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Regression models involving categorical and continuous dependent variables with a study of labour supply.. Lau, Yat Wing 1975-12-31

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REGRESSION MODELS INVOLVING CATEGORICAL  AND  CONTINUOUS DEPENDENT VARIABLES with A STUDY ON  LABOUR SUPPLY OF  MARRIED SOMEN  by IAT B.Sc,  9ING  LAO  U n i v e r s i t y Of B r i t i s h C o l u m b i a ,  A Thesis Submitted The  In P a r t i a l F u l f i l m e n t  Requirements  F o r The  M a s t e r Of i n Commerce and  i n the  Degree  1973  Of  Of  Science  Business Administration  Faculty of  Commerce and  He a c c e p t t h i s required  THE  Business  Administration  t h e s i s as c o n f o r m i n g t o t h e  standard  UNIVERSITY OF B R I T I S H COLUMBIA December,  1975  In  presenting  an  advanced  the I  Library  this  degree shall  f u r t h e r agree  for  scholarly  by  his  of  this  written  at make  that  thesis  it  freely  may  fulfilment  of  of  Columbia,  It  is  British  available  for  for extensive  be g r a n t e d  financial  by  shall  that  not  requirements I  agree  r e f e r e n c e and copying  t h e Head o f  understood  gain  the  of  be a l l o w e d  or  that  study.  this  thesis  my D e p a r t m e n t  copying  for  or  publication  without  my  permission.  of  University  C^Mfj tlZC$ of  British  2075 W e s b r o o k P l a c e V a n c o u v e r , Canada V6T 1W5  Date  partial  permission  purposes  for  in  the U n i v e r s i t y  representatives.  Department  The  thesis  JTfrJ  Columbia  fr®  fa  ASfa  M  t  °  i  Abstract This t h e s i s i s going t o consider the inferences relationships that categorical separately model.  variable.  or  relationships This  a continuous variable  relationships  i n t o two models: a r e g r e s s i o n  by  the  will  kind  economic problem. women.  of  of  model  It  i s applied  i s to  consider  and  model  Press.  Such  consideration. in the  the  analysis  labour  o f an  supply  youngest  child  t h e most s i g n i f i c a n t f a c t o r t o d e t e r m i n e t h e number o f h o u r s  w o r k e d by a m a r r i e d woman, a n d b i r t h g a p i s t h e m a j o r e f f e c t the  of  D a t a a r e p o o l e d f r o m t h e P a n e l S t u d y o f Income  D y n a m i c s 1972. I t i s f o u n d t h a t t h e a g e o f t h e is  least  The p r o b a b i l i t y  Serlove  be g i v e n more c o m p l e x  and a  model a n d a p r o b a b i l i t y  two s t a g e method. method  the  c a n be c o n s i d e r e d  model c a n be e s t i m a t e d by o r d i n a r y  Zellner's  estimated  married  These  The r e g r e s s i o n  squares, is  determine j o i n t l y  about  probability  years o f age.  of  a  wife  having a c h i l d  not older  in  than s i x  Table  of  Contents  Abstract Table List  of of  contents Tables  Acknowledgment  Chapter  I  Introduction  Chapter  II  Basic  The  Model  mathematical  Discrete  model  dependent  variable  regression  Estimation Hypothesis  testing  Polytomous  variable  Chapter  I I I System  Estimation  of Eguations  eguation-by-eguation  D e p e n d e n c e among Further  Chapter  IV  Lagged  groups  discussion  Model  Extensions  variables  model  Model  with  constraints  Model  with  jointly  dependent  Simultanous-eguation fiecursive  Model  model  model  variables  i i i  Chapter  V  A Study  on  Labour  Supply  of  Married  Women  Model D e s c r i p t i o n  37  Introduction  37  Specification  of  models  38  Specification  of  Variables  39  Data  Chapter  restriction  VI  44  Empirical  Results  of  Results  from  the  labour  Results  from  the  probability  Further  estimation  Chapter  regression  Besults  of  single  eguation  Results  of  Z e l l n e r ' s seemingly  Results  of  probability  Least-squares  Appendix  B  Likelihood ratio  Biblography  Model I I  53  estimation least  53 s g u a r e s method  59  estimation test  coefficient  Parameter  estimates  1967  without  -  71  54  86  A  C  of  Conclusion  regression  46 47  functions  Appendix  Appendix  eguations  48  Results  VIII  46  eguation  Empirical  Chapter  VII  Model I  for  88 micro  vector  for labour (Unemploy) *  eguality  90  eguation 92  93  iv  List  I  The P a r a m e t e r  Estimates  II  Mean a n d S t a n d a r d  III  Probability  of  Tables  f o r Labour  Equations  D e v i a t i o n of t h e Model  Function  Estimates  50  I  51  o f t h e Model I  52  IV  A  Parameter  Estimates  f o r the Labour  Equations,  1967  66  IV  B  Parameter  Estimates  f o r the Labour  Equations,  1968  67  IV  C  Parameter  Estimates  f o r the Labour  Equations,  1969  68  IV  D  Parameter  Estimates  f o r the Labour  Equations,  1970  69  IV  E  Parameter  Estimates  f o r the Labour  Equations,  1971  70  V A  Mean a n d S t a n d a r d  D e v i a t i o n o f t h e Model  I I , 1967  71  V B  Mean a n d S t a n d a r d  D e v i a t i o n o f t h e Model  I I , 1968  72  V C  Mean a n d S t a n d a r d  D e v i a t i o n of t h e Model  I I , 1969  73  V D  Mean a n d S t a n d a r d  D e v i a t i o n o f t h e Model  I I , 1970  74  V E  Mean a n d S t a n d a r d  D e v i a t i o n o f t h e Model  I I , 1971  75  VI A  T h e C o m p a r i s o n o f t h e Two S t a g e A i t k e n a n d t h e S i n g l e Equation L e a s t - s g u a r e s E s t i m a t i o n o f t h e Model I I , 1967  76  T h e C o m p a r i s o n o f t h e Two S t a g e A i t k e n a n d t h e S i n g l e Eguation L e a s t - s g u a r e s E s t i m a t i o n o f t h e Model I I , 1968  77  T h e C o m p a r i s o n o f t h e Two S t a g e A i t k e n a n d t h e S i n g l e Eguation L e a s t - s q u a r e s E s t i m a t i o n o f the Model I I , 1969  78  T h e C o m p a r i s o n o f t h e Two S t a g e A i t k e n a n d t h e S i n g l e E q u a t i o n L e a s t - s q u a r e s E s t i m a t i o n o f t h e Model I I , 1970  79  T h e C o m p a r i s o n o f t h e Two S t a g e A i t k e n a n d t h e S i n g l e Eguation Least-sguares E s t i m a t i o n of the Model I I , 1971  80  VI  B  VI C  VI D  VI E  V  VII  A Probability  Function  Estimates  of t h e Model  I I , 1967  81  VII  B Probability  Function  Estimates  o f t h e M o d e l I I , 1968  81  VII  C Probability  Function  Estimates  o f t h e M o d e l I I , 1969  83  VII  D Probability  Function  Estimates  o f t h e M o d e l I I , 1970  84  VII  E Probability  Function  Estimates  of the Model 171,  85  1971  I would l i k e to express my g r a t i t u d e t o my t h e s i s committee members  who  offered  stimulating  me  their  advice,  suggestions and a s s i s t a n c e .  helpful  criticisms,  S p e c i a l thanks  to P r o f e s s o r Press f o r the i d e a which made t h i s t h e s i s h i s encouragement, and f o r the permission to program  to  compute  P r o f e s s o r Berndt labour  supply  information  on  the  data.  use  S p e c i a l thanks  are due  possible,  his  logistic  are a l s o due t o  f o r generously s h a r i n g h i s r e s e a r c h work on the of  married  the  Panel  women,  and  providing  access  to  Data.  I a l s o thank my mathematics  t e a c h e r . P r o f e s s o r Nash, f o r h i s v a l u a b l e comments. I  am  fellowship first  indebted  tc  Doll  f o r providing  me  a  from the Centre f o r T r a n s p o r t a t i o n S t u d i e s during my  year study, and t o  research  Professor  assistanship  U n i v e r s i t y i n my second  Professor from  year.  Press  f o r offering  h i s research  fund  of  Without t h e i r f i n a n c i a l  me  a  Chicago support,  the t h e s i s i n i t s present form c o u l d not have been a c h i e v e d . I  would  also  like  to express my g r a t i t u d e to Miss Nancy  Reid f o r s h a r i n g her e a r l y r e s u l t s which we d i d f o r the Economic and  S t a t i s t i c Workshop at the beginning of t h i s  Norine  Smith,  research  assistant  to  Miss  of P r o f e s s o r Berndt f o r her  help to access data from the Panel Data f i l e , Johnston  year;  and  to  Ms  Valda  f o r her b e a u t i f u l e d i t o r i a l work.  Wing Lau November  o f 1975.  1  Chapter  I  Introduction  This t h e s i s  is  1  relationships that categorical  concerned  making  inferences  variable.  Given the  variable  is  variables;  also  discrete  the  of e x p l a n a t o r y  discrete  related  to  a  set  of  random v a r i a b l e  variables.  For  i s related  example, the  depend upon t h e  Let  than it  6 years.  We  will  i s 1 when a f a m i l y The  case, that  i s for  hushand's  income,  A l s o the  her  call  has  timing a  be  would  her  related  like  to  labour force  activity  older  6  than  parameters  of  parameters  of  z,  of  of  her  has  a  6 years or  of  z,  may  a family  to the  same s e t  know t h e as  joint  of  her  It  shown  conditional  is  that  regression  discrete  individually.  I t i s very easy to extend the  a  dependent  polytomous  dichotomous dependent  variable  variable.  in  not  a  older  and  0  a c t i v i t y i n each relate  not  to  and  older  her  can  so  on.  than  timing not  6  in  being  estimate  the  and  the  variable  regression  problem t o  regression  her  variables.  equations  dependent  a  variable, i f  child we  of  assume t h a t  explanatory  youngest  the  likely  younger,  p r o b a b i l i t y of  w e l l as  to  will  fecundity  having a c h i l d  the  timing  child  labour force  value  us  constraint  e x p e c t e d wage, h e r  years.  the  a family  a child  given  p r o b a b i l i t y of  y e a r s can  youngest c h i l d .  work l e s s h o u r s when s h e  otherwise.  We  of  a  explanatory  activity  age  and  random v a r i a b l e ,  m a r r i e d woman p a r t i c i p a t i n g i n l a b o u r f o r c e  housewife w i l l  about  determine j o i n t l y a continuous v a r i a b l e  continuous  same s e t  with  rather  involve than  a  2  To  consider  model,  the  equations of  model  explanatory  found  than  time  that  apply  such  to estimate  extensions  The  plan  divided first 3,  into part  a n d U.  that  of t h i s  discrete  for  In chapter  testing  a  than the  part  of  the  1967  a  set  set of having  same such  to  equations  a  set of relations  1971.  I t i s  t o estimate the i s more Some  i s as f o l l o w s .  efficient  interesting  2 we d e s c r i b e  i n  independent,  equality  of  In chapter  of regression  eguation-by-equation  regression,  i s  part.  model  and  Chow's  variable  why we  t e s t (1960)  c o e f f i c i e n t s o f two 3 we e x t e n d  prove  parameters  we e x p l a i n  the  2,  regression  dependent  estimating  The  of chapters  i n conditional  discrete In  functions.  composed  the basic  of the parameters  other  This thesis i s  and an a p p l i c a t i o n a l  discussion  parameters  equations. set  to  a  woman i n  of a family  method{1962)  thesis  dependent v a r i a b l e  rather  regression into  this  a theoretical  are  terms  equation-by-equation.  i sa theoretical  regression  logit  of  and t h e  and  F o r example, a t  we may c o n s i d e r  regression  regression  model a r e mentioned.  the estimators  equations  i n  as from  regression  of a married  related  Zellner*s  them  a  a setof  regressions.  Hence,  parameters of c o n d i t i o n a l than  i s  of  regressions  of z, the timing  6 years  periods,  to  into  and t h e p r o b a b i l i t y  variables.  certain  terms  extended  c a n be e x p r e s s e d  variables,  not older  i n  variable  value  activity  explanatory child  be  dependent  force  problem  a set of conditional  t , f o ra given  labour  for  can  including  discrete  year  t h e above  i s  of  prefer used  conditional  basic  model  equations.  The method o f  estimating  mentioned.  Zellner's  seemingly  3  unrelated  regressions  efficient the  estimation.  The  second  problem from  of t h i s  of  labour  Data  i s an  which  are  description under  Centre of  second  empirical  based  on  of our economic  model.  Chapter  of  some  the  1972,  an  testing  interesting  studies 2  the  models,  3.  economic  data used i s  Michigan.  economic  household  status. personal  two  Chapter  5  the r e s u l t s  models i s  chapter 6 gives the  be t h e c o n c l u s i o n  are  collected  of  a r e compounded by and  will  an  which  University  chapter 7 gives 8  The  on c h a n g e i n f a m i l y  chapters  model and  order to get  concerning  m a r r i e d women.  t e c h n i g u e i s m a i n l y on  The  the f i r s t  4 i s t o propose  application  are focused p r i m a r i l y  interview.  in  i s the description  o f Income Dynamics  Research  Data-collection  applied  model.  supply of  the Panel Study  the Survey  Chapter  basic  part  i s  Following  aggregation Mas.  extensions  by  method  the  results  under  of  the  the whole  thesis.  Footnotes 1  me  T h i s i s based  the  preliminary  i n t h e E c o n o m e t r i c s and  University of  on  of B r i t i s h  the I n s t i t u t e  University  of B r i t i s h  Statistics  Columbia.  of A p p l i e d  w o r k d o n e by  this  Reid year at  Miss Reid i s a master's  Mathematics  Columbia.  Workshop  Nancy  and  Statistics  and the  student of  the  4  Chapter  Easic  I  The  mathematical  let  us b e g i n  dependent  the other  1.  For the given  expressed  with  a  very  one  simple  as  model  variable  having  v a l u e o f z, y i s d i s t r i b u t e d  a function  the x's as the  which  o f which i s y, a c o n t i n u o u s  i s z, a d i c h o t o m o u s  Since z i s dichotomous, and  Model  model  variables,  and  II  variable,  normally,  denote the f u n c t i o n a l  two  the value  o f a number o f v a r i a b l e s  we  has  1  0 or  and  is  1  k  x ,  x .  relations  of  y  following:  1 E(y  | z=1)  =  E(y  | z=0)  = g(x  k x )  f (x ,  and  1  There  will  relations. linear. formally  be The  So  a  variety  of  functions  simplest relationship  for  n  observations,  under t h e l i n e a r  (y (i) Jz ( i ) =1)  hypothesis  11 = a x  k x )  ,  to s a t i s f y  between we  y and  the  as: k k + a x (i) + u (i)  ( i ) + ...  k k + b x (i) + v (i)  and  (y (i) f z ( i ) = 0 ) i=1,  = b  x  n or i n vector n o t a t i o n ,  above  x's  w r i t e e a c h o f them  ( i ) + ...  11  the  is more  5  <y (i) | z ( i ) = 1 )  = A'X(i) + u ( i )  and (y(i) | z ( i ) = 0 ) = B«X(i) + v ( i ) 1 k 1 k where A= (a , ...,a ) * , B=(b ,...,b )»,  1 k and X= (x <i),...,x (i) ) • ;  u, and v denote v a r i a b l e s which may take on p o s i t i v e or n e g a t i v e values.  U s u a l l y u and v a r e c a l l e d e r r o r terms  terms. and  disturbance  In order to make the model simple, l e t us f i r s t  v have t h e same d i s t r i b u t i o n .  and  or  normally d i s t r i b u t e d  assume u  We assume u and v are random  with mean  zero,  variance  var (u)  and  zero c o v a r i a n c e , t h a t i s  E[ u (i) ] = 0 ± * j  E [ u ( i ) ,u (j) ] = 0 = var(u)  Hence  A,  B  i = j  and var(u) are unknown parameters.  estimate these parameters s t a t i s t i c a l l y sample  observations,  and  to  test  on  when  z=1,  y  i s distributed  basis  hypotheses  T h e r e f o r e , i f we c o n s i d e r the c o n d i t i o n a l then  the  He may wish to c f our  about  them.  distributions  normally  of  y,  with mean A'X and  variance v a r ( u ) , and when z=0, y i s d i s t r i b u t e d  normally  with  mean B'X and v a r i a n c e v a r ( u ) . For  the  dichotomous  dependent  i n t e r e s t e d i n the p r o b a b i l i t y t h a t z w i l l The of  variable,  1  k , ...,x .  