Александровский филиал Российского Нового Университета Н.А.УЛЬЯНЦЕВ Э к о н о м е т р и к а Краткий курс лекций
PAGE 9
.. 2007 . .. ( )
.
, , , , , . , , , . , , , , .
.
.23 , - 7.
.
.. 41. . 4 1.1. ............ 4 1.2. . . X S2 .. 51.3., , ..7
1.4. .91.5. ..............111.6. ......121.7. ...141.8. 161.8.1. ...161.8.2. , ..181.9. . .241.9.1. 24
2. ..242.1. .242.2. ....262.3. . R2..27 2.4. ..292.5. 312.5.1. ..332.5.2. ...352.6.. .37 2.6.1. . .392.7..42 3. .. 433.1 .................433.1.1. ..443.1.2. .
463.2. ...463.2.1. ( ) ..473.3. . 48 3.4. ...503.4.1. ..503.4.2. 1- .513.4.3. 2 .513.5. 54
3.5.1. ..543.5.2. ...553.5.3. ...56 3.5.4. .573.6. 583.6.1. ...60 4. ........62 ..77 . , , , , - ? , , - , , . , . , , , . , , .
.
, . . ( ) , , , . .1. .
1.1. (). - , . , , , . (, , , ) , , . , 1/2 , 1/6. ( ) (P ) , . P (), , . . , , . , ( ). (). . , ..( ). , , . , , , . , , . , ( ), , , . , () .1.2. . . X S2 .
- , . ( 1/6 ). . x, M(x)., x n (x1 x2 x3 x4 .. xn) xi Pi ,
M(x)= x1 p1 + x2 p2 + .. + xn pn = ( 1.1 ) . x . x , . x , . . (x - ), x. 2 , , , Var(x).
2 = Var(x)= M{(x - )2} (1. 2 )
2 , 2 , . x ,
x = + u, ( 1.3 )
u ( ) ,
u = x - (1.4)
M(u )= 0. (1.5)
x u, , x u.
2 = 2u = M(u2 ) (1.6) , 2 x u. , ( ), . , . . . n , .
, , , , , , .
: 2 :.1.1.
, X = 1/ n( xi )
, 2S2 = 1/( n- 1) { ( xi x )}
X Xi ()
S2
X , x .
, , X :
X = 1/ n (x1 + x2 + x3 + .+ xn ).
, xi = + ui, - x = + u - u - - ui. , , - () u ( ) . (),
,( ) .1.
x , , xi ui .
S2 - , xi - x =( + ui ) ( + u) = ui u. , S2..
1.3., , .
, . , , , (u) .
, , 2 S2 . , . .
, . , () , S2 Q2. , . - .
, , , , () 1.2.
, , . .
, .
, , (), () 1..3. , , . ?.
, . , , . , , . , .
, , , .
n?. Q2/n, , , (1.4). , n Q/n . :
Lim x =
n
, . , (1..5. ).
1.4. .
, , .
. . n (,) :
Cov(x,y)= (xi x ) (yi y )= { (x1 x ) (y1 y ) +.........+ (xn x )(yn y) (1.7) Cov(x, y) cov (x, y ) y .
Var( x ) var (x,).
, .
1. y = v +w Cov(x,y ) = Cov(x,v) + Cov(x,w) (1.8)
2.
y = ar, , Cov(x,y ) = a Cov(x,r) (1.9)
3.
y = , Cov(x,y )= 0 (1.10)
, y .
cov(x,y ) = y = M {(x -x ) (y - y ) } (1.11)
- -x , y .
, , . ,
M {Cov(x,y) } = { cov(x,y ) } (1.12)
, n/(n-1)
, .
. , , . n 1 n, : Var( x ) =( xi x )2 = [( x1 x )2 + ( x2 x )2 + ..... + (xn x )2] (1.13)
:
1. n/(n-1) Cov (x,y), s2 = n/ (n-1)[Var(x )]
, .
1.
y = v +w, Var( y ) = Var( v ) + Var( w ) + 2Cov (v,w) (1.14)
2.
y = z, , Var( y ) = a Var( z ) (1.15)
3. y = , , , Var( y ) = 0 (1.16)
4.
y = v +, , , Var( y ) = Var( v ) (1.17)
, .
