Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Regression models involving categorical and continuous dependent variables with a study of labour supply.. 1975

You don't seem to have a PDF reader installed, try download the pdf

Item Metadata

Download

Media
UBC_1976_A4_6 L38.pdf
UBC_1976_A4_6 L38.pdf [ 4.72MB ]
Metadata
JSON: 1.0093533.json
JSON-LD: 1.0093533+ld.json
RDF/XML (Pretty): 1.0093533.xml
RDF/JSON: 1.0093533+rdf.json
Turtle: 1.0093533+rdf-turtle.txt
N-Triples: 1.0093533+rdf-ntriples.txt
Citation
1.0093533.ris

Full Text

REGRESSION MODELS INVOLVING CATEGORICAL AND CONTINUOUS DEPENDENT VARIABLES w i t h A STUDY ON LABOUR SUPPLY OF MARRIED SOMEN by IAT 9ING LAO B . S c , 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 , 1973 A T h e s i s S u b m i t t e d In P a r t i a l F u l f i l m e n t Of The Requirements For The Degree Of Master Of S c i e n c e i n Commerce and Business Administration i n t h e F a c u l t y o f Commerce and B u s i n e s s A d m i n i s t r a t i o n He a c c e p t t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA December, 1975 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h Co lumb ia , I a g ree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s tudy . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d that c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i thou t my w r i t t e n p e r m i s s i o n . Department o f C^Mfj tlZC$ fr® fa ASfa M t ° The U n i v e r s i t y o f B r i t i s h Co lumbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date JTfrJ i A b s t r a c t T h i s t h e s i s i s g o i n g t o c o n s i d e r t h e i n f e r e n c e s about t h e r e l a t i o n s h i p s t h a t d etermine j o i n t l y a c o n t i n u o u s v a r i a b l e and a c a t e g o r i c a l v a r i a b l e . These r e l a t i o n s h i p s can be c o n s i d e r e d s e p a r a t e l y i n t o two models: a r e g r e s s i o n model and a p r o b a b i l i t y model. The r e g r e s s i o n model can be e s t i m a t e d by o r d i n a r y l e a s t s q u a r e s , o r Z e l l n e r ' s two s t a g e method. The p r o b a b i l i t y model i s e s t i m a t e d by t h e method o f S e r l o v e and P r e s s . Such r e l a t i o n s h i p s w i l l be g i v e n more complex c o n s i d e r a t i o n . T h i s k i n d o f model i s a p p l i e d i n t h e a n a l y s i s of an economic problem. I t i s t o c o n s i d e r t h e l a b o u r s u p p l y o f m a r r i e d women. Data a r e poole d from t h e P a n e l Study of Income Dynamics 1972. I t i s found t h a t t h e age of t h e youngest c h i l d i s t he most s i g n i f i c a n t f a c t o r t o determine t h e number of hours worked by a m a r r i e d woman, and b i r t h gap i s t h e major e f f e c t i n th e p r o b a b i l i t y of a w i f e h a v i n g a c h i l d not o l d e r t h a n s i x y e a r s o f age. T a b l e o f C o n t e n t s A b s t r a c t T a b l e o f c o n t e n t s L i s t o f T a b l e s A c k n o w l e d g m e n t C h a p t e r I I n t r o d u c t i o n C h a p t e r I I B a s i c M o d e l The m a t h e m a t i c a l model D i s c r e t e d e p e n d e n t v a r i a b l e r e g r e s s i o n E s t i m a t i o n H y p o t h e s i s t e s t i n g P o l y t o m o u s v a r i a b l e C h a p t e r I I I S y s t e m o f E g u a t i o n s M o d e l E s t i m a t i o n e g u a t i o n - b y - e g u a t i o n Dependence among g r o u p s F u r t h e r d i s c u s s i o n C h a p t e r IV M o d e l E x t e n s i o n s L a g g e d v a r i a b l e s m o d e l M o d e l w i t h c o n s t r a i n t s M o d e l w i t h j o i n t l y d e p e n d e n t v a r i a b l e s S i m u l t a n o u s - e g u a t i o n m o d e l fiecursive model i i i C h a p t e r V A S t u d y on L a b o u r S u p p l y o f M a r r i e d Women M o d e l D e s c r i p t i o n 37 I n t r o d u c t i o n 37 S p e c i f i c a t i o n o f m o d e l s 38 S p e c i f i c a t i o n o f V a r i a b l e s 39 D a t a r e s t r i c t i o n 44 C h a p t e r V I E m p i r i c a l R e s u l t s o f M o d e l I 46 R e s u l t s f r o m t h e l a b o u r r e g r e s s i o n e g u a t i o n s 46 R e s u l t s f r o m t h e p r o b a b i l i t y e g u a t i o n 47 F u r t h e r e s t i m a t i o n 48 C h a p t e r V I I E m p i r i c a l R e s u l t s o f M o d e l I I 53 B e s u l t s o f s i n g l e e g u a t i o n e s t i m a t i o n 53 R e s u l t s o f Z e l l n e r ' s s e e m i n g l y l e a s t s g u a r e s method 54 R e s u l t s o f p r o b a b i l i t y f u n c t i o n s 59 C h a p t e r V I I I C o n c l u s i o n 86 A p p e n d i x A L e a s t - s q u a r e s e s t i m a t i o n 88 A p p e n d i x B L i k e l i h o o d r a t i o t e s t f o r m i c r o r e g r e s s i o n c o e f f i c i e n t v e c t o r e g u a l i t y 90 A p p e n d i x C P a r a m e t e r e s t i m a t e s f o r l a b o u r e g u a t i o n 1967 - 71 w i t h o u t (Unemploy) * 92 B i b l o g r a p h y 93 i v L i s t o f T a b l e s I The P a r a m e t e r E s t i m a t e s f o r L a b o u r E q u a t i o n s 50 I I Mean and S t a n d a r d D e v i a t i o n o f t h e M o d e l I 51 I I I P r o b a b i l i t y 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 52 IV A P a r a m e t e r E s t i m a t e s f o r t h e L a b o u r E q u a t i o n s , 1967 66 IV B P a r a m e t e r E s t i m a t e s f o r t h e L a b o u r E q u a t i o n s , 1968 67 I V C P a r a m e t e r E s t i m a t e s f o r t h e L a b o u r E q u a t i o n s , 1969 68 IV D P a r a m e t e r E s t i m a t e s f o r t h e L a b o u r E q u a t i o n s , 1970 69 I V E P a r a m e t e r E s t i m a t e s f o r t h e L a b o u r E q u a t i o n s , 1971 70 V A Mean and S t a n d a r d D e v i a t i o n o f t h e M o d e l I I , 1967 71 V B Mean and S t a n d a r d D e v i a t i o n o f t h e M o d e l I I , 1968 72 V C Mean and S t a n d a r d D e v i a t i o n o f t h e M o d e l I I , 1969 73 V D Mean and S t a n d a r d D e v i a t i o n o f t h e M o d e l I I , 1970 74 V E Mean and S t a n d a r d D e v i a t i o n o f t h e M o d e l I I , 1971 75 VI A The 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 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 , 1967 76 VI B The 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 77 VI C The 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 I I , 1969 78 V I D The 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 and 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 , 1970 79 VI E The 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 , 1971 80 V V I I A P r o b a b i l i t y 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 , 1967 81 V I I B P r o b a b i l i t y 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 , 1968 81 V I I C P r o b a b i l i t y 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 , 1969 83 V I I D P r o b a b i l i t y 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 , 1970 84 V I I E P r o b a b i l i t y 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 1 7 1 , 1971 85 I would l i k e to express my gratitude to my thesis committee members who offered me their advice, helpful c r i t i c i s m s , stimulating suggestions and assistance. Special thanks are due to Professor Press for the idea which made t h i s thesis possible, his encouragement, and for the permission to use his l o g i s t i c program to compute the data. Special thanks are also due to Professor Berndt for generously sharing his research work on the labour supply of married women, and providing access to information on the Panel Data. I also thank my mathematics teacher. Professor Nash, for his valuable comments. I am indebted tc Professor Doll for providing me a fellowship from the Centre for Transportation Studies during my f i r s t year study, and to Professor Press f o r offe r i n g me a research assistanship from his research fund of Chicago University i n my second year. Without their f i n a n c i a l support, the thesis i n i t s present form could not have been achieved. I would also l i k e to express my gratitude to Miss Nancy Reid for sharing her early results which we did for the Economic and S t a t i s t i c Workshop at the beginning of t h i s year; to Miss Norine Smith, research assistant of Professor Berndt for her help to access data from the Panel Data f i l e , and to Ms Valda Johnston for her beautiful e d i t o r i a l work. Wing Lau November o f 1975. 1 Chapter I I n t r o d u c t i o n T h i s t h e s i s 1 i s concerned w i t h making i n f e r e n c e s about r e l a t i o n s h i p s t h a t d e t e r m i n e j o i n t l y a c o n t i n u o u s v a r i a b l e and a c a t e g o r i c a l v a r i a b l e . G i v e n the d i s c r e t e random v a r i a b l e , t h e c o n t i n u o u s v a r i a b l e i s r e l a t e d t o a s e t o f e x p l a n a t o r y v a r i a b l e s ; a l s o t h e d i s c r e t e random v a r i a b l e i s r e l a t e d t o t h e same s e t of e x p l a n a t o r y v a r i a b l e s . For example, the t i m i n g o f a 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 a c t i v i t y w i l l l i k e l y depend upon the age o f her youngest c h i l d . L e t us assume t h a t a h ousewife w i l l work l e s s hours when she has a c h i l d not o l d e r than 6 y e a r s . We w i l l c a l l z, a f a m i l y c o n s t r a i n t v a r i a b l e , i f i t i s 1 when a f a m i l y has a c h i l d o f 6 y e a r s o r younger, and 0 o t h e r w i s e . The t i m i n g of her l a b o u r f o r c e a c t i v i t y i n each c a s e , t h a t i s f o r a g i v e n v a l u e of z, may r e l a t e t o her hushand's income, her expected wage, her f e c u n d i t y and so on. A l s o t h e p r o b a b i l i t y o f a f a m i l y h a v i n g a c h i l d not o l d e r t h a n 6 y e a r s can be r e l a t e d t o t h e same s e t o f e x p l a n a t o r y v a r i a b l e s . We would l i k e t o know t h e j o i n t p r o b a b i l i t y o f her t i m i n g i n l a b o u r f o r c e a c t i v i t y as w e l l as her youngest c h i l d not b e i n g o l d e r than 6 y e a r s . I t i s shown t h a t we can e s t i m a t e t h e parameters of c o n d i t i o n a l r e g r e s s i o n e q u a t i o n s and t h e parameters of t h e d i s c r e t e dependent v a r i a b l e r e g r e s s i o n i n d i v i d u a l l y . I t i s v e r y easy t o e x t e n d t h e problem t o i n v o l v e a polytomous dependent v a r i a b l e i n r e g r e s s i o n r a t h e r t h a n a dichotomous dependent v a r i a b l e . 2 To c o n s i d e r t h e above p r o b l e m i n t e r m s o f a r e g r e s s i o n m o d e l , t h e model c a n be e x t e n d e d i n t o a s e t o f r e g r e s s i o n e q u a t i o n s i n c l u d i n g a s e t o f c o n d i t i o n a l r e g r e s s i o n s a n d a s e t o f d i s c r e t e d e p e n d e n t v a r i a b l e r e g r e s s i o n s . F o r e x a m p l e , a t y e a r t , f o r a g i v e n v a l u e o f z , t h e t i m i n g o f a m a r r i e d woman i n l a b o u r f o r c e a c t i v i t y c a n be e x p r e s s e d i n t e r m s o f a s e t o f e x p l a n a t o r y v a r i a b l e s , and t h e p r o b a b i l i t y o f a f a m i l y h a v i n g a c h i l d n o t o l d e r t h a n 6 y e a r s i s r e l a t e d t o t h e same s e t o f e x p l a n a t o r y v a r i a b l e s . H e n c e , we may c o n s i d e r s u c h r e l a t i o n s f o r c e r t a i n t i m e p e r i o d s , s u c h a s f r o m 1967 t o 1971. I t i s f o u n d t h a t t o a p p l y Z e l l n e r * s method{1962) t o e s t i m a t e t h e p a r a m e t e r s o f c o n d i t i o n a l r e g r e s s i o n e q u a t i o n s i s more e f f i c i e n t t h a n t o e s t i m a t e them e q u a t i o n - b y - e q u a t i o n . Some i n t e r e s t i n g e x t e n s i o n s o f t h i s m o d e l a r e m e n t i o n e d . The p l a n o f t h i s t h e s i s i s a s f o l l o w s . T h i s t h e s i s i s d i v i d e d i n t o a t h e o r e t i c a l p a r t and an a p p l i c a t i o n a l p a r t . The f i r s t p a r t i s a t h e o r e t i c a l d i s c u s s i o n composed o f c h a p t e r s 2, 3, and U. I n c h a p t e r 2 we d e s c r i b e t h e b a s i c m o d e l and p r o v e t h a t t h e e s t i m a t o r s o f t h e p a r a m e t e r s i n c o n d i t i o n a l r e g r e s s i o n e q u a t i o n s and t h e p a r a m e t e r s i n d i s c r e t e d e p e n d e n t v a r i a b l e r e g r e s s i o n a r e i n d e p e n d e n t , I n e s t i m a t i n g p a r a m e t e r s o f d i s c r e t e d e p e n d e n t v a r i a b l e r e g r e s s i o n , we e x p l a i n why we p r e f e r l o g i t r a t h e r t h a n o t h e r f u n c t i o n s . Chow's t e s t (1960) i s u s e d f o r t e s t i n g t h e e q u a l i t y o f c o e f f i c i e n t s o f two c o n d i t i o n a l r e g r e s s i o n e q u a t i o n s . I n c h a p t e r 3 we e x t e n d t h e b a s i c model i n t o a s e t o f r e g r e s s i o n e q u a t i o n s . The method o f e s t i m a t i n g e g u a t i o n - b y - e q u a t i o n i s m e n t i o n e d . Z e l l n e r ' s s e e m i n g l y 3 u n r e l a t e d r e g r e s s i o n s method i s a p p l i e d i n o r d e r t o g e t an e f f i c i e n t e s t i m a t i o n . F o l l o w i n g i s t h e d e s c r i p t i o n o f t e s t i n g t h e a g g r e g a t i o n M a s . C h a p t e r 4 i s t o p r o p o s e some i n t e r e s t i n g e x t e n s i o n s o f t h i s b a s i c m o d e l . The s e c o n d p a r t i s an a p p l i c a t i o n c o n c e r n i n g an e c o n o m i c p r o b l e m o f l a b o u r s u p p l y o f m a r r i e d women. The d a t a u s e d a r e 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, w h i c h i s c o l l e c t e d by t h e S u r v e y R e s e a r c h C e n t r e o f t h e U n i v e r s i t y o f M i c h i g a n . D a t a a r e f o c u s e d p r i m a r i l y on c h a n g e i n f a m i l y e c o n o m i c s t a t u s . D a t a - c o l l e c t i o n t e c h n i g u e i s m a i n l y on t h e h o u s e h o l d p e r s o n a l i n t e r v i e w . The e m p i r i c a l s t u d i e s a r e compounded by two m o d e l s w h i c h a r e b a s e d on c h a p t e r s 2 and 3. C h a p t e r 5 i s t h e d e s c r i p t i o n o f o u r e c o n o m i c m o d e l s , c h a p t e r 6 g i v e s t h e r e s u l t s u n d e r t h e f i r s t m odel and c h a p t e r 7 g i v e s t h e r e s u l t s u n d e r t h e s e c o n d m o d e l . C h a p t e r 8 w i l l be t h e c o n c l u s i o n o f t h e w h o l e t h e s i s . F o o t n o t e s 1 T h i s i s b a s e d on t h e p r e l i m i n a r y work done by Nancy R e i d and me i n t h e E c o n o m e t r i c s and S t a t i s t i c s Workshop t h i s y e a r a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a . M i s s R e i d i s a m a s t e r ' s s t u d e n t o f t h e I n s t i t u t e o f A p p l i e d M a t h e m a t i c s and S t a t i s t i c s o f t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a . 