ࡱ> `! #ns=@@{/vlxHFxu enp]Z(Y[e]ZKTuiBR"efIdrJ) (S22hsfs}s%Qbď[U*h>8;(#%Q|=^YWįθ|qDu: ^|Q<7}R&ūoN7 E Mp4<U Ds^ Y磁 C?W_TcH*~~eď'F)&q y0}a:Oc%~F?(RG1|6eɐ ٦/Ak,A+ 9-~ښ~9ڠkWנ=Vr-g_W7JAh~~b#rկ-n .#m*~zc+z7{1{GNTM_!O\U1(0bh8 2ѦO`0RI, p?wb<`ӜL]F y1W0_Ãw.#OrL5=#pc]Z|4M3}4Mˈ/CRch&ܘ*4C},2}=g_晾=4W}5>>-B=ޥŧЋOp!-CZ>oD%OpɁoBLߔ֠1O3uӧZ&JKMIoSAu7x?h{]}1_og-[#|vwϤ3iO9SߘGDΈO窾i=_$:z;$ΈOdKSGO'ջCttaΈwUߏwJmt5"sF|&>>[R>#3BYEWQWׂ/#͞_O#c{3O?6K GhcGGQWw>Gyu;*GbCD2o}ΈB!#~/ne<&ogi2>~G/O2M_Z!-MKq7phklKo~%Z~5N5Mho_5AG'ZrZ\~:iN,?x |i_#=L_{h7&~2z~ xRhP4O`qԏb0pF` R/|ci2uw W߅;9I8#k[a1Z`\U|39R&UPm&Yfs?\S1|nO_(%`!)1aҼ!tYTxcdd``bd``baV d,FYzP1C&,7\A! KA?HZ NĒʂT `Wfrv  ,L ! ~ Ay ߖ_H_4AHq`ZQ bR;|36dfʿp#[ #d + +ss] ߽g 7Ȉ$=R 6Uķ 8ޅƉ;׌LLJ% 93t1al0`'n`!WqiㆍT@Hxcdd``^ @c112BYL%bL0YnB@?6 u jjbnĒʂT+~35;a&br___PPT9 / 0z? %O =K   1. Estimar una relacion entre variables economicas, tal como y = f(x). Explicacion cuantitativa de dicha relacion. 2. Predecir el valor de una variable y, basado en el valor de otra variable x.r@   S  (                                      Gastos Alimenticios Semanales(N         y = Euros gastados en alimentos por semana. x = renta familiar del consumidor por semana. Supongamos que la relacion entre x y el valor esperado de y sea lineal: E(y|x) = f(x) = b1 + b2 x Cada media condicionada E(y|xi) es una funcion de xi. Esta ecuacion se conoce como la recta de regresion poblacional (RRP).@}0P-  1  Q         C    C         H                                    3                             $   & .('!Razones para la existencia de u* N       RImprecision de la teoria economica Datos no disponibles Efecto directo vs efecto indirecto Comportamiento humano es intrinsecamente aleatorio Deficientes variables  proxy Principio de la Parsimonia ( La navaja de Occam ) Omision de variables relevantes Mala especificacion de la forma funcionalh*P#&32I.                                                     )#+%      /+e90,3-4.5/607182,&95:6;7     !  "  /#?@ABCDEG I J K LMNOPQRSTUVWX\_!`"a#b$c%d&f'P  ` ̙33` ` ff3333f` 333MMM` f` f` 3>?" dZ@$z?" dZ@  " @ ` n?" dZ(@   @@``PH    @ ` ` pH @`  H @`   B:(  d  <8c?p4  # B  s *޽h ? a( transparent0 *(  p  01 ?   K-  ZKxaxa1 ? ? K ;Body Text Second Level Third Level Fourth Level Fifth Level    < E  TKxaxa1?7 i Slides created by Lawrence.C.Marsh.1@nd.edu Copyright 1997 John Wiley & Sons, Inc. jj c B  s *h[ ? a(  0(   B  s *h[ ? a(1  0Y(  !  T>xaxa>??N  )Analisis de Regresion Para dos Variables**CB4      H  0޽h ? a(  C(  X$   Z(Kxaxa1 ?@P0<$0 K )  N\Kxaxa? #Objetivo del Analisis de Regresion "$"(,B      H  0޽h ? a(  80( 2 7   Z c>xaxa1 ?`  <$0 K   Z >xaxa ?p   > H  0޽h ? a(%  .    M (   8     `B   0o?  `B   0o?      BCVDEFo?==U?U~FF6<{9i9E\B@d?w~;;zO8w5Ot;1 q  w . dm  @ ; z \  7 9v i  4 s   61FpFUU{|@i ^B  6o?   Zixaxa1?w5 M f(y|x=80)     Zixaxa1?jG  M f(y|x=80)     Zixaxa1?   Ey  XB  01?@ @ |   T4 ixaxa1? w  /  nmy|x=80$C     Zixaxa8c?  