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CeG:b$kOi\e,)b$@* LҪ)b$G@/D۞'O0 E*ob$u:򙜢>Ty93)<2$PC/tƭ0Vfb$:Ԏ\~5´)gb$YEr>g;C1בb$uA ƞqD.b$P hlJ7w+b$v^d]$3*b$t^̬K`m+,Hb$*B*m:d$x)s2$]KOK Ţ]032$_I]퍁E|^?$2$-iTkwX2$xs$UTW2$9By4@;X2$}Y.3#_L2$JUQ!إ2$7/VlX*2$ᙄn5@ke2$Ld#oU@8A2$WIg!Ϩ!:Ϭ2$&.(ѦZ2$ 0Pmj2$82iσJ w)2$=N+LΨL²2$Պh\pקec $ 33f@$g4BdBdv 0RTppp@  <4BdBd 0Lvg4+d+dv 0rp@ pp]ʚ;2Nʚ;<4!d!d{ 0<4dddd{ 02*___PPT9 z? -O =|3/ P>Q?R@SATBUCVDWEXFYGZHA;:6;7<8  !".+/, 1-3 ` ̙33` ` ff3333f` 333MMM` f` f` 3>?" dd@,|?" dd@   " @ ` n?" dd@   @@``PR    @ ` ` p>> me  (    0 B @  B B 4  #    60B$j a . f ` H  0޽h ? ̙33 Default Design0 jb( X   0T~ P    P*    0B     \* d  c $ ?    0E  @  8 cN NN}/kGreW[#j_ ,{Nd\ ,{ Nd\ ,{Vd\ ,{Nd\    6H `P   P*    6~ `   R*  H  0޽h ? ̙338 (    0Z P    V*    0_     X*    6b `P   V*    64f `   X*  H  0޽h ? ̙33f (    6T7Bf) `0 ~Regresion Multiple, b0`      <H=2 S ( Y - b1- b2X2 - b3X3)(-X2) = 0%  % <xB P  5^ & <$E0 p `  7 ^ ' <Et p  `  6 ^ ( <E p `  5^fB ) 6D  8   >  - 6| E  hdSCR db38   . <E` @J =2 S ( Y - b1- b2X2 - b3X3)(-X3) = 0&  / <EdtT 5^ 0 <@E `  7 ^ 1 <@ E  6 ^ 2 <#E  5^fB 3 6DppX2 ? 0@ ` ` A c $A )??  )H  0޽h ? ̙33!   @,9 @A (     0,E  Ordenando las tres ecuaciones: 2H  :8  5    6(2E Bnb1 + b2 SX2 + b3X3 = SY" 2b@ P  0 P    <L=E0 7^Z   <@E 7^Z   <DE P  8 ^ZF    )  x   6DJE   Fb1 SX2 + b2 SX22 + b3 S X2X3 = SX2YH$ 2   <XE  7^Z   <HE 9 ^Z   <^E   9 ^Z{F  p  +  p     6cE   Fb1 SX3 + b2 SX2X3 + b3 S X32 = SX3Y8$ 2   <qE p` 7^Z   <sEp` 9 ^Z   <wEP p, ` 9 ^Z- 8  P  " P     0 |E P@ p ien forma matricial: 2    0|E r  V n SX2 SX3 SX2 SX22 SX2X3 SX3 SX2X3 SX32tJ 2  ,$f   6G  @ [@ 0 p r    p    0 E0  r  b1 b2 b3t 2   6`E0 p `  7^Z   6 4E0  5^   6E0   5^f   6G0 pp    0xE @  5= 2   0E r  SY S X2Y SX3YV 2f !  6G` p  # 0hEP p r (X X)L 28  P  & P  $  <E  Fb( %  <E P p 3^ ' 0hE 0 ^X Y< 2dB , <D@ dB - @ <D@ @ dB 1 <DT j  T dB 2 <DT T  3 6pEP  jCaso 2-variables `   X 6 0p@ X 7 0`P  9 6xE   kCaso 3-variables  `  H  0޽h ? ̙33% >%6%P*>$( PL{{   <E   Xn SY SX3 SX2 SX2Y SX2X3 SX3 SX3Y SX32H @  <0EP < n SX2 SX3 SX2 SX22 SX2X3 SX3 SX2X3 SX32rG  XB  0DP` P  0Ep 5= 2XB  0D  XB  0D  XB   0DPXB   0D PF  P =    `  <@E =  Vb26  <H P  p A^$  0XH 0  5= 2XB  0DP P  0 Hp  b8(Syx2)(Sx32) - (Syx3)(Sx2x3) 2  0 H`  J.(Sx22)(Sx32) - (Sx2x3)2 2  B$H  Rn SX2 SY SX2 SX22 SX2Y SX3 SX2X3 SX3YE >  BT9H   n SX2 SX3 SX2 SX22 SX2X3 SX3 SX2X3 SX32rG  ^B  6D `   6hIHP pp  5= 2^B  6D ^B  6D ^B  6D@  ^B  6D@ L  P =  #  p   <MH =  Vb36  <HRH P  p A^$  6VHP  0 p  5= 2^B   6D   ! 6(\H`   b8(Syx3)(Sx22) - (Syx3)(Sx2x3) 2 " 6hH 0  J.