4 Chapter I I E a s i c Model I The mathematical model l e t us begin with a very simple model which has two dependent v a r i a b l e s , one of which i s y, a continuous v a r i a b l e , and the other i s z, a dichotomous v a r i a b l e having the value 0 or 1. For the given value of z, y i s d i s t r i b u t e d normally, and i s 1 k expressed as a f u n c t i o n of a number of v a r i a b l e s 1 x , x . Since z i s dichotomous, we denote the f u n c t i o n a l r e l a t i o n s of y and the x's as the f o l l o w i n g : 1 k E(y | z=1) = f (x , x ) and 1 k E(y | z=0) = g(x , x ) There w i l l be a v a r i e t y of f u n c t i o n s to s a t i s f y the above r e l a t i o n s . The s i m p l e s t r e l a t i o n s h i p between y and the x's i s l i n e a r . So f o r n o b s e r v a t i o n s , we w r i t e each of them more f o r m a l l y under the l i n e a r hypothesis as: 1 1 k k (y (i) Jz (i) =1) = a x (i) + ... + a x (i) + u (i) and 1 1 k k (y (i) f z ( i ) = 0 ) = b x (i) + ... + b x (i) + v (i) i = 1 , n or i n vec t o r n o t a t i o n , 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 1 k where A= (a , ...,a )*, B=(b ,...,b )», and X= (x <i),...,x (i) ) • ; u, and v denote variables which may take on positive or negative values. Usually u and v are c a l l e d error terms or disturbance terms. In order to make the model simple, l e t us f i r s t assume u and v have the same d i s t r i b u t i o n . We assume u and v are random and normally d i s t r i b u t e d with mean zero, variance var (u) and zero covariance, that i s E[ u (i) ] = 0 E[u(i) ,u (j) ] = 0 ± * j = var(u) i = j Hence A, B and var(u) are unknown parameters. He may wish to estimate these parameters s t a t i s t i c a l l y on the basis cf our sample observations, and to test hypotheses about them. Therefore, i f we consider the conditional d i s t r i b u t i o n s of y, then when z=1, y i s distributed normally with mean A'X and variance var(u), and when z=0, y i s dist r i b u t e d normally with mean B'X and variance var(u). For the dichotomous dependent variable, we may be interested i n the probability that z w i l l have the value 0 or 1. The probability of z being 1 can also be expressed as a function 1 k of X , ...,x . So, 6 1 k P r o b ( z = 1 ) = h (x , x ) S u p p o s e t h a t we w a n t a r e l a t i o n s h i p i n w h i c h P r o b ( z = 1 ) i s a n o n d e c r e a s i n g f u n c t i o n o f t w i t h FI-OOJ^O a n d F(o») = 1 ; f o r t h e i t h o b s e r v a t i o n , P r o b ( z ( i ) = 1) = p ( i ) = F ( t ( i ) ) i = 1 , . . . , n 1 1 k k w h e r e t ( i ) = c x ( i ) + . . . + c x ( i ) , o r i n v e c t o r n o t a t i o n : P r o b ( z ( i ) = 1 ) = p ( i ) = F ( C « X ( i ) ) (1) F ( C » X ( i ) ) i s t a k e n t o b e a c u m u l a t i v e d i s t r i b u t i o n f u n c t i o n 2 . T h e r e f o r e , we k n o w p ( i ) w i l l l i e b e t w e e n 0 a n d 1, a n d p ( i ) i s a n o n d e c r e a s i n g f u n c t i o n o f C * X , b u t may b e d e c r e a s i n g i n s o m e v a r i a b l e s , d e p e n d i n g u p o n t h e s i g n s o f t h e c o m p o n e n t s o f C . T h e r e f o r e , we w i l l f o c u s o u r i n t e r e s t o n t h e e s t i m a t i o n o f C . B e f o r e we s t e p i n t o t h e e s t i m a t i o n o f t h e s e p a r a m e t e r s , we s h o u l d d i s c u s s m o r e a b o u t f u n c t i o n F , b e c a u s e i n t h e h i s t o r y o f s t a t i s t i c s , t h e r e w a s a l o n g a r g u m e n t a b o u t F . I n t h e f o l l o w i n g s e c t i o n , we w i l l d i s c u s s s e v e r a l t r a n s f o r m a t i o n f u n c t i o n s a n d e x p l a i n why t h e l o g i s t i c f u n c t i o n i s c h o s e n . I I D i s c r e t e d e p e n d e n t v a r i a b l e r e g r e s s i o n F a i l u r e o f l i n e a r a p p r o x i m a t i o n t o t h e p r o b a b i l i t y f u n c t i o n : I f we u s e l i n e a r a p p r o x i m a t i o n t o t h e p r o b a b i l i t y f u n c t i o n F ( C ' X ) , t h e n we w i l l o b s e r v e t h a t t h e f u n c t i o n i s w e l l a p p r o x i m a t e d i n t h e c e n t r e , b u t p o o r f o r v e r y l a r g e o r s m a l l 7 v a l u e o f C'X. T h e r e a r e t e c h n i c a l d i f f i c u l t i e s i n u s i n g s t a n d a r d r e g r e s s i o n t e c h n i q u e s on b i n a r y d a t a 3 . F i r s t , f o r g i v e n o b s e r v a t i o n x ( i ) , z ( i ) i s a B e r n o u l l i random v a r i a b l e , so t h a t t h e v a r i a n c e o f t h e j t h d i s t u r b a n c e t e r m d e p e n d s upon j . T h o s e d i s t u r b a n c e t e r m s a r e h e t e r o s c e d a s t i c , t h e r e f o r e , o r d i n a r y l e a s t - s g u a r e s e s t i m a t i o n w i l l g i v e i n e f f i c i e n t e s t i m a t o r s and i m p r e c i s e p r e d i c t i o n s . Z e l l n e r a n d Lee ( 1 9 6 5 ) , G o l d b e r g e r (1964) s u g g e s t e d t h e u s e o f g e n e r a l i z e d l e a s t - s g u a r e s t o remove t h e h e t e r o s c e d a s t i c p r o b l e m , b u t t h i s f a i l e d , b e c a u s e i t i g n o r e d t h e B e r n o u l l i c h a r a c t e r o f t h e e r r o r s , o r d i d n o t g u a r a n t e e t h a t z ( i ) s h o u l d l i e b e t w e e n 0 and 1 f o r a l l i , a n d r e s u l t e d i n some n e g a t i v e v a r i a n c e s . F u r t h e r m o r e , t h e t r a n s f o r m a t i o n on g e n e r a l i z e d l e a s t - s g u a r e s l e d i n t o t h e n u m e r i c a l p r o b l e m t h a t i f t h e i n d e p e n d e n t v a r i a b l e i s l a r g e r t h a n 1, t h e t r a n s f o r m a t i o n i s u n d e f i n e d . Cox (1970) c o n c l u d e s t h a t l i n e a r a p p r o x i m a t i o n t o t h i s f u n c t i o n f a i l s , s i n c e , " b e c a u s e t h e z ( i ) * s a r e n o r m a l l y d i s t r i b u t e d , no method o f e s t i m a t i o n t h a t i s l i n e a r i n t h e z ( i ) * s w i l l i n g e n e r a l be f u l l y e f f i c i e n t . " P r o b i t a n a l y s i s : One r e a s o n a b l e a p p r o a c h i s c a l l e d p r o b i t a n a l y s i s . B l i s s (1934) was t h e f i r s t t o u s e i t . F i n n y ( 1 9 4 7 ) a p p l i e d t h i s method i n a n a l y z i n g g u a n t a l ( b i n a r y ) r e s p o n s e s i n b i o a s s a y , C o r n f i e l d and N a t h a n H a n t a l ( 1 9 5 0 ) a p p l i e d i t i n c a l c u l a t i n g t h e d o s a g e r e s p o n s e c u r v e , and T o b i n ( 1 9 5 5 ) a p p l i e d i t i n e c o n o m i c s u r v e y s . 8 T h i s method a p p l i e s a g r o u p i n g method f o r e s t i m a t i n g t h e e g u a t i o n ( 1 ) . F (t ) i s c o n s i d e r e d as t h e c u m u l a t i v e d i s t r i b u t i o n f u n c t i o n o f t h e s t a n d a r d n o r m a l d i s t r i b u t i o n by u s i n g g r o u p e d d a t a * , Hence p ( i ) i s e s t i m a t e d by s a m p l e p r o p o r t i o n , i . e , p + ( i ) = r ( i ) / n ( i ) i=1,...,n where r ( i ) i s t h e number o f e l e m e n t s i n i t h c e l l h a v i n g v a l u e 1, and n ( i ) i s t h e t o t a l number o f e l e m e n t s i n t h e i t h c e l l . We d e f i n e p r o b i t a s f o l l o w i n g , P r o b i t ( p + ( i ) ) = t + ( i ) + 5 (2) where t + ( i ) i s d e f i n e d by p+ ( i ) = F (t+ ( i ) ) . F i s t h e c u m u l a t i v e s t a n d a r d i z e d n o r m a l d i s t r i b u t i o n . One a d d s 5 i n t h e t r a n s f o r m a t i o n i n o r d e r t o g e t p o s i t i v e v a l u e s f o r t h e t r a n s f o r m e d v a r i a b l e . Hence P r o b i t ( p * ( i ) ) i s n o r m a l l y d i s t r i b u t e d w i t h mean 5 and v a r i a n c e 1. So we c a n a p p l y o r d i n a r y l e a s t - s g u a r e s t o t h e t r a n s f o r m e d d a t a . P u t t i n g i t i n t o a r e g r e s s i o n e g u a t i o n , i t w i l l be P r o b i t ( p * ( i ) ) = C * X ( i ) + e ( i ) i = 1 , ...,n (3) where e ( i ) w i t h z e r o mean, z e r o c o v a r i a n c e , and v a r i a n c e e g u a l t o v a r (e ( i ) ) . F i n a l l y , we n o t e w i t h P r e s s and N e r l o v e (1973) t h a t : " F o r t h i s p r o b i t a n a l y s i s method t o be u s e f u l , t h e r e s h o u l d be s e v e r a l o b s e r v a t i o n s p e r c e l l (n ( i ) > f o r e v e r y i ) . M o r e o v e r , 9 e f f i c i e n c y o f e s t i m a t i o n i s l o s t i n t h e ad hoc p r o c e d u r e a s s o c i a t e d w i t h t h e added 5 i n ( 2 ) . N o t e a l s o t h a t t h e r e a r e c o m p u t a t i o n a l d i f f i c u l t i e s a s s o c i a t e d w i t h t h e use o f t h e i n t e g r a l s i n t h i s p r o c e d u r e . U n e q u a l numbers o f o b s e r v a t i o n s p e r c e l l a r e i n e f f i c i e n t , and c e l l s w i t h one o r z e r o o b s e r v a t i o n s p e r c e l l a r e n o t u s e f u l . " L o g i t a n a l y s i s : A n o t h e r method c a l l e d l o g i t a n a l y s i s was i n t r o d u c e d by B e r k s o n ( 1 9 4 4 ) . U s i n g c e l l f r e q u e n c y , F ( t ) i s c o n s i d e r e d a s t h e c u m u l a t i v e d i s t r i b u t i o n f u n c t i o n o f t h e s t a n d a r d l o g i s t i c d i s t r i b u t i o n f u n c t i o n ; t h a t i s , F ( t ) = 1 / ( 1 + e x p ( - t ) ) where t i s r e a l ; s o P ( i ) = V [ 1+exp {-C'X ( i ) ) ] o r , l o g ( p ( i ) / ( 1 - p ( i ) ) = C«X(i) Now we d e f i n e L o g i t ( p + ( i ) ) a s f o l l o w i n g , L o g i t ( p + ( i ) ) = l o g ( p * ( i ) / ( 1 - p + ( i ) ) ) where p + ( i ) i s e s t i m a t e d by s a m p l e p o r t i o n . C c a n be e s t i m a t e d f r o m r e g r e s s i o n e s t i m a t i o n . B i s h o p ( 1 9 6 9 ) , Goodman (1970) , P r e s s and N e r l o v e ( 1 9 7 3 ) a p p l y t h i s method i n d e a l i n g w i t h c o n t i n g e n c y t a b l e s . 10 O t h e r t r a n s f o r m a t i o n s ; C o l e m a n ( 1 9 6 4 ) h a s p r o p o s e d an e x p o n e n t i a l model by c h o o s i n g p ( i ) = l - e x p ( - X ' C ) The w e akness o f t h i s f u n c t i o n i s t h a t p ( i ) i s n o t c o n s t r a i n e d t o l i e b e t w e e n z e r o and one u n l e s s a l l o f t h e p a r a m e t e r s a r e n o n - n e g a t i v e . Goodman (1972) made a comment a b o u t t h i s t r a n s f o r m a t i o n . He s a i d , " C o l e m a n ' s a r t i c l e d i d n o t show how t o t e s t w h e t h e r h i s m o d e l f i t t h e a c t u a l d a t a , n o r was he a b l e t o measure how w e l l i t f i t . F u r t h e r m o r e , he d i d n o t show how t o t e s t t h e s t a t i s t i c a l s i g n i f i c a n c e o f t h e c o n t r i b u t i o n made by t h e v a r i o u s p a r a m e t e r s i n t h e m o d e l , n o r c o u l d he measure t h e i r c o n t r i b u t i o n ' s m a g n i t u d e . " A n g u l a r t r a n s f o r m a t i o n s a r e v e r y p o s s i b l e c a n d i d a t e s , b u t t h o s e t r a n s f o r m a t i o n s a r e n o t a s s i m p l e a s t h e l o g i s t i c t r a n s f o r m a t i o n . So o u r s e l e c t i o n i s l i m i t e d t o p r o b i t and l o g i t a n a l y s i s . The c h o i c e o f t r a n s f o r m a t i o n : The a b o v e d i s c u s s i o n s h o w s , o u r c h o i c e w i l l be e i t h e r p r o b i t o r l o g i t a n a l y s i s . G u n d e r s o n ( 1 9 7 4 ) , B u s e ( 1 9 7 2 ) , Chambers and C o x ( 1 9 6 7 ) h a v e done some work a b o u t t h i s p r o b l e m . They f o u n d t h a t t h e n u m e r i c a l d i f f e r e n c e between t h e s e two i s v e r y s l i g h t e x c e p t a t t h e two e x t r e m e s . From t h e o p t i m i z a t i o n p o i n t o f v i e w , i t i s s u r e t h a t t h e maximum o f a l o g i s t i c f u n c t i o n i s t h e g l o b a l maximum. I f we c o n s i d e r t h e c o s t o f c o m p u t a t i o n , 11 l o g i t a n a l y s i s i s much b e t t e r t h a n p r o b i t a n a l y s i s . On t h e o t h e r h a n d , t h e r e i s t h e t h e o r e t i c a l a r g u m e n t t h a t t h e p r o b i t t r a n s f o r m a t i o n i s t h e a p p r o p r i a t e one t o use u n d e r t h e h y p o t h e s i s o f l o g - n o r m a l l y d i s t r i b u t e d t o l e r a n c e s . B e r k s o n (1951) i s v e r y d o u b t f u l a s t o t h e v a l i d i t y o f t h i . s h y p o t h e s i s . He s a y s t h a t t h e p r a c t i c e o f i n j e c t i n g an i n t e r p r e t a t i o n o f " t o l e r a n c e " i n t o r e s p o n s e d a t a i s o b j e c t i o n a b l e ; i t c a n be m i s l e a d i n g and h a r m f u l . He e x p l a i n s t h a t i f on t h e o t h e r hand t h e f o r m u l a t i o n i s o n l y t h a t o f a " m a t h e m a t i c a l m o d e l " , t o g u i d e t h e method o f c a l c u l a t i o n , t h e n i t w o u l d seem more o b j e c t i v e and h e u r i s t i c a l l y s o u n d e r n o t t o c r e a t e any h y p o t h e t i c a l t o l e r a n c e s , b u t m e r e l y t o p o s t u l a t e t h a t t h e p r o p o r t i o n o f r e s p o n s e s a f f e c t e d f o l l o w s t h e i n t e g r a t e d n o r m a l f u n c t i o n . F o r t h e s e r e a s o n s , t h e l o g i t a n a l y s i s i s p r e f e r r e d . I l l E s t i m a t i o n We r e c a l l t h e c o n d i t i o n a l d i s t r i b u t i o n s o f y when z=1, i s 1 N(A'X, v a r ( u ) ) , and when z=0, y i s N(B*X, v a r ( u ) ) where A = (a , k 1 k 1 k . . . , a ) ' , B = ( b , b ) • and X =(x , x ) * ; p r o b (z=1) = 1 k F(X'C) where C = ( c , ..., c ) ' . Thus t h e j o i n t d e n s i t y o f y and z w i l l be f ( y , z ) . I f we e x p r e s s t h e j o i n t l i k e l i h o o d f u n c t i o n u s i n g m a t r i x a l g e b r a , t h e n f (Y,Z|P,X) = h ( Y | Z , P , X ) g ( Z | P , X ) (4) 12 Hhere, P= (A* ,B« ,C» , var (u)) •, X i s a nxk matrix, i . e . n observations and k dimensions. For each observation i , h(y(i) |z(i) ,P,X(i) ) = (2lTvar (u) )-o.s e Xp{[y ( i ) - (X (i) »Az (i)+X (i) »B (1-z (i)) p y (2var (u) ) } and, z(i) (1-z(i)) g(z{i) |P,X(i) ) =F(X(i)»C) £ 1-F(X(i) • C) ] Let L(P) be the j o i n t l i k e l i h o o d function, L (P) i s proportional to the products of h and g, that i s n L(P) : TT h(y (i) | z (i) ,P,X <i) ) g (z(i) |P,X(i)) i=1 n log(L) : T. {z (i)log[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} 2 i=1 - (n/2) log (var (u))/2 - (n/2) log (2?T) (5) Estimation of l o g i t parameter: This i s just as Dempster (1972) points out that the joint density of Y and Z i n (4) can be factorized into two functions, h and g, which depend on d i s j o i n t parameter sets. The maximum li k e l i h o o d estimators of a l l the parameters can be found by maximizing these two functions separately. The function g i s a log l i k e l i h o o d from a fixed l o g i t model, and function h i s just a multivariate general linear regression model. There are several methods to estimate l o g i t parameters. 