T `Probabilidad Condicional f(y|x=80) del Gasto Alimenticio dada una renta de x= 80.TT               H  0޽h ? a(  TL$(  $8   $ @ (  $( `B $ 0O8c?( ( `B $ 0O8c?( (  $ c zBCVDEFO8c?==U?U~FF6<{9i9E\B@d?w~;;zO8w5Ot;1 q  w . dm  @ ; z \  7 9v i  4 s   61FpFUU{|@  $  BCVDEFo?==U?U~FF6<{9i9E\B@d?w~;;zO8w5Ot;1 q  w . dm  @ ; z \  7 9v i  4 s   61FpFUU{|@ fB $ 6o?( ( `B  $ 01?@@H fB  $ 6o?l$   $ Zlxaxa1?G Jf(y|x)   $ ZL`ixaxa1?  e M f(y|x=80)    $ Zbixaxa1?g I  Ey   $ Teixaxa1?P  nmy|x=80$C    @   $- fB $ 6o?( ( `B  $ 01?@@H  $ Zjixaxa1? ? N f(y|x=100)    $ TLoixaxa1?   pmy|x=100$ C    $ Zqixaxa8c? Zl <Probabilidad Condicional f(y|x=80) del Gasto Alimenticio dada una renta de x= 80 y x= 100.\\N       H $ 0޽h ? a( f^@( O8 jB  BDo?@``@jB  BDo?@`@  B@8c?P GY    B @8c?  GX  dB  <DԔ?`   B@8c? NE(y|x)  pB  HD8c?p00@pB  HD8c?Ppp@pB  HD8c? @dB  @ <D8c? `0   c BCDEF8c?@ kp dB  @ <D8c? `   B @8c?  Z80 100 120    BCDE4FOԔ? ",B`~NluK* i @     U"    BCDE4FOԔ? #7EjK PjA= @     $   BC DE@F OԔ?+EjGo `lA**q   @     K U jB  BD)? F jB  BD)? p jB  BD)?    BX&@8c?`  P 149 101 65      B\*@8c? P  Distribucion de Y dado X=120*   jB  BD8c? p    <T0@8c?@@ xMedia Condicional (   vB @ ND8c?   B5@8c? P  r Recta de Regresion Poblacional !!   vB @ ND8c?` vB @ ND8c?`0 H  0޽h ? a( . =5@((  ($L 0  (# 0 `B ( 01?4  `B ( 01?00 ^B ( 6>?<4oa  ( Zsixaxa1?7R  G{,  ( ZHixaxa1?: w  vb1@GOOO   ( xBCDEF1?@| XB  ( 01?   ( Zlixaxa1? /  bDx"C    ( Zixaxa1?w  lDE(y|x)"C    ( Zlixaxa1?:u JE(y|x)   ( Z(ixaxa1? {Consumo Medio Y4     ( Z0ixaxa1? W_  g X (renta) &   0 ( Zixaxa1?w  E(y|x)=b1+b2x\C C    ( Zixaxa1?:' rb2=:GOO  ( xBCDEF1?@t ( Z8ixaxa1?. lDE(y|x)"C   ( Zixaxa1?'? bDx"C   ( ZHixaxa8c? l FModelo Econometrico: una relacion lineal entre consumo medio y renta.GG              ( Bi8c? 0  k constante$      ( Bhi8c?p@`  a pendiente     H ( 0޽h ? a(    P` ( hW   <\=@8c? $Especificacion Estocastica de la RRP&%$ 4      A  BI@8c?00 Dado un nivel de renta Xi, el consumo familiar se concentra alrededor del consumo medio de todas las familias con nivel de renta Xi .. Es decir alrededor de su media condicional, E(Y|Xi).@ l 8                                      BT@8c? PP bV La desviacion de un individuo Yi es: ui = Yi - E(Y|Xi) o Yi = E(Y|Xi) + ui o Yi = b1 + b2 X + ui   O O                   O            O                    %     BTo@8c? `0 {Error estocastico &    H  0޽h ? a(    `C ( py00   Zw@xaxa ?! qEl Termino de Error,&    H   ZX{@xaxa1 ?`@,$0 0 y es una variable aleatoria compuesta de dos partes (suponga x fija): I. Componente sistematico: E(y|x) = b1 + b2x Esta es la media de y. II. Componente aleatorio: u = y - E(y|x) = y - b1 - b2x Denominado error. Uniendo E(y|x) y u obtenemos el modelo: y = b1 + b2x + u|(  ! GG  SGG  $G$$$G$$$$$                               i            4 XB  0p?XXB  0p? h H  0޽h ? a(   Z( Ah     68i8c ?`  i   <i8c ? i H  0޽h ? a(S'  '&p88&( C8 0    6@8c ?@ !Recta de Regresion Muestral (RRM) "!@          BC DEFo? @+,^B  6o?&7^B  68c? .C ^B  68c? KKL^B  68c?Q D^B @ 68c?S S ^B  68c?  ;^B  @ 68c?& ^B  68c?||D  # B C(DEFԔ?( @@ ^B  @ 68c?  Z@xaxa8c?q I.(   Z@xaxa8c? x G  I.(   Z @xaxa8c?3N  I.(   Z@xaxa8c?   I.( XB  08c?KXB  08c?KXB  08c? K XB  08c?P KP   Z@xaxa8c?E Ry4      Zl@xaxa8c?W   Ry1      Z$@xaxa8c?  Ry2      Z@xaxa8c?  Ry3      Z@xaxa8c?0u Rx1      T|@xaxa8c?/K t Rx2      T,@xaxa8c?;  Rx3      T@xaxa8c?6{ Rx4      Z@xaxa8c?  +}   Z@xaxa8c? =  G}$   Z@xaxa8c? G  G{  ! Z@xaxa8c?0 G{$  " ZT@xaxa8c?   Vu1$O    # Z@xaxa8c?GA  Vu2$O    $ Z@xaxa8c?Z  Vu3$O    % ZAxaxa8c? + Vu4$O   XB & 08c?CCjXB ' 08c?uueXB ( 08c?  eXB ) 08c?eX * ZAxaxa8c?0E  y = b1 + b2xx0ZC C     + T8Axaxa8c?y #Q Ex   , ZAxaxa8c?mO Ey  jB - BDԔ? @@ jB . BDԔ?0 ` jB / BDԔ?! Q  jB 0 BDԔ? 1 BpA8c?  I^  2 B A8c?l I^  3 BA8c?  I^  4 s BC DEF8c? @ @55D 5 T4"Axaxa8c?* "E(y|x) = b1 + b2xn0ZC C    6 B$*A8c?0  O(RRM)    7 BD.A8c?P0 [(RRP)    8 B1A8c?c0 )Diferentes muestras tienen diferentes RRM.* )O&  H  0޽h ? a(  W( XZhZ   B8?AԔ? P r RRM: Yi = b1 + b2 Xi o Yi = b1 + b2 Xi + ui o Yi = b1 + b2 Xi + ei RRP: Yi = b1 + b2 Xi + ui Yi = estimador de Yi (E(y|xi) bi o bi = estimador de bi    O  O O                                    ZvAxaxa8c?:  U 0Z 2  <zA8c?0P  _Residuo     <8~A8c?   NError    jB  BDo? @ jB  BDo?p @  BA8c?0 K P I^   BA8c?@Q` I^   B܉A8c?@<` I^   BĈA8c?5;U I^   BA8c?7. I^   BA8c?G. I^   BTA8c?   @ I^   BA8c? {  I^   BA8c?p I^ H  0޽h ? a((  . X(P(8B@'( < @w @ Nixaxa8c?. 7Relacion entre y, u y la recta de regresion verdadera.68%h           @ xBCE DEF8c?E @S"& XB @ 08c?s &7s ^B @ 68c? .C ^B @ 68c? KK  @  B CDE F8c?  @w  @  ~BC DEF8c? @  @  ~BC{DEF8c?{@& ' ^B  @@ 68c?   @  ~BC DEF8c? @qy}w XB  @ 0o?E.~ ^B  @@ 68c? @ ZP"ixaxa8c?\ I.(  @ Z#ixaxa8c? F  I.( XB @ 08c?KXB @ 08c? K  @ Z"ixaxa8c?- Ry4     @ Z ixaxa8c? ?  Ry1     @ Zixaxa8c?\ Ry2     @ Z\:>xaxa8c?G Ry3     @ ZP>>xaxa8c?_  Rx1     @ T;>xaxa8c?^ K  jx28    @ Txaxa8c?j  Rx3     @ TLxaxa8c?e  Rx4     @ Zxaxa8c?  +}  @ Z|xaxa8c?N   G}$  !@ Z(xaxa8c?A G{$  "@ Z!xaxa8c?j   Vu1$O    #@ Z%xaxa8c?_ Vu2$O    &@ 3 BCDEF8c?@ DIa XB '@ 08c? uu XB (@ 08c?    XB )@ 08c?  . *@ T\*xaxa8c?0 E(y|x)=b1+b2x`0ZC C    +@ T(1xaxa8c?: ;  Ex   ,@ Z(5xaxa8c?X~ Ey  jB -@ BDo?  dB .@ <Do?T4 X /@ Z|9xaxa8c? J  y = b1 + b2xx0ZC C     0@ BB8c? I^  1@ BF8c?| I^  2@ BJ8c? I^  3@ BM8c? ` k(RRM)6 33   4@ B?jB 9@ BD>?pr :@ HԔ? ;@ B`W8c?  E(y|x2)@   jB <@ BDԔ?r =@ B^jJ?| F    >@ Z`xaxa8c? Vu2$O    ?@ Bd8c?<c ]^*  O  @@ Zxixaxa8c?z hy260Z   A@ BDn8c?}  Y^& pB B@ HD8c?pH @ 0޽h ? a(%, . ++<DDM+(  D D Z0txaxa8c?N   G}$  D ZPxxaxa8c? F  I.(  D Z|xaxa8c?g  +}  D Z@~xaxa8c?2 x  I.( XB D 08c?S&& XB D 08c? &7 XB D 08c? KK XB  D 08c?XB  D 08c? ]XB  D 08c?q||XB  D 0o?E.~   D Zxaxa8c?\ I.(  D Zxaxa8c?N  I.(  D Z xaxa8c?Pt Ry4     D ZԎxaxa8c?U T  Ry1     D Zxaxa8c?  Ry2     D Zxaxa8c?$P ~ i Ry3     D Zxaxa8c?   Rx1     D T\xaxa8c? K   Rx2     D Tܜxaxa8c? #  Rx3     D Txaxa8c?   Rx4     D ZЫxaxa8c? G{  D Zxaxa8c?A G{$  D Ttxaxa8c?`    Vu1$O    D Zxaxa8c?G Vu2$O    D TPxaxa8c??Z  Vu3$O    D ZPxaxa8c?