Var( x ) = Cov (x,x) (1.18)
, . .
.
: xy = cov(x,y)/ var (x ) var (y) = ,y / 2 2y (1.19)
=0 . , - , =1, = -1. : rxy :
(1.20) , r 1, , - -1. r=0. r=0 , =0 . rxy xy. xy rxy 0,3, . 1.5. .
, , , . , , .
:
(1.21)
y, , : 1. + x, ( ) , ; 2. u. .6 .
1 , 2 , 3 , 4 , - . , Q1 Q2 Q3 Q4 . ( u.) , - 1 ,2 , 3 , 4. , , 1.6. , Q1 Q2 Q3 Q4 , .
, , .
, u . u =0, Q . .1.6. .
: .
, 1.7 , (1.21). , . 1.7 .
, , , b.
, . , , , , .
b , . (1. 8.) = 1 R1 , *1, 1. *1 1 1. (1 *1) P1 R1 . 1 . 2 3 4 . , , . , .
S. 1.8. :
S= 21 + 22 + 23 + 24 (1.22)
S b, . S , . S =0, , , 0.
. .1. , : = 3, = 1, = 5 = 2. b
( !.23), , , , . .
= 1, . = 2, ., 1.2.xyy*e
13 +b3 b
25 +2b5 2b
, . 1 , , (3 a b), 2, , (5 a 2b), :
, . , :
, :
, =1 =2. :
.
, , :
, . . 2. , : . , . .
, 1,2,3, , (a+2b) (+3b); .3.xyy*e
13
3-a-b
25(a+2b)5-a-b
36(a+3b);6-a-b
,
:
, =1,67 b=1,50. , :
1.7.
, n . , ,
y = + bx (1.24)
(.. ), . S= 21 + 22 + 23 + 24
, , :
b = Cov (x, y, )/ Var (x ) (1.25) a = y bx (1.26)
.
y y , , x. , *i i *i = i + i (1.27)
*i i - , , . , , , . i . , . (1.14) . Var ( ) = Var (* + )= Var (* ) + Var ( e ) + 2 Cov (*, e ) (1.28)
, , Cov ( y,e ) , .
Var ( ) = Var (* ) + Var ( e ) (1.29)
, Var ( ) : Var (*) Var ( e ) - .
*: Var ( + bx);
Var [- ( + bx)]. .
Var (* ) /Var ( ) , . (R2).
R2= Var (*) /Var (), (1.30)
R2 = 1 - Var () /Var (), (1.31)
(TSS) .
(SS) - (+bx) ( + bx ). ( ) (RSS) .
, R .
R= Var ( + bx ) /Var ( y ) = SS / TSS (1.32)
: TSS = RSS + SS.
R 1. , , , *i= i , i .
Var (* ) =Var (), Var ( )=0 R2= 1.
1.8.
b, , : 1) , ; 2) , Cov ( x, u ), b .
u.
1.8.1
, , . , - ( ).
.
, . , . , , , , . , , . , , . .
? , ( ?). . , . . - :
1) ;
2) ;3) ( ) u .
-, y = + x +u , ,x, u.
- , b y x. , b , . . , , , . , , .. y x. , y., , . , , , , .
1. (u )=0 i.
2. : 2 (ui ) i.
3.
Cov (ui, uj )=0, i j .
4. Cov (uk, k )=0 k.
: , , .
. b :
(1.33)
(1.34)
. , 2u. , , . , , .
2u :
(1.35)
:
(s.d.) ; .
(..) , .
(1.33), (1.34), b .
(1.36)
(1.37)
, :
1. , .2. n, Var ( x ) 2u (, ).3. :
(1.38)4. :
(1.39)1.8.2. ,
, .
. , , , . . ( ) . 00,1 0,05 1 5. , , . . (Q* ), (Q* + ), () .
( ) Q Q* , Q - Q* , Q -= Q* . . , . , 0,95; 0,99 0,999.
, . , , (., ) , ( w., ), :
. = = + w. +u, (1.40)
, u .
, , . , , , , 0 , =1 = 0. , 1, , , 0. , 1. :
0 : =1.
1 : 1. , , .
, .
(0 ) , .
(1) , , .
