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X Y X 2In Y |X N (X, 2In).

OLS = (X X)-1X Y |X N (, 2(X X)-1).

n n n n n p n d n(n - ) N (, ) > 0 1 = n 2 n n d n(n - ) N (0, ), d n(n - ) N (0, ), - Zn Z as Zn Z P lim Zn = Z = 1, n Z p Zn Z Zn Z p lim Zn = Z > 0 lim P { Zn - Z > } = 0, n Z Zn ms Z Zn Z 2 lim E Zn - Z = 0, n Zn Z d d Zn Z Zn DZ DZ Z lim P {Zn z} = P {Z z} n z DZ as ms p {Zn Z Zn Z} Zn Z p d Zn Z Zn Z p d Z Zn Z Zn Z Z Z Z Z Z {Zn} :,,,,...,,..., 1 2 3 4 n 1 Z E(Zn) = 0 V ar(Zn) = n2 ms p Zn 0 Zn 0 1 1 g : Rk k2 Rl l2 Zn as as Zn Z g(Zn) g(Z) p p Zn Z g(Zn) g(Z) ms ms Zn Z g g(Zn) g(Z) d d Zn Z g(Zn) g(Z) Z g Z Un U Vn p d V Un U Vn V d Un + Vn U + V d UnVn UV d -1 Un Vn U-1V P r{det(Un) = 0} = 0 Z Z N (0, 1) p d {Zn} = {Z, Z, Z, Z,...} {Xn} = {Z, -Z, Z, -Z,... } {Zn} Z {Xn} Z {Zn + Xn} = {2Z, 0, 2Z, 0, 2Z,... } Z d k 1 n(Zn - Z) N (0, ) Z = p lim Zn g Rk Rl Z d n(g(Zn) - g(Z)) N (0, GG ), g(z) G = |z=Z z p d x n(x - ) N (0, ) g(x) = x x d -1/2 n(x - ) N (0, Ik) (-1/2) -1/2 = -1 d n(x - ) -1(x - ) 2(k).

(x x) G = | = 2x | = 2 x d n(x x - ) N (0, 4 ).

x1 1 d n - N (0, I2).

x2 2 x1 x1 g = x2 x2 x1 - 1 d N (0, 1), x2 - 2 N (0, 1) x1 x2 1 1 G = =, -, (x1, x2) 2 2 1 ( ) 2 2 1 x1 1 d 0, 1 + 2 n - N.

x2 2 2 2 {Zn} i=1 E|Zi| n 1 as Zi E[Zi].

n i=1 2 i 2 {Zn} i < i=1 i=1 i2 n n 1 1 as Zi - E Zi 0.

n n i=1 i=1 n 1 2 {Zn} Cov(Zi, Zj) = 0 i = j i i=1 i=1 n2 n 0 n n 1 1 p Zi - E Zi 0.

n n i=1 i=1 {Zn} i=1 E[Zi] = V ar[Zi] = 2 n 1 d n Zi - N (0, 2).

n i=1 {Zn} E[Zi] = i i=1 2 V ar[Zi] = i E[|Zi - i|3] = i n ( i)1/i= 0, n ( i )1/2 n i=n (Zi - i) d i= N (0, 1).

n ( i )1/i= d n(Zn - ) N (0, 2).

Zn 2 Zn n(Zn - ) n(Zn - ) d = N (0, 1), d n(Zn - )/ N (0, 1) p / N (0,1) N (0,1) Zn -, Zn + q1-.

q1- 2 n n H0 : = N (0,1) n|Zn - 0|/ > q1- n n 1 p = (Zi - Zn)2 = (Zi - )2 - (Zn - )2, n n i=1 i=p p n (Zi - )2 E[(Zi - )2] = 2 (Zn - )2 i=n Z1 Z2 Z3... Zn n Z1, Z2, Z3,..., ZT Zt Zt, Zt-1,..., Zt-k t k Zt Zt Zt+k k Zt iid t iid(0, 2) AR(1) : zt = zt-1 + t || < MA(1) : zt = t + t- zt = zt-1 + t V ar(zt) = V ar(zt-1) + t zt = z0 + i i= z N (0, 1) zt = z + t t z zt zt = s(, t) + t s(, t) = s(, t + ) zt Yt = f(zt, zt-1...) Yt t zk zt It = {zt, zt-1,...} zt E[zt|It-1] = {Zt}+ t= E[|Zt|] < T as Zt E[Zt] T t= T {Zt}+ t= 2 = E[Zt ] < T d Zt N (0, 2) T t= T {Zt}+ t=+ 2 = Cov[Zt, Zt-j] <.

j=T d T Zt - E[Zt] N (0, 2) T t= T AR(1) xt = xt-1 + t, || < 1, t iid(0, 2).