\ + Vu4$O   XB D 08c? CC XB D 08c? uu XB D 08c?    XB  D 08c?   !D Txaxa8c? N  Ex   "D Zxaxa8c?X~ Ey  8 < (D< &D Zxaxa8c? B0Z  'D Zxaxa8c?< <   )D Zxaxa8c?V I.(  *D Zxaxa8c? l ; I.(  +D ZDxaxa8c?T J I.(  ,D ZLxaxa8c?e  I.(  -D Txaxa8c?5c  Vy1$    .D ZHxaxa8c?  Vy2$    /D Zxaxa8c?Wn  Vy3$    0D Zxaxa8c?t Vy4$    1D Zxaxa8c?93~  G^  2D ZTxaxa8c? {E  G^  3D Zxaxa8c?;  G^  4D ZTxaxa8c?a G^  5D Zxaxa8c? p K e G^  6D ZCxaxa8c?U 0 G G^  7D ZCxaxa8c?`oJ G^  8D Z` Cxaxa8c? ] G^ 8 0P ?D 0 =D <,C8c?0P 9La relacion entre y, y la recta de regresion ajustada.2:%            >D BC8c? 00  B   ` AD c $A ?? ` BD c $A ??8 h  ` CD c $A  ??8 h   ` DD c $A  ??` 9 @  H D 0޽h ? a(   y  ( | 8 8    ZAxaxap?-   G^    ZܧAxaxap?7 !  G^    BA8c?8 Valores de la regresion muestral: yi = b1 + b2xi + u i Recta de regresion muestral: yi = b1 + b2xi * ( (( (( (( (( ((  (( (( (( (( (( (                 XB  0p?@X@T  ZTAxaxap ?P,$0 Valores de la regresion poblacional: yi = b1 + b2x i + ui Recta de regresion poblacional: E(y i|xi) = b1 + b2x i JF0d( (( ((C( ((C( ((  (( ((( ( ((  (( ((C( ((C( ((  (                 H  0޽h ? a($' D$<$,/#(    <A8c?!7  YMCO&   dB  <D8c?   c BCDEF8c?@ u   BxA8c? P  GX    BA8c?0 GY  dB  <DԔ?`dB  <DԔ?` g   BpA8c?iF RRM1:Y1= b1+b2X  g   BdA8c?%e RRM2:Y2= a1+a2X     c B%C%DEFfԔ?%%@z0U   s B%C%DEFfԔ?%%@  s B%C%DEFfԔ?%%@ $  s B%C%DEFfԔ?%%@c  s BCDEFfԔ?@{   c BCDEF8c?@05`pB  HD8c?pB  HD8c?0pB  HD8c?ppB  HD8c? p p d  <8c?``@d  <8c?` d  <8c?P0 ` @d  <8c?    BxB8c? U2"   B B8c?*J ( X-1/2"   BB8c?N > @ U1"   BB8c?  p V-2" r  BlB8c?`P B     B`B8c?;[ Y1&  !  BCDE(F8c?  `0x @   P8  " BB8c?  Z-1&  # B#B8c?Xx U0"  $  BCDE F8c?+@ % B%B8c?p  Z-1& r & Bh,B8c?@pp B   ' B4/B8c?jWw Y1&  (  BCDE F8c?@P @  ) B2B8c?`   -21/2Z  * B9B8c? S^   + BtA8c? S^   , <@B8c?   l^RRM1: Su2 =12 + 12 + 12 + 12 + 22 = 80 0  . <QB8c?   lRRM2: Su2 =22 + 02 + (-1/2)2 + 12+ (-21/2 )2 = 11.5|7  7  / BbB8c?   emenor"  H  0޽h ? a(  ( 2x    T kBxaxap?Pa 0.yi = b1 + b2xi + ui( (($C$ $ ((C$ $( (( (4    `  Z4xBxaxap?  (Minimiza la suma de errores al cuadrado:")$!$h        XB  0p?@X@F + ip  + ip   `DBxaxap? i  RZ ui2 = S(y i - b1 - b2x i )2 = f(b1,b2) .((O((  KB (  ((G$$ ((G$$(  (   (  ( G$$$G$$ ,    `ܘBxaxap?0+ p Ii=1     `Bxaxap?+ p  Gn    ZdBxaxap?@8X J< u i = y i - b1 - b2x i (  ((  (($C$ $ ((C$ $(  ((  XB  0p?X>F J   Pp2    E$GH I`TQp? `T:T`T:T`T`T`T:T`T`T2   # BTENG2 I9Qp? ٫TT`T٫TT`T9`TT`T9`T4J  <@BԔ? `p "Minimos Cuadrados Ordinarios (MCO)# #$4       ZBxaxap? pe Ii=1    ZBxaxap?1 v  Gn  H  0޽h ?/    a(9 y(    Z\Bxaxap?tg , 4Minimiza c.r.a. b1 y b2:$O $(G$$ ($G$$(   eF W  WN  Z(Bxaxap?W Jf(b1,b2) = S(y i - b1 - b2x i )2B&( C$ $$C$ $ KB (  ((C$ $ ((C$ $(  (   (   &   Z\Bxaxap?wM Hi =1    ZBxaxap? En  XB  0p?`h`  ZBxaxap?)  vD = - 2 S (y i - b1 - b2x i ) # C  KB (  ((C$ $ ((C$ $(  (     (  TDxaxap? Fn  @= - 2 S xi (yi - b1 - b2xi ) !  C  KB( (  ( ((C$ $ ((C$ $( ( B        Z,Dxaxap?_  f(.)HC ( C(     Z Dxaxap?L  \b1&$ $ XB  0p?((  ZDxaxap?    