So,  may  be  have the value 0 or 1.  p r o b a b i l i t y of z being 1 can a l s o be expressed X  we  as a f u n c t i o n  6  k  1 Prob(z=1)  Suppose  that  we  nondecreasing ith  want  a  h (x  ,  x  relationship  function  of  t  with  i n  )  which  FI-OOJ^O  Prob(z=1)  is  F(o») = 1 ;  and  a  for  the  observation,  P r o b ( z ( i ) = 1)  where  11  t(i)=c  or  Prob(z(i)=1)  =  p(i)  =  taken know  nondecreasing  p (i)  depending  Therefore,  we  explain  will  we  discuss  we  Discrete  was  will the  of  linear  If  we  use  approximated  i n  the  function  is  w i l l  the  centre,  estimation  these  F.  i n  is  i n  some  of of  C.  C.  parameters, the  In  history  the  a  we of  following  functions  and  chosen.  regression to  observe but  p(i)  components  the  about  2  decreasing  the  of  function .  and  1,  transformation  the  approximation  we  be  because  argument several  and  may  on  F,  (1)  0  estimation  variable  notation:  distribution  of  interest  approximation  linear  but  function  long  vector  between  signs  the  discuss  dependent  then  a  logistic  Failure  F(C'X),  about  i=1,...,n  F(C«X(i))  C*X,  our  into  in  cumulative l i e  of  focus  more  a  will  upon  step  there  why  be  function  variables,  section,  to  F (t ( i ) )  k (i) ,  we  statistics,  =  x  Therefore,  should  p(i)  (i)+...+c  is  Before  =  k  x  F(C»X(i))  II  =  to  probability  the  that  poor  for  function:  probability  the very  function large  function is or  well small  7  value  of  standard given that  C'X.  There  regression  observation the  Those  variance  disturbance  least-sguares imprecise  undefined. this  z(i)*s  will  will  give  Lee ( 1 9 6 5 ) ,  the  this  failed,  e r r o r s , or  0 and  1 for a l l i , the  the  concludes  that  s i n c e , "because method  i n general  be  of fully  1,  j .  and  (1964)  remove  the  i t ignored  the  not  guarantee  that  and  resulted in  some  transformation  on  the  linear  the  to  so  ordinary  Goldberger  numerical  v a r i a b l e i s l a r g e r than  no  upon  estimators  because  did  for  variable,  depends  generalized least-sguares  least-sguares led into  ,  random  term  using  First,  3  inefficient  Furthermore,  function fails,  data .  Bernoulli  Z e l l n e r and  l i e between  Cox (1970)  distributed  binary  in  heteroscedastic, therefore,  problem, but  independent  difficulties  j t h disturbance  of  variances.  generalized the  the  of  on  z(i) i s a  terms are  character  should  negative  of  use  heteroscedastic  z (i)  x(i),  estimation  the  technical  techniques  predictions.  suggested  Bernoulli  are  problem  that i f  transformation  is  approximation  to  z(i)*s  are  normally  estimation that i s linear  in  the  efficient."  Probit analysis: One  reasonable  B l i s s (1934)  was  the  method i n a n a l y z i n g Cornfield  and  dosage response surveys.  approach first  to use  guantal  Nathan curve,  is  i t .  (binary)  Hantal(1950) and  called  probit  Finny(1947) responses  analysis. applied  in  bioassay,  applied i t i n calculating  Tobin(1955)  applied  i t  in  this  the  economic  8  This  method  applies  eguation  (1).  function  of the standard  data*,  F (t)  Hence  a grouping  i s considered  f o r estimating the  as t h e c u m u l a t i v e  normal d i s t r i b u t i o n  p ( i ) i s estimated  p+(i)  method  by  =r(i)/n(i)  of elements i n i t h c e l l  and  i s  number  define  p r o b i t as  t  +  normal  transformation transformed  ordinary  by p+ ( i )  in  order  with  mean  P r o b i t (p* (i))  to  to  get  Hence 5  e (i)  with  = C*X(i)  zero  adds  positive  5  values  P r o b i t (p* (i) )  and  i t will  F i s the  One  variance  least-sguares to the transformed  a regression eguation,  where  1, We  (2)  = F (t+ ( i ) ) .  distribution.  variable.  distributed  value  following,  ( i ) i s defined  standardized  having  of elements i n the i t h c e l l .  Probit (p+(i)) = t + ( i ) + 5  where  grouped  i=1,...,n  i s t h e number the t o t a l  using  by s a m p l e p r o p o r t i o n , i . e ,  where r (i) n (i)  distribution  1.  in  the  for  the  i s  So  data.  cumulative  normally  we c a n  apply  Putting i t into  be  + e (i)  mean, z e r o  i=1,...,n  c o v a r i a n c e , and  (3)  variance  egual  v a r (e ( i ) ) . Finally,  this  we  note  probit analysis  several  observations  with Press  method per  to  a n d N e r l o v e (1973)  be  c e l l (n ( i )  useful,  there  > f o r every  i ) .  that: "For should  be  Moreover,  9  efficiency  of  associated  estimation  with  the  i s  added  computational  difficulties  integrals  i n  this  per  are  cell  observations  Logit  lost  the  ad  hoc  procedure  5 i n ( 2 ) . Note a l s o t h a t t h e r e a r e associated  procedure.  inefficient,  per c e l l  i n  with  the  Unequal numbers o f  and  cells  with  use  of  the  observations  one  or  zero  are not useful."  analysis: Another  method  Berkson(1944).  called  Using  cell  logit  frequency,  cumulative  distribution  distribution  f u n c t i o n ; that i s ,  function  F(t) where t i s r e a l ;  analysis  =  was i n t r o d u c e d  F (t) i s c o n s i d e r e d of  the  standard  as  by the  logistic  1/(1+exp(-t))  so P (i)  =  V [ 1+exp {-C'X ( i ) ) ]  or,  log(p (i)/(1-p(i))  Now  we d e f i n e L o g i t ( p ( i ) ) +  as  = C«X(i)  following,  Logit (p+(i)) = log(p* (i)/(1-p+(i) )) where  p ( i ) i s estimated  from  regression estimation.  and  +  Nerlove(1973)  tables.  apply  by s a m p l e  this  portion.  Bishop (1969),  C c a n be  estimated  Goodman (1970) ,  method i n d e a l i n g w i t h  Press  contingency  10  Other  transformations; Coleman(1964) has proposed  p(i)  The  weakness o f t h i s  lie  between  zero  negative.  whether  m e a s u r e how test the  the  and one u n l e s s  p (i)  statistical  a l l o f t h e parameters a r e non-  made  a  comment  "Coleman's a r t i c l e  Furthermore, significance  Angular  about  this  d i d n o t s h o w how t o n o r was h e a b l e t o  he d i d n o t  show  how  to  o f t h e c o n t r i b u t i o n made by  parameters i n t h e model, nor c o u l d  contribution's  choosing  i s not constrained to  h i s model f i t t h e a c t u a l d a t a ,  well i tf i t .  various  those  function i s that  He s a i d ,  model by  l-exp(-X'C)  Goodman (1972)  transformation. test  =  an e x p o n e n t i a l  he m e a s u r e  their  magnitude."  transformations  transformations  transformation.  are  are very not  So o u r s e l e c t i o n  as  possible candidates, but  simple  i s limited  as  the  logistic  t o p r o b i t and  logit  analysis.  The  choice The  probit and  of transformation: above  or l o g i t  Cox(1967)  found  that  slight  except  discussion analysis.  have the  shows,  work  view,  i ti s sure  the  g l o b a l maximum.  that  about  will  Buse(1972), this  From  t h e maximum  I f we c o n s i d e r  cost  of  either  Chambers They  two i s very  the optimization  of a l o g i s t i c  the  be  problem.  d i f f e r e n c e between t h e s e  a t t h e two e x t r e m e s .  of  choice  Gunderson (1974),  d o n e some numerical  our  point  function i s computation,  11  logit  analysis  is  other  hand, there  transformation hypothesis  He  on  "mathematical would  create the  the  be  the  as  to  practice into  misleading  and  guide  the  method  more o b j e c t i v e  and  tolerances,  For  use  validity  of  thi.s  injecting  an  data He  i s only  is  explains that  of  calculation, sounder  but  postulate  merely  to  follows  reasons,  the  tolerances.  heuristically  affected  these  of  the  under  harmful.  model", to  function.  the  response  formulation  responses  probit  of  the  of  that  the  hand  hypothetical  On  to  other  proportion  normal  that  one  analysis.  distributed  "tolerance"  i t can  seem  any  appropriate  doubtful  of  probit  t h e o r e t i c a l argument  very says  interpretation  i f  than  log-normally  is  objectionable;  better  the  of  hypothesis.  it  i s the i s  B e r k s o n (1951)  that  much  the  the logit  a  then  not  to that  integrated analysis  is  preferred.  Ill  Estimation We  recall  the  c o n d i t i o n a l d i s t r i b u t i o n s of  y  when z=1,  i s  1 N(A'X, v a r ( u ) ) , and k 1 . . . , a ) ' , B = (b, F(X'C) and  where C =  z  function  will using  be  when z=0, y i s N ( B * X , v a r ( u ) ) w h e r e A = (a k 1 k b ) • and X =(x , x ) * ; p r o b (z=1) 1 k  ( c ,  ...,  f(y,z).  matrix  f (Y,Z|P,X)  c )'. If  algebra, =  Thus the we  joint  express the  density  joint  then  h(Y|Z,P,X)g(Z|P,X)  (4)  of  , = y  likelihood  12  Hhere,  P= (A* ,B« ,C» , var (u)) •,  observations  X  and k dimensions.  is  a  nxk  matrix,  i.e. n  For each o b s e r v a t i o n i ,  h ( y ( i ) |z(i) ,P,X(i) ) = (2lTvar (u) ) - o . s p { [ y ( i ) - (X (i) »Az (i)+X (i) »B (1-z ( i ) ) p y ( 2 v a r (u) ) } eX  and, g(z{i) |P,X(i) ) = F ( X ( i ) » C ) Let  L(P)  be  p r o p o r t i o n a l to the  L(P)  the  joint  z(i)  (1-z(i)) £ 1-F(X(i) • C) ]  likelihood  products of h and  function,  L (P)  is  g, that i s  n : TT h(y (i) | z (i) ,P,X <i) ) g (z(i) |P,X(i)) i=1 n  log(L)  : T. {z ( i ) l o g [ F ( X ( i ) «C]+ (1-z (i)) log[ 1-F (X (i) «C) ]} i=1 n -(2var (u))-iZT{y(i)-[A'X(i)z(i)+B«X(i) ( 1 - z ( i ) j} i=1 - (n/2) l o g ( v a r (u))/2 - (n/2) l o g (2?T) (5) 2  Estimation  of l o g i t parameter:  This i s j u s t as Dempster (1972) p o i n t s out density h and  of Y and  Z i n (4) can  estimators  maximizing these two likelihood  of  all  the  joint  functions,  The  maximum  parameters can be found  functions separately.  The  from a f i x e d l o g i t model, and  a m u l t i v a r i a t e general  the  i n t o two  g, which depend on d i s j o i n t parameter s e t s .  likelihood  log  be f a c t o r i z e d  that  function g i s  by a  function h i s just  l i n e a r r e g r e s s i o n model.  There are s e v e r a l methods  to  estimate  logit  parameters.  13  Berkson(1955) whose  introduced  results  are  a  method  asympototically  called  equivalent  likelihood  estimation.  Theil(1970)  generalized  least-sguares  method.  large one  samples  those  observations,  cells  which  and i n Berkson's  Both  method  per c e l l .  Goodman (1972)  estimation  i n  analysis,  smaller  weighted takes  variance  from  least-squares.  the  t h e use o f t h e  and  only  one  found  method or  disadvantage  of  no  likelihood  that  he g o t a  MLE t h a n T h e i l ' s e s t i m a t i o n  One  i n  more t h a n o n e  t h e maximum  he  only  i n Theil's  one r e q u i r e s used  maximum  from  MLE i s t h a t i t  more c o m p u t a t i o n a l t i m e .  This estimation Press  logit  C h i square"  are applicable  contain  observation  to  suggested  or designed experiments, since  deletes  somewhat  "minimum  and  thesis and  adopts uses  the  method  of  maximum  t h e computer program which  Nerlove (1973).  The  method  can  be  likelihood  i s developed  by  summarized  as  follows:  L{g(Z|P,X))  n = TT g(z{i) |P,X(i) ) i=1 =  Define  T  +  a s t h e sum o f X { i ) z ( i ) ,  T+ i s a s u f f i c i e n t X (i)  f o r which  n z ( i ) (1-z(i)) T T £ P ( X ( i ) «C) ] [ 1 - F ( X ( i ) »c) ] i=1  statistic  z(i)=1.  Hence  [ 1 + e x p ( - X ( i ) «C+) j - * X ( i )  where i r u n s f r o m  1 t o n.  f o r C, i . e . T+ i s t h e sum o f t h o s e C+, t h e MLE o f C m u s t  n = T+ = Z l i=1  X(i)z(i)  satisfy  (6)  Note  log(L)  maximum.  i s  globally  program  algorithm  6  Estimation In  get A  A , +  (5) p r o v i d e s  by w h i c h  Press  and  L  can  Nerlove  of function minimization  an  absolute  uses the  (1973)  Davidon matrix.  parameters:  to estimate  A,B a n d v a r (u)  by u s i n g  (1-z(i))z(i)  of A ; since  = 0, t h e n  maximized.  and t h e  = 0, 3 f / 3 8 = 0, a n d 3 f / c D v a r { u ) = 0.  Sf/tDA  be  the inverse o f the information  of conditional  the estimator  from  +  method  by  f o r computing  order  af/5>A  methods  developed  Fletcher-Powell  set  so  = C 1 + e x p ( - X ( i ) «C+) ] - i  a r e many n u m e r i c a l  The  5  Hence,  P+(i)  There  concave ,  we  MLE m e t h o d ,  we  For example,  to  = 0, we c a n s o l v e  have,  &+ = {Xi« X* ]-iX» ' y i  where 1  [x  X  i s nxk m a t r i x , f o r each observation k (i)z(i), x ( i ) z ( i ) ] ' , i = 1 , . . . , n a n d Y» = [  y(n)z(n)]'.  X  2  +  = [ (X ) •(X ) 3~i(X ) » Y 2  i s nxk m a t r i x ,  for  1  £x  , X  (i)  1  =  y ( 1 ) z { 1 ) ,  ,  i ,X  =  Similarly,  B  where  i  1  (i)p-z(i)),  x  k  2  each  2  2  observation  ( i ) ( 1 - z ( i ) ) J«,  i=1,.,.,n  2  and  (i) y  2  =  15  [ y (1) n-z  O))  *  y (n) ( 1 - z (n) ) ] * .  The  estimated variance  will  be  Var+(u)  where  Y  =  +  estimator  ( X ) *A + ( X ) • B + . l  of  +  var (u)  Those  group, B  +  two  we  will  =  are  groups,  a l l z=0,  and  The  not one  of unequal  we  relax  distribution, the  of  the  unbiased  )/(n-k)  +  strange  to  contains  us.  apply ordinary  A).  Var+(u)[ (X ) ' (X ) J " 2  2  A  A and  that  u A,  E a r e same as  8e and  +  and we  may  b e f o r e , but  =  ( Y - (X ) B )  cov(B+)  2  2  +  1  B  1  +  (Y - (X ) B + ) / (n -k) 2  2  2  = V a r + (v) [ ( X ) » ( X ) J - i 2  2  +  each and  are  observe  var+(v)  1  know A+  v have t h e  ( Y - ( X ) A ) ' ( Y - ( X ) A ) / (n *-k) +  on  model:  =  l  ether  1  Var+(u)  1  the  our  l  variances of c o n d i t i o n a l  shown i n A p p e n d i x  and  split  least-sguares  1  condition  I f we  a l l z=1  = V a r + ( u ) [ (XV) » ( X ) ] ~  the  then as  estimations  use  c o v a r i a n c e m a t r i c e s of  cov(B+) =  If  to  same r e s u l t s ( A p p e n d i x  cov(A+)  Estimation  prefer  ( Y - Y + ) * (Y-Y  i f we  get the  are unbiased.  1  then  +  results  sample i n t o  (Y-Y+) (Y-Y+) / n  I f we  2  Var+(u)  contains  =  same that  16  where  n  i s the total  l  number  of observations  when z = 1 , a n d n  2  •=  n - n . 1  IV  Hypothesis  Logistic For fact  model: l a r g e s a m p l e s , we s e t h y p o t h e s e s a b o u t  that  obtained  C+ i s a s y m p t o t i c a l l y n o r m a l . from  C  Also,  any h y p o t h e s i s  under  2  test  of the value  about  C  can  (Appendix B ) .  +  constraints  of  without  the  maximized  value  o f -21og (r) i s d i s t r i b u t e d  tested  by  function  hypothesis  constraints.  g i s t h e number  be  In  being large  where  i,j=i,...,k  The l i k e l i h o o d  of the l i k e l i h o o d  value  freedom;  +  the  matrix i s  I(C ),  2  ratio  the  by u s i n g  the i n v e r s e o f i t sinformation matrix  i j i J = [ a L (c+)/ac ac ] = { a g (c+)/ac ac j  ratio  +  I t s covariance  i(c+)  likelihood the  testing  using  a  ratio, r i s g  maximized  tested tothe samples,  as C h i square w i t h  the  q degrees of  of independent r e s t r i c t i o n s  i n the null  hypothesis.  Conditional  model:  Hypothesis vector.  F o r each  about  A  component  i s  H:  under  A  =  H, we  A ° , where know  A° i s a  qiven  17  i  i  (a+  has  asymptotic  - a"  i  i > V a r ( u ) ) °.  )/(w  q  t - d i s t r i b u t i o n with  5  degrees  l  of  freedom,  in  i i which and  g  when is  g*  > 0;  w  i s the difference  1  z=1,  and  k,  F-test. An  k+1)  The  T  i s an  2  way +  where T  Similarly component  to test  tail  H,  1  1  - 1  variables. test  we  the  F ^ q  1  H:  B+=B°,  1  1  Since  t  2  hypothesis  ) .  