(Sx22)(Sx32) - (Sx2x3)2 22F pp  ; @ b 3 6tHp  (b1 = Y - b2X2 - b3X3 2 4 <XH0 E^(Z 5 <HH S ^4F 6 <H T ^4P 7 <XH 7_( 8 <Hp` 7 _ 9 <pHp@ 8 _^B < 6DԔ^B = 6DԔ@@ > 6ԖH0  lRegla de Cramer: `   H  0޽h ? ̙33!, ++`;K 9+(  /  0|HP /or en forma matricial (no entra en este curso):,0 2P   BdH3 A^$v8 p  Hp   B(H 0 Fb(  6LHP r (X X)L 2  64H` p 0 ^X Y< 2  6H  73x3 2   6H  73x1 2   6XH`   73x1 2  6H@`  5= 2F @  E @    6H@Pp 7==> 2T  P    # @$p   <H  Fb(  <H P  p A^$  6H0PPp 5= 2  60HP  (X X)-1f 2   6xH0 P p z (X Y)T 2  6 Pf @  0  = `  % 6I 0  zVar-cov(b) = B 2 ( <P#I   ;^8 - 0(+IP P  Var (b1) Cov (b1 b2) Cov (b1 b3) Cov (b2 b1) Var (b2) Cov (b2 b3) Cov (b3 b1) Cov (b3 b2) Var (b3)    . 6*I  ;^ / 6THI  ;^ 0 6LI @  ;^ 1 6 QI   D ^& 2 6lUI   ;^ 3 6YI   D ^& 4 68^I   ;^ 5 6bI@   =^ Z 6 6fI@@   =^ Z 7 68kI @  =^ Z 8 6oI ` @  =^ Z 9 6sI `  =^ Z : 6PxIp` 0  =^ Z ; 6|I `  ;^ < 6ЀI0 `  =^ Z` ? 0  8  ` K `" A 6I @` = su2(X X)-1 2 C <(I 1^X I 0  ^B J 6D1H  0޽h ? ̙33]  p )@(    0I `R Jn SX2 SX3 SX2 SX22 SX2X3 SX3 SX3X2 SX32 s38 0  0N @` `  `  0`I@@` ` d = su2> 2  6I ;^`  0  6I0 2-1E 8  @PP @P   6 I   su2 = h 2   <нI` E^(fB  6D ` `T  @ ` # P @` `  6I @ ` LSu2* 2  <I @P  5^  6IP 00 P 7n-3 2  6I  5y 2  6I   5= 2T  @ ` # @@P`  0I @ ` LSu2* 2  6I @P  C^&fB  6D `P`  6tIp0PP 7n-k 2^B  6DԔ  P F  @ `   @  ` ! 0I @ ` LSu2* 2 " 6I @P  C^&XB # 0DP P P^B % 6DԔP P P^B ' 6DP  ( 0xI 0  7k=3 2 ) 0@I p  =# de variables independientes (incluido el termino constante)> >> H  0޽h ? ̙33- }-u-=U( ,( :   0pIPP- #Propiedades de las estimaciones MCO&$ 2#  6    6K MO1. La linea (superficie) de regresion pasa a traves de las medias de Y1, X2, X3fP 2F   t    <K,`  1_  <| K@B 1_  <K@v 1_   0<K 9i.e., 2F  pP N p Pr  6K  (Y = b1 + b2X2 + b3X3 2  <KpP` 7_F  <#K0   T ^4Z  <(K   W ^4Z  <t,Kp` R ^2  <01K   6 _  <4K   8 _  67K 00P 7==> 2  0 ;K  62. 28 @`  &@` L  04>K@`  &Y = Y + b2x2 + b3x3 2  6GK@ 9 ^(   6KK  : _F ! 6 OKpp R ^2 " 6@SK  p S ^2-8 @ P  S @ P Z  6WK@0 P   y = b2x2 + b3x3 2 # <aK    P ^2 $ <lfK   C^& ( 0jK0 0P  5o 2 ) 0,mK 0  63. 28 0 0  , 00  * 0xpK0 0  >Su=0 2 + 6sK P  C^& / 6TxK   SuX2 = SuX3 = 0V 2 0 <~K| V<  C^& 1 <ȂK| W<  C^&F 0 0  3 @0` 4 0K0 0  b SuY=0 2 5 6 K P  5^ 6 0K   64. 2 7 0K@` 65. 2 9 6K ` P  (SuXk=0 )8 2 : < K P  C^&8 @ P U@ `p0 > 6hK ` P  7t = 2fB ? 6D     A 6K P   ` bk - 08 2 B 6K P  l se(bk)D 2 C <K p   5^ D < var(b2) y var(b3) mas grandes. Ademas sera mas dificil estimar los verdaderos valores de b2 y b3 .  2    J +  <$Lpp0 N ^0x  <L@` L ^.  6 L@ Cuanto mayor es la variacion muestral de los valores de X2 o X3, menor es la varianza de b2 y b3 , y las estimaciones son mas precisas. 