f(.)HC ( C(     Z Dxaxap?o .  \b2&$ $ XB  0p? v XB  0p? h l  TDxaxap? * Iguale estas derivadas a cero y resuelva dos ecuaciones con dos incognitas: b1 b2|W1G$$G$$ $                  H  0޽h ? a(H( ''11'( ` ` p  HA)?Z!XB  0p?gXB  0p?  Z`-D 8c 8cp??& jf(.)6        Z2D 8c 8cp?[6 jf(.)6        Z08D 8c 8cp?-". lbi,G    ZdG  ZHD 8c 8c8c?+ {  O.UC0 XB  08c?]  p  ZKDxaxap?tgb , 6Minimizar c.r.a. b1 y b2:r$(C$ $ ((C$ $(   %F 3TV  T3V  ZVDxaxap?3 |Df(.) = S (y i - b1 - b2x i )2#( C$ KB (  ((C$ $ ((C$ $(  (   (  #   Z\fDxaxap?9 V Ht =1    Z,iDxaxap?T  En    ZxlDxaxap? NR  K    ZoDxaxap? ~R  P G   Z`kDxaxap? R  PG   ZqDxaxap? uF  jf(.)6    ZzDxaxap? lO  PG   Z~Dxaxap? O  lbi,G  RB  s *8c?2 W2   ZpDxaxap?|   K<   ZDDxaxap?   K0   ZDxaxap?  K    ZDxaxap? #  P G   ZDxaxap? -  K   ! ZLDxaxap? ]  P G  " Z4Dxaxap?   PG  # ZDxaxap? T  jf(.)6   $ Z̟Dxaxap? jK  PG  % ZDxaxap?   lbi,G  RB & s *8c? 6  ' ZDxaxap?$ i  K>  ( ZԫDxaxap?0 fu  K0  ) Z0Dxaxap?Z   K   * ZDxaxap?Z F  P G  + ZDxaxap?Z i  PG  , Z0Dxaxap?N e  jf(.)6   - ZDxaxap?W  PG  . ZDxaxap?W \ s  lbi,G  RB / s *8c? ]   0 ZDxaxap? s c   K=  1 Z|Dxaxap?    K0 H  0޽h ? a(/  o( AK,nLn _  ZD 8c 8c) ? 5G  CPO para minimizar f(.) HUG G(G N         ZxDxaxap? `L = - 2 S (y i - b1 - b2xi ) = 0 ' C  KB (  (($ $ (($ $( (     <  ZDxaxap?X>6  T= - 2 S xi (yi - b1 - b2xi ) = 0 +  C  KB( ( ( (($ $ (($ $( (   B        ZDDxaxap?o  f(.)FC   C$     ZGxaxap?T jb12C C$ $ XB  0p?  ZGxaxap?v  f(.)FC   C$     ZGxaxap?7  lb24C $ $ XB  0p?ee  T&GxaxaOp? " Cuando estos dos terminos se igualan a cero, b1 y b2 se convierten en b1 y b2 porque ya no representan cualquier valor de b1 y b2 sino los valores especificos que corresponden al minimo de f(.) ..C$ $C$ $$ $$ $.C$ $C$ $>( C$ R                                H  0޽h ? a(  ( HY @  ZX3Gxaxap?  T - 2 S xi (y i - b1 - b2xi ) = 0 + C  KB( (  (  (($ $ (($ $( (    (    XB  0p?h  Z$FGxaxap?ee  $XS xi yi - b1 S x i - b2 S xi = 0 L-KB( ( ( ((C ($ $KBK((  (C ($ $KB $( (   \       6  Z[Gxaxap?hF B S yi - nb1 - b2 S xi = 0 " KB( ((C ($ $ (C ($ $KB $( (   B     XB  0p?` h`   ZjGxaxap?  I2   ZoGxaxap?   H nb1 + b2 S xi = S y i %($ $ (C ($ $KB $( (  KB(  ((2        Z|Gxaxap?   Nb1 S xi + b2 S xi = S xi yi ($ $KBK(( (C ($ $KB $( (  KB( ( ( ((h           Z0Gxaxap? >  I2 >F      2    E$GH I`TQOp? `TDT`TDT`T`T`TDT`T`T 2  # B|TENG IQOp? |T|T`T|T|T`T`T|T`T`T.  DF  q   q2   E$GMH I`TQp? >TBT>TBT`T`T>TBT`T`T q> 2  3 B~TENG HgIQp? ~T~T>T~T~T>T`T~T>T`TN a>F ER*  ER*2   B|TENG IQOp? |T|T`T|T|T`T`T|T`T`TEA2  # E$GH I`TQOp? `TDT`TDT`T`T`TDT`T`TVR*DF %j   R 2   BTENG HgI$Qp? īTTATīTTAT$`TTAT$`T%2  # E$GHD I`TQp? @T.T@T.T`T`T@T.T`T`Tj   ZXGxaxap?] P - 2 S (y i - b1 - b2xi ) = 0 ) C  KB (  (($ $ (($ $( (       H  0޽h ?  a(, #l(    Z$Gxaxap?v8 Q ( XB  0p?g!gXB  0p?g<g  c BC DEF)? @ox zF `^M   ^ 6  ZGxaxap?^M<  8n S xi yi - S xi S yi  6 KBK ( ( ( (C KB $( (C KB( ((N         ZGxaxap?7   .&n S x i - (S xi ) 6 KBK (  ((C KB $( (C &     ZGxaxap?P q  i  I2   ZGxaxap?x p  I2 `B  0p?U U   ZGxaxap?