13 B e r k s o n ( 1 9 5 5 ) i n t r o d u c e d a method c a l l e d "minimum C h i s q u a r e " whose r e s u l t s a r e a s y m p o t o t i c a l l y e q u i v a l e n t t o t h e maximum l i k e l i h o o d e s t i m a t i o n . T h e i l ( 1 9 7 0 ) s u g g e s t e d t h e use o f t h e g e n e r a l i z e d l e a s t - s g u a r e s method. B o t h a r e a p p l i c a b l e o n l y i n l a r g e s a m p l e s o r d e s i g n e d e x p e r i m e n t s , s i n c e i n T h e i l ' s method one d e l e t e s t h o s e c e l l s w h i c h c o n t a i n o n l y one o r no o b s e r v a t i o n s , and i n B e r k s o n ' s method one r e q u i r e s more t h a n one o b s e r v a t i o n p e r c e l l . Goodman (1972) u s e d t h e maximum l i k e l i h o o d e s t i m a t i o n i n l o g i t a n a l y s i s , and he f o u n d t h a t he g o t a somewhat s m a l l e r v a r i a n c e f r o m MLE t h a n T h e i l ' s e s t i m a t i o n f r o m w e i g h t e d l e a s t - s q u a r e s . One d i s a d v a n t a g e o f MLE i s t h a t i t t a k e s more c o m p u t a t i o n a l t i m e . T h i s t h e s i s a d o p t s t h e method o f maximum l i k e l i h o o d e s t i m a t i o n and u s e s t h e c o m p u t e r p r o g r a m w h i c h i s d e v e l o p e d by P r e s s and N e r l o v e ( 1 9 7 3 ) . The method c a n be s u m m a r i z e d a s f o l l o w s : n L { g ( Z | P , X ) ) = T T g ( z { i ) | P , X ( i ) ) i=1 n z ( i ) ( 1 - z ( i ) ) = T T £P(X(i) «C) ] [ 1 - F ( X ( i ) »c) ] i = 1 D e f i n e T + a s t h e sum o f X { i ) z ( i ) , where i r u n s f r o m 1 t o n. T+ i s a s u f f i c i e n t s t a t i s t i c f o r C, i . e . T+ i s t h e sum o f t h o s e X ( i ) f o r w h i c h z ( i ) = 1 . Hence C+, t h e MLE o f C must s a t i s f y n [ 1 + e x p ( - X ( i ) «C+) j - * X ( i ) = T+ = Z l X ( i ) z ( i ) (6) i= 1 N o t e l o g ( L ) i s g l o b a l l y c o n c a v e 5 , s o (5) p r o v i d e s an a b s o l u t e maximum. Hence, P + ( i ) = C 1 + e x p ( - X ( i ) «C+) ] - i T h e r e a r e many n u m e r i c a l methods by w h i c h L c a n be m a x i m i z e d . The p r o g r a m d e v e l o p e d by P r e s s and N e r l o v e ( 1973) u s e s t h e F l e t c h e r - P o w e l l method o f f u n c t i o n m i n i m i z a t i o n a n d t h e D a v i d o n a l g o r i t h m 6 f o r c o m p u t i n g t h e i n v e r s e o f t h e i n f o r m a t i o n m a t r i x . E s t i m a t i o n o f c o n d i t i o n a l p a r a m e t e r s : I n o r d e r t o e s t i m a t e A,B and v a r (u) by u s i n g MLE method, we s e t af/5>A = 0, 3 f / 3 8 = 0, and 3f/cDvar{u) = 0. F o r e x a m p l e , t o g e t A + , t h e e s t i m a t o r o f A ; s i n c e ( 1 - z ( i ) ) z ( i ) = 0, we c a n s o l v e A + f r o m S f / t D A = 0, t h e n we h a v e , &+ = {Xi« X* ]-iX» ' y i where X 1 i s nxk m a t r i x , f o r e a c h o b s e r v a t i o n i , X 1 ( i ) = 1 k [ x ( i ) z ( i ) , x ( i ) z ( i ) ] ' , i = 1 , . . . , n and Y» = [ y ( 1 ) z { 1 ) , , y ( n ) z ( n ) ] ' . S i m i l a r l y , B+ = [ ( X 2 ) • ( X 2 ) 3 ~ i ( X 2 ) » Y 2 where X 2 i s nxk m a t r i x , f o r e a c h o b s e r v a t i o n i , X 2 ( i ) = 1 k £x ( i ) p - z ( i ) ) , x ( i ) ( 1 - z ( i ) ) J«, i = 1 , . , . , n and y 2 = 15 [ y (1) n-z O)) * y (n) (1-z (n) ) ] * . The e s t i m a t e d v a r i a n c e w i l l be V a r + ( u ) = (Y-Y+) 1 (Y-Y+) /n where Y + = ( X l ) *A + + ( X 2 ) • B + . I f we p r e f e r t o use t h e u n b i a s e d e s t i m a t o r o f v a r + ( u ) t h e n V a r + ( u ) = (Y-Y + ) * (Y-Y + ) / ( n - k ) T h o s e r e s u l t s a r e n o t s t r a n g e t o u s . I f we s p l i t o u r s a m p l e i n t o two g r o u p s , one c o n t a i n s a l l z=1 and t h e e t h e r c o n t a i n s a l l z=0, and i f we a p p l y o r d i n a r y l e a s t - s g u a r e s on e a c h g r o u p , we w i l l g e t t h e same r e s u l t s ( A p p e n d i x A ) . 8e know A+ and B + a r e u n b i a s e d . The c o v a r i a n c e m a t r i c e s o f A + and B + a r e c o v (A+) = V a r + ( u ) [ (XV) » ( X 1 ) ] ~ l cov(B+) = V a r + ( u ) [ ( X 2 ) ' ( X 2 ) J " 1 E s t i m a t i o n o f u n e q u a l v a r i a n c e s o f c o n d i t i o n a l m o d e l : I f we r e l a x t h e c o n d i t i o n t h a t u and v have t h e same d i s t r i b u t i o n , t h e n a s shown i n A p p e n d i x A, we may o b s e r v e t h a t t h e e s t i m a t i o n s o f A and E a r e same a s b e f o r e , b u t Var+(u) = ( Y 1 - ( X l ) A + ) ' ( Y 1 - ( X 1 ) A + ) / (n *-k) v a r + ( v ) = ( Y 2 - ( X 2 ) B + ) 1 ( Y 2 - ( X 2 ) B + ) / ( n 2 - k ) cov(B+) = Var+ (v) [ ( X 2 ) » ( X 2 ) J - i 16 where n l i s t h e t o t a l number o f o b s e r v a t i o n s when z = 1 , a n d n 2 •= n - n 1 . IV H y p o t h e s i s t e s t i n g L o g i s t i c m o d e l : F o r 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 C + by u s i n g t h e f a c t t h a t C+ i s a s y m p t o t i c a l l y n o r m a l . I t s c o v a r i a n c e m a t r i x i s o b t a i n e d f r o m t h e i n v e r s e o f i t s i n f o r m a t i o n m a t r i x I ( C + ) , where i j i J i(c+) = [ a 2 L (c+)/ac ac ] = {a 2g (c+)/ac ac j i , j = i , . . . , k A l s o , any h y p o t h e s i s a b o u t C + c a n be t e s t e d by u s i n g a l i k e l i h o o d r a t i o t e s t ( A p p e n d i x B ) . The l i k e l i h o o d r a t i o , r i s t h e r a t i o o f t h e v a l u e o f t h e l i k e l i h o o d f u n c t i o n g m a x i m i z e d u n d e r t h e c o n s t r a i n t s o f t h e h y p o t h e s i s b e i n g t e s t e d t o t h e v a l u e m a x i m i z e d w i t h o u t c o n s t r a i n t s . I n l a r g e s a m p l e s , t h e v a l u e o f -21og ( r ) i s d i s t r i b u t e d as C h i s q u a r e w i t h q d e g r e e s o f f r e e d o m ; g i s t h e number o f i n d e p e n d e n t r e s t r i c t i o n s i n t h e n u l l h y p o t h e s i s . C o n d i t i o n a l m o d e l : H y p o t h e s i s a b o u t A i s H: A = A ° , where A° i s a q i v e n v e c t o r . F o r e a c h component u n d e r H, we know 17 i i i i > (a+ - a" )/(w V a r ( u ) ) °. 5 h a s a s y m p t o t i c t - d i s t r i b u t i o n w i t h ql d e g r e e s o f f r e e d o m , i n i i w h i c h g* > 0; w i s t h e i t h d i a g o n a l e l e m e n t o f ( ( X 1 ) * X 1 ) " " 1 # and g 1 i s t h e d i f f e r e n c e between m, t h e number o f o b s e r v a t i o n s when z=1, and k, t h e number o f i n d e p e n d e n t v a r i a b l e s . S i n c e t 2 i s d i s t r i b u t e d a s F d i s t r i b u t i o n , we c a n t e s t t h e h y p o t h e s i s u s i n g F - t e s t . The r a t i o i s d i s t r i b u t e d w i t h F ^ q 1 ) . An a l t e r n a t i v e way t o t e s t t h e h y p o t h e s i s i s H o t e l l i n g ' s T 2 t e s t . S i n c e (A +-A°) ' ( ( X 1 ) ' X 1 ) - 1 (A+-A 0) / ( v a r ju) ) i s a H o t e l l i n g ' s T 2 where T 2 = kg»F ( k , g i - k + 1 ) / ( g l - k + 1 ) and F* ( k , q 1 - k+1) i s an u p p e r t a i l o f t h e p r o b a b i l i t y f u n c t i o n . S i m i l a r l y f o r B, H: B+=B°, where B° i s g i v e n , f o r e a c h component u n d e r H, we know i i i i (b+ - b° ) / ( r Var ( v ) ) o. s i i i s d i s t r i b u t e d a s t ( q 2 ) where g 2-n-m-k > 0, and r i s t h e i t h d i a g o n a l e l e m e n t o f { ( X 2 ) • ( X 2 ) S i m i l a r l y t h e h y p o t h e s i s c a n be t e s t e d by u s i n g F - t e s t , F { 1 , g 2 } . O r , u s i n g H o t e l l i n g ' s T 2 , H o t e l l i n g ' s T 2 i s (B+-B0) » ( ( X 2 ) ' X 2 ) ~ i (B+-B°)/(var (u) ) , where T 2 = k q 2 F ( k , g 2 - k + 1 ) / { g 2 - k + 1 ) . T e s t i n g e q u a l i t y b e t w e e n two c o n d i t i o n a l d i s t r i b u t i o n s : The c o n d i t i o n a l v a r i a b l e y g i v e n z may have t h e same 18 d i s t r i b u t i o n f o r d i f f e r e n t v a l u e s o f z , s o we a r e g o i n g t o t e s t t h e i r e g u a l i t y . We a p p l y Chow's t e s t { 1 9 6 0 ) t o t e s t H: A=B=F, and r e w r i t e t h e d i s t r i b u t i o n s i n t o l i n e a r m o d e l s . Y M i ) = (y ( i ) |z ( i ) =1) = X M i ) ' A + OB + u ( i ) y 2 ( i ) = (y ( i ) |z ( i ) = 0 ) OA + x 2 ( i ) ' B + v { i ) I n t h e r e , we assume t h e y have e g u a l v a r i a n c e and z e r o c o v a r i a n c e . Under t h e H t h e n Y i = x**F + 0 y2 = x 2 * F + V s o F i s e s t i m a t e d a s F+ = [ ( X i ,X 2) ( X i , X 2 ) * ] - i ( X i , X 2 ) (Y» ,Y 2) ' L e t E= ( U , V ) • t h e n E+'E + = [ ( Y i , Y 2 ) ' - ( X i , X 2 ) VF+]»£ ( Y i , Y 2 ) »-(Xi,X 2) *F+] (7) E + i s e s t i m a t e d f r o m t h e e n t i r e s a m p l e , s o E + , E + h a s r a n k n-k. Und e r t h e a l t e r n a t i v e h y p o t h e s i s A#B, we have U+'U + + V+'V+ = ( Y 1 - X 1 A + ) ' ( Y i - X lA+) + ( Y 2 - X 2 B+) » ( Y 2 - X 2 B + ) (8) U +'U+ h a s r a n k g l and v + ' V + h a s r a n k g 2 . U and V a r e i n d e p e n d e n t , so t h e r a n k o f U*'U++V+'V* i s g i + g 2 = n - 2 k . 19 r Y l - X i F + - , = r ^ - X i A + T + r X l A + - X i F + - , l y z - x z p + J «-X 2-X 2B +-' «-X 2B+-X 2F+J (9) | | ( Y i - X * F * , Y 2 - X 2 F + ) | | 2 = I | (Y l-X* A+, Y 2 - X 2 B + ) I I2 + | | ( X 1 A + - X 1 F + , X 2 B + - X 2 F , + ) J | 2 + c r o s s p r o d u c t t e r m s S i n c e t h e c r o s s p r o d u c t t e r m i s z e r o , s o t h e s g u a r e on t h e l e f t o f (9) i s t h e sum o f s g u a r e s on t h e r i g h t , t h a t i s | I ( Y * - X * F + , Y 2 - X 2 F + ) j | 2 = | | (Y l-XVA + ,Y 2-X 2B+) || 2 + | | ( X i A + - X i F + , X 2 B + - X 2 F + ) | \z (10) o r s a y , Q 1 = Q 2 + Q 3 From t h e e s t i m a t i o n s o f A, B and F, we g e t ( X i ' X i + X 2 f X 2 ) F + = X i ' Y i + X 2 , Y 2 = X*»X»A* + X 2 » X 2 B + w h i c h i m p l i e s B+-F+ = - ( X 2 1 X 2 ) ~ M X l »XM (A+-F+) (11) A+-F+ i s a l i n e a r t r a n s f o r m a t i o n o f 0 and V, s o we s u b s t i t u t e t h e e s t i m a t e d f u n c t i o n s o f A + and F + i n t e r m s o f U and V. Then u n d e r H we w i l l h a v e 20 &+-F+ = - ( X i ' X i + X 2 ' X 2 ) - M X i # X 2 ) « (U,V) • (12) S u b s t i t u t i n g (11) and (12) i n t o e g u a t i o n ( 1 0 ) , we c l a i m Q 3 h a s r a n k k, s i n c e r a n k ( Q 2 ) -= n - 2 k and r a n k ( Q i ) < r a n k (Q 2) + r a n k ( Q 3 ) T h e r e f o r e , t h e H c a n be t e s t e d by F r a t i o F ( k , n - 2 k ) = 1 J . J . X 1 A l z X I I l l l £ i i l X £ B l - X £ F l i l £ l i n r 2 k i ( T | Y * - X * A + | | 2 + J l Y 2 - X 2 B + | | 2 ) k V P o l y t o m o u s V a r i a b l e G e n e r a l i z i n g t h e d i c h o t o m o u s v a r i a b l e t o a p o l y t o m o u s v a r i a b l e i n t h i s model i s v e r y e a s y . The b a s i c s t r u c t u r e on e s t i m a t i o n and h y p o t h e s e s t e s t i n g a r e m o s t l y t h e same, t h e r e f o r e i n t h i s s e c t i o n , we j u s t b r i n g o u t t h e i d e a o f t h i s g e n e r a l i z e d model and i t s p a r a m e t e r s e s t i m a t i o n s ; h y p o t h e s e s t e s t i n g i s o m i t t e d . L e t us assume t h a t t h e c a t e g o r i c a l v a r i a b l e z has more t h a n two c a t e g o r i e s . The d i s t r i b u t i o n o f j j j j y ( i ) g i v e n z ( i ) = a i s n o r m a l , N ( X ( i ) ' S , v a r ( u ) ) , where a i s a s c a l a r , S i s a v e c t o r k x 1 , X ( i ) i s a v e c t o r k x 1 ; i = 1 , . . . , n , n o b s e r v a t i o n s , and j=1#...#g# g p o s s i b l e r e s p o n s e s on z. i j j j p = P r o b ( z ( i ) = a ) = F ( X ( i ) ' R ) j where B i s a v e c t o r k x 1 . D e f i n e a t r a n s f o r m a t i o n t ( i , j ) a s 21 n r j r t ( i , j ) = IT ( z ( i ) - a ) / ( a -a ) r , j = 1 , . . . , g ; i = 1 , . . . , n r=1 j Hence t ( i , j ) = 1 when and o n l y when z ( i ) =a , o t h e r w i s e t ( i , j ) = 0 . Now we d e f i n e , 1 9 j i ) Y t o be a n g x l v e c t o r , Y = (Y , Y ) ; where Y ( t { i , j ) y ( 1 ) , ..., t ( n , j ) y ( n ) ) f o r j=1, i i ) X t o be a b l o c k d i a g o n a l m a t r i x w i t h d i m e n s i o n n g x q k , i . e , * r *1 T X = IX 0 ... 0 j I *2 I I 0 X ... 0 | I ... I I *gJ i-O 0 ... X -« * j f o r e a c h X , * j r 1 k 1 X = | t ( 1 , j ) x (1) ... t (1, j ) x (1) j I . . . I | 1 k | L t ( n , j ) x (n) ... t ( n , j ) x ( n ) J 1 g j i i i ) S t o be a g k x l v e c t o r , S=(S •,...,S » ) * , and S i s kx1 j 1j v e c t o r , S = (s , . . . , s ) ' ; 1 j j i v ) U t o be a n g x l v e c t o r , U= (U U 1 ) » ; U = ( u ( i ) , j u ( n ) ) * , j = 1 # . - • , g . The d i s t u r b a n c e v a r i a n c e - c o v a r i a n c e i s a d i r e c t p r o d u c t o f a gxg d i a g o n a l m a t r i x D and a n x n u n i t m a t r i x 22 1 g I , where d i a q (D) = [ v a r ( u ) , ..., v a r (u ) ] . i j j j q s v) p = P r o b ( z ( i ) = a ) = e x p ( X ( i ) ' R ) / 2 ~ exp (X ( i ) ' R ) , and R = s=1 1 9 3 J 1 3 * {R R » ) ' f o r B = ( r , r )». As b e f o r e , we c a n o b t a i n t h o s e p a r a m e t e r e s t i m a t e s by e s t i m a t i n g two s e p a r a t e d m o d e l s . H e n c e , t h e c o n d i t i o n a l m o d el i s * Y = X S + U S i n c e z ( i ) ( 1 - z ( i ) ) = 0 , t h i s i m p l i e s X and X a r e o r t h o g o n a l when i * j s o j * j * j * j j S = ( (X * X ) - * (X ) * Y j 3 * j 1 J * 3 j Var+ (u ) = (Y - (X ) • S+ ) « (Y - (X ) 15* ) / (n-k) j * j * j 3 Cov (S+ ) = {(X ) • (X ) ) - i V a r + ( u ) I n l o g i s t i c p a r t , t h e maximum l i k e l i h o o d f u n c t i o n i s n g i j q i j J_ L = "TT TT (p ) t ( i , j ) , Z _ P = 1 , Z : t ( 1 , 3 ) = 1 i = 1 j = 1 j=1 .1=1 j n i A l s o 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 be f o u n d by 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 , and R+ must s a t i s f y t h e e q u a t i o n s n i j g i s i n i TL [ e x p ( X »R+ ) / Z I e x p ( X • E+ ) ]X = 21 X t ( i , j ) i = 1 s = 1 i = 1 23 He c l a i m t h e s o l u t i o n t o t h i s p r o b l e m y i e l d s a maximum. S i n c e , q j 3 n g i j l o g ( L ) = 27 T ' r - £ l o g ( ZT exp (X * B ) ) j=1 i=1 j=1 m m a> 2log (1)/SB SB 1 g i j m i i n iZ §X£LX I i i Z 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 ) ] } 2 j=1 i l Hence t h e l o g ( L ) i s c o n c a v e b e c a u s e ( X X ' ) i s p o s i t i v e semi- d e f i n i t e f o r a l l i , 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 . F o o t n o t e s : 1 C o n s t a n t t e r m a, i s c o n s i d e r e d as a p r o d u c t o f a x 1 , where x 1 i s a v a r i a b l e a l w a y s h a v i n g v a l u e 1. 2 J . P r e s s and M. N e r l o v e , 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 - l i n e a r and L o g i s t i c M o d e l s , Dec. 1973, pp.10. 