29  '     <L0  P ^2  <L pn P ^2  0t$L`   68. 2  0(L` p  :ELIMV. 2+  01L  Todos los supuestos para el caso de regresion simple (dos variables) se siguen manteniendo para el caso de regresion multiple. Pero hace falta un supuesto adicional: No existe una relacion lineal exacta entre las variables independientes.P `Gdddh                                                     6+L`P *Propiedades de las estimaciones MCO (cont)++ 2   H  0޽h ? ̙33 H@(  <  0EL 0El significado de los coeficientes de regresin:*1 20^     0NL0 ~`E(Y| X2, X3) = b1 + b2X2 + b3X3 =PLO(Y| X2, X3) v1 2      F `  `&  0`L` D E(Y|X2, X3)b 2  `B  0D  0$hLL ZDX28 2"  0mL @  = b2P 2 ,     0sL P0 @ d: b2@ 2   0HyL ` ocmide el cambio en el valor medio de Y, por un cambio de una unidad en X2, manteniendo X3 constante.Xd 2G         0$LP 5o 28   0L  El efecto  directo de un cambio en una unidad en X2 sobre el valor medio de Y, neto de X3.d] 2 ! %     0 L   :?????? 2XB  0D0 0F `  `  &  0ܕL` D E(Y|X2, X3)b 2  `B  0D  0LL DX38 2 8  0`L @  = b3P 2 B     0tL t  >Los parmetros son desconocidos, as que habr que ESTIMARLOS.?(2?>  H  0޽h ? ̙33   \(    C hAPD:\COURSE\ECON2170\GUJARATI\tab701c.bmp` F         6L   aY. d` `B  0D    6L@   ~sYH `   6L0` P NPY=E(Y|X2,X3)+u =b1 + b2 X2 + b3 X3 + u =) `h`h` ) X2  00  L  @8   #  0    6L 48  ~suH `    6PL @  a^. d` X   0 `  c $A K??  KH  0޽h ? ̙33! !!(( .!(      0LP  {Tasa de inflacion actual(%) 2$  2   6tL0 P  7 Y ^B   6D0 ,   6 L@: DE(Y|X2,X3)n 2  ^B   6D)   6(L2 ZDX28 2   6L |= b2 = -1.3925D 2    6"L>  6*Y = b12 + b22X2 + u2 2J      <"xL`  8 P   <"La! 9 F   <"Ok 9 <^B   6DP p   6E7(OP  (Y = b13 + b23X3 + u3 24      <E7PO  8 <   <E7O` P  9 F   <E7HO  9 <2   6O@`  ^ X3 > 2   6O0P  ^ X2 > ^B   6D  ^B  @ 6D     <#O w B 2 ^B   6DH    6t&O  ZDX38 2   6,O l  = b3= 1.4700D 2      0l0O 00  Tasa de inflacion esperada (%)6    0,6O p  oTasa de Paro(%)$U   0:O `@  X3 = g31 + g32X2 + u23 2AA4      0dDO0  >X2 = g21 + g23X3 + u23 2 E E   J    LF      x    0tPO   ZDX38 2`B   0D   !  0UOx   ZDX28 2 "  0P[Op 0  = g32 = 1.1138B 2E    #   fBTCt DEF<%%D` 4 @d @\ d l t d 8 T % %  { ^   ClgX"dTApR }  -  -`@             tX  $  6Z`OP Efecto directo de X2 `8     %  0,eO P  Efecto indirecto via X3 `8      &  696    $ 0OP . = -1.392472 + (1.470032)(1.11385)/ 2/  "XB $ 0DQQXB $ 0D  $ 6OdT 7 Z $ 6ROdT 7 Z $ 00OP 2 directo +  indirecto  2,    $ 0O@ J= -1.392472 + 1.637395 2  $ 0XO > = 0.244923 2   $ 0O` P  DE(Y|X2)D 2 XB  $ 0D    $ 0O  ZDX28 2 $ 0\O    = b22P 2  $ 0ĮO  T= 0.24490 2 $ 6O  B F   $ P 5N P  $ ` 2 $ 6O P  PX22 fB $B 6D 2 $ 6POP  5Y $ 0@O` P  .E(Y| X2 )= b12+b22X2  2      fB $ 6DԔ  ` $ c $A M??