i s Hotelling's  and  T  i s  0  l  B,  1  observations  = kg»F ( k , g i - k + 1 ) / ( g - k + 1 )  2  ((X ) *X )"" #  (A+-A ) / ( v a r ju) )  of the p r o b a b i l i t y  for  under  can  of of  the hypothesis  (A -A°) ' ( ( X ) ' X )  upper  t h e number  r a t i o i s d i s t r i b u t e d with  Since  Hotelling's  m,  F d i s t r i b u t i o n , we  alternative  test.  between  element  t h e number o f i n d e p e n d e n t  d i s t r i b u t e d as  using  i s the i t h diagonal  F*  2  a  (k,q 1  function.  where  B°  i s given,  f o r each  know  i  i  (b+  -  b°  i i ) / (r V a r ( v ) ) o. s  i i is  distributed  diagonal be  2  element  tested  Hotelling's =  as t ( q ) where g -n-m-k > of  by  using  T  i s  2  2  {(X )• ( X 2  F-test,  2  0,  and  ) S i m i l a r l y  F{1,g }. 2  Or,  r  i s  the  the hypothesis  using  Hotelling's  (B+-B0) » ( ( X ) ' X ) ~ i (B+-B°)/(var (u) ) , w h e r e 2  2  i t h can T , 2  T  2  kq F (k,g -k + 1)/{g -k+1) . 2  2  Testing  The  equality  2  between  conditional  two  conditional  variable  y  given  distributions:  z  may  have  the  same  18  distribution their and  f o rdifferent  eguality.  rewrite  apply  o f z , s o we a r e g o i n g  Chow's t e s t { 1 9 6 0 )  thedistributions into  linear  YMi)  =  (y ( i ) |z ( i ) =1) = X M i ) ' A  y (i)  =  (y ( i ) | z ( i ) = 0 )  2  In  We  values  there,  we  covariance.  assume  they  have  test  H: A=B=F,  models.  +  OA  to test  to  OB  + u (i)  + x (i)'B  + v{i)  2  egual  variance  and  zero  Under t h e H t h e n  Y i = x**F + 0 y2 = x * F + V 2  so  F i sestimated as  F+ = [ ( X i , X ) ( X i , X ) * ] - i ( X i , X ) (Y» , Y ) ' 2  Let  E= ( U , V ) •  2  2  2  then  E+'E + = [ ( Y i , Y ) ' - ( X i , X ) VF+]»£ ( Y i , Y ) » - ( X i , X ) * F + ] 2  E  +  i s estimated  2  from  Under t h e a l t e r n a t i v e  U+'U + + V+'V+ =  U 'U+ h a s +  independent,  rank  2  theentire hypothesis  sample, so E  g  l  and  so t h e rank  v 'V +  +  + ,  E+  (7)  h a s rank n-k.  A#B, we h a v e  ( Y 1 - X 1 A + ) ' ( Y i - X A+) l  2  + ( Y - X B+) » ( Y - X B + ) (8)  has  2  rank  2  2  g .  U  2  2  and  o f U*'U++V+'V* i s g i + g = n - 2 k . 2  V are  19  Y -XiF+-, lyz-xzp+J  =  l  r  r ^ - X i A + T + X A -XiF+-, «-X -X B -' «-X B+-X F+J l  ||(Yi-X*F*,Y -X F+)|| 2  =  2  2  + cross Since  +  || ( X A - X F 1  product  term  i s t h e sum o f s g u a r e s  2  2  +  2  (9)  2  +  1  +  ,X B -X F 2  +  2  ,+  ) J|  2  i s zero,  so t h e sguare  on t h e r i g h t ,  on t h e  left  that i s  2  || (Y -XVA +,Y -X B+) || l  2  terms  | I (Y*-X*F+,Y -X F ) j| =  +  2  +  product  the cross  (9)  of  2  2  2  I | (Y - X * A+, Y - X B ) I I l  +  r  2  2  | | ( X i A + - X i F , X B + - X 2 F + ) | \z  +  2  +  (10)  2  or s a y ,  Q  From  the estimations  (Xi'Xi+X  which  2 f  X )F+ 2  +  2  Q  3  = Xi'Yi  + X  2 ,  Y  2  = X*»X»A* +  X »X B+ 2  2  implies  A+-F+ i s a l i n e a r  under  Q  o f A, B a n d F , we g e t  B+-F+ = - ( X  the  =  1  estimated H  we  will  2 1  X )~MX 2  transformation  functions of A have  +  l  » X M (A+-F+)  o f 0 a n d V,  and F  +  i n terms  (11)  so  we  substitute  o f U a n d V.  Then  20  &+-F+ =  k, s i n c e  (12)  and  -= n - 2 k  rank(Q ) 2  < r a n k (Q )  estimation  has  3  by F  rank(Q ) 3  ratio  1  z  the  + JlY -X B+|| )k  2  2  hypotheses  in this  generalized  section,  model  i s  dichotomous  model i s very  and  testing  Q  2  2  Variable  in this  therefore  claim  1J.J.X Al XIIlll£iilX£Bl-X£Flil£linr2ki  =  Generalizing variable  +  2  (T|Y*-X*A+||  V Polytomous  we  and  t h e H c a n be t e s t e d  F(k,n-2k)  (10),  into eguation  rank(Qi)  Therefore,  (12)  #  (11)  Substituting rank  - ( X i ' X i + X 2 ' X 2 ) - M X i X 2 ) « (U,V) •  and  omitted.  we  variable  easy.  The  testing just bring  i t s parameters Let  us  assume  to  basic  are  a  polytomous  structure  mostly  out the  idea  estimations; that  the  the  on  same,  of  this  hypotheses categorical  variable  z h a s more t h a n two c a t e g o r i e s . The distribution of j j j j y (i) given z ( i ) = a i s n o r m a l , N ( X ( i ) ' S , v a r ( u ) ) , where a i s a scalar,  S  i s a vector  observations,  kx1,  X(i) i s a vector  a n d j=1#...#g# g p o s s i b l e  p  ij  =  Prob(z(i)=a  j  ) =  kx1;  responses  F(X(i)'R  j  i=1,..., on  n, n  z.  )  j where B  i s a vector  kx1.  Define a transformation  t(i,j)  as  21  t(i,j)  n IT  =  r (z(i)-a  j r ) / ( a -a )  r,j=1,...,g;  i=1,...,n  r=1  j H e n c e t ( i , j ) = 1 when a n d o n l y Now i)  when z ( i ) =a  , otherwise  t(i,j)=0.  we d e f i n e , Y  to  be  (t{i,j)y(1),  ii)  a  ngxl  vector,  ..., t ( n , j ) y ( n ) )  X t o be a b l o c k  *  matrix  with  *1  r  =  1  9  (Y ,  Y ) ; where  Y  j  f o r j=1,  diagonal  X  Y =  IX  0 *2  I I 0 I I i-O  X  ngxqk,  i.e,  T 0 j I 0 | I *gJ  ...  ... ...  ...  0  dimension  X  -«  *j for  each  X  ,  X  *j  r  =  | t(1,j) x  1  k  ... t ( 1 , j ) x  (1)  ...  I |  1  Lt(n,j)x  k | (n) ... t ( n , j ) x ( n ) J  g  1  iii)  S t o be a g k x l j  vector,  v e c t o r , S=(S  u  ,...,s  U t o be a n g x l  direct  and  S  i s  kx1  1j  S = (s  (n))*,  j »)*,  •,...,S  )'; j U )»; U  1  iv) j  1  (1) j I  v e c t o r , U= (U  j=1 .-•,g. #  product  1  The d i s t u r b a n c e  o f a gxg d i a g o n a l  =  j (u(i),  variance-covariance  matrix  D and a nxn u n i t  i s  a  matrix  22  I,  w h e r e d i a q (D) = [ v a r ( u  v)  p  ij  j  = Prob(z(i)=a  1 R As  3 »)' f o r B  before,  estimating  we  g ) , ..., v a r (u ) ] .  ) = exp(X(i)'R  9  {R  1  J1 (r ,  = can  two s e p a r a t e d  obtain models.  j  q s ) / 2 ~ e x p (X ( i ) ' R ) , a n d R = s=1 3* r )».  those  parameter estimates  Hence, t h e  conditional  by  model  is  *  Y = X S + U  Since when  z (i) (1-z(i))  = 0,  this  j  j Var+ (u ) =  =  are  orthogonal  * j * j * j j ( (X * X ) - * (X ) * Y  3 *j 1 J *3 (Y - (X ) • S+ ) « (Y - (X ) 5* 1  j C o v (S+ ) = logistic  L =  and X  i * j so  S  In  implies X  n  {(X  * j * j ) • (X ) ) - i V a r + ( u  p a r t , t h e maximum  g  "TT T T  i j (p ) t ( i , j ) ,  i=1j=1  likelihood  q  Z_  P  i j  =  j ) / (n-k) 3  )  function i s  J_  1,  Z: t  j=1  ( 1 , 3 )  =  1  .1=1  j n i we know T = Z » X t ( i , j ) i s s u f f i c i e n t f o r R g i v e n X. So j i=1 t h e MLE o f R c a n b e f o u n d b y m a x i m i z i n g L s u b j e c t t o t h e sum o f j j R f o r a l l j i s 0 , a n d R+ must s a t i s f y t h e e q u a t i o n s Also  n TL i=1  [exp(X  i  j g »R+ ) / Z I e x p ( X s=1  i s • E+ ) ]X  i =  n i 21 X t ( i , j ) i=1  23  He c l a i m t h e s o l u t i o n  log(L)  to this  problem  yields  Since,  q j 3 n g i j 27 T ' r - £ l o g ( ZT e x p (X * B ) ) j=1 i=1 j=1  =  m m a > l o g ( 1 ) / S B SB 2  1  g i j m i i iZ § X £ L X I i i E U- I j J X X H •27 j = 1 g i j m i=1 { 2 e x p [ X «(E - f i ) ] } j=1 n  Z  2  i  Hence t h e l o g ( L ) definite  a maximum.  i s concave because  f o ra l l i,  l  ( X X ' )  i s  positive  semi-  and e x p o n e n t i a l f u n t i o n i s p o s i t i v e .  Footnotes: 1  Constant  is  a variable  2  J.  Press  linear 3  term  always having  a n d M.  Nerlove,  and L o g i s t i c  J.  linear  a, i s c o n s i d e r e d  Press  * Group  data  Nerlove,  Models,  o f a x , where 1  x  1  1.  Univariate  Models, Dec.  a n d M.  and L o g i s t i c  value  as a p r o d u c t  and  Multivariate  Log-  1973, pp.10.  U n i v a r i a t e and M u l t i v a r i a t e L o g -  Dec.  1973,pp.5.  m e a n s many o b s e r v a t i o n s p e r c e l l ,  n » l o g ( L ) = - 2 : { z ( i ) l o g [ F { X ( i ) »C) ] + ( 1 - z ( i ) ) l o g [ 1-F (X ( i ) »C) ]} , F i s i=1 convex  and  function  of  log  function  l o g (F)  i s convex.  convex, but the negative log(L) 6  i s  convex  increasing  so  the  T h e sum o f c o n v e x f u n c t i o n i s concave.  i s concave.  see Box, Davies  a n d S w a n n { 1 9 6 9 ) Ch.4 p p .  38 - 3 9 .  composition functions i s Therefore  24  Chapter I I I  System The  b a s i c model  expanded chapter  in will  Because  proposed  every  dimension.  of the limited  as f o l l o w s :  of  equations  we  will  chapter  model  equations.  This  This  i s  will out  defined  just  discussed  i n the previous chapter.  interest  concentrates involving  variable density  z.  of these  interested example, of  z  will  be  two  are  not  dependent  we may  wish  t o analyze  woman i n l a b o u r  consider effects  distributed  In  then  y  the joint  problem  between y e a r s .  logistic  year  i n the we  joint  are  also  groups.  For  timing  not older  than  will  more  by y e a r  be  and  observe  L e t us s a y , y f o r  so w r i t t e n i n matrix  our  dependent  density of the  i t  we  categorical  between  but  of  as  model,  and  but  of  number  basic  interested  variables,  i n  System  d e n s i t y o f two  effects  system  a b a s i c model  the  only  1967-71,  this  normally,  model.  interesting  certain  f o r c e and h e r c h i l d  o f age w i t h i n a p e r i o d to  some  details.  variable  interaction  interaction i s  we  mention  into  be  the presentation  chapter,  can form  joint  the  interesting the  the  i n  a married  6 years  group  continuous  Here  basic  as a s e t o f r e g r e s s i o n or  groups  on  this  extension, called  a l l the  disjoint  variables  each  can  and the f o l l o w i n g  research,  s e t c a n be p a r t i t i o n e d and  of  b i t more i n t h i s  we  chapter  chapter  discuss a simple  models and l e a v e  equations  previous  scope o f t h i s  model a l i t t l e  following  extension  i n the  Model  discuss several extensions  is  the  of Equations  notation  given they  25  t  t ( i ) i z  t t t i s N (X ( i ) ' A , v a r ( u ) )  t  t { i ) | z (i)=0)  i s N(X  (y  (y  t  t t ( i ) «B , v a r ( v } )  and t Prob(z  t (i)=1)  = F{X  t (i)'C )  t  t ( i ) i s kx1 v e c t o r i = 1 , . . . , n . I n t t o r d e r t o e s t i m a t e t h e j o i n t d e n s i t i e s o f y ( i ) a n d z ( i ) , we t t t t t have t o e s t i m a t e A , B , C . , v a r ( u ) , and v a r ( v ) . L e t us s t a r t  for  t=1,...,d, d groups;  with  I  an easy  Estimation  This assumption those as  method.  eguation-by-equation  method  i s very  that the data  parameters  d  X  separate  simple.  The e s t i m a t i o n i s b a s e d  are independent  c a n be f o u n d  between  on t h e  groups.  Hence,  whole  problem  by c o n s i d e r i n g t h e  b a s i c models, and e s t i m a t i n g those  models one by  one.  In Hence,  most  of  this  kind  following case  II  of  section,  Dependence the  separated  data  among  i s  groups not  d i s c u s s a method  across  groups i s taken  are  correlated.  efficient. which into  In the  handles  the  account.  groups  previous chapter into  across  estimation  we w i l l  when c o r r e l a t i o n  In be  cases,  two p a r t s ,  we a s s e r t t h a t t h e e s t i m a t i o n c a n  because t h e j o i n t  density  function  26  can  be  factorized  parameter  sets.  into  So  i f  we  effects  between  groups,  logistic  part  the  A)  and  Interaction  two  f u n c t i o n s which  wish we  to  effects  consider  will  regression  depend  observe  part  the  those  on  disjoint  interaction  effects  on  the  individually.  between  groups  on  conditional  regressions:  Estimation: There  are  several  kinds  of  g r o u p s on  conditional regressions.  consider  a  chapter  special  4.  different  Let groups  assumption, are  not  the  in  efficient  to  us  This  we  will  section  discuss  the  by  "Estimating  more a b o u t  i t in  t  under  in this  For  proposed  an  Unrelated  generalized g r o u p t , we  t ( i ) «A  terms  eguation-by-eguation  Seemingly  eguations.  i s N(X  only  Hence  an  between we  disturbance  method a p p l i e s A i t k e n ' s  t ( i ) j z (i)=1)  effects  Z e l l n e r (1962) has  efficient.  called  this  correlated.  obtained  whole system of  t (y  In  that  highly  estimators  method  the  and  assume  are  general  Begressions". sguares  one,  interaction  leastknow:  t , var(u  ))  t t t t t (y ( i ) | z <i) = 0) i s N(X (i)«B , v a r (v ) ) t t t h e n (y ( i ) J z ( i ) ) i s distributed normally with mean t t t t t t t t X (i)»A z ( i ) + X ( i ) 'B ( 1 - z ( i ) ) a n d v a r i a n c e v a r [ u ( i ) z ( i ) ] + t t var[ v matrix  ( i ) (1-z algebra  (i) ) ]. then  So  i f we  write i n regression eguation  with  27  t ¥  = X  * t t * t »S + 0  t * t where X i s a nx2k m a t r i x , f o r each row o b s e r v a t i o n i , X (i)= t t t t [ X ( i ) *z ( i ) , X ( i ) * ( 1 - z ( i ) ) ] , Y i s n x 1 v e c t o r o f o b s e r v a t i o n s *t * t t t on t h e t t h g r o u p , U i s nx1 v e c t o r which U = £u (1)2 (1) + t t t t t t v (1) ( 1 - z ( 1 ) ) , u ( n ) z (n) + v (n) ( 1 - z ( n ) ) ] • , a n d t=1, d, d groups. J-1-J  So p u t t h e n t o g e t h e r , r  I ¥ | = J X I 2 | 1 J Y  |  |...|  1 d J L Y -»  i j - 1 - i  * 1  I  |  0 . . . *2 X ...  P  0 . . .  S+ =  Aitken's  d |  X-»«-S-«  generalized  (X'H'HX)-*X'H'Hy =  H i s an o r t h o g o n a l  var(S+)  =  ^  T  • I *d *- 0 (1)  1' d» (Y , . . . , Y ) * X i s a b l o c k - d i a g o n a l m a t r i x , i n w h i c h *1 *d 1* d' *1* i s (X X ) , S = ( S , . . . , S ) ' , U = ( U  diagonal *d* U ) ' t o apply  where  *  | | ... | ]...  *d| |  Y = XS + U  Y =  r  be  0 l | S | + | 0 I | 2 | I *2 0| | S | | U  ...  | «-0  or,  where  i twill  (X'l-iX)-!, Z-  matrix  least-square,  we g e t  (X«E~» X ) - i X'Z--y  such  that  E ( H u u * H')=HXH *=I, a n d  where  1  = V a r - (0) r 11 = |E I ... 1  I I d1 L  =  E  E  1d T I|  •••  I  ... I ...  dd | E IJ  (E+)- ! 1  * i j * i* j i * i* i j s i n c e (n-2k)E+ = ( n - 2 k ) v a r ( U ) = 0 " *U = (Y - X 'S ) ' (Y *j * j * i - X 'S ) , i , j = , . . . , d , w h e r e S i s e s t i m a t e d from the basic  28  models, S  =  (A  estimators  are  coefficients account  of  of zero  equations.  , B  ).  more a  Hence  efficient  Zellner  and  have t h e o p t i m a l  Hypotheses  testing:  homogeneous i n i t e m s  test  on c o e f f i c i e n t s  that  two  are  s = s  several  are considered.  test  statistic  the  2  = ...  =s  takes  occurring i n  other  out  that  these  the  groups  are  vectors.  d  hypothesis, suggested  but only that  the  a F-test as  d ( n - 2 k ) ( S + ) - * D ' r D V a r (S+1D!_/J-1DSJ: 2k(d-1)£Y« (E+)-iII-Y» ( E + J - i I X S + 3  of the restrictions,  ...  I ]0 «-0  procedure  in  this  by u s i n g  D = r l - I 0 IO I-I...  such t h a t  data  ways t o t e s t  c a n be e m p l o y e d  w h e r e D, t h e m a t r i x  the  pointed  One i s a s Z e l l n e r ( 1 9 6 2 )  F (2k ( d - 1 ) ,d ( n - 2 k ) ) =  estimating  regression coefficient  1  There  These  forecasting properties.  of their  H:  in  the Aitken  Huang(1962)  estimators  to  c a n be e s t i m a t e d .  1  because  single equation,  restrictions  We may w i s h  (E+)-  0 0  0 ... 0 ...  0 0  with  dimension  (d-1)xd.  0 0 0 0 |  I I -I 0 i 0 I-IJ  DS=0.  Another  method  i s t o u s e t h e maximum l i k e l i h o o d  ratio  test  29  which  leads to  the  has  been  stated  this  test  in this  If  in simple  from  the  we  the  and  are  the  general  attention  into  contingency this  table  section,  model of  we  several  have been  d o n e by  suggested  by  1)  Hence, the  each one  the  relation and  Nerlove  that  This the  Press  i s assumed  qualitative interest  a  effects  varying  bring  logistic  1  our  model  model. the  of In  general  because  model  vanish,  into  qualitative  will  that:  so  variables  log-linear  The  one  variable,  t h i s r e l a t i o n , nor  Press (1973).  