:
= + +u (1.41)
, . , , 0; 0 : = 0 1 : 0. , . , 4 - .
0 , , , 0 2(u ) /n Var ( x ), , u . , b , 1.9 .
, s.d.(b).= 2(u) /n Var (x ) 0 ( ). , b 0,1. , 1.10. , 0,8 1,2.
. , , , . 1, . , , 1, 0,7, 1. , , 0,0027, .., . , .
1. , .
2. , . , , 1, .. - 1 : 0. , , - ? , =0,7, , , . ?
. 5% 1%. 5%, , b 5%. , 5- .
, b 0 1.96 (s.d.). , , , b 1,96 s.d., 2,5% , - 2,5%. , 1,96 s.d.
, , ,
Z < 1, 96 Z < -1, 96 (1.42)
Z ; Z = b 0 / s.d. ( b) (1.43) :
0 1,96 s.d. ( b) < b < 0 + 1,96 s.d. ( b) (1.44)
b, , = 0. b 5% .
, s.d. ( b) = 0,1, 5%, b 0,196, .. 1,196 0,804. b 0,804 1,196. . 1.11.
, 1%.
0 2,58 s.d. ( b) < b < 0 + 2,58 s.d. ( b) (1.45) , 5% 1% . , 1 2 . , . , .
, , 1 . , 5%, 5% . 1%, 1% .
, 1% .
, , , , , . , . , 2 , . , 1 , .
, .
, b ?
, b . . b , : . Z .. (b ) s.d. ( b) t- ( ):
t = (b 0))/ ..(b) (1.46) , t , t . , .
. .
. 2 , n -2.
t , 1,96 5% -2,58 1% . , , = 0 :
- t . < ( b 0 )/ ..(b) < t . (1.47) t , .
:
, , , ( ) p. = -1,21 + 0,82w., .. ( 0,05) (0,10)
, , , , w -1,0. t- :
t = (b 0 )/ ..(b) = 0,82 1,00/ 0,10 = - 1,8
, , 20 , 18, t 5% 2,101. t- 2,101 2,101, , . = 0,82, 1,00, , .
, , :
b 0 / ..(b) t . < < b + 0 / ..(b) t . (1.48) , , b, , . , , .
5% , 95- , 1% - -99% .
t , 5% ,, 99% 95%. b , 99% 95% , .
:
b 0,093. 0,003, t 5% 2.069. 95% :
0.093- 0,003 2,069 0.093+ 0,003 2,069
, 0,087 0,099
0,099 0,087. , , .
.
, , , .
0 : R2 = 0 ( .. ) F ( ). :
F F ,
( 0 : r = 0 ) r t- :
(1.49) , 5%, t (n-2) . t ( ), , r =0, , ( ).
0 : = 0 0 : R2 = 0 . 0 : r = 0. .
1.9. . = f (x), z = g(x), F (z ) . , .
y= +/x y= x . , . , ( ) ( ). , . z = 1/x, y= +z . , z .
, :y= x u (1.50)
y*=lny, z=lnx *=ln, (1.50), :
y*= * +z (1.50)
y= ax u
lny= lna +lnx +lnu (1.51)
1.9.1.
y= x . , , . :
= dy/dx = (1.52) , :
y=0,01X 0,30 , (1.53) , 0,3. , () 1% () 0,3%.
2. .
2.1. .
, .
, .
: y = +1x1 + 2x2 + 3x3......... + ........... m x m + u (2.1) y = 1x1 + 2x2 + 3x3......... + ........... m x m + u (2.2) . : 1 2 3............ m , .
, , . S, .. :
S = 21 + 22 + 23 + + 2n - i, *, , :
* = + b1 1 + b22 (2.3)i = i * = i - b1 1 - b2 2 (2.4) :
(2.4a) (2.5)
(2.6) , . , , , - TSS, - ESS, - RSS . .
- R2=ESS :TSS, - .
.
( ) ..(i ). :
(2.7)
(2.6)- , . .
2.2. .
. , . ,
(2.7) ,
y = +1z1 + 2z2 + .... (2.8) z1 = x12 , z2 =
. , , x , , , . ,
y= x p , (2.9) y , x - , p , - , , :
lgy= lg + lg x + lgp + lg (2.10) lgy, lg x lgp, lg x , lgp .