$ d  M` $ c $A P??R> PH $ 0޽h ? ̙33~ .&((  ( ( C hAPD:\COURSE\ECON2170\GUJARATI\tab701a.bmp0 ( 6|OP pP `JY=E(Y|X2)+u2 = b12 + b22 X2 + u2 =& `h`h`& X2 ( 0`X ( 0  M ` ( c $A S?? 9 SH ( 0޽h ? ̙33 ,(  , , C hAPD:\COURSE\ECON2170\GUJARATI\tab701c.bmp Pp`X , 0` P  , 6HOP` P X3 = g31 + g32 X2 + U32 = `h` ` , c $A T?? P TH , 0޽h ? ̙33x ( 0(  0t 0 0 O 0Efecto total estimado de X3 sobre E(Y|X2, X3): 1 2   >    0 0TO = = 1.470032 + (-1.392472) (0.369953)> 2> $XB 0 0DQQXB 0 0D  0 6VdT DZ  0 6VdT 7 Z 0 0V ` @ 2 directo +  indirecto  2,    0 0 V m@ J= 1.470032 - 0.515149 2  0 0hV ? = 0.9548828 2   0 00V`   DE(Y|X3)D 2 XB  0 0D `@   0 0V  ZDX38 2 0 0pV @   = b23P 2  0 0X%V  V = 0.9548830 2  0 6`*V  B F   0  5N P  0 ` 2 0 6,V P  PX32 fB 0B 6D 2 0 6 2VP  5Y 0 05V` P  6*Y = b13 + b23 X3 + u3 2J   fB 0 6DԔ  ` 0 c $A V??r` VH 0 0޽h ? ̙33 4-(  4 4 C hAPD:\COURSE\ECON2170\GUJARATI\tab701e.bmp  4 6CV  p  /X2 = g21 + g23 X3 + u23 = `h` X 4 00  ` 4 c $A$ W??@  WH 4 0޽h ? ̙33 x 8(  8 8 C hAPD:\COURSE\ECON2170\GUJARATI\tab701d.bmp0x 8 6@TVS P .Y = b13 + b23 X3 + u3 = ` X 8 0 P P ` 8P c $A% X?? : XH 8 0޽h ? ̙33 WO0#<(  < < 0bV VEjemplo: 2  < 0fV  p@ Y = f ( X2 , X3 , u )L 2    < 0lVp  bProduccion de output 2   < 0qVp ` c input trabajo 2 < 0tVpP A input capital 2 < 0|xVp xY = E(Y|X2, X3) + u = b1 + b2 X2 + b3 X3 + u= 2   ( % < 0TV  GoSuponed que se puede controlar el input capital y queremos medir el impacto del input trabajo sobre el output.$p 2n   XB  < 0D   < 6.VpP  R(X2)2 l  < 0ԒVP V  8Paso I : regresar Y sobre X3 y obtener (regresin corta)d9 2 L  L  `   <#     < 0dV `  7Y = 2 < 6ܟV  9 n^B < 6D ^B < 6D@0^B < 6D@ pwF P d < P d2 < 6VP   5Y2 < 6  @ 60V0 P  B  @ 0V B 2 X @ 03fp X2 @ 0p0 @ ` @ c $A _??pP  _` @ c $A( `?? P $  `` @ c $A# a??P8 a` @ c $A) b??H 8 b` @ c $A* c?? @   c` @ c $A+ f??  fH @ 0޽h ? ̙33^ P ( w   C hAPD:\COURSE\ECON2170\GUJARATI\tab701c.bmp` F         6HV   aY. d` `B  0D    64V@   ~sYH ` ^  6d P 0 4Y = b1 + b2 X2 + b3 X3 + u ` X2  00  L  @8   #  0    6d 48  ~suH `    6Hd @  a^. d` X   0 H  0޽h ? ̙33 `  ( w   C hAPD:\COURSE\ECON2170\GUJARATI\tab703a.bmp k   C hAPD:\COURSE\ECON2170\GUJARATI\tab703b.bmp@ X2  0P `0 X  0 H  0޽h ? ̙33 p ( w   C hAPD:\COURSE\ECON2170\GUJARATI\tab703c.bmp   C hAPD:\COURSE\ECON2170\GUJARATI\tab703d.bmpX2  0 P X  0` H  0޽h ? ̙33 ~v ( w   C hAPD:\COURSE\ECON2170\GUJARATI\tab703e.bmp p  C hAPD:\COURSE\ECON2170\GUJARATI\tab703f.bmp`  ^  6oP ^  6o(    6 d < 1u1 = l1 + l2u2 + v 2 E EM 4    <,d  6 ^  <0dP  P ^2`  c $A g??, d  g  B5d`N Q^   B9d0P Q^   B6d`p> Q^ H  0޽h ? ̙33?    