have  jointly  dichotomous v a r i a b l e s ,  interaction  we  the our  of  and  introduced  continuous i f  standard  groups.  model:  explore  between  taken  asymptotic  qualitative  value.  the  estimators  m o d e l as  probability  discuss neither qualitative  may  applying  aggregation  group,  number  the  particular  analysis  we  any  a  order  basic  Therefore,  of and  and  a  Within  then  no  from i n d i v i d u a l  logit  test  B.  statistically  form  and  this  about  i s  have d c o n d i t i o n a l  case  Nerlove  a l l higher  there  be  variable  unordered,  on  A detail  b e t w e e n g r o u p s on  variables  takes  will  variables.  variables  variable  then  aggregation.  m o d e l we  i d e a of  i s shown i n A p p e n d i x  chapter.  continuous  dichotomous  chapter.  each group can  d qualitative  more  general  parameters estimated  previous  this special  The  i s true,  sample  effects  said,  conditional in  linear  to the  Interaction  in  hypothesis  entire  equivalent  As  i n previous  t h i s hypothesis  bias  B)  same r e s u l t .  which  they i s  30  2)  the  second  independent  3)  the  order  of the values  main  effects  exogenous e x p l a n a t o r y  Parameters  estimating the  IV  the  period  2  i s based  the  various  method.  on t h e F l e t c h e r - P o w e l l  and  the  of the matrix  Davidon  The  method  algorithm  of t h e second  discussion,  i n the regression of time,  handle  we c a n a l s o This  consider  for  derivatives,  consider  correlated.  since  we  i n regression.  t e s t and apply  i n our f i r s t  the For  the observations  solved,  problem  estimates  only  are  can be e a s i l y  autocorrelation  to get better  we  equations  a u t o c o r r e l a t i o n by c o n v e n t i o n a l  procedure*  of  b y t h e maximum l i k e l i h o o d  previous  how  apply  functions  3  correlated.  detect  and  matrix .  serially to  constant  discussion  disturbances each  are  variables.  the inverse  Further  linear  minimization,  information  In  are  algorithm  function  effects  o f any of t h e exogenous v a r i a b l e s .  are estimated  computational of  interaction  are know  We  may  T h e i l BLOS  stage,  then  we  Z e l l n e r ' s method t o g e t o u r r e s u l t .  Furthermore,  i t i s  no  model t o i n v o l v e  polytomous  variables.  chapter  generalize  In  our basic  model i s j u s t a  model  great  trouble  variables  2,  we  from  b i t beyond  have  t o generalize  instead discussed  of  dichotomous  about  dichotomous t o polytomous.  our  basic  model,  so  this  how t o This  everything  31  discussed variables.  i n  this  chapter  Therefore,  are s t i l l  applicable  polytomous g e n e r a l i z a t i o n  t o polytomous  i s omitted.  Footnote: T h i s assumption i n t h e computer program o f N e r l o v e and Press has now been eliminated and i n an updated r e v e r s i o n o f t h e program, h i g h e r order i n t e r a c t i o n e f f e c t s a r e p e r m i t t e d . 1  2  See N e r l o v e  and Press (1973),  A p p e n d i x A, e s p .  See Eox, Davies and Swann(1969), c h . and t h e r e f e r e n c e s c i t e d t h e r e i n . 3  H. Theil, Analysis", J. 1079, 1965. 4  4,  esp.  pp. pp.  92-94. 38-39,  "The Analysis of Disturbances i n Regression Am.. Statist,. Assoc. , v o l . 6 0 , pp. 1067  J. Koerts, "Some Further Regression Analysis", J Am. 169 - 1 8 3 , 1 9 6 2 . ±  Notes on D i s t u r b a n c e E s t i m a t e s i n Statist. Assoc. , v o l . 62, pp.  H. T h e i l , "A S i m p l i f i c a t i o n o f t h e BLOS P r o c e d u r e f o r A n a l y z i n g Regression Disturbances", J . . Am. Statist^ Assoc., v o l . 63, pp. 242 - 2 5 1 , 1 9 6 8 . J, K o e r t s a n d A. P. Procedure", J.. Am. 1236, 1968.  J . A b r a h a m s e , "On t h e P o w e r o f t h e B L O S Statist. Assoc,. , v o l . 6 3 , p p . 1227 -  32  Chapter IV  Model  I  Lagged  v a r i a b l e s model  We s u p p o s e only of  on  X.  on  ...etc.)  t h e dependent  v a r i a b l e s a r e dependent,  t h e c u r r e n t v a l u e o f x , b u t a l s o on t h e p r e v i o u s  economic of this  year.  bear  year  year.  variable,  and  (head's  as well  as the  economic  L e t us c o n s i d e r there  the very  i s  with respect  t o time.  t = a + bw  + b w  wage,  of  the  woman  will  of this  year  simple  case  multicollinearity  i n the  then  t +...  + e  t t t 1 (t-1) l o g (p / ( 1 - p } ) = r + s x + s x + ... where b In  i  = b*(d)  i  r e g r e s s i o n , we  ,  s  i  ,  i=1,2,...,  (t-1)  t = a + bw  (t-1) y  = s * (d)  have  t y  i  + bdw  (t-1) = a + bw  + bdw  + ...  t * e  (t-2) + ...  of  exponentially  So l e t w = x z + x ( 1 - z )  1 (t-1)  t  y  value  wife's  factors  not  depends  factors  We may a s s u m e t h a t a l l t h e c o e f f i c i e n t s  decrease  wife  that a married  upon t h e e c o n o m i c  assume  by  unemployment,  the probability  i s dependent  as l a s t  problem.  factors  Similarly,  a baby  as w e l l one  that  F o r e x a m p l e , t h e number o f h o u r s w o r k e d  the  last  Extensions  (t-1) + e  0<d<1  33  (t-1)  t y  - dy  t (t-1) (e - de )  t + b w +  = a(1-d)  Eguation  (1) c a n b e e s t i m a t e d e a s i l y .  part,  have  we  Similarly,  Model  In only  that  with  a  than  we may k n o w some p a r a m e t e r s  For  working  determine categorical married  and  of  number.  dependent  unordere'd.  We  may  two  events  3,  i n some c a s e s  qualitative  wish  head  and  the  a model w i t h  about  For  to a set of social logistic  polytomous  wife  case must  likelihood  constraints.  relationships  variables,  example,  the joint  factors.  function  variables.  a  may  variables  discrete  to relate  meaning  of parameters  the  woman i s e m p l o y e d o r u n e m p l o y e d ; s h e w i l l  not.  have  We know t h e maximum  inferences  dependent  will  we may c o n s i d e r s u c h  the  t o e s t i m a t e such  consider  jointly  example,  hours  a certain  Model w i t h j o i n t l y  may  t *• s x  c a n be e s t i m a t e d .  domain, o r i n t e r r e l a t i o n s h i p  i s suitable  He  will  function  constraint.  the total  method  i n the l o g i s t i c  constraints  some c a s e s  greater  III  the l o g i t  i n a certain  form  Similarly  t (t-1) (t-1) /(1-p ) ] - d*log[p /(1-p ) J = r(1-d)  t logfp  II  (1)  which  are both  hushand bear  a  of these  As d i s c u s s e d i n  c a n be c o n s i d e r e d a s o f such  of  a baby o r  probability  The s o l u t i o n  that  chapter several function  34  has  been p r o p o s e d  model  with  special  IV  by  Press  jointly  case  of  may  chapter  our  basic  systems.  regression  and  In  sguares  which i s proposed  method  is  P.  S c h m i d t and  likelihood model of  family  E.  by  will  factors,  such depend  and  expenditure  family  going  going  So on  on  i f we  logistic  upon t h e  hours  and  wish  model  considered  the  hours other  i s formed  apply  as  and  a  by  by  system  linear logistic least  has  been  using  This  solved  the case  problems  of  the  can  be  the  expenditure  w o r k e d by  the  w i f e , and  w o r k e d by  wife  factors.  The  joint  will  the  probability  expenditure,  we  may  of  depend  wife of  a  other  probability  whether  by  maximum  example,  on  two  Theil(1962).  Hany s o c i a l  t o know t h e  of  three-stage  i t i s a special  depend  i t s annual  composed i s formed  They a r e  consider  For  the  can  The  model.  v a c a t i o n and  Recursive  we  Press(1973).  i s  system  Zellner  vacation will  model.  V  other  by  and  also  be  system  Strauss(1974).  and  family  baby.  us.  approach  Nerlove  analyzed  the  model  One  r e g r e s s i o n system,  known t o  can  the  model  consider  eguations,  variables  Therefore  3.  simultanous-eguation  equations.  Nerlove(1973).  dependent  Simultanous-eguation  we  and  upon of  bears a  apply  a a  family this  35  This very  i s  the  new s t u d y  kind  of  area.  model.  discrete  most  interesting  T h e r e i s no f o r m a l  In this  variable  = f(X,z)  p = P r o b (z=11 y, X)  Schmidt  a n d R.  Strauss  how  t o maximize t h e l i k e l i h o o d t o be n o v e l  discussion this  m o d e l , a n d we h o p e  If  function.  a s an  in  z i s a categorical  regression  simultaneous either maxi-mum  by  be v e r y  Hence,  three-stage  method  number  of  degrees  sguares  i s an e x t e n s i o n  topic.  freedom of  They  step  apply (2)  system.  consider  following  i n  attacking  direction.  dummy v a r i a b l e s (4) we  or the f u l l  form  can  large.  a  solve  information  information  computational  two-stage  does  The  and  This  full  i s  but  (4)  t o obtain parameter  of  problem,  i t becomes  equations  The  and t h e  Their suggestion  we w i l l  i s an e x p e n s i v e  difficult  this  i n the right  least-squares,  method.  this  (3)  + dy  variable,  regression eguation  likelihood  likelihood will  (2).  J-i  initial  (3), then  w = l o g [ p / ( 1 - p ) ] = C'X  Since  variable  t o compute.  i ti s a step  we r e w r i t e e q u a t i o n  about  (2)  of this  and i s e x p e n s i v e  i s a  That i s  discussed  i tas a s t a r t  c a n be c o n s i d e r e d  This  z=0,1  have  only  seem  literature  = [ 1+exp (-C X-dy)  they  not  considered  proposed.  model, t h e continuous  are inter-dependent.  E(y|X,z)  P.  model  maximum  method, andi t  estimates  when  Three-stage  least-sguares,  the  least-  which  we  36  mentioned the  i n c h a p t e r 3.  I t i s more e f f i c i e n t  than two-stage  disturbances i n various s t r u c t u r a l equations are  Both  m e t h o d s a r e d e s c r i b e d i n many e c o n o m e t r i c  text  1  i f  correlated. books.  Footnote: 1  A.  Zellner  Simultaneous vol.  30,  pp.  and  H.  Estimation 54  - 78,  Theil,  "Three-stage,  of Simultaneous 1962.  (  Least-sguares:  Equations",  Econometrica  37  Chapter  A Study  on L a b o u r  V  Supply  o f M a r r i e d Women  Model D e s c r i p t i o n  I  Introduction The  empirical  especially J.  literature  f o rmarried  K o r b e l (1962),  Hof f e r (1973) ,  female  women, i s n o t m u c h .  J .  R.  on  M i n c e r (1963) ,  Freeman (1973) ,  labour  Related studies are  G. and  supply,  Cain (1966), E.  E-erndt  S.  and  T.  Wales (1974). In  this  participation determination five-year parts: five  of married  the first  yearly.  part  and  Our d a t a i s Research  Centre  were 2500 f a m i l y  unit  was  E.  71  was o f p a r t i c u l a r  and  a n d T.  f o r women  our  from  Panel  units  re-interviewed  As  This study w i l l  t h e second  drawn  labour  situations,  Study  randomly  be d i v i d e d  economic  part the  interest  force and t h e  20 a n d o v e r  using  i s t o study t h e problem University  of  Michigan  o f Income Dynamics (1972). chosen,  and  pointed out this  since  the  i n t o two  problem  a n n u a l l y o v e r t h e 1967-71  W a l e s (1974)  aged  the  i n t h e United States over  i s t o study  There  rate  women i n d i f f e r e n t  1967-71.  data,  Berndt  i s t o observe  of these s i t u a t i o n s  period  years  Survey  chapter, our study  the varied  national  each  family  time  period.  p e r i o d , 1967unemployment  considerably  3.7% i n 1968 a n d 1969 t o 5.7% i n 1 9 7 1 ; f u r t h e r ,  from  3.8%  toward  the  38  end  of  this  eliminating  II  period  an  increasing  discrimination  a g a i n s t working  was  placed  on  women.  S p e c i f i c a t i o n o f models , Although  s t i l l the  have  this the  study  The  i s  built  units  all  the  and  group  two  groups  upon c h a p t e r  which  i n each  of  a family  group  child  was 5 y e a r s  new b o r n  child  units  I I  constraint than  6  the family  assume t h a t  belongs I I .  Group I  the labour force  be l e s s t h a n  those  determined  differently,  II.  of married  contains younger, these  t h e number o f t h e age  family  In here family  i n  no  Then i n 196 9 - 7 1 ,  I o f 1967 a n d  we c a l l  has  child  had  1967-71.  T h e r e f o r e , group  activity  these  that  I , but  a  a  family  child  Hence, group  z - 1 , and group  i n group  o f these two groups  the  period  t o group  z t o b e 1 when a  i n which  year,  year, because  and  y e a r s , and 0 o t h e r w i s e . units  are selected.  Every  that  the  Suppose t h e youngest  o f 1970 a r e n o t d i s j o i n t e d . variable  part,  of 6 years or  f o revery  the time  t o group  belongs  part,  I t i s very obvious  within  family  timing  have a c h i l d  1967,  this  I n the f i r s t  units.  oldi n  family  they  two groups.  i s increasing.  and 1968, t h i s  will  family  i sfixed  1967  all  340 f a m i l y  parts,  second  are d i s j o i n t e d , but i ti s not true  t h e youngest  older  3.  I I contains the rest.  of  group  two  2 and i n t h e  on t h e s e  units  into  structure.  are partitioned into  family  elements  further  be d i v i d e d  upon c h a p t e r  a n a l y s e s a r e based  family  will  same e c o n o m i c a l  model i s b u i l t  model  be  emphasis  I contains  I I i s when z = 0 .  of the  wife  We w i s h  t o observe  i n  group  We I  how t h e  women i n l a b o u r f o r c e  d u e t o t h e same e c o n o m i c  not  will  factors( or  39  independent variables) . that  a  Also  family  has a c h i l d  same e x p l a n a t o r y  variables.  call  the c o n d i t i o n a l  logistic function,  I l l  Specification of  is  1  when  z=1  probability  o f age t o t h e  discussion  labour  will  e g u a t i o n s , and  function.  variable:  a family  has a c h i l d  o f z i s 0.  when a f a m i l y of z  led  because of our lack  work  less  not  not older  than  born c h i l d  statistical  when z = 1 .  because  whose  6 years z  otherwise  i f her c h i l d  like  to lock  after  her  This  i n  the  a  wife  a baby, but also  child  s h e may  defined  z=0.  Furthermore,  of having  age, o r even  of age,  i n 1970 a n d 1 9 7 1 , a n d  work.  who i s n o t o f s c h o o l  value  was  insignificance  of observation  only  i s z,  At t h e b e g i n n i n g  h a s a new into  variable  because o f h e r commitment t o h e r f a m i l y  I,  we  variables  most o f t h e w i v e s d i d n e t work will  6 years  eguations,  study, our categorical  the value  definition models  than  In the following  a probability  dependent  this  otherwise as  to relate the  variables:  Categorical In  not older  regression  the  Dependent  we w i s h  Suppose she has a  child  rather  i s  the timing  i s i n grade  than  to  work  outside.  Continuous Our force  dependent  variable:  continuous  variable  of a married  woman: t h a t  y,  i s her annual  worked  i n the hours.  labour  40  Independent At around of  variables;  the beginning, the following  wife,  head's  the explanatory  economic f a c t o r s : b i r t h income  unemployment o f head,  variables  including  fecundity,  are  centered  gap; predicted  income  from  and t h e r a t i o  of  wage  elsewhere,  incomes  over  needs.  BirthGap  —  B i r t h gap  Birth  gap  minus  actual  family  size  by  couple.  