- . 1927 . , , , (Y), () (L), . , , :Y = AKaL1-a (2.11) , , , . log , , log L, (1 - ), .
L :Y/L=A (K/L)av (2.12)( v). , , :log (Y/L) = log A + a log (/ L) + log v (2.13) (Y), () (L) . 2.3. .
R2.
.
, 1 -1, , 1 -1.
, , () . , . , .. , , .
, , , . , . , . , .
, , : , ; . , , , . , . , .
, R2 , , = *. ( ) , . , 1 2 :
(2.14), , 1 , :
. (2.15) :
(2.16)
(2.14) (2.16) 1 2 , . (2.16) 2 0, , b2 , , . ( , .) , R2 (2.14), (2.16), . , R2 , , . , . , , , . ,- .
1 2 3............ m , . , .
F- , , , F- .
(2.17)- ESS- , RSS- () , k- , .
F , , . F , k , (n-k-1) (F-). F- , . F- , (2.17)) TSS ( ), , , . :
.2.1 , F- , ,
(...) (.)
(..)...,
..
F-
,
207,49369,1869,16/ 5,70=12,1
91,18165,7
F 3 16 1% 5,27; , F- , 12,1, .2.4. , , , .
, . , , .
.
1. , , , .
2. , , .
3. .
, .
.
, .
1. , , , . t .
2. , , . , - .
R2 , . , .
, , . , , , . , , . , , .
- , , . , , .
, . . ,
(2.18) , 1, (z) ,
(2.19.) , .
(2.20) 1. , , a . . , . , , .
, , :
(2.21) - ; - . , , . , () , . (), () () . :
(2.22) , (). , =0, . - (). , . , . ( ). . , ., (), (), , , , :
(2.23) t , , .
, , , , , , :
(2.24) , .2.5. , , , , .
, , , . , ?
, . - , : 1- 0 .
, , , ..
2.2 , 1974 . , . . , , , 2.1, - , - .
D, 1963-1973 .. 1974 -1982..
(2.25).2.2 1963- 1982..
( 1972., . .)196318,5196826,5197333,9197834,8
196419,7196926,7197425,5197932,9
196523,5197022,7197525,4198028,7
196623,6197128,0197631,1198129,6
196722,2197231,6197734,4198229,8
1963 1973.. D=0 :
(2.26) 1974- 1982.. D=1:
(2.27) , . y x . D, 11 , 9 , :
EMBED Equation.3 , 1963 1973., 2.1, 0,57, 1974- 1982..-3,83. t- , 5%. , :
5%.- , pgas, :
. . pgas 1% . R2 0,69 0 ,87. , () , .
R2 , t- . , ( ) .
2.5.1.
: . . :
- - . , 964 :
, , , 3400 7 . . : . :
(2.28) , , . , .. , .
:
(2.29) , , , . .2.2. . , -. , . - D.
(2.30)D=0 D=1 .
y,x :
103. 964 :
, , ?
: ( -0 ); 1; - 2; - 3. -0, . D1 , D2 , D3 :
0 D1 = D2 = D3 =0 1 D1 =1; D2 = D3 =0
0 D2 =1; D1 = D3 =0
0 D3 =1; D1 = D2 =0
:
, .
, . , .
- , . .
, , . :
(2.31)- .
- 1 0 (), - 2 0 , 3 0. 964 , :
.
, 1 127 0, 2- 102, 3 105. , , , .2.5.2.
, , , , 4 . . . , , . , - , , , .. 1 4- 6 . 1- . :
(2.32) D2, D3 D4 , : D2 , , D3=1 3 0 , D4=1 4 0 .
- , . - 2 1- , .
EMBED Equation.3 .
, :
. (..) (0,09) (0,005) (0,09) (0,10) (0,10)
:
2.5.3. .
, ( ), , , . , . () () (D):
(2.33) , , , .
, - D
(2.34)
(2.35) D=0, , .
D=1, , . () , , () , .
, , Dx . 964 :
D=0 D=1, :
( )
( , ).
, , , , , , (12,1 4,0), 5%.2.6..