3 J . P r e s s and M. N e r l o v e , 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 - l i n e a r a n d L o g i s t i c M o d e l s , Dec. 1973, p p . 5 . * G r o u p d a t a means 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 c o n v e x a nd l o g f u n c t i o n i s i n c r e a s i n g s o t h e c o m p o s i t i o n f u n c t i o n o f l o g (F) i s c o n v e x . The sum o f c o n v e x f u n c t i o n s i s c o n v e x , b u t t h e n e g a t i v e c o n v e x f u n c t i o n i s c o n c a v e . T h e r e f o r e l o g ( L ) i s c o n c a v e . 6 s e e B o x , D a v i e s and Swann{1969) Ch.4 pp. 38 - 39. 24 C h a p t e r I I I S y s t e m o f E q u a t i o n s M o d e l The b a s i c model p r o p o s e d i n t h e p r e v i o u s c h a p t e r c a n be e x p a n d e d i n e v e r y d i m e n s i o n . T h i s c h a p t e r a n d t h e f o l l o w i n g c h a p t e r w i l l d i s c u s s s e v e r a l e x t e n s i o n s o f t h i s b a s i c m o d e l . B e c a u s e o f t h e l i m i t e d s c o p e o f t h i s r e s e a r c h , t h e p r e s e n t a t i o n i s a s f o l l o w s : we w i l l d i s c u s s a s i m p l e e x t e n s i o n , c a l l e d s y s t e m o f e q u a t i o n s m o d e l a l i t t l e b i t more i n t h i s c h a p t e r , t h e n i n t h e f o l l o w i n g c h a p t e r we w i l l j u s t m e n t i o n some i n t e r e s t i n g e x t e n s i o n m o d e l s and l e a v e o u t a l l t h e d e t a i l s . S y s t e m o f e q u a t i o n s model i s d e f i n e d a s a s e t o f r e g r e s s i o n o r l o g i s t i c e q u a t i o n s . T h i s s e t c a n be p a r t i t i o n e d i n t o c e r t a i n number o f d i s j o i n t g r o u p s and e a c h g r o u p c a n f o r m a b a s i c m o d e l as we d i s c u s s e d i n t h e p r e v i o u s c h a p t e r . I n t h e b a s i c m o d e l , o u r i n t e r e s t c o n c e n t r a t e s on t h e j o i n t d e n s i t y o f two d e p e n d e n t v a r i a b l e s i n v o l v i n g c o n t i n u o u s v a r i a b l e y and c a t e g o r i c a l v a r i a b l e z. H e r e we a r e n o t o n l y i n t e r e s t e d i n t h e j o i n t d e n s i t y o f t h e s e two d e p e n d e n t v a r i a b l e s , b u t we a r e a l s o i n t e r e s t e d i n t h e i n t e r a c t i o n e f f e c t s b e t w e e n g r o u p s . F o r e x a m p l e , we may w i s h t o a n a l y z e t h e j o i n t d e n s i t y o f t h e t i m i n g o f a m a r r i e d woman i n l a b o u r f o r c e a n d h e r c h i l d n o t o l d e r t h a n 6 y e a r s o f age w i t h i n a p e r i o d 1967-71, b u t i t w i l l be more i n t e r e s t i n g t o c o n s i d e r t h i s p r o b l e m y e a r by y e a r a n d o b s e r v e t h e i n t e r a c t i o n e f f e c t s b e t ween y e a r s . L e t us s a y , y f o r g i v e n z i s d i s t r i b u t e d n o r m a l l y , s o w r i t t e n i n m a t r i x n o t a t i o n t h e y w i l l be 25 t t t t t (y ( i ) i z i s N (X ( i ) ' A , v a r (u )) t t t t t (y { i ) | z ( i ) = 0 ) i s N(X ( i ) «B , v a r ( v }) and t t t P r o b ( z ( i ) = 1 ) = F{X ( i ) ' C ) t t f o r t=1,...,d, d g r o u p s ; X ( 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 ) and z ( i ) , we t t t t t h a v e 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 u s s t a r t w i t h an e a s y method. I E s t i m a t i o n e g u a t i o n - b y - e q u a t i o n T h i s method i s v e r y s i m p l e . The e s t i m a t i o n i s b a s e d on t h e a s s u m p t i o n t h a t t h e d a t a a r e i n d e p e n d e n t b e t w e e n g r o u p s . H e n c e , t h o s e p a r a m e t e r s c a n be f o u n d by c o n s i d e r i n g t h e w h o l e p r o b l e m as d s e p a r a t e b a s i c m o d e l s , and e s t i m a t i n g t h o s e m o d e l s one by one . I n most o f c a s e s , d a t a a c r o s s g r o u p s a r e c o r r e l a t e d . H e n c e , t h i s k i n d o f e s t i m a t i o n i s n o t e f f i c i e n t . I n t h e f o l l o w i n g s e c t i o n , we w i l l d i s c u s s a method w h i c h h a n d l e s t h e c a s e when c o r r e l a t i o n a c r o s s g r o u p s i s t a k e n i n t o a c c o u n t . I I Dependence among g r o u p s I n t h e p r e v i o u s c h a p t e r 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 be s e p a r a t e d i n t o t w o p a r t s , b e c a u s e t h e j o i n t d e n s i t y f u n c t i o n 26 c a n be f a c t o r i z e d i n t o two f u n c t i o n s w h i c h d e p e n d on d i s j o i n t p a r a m e t e r s e t s . So i f we w i s h t o c o n s i d e r t h e i n t e r a c t i o n e f f e c t s b e t w e e n g r o u p s , we w i l l o b s e r v e t h o s e e f f e c t s on t h e l o g i s t i c p a r t and t h e r e g r e s s i o n p a r t i n d i v i d u a l l y . A) I n t e r a c t i o n e f f e c t s b e t w e e n g r o u p s on c o n d i t i o n a l r e g r e s s i o n s : E s t i m a t i o n : T h e r e a r e s e v e r a l k i n d s o f i n t e r a c t i o n e f f e c t s b e t ween g r o u p s on c o n d i t i o n a l r e g r e s s i o n s . I n t h i s s e c t i o n we o n l y c o n s i d e r a s p e c i a l o n e , and we w i l l d i s c u s s more a b o u t i t i n c h a p t e r 4. L e t u s assume t h a t t h e d i s t u r b a n c e t e r m s i n d i f f e r e n t g r o u p s a r e h i g h l y c o r r e l a t e d . Hence u n d e r t h i s a s s u m p t i o n , t h e e s t i m a t o r s o b t a i n e d by an e g u a t i o n - b y - e g u a t i o n a r e n o t i n g e n e r a l e f f i c i e n t . Z e l l n e r (1962) h a s p r o p o s e d an e f f i c i e n t method c a l l e d " E s t i m a t i n g S e e m i n g l y U n r e l a t e d B e g r e s s i o n s " . T h i s method a p p l i e s A i t k e n ' s g e n e r a l i z e d l e a s t - s g u a r e s t o t h e w h o l e s y s t e m o f e g u a t i o n s . F o r g r o u p t , we know: t t t t t (y ( i ) j z ( i ) = 1 ) i s N(X ( i ) «A , v a r ( 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 d i s t r i b u t e d n o r m a l l y w i t h mean t t t t t t t t X (i)»A z ( i ) + X ( i ) 'B ( 1 - z ( i ) ) and v a r i a n c e v a r [ u ( i ) z ( i ) ] + t t v a r [ v ( i ) (1-z ( i ) ) ]. So i f we w r i t e i n r e g r e s s i o n e g u a t i o n w i t h m a t r i x a l g e b r a t h e n 27 t * t t * t ¥ = X »S + 0 t * t where X i s a nx2k m a t r i x , f o r e a c h 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 nx1 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 w h i c h 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 ) ) ]• , and t = 1 , d, d g r o u p s . So p u t t h e n t o g e t h e r , i t w i l l be J - 1 - J r * 1 i j - 1 - i r * ^ T I ¥ | = J X 0 . . . 0 l | S | + | 0 I 2 | 1 *2 I | 2 | I *2 J Y | I P X ... 0| | S | | U |...| | ... | | ... | ]... 1 d J | *d| | d | •I *d L Y -» «-0 0 . . . X - » « - S - « *- 0 o r , Y = XS + U (1) 1' d» where Y = (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 * d i a g o n a l i s (X X ) , S = ( S , . . . , S ) ' , U = ( U * d * U ) ' t o a p p l y A i t k e n ' s g e n e r a l i z e d l e a s t - s q u a r e , we g e t S+ = (X'H'HX)-*X'H'Hy = (X«E~» X ) - i X'Z--y where H i s an o r t h o g o n a l m a t r i x s u c h t h a t E(Huu* H')=HXH *=I, a n d v a r ( S + ) = ( X ' l - i X ) - ! , where Z- 1 = V a r - 1 (0) r 11 1d T = |E I ... E I | I ••• I I d1 ... dd | LE I ... 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 , where S i s e s t i m a t e d f r o m t h e b a s i c 28 m o d e l s , S = (A , B ) . Hence ( E + ) - 1 c a n be e s t i m a t e d . T h e s e e s t i m a t o r s a r e more e f f i c i e n t b e c a u s e i n e s t i m a t i n g t h e c o e f f i c i e n t s o f a s i n g l e e q u a t i o n , t h e A i t k e n p r o c e d u r e t a k e s a c c o u n t o f z e r o r e s t r i c t i o n s on c o e f f i c i e n t s o c c u r r i n g i n o t h e r e q u a t i o n s . Z e l l n e r a n d Huang(1962) p o i n t e d o u t t h a t t h e s e e s t i m a t o r s have t h e o p t i m a l f o r e c a s t i n g p r o p e r t i e s . H y p o t h e s e s t e s t i n g : We may w i s h t o t e s t t h a t t h e d a t a i n t h e g r o u p s a r e homogeneous i n i t e m s o f t h e i r r e g r e s s i o n c o e f f i c i e n t v e c t o r s . 1 2 d H: s = s = ... = s T h e r e a r e s e v e r a l ways t o t e s t t h i s h y p o t h e s i s , b u t o n l y two a r e c o n s i d e r e d . One i s a s Z e l l n e r (1962) s u g g e s t e d t h a t t h e t e s t s t a t i s t i c c a n be e m p l o y e d by u s i n g a F - t e s t a s F (2k (d-1) ,d (n-2k) ) = d (n-2k) (S+)-*D ' r DVar (S+1D!_/J-1DSJ: 2k(d-1)£Y« (E+)-iII-Y» ( E + J - i I X S + 3 where D, t h e m a t r i x o f t h e r e s t r i c t i o n s , w i t h d i m e n s i o n ( d - 1 ) x d . D = r l - I 0 - 0 0 0 IO I - I . . . 0 0 0 | I . . . I ] 0 0 0 ... I -I 0 i «-0 0 0 ... 0 I - I J s u c h t h a t DS=0. A n o t h e r method i s t o use t h e maximum l i k e l i h o o d r a t i o t e s t 29 w h i c h l e a d s t o t h e same r e s u l t . The g e n e r a l i d e a o f t h i s t e s t h a s b e e n s t a t e d i n p r e v i o u s c h a p t e r . A d e t a i l a b o u t a p p l y i n g t h i s t e s t i n t h i s h y p o t h e s i s i s shown i n A p p e n d i x B. I f t h i s h y p o t h e s i s i s t r u e , t h e n t h e r e i s no a g g r e g a t i o n b i a s i n s i m p l e l i n e a r a g g r e g a t i o n . H e n c e , t h e e s t i m a t o r s t a k e n f r o m t h e e n t i r e s a m p l e w i l l be s t a t i s t i c a l l y a s y m p t o t i c e q u i v a l e n t t o t h e p a r a m e t e r s e s t i m a t e d f r o m i n d i v i d u a l g r o u p s . B) I n t e r a c t i o n e f f e c t s b e t ween g r o u p s on l o g i t m o d e l : As we s a i d , e a c h g r o u p c a n f o r m a b a s i c m o d e l as i n t r o d u c e d i n t h e p r e v i o u s c h a p t e r . W i t h i n e a c h g r o u p , we h a v e one c o n d i t i o n a l c o n t i n u o u s v a r i a b l e and one q u a l i t a t i v e v a r i a b l e , so i n t h i s s p e c i a l m o d e l we have d c o n d i t i o n a l c o n t i n u o u s v a r i a b l e s and d q u a l i t a t i v e v a r i a b l e s . T h e r e f o r e , i f t h e q u a l i t a t i v e v a r i a b l e s a r e u n o r d e r e d , t h e n we may e x p l o r e o u r i n t e r e s t i n t o t h e more g e n e r a l c a s e o f any number o f j o i n t l y v a r y i n g d i c h o t o m o u s v a r i a b l e s and t h e p r o b a b i l i t y t h a t a q u a l i t a t i v e v a r i a b l e t a k e s on a p a r t i c u l a r v a l u e . T h i s w i l l b r i n g o u r a t t e n t i o n i n t o t h e r e l a t i o n b e t w e e n t h e l o g - l i n e a r m o d e l o f c o n t i n g e n c y t a b l e a n a l y s i s a nd t h e s t a n d a r d l o g i s t i c m o d e l . I n t h i s s e c t i o n , we d i s c u s s n e i t h e r t h i s r e l a t i o n , n o r t h e g e n e r a l model o f s e v e r a l q u a l i t a t i v e d i c h o t o m o u s v a r i a b l e s , b e c a u s e t h e y h a v e b e e n done by N e r l o v e and P r e s s ( 1 9 7 3 ) . The m o d e l w h i c h i s s u g g e s t e d by N e r l o v e and P r e s s i s assumed t h a t : 1) a l l h i g h e r o r d e r i n t e r a c t i o n e f f e c t s v a n i s h , 1 30 2) t h e s e c o n d o r d e r i n t e r a c t i o n e f f e c t s a r e c o n s t a n t a n d i n d e p e n d e n t o f t h e v a l u e s o f any o f t h e e x o g e n o u s v a r i a b l e s . 3) t h e main e f f e c t s a r e l i n e a r f u n c t i o n s o f t h e v a r i o u s e x o g e n o u s e x p l a n a t o r y v a r i a b l e s . P a r a m e t e r s a r e e s t i m a t e d by t h e maximum l i k e l i h o o d method. The c o m p u t a t i o n a l a l g o r i t h m 2 i s b a s e d on t h e F l e t c h e r - P o w e l l method o f f u n c t i o n m i n i m i z a t i o n , and t h e D a v i d o n a l g o r i t h m f o r e s t i m a t i n g t h e i n v e r s e o f t h e m a t r i x o f t h e s e c o n d d e r i v a t i v e s , t h e i n f o r m a t i o n m a t r i x 3 . IV F u r t h e r d i s c u s s i o n I n t h e p r e v i o u s d i s c u s s i o n , we o n l y c o n s i d e r t h e d i s t u r b a n c e s i n t h e r e g r e s s i o n e q u a t i o n s a r e c o r r e l a t e d . F o r e a c h p e r i o d o f t i m e , we c a n a l s o c o n s i d e r t h e o b s e r v a t i o n s a r e s e r i a l l y c o r r e l a t e d . T h i s c a n be e a s i l y s o l v e d , s i n c e we know how t o h a n d l e a u t o c o r r e l a t i o n p r o b l e m i n r e g r e s s i o n . We may d e t e c t 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 t e s t a n d a p p l y T h e i l BLOS p r o c e d u r e * t o g e t b e t t e r e s t i m a t e s i n o u r f i r s t s t a g e , t h e n we a p p l y Z e l l n e r ' s method t o g e t o u r r e s u l t . F u r t h e r m o r e , i t i s no g r e a t t r o u b l e t o g e n e r a l i z e t h i s m o d e l t o i n v o l v e p o l y t o m o u s v a r i a b l e s i n s t e a d o f d i c h o t o m o u s v a r i a b l e s . I n c h a p t e r 2, we have d i s c u s s e d a b o u t how t o g e n e r a l i z e o u r b a s i c model f r o m d i c h o t o m o u s t o p o l y t o m o u s . T h i s m o d e l i s j u s t a b i t b e y o n d o u r b a s i c m o d e l , s o e v e r y t h i n g 31 d i s c u s s e d i n t h i s c h a p t e r a r e s t i l l a p p l i c a b l e t o p o l y t o m o u s v a r i a b l e s . T h e r e f o r e , p o l y t o m o u s g e n e r a l i z a t i o n i s o m i t t e d . F o o t n o t e : 1 T h i s a s s u m p t i o n i n t h e c o m p u t e r p r o g r a m o f N e r l o v e and P r e s s h a s now been e l i m i n a t e d a n d i n a n u p d a t e d r e v e r s i o n o f t h e p r o g r a m , h i g h e r o r d e r 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 . 2 See N e r l o v e a n d P r e s s ( 1 9 7 3 ) , A p p e n d i x A, e s p . pp. 92-94. 3 See E o x , D a v i e s and S w a n n ( 1 9 6 9 ) , c h . 4, e s p . pp. 3 8 - 3 9 , 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 . 4 H. T h e i l , "The A n a l y s i s o f D i s t u r b a n c e s i n R e g r e s s i o n A n a l y s i s " , J . Am.. S t a t i s t , . A s s o c . , v o l . 6 0 , pp. 1067 1079, 1965. J . K o e r t s , "Some F u r t h e r N o t e s on D i s t u r b a n c e E s t i m a t e s i n R e g r e s s i o n A n a l y s i s " , J± Am. S t a t i s t . A s s o c . , v o l . 6 2 , pp. 169 - 183, 1962. 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 R e g r e s s i o n D i s t u r b a n c e s " , J.. Am. S t a t i s t ^ A s s o c . , v o l . 6 3 , pp. 242 - 2 5 1 , 1968. J, K o e r t s and A. P. J . A b r a h a m s e , "On t h e Power o f t h e BLOS P r o c e d u r e " , J.. Am. S t a t i s t . Assoc,. , v o l . 6 3 , pp. 1227 - 1236, 1968. 