a  i s  i s the total  gap,  the greater  Wife  wg —  avoid  labour is  wage w o u l d having  force  correct  y,  the  the  oriented. to  as soon  a n d T. tend  probabilty They  would  their children  wage w i l l  completed and  decided  on t h e s u r v e y the larger  i s measured  data  the  of like  z=1.  on w o r k i n g  so  birth  than  with  to the a  higher  more and w o u l d  like  Therefore,  t o go b a c k i four  i t  i s  Following  positively  married  assumption  outside,  with  t h e same correlated  women a r e f a m i l y  a t home, o r g i v e n  t o work  t r y  to the  be p o s i t i v e l y c o r r e l a t e d  Many  t o work  according  A wife  worked by t h e w i f e .  assume t h a t  rather  expected  size  z=1.  o r she would  as p o s s i b l e .  the predicted  we c a n a l s o  Expected  Wales (1974).  t o keep  baby,  number o f h o u r s  argument, with  a  family  wage  wage o f t h e w i f e  Berndt  completed  t .  expect that  the p r o b a b i l i t y that  predicted o f E.  i n year  c a n be f o u n d  We w i l l  Wife e x p e c t e d  predicted to  data.  expected  number o f c h i l d r e n  These f i g u r e s  are actual  result  as  number o f c h i l d r e n  they  The  defined  more  care  o r t o work  a few  41  hours  f o r pleasure.  wage  of  the  negatively. between  Head  wife  Hence,  (Wife  introduces  The  will  cases,  correlated  and  Prob(z=1)  v a r i a b l e : the  Head's  i n those  with  wage p l u s  predicted  or  Prob(z=1)  (Wife  wg)  are i n quadratic  (Wife  wg)*  transfer  v a r i a b l e , t h e head's  the  y,  t h e r e l a t i o n s h i p s between  wg)  a new  inc —  Therefore,  i n our  and  y;  shape.  That  models.  income  income  i s  very  similar  to  the  i  predicted  wage  of  the  variable  includes  pensions,  incomes from  workmen's sources  She  a  the  wife  just  necessarily  and  have  a high  case, work  of  i n order z  will  —  The head ( g i v e n  the wife  work  and  more,  head's  i n the  support.  A l l the  called  transfer  income.  guickly.  income work  low  days),  unemployment(given  On  i s high, less,  because a high  and  or  head's  put l e s s  the  that  o r have income  not  higher  family  into  stable.  other  does a  wage.  time  i s low.  may  In  this  her  house  Hence  (Head  t h e unemployment  of the  models.  unemployment  relationship i n  unemployment  or c h i l d  force  a  This  children,  when h e r f a m i l y i n c o m e  will  a baby,  too.  dependent  security,  are  more  to  t o make h e r f a m i l y e c o n o m y  i s included  Unemploy  alimony  family  t r a n s f e r income  the wife  social  labour  the  having  aid  wage  has t o work  mean t h a t  probability  wage,  head's  because  I t v a r i e s i n U-shape  welfare,  would t r y t o r e - e n t e r  hand,  inc)  head's  compensation  except  Usually  wife.  between or  y  between  and  Prob(z=1)  i n weeks), i s unexpected.  and  the  Normally  head's we  would  42  think  when a h u s h a n d  more a n d  leave  the family a f f a i r  negative  result  (Appendix  C).  within  high  in  behind.  our  This  town  to another  own  town.  town  This  reason  will  cause the wife  vacation.  less.  we  that  form.  Therefore,  Fecundit  —  data  Fecundity wife and  at time y,  or  (Fecundit) work a  and  high  2  This  I  on t h e p u l l e d  may  economy be  will  we  add  (Unemploy)  i n the  wives t o  this  work  move f r o m  one  a job i n  his  her j o b .  Sometime  opportunity to  to  work  t h e number o f h o u r s  o f t h e head i n d a y s ,  more o r l e s s 2  bad  workers with  also cause h i s wife using  data  was  explained  for their  would take  distributed  as another  in a  new  we  guadratic  variable.  fecundity  i s an age  t.  get a  be s e a s o n a l  the unemployment  are  Wife's  U n e x p e c t e d l y , we  to lose  p l o t t e d out the data,  against  found  work  because t h e head can not f i n d  have a l o n g e r  wife  to  i s a f a m i l y may  when a p e r s o n c h a n g e s h i s j o b , h e  worked by  may  that i t i s not necessary possible  would have  entire  outcome  Some o f t h e h e a d s  Another  when  model  net because the  1967-71.  ways.  wage s u c h  more.  testing  I t was  the period  following  i s unemployed, the wife  we  consider  between i s added  variable defined  the relationship  fecundity  and  to  bear  between  Prob(z=1)  because a younger  better physical ability probability  a s 45 m i n u s  woman  t o work  more,  i s  has  t h e age  fecundity guadratic.  less  family  but she does  a baby, which f o r c e s  of  her t o  have work  less.  Inc/need  --  ratio  of t o t a l  incomes except  wife's  wage o v e r  needs  43  Incomes income  per needs  minus t h e w i f e ' s  1  Even i f t h i s family  has  less.  ratio  family  P r o b ( z = 1)  size. in  (inc/need)  2  a  likely,  non-linear  divided not  by  family  net  the family  always  mean  real  needs . 2  that  the  t h e n u m b e r o f d e p e n d e n t s may  i f the r a t i o  Most  i n our  Finally  i t does  income, since even  as t h e t o t a l  wage a n d  i s high  a high  Similarly,  large  i s defined  i s low t h a t  this  ratio  pattern,  so  may  be c a u s e d  varies we  be  with  also  y  by or  consider  model.  i n o u r m o d e l s , we  have  11  explanatory  variables.  They a r e : 1.  BirthGap  birth  2.  Wife  predicted  3.  (Wife  4. 5.  wg)  2  Head i n c (Head i n c ) 2 Unemploy  7.  (Unemploy)  8.  Fecundit  9.  (Fecundit)  of Wife  head's  income  square  o f Head i n c  sguare  2  of  fecundity — —  wg plus  income  Unemploy  -—  ratio  square  of t o t a l  incomes except  needs  2  square  inc/need  sample  i s obtained  Data  restriction  Research  transfer  of head  wage o v e r (inc/need)  Survey  2  inc/need  data  square  wife  fecundity  11.  The  wage o f  unemployment  6.  10.  IV  wg  gap  Centre,  Panel  of  from the U n i v e r s i t y S t u d y Of I n c o m e  of  wife's  Michigan  Dynamics  (1972)  44  which  i s based  analysis 1.  of  on  the  husband  all  5 years,  1967-71.  2.  The  head  married  3.  The  husband  the  5  4.  The  wife  was  not  5.  The  head  was  less  6.  The  birth  cases  so  sample. those  that  marriage.  i s  alone  while  Restriction married  to  interesting, the  the  first  i n the  first  least  50  in  our  household  in  time.  350  hours  was  these to  years  old  i n each  of  be  e l i m i n a t e a l l the on  a  c o n s t r a i n t s are some c a s e s  have  children  used the  wife  from  the  dependent  upon  ensure the  stability  or  she  works. to  interesting c h i l d r e n are This  prevent  i s case  millionaire. data  are  not  of  families, The  family  i f the  that  have  earlier  earning.  family Hence,  economy.  wife to  is be  a young g i r l  observations  This  of left  constraint  Constraint 6  meaning.  will  the  usage of  such  eliminate  the c h i l d r e n  o l d enough  Such  biased.  statistical  his  only  the  that of  reliable  to  f a m i l y economy.  most  special  more  for  the  the  years.  In  the  has  in a l l 5  based  In  minus  couple.  or  gap  to  size  1971. 1971.  the  age,  because  positive  in  c o n s t r a i n t s ensure  to  eld  age  expected  used  two  of  in  two  mainly  5 i s used  45  years  family  and  i s valid  an  than  are  responsible  birth  data  following constraints:  than  analysis will  3 i s used  bearing  older  households.  is  child  restrict  were p r e s e n t  f o r at  the  before,  observation  that  gap,  f a i m l y belong  constraint An  our  Hence  expenditure  the  f o r the  family size,  Suppose  head  worked  constraints  married  the  was  wife  He  years.  abnormal  been  and  using  The  These  the  surveys.  m o d e l s by  actual  in  5 annual  i s to  leaves  are  4. i s not  ensure us  with  45  1700  observations  units  f o r the 5 years  period:  that  i s  340  family  in total.  Footnotes:  The  1  total  family net real  income minus t h e c o s t the in  2  family unit, A Panel  The  need Income  Family  of earning  i f there  Study  income  Of I n c o m e  in  Dynamics )  year  as the t o t a l  income, minus h e l p  are c h i l d r e n Dynamics  needs i s adjusted  standard  i s defined  under  18. ( s e e  outside  definition  )  a c c o r d i n g t o t h e US  t.(see  from  real  definition  annual  i n A Panel  living  Study  Of  46  Chapter Empirical  This chapter family the  model 2.  the  have  women  sample i n t o  two  i n the  than  years  of  put  into  we  which  group I I .  observations,  that  z=0)  has  815  observations.  is  much l e s s  II.  This  older  than  I.  has  a  6 years  from  Table  bigger  Suppose  we  wife's  wage  wife; in  (about  then  group  s h o w n on  number o f 4 7%)  supports  Results  From  the  of  put  group  From  regression  that term  the  curve  I , the  by  a  that  the  i n group  wife with  that  in a  I  group  child  not  equations:  the  curve  the  observe  wife  a  eguation  of  equation  of  the  hours  hours  group  let  worked  by  upward  From t h e worked  I  group I I .  and  i s concave  i s concave downward.  know  I;  condition  with  i n group  group  older  the  sguare  find  not  the  II(under  variables fixed  vary  of  condition  explanatory  i t s  split  hours.  labour  than  of  the  t a b l e I I , we  work l e s s  timing  I(under  hours worked  will  340  age  i t into  wife  find  we  we  in  i s  a child  that a  I , we  table I,  and  that  f o r c e , we  the  assumption  labour  the  to  by  and  I I ,  labour  worked  the  constant  i n the  those  age  keep a l l t h e  we  than  our  in total,  Hence, group  885  described  c o n s i d e r i n g the  i s z=1,  has  average,  model  I f a f a m i l y has  z=1)  the  In  I  the  according  that  on  using  5 years.  family.  age,  Model  observations  categories  child  others  of  participating  youngest 6  by  1700  u n i t s i n a l l the  married  Results  i s estimated  We  VI  by  the the but  figures those  47  housewives normally more  having  will  only  ambitious  not  be  when to  children  a f f e c t e d by  their  work.  of  In  the  labour  equation  2)  the  head's  wage p l u s  needs  and,  statistically hours  hand  work of  and  by  in  the  group  sguares of  1 ) , 2)  and  Hypothesis  testing  —  the  eguation  above  has  the  eguations  two  equations.  that  is  reject  II  the  Results  From  far  are  Table  3 ) , and  the  hours  1)  the  5)  we  i n order  From Chow's t e s t ,  the  are  hours.  wife's  wage,  the  of  incomes 3)  are  number  of  variables. by  the  On  the  wife  is  n e e d s , 3)  f e c u n d i t y , 4)  square of  the  two  labour  test find  wife's  that  the  wage.  each now  labour  we  assume  e q u a l i t y of  F (12, 1676)  significance  value  the  equations  s t r u c t u r e , but to  6  wage p l u s t r a n s f e r  observe  we  who  of  ratio  of  head's  the  work  1 ) , 2 ) , and  worked  e q u a l i t y of  95%  the  explanatory  characteristic  beyound  1)  that  age,  older than  find  regression  incomes over  egual,  f o r those  number  of  of  will  a certain  squares  discussion,  hypothesis  from  by  the  i t s own  to  I , we  i n the  the  of  years  They  high, except  up  the  I I ,  ratio  the  6  wages.  t r a n s f e r i n c o m e , 3)  on  income,  In  group  wife  affected  2)  only  4)  significantly  their  than  w i v e s whose c h i l d r e n a r e  significant  worked  other  will  older  wages a r e  Those  years  over  age,  not  =  these 11.7861  ( F = 2 . 3 0 ) , so  we  of e q u a l i t y .  probability  I I I , we  find  the  equation  birth  gap,  the  head's  income.  48  fecundity  and  significant  effect  good  outcome  expected the  incomes  over  sguares  5. and  Further  If  as  we  we  consider  obtain  with  Group  I,  This  i s  variables  the r a t i o  of  turn  incomes  strongly a  out over  wage, do  very as  we  need, sightly  z=1.  of  the  the  head's  income,  unemployment t o speed  up  maximum  likelihood  difficult  to  those  of  the  the  ratio  head,  of  and  the  the convergence rate  in  function;  otherwise  significant  variables  obtain.  from  statistically  the r e s u l t s  variables,  we  those variables.  Begression  a  estimation:  reliable  only  of  of these v a r i a b l e s ,  convergence i s very  o f z=1.  have  the square of the wife's  needs,  the  fecundity  major  Also,  the units  maximizing  which  these  the p r o b a b i l i t y  scale  of  the p r o b a b i l i t y  because  wage  He  I l l  square  on  i n chapter  wife's  affect  the  model  ( y  —  | z =  would The  Labour  of the estimation like  of t h i s  to re-estimate  following  a r e t h e new  the  model model  estimates.  eguation  1 )  HourWork =  1193.0 -  633.7 (Wife  0 . 2 5 x 1 0 - 5 (Head  inc)  2  wg)  +  182.0 ( l i f e  + 5.789 ( i n c / n e e d )  wg)  2  -  0.1772 (Head i n c ) +  - 0.0034 ( i n c / n e e d )  2  49  Group I I , ( y  | z = 0 )  HourWork = 792. 1 - 35.57 (Wife inc)  2  wg)  + 6.197 ( i n c / n e e d )  Logistic  model —  -  0. 1 6 7 6 ( H e a d  inc)  - 0.0032(inc/need)  Probability  +  0 . 2 2 x 1 0 ~ s {Head  + 42.15 (Fecund)  2  eguation  L o g i t {p+ ( i ) ) = - 1 . 8 8 3 - 0.3412 ( B i r t h G a p ) +  0.0494 (Head  0.0037 ( F e c u n d )  inc) 2  -  + 0.5426 (Wife  0.1616 ( i n c / n e e d )  wg) +  - 0.1556 ( W i f e  wg)  0. 1 8 9 5 ( F e c u n d i t )  2  -  Table Parameter  Estimates  Group I ( y J z =  Variable  1 )  I  f o r Labour  Equations  Group I I ( y | z = 0 )  Whole s a m p l e y unconditional  constant  1699.6306* (73.384)  472.9470* (21.344)  1180.0698* (71.507)  BirthGap  -44.7253 (1.378)  -36.7626 (1.708)  6.6327 (0.392)  Wife  wg  - 7 0 4 . 2 5 4 3* (3.274)  364.1801 (1.774)  -130.0699 (0.863)  wg)2  190.1427* (3.896)  -114.0060* (2.582)  31.8822 (0.958)  -0.2082* (14.829)  -0.1744* (13.536)  -0.1949* (21.289)  0.2918x10-5* (9.931)  0.2348x10-5* (7.248)  0.2617x10-5* (13.436)  3.0143 ( 1 . 176)  -0.7786 (0.275)  1.4487 (0.749)  -0.0220 (1.403)  -0.0123 (0.666)  -0.0188 (1.547)  6.6705* (8.978)  6.1337* (14.408)  6.1123* (17.821)  -0.4145x10-2* (4.455)  -0.3154x10-2* (8.795)  -0.3184x10-2* (9.775)  -12.6039 (0.629)  43.2141* ( 3 . 130)  5.2290 (0.469)  (Fecundit) 2  -0.5377 (0.785)  -1.7835* ( 3 . 309)  -1.1625* (2.833)  Observations  885  815  1700  R2  0.2538  0.3201  0.2789  (Wife  Head i n c (Head i n c ) 2  Unemploy (Unemploy) 2  inc/need (inc/need) 2  Fecundit  *  significant  Asymtotic  level  t values  o f 5% are i n  under  H:  parameters =  parentheses.  0.0  Table Mean a n d  Standard Deviation  Group I ( y | z =  Variable  I I  1 )  of the Model  I  Group I I ( y | z = 0 )  Whole sampl y unconditi  HourWork  771.519 (792.743)  1133.79 ( 7 6 1 . 958)  945.196 (798.696)  BirthGap  0.3785 (0.7627)  0.6025 (1.4130)  0.4859 (1.1277)  Wife  wg  1.8497 (0.5728)  1.9598 (0.6316)  1.9025 (0.604)  (Wife  wg)  3.7491 (2.4977)  4.2393 (2.8939)  3.9841 (2.705)  8297.75 (4569.88)  8933.23 (5243.44)  8602.40 ( 4 9 1 3 . 16)  0.8 97x10« (0.209x109)  0.107x109 (0. 175x109)  0.981x108 (0. 194x109)  5.1388 (21.5523)  4.1931 (19.8563)  4.6854 (20.7558)  490.385 ( 3 5 0 9 . 11)  411.371 (3028.43)  452.505 (3286.73)  285.906 ( 1 3 2 . 113)  332.063 (173.459)  308.034 (155.012)  0.992x105 (0.108x106)  0.140x10* (0. 184x106)  0. 