8 ( 48H( 8  8 s *Dd3 -Contraste de hipotesis en regresion multiple:*. 2-  6     8 0\Jd  ,1. Hipotesis sobre coeficientes individuales- 2-P     8 0Od  42. Significatividad global de todos los coeficientes5 25P  T 8 0dUd <3. Restriciones sobre variables (aadir o quitar): Xk = 0 ?2= 25 b   8 0]d4 Vz4. Hipotesis sobre varios coeficientes b2 + b3 = 1h> 2  P   8 0edP   Ey6. Contrastar la estabilidad del modelo de regresion -- en el tiempo -- en diferentes secciones cruzadas z z       % 8 0md@  95. Contrastar la forma funcional del modelo de regresion.: 2:P   H 8 0޽h ? ̙33  N F *<` ( : <v < 6wd 0@  PModelo: PCE = b0 + b1PD1 + b2TIME + u) 2# < 64dp   t Y = 53.163 + 0.726 X2 + 2.763 X38! 2  < <   8 p @0 @ 0 tc ==> rechazar HoP& 2   >k8  `6 @ `&m @ 0n0 `6 WRespuesta : Si, ya que es estadisticamente significativo (diferente de cero).X 2@l      @ 6n`   @x  @ 6x n0 'AManteniendo X3 constante: Tiene X2 algun efecto sobre Y en media?hB 2   Z    2 @ <n 11L `   @# P `@ !@ 0n`   >DY 2`B "@ 0D @  #@ 0|n  NDX2, 2 $@ 0n@  b= b2 = 0?4 2` %@ c $A ,?? h 8 ,H @ 0޽h ? ̙333  #Dg(  D2 D 6*n 12 D 0-np %AManteniendo X2 constante: Tiene X3 algun efecto sobre Y en media?TB 2   l   nF p D 0`P D 0`7n xH0 : b3 = 0F 2 D 6<=n p0 2 ^  D 0@n xH1 : b3 0F 2  D 6(Gn  2 ^F `   D  @  D 0HKn`   >DY 2`B  D 0D @  D 0`8n  NDX3, 2 D 0tRn@  b= b3 = 0?4 2i8   0  D`0@  D 6 | tc | ==> rechazar Hoh. 2    >  4F  `P  D  p*  D 0Pn0 `P  8Respuesta: Si, es estadisticamente significativo.9 2*H    !D 6n`   >  "D B n`s 41. Contraste sobre un coeficiente individual (cont) 5 25, ` #D c $A -??P ` -H D 0޽h ? ̙33  T L  1LP ( X"@xv0 L~ L s *|n` $D2. Contrastar la significatividad global de 2 y 3 en la regressionE 2*$ $ , $h L 0n0 0 0Y = b1 + b2X2 + b3X3 + u 28  0 L` 8 L 6n  (H0 : b2 = 0, b3 = 0, 2  ` L 6Tn 0 (H1 : b2 0 o b3 0 2 $ L 0n0`  1. Calcular y obtener el estadistico F 2. Buscar el valor critico Fc (F c a, k-1, n-k) 3. Comparar F y Fc , y si F > Fc ==> rechazar H0 C              .L <8nj f  B    ` /L c $A 4?? p  4` 1L <n0  SCRr= suma de cuadrados de los residuos del modelo restringido SCRnr=suma de cuadrados de los residuos del modelo no-restringido q=grados de libertad del numerador=glr-glnr n-k=grados de libertad del denominador=glnrl>b* ;  `    '   H L 0޽h ? ̙33 1HP"( v@ H H 0np fPaso I: recordad 2` H <n (y = b2x2 + b3x3 + u H <n@p`  E^(x H <r@Z J  P ^2  H <hrfB 2  P ^20  H < r@ RS y2 = b2 S x2 y + b3 S x3 y + S u2V*   H <rP C^& H <|rs c P C^& H <T!r Z T ^4xd H <76@d H <76 d H <76@? H 