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Modelo de regresion lineal (en los parametros): y = b1 + b2x2 +& + k xk + u 2. Muestreo aleatorio: {(yi, xi); i=1, & , n} muestra aleatoria del modelo poblacional 3. Media condicional de u es cero. El error u, tiene un valor esperado de cero, dado cualquier valor de las variables independientes, E(u| x1, x2, & , xk) = 0 4. Variacion muestral en las variables independientes y no existencia de relacion lineal exacta alguna entre ellas 5. Homocedasticidad o igual varianza de los ui dado cualquier valor de las variables independientes, var(u|x1, & , xk ) = s2 > $OGOOOGOOOOO&O.O&O.O&OO6_OO5O G                                                                                        H  0޽h ? a(a    0   ( ̙33    s *$`-3 Errores de Especificacion, 2 4    Xr  0H(p  s *e-y bP(i) Omision de una variable importante (ii) Inclusion de una variable irelevanteTQ 2"$                r  0n-0  NConsecuencias de la omision de una variable importante en el modelo verdadero:O 2O              0t-   Modelo Verdadero :  24       0y- p@  lFY = b1 + b2 X2 + b3 X3 + u (M1)$ 2  $ q  0-  5Modelo Estimado: (Omision de una variable importante)66 $              0- p   $BY = b1 + b2 X2 + v (M2)" 2 " H  0޽h ? ̙33z *"@ $( : $x  $ 0l9   Por ejemplo, pensad que la variable salario viene determinada por Sala=1+ 2 Educ + 3 Habi + u. (M1) Dado que la variable Habilidad no se observa, decidimos estimar el modelo Sala=1+ 2 Educ + v, (M2) donde v = 3 Habi + u. Estimando la pendiente 2 por MCO en el modelo (M2) FC&KP 0$,$$, $, $  N $,$$,$$   $,$ $,   O    b    <   f  $ s *A ?? @    $ <(QU eM `Sustituyendo yi por su expresion en el verdadero modelo (M1) se obtiene el siguiente numerador:La  )%L      ?  H $ 0޽h ? ̙33W  (  f  s *A ??#@\   03U donde dm2 es el denominador de . Dividiendo por dm2 , tomando esperanzas condicionales con respecto a los valores de las variables independientes y recordando que E(ui)=0 , obtenemos , p  2    `  c $A ?? x  f  s *A ??@PR    0U  OEn general no sera igual a , es decir sera sesgado para . es lo que se llama el sesgo de variables omitidas.L  '  r  r  6A ?   r   6A ? 0   r   6A ?   x   <A ??P @x l   0A ?  B H  0޽h ? ̙33  \T&(  ~  s *U P  U  f  % #"000@   <f?p  tSesgo Positivo   @`   <8sf? p tSesgo Negativo   @`   <Hf?  3<0. ( $    @`   <{f?p@  tSesgo Negativo   @`  <f?@p  tSesgo Positivo   @`   <f?@  3>06 ( $    @`  <<f?p@ Corr(x2, x3)<08&     @`  <xUf?p@ Corr(x2, x3)>08&     @`  6CUf?@ T  @`fB   6fo ?`B  0f1 ?@@`B  0f1 ?  fB  6fo ?fB  6fo ?@`B  0f1 ?`B  0f1 ?ppfB  6fo ?fB  6fo ?fB  6fo ?@x $ <A ??Pp  & <PU <En clase discutiremos el ejemplo de la ecuacion de salarios.==332'      H  0޽h ? ̙33 ;3  (    0X{U DCon respecto a la varianza, se puede demostrar que en el modelo (M1):E'D  f  s *A ?f?php   0,U  Epara j=1, 2, 3, donde es el R-cuadrado de regresar xj sobre todas las demas otras variables independientes (e incluyendo un termino constante).>; ZL;    C  r  6A ?0   <U   "   0 @  qEn el modelo (M2)$  f  s *A ?f? @   <UZz ?Comparando estas varianzas sacad vuestras propias conclusiones.@@33?  H  0޽h ? ̙33 @+(    s *?   H 2  o  08  QConsecuencias de la inclusion de una variable irrelevante en el modelo verdadero:&R 2Qt              01 @  (1 - R2nuevo )/ n-mJ 2      0 0D1 P0   (m-k) ( = 2)* 24   f 0 s *hI100 P $Si F* > Fc ==> rechazar H0v% 2    N       ` 0 c $A ?? 