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y Fp(Y)19 b Fr(,)35 b(then)f(in)n(terv)n(ening)f Fp(X)40 b Fr(will)34 b(c)n(hange)e(the)i(v)-5 b(alue)34 b(of)f Fp(Y)53 b Fr(in)30 5790 y(a)37 b(certain)g(w)n(a)n(y)-7 b(,)40 b(but)e(in)n(terv)n(ening)f Fp(Y)56 b Fr(should)38 b(not)f(c)n(hange)2317 1665 y gsave currentpoint currentpoint translate -90 neg rotate neg exch neg exch translate 2317 1665 a @beginspecial 60 @llx 135 @lly 400 @urx 450 @ury 2040 @rwi @setspecial %%BeginDocument: ins.ps %!PS-Adobe-3.0 %%Title: Microsoft PowerPoint - ins.ppt %%Creator: Windows NT 4.0 %%CreationDate: 11:12 3/7/2001 %%Pages: (atend) %%BoundingBox: 9 19 784 593 %%LanguageLevel: 1 %%DocumentNeededFonts: (atend) %%DocumentSuppliedFonts: (atend) %%EndComments %%BeginProlog %%BeginResource: procset NTPSOct95 /NTPSOct95 100 dict dup begin/bd{bind def}bind def/ld{load def}bd/ed{exch def} bd/a{currentpoint}bd/c/curveto ld/d/dup ld/e/eofill ld/f/fill ld/tr/translate ld/gr/grestore ld/gs/gsave ld/j/setlinejoin ld/L/lineto ld/M/moveto ld/n /newpath 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4225 y Fk(j)p Fp(\017)1045 4237 y Fn(X)1108 4225 y Fp(;)c(X)7 b Fr(])30 4375 y(=)23 b Fp(s)p Fr(\()p Fp(X)7 b Fr(\))18 b(+)g(E)o([)p Fp(\017)511 4387 y Fn(Y)569 4375 y Fk(j)p Fp(\017)626 4387 y Fn(X)689 4375 y Fr(])23 b(=)f Fp(s)p Fr(\()p Fp(X)7 b Fr(\))18 b(+)h Fp(h)p Fr(\()p Fp(\017)1217 4387 y Fn(X)1279 4375 y Fr(\).)30 4524 y(An)33 b(additiv)n(e)f(regression)e(metho)r(d)j (will)g(giv)n(e)e(the)i(estimate)30 4624 y(of)28 b Fp(s)p Fr(\()p Fp(X)7 b Fr(\))27 b(and)g Fp(h)p Fr(\()p Fp(\017)606 4636 y Fn(X)669 4624 y Fr(\))h(sim)n(ultaneously)-7 b(.)1282 4594 y Fo(2)30 4863 y Fq(2)112 b(PRIOR)36 b(W)m(ORK)30 5067 y Fr(There)25 b(ha)n(v)n(e)g(b)r(een)h(in)n(tensiv)n(e)g(studies)g (of)g(the)g(use)g(of)f(instru-)30 5167 y(men)n(ts)j(in)g(econometric)f (literature.)36 b(Muc)n(h)28 b(w)n(ork)e(has)i(b)r(een)30 5266 y(fo)r(cusing)21 b(on)f(the)i(study)f(of)f(the)i(e\016ciency)e(of) h(an)g(instrumen)n(t)30 5366 y(\(see)30 b(Nelson)f(et)h(al)g(\(1990\),) 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1768 y(tels)21 b(\(1991\),)f(Hec)n(kman)g (\(1995\),)h(and)g(Bound)f(et)h(al)f(\(1995\).\).)2130 1867 y(Therefore,)j(ev)n(en)f(in)h(the)g(case)f(where)h(w)n(e)f(do)h (ha)n(v)n(e)e(some)i(do-)2130 1967 y(main)j(kno)n(wledge)g(to)g(help)h (us)f(iden)n(tify)h(instrumen)n(ts,)g(some)2130 2066 y(kind)32 b(of)g(testing)g(pro)r(cedure)f(is)g(still)i(useful)f(in)g (that)g(it)g(can)2130 2166 y(serv)n(e)26 b(as)g(a)g(double)h(c)n(hec)n (k.)36 b(It)27 b(is)g(the)g(goal)f(of)h(this)g(pap)r(er)f(to)2130 2266 y(\014nd)f(out)g(whether,)g(under)g(certain)f(reasonable)e (conditions,)2130 2365 y(w)n(e)27 b(can)g(test)h(whether)g(a)f(v)-5 b(ariable)27 b(is)g(an)g(instrumen)n(t.)2130 2600 y Fq(3)112 b(APPR)m(O)m(A)m(CH)2130 2800 y Fr(Within)24 b(the)f(framew)n(ork)e(of) i(linear)f(mo)r(dels,)i(the)f(fact)g(that)g(a)2130 