6&rA (SCT = SCE + SCRH) 2~ /H s *-r` $D2. Contrastar la significatividad global de 2 y 3 en la regressionE 2*$ $ , $` 0H c $A" 5??P*  5 1H 06r` 0  y= 32982.55/ 6.430 = 5128.870   H H 0޽h ? ̙33  X P  ` ( : `F  0 ` p ` 0?r  (H0 : b2 = 0, b3 = 0,b 2 ` 0Hr 0 (H1 : b2 0 ; b3 0n 2 ` 0Or -Fc0.01, 2, 12 = 6.93 Fc0.05, 2, 12 = 3.89t. 2    f ` 0tXr  >Regla de decision: Dado que F* > Fc ==> rechazar Hon? 2   6 ` 0(arp 0  6Respuesta : Las variables X2 y X3 no son irrelevantes.87 2$  &  ` 6frP P ,Buscar el valor critico en la tabla de la F- -`T        ` 60lr`@p 2DY = b1 + b2 X2 + b3 X3 + u #  ` 0$xr' >   ` BT{r XPaso II:  Xr ` 0 @ ` Byr Y Paso III:   H ` 0޽h ? ̙33& &&EaT0  &( : T  T s *܇rp3  Relacion importante entre R2 y FL! 2    6 8 Pp  BTpP  T 6rP0  CF =$ 2@  p Tp0 fB  T 6D   T 6r p <SCE/ k-1 2   T 6Hr  <SCR/ n-k 2  T 0tr0 P  6 = 2`B T 0D   T 0r p  = SCE (n-k) 2  T 0,r   = SCR (k-1) 2  8 `  AT@   T 0Xr0 5= 2`B !T 0D00 0 "T 0r``  ;SCT-SCE 2 #T 0Tr 0 7SCE 2f@ ` `  0T` ` `B $T 0D 0 0 %T 0r 0 0 7n-k 2 &T 0 r `0  7k-1 2` 'T 0` p`  -T 0r& hF  7SCT 2*8 P`  @T  (T 0Hr0  5= 2`B *T 0Dp0 p +T 0rP0 p ;SCE/SCT 2 ,T 0rP@0 `  7SCE 2`B .T 0DP0 0 0  /T 0XrP  71 - 2tN ` `  1T ` P` `B 2T 0D 0 0 3T 0hr 0 0 7n-k 2 4T 0r `0  7k-1 2` 5T 0` p` k8  `  DT @`  6T 6lr 0  5= 2fB 7T 6D p  8T 6r   PR22 2  9T 6dr@ @`  T1 - R22 2 zT ` `  :T#  ` `B ;T 0D 0 0 T 0` p` 8   0 KT @` ` CT 6,rP 0p 5= 2fB ET 6D   FT 6r  X R2 / (k-1)2 2   GT 6rP 0 Z (1-R2) / n-k2 2  HT 6r @g AF$ 2 JT 6xr P  J  d F p0 `  LT  0 MT 0(rp RR2 =2 2 `B NT 0D 0  OT 0$t @0 `  B(k-1)F + (n-k) 2 PT 08t@  ;(k-1) F 2X QT 0 @ WT 0t @  Y Reverso : 2 1F    XT ` P  YT 0@ t   xCaso de tres variables : 2$ N  @  ZT  @  [T 0pt 7F = 2`B \T 0D@@ @ ]T 0t  @ TR2 / 22 2  ^T 0tp  Z (1-R2) / n-32 2 X _T 00`  ^B `T 6D P ` ^B aT@ 6D@ H T 0޽h ? ̙33& |&t&>Pd$ %(   d 8 @ -d d 6 t RR2 =2 2 fB d 6Dp`` d 6$tp@` 7SCE 2 d 6d(tp`P 7SCT 2  d 6+t  9= 1 - 2fB  d 6D` ` `  d 6t/t @ ` 7SCR 2  d 62t `  7SCT 2  d 65t  9= 1 - 2@  @ 'd @fB d 6D `` d 69t @` XSu26 2 d 6(>t ` \Sy2: 2 d <lBt@@ C^&L ` d# 0 d 0Ft@` VR2 = 1 -2 2  d 6Kt@ 5_^B d 6DP`P d 6tOt0pP Vs26 2  d 6@Tt0@P _SY2@ 2   !d <Yt`  C^&n8  0 ;d 0m@   d 0fB d 6D`  d 6hDt  zSu2 / (n-k)H 2 d 6ct  Sy2 / (n-1)X 2@ ` d` d 6lit@` VR2 = 1 -2 2  d 6ot@ 5_ "d 6pqtPP C^&,8   =d N ` #d  p  $d 0ut@` VR2 = 1 -2 2  %d 6zt@ 5_ N  @ (d  `B )d 0D `` *d 0|t @` XSu26 2 +d 0,t ` \Sy2: 2 ,d 6Ȇt@@ C^&@   4d `B .d 0D p  /d 0t0  9(n-1) 2 0d 0t0  9(n-k) 2 7d 6xt  p  7n-1 2]8   Nd  T 0   9d#   `  2d 6t   wR2 = 1 - (1-R2)L 2    3d <d 6tP  R2 R2Z 2 ?d <ܨt @P  6 _ Fd 0Pt  7k : 2" Gd 0xtpl 