1 1 9 x 1 0 6 (0.151x106)  15.9503 (5.8996)  11.1767 (6.9893)  13.6618 (6.871)  289.177 (175.181)  173.709 (189.114)  233.821 (190.871)  885  815  1700  2  Head i n c (Head i n c ) 2 Unemploy (Unemploy)  2  inc/need  (inc/need)  2  Fecundit (Fecundit)  2  Observation Standard  deviations  are i n  parentheses.  Table I I I  Probability  Function  E s t i m a t e s o f t h e Model I  Coefficient  As ym  constant  -1 . 9 4 2 2 2 *  0. 3 0 7 9  6. 3 0 8  BirthGap  -0 . 3 3 5 2 6 *  0. 0 4 1 2  8. 134  Wife  0. 5 0 0 6 3 *  0. 2 5 5 7  1. 9 5 8  -0 . 1 4 7 1 4 *  0. 0 5 6 4  2. 60 9  0. 0 6 9 8 1 *  0. 0 1 5 8  4. 4 2 9  -0 . 0 0 0 5 0  0. 0 0 0 3  1. 5 0 5  0. 2 0 5 5 8  0. 3 2 5 7  0. 6 3 1  -0 . 0 9 5 7 7  0. 1 9 9 9  0. 4 7 9  -0 . 1 7 2 3 9 *  0. 0 7 4 7  2. 3 0 7  0. 0 0 0 1 9  0. 0 0 8 4  0. 0 2 2  0. 1881 1*  0. 0 2 1 6  8. 7 2 6  -0 . 0 0 3 5 7 *  0. 0 0 0 8  4. 6 2 7  Variable  (Wife  wg wg)  2  Head i n c (Head i n c ) 2 Unemploy (Unemploy)  2  inc/need (inc/need)  2  Fecundit (Fecundit)  Log  2  of likelihood  * significant  function  level  stdv  = -947.522  Asym  after  11  o f 5% u n d e r H: p a r a m e t e r s =  iterations. 0.0  note: 1 Head  inc =  1 inc/need  $1,000  = 100  1 Unemploy = 1  (unemploy)  1 1  (Head i n c )  (inc/need)  100 d a y s o f h e a d ' s 2  =  10,000  days.  2  =  2  =  $1,000,000  10,000  unemployment  53  Chapter Empirical Before annual  hours  general, age  we  worked  those  will  wives  work  In  comparing  in  group  group  begin  I I .  R e s u l t s of  by  wives.  From t a b l e  whose c h i l d r e n  fecundities younger  These  Model  II  to analyze the r e s u l t ,  more h o u r s .  I are  VII  The  we  and  V,  are older  range  i s from  know t h a t their  on  we  consider  observe  than  6  with the  less  in  years  of  of  57.95%*.  those  than  findings  the  that  30.17% t o  the average  wages a r e  are c o n s i s t e n t  l e t us  wives  those  in  Berndt  and  wales (1974).  I  R e s u l t s of From  income,  table and  significant significant rest we  of  the  IV,  the  that  head  and  will  find  is  upward.  to  us  find  i n most o f in  any  that  of  the  the  years,  year.  The  functional  they  have  variables  i n 1968,  the  An  increases  i n two  that the  variable  vary  curve  extremes. will  be  will  i s  significance  Suppose  1968  upward  gap  the  with  we  A downward meaningful  If  we  will  keep  a l l  1969;  we  then in  us that  curve only  the  of  give  point out  of  unemployment  downward, but will  not  years.  year-by-year,  except  i s concave  or  birth  of  transfer  statistically  in different  shapes.  compare  curve  upward  the  degrees  constant  and  Downward  but  plus  needs are  structures  different  i t s sguare,  h e a d ' s wage  incomes over  the  interpretations. big  we  ratio  explanatory  the  eguation estimation  the explanatory v a r i a b l e s  compare  observe  single  i t  different there  will in  1969  a  are  indicate certain  54  domain,  because  dependent  each  testings —  the equality  hypothesis  eguations  hypotheses  point are  relation  for  what t h e v a l u e  a 5$ c r i t i c a l  are  acceptable;  not acceptable.  we c a n a g g r e g a t e  single  that  a l l the year  using  equation,  obtain  the  same  data  o f t h e two  a l l other  and  a linear  least  i s not a p p r o p r i a t e . m e t h o d , we  labour  years  a  the  u s t h a t f o r 1967 o r express  the  labour  function, regardless of  squares  method  the weighting  The r e a s o n  i n  i s that  use t h e l e a s t - s q u a r e s  a  i n an  procedure  the parameter e s t i m a t e s , but a l l v a r i a b l e s are qiven  weiqht.  This  weiqht  we  the  two  stage  find  that the  use  Z e l l n e r {1962);  then  obtained  at least  squares.  i n  h e t e r o s c e d a s t i c i t y i s present, then  to  obtained  eguations  of zi s .  equation-by-equation  Suppose  of the  p o i n t , a n d i n 1970 u s i n g  These t e l l  Results of Z e l l n e r ' s seemingly If  o f t h e two l a b o u r  t e s t i n g s of the equality  i n 1967 u s i n g  1% c r i t i c a l  II  values  year;  The  1970  i n negative  variable.  Hypothesis in  we a r e n o t i n t e r e s t e d  are  we  i s unsatisfactory i n A i t k e n method regression  our  as i n t r o d u c e d parameters  a s y m p t o t i c a l l y more e f f i c i e n t  by a n e q u a t i o n - b y - e q u a t i o n  method  usinq  sample.  than  ordinary  by so  those least-  55  The  comparison  from  single In  the is  of  eguation  table  deviations from  10% t o  quadratic  those  from  method  we  the r e s u l t s  20%.  He Such  ordinary  but  some  of  the  wife's  For  i f  we  keep  equation  from  two  the  result  wives  in  wage a n d  the  h a n d , i f we  use  the  we  unemployment from from is  the in  tested  results two  a  this  their  from  similar table.  stage estimation, It  E,  in  i s  expected  the  but  H e n c e , i f we  adopt  will  those  say  that  will  work  more i n b o t h  difference  the  can  none o f t h e h y p o t h e s e s  head's  easily  significant  we  extremes,  the be  be  ether  estimation,  in  the  not  On  will  like  the  downward,  stage  not  fixed  from  two  Another  different  find  wages.  examples  labour  unemployment.  the  Such  ordinary  variables  will  upward.  result  economical  there are head's  we  the  the  t h a n 6 y e a r s o f age  of other years.  i s acceptable.  that  e q u a t i o n , we  that those wives find  we  Aitken's  of the  the curve i s concave  are older  work more by  likewise,  so  affect  the  i t s square,  to  because  stage  a l lthe independent  stimulated  observe  not  VI  wage a n d  single  result  the r e s u l t  Suppose  s t a g e method i t i s concave from  possible  I n t h e two  do  table  method t h a t  whose c h i l d r e n  will  different  are  range  in  there are apperently  than  do.  interpretation  single  results  reduction  Most of t h e changes  wife's  the  e s t i m a t o r s a r e more e f f i c i e n t t h a n  different  II  the  that  The  weights to the sample,  group  except  with  for  differences  different  interpretation,  example,  method  show a s i g n i f i c a n t r e d u c t i o n  least-sguares.  somewhat  of  find  stage Aitken's  least-sguares.  equations  this  of estimated parameters.  assign  be  from  estimation  VI,  forms.  e x p e c t t h e two  should  the r e s u l t s  found  difference of  single  eguality equation  56  estimation  Testing  w h e r e we  for  every  we  are  H:  use  labour  coeff  =  labour  an  estimated  coeff  testing  group of  Therefore,  involved  The  67  i n s o f a r as  the  conclude  reduction  i n the  deviations of  Zellner*s  two  be  summarized  as  that  the  data  in  coefficient  =  ...  the  =  the  coeff  method ratio  t e s t we =  that  there  71  suggested test.  find  3.4044 =  of  Here  that and  58.4047. i s an  by  for  we the  for  the  Both  are  aggregation  aggregation.  eguations:  discussion  can  consider  regression  I I , F(48,1640)  about  stages  l e t us  asserts  likelihood  From  in single linear  labour  68  I , F (48,1640)  group we  of  methods:  approach. of  model,  is  F - t e s t , and  equation 2  The  two  F-test  rejected . bias  of  Model I I :  logistic  homogeneous That  equation  for  testing hypothesis  concerned.  using  the  Bias  i n t o the Our  are  There are Zellner  go  bias.  year  vectors  h a v e some s t a t i s t i c a l s i g n i f i c a n c e .  Aggregation  Before aggregation  do  shows  estimation follows:  that  estimated method.  there  is a  parameters The  estimated  significant by  using  eguations  57  Group  I, ( y  I z =  1 )  1967: HourWork =  -24.88  + 206.3 ( B i r t h G a p )  0.2742 (Head  inc)  0.5963 ( U n e m p l o y )  0.621x10-» (Head +  2  2 0 . 42 ( F e c u n d i t ) -  + 360.0 (Wife inc)  wg) +  2  10.54 ( i n c / n e e d )  - 50.99 ( R i f e  -  wg)  -  2  3 7 . 26 ( U n e m p l o y ) 0.0095(inc/need)  +  2  1.512 ( F e c u n d i t ) *  1968: HourWork =  719.5  + 58.64(BirthGap)  0.1846 (Head  inc)  0.0856 (Onemploy)  -  wg)  0.242x10~s(Head  +  2  4.671 ( F e c u n d i t )  +  + 54.45(Wife  6.638 ( i n c / n e e d )  1.278 ( F e c u n d i t )  + 8.394 ( W i f e  inc)  -  2  + 6.746 ( U n e m p l o y ) -  2  -  wg)  0.0031 ( i n c / n e e d )  +  2  2  1969: HourWork =  1074.0  + 4 4 . 85 ( B i r t h G a p )  -  240.5 (Wife  0 . 2 0 5 4 ( H e a d i n c ) + 0. 2 6 7 x 1 0 - s ( H e a d i n c )  -  2  0. 1 8 4 6 ( U n e m p l o y ) 52.76 ( F e c u n d i t )  +  2  9.697 ( i n c / n e e d )  + 0.3216 ( F e c u n d i t )  wg)  + 81.98 (Wife  wg)  2  5. 1 5 3 9 ( O n e m p l o y )  -  0.0073 ( i n c / n e e d )  +  2  -  2  1970: HourWork =  1413.0 - 92.92 ( B i r t h G a p ) -  0.2830(Head  inc)  0.0290 ( U n e m p l o y )  2  6 5 3 . 0 («ife wg)  +  157.1 ( W i f e  wg)  +  0.470x10~ (Head  inc)  2  +  10.68 ( i n c / n e e d )  -  0.0074 ( i n c / n e e d )  5  2  + 4.514(Unemploy) 2  -  58  15-17 ( F e c u n d i t )  - 0.8335 ( F e c u n d i t )  2  1971: Hourwork =  1330.0 -  58.39 ( B i r t h G a p )  -  4 7 3 . 8 ( W i f e wg)  0.2869 (Head i n c ) + 0 . 6 4 8 x 1 0 ~ s (Head 0.0239 (Unemploy)2 19.41 ( F e c u n d i t )  Group  I I , ( y  +  -  8.738 ( i n c / n e e d )  0.2104(Fecundit)  | z =  inc)  +  +  2  -  130.3 ( W i f e  wg)2  -  4.826 (Unemploy) 0.0062(inc/need)2  -  +  -  2  0 )  1967: HourWork =  1704.0 + 6.748 ( B i r t h G a p )  0.2643 (Head  - 789. 1 (Wife  wg)  100.5(Wife wg)  inc)  +  0.433x10~s(Head  inc)  2  0.0102(Unemploy) 2  +  10.37(inc/need)  -  0.0067(inc/need)2  29. 1 1 ( F e c u n d i t )  -  1. 6 9 2 ( F e c u n d i t )  -  2  1.174 ( U n e m p l o y )  +  2  1968: HourWork =  1207.0 -  11.95 ( B i r t h G a p )  -  524.5 ( H i f e  0.2294 (Head i n c ) + 0 . 3 6 3 x 1 0 - 5 (Head i n c )  2  0.0541 ( U n e m p l o y ) 23.43 ( F e c u n d i t )  1969: HourWork  2  +  +  11.08 ( i n c / n e e d )  0.2166(Fecundit)  2  wg) -  + 7 3 . 1 8 ( W i f e wg)  2  0.2577 (Unemploy) 0.0077 ( i n c / n e e d )  +  2  -  59  =  -61.86  - 53.81 ( B i r t h G a p )  0.1750 (Head  inc)  0.0385 ( U n e m p l o y ) 75. 9 0 ( F e c u n d i t )  2  + 4 7 9 . 7 ( W i f e wg)  -  135.4(Wife wg)  +  0.257x10-s(Head  inc)  2  +  7.854 ( i n c / n e e d )  -  0.0050 ( i n c / n e e d )  - 3.208 ( F e c u n d i t )  -  2  + 6.129(Unemploy)  +  2  2  1970: HourWork =  47.04  - 34.05 (BirthGap)  + 590.2 ( w i f e  0. 1 5 4 8 ( H e a d i n c ) + 0 . 1 9 9 x 1 0 ~ s ( H e a d 0.0035 (Unemploy)  5.528 ( i n c / n e e d )  2  7 4 . 95 ( F e c u n d i t )  inc)  - 3.702 ( F e c u n d i t )  wg) -  2  -  -  146.9 ( W i f e  wg)  2  -  3.178 (Unemploy)  0.0025 ( i n c / n e e d )  +  2  2  1971: HourWork = 839.7  -  112. 1 ( B i r t h G a p )  0. 1 3 6 8 ( H e a d  -  104.1 ( W i f e  wg)  + 8.551 ( W i f e  inc)  +  0.176x10-s{Head  inc)2  0.0163 (Unemploy)2  +  4.813 ( i n c / n e e d )  -  29.40 ( F e c u n d i t )  III  Results  -  From  table  all  the  probability  and  1969  transfer  V I I we  find  affected  by  income, f e c u n d i t y  gap,  + 3.280(Unemploy)  -  0.0021 ( i n c / n e e d )  +  2  functions  the birth  functions.  are i n t e r e s t i n g .  1969, t h e f u n c t i o n  -  1. 0 3 0 ( F e c u n d i t ) 2  of p r o b a b i l i t y  significantly  wg) 2  birth  the  over  needs, f e c u n d i t y ,  The  i s very  significant  r e s u l t s i n the years  i n 1968  I n 1968, t h e p r o b a b i l i t y f u n c t i o n the birth and  i s affected  the  gap  the by  gap, t h e head's  square  most  wage  plus  fecundity.  In  of t h e v a r i a b l e s , such  head's unemployment, the wife's  of  i s  wage a n d  the ratio the  as  of incomes  sguares  of  the  60  head's  unemployment  surprised not in  t h e wage's wage.  us i s t h a t t h e h e a d ' s  significantly  affect  most o f t h e y e a r s  draw any f r u i t f u l and  and  thing  wage p l u s t r a n s f e r  the probability  except  One  has  incomes  does  f u n c t i o n t h a t happens  1968, and 1971.  c o n c l u s i o n from  which  Moreover,  the results  we d o n o t  of the years  1970  1971.  Test  f o r aggregation In  part,  considering  we f i n d  s q u a r e (48) that  there  =  test  of  0.39x10 ) 9  probability  as  of the aggregation  maximum that  i s no a q g r e g a t i o n  He c o n c l u d e year  the  the ratio  The e s t i m a t e d  each  bias;  likelihood  bias f o r this  i s  we c a n n o t a c c e p t  so  b i g (Chi  the hypothesis  bias.  functions:  the estimation  of  probability  functions  for  follows.  1967:  L o g i t (p+ ( i ) ) = - 2 . 4 2 8 - 0.4167 ( B i r t h G a p ) +  + 0.3549 (Wife  0.0651 (Head i n c ) + 0 . 4 8 8 x 1 0 - * ( H e a d i n c )  0.0005 ( u n e m p l o y ) 0.2835 ( F e c u n d i t )  2  -  0.0221 ( i n c / n e e d )  - 0. 0 0 7 3 ( F e c u n d i t )  -  wg) 2  - 0.0962 ( W i f e  wg)  + 0.0283 (Onemploy) 0.0281 ( i n c / n e e d )  2  2  +  2  1968: Logit (p+(i))  = - 3.055  - 0.4486 ( B i r t h G a p )  +  1.002 ( W i f e  wg) -  0.2201 ( W i f e  wg)  2  61  +  0.1150 (Head  inc)  0.0001 ( U n e m p l o y ) 0.2486 ( F e c u n d i t )  -  2  -  0.0005 (Head  inc)  0.3319 ( i n c / n e e d )  - 0. 0 0 4 9 ( F e c u n d i t )  - 0.0109(Unemploy) +  2  -  0.0028 ( i n c / n e e d )  +  2  2  1969: L o g i t (p+ ( i ) ) =  -2.390 -0.4669 ( B i r t h G a p )  0.0717{Head  inc)  0.0011 (Unemploy)  0.0003(Head  -  2  0.1198 ( F e c u n d i t )  -  + 1.671 ( H i f e wg) inc)  +  2  0.4832 ( i n c / n e e d )  - 0.0002 ( F e c u n d i t )  - 0.3913(Wife  +  wg)  +  2  0.0534(Unemploy)  0.0289 ( i n c / n e e d )  +  2  2  1970: Logit =  (p+(i))  -2.035 -  0. 2 9 6 8 ( B i r t h G a p )  + 0.0887 (Head i n c )  -  1.258 ( H i f e wg)  0.0017 (Head  0.411x10-* (Unemploy)  -  2  0.0776 ( F e c u n d i t )  +  inc)  2  0. 2 7 5 5 ( i n c / n e e d )  + 0.00 16 ( F e c u n d i t )  +  - 0.3676 ( H i f e  wg)  2  0.0016 (Unemploy)  + 0. 0 1 4 4 ( i n c / n e e d )  + +  2  2  1971; L o g i t (p+ ( i ) ) =  -1.297 - 0.1922 ( B i r t h G a p )  + 0.2137(Wife  + 0.1593(Head i n c ) -  0.0059(Head  0.340x10-* (Unemploy)  -  2  0.0500 ( F e c u n d i t )  IV  Further If  inc)  2  0. 1 8 3 0 ( i n c / n e e d )  + 0.0023 ( F e c u n d i t )  wg) -  - 0.1016(Wife  wg)  0.0049(Unemploy)  + 0.0075 ( i n c / n e e d )  2  2  + +  2  estimation  we c o n s i d e r  those  statistically  significant  variables  62  which as  we  obtain  reliable  only  with  follows:  variables,  those  we  of the estimation  would l i k e  variables.  The  of this  to re-estimate  new . e s t i m a t e s  are  the  model model  shown  as  3  Regression Group  from t h e r e s u l t s  model  I, ( y  —  | z =  labour  eguation  1 )  1967: Hourwork 592.6  -  0.2331(Head  45.87(Unemploy)  -  0.0064 ( i n c / n e e d )  inc)  0.7770(Unemploy)  +  0.51x10~ (Head  2  •  5  inc)  2  +  8.262(inc/need)  2  1968: HourWork =  1768.0  inc)  + 187. 8 ( W i f e  wg)  2  - 0.0763 (Head i n c ) + 0.14x10-s (Head  2  1969: HourWork 732.6  +  51.77 ( W i f e  0.13x10-5(Head  inc)  2  wg)  -  2  0. 1448 ( H e a d  inc)  +  + 3.327(inc/need)  1970: HourWork 640.8  -  7.428(inc/need)  0.2325 (Head  inc)  - 0.0048(inc/need)  + 2  0.37x10-s (Head i n c )  2  +  63  1971: HourWork =  2 9 1 . 1 - 0. 1 4 5 9 ( H e a d i n c ) + 7 . 1 0 0 ( i n c / n e e d )  Group  I I , ( y  - 0.0050 ( i n c / n e e d )  2  | z = 0 )  1967: HourWork HourWork =  455.6  -  0. 196 ( H e a d  8.959 ( i n c / n e e d )  inc)  +  - 0.006 1 ( i n c / n e e d )  0.26x10~s (Head  inc)  +  2  2  1968: HourWork =  3 3 2 . 5 - 0. 