T T xt-1xt xt-1t t=2 t= = = +.

T T x2 xt-1 t-t=2 t=T p xt-1t E[xt-1t] = 0, T - t=T p x2 E[x2 ].

t-1 t-T - t= p T xt-1t T t=T -T ( - ) =.

T x2 T - t-t-T -p T T 1 x2 E[x2 ] t-1 t-t-T -1 T -n xt-1t It-1 = {xt-2t-1, xt-3t-2...}.

E[xt-1t|It-1] = E[E[xt-1t|xt-1, xt-2t-1...]|It-1] = 0.

xt-1t T d xt-1t N (0, E[x2 2]).

t-1 t T - t= E[x2] = V ar[xt] = 2V ar[xt-1] + 2 = t 1- d T ( - ) N (0, 1 - 2).

T ( - ) d N (0, 1).

1 - 1 - 2 1 - CI = - 1.96 ; + 1.96.

T T Zt E[ZtZt-j] = 0 j > 0 2 = E[Zt ] T T 1 V ar Zt = V ar Zt = T T t=1 t== [T V ar(Zt) + (T - 1)Cov(Zt, Zt+1) + (T - 1)Cov(Zt, Zt-1) + T + (T - 2)Cov(Zt, Zt+2) + (T - 2)Cov(Zt, Zt-2) +... + + + Cov(Z1, ZT ) + Cov(ZT, Z1)] Cov(Zt, Zt-j).

T j= MA(1) zt = t + t-1, t iid(0, 2).

V ar(zt) = (1 + 2)2, Cov(zt, zt-1) = 2, Cov(zt, zt-j) = 0, j > 1.

+ Cov(zt, zt-j) = (1 + 2)2 + 22 = (1 + )22.

j=T d zt N (0, (1 + )22).

T t= zt It = {zt-1, zt-2, zt-3...} E[zt|zt-1, zt-2,...] = t-1 = T T -1 T 1 = (Zt - Z)(Zt - Z) + {(Zt - Z)(Zt-j - Z) + (Zt - Z)(Zt+j - Z) }.

T T t=1 j=1 t=j+p m << T T m m/T min(T,T +j) m |j| NW = 1 - (Zt - Z)(Zt-j - Z).

m + 1 T j=-m t=max(1,1+j) m 1/T m = 4.

m zt = t + t-T d zt N (0, (1 + )22).

T t=p p 2 z = (1 + ) z = V ar(zt) + 2Cov(zt, zt-1) z = V ar(zt)+2Cov(zt, zt-1) T T 1 V ar(zt) = zt, Cov(zt, zt-1) = ztzt-1.

T T t=1 t= min(T,T +j) m |j| z = 1 - ztzt-j.

m + 1 T j=-m t=max(1,1+j) Xt Xt = Xt-1 + t, X0 = 0, t iid(0, 2).

T 1 T 2 T - t + Xt = + T -1 + + t + + 1.

T T T T t=T 2 2 1 T - 1 2 V ar Xt = 2 1 + + + +, T T T T t=T 1 (T + 1)(2T + 1) V ar Xt = 2 = O(T ).

T 6T t=T T T 1 1 p p p Xt V1, Xt-1t V2, Xt-1 V3, 3/2 T T T t=1 t=1 t= V1, V2, V Vp T ( - 1).