0  0 <6 P (  P8   P$ P s * 1 jH0 : efecto conjunto de X5 y X6, i.e., b5 = b6 = 06 2    \        P 601[0 F    P 61  0 F   r P 6A ??    P 0,1pP p  fModelo Nuevo (n)`   R2  P s *  p H P 0޽h ? ̙33; T{( uxv0 T T s *1p- oContraste de WALD ( 2      8 @  TP   T s *(1   f(0.822750 - 0.601277) /2 2 w @ @  T@  T s *p1Pp  V0.017725 2   g@ @  T@  T s *1@p   nF* =: 2   `B T 0D K T s *`1  (R2n - R2v) / #restricionesx 2    2      T s *1P    (1-R2n) / n-kJ 2      T s *<1 p   O= 2 `B T 0D    T s * 1   a(1 - 0.822750) / 10 2   T s *10  O= 2 `B  T 0Dp p  T s *L1 P  h W 0.1107365 2    T s *@1  U= 6.247 2 | T s *1  "Como F* > Fc ==> rechazar H0 2   @       T s *H1P p Fc0.05, 2, 10 = 4.10J 2    F  T 0$ T s *6 jH0 : efecto conjunto de X5 y X6, i.e., b5 = b6 = 06 2    \        T 0 6[0 F    T 0 6  0 F   8 @0   T` pp  T s *6@`   4DH0 : no efecto de X5 i.e., b5 = 0# 2  B      T 0@6 0  B`  T s *60 %!Como t* < tc ==> no rechazar H0" 2   @       2 T 0%6p I1`  2 T 0(6`   I2`  H T 0޽h ? ̙33  < 4  X (  X X 0/6   oF* = : 2   ^B X 6D @ @ X 046` @  (R2nr - R2r) / k-mx 2    2      X 0=6    1-R2nr/ n-kJ 2  $    X 0B6 @  O= 2 ^B X 6D    X 0,G6` 0  g(0.822632 - 0.822750) /-1 2   X 0 K6 `  d(1-0.822625) / (16- 6) 2   X s *E6`p  W = 0.00643 2     X s *L6 ` Fc 0.05, 1, 10 = 4.96J 2     X C bAJC:\COURSE\ECON2170\GUJARA\ex832a.bmp` H X 0޽h ? ̙33<     hp (  h h 0[6J ,Ejemplo sobre como contrastar la estabilidad o constancia de los parametros del modelo ( Test de Chow  )Ni 2a                      P h 0e6@ .Procedimiento para llevar a cabo el contraste:/ 2/            8 \  h`Pa h 0k6< %9H0 : no cambio estructural o constancia de los parametros:: 2 7               h 0Hs6@ \ H1 : si lo hay: 2 &   m h 06P " 1. Dividir la muestra de N observaciones en dos grupos - grupo1 consistente en las primeras n1 observaciones. - group 2 consistente en las restantes n2 = N - n1 observaciones. 2` ;                                       H h 0޽h ? ̙33     <! ( : < < s *č6P ?I2. MCO en las dos submuestras separadamente para obtener SCR1 y SCR2 J 9   N    '   < s *ԗ6 QI3. MCO en toda la muestra con todas las N observaciones para obtener SCRRHJ E             88   <   < s *\60P 4. Calcular F* = : 2 (    `B < 0DV < s *6 (SCRR - SCR1 - SCR2) / k 2     & < s *6p (SCR1 + SCR2) / N-2kt 2    \ < 0\6 P`  :5. Calcular F* y Fca, k, N-2k\ 2    B      i  < 06 2  !Si F* > Fc ==> rechazar H0v" 2   N        < 06b p 8En este caso habria un cambio estructural en la muestra.9 29              H < 0޽h ? ̙33 ?7G@(  @ @ s *x6P Cambio Estructural:Test de CHOW@ ( 2      G@ C dA LC:\COURSE\ECON2170\GUJARA\tab808i.bmppH @ 0޽h ? ̙33| ,$l(  l l 0060 .Scatter plot de las variables Income y Savings//`&    l C dA LC:\COURSE\ECON2170\GUJARA\tab808f.bmp0H l 0޽h ? ̙33 woD( ]%] D D s *6 0@ nSCRR: 2   r  D 6A ??p R2  D s * 0 R  D s *@@  D 0t6 0 qToda la muestra`&   H D 0޽h ? ̙33  H (  Xh H> H s *6PP  "Y = a1 + a2X + u1 2   H s *6`   nSCR1: 2   r H 6A ?? R H s * H 089  Sub-muestra n1F `dl`&   H H 0޽h ? ̙33 KC0L( : L< L s * 9P@   Y = b1 + b2X +u2 2   L s *d9 p  nSCR2: 2   [ L s *P9@ 'Conclusion: F* > Fc ==> rechazar H0v( 2   @     r L 6A ??P R L s *p L 0 9P  Sub-muestra n2F `dl`&   R2 L s *  H L 0޽h ? ̙33  n f @| (  |` | c $A ??X   | 0,,9  Resultados Empiricos:`4      XB | 0D  %8  p  |p @  | 019   oF* = : 2   fB | 6D  \  | 0-9    (SCRR - SCR1 - SCR2) / k 2     ,  | 0?9P p (SCR1 + SCR2) / N-2kt 2      | 0G9  0   I=`  XB  | 0D 0   | 0K9p @`  a(.1891 - .1412 -.1660)/ 2`   | 0O9  b(.1412 + .1660) / (18-2*2)`   | 0`S9@` P F* = 5.05 `  " | 0W9  (Fc (a, k, n-2k) = ?