6# de variables independientes mas el termino constante7 27P  Hd 0pt   7n : 2 Id 0Ըt@0  h# de observaciones 2  Jd 0t 07 V0 < R2 < 1, 2  Od s *Xt` R2 y el R2 ( ) ajustadoR     ` Pd c $A! 8?? x 8H d 0޽h ? ̙33+  &l@S(  l l s *Htp  zSignificatividad global:, 2   @ l 0Dt @ *H0 : b2 = b3 = b4 = 0 2  l 0@tp  !H1 : al menos un coef no es cero.2" 2  >, l 0t  <b2 0 , or b3 0 , or b4 0t 2   l 64tz*  RF* =2 2 8  P  #l P fB l 6D0 j 1  l 6lt^ 0  > 0.9710 / 3 2   l 64t0 P  B(1-0.9710) /16 28 0 P   lP   l 6t   5= 2fB  l 6D00 0   l 6 2   l 6w00 P P  g (1-R2) / n- k> 2 ~8  P  l p l 0w  Fc(0.05, 4-1, 20-4) = 3.242 2 fB l 6D ` fB l 6D ` fB l 6Dp p`  l 0 w0 P  Da& 2 l 0Hw00 P  7k-1 2 l 0w0 P  7n-k 2\ l 0 w 0  BComo F* > Fc ==>\ rechazar H0.j" 2   ,  l 68wP   Z F* = 179.13 2   | !l C TA <C:\COURSE\ECON2170\ex718a.bmp   X "l 00 X2 $l 0D  X %l 04,X &l 0 0P H l 0޽h ? ̙33    0  O ( < M  0L)wjJ ?3. Contrastar la adicion de una variable al modelo de regresion(@ 2?b     0/w@` zModelo viejo :$ 2$L  04w 2Y = b1 + b2 X2 + u1 2  6@>w  Calcula R2v o SCEv Z 2     j  0(Ew` <Ahora considerad una nueva variable X3, es relevante o no?>= 2$ l    0Lw00  nModelo nuevo : 2$  0Pw   ,6Y = b1 + b2 X2 + b3 X3 + u2 2N  6\w   Calcula R2n or SCRn y SCRnt 2     6<  0ew0 0 |H0 : b3 = 0, X3 no es relevante H1 : b3 0, X3 es relevante? 2  P  H  0޽h ? ̙33 @ (  S  0yw =Pasos del test para ver si X3 es relevante o no en el modelo @> 2 !R     0w@  1. Calcular el estadistico F 2,  8 V V   0wp ;(m = 1) 2@  V  0|w RF* =2 2 `B  0D```I  0dw` ;(SCEn - SCEv) / # de variables adicionales L< 2  06    0w SCRn / gl del modelo nuevo2 2 H   0w  9(n-3) 2D  0wx  <2. Comparad F* y Fc(a, 1, n-3)j 2       0|w y  3. Regla de decision: Si F* > Fc ==> rechazar H0 : b3 = 0 es decir, X3 es una variable relevante para aadir al modelo. 2   2   6w: 8F =  2^B  6Dp   6`w   (R2n - R2v) / glh 2      6w  (1 - R2n) / gl@ 2   dB @ <D dB  <Dj   6w  n-k (en el modelo nuevo) 2,   6$wpP D(2   6`w 0  ~# de nuevos regresores 2,   <w @p {(aadir o eliminar),X  6zp v @\H0 : aadir X4 no es conveniente b4 = 0/   >H  0޽h ? ̙33    p x+ ( :  x x 0dNz` !Aadir una variable irelevante X4," 2  $   x 0Sz@F l Modelo viejo 2 $ x C dALC:\COURSE\ECON2170\GUJARA\tab701b.bmp  x 0Wz ^^H0 : aadir X4 (TIME) no es conveniente, b4 = 00 2   >   x 6 no se rechaza H0.% 2  ,   | C dALC:\COURSE\ECON2170\GUJARA\tab701a.bmpPp@  | 0zG (Aadir una variable irelevante X4(TIME):,) 2  $   X2 | 0 .X | 0 6 H | 0޽h ? ̙33  !