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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(1972), " A Technical V a r i a b l e s as A p p l i e d i n t h e S o c i a l Research Project of the Alberta C o u n c i l , E d m o n t o n , A l b e r t a , 1972. Beport on B i n a r y Dependent Sciences", A Commissoned Human Besources Research C h a m b e r s , E. A., a n d D. B. C o x , ( 1 9 6 7 ) , " D i s c r i m i n a t i o n between Alternative B i n a r y B e s p o n s e M o d e l s " , B i p m e t r i k a ^ V o l . 54, 1967, pp. 573-578. Chow, G. C , (1960), "Tests of Eguality Between Sets of Coefficients i n Two L i n e a r B e g r e s s i o n s " , E c o n o m e t r i c a Vol.28, No.3, J u l y 1960, p p . 5 9 1 - 6 0 5 . x C o r n f i e l d , J . a n d N. M a n t e l , ( 1 9 5 0 ) , "Some New Aspects of the Application o f Maximum Likelihood to the Calculation ofthe Dosage Besponse C u r v e " , Journal o f t h e American Statistical 94 issociatiojQ C o x , D. 1970. V o l . 45, x S. (1970), Dempeter, Analysis", 346. A. P., Journal 1950, The pp. Analysis 181-210, of Binary. J ) a t a (1971), "An Overview of M u l t i v a r i a t e Analysis Dempster, A, P., (1972), "Aspects of Model", M u l t i v a r i a t e Analysis I I I Edited A c a d e m i c ' p r e s s , 1972, pp. 1 2 9 - 1 4 2 . X Finney, D. J., Press, Cambridge, Goldberger, 1964. A. S., (1947), England, (1964), of I x Methuen, x Multivariate 1971, pp. Data 316- the Multinomial Logit b y P. B. Krishnaiah, P r o b i t A n a l y s i s ^ Cambridge 1947. (3rd e d i t i o n , 1971). Econometric London, Theory x University W i l e y , New York, Goodman, L. A., (1970), "The Multivariate Analysis of Q u a l i t a t i v e D a t a : I n t e r a c t i o n s Among M u l t i p l e Classifications", J o u r n a l o f t h e A m e r i c a n S t a t i s t i c a l A s s o c i a t i o n , V o l . 6 5 , No.329, M a r c h ~ 9 7 0 , pp. 226-256. G o o d m a n , L. A., to the Analysis Review, Vol.37, ( 1 9 7 2 ) , "A M o d i f i e d M u l t i p l e R e g r e s s i o n A p p r o a c h of Dichotomous V a r i a b l e s " , American S o c i o l o g i c a l 1972, p p . 2 8 - 4 6 . Gunderson, M., (1974), "Betention of Trainees — A Study With Dichotomous Dependent V a r i a b l e s " , Journal of Econometrics 2 1974, p p . 7 9 - 9 3 . X Johnson, 1972. J., N e r l o v e , M. LcgzLinear Corporation (1972), Econometric Methods, 2nd E d i t , Mcgraw a n d S. J . P r e s s , (1973), U n i v a r i a t e and and Logistic Models, Santa Monica, R e p o r t R-1306, 1973. Bill, Multivariate C a l i f . : Rand N e t e r , J . , a n d E. S. M a y n e s , ( 1 9 7 0 ) , "On t h e A p p r o p r i a t e n e s s of the Correlation Coefficient with a 0, 1 D e p e n d e n t V a r i a b l e " , J g u r n a 1 c f t h e A m e r i c a n S t a t i s t i c a l A s s o c i a t i o n , , V o l . 6 5 , No.330, J u n e ~ 1 9 7 0 , ppT 501-509. P r e s s , S. J., (1972), Applied R i n e h a r t a n d W i n s t o n , New Y o r k , Multivariate 1972. Analysis, Holt, Schmidt, P. a n d P. S t r a u s s , (1974), " E s t i m a t i o n of Models with J o i n t l y Dependent Q u a l i t a t i v e V a r i a b l e s : A Simultaneous Logit Approach", U n i v e r s i t y of C a r o l i n e , 1974. Schmidt, P. And P. S t r a u s s , (1975), " E s t i m a t i o n o f Models with J o i n t l y Dependent Q u a l i t a t i v e V a r i a b l e s : A Simultaneous Logit Approach", Econometrica V o l . 4 3 , No.4, J u l y 1975, pp. 745-755. x Survey Research Centre, University of Michigan (1972), A Panel 95 Study o f Income D y n a m i c s , Ann A r b o r , 1 9 7 2 . T h e i l , H. ( 1 9 6 9 ) , "A M u l t i n o m i a l E x t e n s i o n o f t h e Model", International Economic Review, Vol.10, 1 9 6 9 , pp.~251-259T Linear Logit No.3, October T h e i l , H., ( 1 9 7 0 ) , "On t h e E s t i m a t i o n o f R e l a t i o n s h i p s I n v o l v i n g Q u a l i t a t i v e V a r i a b l e s " , American J o u r n a l o f S o c i o l o g y ^ Vol.76, 1970, pp. 103-154. Theil, H. (1970), Principles W i l e y and Sons, 1970. of Econometrics^ New Y o r k : John Tcbin, J . , (1955), "The A p p l i c a t i o n of Multivariate Probit Analysis t o Economic Survey Data", Cowles Foundation D i s c u s s i o n P a p e r No. 1, D e c e m b e r 1, 1 9 5 5 . W o n n a c o t t , H. J , A n d T. H. W o n n a c o t t , ( 1 9 7 0 ) , E c o n o m e t r i c s ^ J o h n W i l e y and Sons I n c . , 1970. Zellner, A. a n d H. Theil, (1962), "Three-Stage Sguares: Simultaneous Estimation of Simultaneous Eguations", Econometrica , V o l . 3 0 , No. 1, J a n u a r y 1 9 6 2 , p p . 5 4 - 7 8 . Zellner,, A., (1962), "An Efficient Method of Estimating Seemingly U n r e l a t e d R e g r e s s i o n s and T e s t s f o r A g g r e g a t i o n B i a s " , J o u r n a l of American S t a t i s t i c a l A s s o c i a t i o n ^ Vol.57, 1962, pp. 348-368. " Zellner, A. a n d D. S. Huang, (1962), " F u r t h e r P r o p e r t i e s o f Efficient Estimators for Seemingly Unrelated Regression Eguations", International Economic Review, Vol.3, No.3, September 1962, pp. 300-313. Zellner, A., (1963), "Estimators for Seemingly Unrelated R e g r e s s i o n E g u a t i o n s : Some E x a c t F i n i t e S a m p l e R e s u l t s " , J o u r n a l of A m e r i c a n S t a t i s t i c a l A s s o c i a t i o n , V o l . 5 8 , 1 9 6 3 , pp. 977-992. Zellner, A. a n d T. H. 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Regression models involving categorical and continuous dependent variables with a study of labour supply.. Lau, Yat Wing 1975
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Title | Regression models involving categorical and continuous dependent variables with a study of labour supply of married women |
Creator |
Lau, Yat Wing |
Date | 1975 |
Date Issued | 2010-02-02T22:58:46Z |
Description | This thesis is going to consider the inferences about the relationships that determine jointly a continuous variable and a categorical variable. These relationships can be considered separately into two models: a regression model and a probability model. The regression model can be estimated by ordinary least squares, or Zellner's two stage method. The probability model is estimated by the method of Serlove and Press. Such relationships will be given more complex consideration. This kind of model is applied in the analysis of an economic problem. It is to consider the labour supply of married women. Data are pooled from the Panel Study of Income Dynamics 1972. It is found that the age of the youngest child is the most significant factor to determine the number of hours worked by a married woman, and birth gap is the major effect in the probability of a wife having a child not older than six years of age. |
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Thesis/Dissertation |
Type |
Text |
Language | eng |
Collection |
Retrospective Theses and Dissertations, 1919-2007 |
Series | UBC Retrospective Theses Digitization Project |
Date Available | 2010-02-02 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0093533 |
URI | http://hdl.handle.net/2429/19565 |
Degree |
Master of Science in Business - MScB |
Program |
Business Administration |
Affiliation |
Business, Sauder School of |
Degree Grantor | University of British Columbia |
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UBCV |
Scholarly Level | Graduate |
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