V T F (x) n T Fn(x) = I(Xi x) i=n F (x) n x1 1 x2 =, =.

y1 2 y2 y x yi = xi + i x1y1 + x2y2 1 2 + 2 1 = = =.

x2 + x2 12 + 22 1 (1, 2) 1/(x, y) = (2, 1) 1/ (1, 2), (1, 2) 1/ (2, 1), (2, 1) 1/(x1, y1), (x2, y2) = (1, 2), (2, 1) 1/ (2, 1), (1, 2) 1/ 1/2 1/ 2 = 4/5 1/ 2 1/ n nn n B b = 1, 2,..., B (z1; z2;... ; zn)b (z1;... ; zn) b = ((z1;... ; zn)b) 1,..., B q, q1- [B ], [B(1- )+1] 1 1 = {b = b }B b= q/2, q1-/ CIE = [q/2, q1-/2].

= - {b = b - }B b= q/2, q1-/ CIH = [ - q1-/2, - q/2].

- se() B b - se(b ) b= q/2, q1-/ CIt = [ - se()q1-/2, - se()q/2].

B | - | |b - | se() se(b ) b= q CI|t| = [ - se()q1-, + se()q1-].

- CI|t| E[] =.

Bias = E[] -.

= - Bias.

B Bias = E[b ] - = b -.

B b=B = - b - = 2 -.

B b= H0 : = Ha : > 0.

- = se() B b - b = q1-.

se(b ) b= - H0 > q1-.

se() Ha : = 0.

| - | =.

se() B |b - | b = q1-.

se(b ) b=| - 0| H0 > q1-.

se() H0 : = - = ( - ) V ( - ).

B -b = (b - ) V (b - ) q1-.

b=- H0 0 = ( - 0) V ( - 0) > q1-.

H0 : R = r R = (R - r) (RVR )-1(R - r).

B b = (b - ) R (RV R )-1R(b - ) q1-.

b= H0 = (R - r) (RVR )-1(R - r) > q1-.

F(x) F (x) F(x) F (x) - =.

se() d N (0, 1) F(x) F (x) (x) h1(x, F ) h2(x, F ) F(x) = (x) + + + O, n n n3/h1(x, F ) h2(x, F ) F (x) = (x) + + + O.

n n n3/ h1(x, F ) x F h2(x, F ) x F h1(x, F ) 1 (x) - F(x) = + O = O, n n n h1(x, F ) - h1(x, F ) 1 F (x) - F(x) = + O = O.

n n n h1(x, F ) - h1(x, F ) n n n F (x) - F (x) = n 1[xi x] - E [1[xi x]] n i=d N (0, P {xi x}P {xi > x}).

d = n( - ) N (0, V).

(x, V) V h1(x, F ) F(x) = (x, V) + + O, n n h1(x, F ) F (x) = (x, V ) + + O.

n n h1(z, F ) 1 (x, V) - F(x) = - + O = O, n n n 1 F (x) - F(x) = (x, V ) - (x, V) + O = O.

n n | - | d = |N (0, 1)|.

se() - 2h2(x, F ) F(x) = P r{-x x} = 2(x) - 1 + + O, n n3/se() 2h2(x, F ) F (x) = 2(x) - 1 + + O.

n n3/2(x) - 1 - F(x) = O, n 2 1 F (x) - F(x) = h2(x, F ) - h2(x, F ) + O = O.

n n3/2 n3/y = x + e, E[e|x] = 0, {(xi, yi)} iid.

{(xi, yi)}n n (x, yi ) i=1 i ei = yi - x i {(xi, ei)}n i= n (x, e) i i yi = x + e i i x {xi}n e {ei}n i i=1 i i= ei N (0, 2) x i N (0, 2) {yt}T t= l y1,..., yl y2,..., yl+1 y3,..., yl+2 T -l+ yT -l+1,..., yT {yt}T t= y1,..., yl yl+1,..., y2l T yl[ ]-l+1,..., yl[ ] T T l l l p (1 - p) p (X, Y ) f(X,Y )(x, y) + + f(X,Y )(x, y)dxdy = 1.

- (X, Y ) [a, b] [c, d] d b P r{a X b, c Y d} = f(X,Y )(x, y)dxdy.

c a X (X, Y ) + fX(x) = f(X,Y )(x, y)dy.