32 C h a p t e r I V M o d e l E x t e n s i o n s I L a g g e d v a r i a b l e s m o d e l We s u p p o s e t h a t t h e d e p e n d e n t v a r i a b l e s a r e d e p e n d e n t , n o t o n l y on 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 v a l u e o f X. 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 by w i f e d e p e n d s o n t h e e c o n o m i c f a c t o r s ( h e a d ' s u n e m p l o y m e n t , w i f e ' s wage, . . . e t c . ) o f t h i s y e a r a s w e l l a s t h e e c o n o m i c f a c t o r s o f t h e l a s t y e a r . S i m i l a r l y , t h e p r o b a b i l i t y t h a t a m a r r i e d woman w i l l b e a r a b a b y i s d e p e n d e n t upon t h e e c o n o m i c f a c t o r s o f t h i s y e a r a s w e l l as l a s t y e a r . L e t us c o n s i d e r t h e v e r y s i m p l e c a s e o f one v a r i a b l e , a n d assume t h e r e i s m u l t i c o l l i n e a r i t y i n t h e p r o b l e m . We may assume t h a t a l l t h e c o e f f i c i e n t s e x p o n e n t i a l l y d e c r e a s e w i t h r e s p e c t t o t i m e . So l e t w = x z + x ( 1 - z ) t h e n t t 1 (t-1) t y = a + bw + b w +... + e t t t 1 ( t - 1 ) l o g (p / ( 1 - p }) = r + s x + s x + ... where i i i i b = b*(d) , s = s * (d) , i = 1 , 2 , . . . , 0<d<1 I n r e g r e s s i o n , we h a v e t t (t-1) t y = a + bw + bdw + ... * e (t-1) ( t - 1 ) ( t - 2 ) (t-1) y = a + bw + bdw + ... + e 33 t ( t - 1 ) t t ( t - 1 ) y - dy = a(1-d) + b w + (e - de ) (1) E g u a t i o n (1) c a n be e s t i m a t e d e a s i l y . S i m i l a r l y i n t h e l o g i s t i c p a r t , we have t t ( t - 1 ) (t-1) t l o g f p / ( 1 - p ) ] - d * l o g [ p / ( 1 - p ) J = r ( 1 - d ) *• s x S i m i l a r l y , t h e l o g i t f u n c t i o n c a n be e s t i m a t e d . I I M o d e l w i t h c o n s t r a i n t s I n some c a s e s we may know some p a r a m e t e r s w i l l h a v e meaning o n l y i n a c e r t a i n d o m a i n , o r i n t e r r e l a t i o n s h i p o f p a r a m e t e r s may f o r m a c o n s t r a i n t . F o r e x a m p l e , we may c o n s i d e r s u c h a c a s e t h a t t h e t o t a l w o r k i n g h o u r s o f t h e head and t h e w i f e must g r e a t e r t h a n a c e r t a i n number. We know t h e maximum l i k e l i h o o d method i s s u i t a b l e t o e s t i m a t e s u c h a model w i t h c o n s t r a i n t s . I I I M o d e l w i t h j o i n t l y d e p e n d e n t v a r i a b l e s He may c o n s i d e r i n f e r e n c e s a b o u t r e l a t i o n s h i p s t h a t d e t e r m i n e j o i n t l y d e p e n d e n t d i s c r e t e v a r i a b l e s , w h i c h a r e b o t h c a t e g o r i c a l and uno r d e r e ' d . F o r e x a m p l e , t h e h u s h a n d o f a m a r r i e d 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 b e a r a baby o r w i l l n o t . We may w i s h t o r e l a t e t h e j o i n t p r o b a b i l i t y o f t h e s e two e v e n t s t o a s e t o f s o c i a l f a c t o r s . As d i s c u s s e d i n c h a p t e r 3, i n some c a s e s l o g i s t i c f u n c t i o n c a n be c o n s i d e r e d a s s e v e r a l q u a l i t a t i v e p o l y t o m o u s v a r i a b l e s . The s o l u t i o n o f s u c h f u n c t i o n 34 h a s been p r o p o s e d by P r e s s and N e r l o v e ( 1 9 7 3 ) . T h e r e f o r e t h e m o d e l w i t h j o i n t l y d e p e n d e n t v a r i a b l e s c a n be c o n s i d e r e d a s a s p e c i a l c a s e o f c h a p t e r 3. I V S i m u l t a n o u s - e g u a t i o n m o d e l we may c o n s i d e r o u r b a s i c m o d e l i s composed o f two s i m u l t a n o u s - e g u a t i o n s y s t e m s . One s y s t e m i s f o r m e d by l i n e a r r e g r e s s i o n e g u a t i o n s , and t h e o t h e r s y s t e m i s f o r m e d by l o g i s t i c e q u a t i o n s . I n r e g r e s s i o n s y s t e m , we c a n a p p l y t h r e e - s t a g e l e a s t s g u a r e s w h i c h i s p r o p o s e d by Z e l l n e r and T h e i l ( 1 9 6 2 ) . T h i s method i s known t o u s . The l o g i s t i c s y s t e m h a s been s o l v e d by P. S c h m i d t and E. S t r a u s s ( 1 9 7 4 ) . They a r e u s i n g t h e maximum l i k e l i h o o d a p p r o a c h and c o n s i d e r i t i s a s p e c i a l c a s e o f t h e m o d e l o f N e r l o v e and P r e s s ( 1 9 7 3 ) . Hany s o c i a l p r o b l e m s c a n be a n a l y z e d by s u c h m o d e l . F o r e x a m p l e , t h e e x p e n d i t u r e o f a f a m i l y w i l l d e p e n d upon t h e h o u r s w orked by t h e w i f e , and o t h e r f a c t o r s , and a l s o t h e h o u r s w o r k e d by w i f e w i l l d e p e n d upon f a m i l y e x p e n d i t u r e and t h e o t h e r f a c t o r s . The p r o b a b i l i t y o f a f a m i l y g o i n g on v a c a t i o n w i l l d e p e n d on w h e t h e r t h e w i f e b e a r s a b a b y . So i f we w i s h t o know t h e j o i n t p r o b a b i l i t y o f a f a m i l y g o i n g on v a c a t i o n and i t s a n n u a l e x p e n d i t u r e , we may a p p l y t h i s m o d e l . V R e c u r s i v e model 35 T h i s i s t h e most i n t e r e s t i n g m o d e l p r o p o s e d . T h i s i s a v e r y new s t u d y a r e a . T h e r e i s no f o r m a l l i t e r a t u r e a b o u t t h i s k i n d o f m o d e l . I n t h i s m o d e l , t h e c o n t i n u o u s v a r i a b l e a n d t h e d i s c r e t e v a r i a b l e a r e i n t e r - d e p e n d e n t . T h a t i s E ( y | X , z ) = f ( X , z ) z=0,1 (2) p = P r o b (z=11 y, X) = [ 1+exp (-C X-dy) J-i (3) P. S c h m i d t a n d R. S t r a u s s h a v e d i s c u s s e d t h i s p r o b l e m , b u t t h e y o n l y c o n s i d e r e d i t a s a s t a r t o f t h i s t o p i c . They c o n s i d e r how t o m a x i m i z e t h e l i k e l i h o o d f u n c t i o n . T h e i r s u g g e s t i o n d o e s n o t seem t o be n o v e l and i s e x p e n s i v e t o c o m p u t e . The f o l l o w i n g d i s c u s s i o n c a n be c o n s i d e r e d a s an i n i t i a l s t e p i n a t t a c k i n g t h i s m o d e l , and we hope i t i s a s t e p i n t h e r i g h t d i r e c t i o n . I f we r e w r i t e e q u a t i o n ( 3 ) , t h e n i t becomes w = l o g [ p / ( 1 - p ) ] = C'X + dy (4) S i n c e z i s a c a t e g o r i c a l v a r i a b l e , we w i l l a p p l y dummy v a r i a b l e s i n r e g r e s s i o n ( 2 ) . Hence, e q u a t i o n s (2) and (4) f o r m a s i m u l t a n e o u s r e g r e s s i o n e g u a t i o n s y s t e m . T h i s we c a n s o l v e e i t h e r by t h r e e - s t a g e l e a s t - s q u a r e s , o r t h e f u l l i n f o r m a t i o n maxi-mum l i k e l i h o o d m e t hod. The f u l l i n f o r m a t i o n maximum l i k e l i h o o d method i s an e x p e n s i v e c o m p u t a t i o n a l method, and i t w i l l be v e r y d i f f i c u l t t o o b t a i n p a r a m e t e r e s t i m a t e s when t h e number o f d e g r e e s o f f r e e d o m i s l a r g e . T h r e e - s t a g e l e a s t - s g u a r e s i s an e x t e n s i o n o f t w o - s t a g e l e a s t - s g u a r e s , w h i c h we 36 m e n t i o n e d i n c h a p t e r 3. I t i s more e f f i c i e n t t h a n t w o - s t a g e 1 i f t h e d i s t u r b a n c e s i n v a r i o u s s t r u c t u r a l e q u a t i o n s a r e c o r r e l a t e d . B o t h methods 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 t e x t b o o k s . F o o t n o t e : 1 A. Z e l l n e r and H. T h e i l , " T h r e e - s t a g e , L e a s t - s g u a r e s : S i m u l t a n e o u s E s t i m a t i o n o f S i m u l t a n e o u s E q u a t i o n s " , E c o n o m e t r i c a v o l . 3 0, pp. 54 - 7 8 , 1962. ( 37 C h a p t e r V A S t u d y on L a b o u r S u p p l y o f M a r r i e d Women Mod e l D e s c r i p t i o n I I n t r o d u c t i o n The e m p i r i c a l l i t e r a t u r e on f e m a l e l a b o u r s u p p l y , e s p e c i a l l y f o r m a r r i e d women, i s n o t much. R e l a t e d s t u d i e s a r e J . K o r b e l ( 1 9 6 2 ) , J . M i n c e r (1963) , G. C a i n ( 1 9 6 6 ) , S. Hof f e r (1973) , R. Freeman (1973) , and E. E-erndt and T. Wales ( 1 9 7 4 ) . I n t h i s c h a p t e r , o u r s t u d y i s t o o b s e r v e t h e l a b o u r f o r c e p a r t i c i p a t i o n o f m a r r i e d women i n d i f f e r e n t s i t u a t i o n s , a n d t h e d e t e r m i n a t i o n o f t h e s e s i t u a t i o n s i n t h e U n i t e d S t a t e s o v e r t h e f i v e - y e a r p e r i o d 1 9 6 7 - 7 1 . T h i s s t u d y w i l l be d i v i d e d i n t o two p a r t s : t h e f i r s t p a r t i s t o s t u d y o u r e c o n o m i c p r o b l e m u s i n g f i v e y e a r s d a t a , and t h e s e c o n d p a r t i s t o s t u d y t h e p r o b l e m y e a r l y . Our d a t a i s drawn f r o m t h e U n i v e r s i t y o f M i c h i g a n S u r v e y R e s e a r c h C e n t r e P a n e l S t u d y o f Income D y n a m i c s ( 1 9 7 2 ) . T h e r e were 2500 f a m i l y u n i t s r a n d o m l y c h o s e n , and e a c h f a m i l y u n i t was r e - i n t e r v i e w e d a n n u a l l y o v e r t h e 1967-71 t i m e p e r i o d . As E. B e r n d t and T. Wales (1974) p o i n t e d o u t t h i s p e r i o d , 1967- 71 was o f p a r t i c u l a r i n t e r e s t s i n c e t h e n a t i o n a l unemployment r a t e f o r women a g e d 20 and o v e r v a r i e d c o n s i d e r a b l y f r o m 3.8% and 3.7% i n 1968 and 1969 t o 5.7% i n 1971; f u r t h e r , t o w a r d t h e 38 end o f t h i s p e r i o d an i n c r e a s i n g e m p h a s i s was p l a c e d on e l i m i n a t i n g d i s c r i m i n a t i o n a g a i n s t w o r k i n g women. I I S p e c i f i c a t i o n o f m o d e l s , A l t h o u g h t h i s s t u d y w i l l be d i v i d e d i n t o two p a r t s , t h e y s t i l l h a v e t h e same e c o n o m i c a l s t r u c t u r e . I n t h e f i r s t p a r t , t h e model i s b u i l t upon c h a p t e r 2 and i n t h e s e c o n d p a r t , t h e model i s b u i l t upon c h a p t e r 3. 340 f a m i l y u n i t s a r e s e l e c t e d . The a n a l y s e s a r e b a s e d on t h e s e f a m i l y u n i t s . E v e r y y e a r , t h e s e f a m i l y u n i t s a r e p a r t i t i o n e d i n t o two g r o u p s . Group I c o n t a i n s a l l t h e f a m i l y u n i t s w h i c h h a v e a c h i l d o f 6 y e a r s o r y o u n g e r , and g r o u p I I c o n t a i n s t h e r e s t . I t i s v e r y o b v i o u s t h a t t h e s e two g r o u p s a r e d i s j o i n t e d , b u t i t i s n o t t r u e t h a t t h e number o f e l e m e n t s i n e a c h g r o u p i s f i x e d f o r e v e r y y e a r , b e c a u s e t h e age o f t h e y o u n g e s t c h i l d i s i n c r e a s i n g . S u p p o s e t h e y o u n g e s t c h i l d o f a f a m i l y was 5 y e a r s o l d i n 1967, and t h e f a m i l y had no f u r t h e r new b o r n c h i l d w i t h i n t h e t i m e p e r i o d 1 9 67-71. Then i n 1967 and 1968, t h i s f a m i l y b e l o n g s t o g r o u p I , b u t i n 196 9-71, t h i s f a m i l y b e l o n g s t o g r o u p I I . T h e r e f o r e , g r o u p I o f 1967 and g r o u p I I o f 1970 a r e n o t d i s j o i n t e d . I n h e r e we c a l l a f a m i l y c o n s t r a i n t v a r i a b l e z t o be 1 when a f a m i l y has a c h i l d n o t o l d e r t h a n 6 y e a r s , and 0 o t h e r w i s e . H e n c e , g r o u p I c o n t a i n s a l l t h e f a m i l y u n i t s i n w h i c h z - 1 , and g r o u p I I i s when z=0. We assume t h a t t h e l a b o u r f o r c e a c t i v i t y o f t h e w i f e i n g r o u p I w i l l be l e s s t h a n t h o s e i n g r o u p I I . We w i s h t o o b s e r v e how t h e t i m i n g o f t h e s e two g r o u p s o f m a r r i e d women i n l a b o u r f o r c e w i l l be d e t e r m i n e d d i f f e r e n t l y , due t o t h e same e c o n o m i c f a c t o r s ( o r 39 i n d e p e n d e n t v a r i a b l e s ) . A l s o we w i s h t o r e l a t e t h e p r o b a b i l i t y t h a t a f a m i l y h a s a c h i l d n o t o l d e r t h a n 6 y e a r s o f age t o t h e same e x p l a n a t o r y v a r i a b l e s . I n t h e f o l l o w i n g d i s c u s s i o n we w i l l c a l l t h e c o n d i t i o n a l r e g r e s s i o n e g u a t i o n s , l a b o u r e g u a t i o n s , and t h e l o g i s t i c f u n c t i o n , a p r o b a b i l i t y f u n c t i o n . I l l S p e c i f i c a t i o n o f v a r i a b l e s D e p e n d e n t v a r i a b l e s : C a t e g o r i c a l d e p e n d e n t v a r i a b l e : I n t h i s s t u d y , o u r c a t e g o r i c a l v a r i a b l e i s z, whose v a l u e i s 1 when a f a m i l y h a s a c h i l d n o t o l d e r t h a n 6 y e a r s o f a g e , o t h e r w i s e t h e v a l u e o f z i s 0. At t h e b e g i n n i n g z was d e f i n e d a s z=1 when a f a m i l y h a s a new b o r n c h i l d o t h e r w i s e z=0. T h i s d e f i n i t i o n o f z l e d i n t o s t a t i s t i c a l i n s i g n i f i c a n c e i n t h e mo d e l s b e c a u s e o f o u r l a c k o f o b s e r v a t i o n i n 1970 and 1 9 7 1 , and most o f t h e w i v e s d i d n e t work when z = 1 . F u r t h e r m o r e , a w i f e w i l l work l e s s n o t o n l y b e c a u s e o f h a v i n g a b a b y , b u t a l s o b e c a u s e o f h e r commitment t o h e r f a m i l y work. Suppose s h e h a s a c h i l d who i s n o t o f s c h o o l a g e , o r e v e n i f h e r c h i l d i s i n g r a d e I , s h e may l i k e t o l o c k a f t e r h e r c h i l d r a t h e r t h a n t o work o u t s i d e . C o n t i n u o u s d e p e n d e n t v a r i a b l e : Our c o n t i n u o u s v a r i a b l e y, i s t h e t i m i n g i n t h e l a b o u r f o r c e o f a m a r r i e d woman: t h a t i s h e r a n n u a l w o r k e d h o u r s . 