. , . , . , . , . :
y = +x + u , :
1. (ui ) = 0, ;
2. 2 (ui ) = 0. ;
3. Cov (ui ,uj ) = 0, i = j;
4. Cov (i ,ui ) = 0, .
, , .
, .
, (. 2.3.).
, .
, . ,
:
: 2 (ui ) = 2 : 2 (ui ) = i2 , (.2.4.) - , .
.
. , , 2.4., ( 1 2.3), = + , , , . , ( n 2.3), . , , . , , , , , . . 2.6.1. . .
, . , . :
; ; .
, , . .
, , , , . , :
(2.36) Di -
, 0, 0 1/ ( n-1 ) . 5%, 1,96, 1%, 2,58. , .
, , ( ) (GDP)
, 1 . . 67 .. ( - ) :
. 2,58, , , . , , u , . n , ; ( n-2) . , u ,
, i i , , i= + xi (2.37) , , xi i. , i. , 0. ?
- u i. , , . :
(2.38)
(2.39) - , I
. . , . , , .
, z zi = - . z :
(2.40) , .
, , . , :
(2.41) , .
2.7.
- Cov (ui ,uj ) = 0, i = j; . .
.
, .
( 2.5)
- , (-2.6). , .
, :
Uk+1 = p uk + ek+1, p - , ek+1 -
, :
(2.42)
ek - .
, =0 DW . , DW , . - 1 -1, DW 0 4.
.
- , :
1. .
2. .
3. ek ek-1 Uk+1 = p uk + ek+1, ek-1, .4. yk = +xk + uk yk = gk +xk + uk , yk = yk - yk-1 , xk = xk - xk-1 gk =1 - ,
.5. , .
3. . 3.1 .
(...). :
X11(t) X12(t) X13(t) -------- Xnm (...) X21 (t) X1 2(t) X13(t) -------- Xnm(t) (3.1) ------------------------------------------ Xn1 (t) Xn 2(t) Xn3(t) -------- Xnm(t) Xnm - n - , n - t. , , t , . , ( .. ... ), . X(t1),(X t2), (X t3), .. (X tn) (t) t1 t2 t3 t4 tn. .. t2 t1 = t3 t2 =.. tn tn - 1 = , - . : (1) (2) (3).. (N). 3.1.1. . , , .
1 2 ..n , .
1. .
2. , ..
P { (X(t1) < X } P { (X(t2) < X }, t1 t2 (74) , , . , , :
. , ( ) X(t). f(t ), . .., . - (t). . . ( ), , , . (t). .
. () , . . ( , ) (t).
:
1. ( ) X(t).
2. , X(t) , , , . , , X(t) :
X(t) = X() f + X() (t) + X() (t) + (t), (3.2) t = 1,2.N , , , . , .
. , .
3.1. X(t) , 96 1963 1970 ., (.. t= 1,2.96; )
, . . 3.2. 1960 1971 ..(; , N = 48 ).
.3.3 56 ( 1884 1939..). , N = 56.
, ( , ).3.1.2. .
. X(t) , n X(t1 ),( X t2), (X t3), .( X tn ) n X(t1 + ),( X t2 + ), (X t3 + ), .( X tn + ), n , t1, t2...tn, . , t, , t , :
(t ) = a (3.3) D ( t ) = [ (t ) ]2 = 2 (3.4) X(t) , D ( t ) . 2 :
- (3.5) (3.6) ( ), - t.
, , .
.
3.2 .
3.2.1. , .
(t ) (t1 ) t. , (t ) : ( ) = cov[ (t ), X(t1 + ) ] = [ (t ) - , X(t1 + ) - ] (3.7) - ( ) . ( ) - :
(3.8) =1,2,. N
= , (0 )
(3.9) , . .
(3.10) r() r(). .
:
(3.11)3.2.2. ( ) .
, .
1- (t ), (t + 2), , , ( .. (t + 1) = )
(3.12) r02 = r([ (t ), X(t1 + 2 ) ]) (3.13) r01 = r([ (t ), X(t1 + 1 ) ]) (3.14) r21 = r([ (t+2 ), X(t1 + 1 ) ]) (3.15) r02= r(2), r01= r(1), r21= r(-1) = r(1), (3.16) . 2 - ( . . 3- , , 2- ) :
(3.17) .