2900 y(v)-5 b(ariable)23 b(is)g(an)g(instrumen)n(t)h(imp)r(oses)g(no)f (constrain)n(t)f(on)i(the)2130 2999 y(join)n(t)34 b(marginal)f (distribution)h(of)g(the)g(observ)n(ed)e(v)-5 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b(is)g(a)f(semi-instrumen)n(t)h(for)2130 1568 y Fp(X)34 b Fr(and)27 b Fp(Y)19 b Fr(.)2130 1718 y(Note)28 b(that)g(b)r(ecause)f Fp(s)p Fr(\()p Fp(X)7 b Fr(\))27 b(is)h(linear,)f(w)n(e)g(ha)n(v)n(e:)2130 1867 y(E[)p Fp(Y)18 b Fk(j)p Fp(Z)2356 1879 y Fo(1)2393 1867 y Fp(;)c(Z)2487 1879 y Fo(2)2524 1867 y Fp(;)g(\017)2595 1879 y Fn(X)2658 1867 y Fr(])2130 2017 y(=)23 b(E)o([)p Fp(cX)i Fr(+)18 b Fp(d)h Fr(+)f Fp(\017)2689 2029 y Fn(Y)2764 2017 y Fr(+)g Fp(g)2887 2029 y Fo(1)2924 2017 y Fr(\()p Fp(Z)3013 2029 y Fo(1)3050 2017 y Fr(\))h(+)f Fp(g)3224 2029 y Fo(2)3261 2017 y Fr(\()p Fp(Z)3350 2029 y Fo(2)3387 2017 y Fr(\))p Fk(j)p Fp(Z)3499 2029 y Fo(1)3536 2017 y Fp(;)c(Z)3630 2029 y Fo(2)3667 2017 y Fp(;)g(\017)3738 2029 y Fn(X)3801 2017 y Fr(])2130 2166 y(=)23 b Fp(cf)2295 2178 y Fo(1)2331 2166 y Fr(\()p Fp(Z)2420 2178 y Fo(1)2458 2166 y Fr(\))s(+)s Fp(g)2601 2178 y Fo(1)2638 2166 y Fr(\()p Fp(Z)2727 2178 y Fo(1)2764 2166 y Fr(\))s(+)s Fp(cf)2944 2178 y Fo(2)2981 2166 y Fr(\()p Fp(Z)3070 2178 y Fo(2)3107 2166 y Fr(\))s(+)s Fp(g)3250 2178 y Fo(2)3287 2166 y Fr(\()p Fp(Z)3376 2178 y Fo(2)3413 2166 y Fr(\))s(+)s Fp(c)14 b(\017)3600 2178 y Fn(X)3666 2166 y Fr(+)s Fp(h)p Fr(\()p Fp(\017)3848 2178 y Fn(X)3911 2166 y Fr(\))s(+)s Fp(d)p Fr(.)2130 2316 y(That)26 b(is,)h(E)o([)p Fp(Y)19 b Fk(j)p Fp(Z)2668 2328 y Fo(1)2705 2316 y Fp(;)14 b(Z)2799 2328 y Fo(2)2836 2316 y Fp(;)g(\017)2907 2328 y Fn(X)2969 2316 y Fr(])27 b(is)f(additiv)n(e)g(in)g Fp(Z)3570 2328 y Fo(1)3607 2316 y Fr(,)h Fp(Z)3714 2328 y Fo(2)3750 2316 y Fr(,)g(and)f Fp(\017)3994 2328 y Fn(X)4057 2316 y Fr(,)2130 2415 y(where)h Fp(Z)2427 2427 y Fo(1)2464 2415 y Fr(,)h Fp(Z)2572 2427 y Fo(2)2609 2415 y Fr(,)f(and)h Fp(\017)2855 2427 y Fn(X)2945 2415 y Fr(are)f(indep)r(enden)n(t.)2130 2565 y(Let)34 b(E[)p Fp(g)2405 2577 y Fo(1)2441 2565 y Fr(\()p Fp(Z)2530 2577 y Fo(1)2568 2565 y Fr(\))23 b(+)f Fp(g)2750 2577 y Fo(2)2787 2565 y Fr(\()p Fp(Z)2876 2577 y Fo(2)2913 2565 y Fr(\))p Fk(j)p Fp(f)3009 2577 y Fo(1)3046 2565 y Fr(\()p Fp(Z)3135 2577 y Fo(1)3172 2565 y Fr(\))h(+)f Fp(f)3355 2577 y Fo(2)3392 2565 y Fr(\()p Fp(Z)3481 2577 y Fo(2)3519 2565 y Fr(\))33 b(=)g Fp(Z)6 b Fr(])34 b(=)f Fp(t)p Fr(\()p Fp(Z)6 b Fr(\).)2130 2664 y(W)-7 b(e)28 b(also)e(ha)n(v)n(e:)2130 2814 y(E[)p Fp(Y)18 b Fk(j)p Fp(Z)q(;)c(\017)2428 2826 y Fn(X)2491 2814 y Fr(])23 b(=)g(E)o([)p Fp(cX)i Fr(+)18 b Fp(d)p Fk(j)p Fp(Z)q(;)c(\017)3112 2826 y Fn(X)3175 2814 y Fr(])k(+)h(E)o([)p Fp(\017)3413 2826 y Fn(Y)3470 2814 y Fk(j)p Fp(Z)q(;)14 b(\017)3622 2826 y Fn(X)3685 2814 y Fr(])19 b(+)f Fp(t)p Fr(\()p Fp(Z)6 b Fr(\))2130 2963 y(=)23 b Fp(cZ)h Fr(+)18 b Fp(t)p Fr(\()p Fp(Z)6 b Fr(\))18 b(+)g Fp(c)c(\017)2760 2975 y Fn(X)2841 2963 y