`~   cb2 =,$ $   ?  Z@Gxaxap?{  3 b1 = y - b2 xr$ $ ( C $ $(  XB  0p?  n XB  0p? j   <GԔ?  Despejando las dos incognitas N          B G8c?0 *= S yi = S xi yi  KB( ( KB( ( ( (B    d  <8c?3pR.  BH8c?   8n S xi S xi S xi2    ( $C (KB $( (KBK(( (KB $( ( (( (L       d ! <8c?  " BH8c? 6 a b1 b2X   $$  $  d # <8c?  H  0޽h ? a(T@ @?WX?(     ZHxaxa8c?. K{CH   ZHxaxa8c?  K{C   ZPGxaxa8c?x0 I.(   Zl!Hxaxa8c? F  I.(   ZHxaxa8c?2 x  I.( XB  08c?S&& XB  08c? &7 XB  08c?KK XB  08c?XB  08c? ]XB  08c?q||XB  0o?E.~   Z4)Hxaxa8c?\ I.(   Z-Hxaxa8c?pH ( I.(   Z0Hxaxa8c?Pt Ry4      Z4Hxaxa8c?U T  Ry1      Zx8Hxaxa8c? 3 Ry2      Z@ G*  4 ZpHxaxa8c? G*  5 ZHxaxa8c?X G* ZF ~  6 ~ #N ~  7 ~  8 ZDHxaxa8c?  Re1     9 ZHxaxa8c?~ G^  : Z ZHxaxa8c? G^  ? ZHxaxa8c?U G* ZF  n @  n#N  n A  n B ZHxaxa8c?) n Ry2     C ZHxaxa8c?l $ G^  D ZHxaxa8c?   G* ZF r  ; E r ;#N r  ; F r  ; G Z Hxaxa8c?w  ; Re3     H ZHxaxa8c?r M  G^  I ZDHxaxa8c?   G*  J ZpHxaxa8c?6  G* TF  8 K  8N  8 L  8 M Z Hxaxa8c? 8 Re4     N TPHxaxa8c? q G^  O ZHxaxa8c?  G*  P ZHxaxa8c? G*  Q ZHxaxa8c?, )  M{C,  R ZlHxaxa8c?D" M{C,  S s BC DEF>? @ (  T s BC DEF>? @ U s BC DEF>? @`  m V s BC DEF>? @p`}} W ZHxaxa8c?@ G^ H  0޽h ? a(    # ( O8 #  ZHxaxaO> ?0 /Supuestos del Modelo de Regresion Lineal SimpleN          Z$Hxaxa1 ?0P,$0 1. Modelo de regresion lineal: (Lineal en los parametros) y = b1 + b2x + u 2. Muestreo aleatorio: {(yi, xi); i=1, & , n} muestra aleatoria del modelo poblacional 3. Media condicional de u es cero, E(ui| xj) = 0 4. Variacion muestral en la variable independiente 5. Homocedasticidad o igual varianza de ui, var(ui|xj) = s2L OGOOOGOOOO6$$O $$$ $$[O$$$$$O $$ $$$G$$J                                                   H  0޽h ? a( 0_(   ^B  6o?   ^B  6o?   XB  01? l XB  01? D , XB  0p?   Z"Ixaxa1?AV I.6   Z$&Ixaxa1?  m H  I.6   Tp)Ixaxa1? E  `xi      Z -Ixaxa1?:   _x1=80*     Zd1Ixaxa1?:   `x2=100*   ^B  6o?\    `<6Ixaxa1?y  `yi      Z:Ixaxa1?/ yf(yi)* &     N?Ixaxa8c?  wULas varianzas de yi en dos niveles distintos de renta familiar, x i , son identicas.VV /                     `;KIxaxa1?ip  ;gasto    NNIxaxa>?N vCaso Homocedastico&    F s   s `B  01?  `B  01?   s BC,DEF1?==+/<FS\f|i]r:vyvrpeU\)PF5*g6  hI / 2 Ji2%^3BR`mEvv'B`z{qe^Q>3 {|@s F j  j`B  01?j`B  01?w  s BC,DEF1?==+/<FS\f|i]r:vyvrpeU\)PF5*g6  hI / 2 Ji2%^3BR`mEvv'B`z{qe^Q>3 {|@l}  Z`VIxaxa8c?` ~  ;renta  H  0޽h ? a( E=@!!(  ^B  6o?   ^B  6o?   XB  01?  XB  01? D , XB  0p?      Z^Ixaxa1?h35 S  I.H   Z bIxaxa1? ?]G  ]x t*      ZdfIxaxa1? gO  Rx1      ZjIxaxa1? 5 O  Rx2    ^B  6o?    `:oIxaxa1?|  `yi      ZsIxaxa1?/ yf(yi)* &     NTyIxaxao?  ,:La varianza de yi aumenta con la renta de la familia xi.@; &                `<āIxaxa1?qx  ;gasto    NIxaxa))?N xCaso Heterocedastico& O   XB  01?    ZIxaxa1?  O  Rx3      ZĎIxaxa1?E G  I.H   ZIxaxa1?EG I.H F       `B  01? `B  01?    c zBC,DEF1?==+/<FS\f|i]r:vyvrpeU\)PF5*g6  hI / 2 Ji2%^3BR`mEvv'B`z{qe^Q>3 {|@  FF T   T `B  01?y! `  s 0BCEDELFT1?D:)tU0nFo&:'(@5z    s BwC7DEF1?