3.3. .3.3.1. .
f , (t) (t)
X(t) = X() f + X() (t) + X() (t) + (t), (3.18) :
1. ( t) ; H0 : (t) = = const (3.19)
H1 : (t) = const (3.20)2. ( ) X(t) = X() f + X() (t) + X() (t), .. (t), X(t) ,. H0 , .
, ..
(1) (2) (3).. (n). (3.21) ( ) (1) (2) (3).. (n). n , n , X(t) +, X(t) > nmed - , X(t) > nmed , nmed . , . , (n) (n) . - . , (1) (2) (3).. (n) , ( .. ), + - . .. + -, , , . , . (n), (n). H0 : (t) = = const ;
(3.22) , , 0,05 0,0975 ( , , ). ( ) , , .
, + -, .
, (1) (2) (3).. (n)
i +, (i +1) - i > 0, - (i +1) - i < 0 ( , ). , ( ), ( ). : , , , .
(n), (n) - , , + .
:
(3.23)
, H0 .
.
, .
( ) , X(t) =X() f + X() (t) +X() (t),
t = 1,2,..n.. X(t) = f (t) + (t), f (t) -, (t) .
, ( 3.2 ), , t, ..
(3.24) - ( . , . ( ) , .
f (t) (t) ( .. ) .
. X(t) Q2 , 1 + 2 +3 +.. (N)/ N , Q2 / N . (t) : (3.25) - , , .. t m + 1 n-m , ( ).3.4.
3.4.1.
, (t) .. , .
X(t) =X() f + X() (t) +X() (t), t = 1,2,..n (3.26).. X(t) = f (t) + (t), f (t) , (t) -
, .
() (t), , ( ) .. 0., .. (t) 0. (t) :
(t) 0, (3.27)[ (t), (t )] = Q0 2 =0, 0, (3.28)
, Q0 2 , t.
, , : m (t)= (t) + 1 (t-1) + 2 (t-2) + .. = 1, (t - k)], (3.29) k =0 m
0 = 1, 2 k <
k =0
, , . , (t) , ( ), ..
(t) = 1 (t-1) + 2 2 (t-2) + . (t). = k (t - k) + (t). (3.30) k =1 1, 2, 3 , (t). .
3.4.2. 1- 1- , j, , 0 :(t) = 1 (t-1) + (t), (3.31) - , < 1 , (t) . ( .. , (3.27 3.28). . :
. (3.32) = .
. 3.4.3. 2 2 ( ), , j , , 0. :
(t) = 1 (t-1) + 2 2 (t-2) + (t), (3.33) (1), (2) .
1 2 :1 = r(1) [ 1- r(2) ] : [1- r2 (1)] (3.34)2 = r(2) r2 (1) ] : [1- r2 (1)] (3.35) (3.34,3.35) r(1) r(2):
(3.36)
(103).
.
1- 2
1
< 1 (3.37) 2 , 1 , 2 : 1 j =1,2,.......p.
- 1 2 3 .1 2 3 Q0 2 ( 1 )
(3.57) , ( 1 ) ( .. < 1 ). ( 2 ) ( 1, 2 ) .
= - 1 ( 1- 2 ) : ( 1 + 21 + 2 1 ) (3.58) = - 2 : ( 1 + 21 + 2 1 ) (3.59) 3. 5.
3 5. 1.
, . X(t) , , , , , . , (t) , X(t) f(t), ( ) . X(t). ( ) (p, ,k ) . X(t), t = 1,2........ N, :
) f, ( - t ) k-1(k< 1) .) X(t), t = 1,2........, N-k , X(t) - , (p, ). 3.5.2. . X(1), X(2), X(3), ............ X(N), (3.60)
y(1), y (2), y (3), ............ y (N), , y(t) X(t), X(t - 1), X(t 2) , t +1, < N. , :
y(t)= 0 + k (t - k) + (t). (3.61) k =0 (t) , t = 1,2,.. N Q0 2 = D (t) - ( X(t), X(t-1),....... X(t- T), ; 0 , 0, 1, T .
. , ( ) t k (k > 0), , k (). k k , . .
, , . . , (.3.4, ) (.20). .3.4.3.5 , , , .
. , (t - ) (t) T , , . ( T = ) . B= k , .. Lim k = B