Fr(+)k Fp(h)p Fr(\()p Fp(\017)3038 2975 y Fn(X)3101 2963 y Fr(\))h(+)f Fp(d)p Fr(.)2130 3113 y(That)30 b(is,)g(E[)p Fp(Y)18 b Fk(j)p Fp(Z)q(;)c(\017)2747 3125 y Fn(X)2810 3113 y Fr(])30 b(is)g(additiv)n(e)f(in)h Fp(Z)36 b Fr(and)30 b Fp(\017)3659 3125 y Fn(X)3722 3113 y Fr(,)g(where)f Fp(Z)2130 3212 y Fr(and)e Fp(\017)2325 3224 y Fn(X)2416 3212 y Fr(are)g(indep)r(enden)n(t.)2130 3362 y(No)n(w)f(if)h Fp(Z)33 b Fr(is)26 b(a)g(semi-instrumen)n(t,)h (i.e.,)g Fp(g)3464 3374 y Fo(1)3501 3362 y Fr(\()p Fp(Z)3590 3374 y Fo(1)3627 3362 y Fr(\))17 b(+)f Fp(g)3797 3374 y Fo(2)3834 3362 y Fr(\()p Fp(Z)3923 3374 y Fo(2)3960 3362 y Fr(\))23 b(=)2130 3461 y Fp(aZ)h Fr(+)18 b Fp(b)p Fr(,)27 b(w)n(e)h(will)f(ha)n(v)n(e:)2130 3611 y(E[)p Fp(Y)18 b Fk(j)p Fp(Z)q(;)c(\017)2428 3623 y Fn(X)2491 3611 y Fr(])23 b(=)g(\()p Fp(c)18 b Fr(+)g Fp(a)p Fr(\))p Fp(Z)25 b Fr(+)18 b Fp(c)c(\017)3119 3623 y Fn(X)3200 3611 y Fr(+)k Fp(h)p Fr(\()p Fp(\017)3397 3623 y Fn(X)3460 3611 y Fr(\))g(+)g Fp(b)h Fr(+)f Fp(d)p Fr(.)2130 3760 y(Therefore,)35 b(to)g(test)g(whether)f Fp(Z)40 b Fr(is)35 b(a)f(semi-instrumen)n(t)g(is)2130 3860 y(equiv)-5 b(alen)n(t)27 b(to)h(test)g(whether)f Fp(Y)47 b Fr(is)27 b(linear)g(in)h Fp(Z)6 b Fr(!)2130 4009 y(The)36 b(follo)n(wing)e(algorithm,)j(whic)n (h)e(is)h(a)f(mo)r(di\014ed)h(double)2130 4109 y(instrumen)n(ts)45 b(testing)h(algorithm,)j(can)c(b)r(e)h(used)f(to)h(test)2130 4208 y(whether)e(t)n(w)n(o)g(semi-instrumen)n(ts)g Fp(Z)3358 4220 y Fo(1)3440 4208 y Fr(and)g Fp(Z)3675 4220 y Fo(2)3756 4208 y Fr(ha)n(v)n(e)g(the)2130 4308 y(same)33 b(linear)g(co)r (e\016cien)n(ts,)h(assuming)f(the)h(direct)f(e\013ect)h(of)2130 4408 y Fp(X)g Fr(on)27 b Fp(Y)47 b Fr(is)27 b(linear)g(in)h Fp(X)7 b Fr(:)2190 4631 y(1.)41 b(Use)22 b(additiv)n(e)g(regression)e (to)i(regress)e Fp(X)29 b Fr(on)22 b Fp(Z)3771 4643 y Fo(1)3830 4631 y Fr(and)g Fp(Z)4043 4643 y Fo(2)2296 4731 y Fr(to)28 b(get)f(E)o([)p Fp(X)7 b Fk(j)p Fp(Z)2771 4743 y Fo(1)2808 4731 y Fp(;)14 b(Z)2902 4743 y Fo(2)2939 4731 y Fr(].)2190 4896 y(2.)41 b(Let)20 b Fp(Z)29 b Fr(=)23 b(E)o([)p Fp(X)7 b Fk(j)p Fp(Z)2846 4908 y Fo(1)2883 4896 y Fp(;)14 b(Z)2977 4908 y Fo(2)3013 4896 y Fr(].)35 b(Regress)19 b Fp(Y)39 b Fr(on)20 b Fp(Z)26 b Fr(using)20 b(linear)2296 4996 y(regression,)25 b(let)2836 4975 y(^)2823 4996 y Fp(Y)2871 5008 y Fn(l)2925 4996 y Fr(b)r(e)j(the)g(estimation)f (of)g Fp(Y)19 b Fr(.)2190 5162 y(3.)41 b(Regress)20 b Fp(Y)41 b Fr(on)21 b Fp(Z)2847 5174 y Fo(1)2906 5162 y Fr(and)g Fp(Z)3118 5174 y Fo(2)3177 5162 y Fr(using)g(additiv)n(e)g (regression,)p 2130 5214 V 2130 5292 a Fl(t)n(w)n(een)29 b(E[)p Fj(Y)17 b Fd(j)p Fj(X)2570 5300 y Fm(1)2606 5292 y Fj(;)c(X)2703 5300 y Fm(2)2738 5292 y Fl(])29 b(and)g(E[)p Fj(Y)17 b Fd(j)p Fj(X)3160 5300 y Fm(1)3195 5292 y Fl(])j(+)f(E[)p Fj(Y)e Fd(j)p Fj(X)3534 5300 y Fm(2)3570 5292 y Fl(])29 b(as)h(the)e(testing)2130 5375 y(statistic,)f(and)f(reject)g(the)f (additivit)n(y)g(assumption)g(if)i(these)f(t)n(w)n(o)2130 5458 y(estimations)37 b(di\013er)f(signi\014can)n(tly)-6 b(.)66 b(It)36 b(is)h(easy)g(to)f(see)h(that)f(in)2130 5541 y(order)i(for)g(E[)p Fj(Y)17 b Fd(j)p Fj(X)2691 5549 y Fm(1)2726 5541 y Fl(])25 b(+)g(E[)p Fj(Y)17 b Fd(j)p Fj(X)3076 5549 y Fm(2)3112 5541 y Fl(])37 b(to)h(b)r(e)f(a)h (consisten)n(t)f(estima-)2130 5624 y(tor)24 b(of)h(E[)p Fj(Y)17 b Fd(j)p Fj(X)2557 5632 y Fm(1)2593 5624 y Fj(;)c(X)2690 5632 y Fm(2)2725 5624 y Fl(],)25 b(w)n(e)f(need)g(not)g(only)g(the)f (additiv)n(e)h(assump-)2130 5707 y(tion,)e(but)d(also)i(the)e (assumption)h(that)f(E[)p Fj(f)3385 5715 y Ff(i)3412 5707 y Fl(\()p Fj(X)3505 5715 y Ff(i)3532 5707 y Fl(\))p Fd(j)p Fj(X)3646 5715 y Ff(j)3679 5707 y Fl(])j(=)f(0,)h(where)2130 5790 y Fj(i;)13 b(j)26 b Fd(2)c(f)p Fl(1)p Fj(;)14 b Fl(2)p Fd(g)p Fj(;)g(i)21 b Fd(6)p Fl(=)g Fj(j)t Fl(.)p eop %%Page: 6 6 6 5 bop 196 323 a Fr(let)329 302 y(^)316 323 y Fp(Y)364 335 y Fn(a)432 323 y Fr(b)r(e)28 b(the)g(estimation)f(of)h Fp(Y)19 b Fr(.)90 499 y(4.)41 b(The)28 b(test)f(statistic)h(is)f(the)h (sum)f(of)h(square)e(of)1707 478 y(^)1694 499 y Fp(Y)1742 511 y Fn(a)1801 499 y Fk(\000)1896 478 y Fr(^)1883 499 y Fp(Y)1931 511 y Fn(l)1957 499 y Fr(.)196 598 y(If)c(its)g(v)-5 b(alue)22 b(is)g(small,)h(then)f Fp(Z)1146 610 y Fo(1)1205 598 y Fr(and)g Fp(Z)1418 610 y Fo(2)1477 598 y Fr(ha)n(v)n(e)e(the)j (same)196 698 y(linear)k(co)r(e\016cien)n(t.)30 928 y(T)-7 b(o)26 b(test)h(the)g(ab)r(o)n(v)n(e)e(algorithm,)h(w)n(e)h(generate)e (t)n(w)n(o)h(samples)30 1028 y(in)i(the)g(follo)n(wing)e(w)n(a)n(y:)30 1177 y(F)-7 b(or)27 b(sample)g(1,)g(w)n(e)h(ha)n(v)n(e:)30 1327 y Fp(X)h Fr(=)23 b Fp(Z)279 1297 y Fo(2)273 1347 y(1)334 1327 y Fr(+)18 b Fp(Z)480 1297 y Fo(2)474 1347 y(2)536 1327 y Fr(+)g Fp(\017)653 1339 y Fn(X)30 1476 y Fp(Y)42 b Fr(=)22 b Fp(X)j Fr(+)18 b Fp(\017)418 1488 y Fn(Y)30 1626 y Fr(where)27 b(E[)p Fp(\017)384 1638 y Fn(Y)441 1626 y Fk(j)p Fp(\017)498 1638 y Fn(X)561 1626 y Fr(])c(=)f Fp(\017)728 1595 y Fo(2)728 1648 y Fn(X)791 1626 y Fr(.)30 1775 y(F)-7 b(or)25 b(sample)g(2,)g(w)n(e)g (use)h(the)g(same)e Fp(Z)1207 1787 y Fo(1)1244 1775 y Fr(,)i Fp(Z)1350 1787 y Fo(2)1387 1775 y Fr(,)g Fp(\017)1470 1787 y Fn(X)1533 1775 y Fr(,)g Fp(X)7 b Fr(,)25 b(and)h Fp(\017)1900 1787 y Fn(Y)1957 1775 y Fr(,)30 1875 y(but)i(w)n(e)f(let)h Fp(Y)42 b Fr(=)23 b Fp(X)h Fr(+)18 b Fp(Z)841 1844 y Fo(2)835 1895 y(2)897 1875 y Fr(+)g Fp(\017)1014 1887 y Fn(Y)1071 1875 y Fr(.)1131 1844 y Fo(8)30 2024 y Fr(Using)34 b(the)h(ab)r(o)n(v)n(e)e(algorithm,)j(for)e(a)g(sample)g(size)g(of)g (200,)30 2124 y(w)n(e)f(the)g(sum)g(of)g(square)f(of)966 2103 y(^)954 2124 y Fp(Y)1002 2136 y Fn(a)1064 2124 y Fk(\000)1163 2103 y Fr(^)1151 2124 y Fp(Y)1199 2136 y Fn(l)1257 2124 y Fr(is)h(only)g(1.695891)d(for)30 2223 y(the)h(\014rst)f(sample,)g(and)g(791.5779)d(for)j(the)h(second)e (sample.)30 2323 y(This)f(suggests)e(that)i(our)f(algorithm)f(is)h(v)n (ery)g(p)r(o)n(w)n(erful.)30 2566 y Fq(5)112 b(DISCUSSION)38 b(AND)e(FUTURE)198 2682 y(W)m(ORK)30 2888 y Ft(5.1)94 b(THREE)31 b(ASSUMPTIONS)30 3068 y Fr(The)36 b(semi-instrumen)n(t)h (testing)f(algorithm)f(assumes)h(that)30 3168 y(the)47 b(\014rst)g(four)f(semi-instrumen)n(tal)h(conditions)f(are)g(sat-)30 3267 y(is\014ed.)115 b(While)54 b(in)f(general)f(w)n(e)h(cannot)g(test) 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y(abilit)n(y)h(to)h(one)f (elemen)n(t)g(in)h(the)g(space,)f(i.e.,)i(the)f(structure)p 30 5372 499 4 v 134 5426 a Fm(8)168 5458 y Fl(The)20 b(distribution)f(of)h(these)f(v)l(ariables)h(are:)32 b Fj(Z)1522 5466 y Fm(1)1576 5458 y Fl(and)18 b Fj(Z)1770 5466 y Fm(2)1824 5458 y Fl(b)r(oth)30 5541 y(are)h(uniform)g(b)r(et)n (w)n(een)g(0)g(and)g(4.)32 b(There)19 b(is)h(also)g(a)f(laten)n(t)g(v)l (ariable)30 5624 y Fj(T)33 b Fl(that)22 b(is)g(uniform)g(b)r(et)n(w)n (een)g(0)h(and)f(2.)34 b Fj(\017)1253 5632 y Ff(X)1333 5624 y Fl(is)23 b(the)e(sum)h(of)h Fj(T)32 b Fl(and)30 5707 y(a)23 b(normal)f(noise)i(with)f(v)l(ariance)g(0.25,)i Fj(\017)1220 5715 y Ff(Y)1296 5707 y Fl(is)e(the)g(sum)e(of)j Fj(T)1800 5675 y Fm(2)1856 5707 y Fl(and)30 5790 y(a)i(normal)f(noise)i (with)f(v)l(ariance)g(0.25.)2130 323 y Fr(where)k(b)r(oth)i Fp(Z)2630 335 y Fo(1)2698 323 y Fr(and)f Fp(Z)2920 335 y Fo(2)2988 323 y Fr(are)f(instrumen)n(ts,)i(whic)n(h)f(means)2130 423 y(that)25 b(b)r(oth)g Fp(Z)2557 435 y Fo(1)2618 423 y Fr(and)g Fp(Z)2834 435 y Fo(2)2895 423 y Fr(ha)n(v)n(e)f(linear)g(co) r(e\016cien)n(ts)g(0.)35 b(On)25 b(the)2130 522 y(other)f(hand,)i(if)g (a)e(semi-instrumen)n(t)h(is)g(not)g(an)g(instrumen)n(t,)2130 622 y(there)h(is)f(no)h(sp)r(eci\014c)g(reason)e(to)i(b)r(eliev)n(e)g (that)g(its)g(linear)f(co-)2130 722 y(e\016cien)n(t)j(should)f(tak)n(e) g(an)n(y)g(sp)r(eci\014c)h(non-zero)d(v)-5 b(alue.)2130 871 y(W)e(e)23 b(mak)n(e)e(a)h(third)g(assumption)g(in)h(an)f(e\013ort) g(to)g(mo)r(dify)h(the)2130 971 y(double)c(instrumen)n(ts)g(testing)g (algorithm)f(so)g(that)h(it)h(has)e(suf-)2130 1070 y(\014cien)n(t)33 b(p)r(o)n(w)n(er:)46 b(W)-7 b(e)33 b(assume)f(that)h(the)g(direct)g (e\013ect)g(of)g Fp(X)2130 1170 y Fr(on)23 b Fp(Y)42 b Fr(is)23 b(a)g(linear)g(function)h(of)f Fp(X)7 b Fr(.)35 b(W)-7 b(e)24 b(notice)f(that)h(this)f(is)g(a)2130 1269 y(rather)c(strong)g(assumption)h(for)f(an)h(additiv)n(e)g(mo)r(del.)34 b(More-)2130 1369 y(o)n(v)n(er,)21 b(b)r(ecause)g(curren)n(tly)g(w)n(e) g(do)g(not)h(ha)n(v)n(e)e(a)h(suitable)h(addi-)2130 1469 y(tiv)n(e)d(regression)f(metho)r(d)i(for)f(the)h(semi-instrumen)n(t)f (testing,)2130 1568 y(w)n(e)30 b(also)f(ha)n(v)n(e)f(to)i(assume,)g (without)h(an)n(y)e(testing,)i(that)f Fp(Z)4043 1580 y Fo(1)2130 1668 y Fr(and)d Fp(Z)2348 1680 y Fo(2)2412 1668 y Fr(are)f(semi-instrumen)n(ts.)36 b(Nev)n(ertheless,)26 b(the)h(mo)r(di-)2130 1768 y(\014ed)19 b(double)g(instrumen)n(ts)g (testing)g(algorithm)f(is)h(still)h(useful:)2130 1867 