1 3 0 3 ( H e a d i n c ) + 8. 5 6 1 ( i n c / n e e d )  - 0.0060 ( i n c / n e e d )  2  1969 HourWork = 6 5 5 . 8 - 0. 1 0 4 6 ( H e a d i n c ) + 6 . 0 9 6 ( i n c / n e e d )  - 0.004 ( i n c / n e e d )  2  1970: HourWork 649.7  -  0. 1 5 2 1 ( H e a d  5.469(inc/need) 3.204(Fecundit)  1971:  2  inc)  0.0028(inc/need)  + 2  0 . 2 0 x 1 0 ~ s (Head +  inc)  62.44(Fecundit)  2  +  6a  HourWork 919.0  -  0,131(Head  4.425(inc/need)  logistic  model  inc)  - 0.0021(inc/need)  —  Probability  +  0.16x10~«(head  inc)  2  +  2  eguation  1967: l o g i t (p+(i)) -1.475  -  0.0058 ( F e c u n d i t )  0.4 866 ( B i r t h G a p )  +  0.2291 ( F e c u n d i t )  2  1968: Logit (p+(i))  = -1.004 - 0 . 6 2 0 8 ( B i r t h G a p )  + 0.0972 ( F e c u n d i t )  1969: Logit (p+(i)) = -1.086 - 0.5104 ( B i r t h G a p ) -  11.56(Unemploy)  + 0.0100 ( F e c u n d i t )  2  1970:  Logit (pMi))  = 0.3183 - 0.0849 ( W i f e  wg)  2  1971: L o g i t (p+(i))  footnote:  = 0.2879 - 0.0363 (Head i n c )  + 5.609 ( U n e m p l o y )  65  i  1967  1971  -  z Under  3  -  1968 - 5 1 . 2 0 % ,  1969 - 5 7 . 9 5 3 ,  1970  -  46.36%,  47.41%  5$ c r i t i a l  Some v a r i a b l e s ,  estimated in  30.17%,  points the value  which  t h e model  re-estimation.  statistically  were  with  of F(48,1640)  statistically  significant.  considered  1.49.  significant  a l l the variables,  «e  i s  such  were  not  when  we  significant  variables  as  not  Table Parameter  Estimates  Group I ( y | z =  Variable  1 )  IV A  f o r Labour Equations  Group ( y I  I I z =  0  1967  Hhole sample unconditional  )  constant  1558.7693* (30.861)  1267.7070* (25.414)  1556.3296* (42.271)  BirthGap  111.2579 (1.169)  -3.7038 (0. 088)  29.7965 (0.806)  Hife  -704.1122 (1.576)  -184.1179 (0.413)  -471.3710 (1.463)  176.6363 (1.916)  -18.0680 ( 0 . 199)  90.6479 (1.369)  -0.2652* (6.312)  -0.2698* (6.912)  -0.2926* (10.685)  (Hife  wg wg)  2  Head i n c  Head  inc  0.5668x10-5* (4.666)  2  Onemploy (Unemploy)  2  inc/need  0.4198x10-5* (3.045)  0.5784x10-5* (6.893)  42.6361* (3.348)  -1.5681 (0.225)  5.1312 (1.044)  -0.7200* (3.366)  -0.0193 (0.547)  -0.0531* (1.967)  8.9754* (4.273)  10.1893* (5.604)  10.1931* (7.500)  -0.6520x10~ * (2.683)  -0.7719x10~ (4.114)  -4.2753 (0.086)  12.6560 (0.291)  -0.6339 (0.020)  -0.7383 (0.4665)  -1.3261 (0.945)  -1.0798 (1.032)  Observations  189  151  340  R  0.3444  0.4646  0.3493  (inc/need)  -0.7210x10(2.384)  2  Fecundit  (Fecundit)  *  2  2  significant level  Asymtotic  t values  Chow t e s t  F (12,316)  2  *  2  o f 5% u n d e r  H:  parameters.=  are i n parentheses. =  2.2980  0.0  2  Table IV B Parameter Estimates f o r l a b o u r Equations 1 9 6 8 Group I ( y | z  Variable  =  1 )  Group I I ( y | z =  0 )  Whole sample unconditional  constant  2402.0710* (49.929)  1344.£320* ( 2 8 . 548)  1577.3882* (45.707)  BirthGap  -77.0932 (0.850)  - 3 0 . 6 823 (0.711)  30.2229 (0.864)  Wife wg  -1036.6126* (2.229)  -542.6887 (1.142)  -537.8107 (1.651)  251.1645* (2.505)  66.3974 (0. 671)  112.4804 (1.620)  -0.1902* (6.316)  -0.2265* (5.276)  -0.2213* (1 1.508)  0.1797x10-5* (2.249)  0.3386x10-5* (2.096)  0.3169x10-5* (7.597)  1.8356 (0.147)  21.6627 (1.270)  10.3385 (1.175)  -0.0231 (0.158)  -0.2880 (1.040)  -0.1266 (1.103)  4.0753 (1. 904)  11.2963* (7.345)  8.6154* (7.627)  0.9217x10-3 (0.291)  -0.8178x10~ * (4.919)  -0.5580x10- * (4.094)  -3.7857 (0.082)  -41.8059 (1.098)  -19.3969 (0.693)  -0.9524 (0.628)  1.1036 ( 0 . 822)  -0.4556 (0.474)  Observations  181  159  340  R  0.3135  0.4371  0.3606  (Wife wg)  2  Head i n c (Head inc)  z  Unemploy (Unemploy)  2  inc/need (inc/need)  2  Fecundit (Fecundit)  2  2  * significant  2  2  l e v e l o f 5% under H: parameters = 0.0  Asymtotic t values are i n parentheses. Chow t e s t F ( 1 2 , 3 1 6 )  =  7.3582  Table Parameter  Estimates  Group I ( y | z  Variable  =  1 )  IV  C  f o r Labour  Group I I ( y | z =  Equations  0 )  1969  Whole sample unconditional  constant  1910.3313* (37.255)  24.0713 (0.4852)  949.3052* (25.880)  BirthGap  -51.3668 (0.618)  -67.4059 ( 1 . 357)  2.9322 (0.079)  Wife  -828.3737 (1.377)  635.7993 (1. 183)  -58.3414 (0.154)  224.4707 ( 1 . 575)  -182.5847 ( 1 . 594)  9.3886 (0.111)  -0. 1907* (6.379)  -0. 1901* (4.018)  -0.1936* (9.857)  0.2277x10-5*  0.3083x10-5  0.2364x10-s*  (Wife  wg  wg)  2  Head i n c  (Head  i n c )  2  (1.802)  (6.149)  -11.4448 (0.698)  2.2814 (0.257)  0.4696 (0.078)  0.2350 (0.608)  -0.0215 ( 0 . 367)  0.5634x10" (0.126)  7.3298* (3.270)  7.5424* ( 5 . 102)  7.7892* (7.103)  (3.511) Unemploy  (Unemploy)  2  inc/need  2  -0.5037x10- * (3.351)  -0.5187x10- * (4.165)  -40.0683 (0.873)  70.3725 (1.738)  10.5490 (0.380)  0.0498 (0.003)  -2.7817 (1.667)  -1.6838 (1.648)  Observations  178  162  34 0  B  0.2817  0.3325  0.2991  (inc/need)  -0.4885x10(1.582)  2  Fecundit  (Fecundit)  *  2  2  significant  level  Asymtotic  t  Chow  F (12,316)  test  values  of  5%  are i n =  2  2  under  H:  parameters  parentheses.  3.6634  2  =  0.0  Jable Parameter  Estimates  Group I ( y 1 z =  Variable  1 )  IV  D  f o r Labour  Group I I ( y | z =  lauations 1 9 7 0  0  )  Hhole sample unconditional  constant  1680.9971* (32.395)  52.8269 (1.074)  1033.5461* (28.059)  BirthGap  -113.2762 (1.889)  -33.8650 (0.592)  -36.6936 (0.934)  life  -736.9992 (1.174)  704.6772 ( 1 . 504)  1.1532 (0.003)  167.3082 (1.055)  -178.8685 (1.760)  -15.3298 (0.187)  -0.2874* (5.622)  -0.1588* (6.260)  -0.2036* (9.236)  0.4527x10-s*  0.2108x10-5*  0.2823x10-5*  (2.514)  (3.881)  (5.415)  5.7146 (1.249)  0.1341 (0.015)  -0.8891 (0.247)  -0.0382 ( 1 . 4 8 6)  -0.0872 ( 0 . 953)  -0.6823X10(0.317)  9.3285* (5.651)  5.3371* (6.407)  6.2063* (8.751)  (Hife  Head  wg  wg)  2  inc  (Head i n c ) 2  Unemploy  (Unemploy) 2  inc/need  (inc/need) 2  2  -0.6144x10- * (3.468)  -0.2656x10~ * ( 4 . 236)  -0.3129x10- * (5.277)  4.4495 (0.097)  75.9236* (2.161)  28.2306 (1.085)  (Fecundit) 2  -1.385 8 (0.841)  -3.5510* (2. 225)  -2.3265* (2.264)  Observations  173  167  340  R  0.3320  0.3444  0.3088  2  Fecundit  *  2  significant  level  Asymtotic  t  Chow  F(12,316)  test  values  of  5%  are  in  =  2  under  H:  parameters  parentheses.  2.7094  2  = 0.0  70  Table Parameter  Estimates  Group I { Y I z =  Variable  1 )  IV E  f o r Labour  Group I I ( y l z =  Equations  o )  1.97J.  Whole sample unconditional  constant  1496.4514* (27.655)  587.5898* (11.748)  1151.3694* (30.555)  BirthGap  -62.9480 (1.022)  -89.8988 (1.520)  -53.3046 (1.311)  Wife  -896.2939 (1.471)  205.7050 (0.358)  -298.0087 (0.721)  224.4905 (1.448)  -75.6943 (0.558)  66.7110 (0.662)  -0.2370* (3.256)  -0.1402* ( 5 . 117)  - 0 . 1772* (7.926)  0.1815x10-5* (3.034)  0.2457x10-s* (4.481)  4. 1 7 3 5 (0.840)  -3.0460 (0.471)  3.3144 (0.882)  -0.0218 (0.749)  0.0274 (0. 569)  -0.0183 (0.761)  8.4099* (4.689)  4.8945* (5.095)  5.4899* (7.327)  (Wife  wg  wg)  2  Head i n c  (Head  inc)  2  Unemploy  (Unemploy)  2  inc/need  0.3970x1C(1.050)  s  (2.940)  -0.2270x10- * (3. 329)  -0.2643x10" * (4.454)  -8.1950 (0.178)  36.3512 (1.086)  19.9307 (0.811)  -0.4393 (0.251)  - 1 . 1797 (0.697)  -1.6618 (1.580)  Observations  164  176  340  R  0.2811  0.2361  0.2394  (inc/need)  2  Fecundit  (Fecundit)  2  2  * significant  -0.6105X10" * 2  level  Asymtotic  t values  Chow t e s t  F(12,316)  2  o f 5% u n d e r are i n =  H: p a r a m e t e r s  parentheses.  5.5301  2  =0.0  Mean a n d S t a n d a r d Group ( y I  Variable  Deviation  I z = 1 )  o f t h e Model I I  A  1967  Group I I ( y I z = 0 )  Whole sample y uncondition  HourWork  774.894 (832. 148)  1008.69 (806.400)  878.726 (827.825)  BirthGap  0.4180 (0.6013)  1.0331 (1. 6183)  0.6912 (1.2054)  Wife  2.0312 (0. 6498)  2.08863 (0.6742)  2.0567 (0.6604)  4.5457 (3.0821)  4.8139 (3.2770)  4.6648 (3.1682)  7812.65 (4311 .02)  7 5 2 2 . 16 (5073.70)  7683.64 (4660.26)  0.7952x108 (0.1608x10*)  0.8215x108 (0. 1333x109)  0.8069x108 (0.1490x109)  3.2090 (10.6461)  4.9735 (25.3447)  3.9927 (18.6509)  123.038 (630.650)  662.838 (5019.12)  362.773 (3382.22)  242.873 (116.078)  265.384 (146.320)  252.871 (130.657)  0.7239x105 (0.8825x105)  0.9170x105 (0.1078x106)  0.8096x105 (0.9776x105)  16.1481 (6.0053)  15.0530 (7.5189)  15.6618 (6.7313)  296.635 (190.053)  282.748 (238.273)  290.468 (212.603)  189  151  340  wg  (Wife w g )  2  Head i n c (Head i n c ) 2 Unemploy (Unemploy)  2  inc/need (inc/need)  2  Fecundit (Fecundit)  2  Observations  Standard  deviations  are i n parentheses.  ll^le Mean a n d S t a n d a r d Group I ( y | z =  Variable  V B  D e v i a t i o n of t h e Model  1 )  I I  X  1968  Group I I ( y J z = 0 )  Whole sample y uncondition  HourWork  749.293 (756.947)  1132.99 (763.656)  928.726 (782.811)  BirthGap  0.3812 (0.6178)  0.7862 ( 1 . 5482)  0.5706 (1.1665)  life  wg  1.9666 (0.5927)  2.0144 (0.6335)  1.9889 (0.6117)  (Wife  wg)  4.2169 (2.6983)  4.4564 (2.9939)  4.3289 (2.8387)  8552.08 (5424.92)  8499.53 (4702.69)  8527.51 (5092.60)  0.1024x10* (0.2962x10 )  0.9422x108 (0. 1099x10*)  0.9858x10« (0.2285x109)  2.8488 (12.6523)  2.6376 (11.3457)  2.7500 (12.0418)  167.313 (1072.79)  134.872 (697.642)  152.142 (915.526)  264.033 (121.435)  303.673 (149.635)  282.571 (136.594)  0.8438x105 (0. 1080x10*)  0.1145x106 (0. 1231x106)  0.9845x105 (0. 1161x10*)  16.5580 (5.8900)  12.5031 (6.9917)  14.6618 (6.7313)  308.669 (183.222)  204.906 ( 2 0 3 . 140)  260.144 (199.365)  181  159  340  2  Head i n c (Head  inc)  2  9  Unemploy (Unemploy)  2  inc/need (inc/need)  2  Fecundit  (Fecundit)  2  Observations  Standard  deviations are i n  parentheses.  Mean a n d S t a n d a r d Group I ( y I z =  Variable  D e v i a t i o n c f the Model  1 )  I I  X  1969  Group I I ( y J z = 0 )  Whole sample y uncondition  HourWork  756.927 (781.734)  1195.58 (745.969)  965.932 (794.667)  BirthGap  0.3483 (0.6656)  0,5988 (1.4764)  0.4677 (1.1323)  Wife  wg  1.8301 (0.5237)  1.9638 (0.6173)  1.8938 (0.5733)  (Wife  wg)  3.6218 (2.2029)  4.2351 (2.8226)  3.9140 (2.5321)  8609.85 (5525.41)  9097.13 (4544.49)  8842.02 (5080.19)  0.1045x109 (0.3004x109)  0. 1 0 3 3 x 1 0 9 (0.1135x109)  0.1039x109 (0.2307x109)  2.6938 (8.6431)  2.7284 (15.9771)  2.7103 (12.6585)  81.5407 (377.597)  261.136 (2441.43)  167.112 (1706.86)  284.146 (123.874)  328.210 (144.391)  3 0 5 . 141 (135.644)  0.9600x10s (0. 1017x10*)  0.1284x106 (0.1273x106)  0. 1 1 1 5 x 1 0 6 (0.1156x106)  16.0955 (5.9296)  10.9877 (6.5579)  13.6618 (6.7313)  (Fecundit) 2  294.028 (176.428)  163.469 (172.537)  231.821 ( 1 8 6 . 158)  Observations  178  162  340  2  Head i n c (Head i n c )2 Unemploy (Unemploy) 2 inc/need (inc/need) 2 Fecundit  Standard  deviations are i n  parentheses.  Table Mean a n d S t a n d a r d  V D  D e v i a t i o n of the Model I I  X  1970  Group I ( y | z = 1 )  Group I I ( y I z = 0 )  Whole sample y uncondition  HourWork  781.601 (807.945)  1143.93 (758. 429)  959.571 (803.603)  BirthGap  0.3757 (0.9357)  0.3713 (1.1745)  0.3735 (1.0581)  Wife  wg  1.7079 (0.46924)  1.9147 ( 0 . 6349)  1.8095 (0.56554)  wg) z  3.1358 (1.8479)  4.0669 (2.8769)  3.5931 (2.4500)  8253.68 (3700.80)  9687.73 (5921.69)  8958.05 (4963.42)  0.8174x108 (0.9086x108)  0.1287x109 (0.2492x109)  0.1048x109 (0.1875x109)  9.2269 (32.2653)  3.3802 (15.3756)  6.3552 (25.5447)  1120.17 (5777.52)  246.421 (1535.59)  691.002 (4275.75)  312.04 0 (140.308)  364.401 (188.303)  337.759 (167.447)  0.1169x10* (0.1181x10*)  0.1680x106 (0.2356x106)  0. 1 4 2 0 x 1 0 6 (0.1868x106)  15.8150 (5.8220)  9.3952 (6. 0260)  12.6618 (6.7313)  283.815 (164.463)  124.365 (141.679)  205.497 (172.990)  173  167  340  Variable  (Wife  Head i n c (Head i n c ) 2 Unemploy (Unemploy) 2  inc/need (inc/need) 2  Fecundit (Fecundit) 2 i  Observations  Standard  deviations are i n  parentheses.  Table  Mean a n d S t a n d a r d Group I ( y I z =  Variable  V E  D e v i a t i o n of the Model I I  1 )  X  2211  Group I I { y | z = 0 )  Whole sample y uncondition  HourWork  797.360 (789.206)  1175.35 (734.960)  993.026 (783.650)  BirthGap  0.3659 (0.9530)  0.2898 ( 1 . 1064)  0.3265 (1.0345)  life  wg  1.6824 (0.5206)  1.8392 (0.5805)  1.7636 (0.5572)  wg) 2  3.09972 (2.0359)  3.7178 (2. 4179)  3.4197 (2.2598)  8283.83 (3356.62)  9668.88 (5530.66)  9000.80 (4657.06)  0.7982x108 (0.6195x108)  0.1239x109 (0. 2085x109)  0.1026x109 (0.1574x109)  8.2317 (31.4318)  7.0483 (26.3531)  7.6191 (28.8773)  1049.70 (5537.04)  740.217 (3553.56)  889.495 (4498.73)  333.982 (140.231)  387.778 (199.672)  361.829 (175.389)  0.1311x10* (0.1130x10*)  0. 1 9 0 0 x 1 0 * (0.2454x10*)  0.1616x10* (0.1952x10*)  15.0366 (5.7939)  8.5171 (5.9904)  11.6618 (6.7313)  (Fecundit) 2  259.463 (154.686)  108.222 (126.997)  181.174 (159.872)  Observations  164  176  340  (Wife  Head i n c (Head i n c ) 2 Unemploy (Unemploy) 2  inc/need (inc/need) 2 Fecundit  Standard  deviations are i n  parentheses.  76  Table The  VI A  C o m p a r i s o n o f t h e Two S t a c j e A i t k e n a n d t h e S i n g l e E q u a t i o n l e a s t - s q u a r e s E s t i m a t i o n o f t h e M o d e l I I 1967 X  ( y 1 2 = 1 ) Single Stage  ( y l z = 0 ) S i n g l e Eqn Staqe  Variable  Two  constant  -24.8795 (536.694)  1558.7693 (602.6486)  1704.61 (555.908)  1267.707 (599.956)  BrithGap  206.277 (79.5456)  111.2579 (95.1705)  6.7483 (40.4629)  -3.7038 (42.2918)  Wife  360.029 (402.916)  -704.1122 (446.8012)  -789.138 (413.982)  -184.1179 (445.6689)  -50.9948 (83.6112)  176.6363 (92.1762)  100.529 (84.9228)  -18.0680 (90.5690)  -0.2742 (0.03637)  -0.2652 (0.0420)  -0.2643 (0.0351)  -0.2698 (0.0390)  0.621x10-s (0.1x10- )  0.567x10-s  0.433x10-5  0.420x10-5  (0.1x10-5)  (0.1x10-5)  37.2640 (9. 8779)  42.6361 (12.7363)  - 1 . 1740 (6.1164)  -1.5681 (6.9694)  -0.5963 (0. 1677)  -0.7200 (0.2139)  -0.0102 (0.0309)  -0.0193 (0.0352)  10.5357 (1.7873)  8.9754 (2.1005)  10.3676 (1.6288)  10.1893 (1.8183)  -0.0095 (0.0025)  -0.0072 (0.0030)  -0.0067 (0.0022)  -0.0065 (0.0024)  20.4197 ( 4 4 . 1752)  -4.2753 (49.6485)  29.1138 (40.8465)  12.6560 (43.4798)  -1.5124 (1.4128)  -0.7383 ( 1 . 5 827)  -1.6923 (1.3143)  -1.3261 (1.4035)  wg  (Hife  wg)  2  Head i n c (Head i n c ) 2  s  Unemploy (Unemploy)  2  inc/need (inc/need)  2  Fecundit (Fecundit)  Standard Chow Two  2  deviations  are i n parentheses.  Test: Stage,  Single  Eqn  F(12,316)  =  3.4416  Egn, F(12,316)  =  2.2980  Two  (0.1x10-5)  J  Table The  77  VI B  C o m p a r i s o n o f t h e Two S t a g e A i t k e n a n d t h e S i n g l e E g u a t i o n l e a s t - s g u a r e s E s t i m a t i o n o f t h e M o d e l I I 1968 X  ( y Stage  2 = 1 ) Single  Egn  ( y Stage  2 = 0 ) Single  Egn  Variable  Two  constant  719.489 (497.420)  2402.071 (617.6222)  1207.37 (506.332)  1344.832 (580.5073)  BrithGap  58.6397 (67.7885)  -77.0932 (90.6618)  -11.9464 (39.5871)  -30.6823 (43. 1462)  Hife  wg  54.4460 (374.372)  -1036.6126 (465.0873)  -524.545 (413.052)  -542.6887 (475.2111)  (Hife  wg)  8.3943 (81.0106)  251.1645 (100.2469)  73.1776 (86.7298)  66.3974 (98.8818)  -0.1846 (0.0229)  -0.1902 (0.0301)  -0.2294 (0.0358)  -0.2265 (0.0429)  0.242x10-5 (0.6x10-*)  0. 1 8 0 x 1 0 - 5  0.363x10-5  0.339x10-5  (0.8x10-6)  (0. 1 x 1 0 - 5 )  (0.2x10-5)  6.7462 (8.6091)  1.8356 (12.4788)  -0.2577 (13.3125)  21.6627 (17. 0604)  -0.0856 (0.0990)  -0.0231 (0. 1458)  0.0541 (0.2178)  -0.2880 (0.2769)  6.6379 (1.5732)  4.0753 (2.1409)  11.0835 (1.2870)  11.2963 (1.5380)  -0.0031 (0.0023)  0.0009 (0.0032)  -0.0077 (0.0014)  -0.0082 (0.0017)  4.6712 (38.6399)  -3.7857 (46.3501)  -23.4310 (34.3664)  -41.8059 (38.0776)  -1.2779 (1.2737)  -0.9524 (1.5159)  0.2166 (1.2025)  1.1036 (1.3420)  2  Head i n c (Head  inc)  2  Unemploy (Unemploy)  2  inc/need (inc/need)  2  Fecundit (Fecundit)  Standard Chow Two  2  deviations  are i n parentheses.  Test: Stage,  Single  F(12,316)  =  9.2483  Egn, ? (12,316)  =  7.3582  Two  78  Table  The  VI C  C o m p a r i s o n o f t h e Two S t a g e A i t k e n a n d t h e S i n g l e E q u a t i o n least-squares E s t i m a t i o n o f t h e Model II~~J969~  ( y 1 2 = 1 ) Single Stage  ( y I z = 0 ) S i n g l e Eqn Stage  Variable  Two  constant  1074.15 ( 5 8 4 . 199)  1910.3313 (685.601)  -61.8565 (515.343)  24.0713 (606.944)  BrithGap  44.8480 (69.4932)  -51.3668 ( 8 3 . 1572)  -53.8075 (43.6123)  -67.4059 (49.6914)  Hife  wg  -240.4760 (501.982)  -828.3737 (601.6902)  479.664 (446.460)  635.7993 (537.3614)  wg)2  81.9762 (118.436)  224.4707 (142.5031)  -135.453 (95.4104)  -182.5847 (114.5202)  -0.2054 (0.0246)  - 0 . 1907 (0.0299)  - 0 . 1750 (0.0387)  -0.1901 (0.0473)  0.267x10-s  0.228x10-s  (0.5x10-6)  0.257x10~s (0.1x10-s)  0.308x10-5  (0.6x10-6)  (Wife  Head i n c  (Head  inc)2  Unemploy  (Unemploy) 2  inc/need  (inc/need)  2  Fecundit  (Fecundit)2  Standard Chow Two  (0.2x10-5)  -11.4448 (16.3863)  6.1288 (7.0696)  2.2814 (8.8619)  0.1846 (0.3000)  0.2350 (0.3862)  -0.0385 (0.0462)  -0.0215 (0.059)  9.6 972 (1.8008)  7.3298 (2.2416)  7.8545 (1.2175)  7.5424 (1.4784)  -0.0073 (0.002)  -0.0049 (0.0031)  -0.0050 . (0.0013)  -0.0050 (0.0015)  -52.7559 (40.7958)  -40.0683 (45. 8751)  75.8987 (35.6987)  70.3725 (40.4924)  0.3216 (1.4042)  0.0498 (1.5686)  -3.2084 (1.4490)  -2.7817 (1.6690)  are i n parentheses.  Test:  Single  Two  -5.1539 (12.7613)  deviations  Stage,  Eqn  F(12,316)  =  7.7696  Egn, F (12,316)  =  3.6634  79  Table The  VI D  C o m p a r i s o n o f t h e Two S t a g e A i t k e n a n d t h e S i n g l e E g u a t i o n L e a s t - s g u a r e s E s t i m a t i o n o f t h e M o d e l I I J.970 X  ( Y 1 z = 1 ) S i n g l e Egn Stage  ( y l z = 0 ) S i n g l e Egn Stage  Variable  Two  constant  1412.61 (559.445)  1680.997 (656.8402)  4 7.0384 (441.321)  52.8269 (517.0958)  BrithGap  -92.9189 (52.7555)  -113.2762 (59.9558)  -34.0479 (51.4447)  - 3 3 . 8650 (57.1928)  Wife  -652.953 (523.368)  -736.9992 (627.373)  590.173 ( 3 8 7 . 140)  704.6772 (468.432)  157.065 (132.528)  167.3082 (158.5716)  -146.911 (84.1822)  -178.8685 (101.6526)  -0.2830 (0.0422)  -0.2874 (0.0511)  - 0 . 1548 (0.0214)  -0.1588 (0.0254)  0.470x10-s  0.453x10-5  wg  (Wife  wg)  2  Head i n c (Head i n c ) 2 Onemploy (Onemploy) 2 inc/need (inc/need)  2  Fecundit (Fecundit)  Standard Chow Two  2  (0.2x10-5)  0. 1 9 9 x 1 0 - 5 (0.5x10-6)  0.211x10-5  (0.2x10-5) 4.5144 (3.5196)  5.7146 (4.5743)  - 3 . 1778 (6. 8682)  0.1341 (9.1211)  -0.0290 (0.0199)  -0.0382 (0.0257)  -0.0035 (0.0687)  -0.0872 (0.0915)  10.6796 (1.3660)  9.3285 (1.6508)  5.5279 (0.6995)  5.3371 (0.8330)  -0.0074 (0.0015)  -0.0061 (0.0018)  -0.0025 (0.0005)  -0.0027 (0.0006)  -15.1719 (41.0756)  4.4495 (45.8121)  74.9506 (32.0058)  75.9236 (35.1332)  -0.8335 ( 1 . 4 8 7 8)  -1.3858 (1.6470)  -3.7018 ( 1 . 4362)  -3.5510 (1.5960)  deviations are i n  parentheses.  Test: Stage,  Single  Two  F (12,316)  =  26.339  Egn, F(12 316)  =  2.7094  f  (0.5x10-6)  Table  The  VI E  C o m p a r i s o n o f t h e Two S t a g e A i t k e n a n d t h e S i n g l e E g u a t i o n L e a s t - s q u a r e s E s t i m a t i o n o f t h e M o d e l Iljt 197J  ( y 1 2 = 1 ) Single Stage  ( y I z = 0 ) S i n g l e Egn Stage  Variable  Two  constant  1330.05 ( 5 8 6 . 100)  1496.4514 (664.1467)  839.732 (493.843)  587.5898 (566.7747)  BrithGap  -58.3862 (56.9849)  -62.948 (61.6013)  -112.064 (54.4209)  -89.8988 (59.1419)  Wife  -473.837 (531.652)  896.2939 (609.2837)  -104.060 (491.017)  205.7050 (575.294)  130.288 (135.215)  224.4905 (155.0258)  8.5510 (116.298)  -75.6943 (135.7505)  -0.2869 (0.0619)  -0.2370 (0.0728)  -0.1368 (0.0241)  -0.1402 (0.0274)  0.648x10-5 (0,3x10-5)  0.397x10-5 (0.4x10-5)  0. 1 7 6 x 1 0 - 5 (0.5x10-*)  0. 1 8 2 x 1 0 - 5 (0.6x10-6)  4.8257 ( 4 . 1312)  4.1735 (4.9665)  3.2804 ( 5 . 1358)  -3.0460 (6.4734)  -0.0239 (0.0240)  -0.0218 (0.0291)  -0.0163 (0.0380)  0.0274 (0.0481)  8.7383 (1.5652)  8.4099 (1.7937)  4.8126 (0.8318)  4.8945 (0.9606)  -0.0062 (0.0018)  -0.0061 (0.0021)  -0.0021 (0.0006)  -0.0023 (0.0007)  -19.4112 (42.7355)  - 8 . 1950 (46.0473)  29.3953 (30.9769)  36.3512 (33.4791)  -0.2104 (1.6331)  -0.4393 (1.7509),  -1.0301 (1.5451)  -1.1797 (1.6928)  wg  (Wife  wg)  2  Head i n c (Head i n c ) 2  Onemploy  (Onemploy)  2  inc/need  (inc/need)  2  Fecundit  (Fecundit)  Standard Chow Two  2  deviations  are i n  parentheses.  Test: Stage,  Single  Egn  F (12,316)  =  48.711  Egn, F(12,316)  =  5.5301  Two  Table Probability  Function  VII A  Estimates  of the Model  I I  X  Asym t - :  Variable  Coefficient  Asym  constant  -2.42797*  0.74520  3.25817  BirthGap  -0.41671*  0.09670  4.30938  Wife  wg  0.354 86  0.54061  0.65641  (Wife  wg)  -0.09620  0.11071  0.86890  0.06513  0.04625  1.40828  0.488x10-*  0.00135  0.03629  0.02830  0.01896  1.49258  -0.00046.  0.00031  1.48280  -0.02214  0.22531  0.09826  -0.02807  0.03065  0.91577  0.28355*  0.05580  5.08141  -0.00735*  0.00177  4. 1 4 3 5 7  2  Head i n c (Head i n c ) 2 Unemploy (Unemploy)  2  inc/need (inc/need)  2  Fecundit (Fecundit)  Log  2  of l i k e l i h o o d  * significant  function  i n 95% under  stdv  = -191.969 a f t e r H:  10  iterations.  parameter estimates  note: 1 Head  i n c = $1,000  1 inc/need  = 100  1  (Head i n c )  1  (inc/need)  =$1,000,000  2  2  =  J967  10,000  =0.0  Table Probability  Function  VII B  Estimates  of t h e Model I I  X  Asym t - :  Variable  Coefficient  fisym  constant  -3,05 5 36*  0.80555  3.79287  BirthGap  -0.44 8 6 1 *  0.10617  4.22535  Wife  1.00167  0.60746  1.64895  -0.22005  0.12970  1.69655  0.11504*  0.03902  2.94809  -0.00055  0.00090  0.61233  -0.01095  0.01661  0.65922  0.00015  0.00022  0.66373  -0.33192  0.24119  1.37618  -0.00278  0.03091  0.08988  0.24864*  0.05810  4.27937  -0.00489*  0.00196  2.49738  (Wife  wg wg)  2  Head i n c (Head i n c ) 2 Unemploy (Unemploy)  2  inc/need (inc/need)  2  Fecundit (Fecundit)  Log  2  of l i k e l i h o o d  * significant  function  i n 95$ u n d e r  stdv  = -176.713 a f t e r H:  13  iterations.  parameter estimates  note: 1 Head i n c = $1,000 1 inc/need  = 100  1  (Head i n c )  1  (inc/need)  =$1,000,000  2  2  =  1968  10,000  =0.0  I § b l § ill C ££2babilitv. F u n c t i o n E s t i m a t e s o f t h e M o d e l I I  X  Variable  Coefficient  Asym stdv  Asym t-:  constant  -2.39040*  0.77684  3. 07709  BirthGap  -0.46689*  0.10790  4. 32698  H i f e wg  1.67102*  0.69282  2. 41191  -0.39129*  0.15509  2. 52292  0.07168  0.03768  1. 90244  -0.00032  0.00088  0. 36407  0.05337*  0.02657  2. 00867  -0.00109*  0.00054  2. 03169  -0.48319*  0.20561  2. 35006  0.02887  0.0 2326  1. 24127  0.11981*  0.05352  2. 23850  -0.00025  0.00204  0. 12416  (Hife  wg)  2  Head i n c (Head i n c ) 2 Unemploy (Unemploy) 2 inc/need (inc/need) 2 Fecundit (Fecundit) 2  1969  Log of l i k e l i h o o d f u n c t i o n = -174.969 a f t e r 20 i t e r a t i o n s . * s i g n i f i c a n t i n 9555 under H: parameter e s t i m a t e s = 0.0  note:  1 Head i n c = $1,000  1 (Head i n c )  2  1 inc/need = 100  1 (inc/need)  2  =$1,000,000 = 10,000  Table  Probability  function  VII D  Estimates  of t h e Model I I  Coefficient  Asym  constant  -2.03529*  0.76525  2.65963  BirthGap  -0.29682*  0.09165  3.23849  Wife  wg  1.25770  0.73180  1.71864  wg) 2  -0.36765*  0.17337  2.12059  0.08875  0.05830  1.52238  -0.00170  0.00210  0.81031  0.00164  0.00909  0.18042  0.411x10-*  0.834x10-*  0.49300  0.15846  1.73869  0.01443  0.01493  0.96648  0.07757  0.05195  1.49326  0.00157  0.00212  0.73948  Head i n c (Head i n c ) 2 Unemploy (Unemploy) 2  -0.27551  inc/need (inc/need) 2 Fecundit (Fecundit) 2  Log  of likelihood  * significant  /  function  stdv  = -172.197 a f t e r  i n 9 5 % u n d e r H:  15  iterations.  parameter estimates  note: 1 Head i n c = $1,000 1 inc/need  = 100  1  (Head i n c ) 2  1  (inc/need)2 =  1970  Asym t - i  Variable  (Wife  X  =$1,000,000 10,000  =0.0  Table Probability  Function  VII E  Estimates  of the J o d e l I I  Coefficient  Asym  constant  -1.29702  0.72149  1.79768  BirthGap  -0.19225*  0.07349  2.61615  Wife  0.21367  0.71663  0.29816  -0.10158  0.17430  0.58281  0.15931*  0.07094  2.24571  -0.00592  0.00315  1.87641  -0.00493  0.00689  0.71503  0.340x10-*  0.459x10-*  0.74210  -0.18304  0.15392  1.18917  0.00748  0.01437  0.52037  0.04998  0.04688  1.06614  0.00228  0.00206  1.10551  (Wife  wg)  2  Head i n c (Head i n c )2 Onemploy (Onemploy)  2  inc/need (inc/need)  2  Fecundit (Fecundit)  Log  2  of l i k e l i h o o d  * significant  function  i n 95% under  stdv  = -180.441 a f t e r  11  iterations.  H: p a r a m e t e r e s t i m a t e s  note: 1 Head i n c = $1,000 1 inc/need  = 100  1  (Head i n c )  1  (inc/need)  =$1,000,000  2  2  =  1971.  Asym t - :  Variable  wg  X  10,000  =  0.0  86  Chapter  VIII  Conclusion  The  proposed  continuous model  dependent  into  model.  basic  a simple  as  a  a system  regression system  method.  eguations  method i n o r d e r  be e s t i m a t e d  and  this  eguations,  to  be  dependent  eguation  of  verification,  i s  economic of  Those  this  common i n s o c i a l  The  discrete  science, although  model i n t h i s  age o f t h e i r  models  married worked  youngest c h i l d  be  of The  model I t i s  extended  the  by  regression  t o be a m o d e l be  are  not  a  with  simultanous-  solvable.  been  One  solved  with  The b a s i c m o d e l i s more  i s a study  women f r o m by m a r r i e d very  a system  Press.  t h e r e i s n o t much  thesis  extended  by Z e l l n e r ' s t w o  i n  or to  t h e r e c u r s i v e model.  t h e number o f h o u r s  can  constraints,  has  i s  eguations.  and  model  which  model be  The p r o b a b i l i t y  Nerlove  extended  least-  by u s i n g t h e  into  of logistic  variables,  model,  American  eguation  model  of disturbances  a model w i t h  model.  extension  of  eguation  by o r d i n a r y  c a n be e s t i m a t e d  basic  the correlation  the  a system  and  by s e p a r a t i n g t h e  i t c a n be s e p a r a t e d  by t h e method  that,  the basic  t o gain e f f i c i e n c y .  considering  jointly  discrete  f u n c t i o n and e s t i m a t e d While  of equations,  interesting  that  i s estimated  that the probability  of regression equations  supply  involves  r e g r e s s i o n model and a p r o b a b i l i t y  logistic  maximum . l i k e l i h o o d  can  variables,  I t i s suggested  formulated  stage  which  T h e r e g r e s s i o n m o d e l c a n be e s t i m a t e d  squares.  into  model,  literature.  on t h e  1967 t o 1971.  labour We  find  women i s a f f e c t e d  much, a n d s l i g h t l y  by  by  their  87  head's  income.  There i s some e f f e c t from  other s o c i a l f a c t o r s ,  such as the head's unemployment, and the r a t i o of needs, yet the s i g n i f i c a n c e of these f a c t o r s year. of  the  The  b i r t h gap  wife  having  incomes  varies  has a s i g n i f i c a n t e f f e c t on the a  c h i l d . not  T h e r e f o r e , the r e s u l t s t e l l the labour market i s g u i t e  older  from  over  year to  probability  than 6 years of  age.  us that the married woman's r o l e dependent upon her f a m i l y  planning.  in  88  Appendix A  Least-Sguares  Estimation  Here we f o l l o w a l l the n o t a t i o n s Therefore  the  conditional  defined  regression  will  i n chapter  2.  be  as  written  following: ryi,  z]  Y  = [X»,X*J  r  A  0-, + £ u , v ]  Re  «-0 B-»  will call  Y = £Y», Y ] , X  =  2  generalized  and  £X» X*], #  E  £u,v].  =  From  m u l t i v a r i a t e r e g r e s s i o n , we know 1  (X X)~»X«Y  A+ =  (X»X)  =  r  (Xi) S f X  X ] =  1  2  L ( X 2 ) »J  (X»X)"  =  1  f  r  (X  1 1  (X ) ' (X ) 1  r  *1  1  ~i  0  (X ) • ( X 2 ) J  0  X )-  »-  1  2  0 T (X 'X2)-u  0  2  so r  A+  I  0 T =  0 -B+-»  r  (xa»Xi)-i «•  -, r X i ' Y  0  0  ( X  2  « X 2 ) -  1  >-  J  1  0  T  X »Y2J  0  2  Hence, A+ = £ X • X J 1  1  - 1  B+ = £ X «X ] - i 2  2  Cov+(u,v) = Var+(u) L 0 r  (X ) ' Y 1  (X2) «  Y  1  2  0 ^ =E+E + «/n Var (v)J +  Since, E+ = £ Y > — X * A +  Y2-X B+] 2  then, Cov+(U,V) = 1 ( Y » - X i A + ) ' ( Y * - X * A * ) n M Y -X B+) • (Y*-X*A+) R  2  =  (Y -X A+) • (Y -X B+) (Y -X B+) • (Y -X B+) 1  1  2  ( Y 2 - X 2 B +)  0  o f V a r ( U ) and V a r ( V ) , +  2  2  2  0  1  unbiased e s t i m a t o r s  2  2  (Y —X*A+) ' (Y»-X»A+)/n  r  i-  For  2  +  J  T MY -X2B+)/nJ 2  we have  89  Var+(u)  =  ( Y 1 - X 1 A + ) * (Yi-X^A+J / (n -k).  var (v)  =  (Y2-X2B+) » (Y2-X2B+)/ (n2-k)  +  where  n  1  i s the t o t a l  J  number o f o b s e r v a t i o n s  when z = 1 ; n  Footnote 1  Press,  J . , Applied  Multivariate  Analysis,  1972, pp.220.  2  =  n-  90  Appendix  Likelihood  Ratio  Test  Coefficient  B  f o r Micro  Vector  Regression  Eguality  1 Under system  the hypothesis  of eguations  of chapter  3 , H:  S  d =.,.=  S ,  The  c a n be w r i t t e n a s  r  I  • i I. I I • i I d| Ly  J  XS8  +  I . .I I* I I. i I d|  I • I I. I I. I I dj  LX  LU  J  J  or, Y* = We  define  Let  TY* =  function,  X*W  + U*  a transformation Y ° , TX* = L(U°),  T, s u c h  X ° , and  under  (B1)  TU*  =  that  E(TU*U*»T«)  U°.  Then  the  =  var(U*)I. likelihood  the hypothesis i s  -dk/2 L(U*)  The  maximum  =  e x p (-U° • U V ( 2 v a r (u°)) )  (2 v a r ( U O ) )  likelihood  Var+(U*)  estimators  =  UO  =  (YO-xow*)  + ,  f o r equation  (B2)  (B1) a r e  UO /dk +  ' (Y0-XQW+)/dk  and W+  Hence then  i f we  =  {XO«XO)-iX  rewrite  0 1  YO  eguation  (B2) i n t e r m s  -dk/2 I (U*+)  =  (2 v a r + ( U * ) )  e x p (-dk/2)  of these  estimators,  91  Likewise, 3  by  0+  then  we  T and  transform express  the  the  variables  maximum  i n eguation  likelihood  (1)  of  chapter  f u n c t i o n i n terms  of  -dk/2 L(U+) So,  the  =  {2 V a r ( U + ) )  estimated r  likelihood  /I  e x p {-dk/2) ratio, r i s  =  L (D*+)  =  -dk/2 [ V a r + ( u * ) / V a r + (u) ]  (U+)  or, -21og(r) Hhich  =  d k l o g [ V a r + { u * ) / V a r * {u) ]  i s asymptotically distributed  degree of  freedom.  as  Chi sguares  with  (d-1)n  92  Appendix Parameter  Estimates  f o r Labour Equation  without  (Onemploy)  When u n e m p l o y m e n t o f t h e h e a d relationship regression We  with  the  m o d e l , we  annual  found  tabulated the r e s u l t s  was  hours  that they  1967 -  1 )  71  2  considered worked  as  a  by t h e w i f e  are negatively  of model I as Group I ( y | z =  Variable  C  linear i n the  correlated.  following: Group I I ( y j z =  0  constant  1728.3230 (4.0485)  478.4102 (1.2482)  BirthGap  -42.8763 (1.7460)  -36.5538 (2.8869)  Wife  -714.5180 (11.0354)  360.3229 (3.0861)  192.8221 (15.6143) -0.2069 (217.8773)  -113.3932 (6.6019) -0.1740 (182.9149)  0.2892x10-5 (97.1418)  0.2337x10-5 (52.2196)  Unemploy  -0.2405 (0.0486)  -2.5008 (4.7875)  inc/need  6.5660 (78.8038)  6.1298 (207.5199)  -0.0040 (19.0210)  -0.0032 (77.3457)  -13.2484 (0.4368)  43.1432 (9.7703)  -0.5110 (0.5569)  -1.7794 (10.9088)  observation  885  815  R  0.2521  0.3197  (Wife  wg wg)  2  Head i n c (Head  inc)  (inc/need)  2  2  Fecundit (Fecundit)  2  2  Asymtotic  t values  are i n  parentheses.  93  Eibliography  Berkson, J . , (1951), "Why I Prefer B i o m e t r i c s ^ December 1951, pp. 327-339.  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