econometrics

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