y(It)h(pro)n(vides)e(a)i(double)f(instrumen)n(ts)h(test)g(for)f(linear) g(mo)r(dels.)2130 2089 y Ft(5.2)94 b(FUTURE)33 b(W)m(ORK)2130 2270 y Fr(T)-7 b(o)23 b(mak)n(e)g(the)h(semi-instrumen)n(t)f(testing)g (p)r(o)n(w)n(erful,)h(w)n(e)f(will)2130 2370 y(con)n(tin)n(ue)31 b(to)g(lo)r(ok)f(for)h(some)f(additiv)n(e)h(regression)e(metho)r(d)2130 2469 y(that)k(is)g(suitable)g(for)g(the)g(case)f(where)h(the)g (predictors)f(are)2130 2569 y(indep)r(enden)n(t.)j(Alternativ)n(e,)20 b(w)n(e)e(ma)n(y)g(also)f(try)h(to)h(\014nd)g(some)2130 2669 y(new)31 b(w)n(a)n(y)e(of)i(generating)e(b)r(o)r(otstrap)h (samples)g(so)g(that)h(the)2130 2768 y(p)r(erformance)25 b(of)h(curren)n(tly)f(a)n(v)-5 b(ailable)24 b(regression)g(metho)r(ds) 2130 2868 y(will)32 b(not)f(get)g(impro)n(v)n(ed)f(for)h(the)h(b)r(o)r (otstrap)f(samples.)48 b(W)-7 b(e)2130 2968 y(will)41 b(also)e(need)i(to)f(get)h(the)f(asymptotic)g(distribution)h(of)2130 3067 y(the)32 b(test)f(statistic)h(for)e(the)i(mo)r(di\014ed)g(double)f (instrumen)n(ts)2130 3167 y(testing.)2130 3378 y Ft(Reference)2130 3548 y Fr([1])56 b(Bartels,)62 b(L.)57 b(\(1991\),)62 b(\\Instrumen)n(tal)55 b(and)h(`Quasi-)2130 3648 y(Instrumen)n(tal')27 b(V)-7 b(ariables,")26 b(in)i Fg(A)n(meric)l(an)i(Journal)g(of)h(Po-) 2130 3748 y(litic)l(al)g(Scienc)l(e,)g(35)p Fr(,)d(pp.)37 b(777-800.)2130 3897 y([2])i(Bound,)i(J.,)h(Jaeger,)e(D.,)i(Bak)n(er,)e (R.)g(\(1995\),)g(\\Prob-)2130 3997 y(lems)j(with)g(Instrumen)n(tal)g (V)-7 b(ariables)41 b(Estimation)i(When)2130 4096 y(the)28 b(Correlation)d(Bet)n(w)n(een)h(the)i(Instrumen)n(ts)f(and)g(the)h(En-) 2130 4196 y(dogenous)k(Explanatory)g(V)-7 b(ariable)33 b(is)g(W)-7 b(eak,")33 b(in)h Fg(Journal)2130 4296 y(of)h(the)g(A)n (meric)l(an)g(Statistic)l(al)g(Asso)l(ciation,)i(V)-6 b(ol.)53 b(90)p Fr(,)36 b(pp.)2130 4395 y(443-450.)2130 4545 y([3])k(F)-7 b(ennessey)g(,)42 b(J.,)h(d'Amico,)h(R.)c(\(1980\),)i (\\Collinearit)n(y)-7 b(,)2130 4644 y(Ridge)26 b(Regression,)f(and)h (In)n(v)n(estigator)e(Judgmen)n(t,")i(in)g Fg(So-)2130 4744 y(ciolo)l(gic)l(al)32 b(Metho)l(ds)f(and)g(R)l(ese)l(ar)l(ch)f(8)p Fr(,)e(pp.)37 b(309-340.)2130 4893 y([4])k(Gozalo,)j(P)-7 b(.,)45 b(Lin)n(ton,)g(O.)c(\(2001\))g(\\T)-7 b(esting)41 b(Additiv-)2130 4993 y(it)n(y)34 b(in)h(Generalized)f(Nonparametric)f (Regression)f(Mo)r(dels)2130 5093 y(with)h(Estimated)f(P)n(arameters,") d(forthcoming)j(in)g Fg(Journal)2130 5192 y(of)f(Ec)l(onometrics)p Fr(.)2130 5342 y([5])g(Hastie,)g(T.,)h(Tibshirani,)f(R.)h(\(1990\),)e Fg(Gener)l(alize)l(d)k(A)l(d-)2130 5441 y(ditive)d(Mo)l(dels)p Fr(.)38 b(New)28 b(Y)-7 b(ork)27 b(:)37 b(Chapman)27 b(and)h(Hall.)2130 5591 y([6])39 b(Hec)n(kman,)i(J.)d(\(1995\),)j (\\Instrumen)n(tal)d(V)-7 b(ariables:)58 b(A)2130 5690 y(Cautionary)44 b(T)-7 b(ale,")44 b Fg(NBER)j(T)-6 b(e)l(chnic)l(al)47 b(Working)g(Pap)l(er)2130 5790 y(No.)39 b(185)p Fr(.)p eop %%Page: 7 7 7 6 bop 30 323 a Fr([7])37 b(Magdalinos,)h(M.)f(\(1994\),)h(\\T)-7 b(esting)37 