\`hh h`  T | 04]9  (Fc0.01 =5.56; Fc0.10=2.52 Fc0.05=3.34; )`hl `hl`hl` ) H | 0޽h ? ̙33 PC( * `  c $A ??  I  0k9 0 QCreacion de variables dummies dummy = 0 para 1946-1954 dummy = 1 para 1955-1963&Rb4`N !        C dALD:\COURSE\ECON2170\GUJARA\tab808j.bmpFdB  <D>  R  s *p 0` H  0޽h ? ̙33J `(  X   C dA LD:\COURSE\ECON2170\GUJARA\tab808k.bmpP@0R  s * @ dB  <D) H  0޽h ? ̙33& p pf(  p< p 08|9p@ 8Contraste de Cambio Estructural usando variables dummies99`h           p 09@ 'Var dummy generada de la forma anterior((`4      p C dA LC:\COURSE\ECON2170\GUJARA\tab808g.bmp@p@^B  p 6D0 P  p 09 0 wMira el estadistico t`&    ^B  p 6D0 p0 R2 p s *@p^ R p s *^ H p 0޽h ? ̙33P (    C dALD:\COURSE\ECON2170\GUJARA\tab808l.bmpp^B  6DԔ0 0 ^B  6DԔ0 `0 H  0޽h ? ̙33 xp ( p   C dALD:\COURSE\ECON2170\GUJARA\tab808m.bmp X2  0` 0 X2  0~ >   69d $ vCambia la constante?`&     dB @ <D`4   69T DP vCambia la pendiente?`&     dB  <D@ 0`dB  <Dp D dB  @ <D$` H  0޽h ? ̙33 (  C  0h9  ?Como leer los resultados de la regresion con variables dummies:&@>``Z            09  5Para el periodo de 1946-1954: Savings = -0.2662 + 0.0470 Income Para el periodo de 1955-1963: savings = (-0.2662 - 1.4839) + (0.0470 + 0.1034) = -1.7501 + 0.1504 Income`@  ?  g H  0޽h ? ̙33$0 t(  ^ S      # l9e9e ?   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Zw.xaxa1 ?0P,$0 p z1. Modelo de regresion lineal (en los parametros): y = b1 + b2x2 +& + k xk + u 2. Muestreo aleatorio: {(yi, xi); i=1, & , n} muestra aleatoria del modelo poblacional 3. Media condicional de u es cero. El error u, tiene un valor esperado de cero, dado cualquier valor de las variables independientes, E(u| x1, x2, & , xk) = 0 4. Variacion muestral en las variables independientes y no existencia de relacion lineal exacta alguna entre ellas 5. Homocedasticidad o igual varianza de los ui dado cualquier valor de las variables independientes, var(u|x1, & , xk ) = s2 . $OGOOOGOOOOO&O.O&O.O&OO6_OO5O G                                                                                        H  0޽h ? a( 4,P $( : $  $ 01  "8Por ejemplo, pensad que la variable salario viene determinada por Log(Sala)=1+ 2 Educ + 3 Habi + u. (M1) Dado que la variable Habilidad no se observa, decidimos estimar el modelo Log(Sala)=1+ 2 Educ + v, (M2) donde v = 3 Habi + u. Estimando la pendiente 2 por MCO en el modelo (M2) FC+K0P 0 $,$$, $, $  N $,$$,$$   $,$ $,   T    h  (  >  f  $ s *A ?? @    $ <1 eM `Sustituyendo yi por su expresion en el verdadero modelo (M1) se obtiene el siguiente numerador:La  )%L      ?  H $ 0޽h ? ̙33 A9p (    0h91 DCon respecto a la varianza, se puede demostrar que en el modelo (M1):E'D  f  s *A ?f?php   0?1  Epara j=1, 2, 3, donde es el R-cuadrado de regresar xj sobre todas las demas otras variables independientes (e incluyendo un termino constante).>; ZL;    C  x  <A ??0   <H1   "   0HK1 @  qEn el modelo (M2)$  f  s *A ?f? @   <P1ZKz ?Comparad estas varianzas y sacad vuestras propias conclusiones.@@33?  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