++C3|@yZ:!rM- ,Kq";[{A{2Dmj 9d !,04(2@6R2](m%vWX@T 9FF  <_    @@ `B  01? ;_ `  s 0BfCDELFT1?:Y~s[7sJ*6BPVd_3e'(@ w[U     s BCDEF1?++L{$J.;L`wsU61a:qQzhWG!6N#w :^wZ;WX@Z<x ! ZIxaxa8c?' iZE  ;renta  H  0޽h ? a( rjP( hW   ZIxaxaO> ?` TSupuestos del MRLSF (continua)     BXI8c? r6. No autocorrelacion entre los errores. cov(ui,uj|xi ,xj) = 0, para todo ij R$$$O$$$O$$$$$$$$ *                        H  0޽h ? a(] ! ##t(  t t TCxaxa)?esG a elasticidades B  @ 8 4 t$ t ZHCxaxa8c?  Cambio porcentual en y4      t Z"Cxaxa8c?  hCambio porcentual en x  `B t 08c?%  t Z&Cxaxa8c?E lh =2CC  t Z+Cxaxa8c? W  T=    t Z/Cxaxa8c? 4 |Dx/x8C    t Z$5Cxaxa8c?   |Dy/y8C  `B  t 08c?   t Z;Cxaxa8c?W T=     t Z?Cxaxa8c?  Dy x8C    t ZECxaxa8c?   Dx y8C  `B t 08c?(G`B t 08c?dXB t 0p?h~ t ZLCxaxa8c? Bd  :Usando calculo, podemos obtener la elasticidad en un punto;;            t ZSCxaxa8c? k,   h = lim< CC   t ZYCxaxa8c? b %  T=     t Z<^Cxaxa8c?O    Dy x8C    t ZcCxaxa8c?` ~    Dx y8C  XB t 08c?~ ~ XB t 08c? B   t ZjCxaxa8c?[    y x8C    t ZpCxaxa8c?l ~    x y8C  XB t 08c?  XB t 08c? 0 XB t 08c? O O XB t 08c?x S x XB t 08c? S XB  t 08c?z   XB !t 08c?u u XB "t 08c?  " #t Z$yCxaxa8c? 0  Dx 0NCC   H t 0޽h ? a(". YQ`x(  db xn x ZĿIxaxap?g ,E(y) = b1 + b2 x0Z ,G,, ,,G,, ,,  68 9E  xE9 @ 9E)  x9E)  x ZIxaxap?pE  l E(y)&0ZC,,   x ZIxaxap?0  fx&0ZC,,  `B x 0p?9 )  0 x ZIxaxap? 5  = b2^0Z,C,,G,, ,   x NIxaxa)?w  aplicando elasticidades B4     2 8  B x B  x ZLIxaxap?1  l E(y)&0ZC,,    x ZIxaxap?  fx&0ZC,,  `B  x 0p? V .  x ZIxaxap?   = b2^0Z,C,,G,, ,  x ZIxaxap?  b h = &0ZC,, `B x 0p?   x ZIxaxa8c?_ B HE(y),   x ZpIxaxa8c?K m  Ex,  `B x 0p?% i  x ZIxaxa8c?  HE(y),   x Z`Ixaxa8c?  Ex,  XB x 0p?hXB x 0p?` h` H x 0޽h ? a(#. PHp+,|(  | | T@Kxaxa)?k5  estimando elasticidades <@       8 A |A | ZIxaxap?[X+  fy&0ZC,,   | ZHKxaxap?(( fx&0ZC,,  `B | 0p?!!6 | ZKxaxap? Gp  = b2Z0Z,C,,,, ,   | ZKxaxap?S b h = &0ZC,, `B | 0p?  | ZKxaxa8c? Ey,    | Z#Kxaxa8c?G% Ex,  `B  | 0p? & &  | Z!Kxaxa8c?   Ey,    | Z@*Kxaxa8c? A  Ex,  `B | 08c?  `B | 08c?  `B | 08c?"uu`B | 08c?" | Z-Kxaxa8c?L G^  XB | 0p?h8 r%  |%r  | Z1Kxaxa1?rA  'yt = b1 + b2 x t = 4 + 1.5 x t(( ((,,,,,( (( ( #  | Z\=Kxaxa1?%z G^  '8 + i  | +i w | ZAKxaxa1?+ i  'x = 8 = aos medios de experiencia@(( ("  Z         `B | 08c?oh h F 6Y   ,| Y 6  |  `xHKxaxa1?6Y   FNy = 10 = salario por hora mediol(( (( (    h        fB | 68c?u  XB | 0p? h c 8   ] +| ] | Z TKxaxa8c?  _= 1.5 = 1.20Z,   | ZXKxaxa8c?  G8,  !| Z[Kxaxa8c?S n%L H10, 6 "| Z _Kxaxap?   = b2Z0Z,C,,,, ,   #| ZdKxaxap?  fT Ph0ZC, `B $| 0p?  %| Z iKxaxa8c?] Ey,   &| Z\lKxaxa8c?~  Ex,  `B '| 08c?H`B (| 08c?] H]  )| Z(pKxaxa8c?V; @ G^  `B *| 0p?yH | 0޽h ? a( $. ,(    TCxaxa)?\1 ^ Prediccion  <   I8 a a   ZCxaxa1?a yt = 4 + 1.5 x t@( ( ( (    ZCxaxa1?I G^  >  ZCxaxa1?G "La ecuacion estimada de regresion:## Z        I  ZCxaxa1? x t = aos de experiencia@( (  @      8       e  Z Cxaxa1?   yt = salario previstoJ( (( ( N         ZCxaxa1?%  G^  XB  0p? h T8 ~   p Ps 6   Z(Cxaxa1?   `Si x t = 2 aos, entonces yt = 7.00 por hora.1 ( (  ( ((  $ $                ZlCxaxa1?X ~  <   .  Z̺Cxaxa1? P `Si x t = 3 aos, entonces yt = 8.50 por hora.1 ( (  ( ((  $ $               Z@Cxaxa1? M 7 $ G^    <0C8c?p @ 0  U^(2   H  0޽h ? a( %. (    TLCxaxap?-l p modelos log-log <&      ZCxaxap?g .ln(y) = b1 + b2 ln(x)0Z ,G,, ,,G,, ,,4    8 U   U @ U  U   Z\Cxaxap?  | ln(y)&0ZC,,&     ZCxaxap? U  fx&0ZC,,  `B  0p?  @  E    E   ZCxaxap?   | ln(x)&0ZC,,&      Z|Cxaxap? E  fx&0ZC,,  `B   0p? { 0   Z\Cxaxap?r @ u  = b2^0Z,C,,G,, ,  8 q ~ q ~  ZhCxaxap?q % fy&0ZC,,    ZExaxap?A fx&0ZC,,  `B  0p?::0  Z$Exaxap? a c  = b2^0Z,C,,G,, ,   Z Exaxap?5 WE K10Z,   Z(Exaxap?5!W Iy0Z,  `B  0p?:N:@  q ~  q ~  Z|Exaxap?q u% fx&0ZC,,    ZlExaxap?AU fx&0ZC,,  `B  0p?:~:  ZExaxap? E K10Z,   Z0Exaxap? ! Ix0Z,  `B  0p? ::XB  08c? p XB  08c?pH  0޽h ? a(4 &.  ''\(   8      Za>xaxap?g fy&0ZC,,    Zdf>xaxap?wG fx&0ZC,,  `B  0p?p0  Zj>xaxap?7  = b2^0Z,C,,G,, ,   Zq>xaxap?' K10Z,   Zt>xaxap?k Iy0Z,  `B  0p?@ `  `    ZTx>xaxap?G fx&0ZC,,     Z(}>xaxap?' fx&0ZC,,  `B   0p?P    Z\>xaxap?' K10Z,    Z>xaxap?k Ix0Z,  `B  0p?`^8 g  g 0  Z>xaxap?' W  = b2^0Z,C,,G,, , @ g` g`  Z>xaxap?gW fy&0ZC,,    ZD>xaxap?g77 fx&0ZC,,  `B  0p?0`0  Zx>xaxap?; Ix0Z,    Z>xaxap? Iy0Z,  `B  0p?00XB  08c?0 p0 J  Z<>xaxap? u  &elasticidad de y con respecto a x:B' , , ,,4      8 w e[ & we[f@  e[ $ e[0  Z>xaxap? e{ = b2^0Z,C,,G,, , @  [ # [  ZX>xaxap?  fy&0ZC,,    Z>xaxap? [ fx&0ZC,,  `B  0p?    Z<>xaxap?7 Y  Ix0Z,   ! Z>xaxap?7 Y ; Iy0Z,  `B " 0p? P  % Z>xaxap?w k d h =&0ZC,, XB ' 08c?pH  0޽h ? a(N0 (    # lKxaxa1 ? ?  K "    H1 ?    KH  0h[ ? a(0 |( 9v i  R  3     K  C 4K ?  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Estimar una relacion entre variables economicas, tal como y = f(x). Explicacion cuantitativa de dicha relacion. 2. Predecir el valor de una variable y, basado en el valor de otra variable x.r@   S  (                                      Gastos Alimenticios Semanales(T         y = Euros gastados en alimentos por semana. x = renta familiar del consumidor por semana. Supongamos que la relacion entre x y el valor esperado de y sea lineal: E(y|x) = f(x) = b1 + b2 x Cada media condicionada E(y|xi) es una funcion de xi. 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Estimar una relacion entre variables economicas, tal como y = f(x). Explicacion cuantitativa de dicha relacion. 2. Predecir el valor de una variable y, basado en el valor de otra variable x.r@   S  (                                      Gastos Alimenticios Semanales(N         y = Euros gastados en alimentos por semana. x = renta familiar del consumidor por semana. Supongamos que la relacion entre x y el valor esperado de y sea lineal: E(y|x) = f(x) = b1 + b2 x Cada media condicionada E(y|xi) es una funcion de xi. 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