b(Instrumen)n(t)g(Ad-)30 423 y(missibilit)n(y:)76 b(Some)48 b(Re\014ned)f(Asymptotic)h(Results",)k(in)30 522 y Fg(Ec)l(onometric)l(a)31 b(V)-6 b(ol.)39 b(62)p Fr(,)28 b(pp.)37 b(373-403)30 672 y([8])26 b(Nelson,)h(C.,)g(Startz,)f (R.)h(\(1990\),)f(\\The)g(Distribution)h(of)30 771 y(the)k(Instrumen)n (tal)g(V)-7 b(ariables)29 b(Estimator)h(and)h(Its)g(t-Ratio)30 871 y(when)39 b(the)g(Instrumen)n(t)f(is)h(a)f(P)n(o)r(or)f(One,")h(in) g Fg(Journal)i(of)30 971 y(Business,)30 b(vol.)40 b(63)p Fr(,)28 b(S125-S140.)30 1120 y([9])40 b(New)n(ey)-7 b(,)42 b(W.)f(\(1985\),)h(\\Generalized)c(Metho)r(d)j(of)f(Mo-)30 1220 y(men)n(ts)24 b(Sp)r(eci\014cation)g(T)-7 b(esting,")23 b(in)h Fg(Journal)i(of)h(Ec)l(onomet-)30 1319 y(rics,)k(V)-6 b(ol.)38 b(29)p Fr(,)29 b(pp.229-256.)30 1469 y([10])20 b(New)n(ey)-7 b(,)22 b(W.,)h(P)n(o)n(w)n(ell,)d(J.,)i(V)-7 b(ella,)22 b(F.)f(\(1999\),)g(\\Nonpara-)30 1568 y(metric)35 b(Estimation)f(of)h(T)-7 b(riangular)33 b(Sim)n(ultaneous)h(Equa-)30 1668 y(tions)27 b(Mo)r(dels,")g(in)h Fg(Ec)l(onometric)l(a)j(V)-6 b(ol.)39 b(67)p Fr(,)29 b(pp.565-603.)30 1817 y([11])42 b(P)n(earl,)j(J.)e(\(1995\))f(\\On)g(the)h(T)-7 b(estabilit)n(y)43 b(of)g(Causal)30 1917 y(Mo)r(dels)37 b(with)h(Laten)n(t)f(and)g (Instrumen)n(tal)g(V)-7 b(ariables,")36 b(in)30 2017 y(P)-7 b(.)28 b(Besnard)f(and)i(S.)f(Hanks)g(\(Eds.\),)h Fg(Unc)l(ertainty)h(in)h(A)n(rti-)30 2116 y(\014cial)i(Intel)t(ligenc)l (e)g(11)p Fr(,)f(pp.435-443.)41 b(San)30 b(F)-7 b(rancisco,)30 b(CA:)30 2216 y(Morgan)c(Kaufmann)30 2365 y([12])j(Shea,)h(J.)f (\(1997\),)g(\\Instrumen)n(t)g(Relev)-5 b(ance)29 b(in)h(Multi-)30 2465 y(v)-5 b(ariate)25 b(Linear)h(Mo)r(dels:)36 b(A)26 b(Simple)h(Measure,")e(in)h Fg(R)l(eview)30 2565 y(of)31 b(Ec)l(onomics)g(and)f(Statistics)f(V)-6 b(ol.)39 b(79)p Fr(,)29 b(pp:)37 b(348-352.)30 2714 y([13])28 b(Spirtes,)g(P)-7 b(.,)28 b(Glymour,)g(C.)g(and)h(Sc)n(heines,)f(R.)g(\(2000\).)30 2814 y Fg(Causation,)f(Pr)l(e)l(diction,)h(and)d(Se)l(ar)l(ch)p Fr(,)g(2nd)d(ed.)35 b(New)23 b(Y)-7 b(ork,)30 2913 y(N.Y.:)38 b(MIT)27 b(Press.)30 3063 y([14])18 b(W)-7 b(u,)21 b(D.)e(\(1973\),)h (\\Alternativ)n(e)d(T)-7 b(ests)19 b(of)f(Indep)r(endence)30 3162 y(Bet)n(w)n(een)31 b(Sto)r(c)n(hastic)g(Regressors)e(and)i (Disturbances,")g(in)30 3262 y Fg(Ec)l(onometric)l(a)g(V)-6 b(ol.)39 b(41)p Fr(,)28 b(pp.733-750.)30 3411 y([15])35 b(Ziv)n(ot,)h(E.,)h(Startz,)h(R.,)f(Nelson,)h(C.,)f(\(1998\),)f(\\V)-7 b(alid)30 3511 y(Con\014dence)30 b(In)n(terv)-5 b(als)30 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Fp(a\017)962 5453 y Fn(X)1050 5441 y Fr(+)g Fp(b)p Fr(,)41 b(i.e.,)g(E)o([)p Fp(Y)19 b Fk(j)p Fp(X)r(;)14 b(\017)1721 5453 y Fn(X)1783 5441 y Fr(])38 b(is)g(a)30 5541 y(linear)31 b(com)n(bination)h(of)g(a)f (univ)-5 b(ariate)32 b(function)h(of)f Fp(X)38 b Fr(and)30 5641 y(a)27 b(univ)-5 b(ariate)27 b(function)h(of)g Fp(\017)943 5653 y Fn(X)1006 5641 y Fr(.)30 5790 y(T)-7 b(o)27 b(sho)n(w)g(the)h (con)n(v)n(erse,)d(supp)r(ose:)2130 323 y(E[)p Fp(Y)18 b Fk(j)p Fp(X)30 b Fr(=)22 b Fp(x;)14 b(\017)2603 335 y Fn(X)2689 323 y Fr(=)23 b Fp(u)p Fr(])g(=)f Fp(s)2997 335 y Fo(1)3035 323 y Fr(\()p Fp(x)p Fr(\))d(+)f Fp(h)3296 335 y Fo(1)3333 323 y Fr(\()p Fp(u)p Fr(\).)2130 472 y(Let)28 b Fp(g)2319 484 y Fo(0)2356 472 y Fr(\()p Fp(x)19 b Fk(\000)f Fp(u)p Fr(\))23 b(=)f(E[)p Fp(g)s Fr(\()p Fp(Z)6 b Fr(\))p Fk(j)p Fp(f)j Fr(\()p Fp(Z)d Fr(\))23 b(=)f Fp(x)d Fk(\000)f Fp(u)p Fr(],)27 b(w)n(e)h(ha)n(v)n(e:)2140 614 y Fp(@)5 b(g)2229 626 y Fo(0)2265 614 y Fr(\()p Fp(x)20 b Fk(\000)e Fp(u)p Fr(\))p 2140 651 387 4 v 2285 727 a Fp(@)5 b(x)2560 670 y Fr(=)22 b Fp(g)2690 635 y Fi(0)2687 690 y Fo(0)2724 670 y Fr(\()p Fp(x)d Fk(\000)f Fp(u)p Fr(\))23 b(=)3129 614 y Fp(d)p 3106 651 91 4 v 3106 727 a(dx)3206 670 y Fr(\()p Fp(s)3277 682 y Fo(1)3315 670 y Fr(\()p Fp(x)p Fr(\))c Fk(\000)f Fp(s)p Fr(\()p Fp(x)p Fr(\)\))2153 851 y(=)-14 b Fk(\))51 b Fp(g)2381 817 y Fi(0)2378 872 y Fo(0)2414 851 y Fr(\()p Fp(x)20 b Fk(\000)e Fp(u)p Fr(\))27 b(is)h(constan)n(t)e(in)i Fp(u)2153 1001 y Fr(=)-14 b Fk(\))51 b Fp(g)2381 971 y Fi(0)2378 1021 y Fo(0)2442 1001 y Fr(is)28 b(a)f(constan)n(t)2153 1150 y(=)-14 b Fk(\))51 b Fp(g)2378 1162 y Fo(0)2442 1150 y Fr(is)28 b(a)f(linear)g(function)2153 1300 y(=)-14 b Fk(\))46 b Fp(g)s Fr(\()p Fp(Z)6 b Fr(\))23 b(=)f Fp(af)9 b Fr(\()p Fp(Z)d Fr(\))19 b(+)f Fp(b)p Fr(,)27 b(where)g Fp(a;)14 b(b)27 b Fr(are)g(constan)n(ts.)2130 1449 y Ft(Pro)s(of)32 b(for)g(Theorem)e(2.)2130 1599 y Fr(Let)37 b Fp(L)2345 1611 y Fo(1)2419 1599 y Fr(b)r(e)h(the)g(linear)e(co)r (e\016cien)n(t)h(of)g Fp(Z)3493 1611 y Fo(1)3530 1599 y Fr(,)j Fp(L)3650 1611 y Fo(2)3724 1599 y Fr(the)e(linear)2130 1698 y(co)r(e\016cien)n(t)21 b(of)g Fp(Z)2656 1710 y Fo(2)2693 1698 y Fr(,)h(and)f Fp(\026)2943 1710 y Fn(L)2989 1718 y Fc(1)3046 1698 y Fr(the)g(distribution)g(of)g Fp(L)3773 1710 y Fo(1)3810 1698 y Fr(.)35 b(Then:)2130 1896 y Fp(P)12 b Fr(\()p Fp(L)2284 1908 y Fo(1)2344 1896 y Fr(=)23 b Fp(L)2489 1908 y Fo(2)2548 1896 y Fr(=)g(0)p Fk(j)p Fp(L)2758 1908 y Fo(1)2817 1896 y Fr(=)g Fp(L)2962 1908 y Fo(2)2999 1896 y Fr(\))g(=)3152 1839 y Fp(P)12 b Fr(\()p Fp(L)3306 1851 y Fo(1)3366 1839 y Fr(=)22 b Fp(L)3510 1851 y Fo(2)3570 1839 y Fr(=)h(0\))p 3152 1876 580 4 v 3228 1952 a Fp(P)12 b Fr(\()p Fp(L)3382 1964 y Fo(1)3442 1952 y Fr(=)22 b Fp(L)3586 1964 y Fo(2)3623 1952 y Fr(\))3765 1896 y(=)g(1,)27 b(for:)2130 2098 y Fp(P)12 b Fr(\()p Fp(L)2284 2110 y Fo(1)2344 2098 y Fr(=)23 b Fp(L)2489 2110 y Fo(2)2525 2098 y Fp(;)14 b(L)2619 2110 y Fo(1)2679 2098 y Fk(6)p Fr(=)23 b(0\))g(=)2951 2031 y Fb(R)2990 2128 y Fa(R)p Fi(nf)p Fo(0)p Fi(g)3185 2098 y Fp(P)12 b Fr(\()p Fp(L)3339 2110 y Fo(2)3399 2098 y Fr(=)23 b Fp(l)3512 2110 y Fo(1)3549 2098 y Fr(\))p Fp(d\026)3674 2110 y Fn(L)3720 2118 y Fc(1)3757 2098 y Fr(\()p Fp(l)3814 2110 y Fo(1)3851 2098 y Fr(\))g(=)g(0)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF