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i n n i=1 i= C IV l > k Qzz z x x = z + u, E[zu ] = 0.

E[z(x - z) ] = 0 = (E[zz ])-1E[zx ] = Q-1Qzx.

zz y = (z + u) + e = (z) + v, v = e + u.

E[zv] = E[z(e + u )] = 0, = (E[z(z) ])-1E[zy] = (QxzQ-1Qzx)-1QxzQ-1Qzy.

zz zz --1 -n n n n n n 2SLS = xizi zizi zix i xizi zizi ziyi, i=1 i=1 i=1 i=1 i=1 i=2SLS = (X Z(Z Z)-1Z X)-1X Z(Z Z)-1Z Y.

p d 2SLS, n(2SLS - ) N (0, V2SLS), V2SLS = (QxzQ-1Qzx)-1QxzQ-1Qe2 Q-1Qzx(QxzQ-1Qxz)-1.

zz zz zz zz zz E[e2|z] = 2 = const Qe2 = 2Qzz zz V2SLS = 2(QxzQ-1Qzx)-1.

zz Qxz k rank(Qxz) = k.

2SLS -1 -n n n n 2SLS = ix i iyi, i = xjzj zjzj zi.

i=1 i=1 j=1 j= i Qzx k l = k = 1, y = x + e, E[e|z] = 0, E[xz] = 0.

Qxz n n 1 d d ziei N (0, Qz e2), zixi N (0, Qz x2).

2 n n i=1 i= ziei d ziyi i n IV = = + + D, zixi zixi i n D H0 : E[xz] = x z l = k -1 - IV = zix i ziyi, IV = zi x zi yi.

i -1 -VIV = n zix i zizie2 xizi, i -1 - VI = n zi x zi zi e2 xzi.

V i i i l > k E[ze] = n ziei = 0.

n i=-2SLS = (...)-1 xizi zizi ziyi, - 2SLS = (...)-1 xzi zi zi zi yi - ziei.

i -1 -V2SLS = n(...)-1 xizi zizi zizie2 zizi zix i(...)-1, i -1 - V2SLS = n(...)-1 xzi zi zi uu zi zi zi x (...)-1.

i i i i n u = zi e - zjej i i j=n yt = x t + et, E[et|It-1] = 0, It-1 = {yt-1, yt-2... ; xt, xt-1,...}.

zt = (yt-1, yt-2,..., yt-l, x t, x t-1,..., x t-l ).

y x It-E[et|zt] = E[E[et|It-1]|zt] = 0.

2SLS yt = x t + et, E[et|It-q] = 0, zt = {yt-q,..., yt-l, x t,..., x t-l }.

y x E[y|x] = g(x, ) g(, ) g(x, ) x x z E[y|z] = z g(x, ) = 0 + 1x1 + 2x2 + 3x1x2 + 4x2 + 5x2.

1 z = (1, x1, x2, x1x2, x2, x2).

1 g(x, ) = 0 + 1x + 2x2 + + pxp.

z = (1, x,..., xp).

x g(x, ) = 1 + 22x + + ppxp-1.

x x x x x, x2,..., xp-1 1, 2,..., p y = AKL1- exp(e), E[e|A, K, L] = 0.

E[log Y | log A, log K, log L] = log A + log K + (1 - ) log L.

E[y|x] = g(x ) = z z(x) x g(x, ) = 1 + 1 + 3x g(x, ) = 1 + 2e x g(x, ) = (1 + 2x1)1[x2 3] + (4 + 5x1)1[x2 > 3] g(x, ) g(x, ) = g(x, ) g(x, ) = x g(x) = x, = arg minE[(y - g(x, b))2].

b n = arg min (yi - g(xi, b))2.

b n i=n (yi - g(xi, ))g(xi, ) = 0.

n i= = (1, 2), g(x, ) = 1x(2).

2 g(x, ) = 1 + 2e x.