%p0 6(  pV p 0hzp  @Minimos Cuadrados restringidos: Rendimientos constantes a escala&A @l     p 0z  gProduccion Cobb-Douglas 2  p 0z  <Y = b1 X2b2 X3b 3 eu 2,  p 0z   b2 + b3 = 1V 2^B p 6D @ @^B p 6Dp@ @0  p 0z`@P b2 = 1 - b3V 2  p 0zpp b3 = 1 - b2V 2  p 0$~@ 2Zln Y = b1 + b2 ln X2 + b3 ln X3 + u. 2   p 0\ ~` wModelo sin restringir(2$ 8    p@`   p 0 ~@   .Vln Y = b1 + ( 1 - b3 ) ln X2 + b3 ln X3 + u, 2  p 0~ p  6=> 28 P p  p   p 6~@P p  .Vln Y = b1 + ln X2 + b3 ( ln X3 - lnX2 ) + u, 2 p 64&~P pp  6=> 28    p 0  p 6\)~@   .Z(ln Y - ln X2) = b1 + b3 ( ln X3 - lnX2 ) + u. 2   p 6 4~ p  6=> 2H p 67~ @  Fln( ) = b1 + b3 ln ( ) + u$ 2   p 6p?~ p 6=> 2XB p 0D 0   p 04C~` 0   5Y 2 p 0G~P p PX22 2 XB p 0D @  p 0K~` @  Px32 2   p 0P~P @ p PX22 2 X !p 0   "p 0U~ ` tModelo restringido(2$ , #p 0Z~0@  *Y* = b 1 + b 3 X* + u 2  $p C BC=DE(F V}]= @   S@' %p s *(d~& 34. Contrastes de hipotesis con varios coeficientes:&4 1P   H p 0޽h ? ̙33i (    C dALC:\COURSE\ECON2170\GUJARA\tab703a.bmp@@ _   C dALC:\COURSE\ECON2170\GUJARA\tab703b.bmp@@ @   0o~p `0   H0 : b2 + b3 = 1| 2   0@w~  bModelo sin restringir lnY = b1+ b2lnX2+b3lnX3 + u2 Z``8      6d~  ^Modelo restringido ln(Y/X2) = b 1+ b 3ln(X3/X2)0 Z`h`8      6h~ @ CF =$ 2^B  6D@@  6̔~ 0  z(SCRR - SCRNR) / mL 2    6~@8 ](SCRNR) / n - k2 2 dB  @ <D` @   6Р~p v  '# de restriciones en modelo restringido((2(, dB  <D 0 `  6~  ( # de variables en modelo sin restringir2)(2',   6D~   #SCRNR = 0.067158 SCRR = 0.091475J$     6 ~ @ = F = 4.345 2 X2  0~  X2  0    0~p "Fc(a, m, n-k) = ?F 2H  0޽h ? ̙331xUkAۏt`ZT"֘^TP[`jh+^]Y1|lKԃgţI/z& z&ldTyޛyۇOQ"k ՘bD7f nжoh{6ߣyryd 1Y ~K{h/5knšDjEAvWd& n `U5e ŗKה)Hl*S4_^򆺷 e-5-6@wJ͓i /l .e% R-QYH"C7N%Ϝ8[23i+VqvٜM/3ŊAJ)rY:iKv$qe<*@1#!udIwVFp~KNh؂I1dG@;cH[}h\e_L i՚!JZ^vV@_z÷uGآUsYyz|Cuj8&sQzzXٙ0-Q[?<~j{EyVj|f(o Rc\5 MX=;Gթ~ZOcQM}.*KK^%F~Q6FׅЅ!*'obXh z+{X:wOr%mY/Bbhjz  =B;N2<$^[ޤs@/_(mvi"ψ#Y֦](/Y̲Xo3V'NuA"W_cA0Isjv3ԫlbz·an4mg-K7xVKkQ>5IhZ2Tp# ])*R54Nm:yL X\\-KۥK`W>sgHTz;9F߻GφS%=`(J{h>)z0o,x<2i1 ~KckBtm}}.4細\%%?T@tm=i?/ OCbSF: mXSϤ.i watc1Q{H¿3,U [u*%ʡR-! kGv%.xdܒM;[l(űl"cN+*#5N['q,9e49/3׮0&4 i`*YB!J!磘9!aBq)1NPkXg/ KrV15qŹZnUEUַX?V䜓Us[n#XLG1BV[nܮ=O;vu++$3Da3g<oWj[!T݇:k!pV!ߜ\!\SnU%Y?,x{{kBh^i\=bnZ[a>J,s3hf=W8GL,$n4;&]1feetWh)OS44]+y 2^ э[KbLu[لh?iaVHӴjq-6xVkA344-VJQY*xCV)m+M7[cnU'= x=z(XBTѤ1@w9yWd^W-ε1h&jX :3 P:Q߻w(6kHYKl-[Z]SjXJ+Xj54V495Nr&2iz=ペ;w! r[6}$w) d g2FOVIːHUL( aF s#qU׈@rH"4 #51>c8?!2gv {FJ*č7oGC6MNC'Q\/}?e+fπo>o.b"UCyOX6ML\9ߋ#\6R5jKˆ%'Hǀši؉'5ʎCѯXf Q.Niz$phtw j(4EєR55gl1TM&@+F5.ʀ_qF-qSIV ?x˕@ n7?V;5H,8-{Ա7zvֹxҔ"|sܙ^;ݙ3Iקgyz8{Vq07N`ʹq>uG=PEtO t^8xyng'G.ف{뒟5vNG_KƬoh}947X3ioYӣE~{]Ǧ?z [kI)Ml_84CMn=ꚽ-EްKnٔ][_}53~Y;| U? 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