Y X = x f(X,Y )(x, y) fY |X=x(x, y) =.

fX(x) Y [c, d] d P r{c Y d | X = x} = fY |X=x(x, y)dy, c d b fX,Y (x, y)dxdy c a P r{c Y d | a X b} =.

b fX(x)dx a Y X = x + E[Y | X = x] = yfY |X=x(x, y)dy.

E[Y | X] X E[h(X, Y )] = E[E[h(X, Y ) | X]], h(X, Y ) (X, Y ) + + + + h(x, y)f(X,Y )(x, y)dxdy = h(x, y)fY |X(x, y)dy fX(x)dx.

- - - X Y Y X E[Y | X] g(X) MSP E = E[(Y - g(X))2] = E[(Y - E[Y |X] + E[Y |X] - g(X))2] = = E[(Y - E[Y |X])2] + E[(E[Y |X] - g(X))2] E[(Y - E[Y |X])2] g(X) = E[Y |X] e = Y E[Y |X].

E[e|X] = 0 E[e] = 0 E[eh(X)] = 0.

Y X X g(X) = a + bX.

Y X Cov(X, Y ) BLP (Y |X) = + X, =, = E[Y ] - E[X].

V ar(X) MSP E = E[(Y - a - bX)2] min a,b -E[2(Y - a - bX)] = 0, -E[2(Y - a - bX)X] = 0.

E[Y |X] BLP (Y |X) MSAE = E[(E[Y |X] - a - bX)2] min.

a,b u = Y - BLP (Y |X) E[u] = 0, E[uX] = 0.

E[Y |X] X E[Y |X] = BLP (Y |X).

Y Y Y XY N,.

X X XY Y x - X 2 y - Y 2 (x - X)(y - Y ) + - 1 Y XY - X f(X,Y )(x, y) = exp.

2(1 - 2) 2XY 1 - X N (X, X).

Y |X = x Y Y |X = x N Y + (x - X), Y (1 - 2).

X E[Y |X] = BLP [X|Y ].

Y X = 0 Y X Y Y Y XY A N A, A A.

X X XY Y A 2 Y Y N (, ), k 1 k k Y 1 (y - ) -1(y - ) fY (y) = exp -.

(2)k/2||1/2 Y Y1 1 11 Y = N, = N (, ).

Y2 2 21 Y1 N (1, 11) Y2|Y1 = y1 N (2 + B (y1 - 1), 22 - B 11B) B = - 12 = 0 Y1 Y g + HY N (g + H, HH ) g H X FX(x) FX(x) n Fn(x) = I[xi x].

n i=+ = E[X] = xdF (x).

n + = xdFn(x) = xi.

n i=y = x + e, E[ex] = 0.

E[(y - x )x] = = (E(xx ))-1E(xy).

-n n 1 = xix i xiyi.

n n i=1 i=E[y|x] = x.

= arg minE[(y - x b)2].

b n = arg min (yi - x ib)2.

b n i= (y, x) y x y x x E[y|x] Med[y|x] q[y|x] Mode[y|x] e = y - E[y|x]. E[e|x] = 0, E[eh(x)] = 0 h(x) E[e] = 0 x e y = E[y|x] + e, E[e|x] = 0.

{(yi, xi)}n i= (y, x) E[y|x] E[y|x] x x E[y|x] = g(x, ) Rk g(x, ) E[y|x] E[y|x] E[y|x] = x y = x + e, E[e|x] = 0, {(yi, xi)}n iid.

i= E[xx ] = arg minE[(y - x b)2].

b -n n n 1 1 = arg min (yi - x ib)2 = xix i xiyi.

b n n n i=1 i=1 i=-n n 1 = + xix i xiei.

n n i=1 i=p -n n 1 d n( - ) = xix i xiei N 0, Q-1Qe2 Q-1, xx xx xx n n i=1 i=Qxx = E[xx ], Qe2 = E[e2xx ] = V ar[xe].

xx n p xix i E[xx ] = Qxx, n i=n p xiei E[xe] = E[xE[e|x]] = 0, n i=n d xiei N (0, V ar[xe]) = N (0, Qe2 ), xx n i= E[e2|x] = 2 = const. Qe2 = 2Qxx xx d n( - ) N 0, 2Q-1.

xx n n 1 p p 2 = (yi - x i)2 2, Qxx = xix i Qxx.