40 I n d e p e n d e n t v a r i a b l e s ; At t h e b e g i n n i n g , t h e e x p l a n a t o r y v a r i a b l e s a r e c e n t e r e d a r o u n d t h e f o l l o w i n g e c o n o m i c f a c t o r s : b i r t h g a p ; p r e d i c t e d wage o f w i f e , h e a d ' s i n c o m e i n c l u d i n g i n c o m e f r o m e l s e w h e r e , unemployment o f h e a d , f e c u n d i t y , and t h e r a t i o o f i n c o m e s o v e r n e e d s . B i r t h G a p — B i r t h gap B i r t h gap i s d e f i n e d a s e x p e c t e d c o m p l e t e d f a m i l y s i z e m i nus a c t u a l number o f c h i l d r e n i n y e a r t . E x p e c t e d c o m p l e t e d f a m i l y s i z e i s t h e t o t a l number o f c h i l d r e n e x p e c t e d and d e c i d e d by a c o u p l e . T h e s e f i g u r e s c a n be f o u n d on t h e s u r v e y d a t a s o t h e y a r e a c t u a l d a t a . We w i l l e x p e c t t h a t t h e l a r g e r t h e b i r t h g a p , t h e g r e a t e r t h e p r o b a b i l i t y t h a t z=1. W i f e wg — W i f e e x p e c t e d wage The p r e d i c t e d wage o f t h e w i f e i s m e a s u r e d a c c o r d i n g t o t h e r e s u l t o f E. B e r n d t and T. W a l e s ( 1 9 7 4 ) . A w i f e w i t h a h i g h e r p r e d i c t e d wage w o u l d t e n d t o k e e p on w o r k i n g more and w o u l d t r y t o a v o i d h a v i n g a b a b y , o r she w o u l d l i k e t o go b a c k t o t h e l a b o u r f o r c e a s s o o n as p o s s i b l e . T h e r e f o r e , i f o u r a s s u m p t i o n i s c o r r e c t t h e p r e d i c t e d wage w i l l be p o s i t i v e l y c o r r e l a t e d w i t h y, t h e number o f h o u r s worked b y t h e w i f e . F o l l o w i n g t h e same a r g u m e n t , we c a n a l s o assume t h a t i t i s p o s i t i v e l y c o r r e l a t e d w i t h t h e p r o b a b i l t y o f z=1. Many m a r r i e d women a r e f a m i l y o r i e n t e d . They w o u l d l i k e t o work a t home, o r g i v e n more c a r e t o t h e i r c h i l d r e n r a t h e r t h a n t o work o u t s i d e , o r t o work a few 41 h o u r s f o r p l e a s u r e . T h e r e f o r e , i n t h o s e c a s e s , t h e p r e d i c t e d wage o f t h e w i f e w i l l c o r r e l a t e d w i t h y, o r P r o b ( z = 1 ) n e g a t i v e l y . H e n c e , t h e r e l a t i o n s h i p s b e t w e e n ( W i f e wg) a n d y; between ( W i f e wg) and P r o b ( z = 1 ) a r e i n q u a d r a t i c s h a p e . T h a t i n t r o d u c e s a new v a r i a b l e : t h e ( W i f e wg)* i n o u r m o d e l s . Head i n c — Head's wage p l u s t r a n s f e r i n c o m e The v a r i a b l e , t h e h e a d ' s i n c o m e i s v e r y s i m i l a r t o t h e i p r e d i c t e d wage o f t h e w i f e . I t v a r i e s i n U-shape t o o . T h i s v a r i a b l e i n c l u d e s h e a d ' s wage, a i d t o d e p e n d e n t c h i l d r e n , p e n s i o n s , i n c o m e s f r o m w e l f a r e , s o c i a l s e c u r i t y , unemployment o r workmen's c o m p e n s a t i o n and a l i m o n y o r c h i l d s u p p o r t . A l l t h e s o u r c e s e x c e p t t h e h e a d ' s wage a r e c a l l e d t r a n s f e r i n c o m e . U s u a l l y a w i f e has t o work more when h e r f a m i l y i n c o m e i s l o w . She w o u l d t r y t o r e - e n t e r l a b o u r f o r c e g u i c k l y . On t h e o t h e r h a n d , j u s t b e c a u s e t h e f a m i l y i n c o m e i s h i g h , t h a t d o e s n o t n e c e s s a r i l y mean t h a t t h e w i f e w i l l work l e s s , o r have a h i g h e r p r o b a b i l i t y o f h a v i n g a b a b y , b e c a u s e a h i g h i n c o m e f a m i l y may h a v e a h i g h t r a n s f e r i n c o m e and a l o w h e a d ' s wage. I n t h i s c a s e , t h e w i f e w i l l work more, and p u t l e s s t i m e i n t o h e r h o u s e work i n o r d e r t o make h e r f a m i l y economy s t a b l e . Hence (Head i n c ) z i s i n c l u d e d i n t h e m o d e l s . Unemploy — h e a d ' s unemployment The r e l a t i o n s h i p b e t w e e n y and t h e unemployment o f t h e head ( g i v e n i n d a y s ) , o r between P r o b ( z = 1 ) and t h e h e a d ' s u n e m p l o y m e n t ( g i v e n i n w e e k s ) , i s u n e x p e c t e d . N o r m a l l y we w o u l d 42 t h i n k when a h u s h a n d i s u n e m p l o y e d , t h e w i f e w o u l d have t o work more and l e a v e t h e f a m i l y a f f a i r b e h i n d . U n e x p e c t e d l y , we g e t a n e g a t i v e r e s u l t i n t e s t i n g o u r model I on t h e p u l l e d d a t a ( A p p e n d i x C ) . I t was n e t b e c a u s e t h e e n t i r e economy was bad w i t h i n t h e p e r i o d 1 9 6 7 - 7 1 . T h i s outcome may be e x p l a i n e d i n t h e f o l l o w i n g ways. Some o f t h e h e a d s may be s e a s o n a l w o r k e r s w i t h h i g h wage s u c h t h a t i t i s n o t n e c e s s a r y f o r t h e i r w i v e s t o work more. A n o t h e r p o s s i b l e r e a s o n i s a f a m i l y may move f r o m one town t o a n o t h e r town b e c a u s e t h e head c a n n o t f i n d a j o b i n h i s own t o w n . T h i s w i l l c a u s e t h e w i f e t o l o s e h e r j o b . Sometime when a p e r s o n c h a n g e s h i s j o b , he w o u l d t a k e t h i s o p p o r t u n i t y t o have a l o n g e r v a c a t i o n . T h i s w i l l a l s o c a u s e h i s w i f e t o work l e s s . when we p l o t t e d o u t t h e d a t a , u s i n g t h e number o f h o u r s w o r k e d by w i f e a g a i n s t t h e unemployment o f t h e head i n d a y s , we f o u n d t h a t d a t a a r e d i s t r i b u t e d more o r l e s s i n a g u a d r a t i c f o r m . T h e r e f o r e , we add ( U n e m p l o y ) 2 a s a n o t h e r new v a r i a b l e . F e c u n d i t — W i f e ' s f e c u n d i t y F e c u n d i t y i s an age v a r i a b l e d e f i n e d a s 45 m i n u s t h e age o f w i f e a t t i m e t . we c o n s i d e r t h e r e l a t i o n s h i p b e t w e e n f e c u n d i t y and y, o r b e t w e e n f e c u n d i t y and P r o b ( z = 1 ) i s g u a d r a t i c . ( F e c u n d i t ) 2 i s added b e c a u s e a y o u n g e r woman h a s l e s s f a m i l y work and b e t t e r p h y s i c a l a b i l i t y t o work more, b u t s h e d o e s h a v e a h i g h p r o b a b i l i t y t o b e a r a b a b y , w h i c h f o r c e s h e r t o work l e s s . I n c / n e e d -- r a t i o o f t o t a l i n c o m e s e x c e p t w i f e ' s wage o v e r n e e d s 43 I n c o m e s p e r n e e d s i s d e f i n e d a s t h e t o t a l f a m i l y n e t r e a l i n c o m e 1 minus t h e w i f e ' s wage and d i v i d e d by t h e f a m i l y n e e d s 2 . Even i f t h i s r a t i o i s h i g h i t d o e s n o t a l w a y s mean t h a t t h e f a m i l y h a s a h i g h i n c o m e , s i n c e t h e number o f d e p e n d e n t s may be l e s s . S i m i l a r l y , e v e n i f t h e r a t i o i s low t h a t may be c a u s e d by l a r g e f a m i l y s i z e . Most l i k e l y , t h i s r a t i o v a r i e s w i t h y o r P r o b ( z = 1) i n a n o n - l i n e a r p a t t e r n , s o we a l s o c o n s i d e r ( i n c / n e e d ) 2 i n o u r m o d e l . F i n a l l y i n o u r m o d e l s , we have 11 e x p l a n a t o r y v a r i a b l e s . They a r e : 1. B i r t h G a p b i r t h gap 2. W i f e wg p r e d i c t e d wage o f w i f e 3. ( W i f e w g ) 2 s q u a r e o f W i f e wg 4. Head i n c h e a d ' s i n c o m e p l u s t r a n s f e r i n c o m e 5. (Head i n c ) 2 s q u a r e o f Head i n c 6. Unemploy unemployment o f h e a d 7. (Unemploy) 2 s g u a r e o f Unemploy 8. F e c u n d i t f e c u n d i t y 9. ( F e c u n d i t ) 2 f e c u n d i t y s q u a r e 10. i n c / n e e d — — -— r a t i o o f t o t a l i n c o m e s e x c e p t w i f e ' s wage o v e r n e e d s 11. ( i n c / n e e d ) 2 s q u a r e o f i n c / n e e d IV D a t a r e s t r i c t i o n The d a t a s a m p l e i s o b t a i n e d f r o m t h e U n i v e r s i t y o f M i c h i g a n S u r v e y R e s e a r c h C e n t r e , P a n e l S t u d y Of Income D y n a m i c s (1972) 44 w h i c h i s b a s e d on 5 a n n u a l s u r v e y s . He r e s t r i c t d a t a i n o u r a n a l y s i s o f t h e m o d e l s by u s i n g t h e f o l l o w i n g c o n s t r a i n t s : 1. The h u s b a n d and w i f e were p r e s e n t i n t h e h o u s e h o l d i n a l l 5 y e a r s , 1967-71. 2. The h e a d was m a r r i e d f o r t h e f i r s t t i m e . 3. The h u s b a n d worked f o r a t l e a s t 350 h o u r s i n e a c h o f t h e 5 y e a r s . 4. The w i f e was n o t o l d e r t h a n 45 y e a r s o l d i n 1971. 5. The h e a d was l e s s t h a n 50 y e a r s o f age i n 1971. 6. The b i r t h g a p , t h e f a m i l y e x p e c t e d s i z e minus t h e a c t u a l f a m i l y s i z e , was p o s i t i v e i n a l l 5 y e a r s . T h e s e c o n s t r a i n t s a r e used t o e l i m i n a t e a l l t h e s p e c i a l c a s e s so t h a t o u r a n a l y s i s w i l l be b a s e d on a more r e l i a b l e s a m p l e . Suppose t h e f i r s t two c o n s t r a i n t s a r e used t o e l i m i n a t e t h o s e a b n o r m a l h o u s e h o l d s . I n some c a s e s t h e w i f e w i l l h a v e been m a r r i e d b e f o r e , and have c h i l d r e n f r o m t h e e a r l i e r m a r r i a g e . Hence t h e s e two c o n s t r a i n t s e n s u r e t h a t t h e c h i l d r e n i n t h e f a i m l y b e l o n g t o t h e c o u p l e . I n most o f t h e f a m i l i e s , t h e head i s r e s p o n s i b l e f o r t h e f a m i l y economy. The f a m i l y e x p e n d i t u r e i s m a i n l y d e p e n d e n t upon h i s e a r n i n g . H e n c e , c o n s t r a i n t 3 i s u s e d t o e n s u r e t h e s t a b i l i t y o f f a m i l y economy. An o b s e r v a t i o n i s v a l i d o r i n t e r e s t i n g o n l y i f t h e w i f e i s o f c h i l d b e a r i n g a g e , o r t h e c h i l d r e n a r e n o t o l d enough t o be l e f t a l o n e w h i l e s h e w o r k s . T h i s i s t h e u s a g e o f c o n s t r a i n t 4. R e s t r i c t i o n 5 i s used t o p r e v e n t c a s e s u c h t h a t a y o u n g g i r l i s m a r r i e d t o an e l d m i l l i o n a i r e . S u c h o b s e r v a t i o n s a r e n o t i n t e r e s t i n g , b e c a u s e d a t a a r e b i a s e d . C o n s t r a i n t 6 i s t o e n s u r e t h a t t h e b i r t h gap h a s s t a t i s t i c a l m e a n i n g . T h i s l e a v e s us w i t h 45 1700 o b s e r v a t i o n s f o r t h e 5 y e a r s p e r i o d : t h a t i s 340 f a m i l y u n i t s i n t o t a l . F o o t n o t e s : 1 The t o t a l f a m i l y n e t r e a l i n c o m e i s d e f i n e d a s t h e t o t a l r e a l i n c o m e m i n u s t h e c o s t o f e a r n i n g i n c o m e , minus h e l p f r o m o u t s i d e t h e f a m i l y u n i t , i f t h e r e a r e c h i l d r e n u n d e r 18. ( s e e d e f i n i t i o n i n A P a n e l S t u d y Of Income D y n a m i c s ) 2 The F a m i l y n e e d s i s a d j u s t e d a c c o r d i n g t o t h e US a n n u a l l i v i n g n e ed s t a n d a r d i n y e a r t . ( s e e d e f i n i t i o n i n A P a n e l S t u d y Of Income D y n a m i c s ) 46 C h a p t e r VI E m p i r i c a l R e s u l t s o f M o d e l I T h i s model i s e s t i m a t e d by u s i n g t h e m o d el d e s c r i b e d i n c h a p t e r 2. We h a v e 1700 o b s e r v a t i o n s i n t o t a l , t h a t i s 340 f a m i l y u n i t s i n a l l t h e 5 y e a r s . I n c o n s i d e r i n g t h e t i m i n g o f t h e m a r r i e d women p a r t i c i p a t i n g i n t h e l a b o u r f o r c e , we s p l i t t h e s a m p l e i n t o two c a t e g o r i e s a c c o r d i n g t o t h e age o f t h e y o u n g e s t c h i l d i n t h e f a m i l y . I f a f a m i l y h a s a c h i l d n o t o l d e r t h a n 6 y e a r s o f a g e , w h i c h i s z = 1 , we p u t i t i n t o g r o u p I ; t h e o t h e r s we p u t i n t o g r o u p I I . H ence, g r o u p I ( u n d e r t h e c o n d i t i o n t h a t z=1) h a s 885 o b s e r v a t i o n s , and g r o u p I I ( u n d e r t h e c o n d i t i o n t h a t z=0) has 815 o b s e r v a t i o n s . From t a b l e I I , we o b s e r v e t h a t on t h e a v e r a g e , t h e number o f h o u r s w o r k e d by a w i f e i n g r o u p I i s much l e s s ( a b o u t 4 7%) t h a n t h o s e w o r ked by a w i f e i n g r o u p I I . T h i s s u p p o r t s o u r a s s u m p t i o n t h a t a w i f e w i t h a c h i l d n o t o l d e r t h a n 6 y e a r s o f age w i l l work l e s s h o u r s . I . R e s u l t s f r o m t h e l a b o u r r e g r e s s i o n e q u a t i o n s : From T a b l e I , we f i n d t h a t t h e l a b o u r e g u a t i o n o f g r o u p I has a b i g g e r c o n s t a n t t e r m t h a n t h e e q u a t i o n o f g r o u p I I . S u p p o s e we keep a l l t h e e x p l a n a t o r y v a r i a b l e s f i x e d and l e t t h e w i f e ' s wage and i t s s g u a r e v a r y w i t h t h e h o u r s w o r k e d by t h e w i f e ; t h e n we f i n d i n g r o u p I , t h e c u r v e i s c o n c a v e upward b u t i n g r o u p I I , t h e c u r v e i s c o n c a v e downward. From t h e f i g u r e s shown on t a b l e I , we know t h a t t h e h o u r s w orked by t h o s e 47 h o u s e w i v e s h a v i n g c h i l d r e n n o t o l d e r t h a n 6 y e a r s o f a g e , n o r m a l l y w i l l n o t be a f f e c t e d by t h e i r wages. They w i l l work more o n l y when t h e i r wages a r e h i g h , e x c e p t f o r t h o s e who a r e a m b i t i o u s t o work. Those w i v e s whose c h i l d r e n a r e o l d e r t h a n 6 y e a r s o f a g e , w i l l work o n l y up t o a c e r t a i n number o f h o u r s . I n t h e l a b o u r e q u a t i o n o f g r o u p I , we f i n d t h a t 1) w i f e ' s wage, 2) t h e h ead's wage p l u s t r a n s f e r i n c o m e , 3) t h e r a t i o o f i n c o m e s o v e r n e e d s a n d , and 4) t h e s q u a r e s o f 1 ) , 2 ) , and 3) a r e s t a t i s t i c a l l y s i g n i f i c a n t i n t h e r e g r e s s i o n o f t h e number o f h o u r s worked by t h e w i f e on t h e e x p l a n a t o r y v a r i a b l e s . On t h e o t h e r hand i n g r o u p I I , t h e h o u r s worked by t h e w i f e i s s i g n i f i c a n t l y a f f e c t e d by 1) t h e h e a d ' s wage p l u s t r a n s f e r i n c o m e , 2) t h e r a t i o o f i n c o m e s o v e r n e e d s , 3) f e c u n d i t y , 4) t h e s g u a r e s o f 1 ) , 2) and 3 ) , and 5) t h e s q u a r e o f t h e w i f e ' s wage. H y p o t h e s i s t e s t i n g — t h e e q u a l i t y o f two l a b o u r e q u a t i o n s I n t h e a b o v e d i s c u s s i o n , we o b s e r v e t h a t e a c h l a b o u r e g u a t i o n h a s i t s own c h a r a c t e r i s t i c s t r u c t u r e , b u t now we assume t h e e g u a t i o n s a r e e g u a l , i n o r d e r t o t e s t t h e e q u a l i t y o f t h e s e two e q u a t i o n s . From Chow's t e s t , we f i n d F ( 1 2, 1 6 7 6 ) = 1 1 . 7 8 6 1 t h a t i s f a r b e y o u n d 95% s i g n i f i c a n c e v a l u e ( F = 2 . 3 0 ) , so we r e j e c t t h e h y p o t h e s i s o f e q u a l i t y . I I R e s u l t s f r o m t h e p r o b a b i l i t y e q u a t i o n From T a b l e I I I , we f i n d t h e b i r t h gap, t h e h e a d ' s i n c o m e . 