1 = (1, 2), 2 = 3 x(2) = (1, e x).

n = arg min min (yi - 1xi(2))2.

n 2 1 i= 2 1(2) = (X (2)X(2))-1X (2)Y, X(2) = (x1(2),..., x2(2)).

n 2 = arg min (yi - 1xi(2))2.

n i= 2 [2, 2] 2 1(2) n (yi - 1(2) xi(2))i=n j j+ |j+1-j| < n (yi - g(xi, j) - g(xi, j)(j+1 - j))g(xi, j) 0.

n i=-n n dj = g(xi, j)g(xi, j) g(xi, j)(yi - g(xi, j)), i=1 i=j+1 = j + dj.

dj j [0, 1] j+1 = j + jdj.

b = g(x, ) = g(x, b) E[(y - g(x, b))2] = E[(y - g(x, ))2] + E[(g(x, ) - g(x, b))2] Qxx = E[xx ] = b E[(x - x b)2] = ( - b) Qxx( - b) > 0.

x = x b g(x, ) = 1 + 2e +3x = 1 + elog 2+4+3x.

2 {zi()}n i=n n 1 p sup zi() - p lim zi() 0.

n n i=1 i=p {zi()}n n i=n n 1 p zi(n) p lim zi().

n n i=1 i=n n 1 zi(n) - p lim zi() n n i=1 i=n n n n 1 1 1 zi(n) - p lim zi() + p lim zi() - p lim zi() n n n n i=1 i=1 i=1 i=n n n n n n 1 1 1 sup zi() - p lim zi() + p lim zi() - p lim zi() n n n n i=1 i=1 i=1 i=n p 0.

n n 1 p p Qe2 = xix ie2 Qe2, Qgg = g(xi, )g(xi, ) Qgg, xx xx i n n i=1 i= Qgg = E[g(x, )g(x, ) ].

g(x, ) b g(xi, ) (yi - g(xi, ))2, g(xi, )g(xi, ), (yi - g(xi, )) ;

Qgg = E[g(x, )g(x, ) ] Qe2 = E[g(x, )g(x, ) e2] gg p d, n( - ) N (0, Q-1Qe2 Q-1).

gg gg gg > n n n 1 (yi - g(xi, ))2 < (yi - g(xi, ))2 +, n n i=1 i=n (yi-g(xi, b))2.

i=n (yi - g(xi, ))n E[(yi - g(xi, ))2] < (yi - g(xi, ))2 +.

n i=n (yi - g(xi, ))2 < E[(yi - g(xi, ))2] +.

n i=E[(y - g(xi, ))2] < E[(y - g(xi, ))2] +.

N() inf E[(y - g(x, b))2] > E[(y - g(x, ))].

bN()c = inf E[(y - g(x, b))2] - E[(y - g(x, ))], bN()c E[(y - g(x, ))2] < inf E[(y - g(x, b))2], bN()c p N() n (yi - g(xi, ))g(xi, ) + n i=n 1 g(xi, ) + (yi - g(xi, )) - g(xi, )g(xi, ) ( - ) = 0, n i= -n 1 g(xi, ) n( - ) = (yi - g(xi, )) - g(xi, )g(xi, ) n i=n d (yi - g(xi, ))g(xi, ) n i=-g(x, ) d - E (yi - g(xi, )) - g(x, )g(x, ) N (0, Qe gg) = N Q-1Qe qqQ-1.

gg gg E[e2|x] = 2 = const.

d Qe2 = 2Qgg n( - ) N (0, 2Q-1).

gg gg E[eg(x, )] = g(x, ) E e = 0.

2(x) n 1 g(xi, ) (yi - g(xi, )) = 0.

n 2(xi) i= n 1 (yi - g(xi, b)) = arg min, b n 2(xi) i= p d, n( - ) N (0, Q-1), gg g(x, )g(x, ) gg Q = E.

2(x) n (yi - g(xi, IV ))zi = 0, n i= zi xi k 1 x i + ei 0, yi = ei|xi N (0, 1).

0, E[y|x] = P {x + e 0|x} = P {e -x |x} = (x ).

n = arg min (yi - (x ib))b n i= p d, n( - ) N (0, Q-1Qe2 Q-1), gg gg gg g(x, ) = f(x )x, Qgg = E[f(x )2xx ], Qe2 = E[f(x )2(y - (x ))2xx ].

gg n 1 (yi - (x ib)) = arg min, b n (x i)(1 - (x i)) i=2(x) = V ar[y|x] = (x )(1 - (x )) = const.

- f(x )2xx p d, n( - ) N 0, E.

(x )(1 - (x )) y = (1 + 2x) + (3 + 4x) + e, E[e|x] = 0.

1 + ex 3 = 4 = 2 yt = 0 + x t1 + t + et, E[et|It-1] = 0, E[e2|It-1] = t = 0 + 1e2.

t t- H0 : 1 = = (1, 2) 1 W (2) sup W = supW (2)

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