n n i=1 i=n (yi - x i)2 = n i=n n n 1 1 = (yi - x i)2 + (x i - x i)2 + (yi - x i)(x i - x i) = n n n i=1 i=1 i=n n n 1 1 = (yi - x i)2 + ( - ) xix i ( - ) + (yi - x i)x i( - ).

n n n i=1 i=1 i=n p (yi - x i)2 2, n i=n p ( - ) xix i ( - ) 0, n i=n p (yi - x i)x i( - ) 0.

n i=n p (yi - x i)2 2.

n i= Qexx n p 2 Qe xx = xix i(yi - x i)2 Qe xx.

n i=V = Q-1Qe2 Q-1.

xx xx xx j se(j) = V.

n jj j - j d tj = N (0, 1).

se(j) h() = l k -d W = h() HVH h() 2(l), h() h() H =, H =.

X = (x1, x2,..., xn), y = (y1, y2,..., yn), = (e1, e2,..., en).

y = X +, E[|X] = 0.

= (X X)-1X y = + (X X)-1X.

E[|X] = + (X X)-1X E[|X] =.

V ar[|X] = (X X)-1X X(X X)-1, = V ar[y|X] = E[ |X].

E[y|X] = X A(X)y A(X) k n X A(X) = (X X)-1X.

E[y|X] = X A(X)y A(X) kn X A(X)X = Ik A(X)X = (X X)-1X X = Ik.

V ar[A(X)y|X] = A(X)A(X) V ar[A(X)y|X] = A(X)y, A(X) = (X -1X)-1X -1.

V ar[|X] = (X -1X)-1.

A(X)X = Ik. A(X) A(X)X = Ik (A(X) - A(X))X = 0, (A(X) - A(X))A(X) = (A(X) - A(X))-1X(X -1X)-1 = 0.

V ar[A(X)Y |X] = A(X)A(X) = = (A(X) - A(X) + A(X))(A(X) - A(X) + A(X)) = = (A(X) - A(X))(A(X) - A(X)) + V ar[A(X)Y |X] V ar[|X].

E[y|X] = X = (X -1X)-1X -1y = 2(X X)-1 2(X X)- (X X)-1X X(X X)-1 (X -1X)- W LS = (X W X)-1X W Y, W -n n 1 xix i 1 xiei = +.

n 2(xi) n 2(xi) i=1 i=n 1 xix i xx p Qxx/2 = E, n 2(xi) 2(x) i=n 1 xiei xe p E = 0, n 2(xi) 2(x) i=n 1 xiei d N 0, Qxx/.

n 2(xi) i=xe xx E = E E[e2|x] = Qxx/2.

2(x) 2(x) 2Q-1 Q-1 = 2Q-xx xx xx/ Q-1Qe2 Q-1 Q-xx xx xx xx/ -n n 1 IV = zix i ziyi, n n i=1 i= zi = f(xi) f : Rk Rk zi = xi zi = xi/2(xi).

-n n 1 IV = zix i ziyi.

n n i=1 i=Vzz = Q-1Qe2 Q-1, zz zx xz Qzx = E[zx ] Qe2 = E[zz e2] = E[zz 2(x)].

zz Q-xx/xx -Vzz - Q-1 = (E[zx ])-1 E[zz 2(x)] (E[xz ])-1 - E = xx/2(x) = (E[vu ])-1 E[vv ] (E[uv ])-1 - (E[uu ])-1 = = (E[vu ])-1 E[vv ] - E[vu ] (E[uu ])-1 E[uv ] (E[uv ])-1 = = (E[vu ])-1 E[ww ] (E[uv ])-1 0.

v = z(x) u = x/(x) w = v-E[vu ](E[uu ])-1u n x 1 xi E e = 0 (yi - x i) = 0.

(x) n (xi) i= 2(xi) 2(x) x 2(x) = E[e2|x] = z, z x z = xe2 = z +, E[|z] = 0.

ei = yi - x i = ei + x i( - ), -p = zizi zie2 + 2 zix i( - )ei + zi(x i( - ))2.

i i i i i 2(xi) = zi, -n n xix i xiyi F = = (X -1X)-1X -1y.

2(xi) 2(xi) i=1 i= ei i = 1,..., n 2(xi) -n n xix i xiyi F = = (X -1X)-1X -1y.

2(xi) 2(xi) i=1 i= 2(xi) < > 0 2(xi) = max(zi, ) 2(xi) < n 2(xi) = zj 2(xi) < j=n F d n(F - ) N (0, Q-1 ).

xx/-n xix i V = n.

2(xi) i= F d n(F - ) N 0, Q-1 Qxx/4 Q-1, xx/2 e2 xx/xx xx Qxx/2 = E, Qxx/4 = E e2.

ez (z )-1 -xix i xix i xix i V = n e2.

zi (zi)2 i zi i i i = V ar[y|X] = (X X)-1X y V ar[|X] = (X X)-1X X(X X)-1, = (X -1X)-1X -1y V ar[|X] = (X -1X)-1.

yt = x t + et, E[et|It-1] = 0, E[e2|It-1] = 2(It-1), t {(xt, yt)}T t=It-1 = {yt-1, yt-2,... ; xt, xt-1,...}.

AR(p) xt = (yt-1, yt-2,..., yt-p) st+1 - st = + (ft - st) + et, E[et|It-1] = 0, ft st t+1 = + it + et, E[et|It-1] = 0, t+1 it E[Y |X] X -T T p = xtx t xtyt.

t=1 t=E[xtet] = E[E[xtet|It-1]] = E[xtE[et|It-1]] = 0.

d T ( - ) N (0, V), V = Q-1Qe2 Q-1, xx xx xx Qxx = E[xtx t] Qe2 = E[xtx te2]. Qexx xx t E[xtetx t-jet-j] = E[E[xtetx t-jet-j|It-1]] = E[xtE[et|It-1]x t-jet-j] = 0.

-T T xt xt = x t yt.

2(It-1) 2(It-1) t=1 t= 2(It-1) It-yt = x t + et, E[et|It-q] = 0, E[e2|It-q] = 2(It-q), t It-q = {yt-q, yt-q-1,... ; xt, xt-1,...} ARMA(p, q) xt = (yt-q, yt-q-1,..., yt-q-p+1) st+q - st = + (ft,q - st) + et, E[et|It-q] = 0, ft,q q st t+q = + it,q + et, E[et|It-q] = 0, t+q it,q q -T T = xtx t xtyt.

t=1 t= d T ( - ) N (0, V), V = Q-1Qe2 Q-1, xx xx xx Qexx q-Qe2 = E[xtx te2] + (E[xtx t-jetet-j] + E[xtx t+jetet+j]).

xx t j= 2q - 1 j > q - E[xtx t-jetet-j] = E[E[xtXt-jetet-j|It-q]] = E[xtx t-jE[et|It-q]et-j] = 0.

V -T T xt xt = x t yt.

2(It-1) 2(It-1) t=1 t= E[y|x] E[y|x] = (x) x x = x + u u x y y = (x) + e = (x - u) + e = x + v, v = e - u, p E[xv] = E[(x + u)(e - u )] = -E[uu ] = 0.

: Q = -1P + e : Q = 2P + ee iid(0, I2).

eE[e1P ] = 0, E[e2P ] = 0.

Q P E[QP ] p.

E[P ] Q 1 2 1 e=, P 1 + 2 1 -1 e2 - 1 E[QP ] =, E[P ] =.

1 + 2 1 + 1 E[QP ] 2 - p =.

E[P ] x y = x + e E[e|x] = z y = x + e E[e|z] = 0 z y = x + e, E[e|x] = z = x l k Qzz = E[zz ] E[ze] = 0.

n zi(yi - x iIV ) = 0, n i=-n n IV = zix i ziyi.

i=1 i=IV = (Z X)-1Z Y, Z = (z1, z2,..., zn).

E[IV |Z, X] = (Z X)-1Z E[Y |Z, X] =, d n(IV - ) N (0, V), V = Q-1Qe zzQ-1, zx xz Qzx = E[zx ] Qe2 = E[e2zz ] zz Qzx Qzx, Qezz n n 1 Qzx = zix i, Qe zz = zizie2, ei = yi - x iIV.

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