48 f e c u n d i t y a nd t h e s q u a r e o f f e c u n d i t y have a s t r o n g l y s i g n i f i c a n t e f f e c t on t h e p r o b a b i l i t y o f z=1. T h i s i s a v e r y good outcome b e c a u s e t h e s e m a j o r v a r i a b l e s t u r n o u t as we e x p e c t e d i n c h a p t e r 5. A l s o , t h e r a t i o o f i n c o m e s o v e r n e e d , t h e w i f e ' s wage a n d t h e s q u a r e o f t h e w i f e ' s wage, do s i g h t l y a f f e c t t h e p r o b a b i l i t y o f z=1. He s c a l e t h e u n i t s o f t h e h e a d ' s i n c o m e , t h e r a t i o o f i n c o m e s o v e r n e e d s , t h e unemployment o f t h e h e a d , and t h e s g u a r e s o f t h e s e v a r i a b l e s , t o s p e e d up t h e c o n v e r g e n c e r a t e i n m a x i m i z i n g t h e maximum l i k e l i h o o d f u n c t i o n ; o t h e r w i s e c o n v e r g e n c e i s v e r y d i f f i c u l t t o o b t a i n . I l l F u r t h e r e s t i m a t i o n : I f we c o n s i d e r t h o s e s t a t i s t i c a l l y s i g n i f i c a n t v a r i a b l e s w h i c h we o b t a i n f r o m t h e r e s u l t s o f t h e e s t i m a t i o n o f t h i s model a s r e l i a b l e v a r i a b l e s , we w o u l d l i k e t o r e - e s t i m a t e t h e model o n l y w i t h t h o s e v a r i a b l e s . The f o l l o w i n g a r e t h e new e s t i m a t e s . B e g r e s s i o n model — L a b o u r e g u a t i o n G r o u p I , ( y | z = 1 ) HourWork = 1193.0 - 633.7 ( W i f e wg) + 182.0 ( l i f e w g ) 2 - 0.1772 (Head i n c ) + 0 . 2 5 x 1 0 - 5 (Head i n c ) 2 + 5.789 ( i n c / n e e d ) - 0.0034 ( i n c / n e e d ) 2 49 Group I I , ( y | z = 0 ) HourWork = 7 9 2 . 1 - 35.57 ( W i f e wg) - 0. 1676 (Head i n c ) + 0.22x10~s {Head i n c ) 2 + 6.197 ( i n c / n e e d ) - 0 . 0 0 3 2 ( i n c / n e e d ) 2 + 42.15 (Fecund) L o g i s t i c model — P r o b a b i l i t y e g u a t i o n L o g i t {p+ ( i ) ) = -1.883 - 0.3412 ( B i r t h G a p ) + 0.5426 ( W i f e wg) - 0.1556 ( W i f e w g ) 2 + 0.0494 (Head i n c ) - 0.1616 ( i n c / n e e d ) + 0. 1895 ( F e c u n d i t ) - 0.0037 ( F e c u n d ) 2 T a b l e I P a r a m e t e r E s t i m a t e s f o r L a b o u r E q u a t i o n s Group I Group I I Whole s a m p l e V a r i a b l e ( y J z = 1 ) ( y | z = 0 ) y u n c o n d i t i o n a l c o n s t a n t 1 699.6306* (73.384) 4 7 2 . 9 4 7 0 * (21.344) 1180.0698* (71.507) B i r t h G a p - 4 4 .7253 (1.378) - 36.7626 (1.708) 6.6327 (0.392) W i f e wg -704.254 3* (3.274) 364.1801 (1.774) -130.0699 (0.863) ( W i f e wg)2 1 9 0 . 1 4 2 7 * (3.896) - 1 1 4 . 0 0 6 0 * (2.582) 31.8822 (0.958) Head i n c - 0 . 2 0 8 2 * (14.829) - 0 . 1 7 4 4 * (13.536) - 0 . 1 9 4 9 * (21.289) (Head i n c ) 2 0 . 2 9 1 8 x 1 0 - 5 * (9.931) 0 . 2 3 4 8 x 1 0 - 5 * (7.248) 0 . 2 6 1 7 x 1 0 - 5 * (13.436) Unemploy 3.0143 ( 1 . 176) -0.7786 (0.275) 1.4487 (0.749) (Unemploy) 2 -0.0220 (1.403) -0.0123 (0.666) -0.0188 (1.547) i n c / n e e d 6.6705* (8.978) 6.1337* (14.408) 6.1123* (17.821) ( i n c / n e e d ) 2 - 0 . 4 1 4 5 x 1 0 - 2 * (4.455) - 0 . 3 1 5 4 x 1 0 - 2 * (8.795) - 0 . 3 1 8 4 x 1 0 - 2 * (9.775) F e c u n d i t -12.6039 (0.629) 4 3 . 2 1 4 1 * (3 . 130) 5.2290 (0.469) ( F e c u n d i t ) 2 -0.5377 (0.785) - 1 . 7 8 3 5 * (3 . 309) - 1 . 1 6 2 5 * (2.833) O b s e r v a t i o n s 885 815 1700 R2 0.2538 0.3201 0.2789 * s i g n i f i c a n t l e v e l o f 5% u n d e r H: p a r a m e t e r s = 0.0 A s y m t o t i c t v a l u e s a r e i n p a r e n t h e s e s . T a b l e I I Mean and S t a n d a r d D e v i a t i o n o f t h e M o d e l I V a r i a b l e G r o up I ( y | z = 1 ) Group I I ( y | z = 0 ) Whole s a m p l y u n c o n d i t i HourWork 771.519 (792.743) 1133.79 (761 . 958) 945.196 (798.696) B i r t h G a p 0.3785 (0.7627) 0.6025 (1.4130) 0.4859 (1.1277) W i f e wg 1.8497 (0.5728) 1.9598 (0.6316) 1.9025 (0.604) ( W i f e wg) 2 3.7491 (2.4977) 4.2393 (2.8939) 3.9841 (2.705) Head i n c 8297.75 (4569.88) 8933.23 (5243.44) 8602.40 (4913. 16) (Head i n c ) 2 0.8 97x10« (0.209x109) 0.107x109 ( 0 . 175x109) 0.981x108 ( 0 . 194x109) Unemploy 5.1388 (21.5523) 4.1931 (19.8563) 4.6854 (20.7558) (Unemploy) 2 490.385 (3509. 11) 411.371 (3028.43) 452.505 (3286.73) i n c / n e e d 285.906 (132. 113) 332.063 (173.459) 308.034 (155.012) ( i n c / n e e d ) 2 0.992x105 (0.108x106) 0.140x10* (0 . 184x106) 0. 119x106 (0.151x106) F e c u n d i t 15.9503 (5.8996) 11.1767 (6.9893) 13.6618 (6.871) ( F e c u n d i t ) 2 289.177 (175.181) 173.709 (189.114) 233.821 (190.871) O b s e r v a t i o n 885 815 S t a n d a r d d e v i a t i o n s a r e i n p a r e n t h e s e s . 1700 T a b l e I I I P r o b a b i l i t y 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 V a r i a b l e C o e f f i c i e n t As ym s t d v Asym c o n s t a n t -1 . 9 4 2 2 2 * 0. 3079 6. 308 B i r t h G a p -0 . 3 3 5 2 6 * 0. 0412 8. 134 W i f e wg 0. 5 0 0 6 3 * 0. 2557 1. 958 ( W i f e w g ) 2 -0 .14714* 0. 0564 2. 60 9 Head i n c 0. 0 6 9 8 1 * 0. 0158 4. 429 (Head i n c ) 2 -0 .00050 0. 0003 1. 505 Unemploy 0. 20558 0. 3257 0. 631 (Unemploy) 2 -0 .09577 0. 1999 0. 479 i n c / n e e d -0 . 17239* 0. 0747 2. 307 ( i n c / n e e d ) 2 0. 00019 0. 0084 0. 022 F e c u n d i t 0. 1881 1* 0. 0216 8. 726 ( F e c u n d i t ) 2 -0 .00 3 5 7 * 0. 0008 4. 627 L o g o f l i k e l i h o o d f u n c t i o n = -947.522 a f t e r 11 i t e r a t i o n s . * s i g n i f i c a n t l e v e l o f 5% u n d e r H: p a r a m e t e r s = 0.0 n o t e : 1 Head i n c = $1,000 1 (Head i n c ) 2 = $1,000,000 1 i n c / n e e d = 100 1 ( i n c / n e e d ) 2 = 10,000 1 Unemploy = 100 d a y s o f h e a d ' s unemployment 1 ( u n e m p l o y ) 2 = 10,000 d a y s . 53 C h a p t e r V I I E m p i r i c a l R e s u l t s o f M o d e l I I B e f o r e we b e g i n t o a n a l y z e t h e r e s u l t , l e t us c o n s i d e r t h e a n n u a l h o u r s w o rked by w i v e s . From t a b l e V, we o b s e r v e t h a t i n g e n e r a l , t h o s e w i v e s whose c h i l d r e n a r e o l d e r t h a n 6 y e a r s o f age w i l l work more h o u r s . The r a n g e i s f r o m 3 0 . 1 7 % t o 5 7 . 9 5 % * . I n c o m p a r i n g f e c u n d i t i e s we know t h a t on t h e a v e r a g e t h o s e w i v e s i n g r o u p I a r e y o u n g e r and t h e i r wages a r e l e s s t h a n t h o s e i n g r o u p I I . T h e s e a r e c o n s i s t e n t w i t h t h e f i n d i n g s o f B e r n d t and w a l e s ( 1 9 7 4 ) . I R e s u l t s o f s i n g l e e g u a t i o n e s t i m a t i o n From t a b l e I V , we f i n d t h a t t h e h e a d ' s wage p l u s t r a n s f e r i n c o m e , and t h e r a t i o o f i n c o m e s o v e r n e e d s a r e s t a t i s t i c a l l y s i g n i f i c a n t i n most o f t h e y e a r s , b u t t h e b i r t h gap i s n o t s i g n i f i c a n t i n any y e a r . The d e g r e e s o f s i g n i f i c a n c e o f t h e r e s t o f t h e e x p l a n a t o r y v a r i a b l e s v a r y i n d i f f e r e n t y e a r s . I f we compare t h e f u n c t i o n a l s t r u c t u r e s y e a r - b y - y e a r , we w i l l o b s e r v e t h a t t h e y h a v e d i f f e r e n t s h a p e s . S u p p o s e we k e e p a l l t h e e x p l a n a t o r y v a r i a b l e s c o n s t a n t e x c e p t t h e unemployment o f t h e head and i t s s g u a r e , and compare 1968 w i t h 1969; t h e n we w i l l f i n d i n 1968, t h e c u r v e i s c o n c a v e downward, b u t i n 1969 i t i s u p w a r d . Downward o r upward w i l l g i v e us d i f f e r e n t i n t e r p r e t a t i o n s . An upward c u r v e w i l l p o i n t o u t t h a t t h e r e a r e b i g i n c r e a s e s i n two e x t r e m e s . A downward c u r v e w i l l i n d i c a t e t o us t h a t t h e v a r i a b l e w i l l be m e a n i n g f u l o n l y i n a c e r t a i n 54 d o m a i n , b e c a u s e we a r e n o t i n t e r e s t e d i n n e g a t i v e v a l u e s o f t h e d e p e n d e n t v a r i a b l e . H y p o t h e s i s t e s t i n g s — t h e e q u a l i t y o f t h e two l a b o u r e g u a t i o n s i n e a c h y e a r ; The h y p o t h e s i s t e s t i n g s o f t h e e q u a l i t y o f t h e two l a b o u r e g u a t i o n s i n 1967 u s i n g a 5$ c r i t i c a l p o i n t , a n d i n 1970 u s i n g a 1% c r i t i c a l p o i n t a r e a c c e p t a b l e ; i n a l l o t h e r y e a r s t h e h y p o t h e s e s a r e n o t a c c e p t a b l e . T h e s e t e l l u s t h a t f o r 1967 o r 1970 we c a n a g g r e g a t e a l l t h e d a t a and e x p r e s s t h e l a b o u r r e l a t i o n f o r t h a t y e a r u s i n g a l i n e a r f u n c t i o n , r e g a r d l e s s o f what t h e v a l u e o f z i s . I I R e s u l t s o f Z e l l n e r ' s s e e m i n g l y l e a s t s q u a r e s method I f h e t e r o s c e d a s t i c i t y i s p r e s e n t , t h e n t h e w e i g h t i n g i n a s i n g l e e q u a t i o n , i s n o t a p p r o p r i a t e . The r e a s o n i s t h a t i n an e q u a t i o n - b y - e q u a t i o n method, we use t h e l e a s t - s q u a r e s p r o c e d u r e t o o b t a i n t h e p a r a m e t e r e s t i m a t e s , b u t a l l v a r i a b l e s a r e q i v e n t h e same w e i q h t . T h i s w e i q h t i s u n s a t i s f a c t o r y i n o u r s a m p l e . S u p p o s e we use t h e two s t a g e A i t k e n method as i n t r o d u c e d by Z e l l n e r { 1 9 6 2 ) ; t h e n we f i n d t h a t t h e r e g r e s s i o n p a r a m e t e r s s o o b t a i n e d a r e a t l e a s t a s y m p t o t i c a l l y more e f f i c i e n t t h a n t h o s e o b t a i n e d by an e q u a t i o n - b y - e q u a t i o n method u s i n q o r d i n a r y l e a s t - s q u a r e s . 55 The c o m p a r i s o n o f t h e r e s u l t s f r o m t h i s method w i t h t h e r e s u l t s f r o m s i n g l e e g u a t i o n e s t i m a t i o n I n t a b l e V I , t h e r e s u l t s show a s i g n i f i c a n t r e d u c t i o n i n t h e d e v i a t i o n s o f e s t i m a t e d p a r a m e t e r s . The r a n g e f o r r e d u c t i o n i s f r o m 10% t o 20%. He f i n d t h a t t h e r e a r e a p p e r e n t l y d i f f e r e n t q u a d r a t i c f o r m s . S u c h d i f f e r e n c e s a r e p o s s i b l e b e c a u s e we e x p e c t t h e two s t a g e A i t k e n ' s 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 t h o s e f r o m o r d i n a r y l e a s t - s g u a r e s . I n t h e two s t a g e A i t k e n ' s method we a s s i g n d i f f e r e n t w e i g h t s t o t h e s a m p l e , s o t h e r e s u l t s h o u l d be somewhat d i f f e r e n t t h a n t h e r e s u l t o f t h e o r d i n a r y l e a s t - s g u a r e s . Most o f t h e c h a n g e s do n o t a f f e c t t h e e c o n o m i c a l i n t e r p r e t a t i o n , b u t some do. S u p p o s e t h a t i n t h e l a b o u r e q u a t i o n s o f g r o u p I I i n t a b l e V I E, t h e r e a r e d i f f e r e n t i n t e r p r e t a t i o n o f t h e w i f e ' s wage and t h e h e a d ' s unemployment. F o r e x a m p l e , i f we keep a l l t h e i n d e p e n d e n t v a r i a b l e s f i x e d e x c e p t t h e w i f e ' s wage and i t s s q u a r e , we w i l l f i n d f r o m t h e s i n g l e e q u a t i o n method t h a t t h e c u r v e i s c o n c a v e downward, b u t f r o m two s t a g e method i t i s c o n c a v e u p w a r d . H e n c e , i f we a d o p t t h e r e s u l t f r o m t h e s i n g l e e q u a t i o n , we w i l l s a y t h a t t h o s e w i v e s whose c h i l d r e n a r e o l d e r t h a n 6 y e a r s o f age w i l l n o t be s t i m u l a t e d t o work more by t h e i r e x p e c t e d wages. On t h e e t h e r h a n d , i f we u s e t h e r e s u l t f r o m t h e two s t a g e e s t i m a t i o n , we w i l l o b s e r v e t h a t t h o s e w i v e s w i l l work more i n b o t h e x t r e m e s , l i k e w i s e , we f i n d a s i m i l a r d i f f e r e n c e i n t h e h e a d ' s unemployment f r o m t h i s t a b l e . S u c h e x a m p l e s c a n be e a s i l y f o u n d f r o m t h e r e s u l t s o f o t h e r y e a r s . A n o t h e r s i g n i f i c a n t d i f f e r e n c e i s i n two s t a g e e s t i m a t i o n , none o f t h e h y p o t h e s e s o f e g u a l i t y t e s t e d i s a c c e p t a b l e . I t i s n o t l i k e t h e s i n g l e e q u a t i o n 56 e s t i m a t i o n where we do 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 . T e s t i n g f o r A g g r e g a t i o n B i a s f o r M o d e l I I : B e f o r e we go i n t o t h e l o g i s t i c m o d e l , l e t us c o n s i d e r t h e a g g r e g a t i o n b i a s . Our t e s t i n g h y p o t h e s i s a s s e r t s t h a t d a t a i n e v e r y y e a r a r e homogeneous i n s o f a r as r e g r e s s i o n c o e f f i c i e n t v e c t o r s a r e c o n c e r n e d . T h a t i s H: c o e f f o f 67 = c o e f f o f 68 = ... = c o e f f o f 71 T h e r e a r e two t e s t i n g m e t h o d s : t h e method s u g g e s t e d by Z e l l n e r u s i n g an F - t e s t , and t h e l i k e l i h o o d r a t i o t e s t . H ere we use t h e F - t e s t a p p r o a c h . From t h e t e s t we f i n d t h a t f o r t h e l a b o u r e q u a t i o n o f g r o u p I , F (48,1640) = 3.4044 and f o r t h e l a b o u r e q u a t i o n o f g r o u p I I , F ( 4 8 , 1 6 4 0 ) = 5 8 . 4 0 4 7 . B o t h a r e r e j e c t e d 2 . T h e r e f o r e , we c o n c l u d e t h a t t h e r e i s an a g g r e g a t i o n b i a s i n v o l v e d i n s i n g l e l i n e a r a g g r e g a t i o n . The e s t i m a t e d l a b o u r e g u a t i o n s : The a b o u t d i s c u s s i o n shows t h a t t h e r e i s a s i g n i f i c a n t r e d u c t i o n i n t h e d e v i a t i o n s o f e s t i m a t e d p a r a m e t e r s by u s i n g Z e l l n e r * s two s t a g e s e s t i m a t i o n method. The e s t i m a t e d e g u a t i o n s c a n be s u m m a r i z e d a s f o l l o w s : 57 G r o u p I , ( y I z = 1 ) 1967: HourWork = -24.88 + 206.3 ( B i r t h G a p ) + 360.0 ( W i f e wg) - 50.99 ( R i f e w g ) 2 - 0.2742 (Head i n c ) 0.621x10-» (Head i n c ) 2 + 37. 26 (Unemploy) 0.5963 ( U n e m p l o y ) 2 + 10.54 ( i n c / n e e d ) - 0 . 0 0 9 5 ( i n c / n e e d ) 2 + 20. 42 ( F e c u n d i t ) - 1.512 ( F e c u n d i t ) * 1968: HourWork = 719.5 + 5 8 . 6 4 ( B i r t h G a p ) + 5 4 . 4 5 ( W i f e wg) + 8.394 ( W i f e w g ) 2 - 0.1846 (Head i n c ) + 0 . 2 4 2 x 1 0 ~ s ( H e a d i n c ) 2 + 6.746 (Unemploy) - 0.0856 (Onemploy) 2 + 6.638 ( i n c / n e e d ) - 0.0031 ( i n c / n e e d ) 2 + 4.671 ( F e c u n d i t ) - 1.278 ( F e c u n d i t ) 2 1969: HourWork = 1074.0 + 44. 85 ( B i r t h G a p ) - 240.5 ( W i f e wg) + 81.98 ( W i f e w g ) 2 - 0.2054 (Head i n c ) + 0. 2 6 7 x 1 0 - s (Head i n c ) 2 - 5. 1539 (Onemploy) + 0. 1846 ( U n e m p l o y ) 2 + 9.697 ( i n c / n e e d ) - 0.0073 ( i n c / n e e d ) 2 - 52.76 ( F e c u n d i t ) + 0.3216 ( F e c u n d i t ) 2 1970: HourWork = 1413.0 - 92.92 ( B i r t h G a p ) - 653.0 («ife wg) + 157.1 ( W i f e w g ) 2 - 0.2830(Head i n c ) + 0 . 4 7 0 x 1 0 ~ 5 ( H e a d i n c ) 2 + 4.514(Unemploy) - 0.0290 ( U n e m p l o y ) 2 + 10.68 ( i n c / n e e d ) - 0.0074 ( i n c / n e e d ) 2 58 15-17 ( F e c u n d i t ) - 0.8335 ( F e c u n d i t ) 2 1 9 7 1 : H o u r w o r k = 1330.0 - 58.39 ( B i r t h G a p ) - 473.8 ( W i f e wg) + 130.3 ( W i f e wg)2 - 0.2869 (Head i n c ) + 0.648x10~s (Head i n c ) 2 + 4.826 (Unemploy) 0.0239 ( U n e m p l o y ) 2 + 8.738 ( i n c / n e e d ) - 0 . 0 0 6 2 ( i n c / n e e d ) 2 - 19.41 ( F e c u n d i t ) - 0 . 2 1 0 4 ( F e c u n d i t ) 2 Group I I , ( y | z = 0 ) 1967: HourWork = 1704.0 + 6.748 ( B i r t h G a p ) - 789. 1 ( W i f e wg) + 1 0 0 . 5 ( W i f e w g ) 2 - 0.2643 (Head i n c ) + 0 . 4 3 3 x 1 0 ~ s ( H e a d i n c ) 2 - 1.174 (Unemploy) - 0.0102(Unemploy) 2 + 1 0 . 3 7 ( i n c / n e e d ) - 0 . 0 0 6 7 ( i n c / n e e d ) 2 + 29. 1 1 ( F e c u n d i t ) - 1. 692 ( F e c u n d i t ) 2 1968: HourWork = 1207.0 - 11.95 ( B i r t h G a p ) - 524.5 ( H i f e wg) + 7 3 . 1 8 ( W i f e wg) 2 - 0.2294 (Head i n c ) + 0 . 3 6 3 x 1 0 - 5 (Head i n c ) 2 - 0.2577 (Unemploy) + 0.0541 ( U n e m p l o y ) 2 + 11.08 ( i n c / n e e d ) - 0.0077 ( i n c / n e e d ) 2 - 23.43 ( F e c u n d i t ) + 0 . 2 1 6 6 ( F e c u n d i t ) 2 1969: HourWork 59 = -61.86 - 53.81 ( B i r t h G a p ) + 479.7 ( W i f e wg) - 1 3 5 . 4 ( W i f e w g ) 2 - 0.1750 (Head i n c ) + 0 . 2 5 7 x 1 0 - s ( H e a d i n c ) 2 + 6.129(Unemploy) - 0.0385 ( U n e m p l o y ) 2 + 7.854 ( i n c / n e e d ) - 0.0050 ( i n c / n e e d ) 2 + 75. 9 0 ( F e c u n d i t ) - 3.208 ( F e c u n d i t ) 2 1970: HourWork = 47.04 - 34.05 ( B i r t h G a p ) + 590.2 ( w i f e wg) - 146.9 ( W i f e w g ) 2 - 0. 1548 (Head i n c ) + 0.199x10~s (Head i n c ) 2 - 3.178 (Unemploy) 0.0035 (Unemploy) 2 5.528 ( i n c / n e e d ) - 0.0025 ( i n c / n e e d ) 2 + 74. 95 ( F e c u n d i t ) - 3.702 ( F e c u n d i t ) 2 1971: HourWork = 839.7 - 112. 1 ( B i r t h G a p ) - 104.1 ( W i f e wg) + 8.551 ( W i f e wg) 2 - 0. 1368 (Head i n c ) + 0 . 1 7 6 x 1 0 - s { H e a d i n c ) 2 + 3.280(Unemploy) - 0.0163 ( U n e m p l o y ) 2 + 4.813 ( i n c / n e e d ) - 0.0021 ( i n c / n e e d ) 2 + 29.40 ( F e c u n d i t ) - 1. 030 ( F e c u n d i t ) 2 I I I R e s u l t s o f p r o b a b i l i t y f u n c t i o n s From t a b l e V I I we f i n d t h e b i r t h gap i s v e r y s i g n i f i c a n t i n a l l t h e p r o b a b i l i t y f u n c t i o n s . The r e s u l t s i n t h e y e a r s 1968 and 1969 a r e i n t e r e s t i n g . 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 i s s i g n i f i c a n t l y a f f e c t e d by t h e b i r t h g a p , t h e h e a d ' s wage p l u s t r a n s f e r i n c o m e , f e c u n d i t y and t h e s q u a r e o f f e c u n d i t y . I n 1 9 6 9 , t h e f u n c t i o n i s a f f e c t e d by most o f t h e v a r i a b l e s , s u c h a s t h e b i r t h g a p , t h e h e a d ' s u n e m p l o y m e n t , t h e r a t i o o f i n c o m e s o v e r n e e d s , f e c u n d i t y , t h e w i f e ' s wage and t h e s g u a r e s o f t h e 60 h e a d ' s unemployment and t h e wage's wage. One t h i n g w h i c h h a s s u r p r i s e d us i s t h a t t h e h e a d ' s wage p l u s t r a n s f e r i n c o m e s d o e s n o t s i g n i f i c a n t l y a f f e c t t h e p r o b a b i l i t y f u n c t i o n t h a t h a p p e n s i n most o f t h e y e a r s e x c e p t 1968, and 1971. M o r e o v e r , we do n o t draw any f r u i t f u l c o n c l u s i o n f r o m t h e r e s u l t s o f t h e y e a r s 1970 and 1971. T e s t f o r a g g r e g a t i o n b i a s ; I n c o n s i d e r i n g t h e t e s t o f t h e a g g r e g a t i o n b i a s f o r t h i s p a r t , we f i n d t h e r a t i o o f maximum l i k e l i h o o d i s s o b i g ( C h i s q u a r e (48) = 0 . 3 9 x 1 0 9 ) t h a t we c a n n o t a c c e p t t h e h y p o t h e s i s t h a t t h e r e i s no a q g r e g a t i o n b i a s . The e s t i m a t e d p r o b a b i l i t y f u n c t i o n s : He c o n c l u d e t h e e s t i m a t i o n o f p r o b a b i l i t y f u n c t i o n s f o r e a c h y e a r as f o l l o w s . 1967: L o g i t (p+ ( i ) ) = -2.428 - 0.4167 ( B i r t h G a p ) + 0.3549 ( W i f e wg) - 0.0962 ( W i f e wg) 2 + 0.0651 (Head i n c ) + 0 . 4 8 8 x 1 0 - * ( H e a d i n c ) 2 + 0.0283 (Onemploy) - 0.0005 ( u n e m p l o y ) 2 - 0.0221 ( i n c / n e e d ) - 0.0281 ( i n c / n e e d ) 2 + 0.2835 ( F e c u n d i t ) - 0. 0073 ( F e c u n d i t ) 2 1968: L o g i t ( 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 w g ) 2 61 + 0.1150 (Head i n c ) - 0.0005 (Head i n c ) 2 - 0 . 0 1 0 9 ( U n e m p l o y ) + 0.0001 ( U n e m p l o y ) 2 - 0.3319 ( i n c / n e e d ) - 0.0028 ( i n c / n e e d ) 2 + 0.2486 ( F e c u n d i t ) - 0. 0049 ( F e c u n d i t ) 2 1969: L o g i t (p+ ( i ) ) = -2.390 -0.4669 ( B i r t h G a p ) + 1.671 ( H i f e wg) - 0 . 3 9 1 3 ( W i f e w g ) 2 + 0.0717{Head i n c ) - 0.0003(Head i n c ) 2 + 0.0534(Unemploy) 0.0011 (Unemploy) 2 - 0.4832 ( i n c / n e e d ) + 0.0289 ( i n c / n e e d ) 2 + 0.1198 ( F e c u n d i t ) - 0.0002 ( F e c u n d i t ) 2 1970: L o g i t ( p + ( i ) ) = -2.035 - 0. 2968 ( B i r t h G a p ) + 1.258 ( H i f e wg) - 0.3676 ( H i f e wg) 2 + 0.0887 (Head i n c ) - 0.0017 (Head i n c ) 2 + 0.0016 (Unemploy) + 0.411x10-* (Unemploy) 2 - 0. 2755 ( i n c / n e e d ) + 0. 0144 ( i n c / n e e d ) 2 + 0.0776 ( F e c u n d i t ) + 0.00 16 ( F e c u n d i t ) 2 1971; L o g i t (p+ ( i ) ) = -1.297 - 0.1922 ( B i r t h G a p ) + 0 . 2 1 3 7 ( W i f e wg) - 0 . 1 0 1 6 ( W i f e w g ) 2 + 0 . 1 5 9 3 ( H e a d i n c ) - 0.0059(Head i n c ) 2 - 0.0049(Unemploy) + 0.340x10-* (Unemploy) 2 - 0. 1830 ( i n c / n e e d ) + 0.0075 ( i n c / n e e d ) 2 + 0.0500 ( F e c u n d i t ) + 0.0023 ( F e c u n d i t ) 2 I V F u r t h e r e s t i m a t i o n I f we c o n s i d e r t h o s e s t a t i s t i c a l l y s i g n i f i c a n t v a r i a b l e s 62 w h i c h we o b t a i n f r o m t h e r e s u l t s o f t h e e s t i m a t i o n o f t h i s model a s r e l i a b l e v a r i a b l e s , we w o u l d l i k e t o r e - e s t i m a t e t h e model o n l y w i t h t h o s e v a r i a b l e s . The new . e s t i m a t e s a r e shown a s f o l l o w s : 3 R e g r e s s i o n m o d e l — l a b o u r e g u a t i o n G r oup I , ( y | z = 1 ) 1967: H o u r w o r k 592.6 - 0.2331(Head i n c ) + 0 . 5 1 x 1 0 ~ 5 ( H e a d i n c ) 2 + 45.87(Unemploy) - 0 . 7 7 7 0 ( U n e m p l o y ) 2 • 8 . 2 6 2 ( i n c / n e e d ) 0.0064 ( i n c / n e e d ) 2 1968: HourWork = 1768.0 + 187. 8 ( W i f e wg) 2 - 0.0763 (Head i n c ) + 0.14x10-s (Head i n c ) 2 1969: HourWork 732.6 + 51.77 ( W i f e w g ) 2 - 0. 1448 (Head i n c ) + 0 . 1 3 x 1 0 - 5 ( H e a d i n c ) 2 + 3 . 3 2 7 ( i n c / n e e d ) 1970: HourWork 640.8 - 0.2325 (Head i n c ) + 0.37x10-s (Head i n c ) 2 + 7 . 4 2 8 ( i n c / n e e d ) - 0 . 0 0 4 8 ( i n c / n e e d ) 2 63 1 9 7 1 : HourWork = 2 9 1 . 1 - 0. 1459 (Head i n c ) + 7.100 ( i n c / n e e d ) - 0.0050 ( i n c / n e e d ) 2 Group I I , ( y | z = 0 ) 1967: HourWork HourWork = 455.6 - 0. 196 (Head i n c ) + 0.26x10~s (Head i n c ) 2 + 8.959 ( i n c / n e e d ) - 0.006 1 ( i n c / n e e d ) 2 1968: HourWork = 332.5 - 0. 1303 (Head i n c ) + 8. 561 ( i n c / n e e d ) - 0.0060 ( i n c / n e e d ) 2 1969 HourWork = 655.8 - 0. 1046 (Head i n c ) + 6.096 ( i n c / n e e d ) - 0.004 ( i n c / n e e d ) 2 1970: HourWork 649.7 - 0. 1521 (Head i n c ) + 0.20x10~s (Head i n c ) 2 + 5 . 4 6 9 ( i n c / n e e d ) - 0 . 0 0 2 8 ( i n c / n e e d ) 2 + 6 2 . 4 4 ( F e c u n d i t ) 3 . 2 0 4 ( F e c u n d i t ) 2 1 9 7 1 : 6a HourWork 919.0 - 0,131(Head i n c ) + 0 . 1 6 x 1 0 ~ « ( h e a d i n c ) 2 + 4 . 4 2 5 ( i n c / n e e d ) - 0 . 0 0 2 1 ( i n c / n e e d ) 2 l o g i s t i c model — P r o b a b i l i t y e g u a t i o n 1967 : l o g i t ( p + ( i ) ) -1.475 - 0.4 866 ( B i r t h G a p ) + 0.2291 ( F e c u n d i t ) 0.0058 ( F e c u n d i t ) 2 1968: L o g i t ( 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: L o g i t ( p + ( i ) ) = -1.086 - 0.5104 ( B i r t h G a p ) + 0.0100 ( F e c u n d i t ) + 5.609 (Unemploy) - 1 1 . 5 6 ( U n e m p l o y ) 2 1970: L o g i t ( p M i ) ) = 0.3183 - 0.0849 ( W i f e wg) 2 1971: L o g i t ( p + ( i ) ) = 0.2879 - 0.0363 (Head i n c ) f o o t n o t e : 65 i 1967 - 3 0 . 1 7 % , 1968 - 5 1 . 2 0 % , 1969 - 5 7 . 9 5 3 , 1970 - 4 6 . 3 6 % , 1971 - 4 7 . 4 1 % z U n d e r 5$ c r i t i a l p o i n t s t h e v a l u e o f F ( 4 8 , 1 6 4 0 ) i s 1.49. 3 Some v a r i a b l e s , w h i c h were s t a t i s t i c a l l y s i g n i f i c a n t when we e s t i m a t e d t h e model w i t h a l l t h e v a r i a b l e s , were n o t s i g n i f i c a n t i n r e - e s t i m a t i o n . «e c o n s i d e r e d s u c h v a r i a b l e s a s n o t s t a t i s t i c a l l y s i g n i f i c a n t . T a b l e I V A P a r a m e t e r E s t i m a t e s f o r L a b o u r E q u a t i o n s 1967 Group I Group I I H h o l e s a m p l e V a r i a b l e ( y | z = 1 ) ( y I z = 0 ) u n c o n d i t i o n a l c o n s t a n t 1 5 5 8 . 7 6 9 3 * (30.861) 1 2 6 7 . 7 0 7 0 * (25.414) 1556.3296* (42.271) B i r t h G a p 111.2579 (1.169) - 3.7038 (0. 088) 29.7965 (0.806) H i f e wg -704.1122 (1.576) -184.1179 (0.413) - 4 71.3710 (1.463) ( H i f e w g ) 2 176.6363 (1.916) -18.0680 (0. 199) 90.6479 (1.369) Head i n c - 0 . 2 6 5 2 * (6.312) - 0 . 2 6 9 8 * (6.912) - 0 . 2 9 2 6 * (10.685) Head i n c 2 0 . 5 6 6 8 x 1 0 - 5 * 0 . 4 1 9 8 x 1 0 - 5 * (4.666) (3.045) 0 . 5 7 8 4 x 1 0 - 5 * (6.893) Onemploy 4 2 . 6 3 6 1 * (3.348) -1.5681 (0.225) 5.1312 (1.044) ( U n e m p l o y ) 2 - 0 . 7 2 0 0 * (3.366) -0.0193 (0.547) - 0 . 0 5 3 1 * (1.967) i n c / n e e d 8.9754* (4.273) 10.1893* (5.604) 1 0 . 1 9 3 1 * (7.500) ( i n c / n e e d ) 2 - 0 . 7 2 1 0 x 1 0 - (2.384) 2 * - 0 . 6 5 2 0 x 1 0 ~ 2 * (2.683) - 0 . 7 7 1 9 x 1 0 ~ 2 (4.114) F e c u n d i t -4.2753 (0.086) 12.6560 (0.291) - 0 . 6 3 3 9 (0.020) ( F e c u n d i t ) 2 - 0 . 7 3 8 3 (0.4665) -1.3261 (0.945) -1.0798 (1.032) O b s e r v a t i o n s 189 151 340 R 2 0.3444 0.4646 0.3493 * s i g n i f i c a n t l e v e l o f 5% u n d e r H: p a r a m e t e r s . = 0.0 A s y m t o t i c t v a l u e s a r e i n p a r e n t h e s e s . Chow t e s t F (12,316) = 2.2980 Table IV B Parameter Estimates for labour Equations 1968 Variable constant BirthGap Wife wg (Wife wg) 2 Head inc (Head inc) z Unemploy (Unemploy) 2 inc/need (inc/need) 2 Fecundit (Fecundit) 2 Observations R2 Group I ( y | z = 1 ) 2402.0710* (49.929) -77.0932 (0.850) -1036.6126* (2.229) 251.1645* (2.505) -0.1902* (6.316) 0.1797x10-5* (2.249) 1.8356 (0.147) -0.0231 (0.158) 4.0753 (1. 904) 0.9217x10-3 (0.291) -3.7857 (0.082) -0.9524 (0.628) 181 0.3135 Group II ( y | z = 0 ) 1 3 4 4 . £ 3 2 0 * (28. 548) -30.6 823 (0.711) -542.6887 (1.142) 66.3974 (0. 671) -0.2265* (5.276) 0.3386x10-5* (2.096) 21.6627 (1.270) -0.2880 (1.040) 11.2963* (7.345) - 0 . 8 1 7 8 x 1 0 ~ 2 * (4.919) -41.8059 (1.098) 1.1036 (0. 822) 159 0.4371 Whole sample unconditional 1577.3882* (45.707) 30.2229 (0.864) -537.8107 (1.651) 112.4804 (1.620) -0.2213* (1 1.508) 0.3169x10-5* (7.597) 10.3385 (1.175) -0.1266 (1.103) 8.6154* (7.627) - 0 . 5 5 8 0 x 1 0 - 2 * (4.094) -19.3969 (0.693) -0.4556 (0.474) 340 0.3606 * s i g n i f i c a n t l e v e l of 5% under H: parameters = 0.0 Asymtotic t values are in parentheses. Chow test F (12,316) = 7.3582 T a b l e IV C P a r a m e t e r E s t i m a t e s f o r L a b o u r E q u a t i o n s 1969 V a r i a b l e G roup I ( y | z = 1 ) Group I I ( y | z = 0 ) Whole s a m p l e u n c o n d i t i o n a l c o n s t a n t B i r t h G a p W i f e wg ( W i f e w g ) 2 Head i n c (Head i n c ) 2 Unemploy (Unemploy) 2 i n c / n e e d ( i n c / n e e d ) 2 F e c u n d i t ( F e c u n d i t ) 2 1910.3313* (37.255) -51.3668 (0.618) -828.3737 (1.377) 224.4707 (1. 575) -0. 1907* (6.379) 0.2277x10-5* (3.511) -11.4448 (0.698) 0.2350 (0.608) 7.3298* (3.270) - 0 . 4 8 8 5 x 1 0 - 2 (1.582) -40.0683 (0.873) 0.0498 (0.003) 24.0713 (0.4852) -67.4059 (1. 357) 635.7993 (1. 183) -182.5847 (1. 594) -0. 1901* (4.018) 0.3083x10-5 (1.802) 2.2814 (0.257) -0.0215 (0. 367) 7.5424* (5. 102) - 0 . 5 0 3 7 x 1 0 - 2 * (3.351) 70.3725 (1.738) -2.7817 (1.667) 949.3052* (25.880) 2.9322 (0.079) -58.3414 (0.154) 9.3886 (0.111) -0.1936* (9.857) 0.2364x10-s* (6.149) 0.4696 (0.078) 0.5634x10" 2 (0.126) 7.7892* (7.103) - 0 . 5 1 8 7 x 1 0 - 2 * (4.165) 10.5490 (0.380) -1.6838 (1.648) O b s e r v a t i o n s B 2 178 0.2817 162 0.3325 34 0 0.2991 * s i g n i f i c a n t l e v e l o f 5% u n d e r H: p a r a m e t e r s = 0.0 A s y m t o t i c t v a l u e s a r e i n p a r e n t h e s e s . Chow t e s t F (12,316) = 3.6634 J a b l e IV D P a r a m e t e r E s t i m a t e s f o r L a b o u r l a u a t i o n s 1 9 7 0 V a r i a b l e Group I ( y 1 z = 1 ) Group I I ( y | z = 0 ) H h o l e sample u n c o n d i t i o n a l c o n s t a n t 1680.9971* (32.395) 52.8269 (1.074) 1033.5461* (28.059) B i r t h G a p -113.2762 (1.889) -33.8650 (0.592) -36.6936 (0.934) l i f e wg -736.9992 (1.174) 704.6772 (1. 504) 1.1532 (0.003) ( H i f e w g ) 2 167.3082 (1.055) -178.8685 (1.760) -15.3298 (0.187) Head i n c -0.2874* (5.622) -0.1588* (6.260) -0.2036* (9.236) (Head i n c ) 2 0.4527x10-s* (2.514) 0 . 2 1 0 8 x 1 0 - 5 * (3.881) 0 . 2 8 2 3 x 1 0 - 5 * (5.415) Unemploy 5.7146 (1.249) 0.1341 (0.015) -0.8891 (0.247) (Unemploy) 2 -0.0382 (1.48 6) -0.0872 (0. 953) -0.6823X10- 2 (0.317) i n c / n e e d 9.3285* (5.651) 5.3371* (6.407) 6.2063* (8.751) ( i n c / n e e d ) 2 - 0 . 6 1 4 4 x 1 0 - 2 * (3.468) - 0 . 2 6 5 6 x 1 0 ~ 2 * (4. 236) - 0 . 3 1 2 9 x 1 0 - 2 * (5.277) F e c u n d i t 4.4495 (0.097) 75.9236* (2.161) 28.2306 (1.085) ( F e c u n d i t ) 2 -1.385 8 (0.841) -3.5510* (2. 225) -2.3265* (2.264) O b s e r v a t i o n s 173 167 340 R 2 0.3320 0.3444 0.3088 * s i g n i f i c a n t l e v e l o f 5% u n d e r H: p a r a m e t e r s = A s y m t o t i c t v a l u e s a r e i n p a r e n t h e s e s . Chow t e s t F(12,316) = 2.7094 0.0 70 T a b l e IV E P a r a m e t e r E s t i m a t e s f o r L a b o u r E q u a t i o n s 1.97J. V a r i a b l e G r o u p I { Y I z = 1 ) Group I I ( y l z = o ) Whole s a m p l e u n c o n d i t i o n a l c o n s t a n t B i r t h G a p W i f e wg ( W i f e w g ) 2 Head i n c (Head i n c ) 2 Unemploy ( U n e m p l o y ) 2 i n c / n e e d ( i n c / n e e d ) 2 F e c u n d i t ( F e c u n d i t ) 2 1 4 9 6 . 4 5 1 4 * (27.655) - 62.9480 (1.022) -896.2939 (1.471) 224.4905 (1.448) - 0 . 2 3 7 0 * (3.256) 0 . 3 9 7 0 x 1 C - s (1.050) 4. 1735 (0.840) -0.0218 (0.749) 8.4099* (4.689) -0.6105X10" 2* (2.940) -8.1950 (0.178) -0.4393 (0.251) 5 8 7 . 5 8 9 8 * (11.748) -89.8988 (1.520) 205.7050 (0.358) - 7 5.6943 (0.558) - 0 . 1 4 0 2 * (5. 117) 0 . 1 8 1 5 x 1 0 - 5 * (3.034) -3.0460 (0.471) 0.0274 (0. 569) 4.8945* (5.095) - 0 . 2 2 7 0 x 1 0 - 2 * (3 . 329) 36.3512 (1.086) - 1 . 1797 (0.697) 1151.3694* (30.555) - 5 3 . 3 0 4 6 (1.311) -298.0087 (0.721) 66.7110 (0.662) - 0 . 1772* (7.926) 0 . 2 4 5 7 x 1 0 - s * (4.481) 3.3144 (0.882) -0.0183 (0.761) 5.4899* (7.327) - 0 . 2 6 4 3 x 1 0 " 2 * (4.454) 19.9307 (0.811) -1.6618 (1.580) O b s e r v a t i o n s 164 R 2 0.2811 176 0.2361 340 0.2394 * s i g n i f i c a n t l e v e l o f 5% u n d e r H: p a r a m e t e r s = 0 . 0 A s y m t o t i c t v a l u e s a r e i n p a r e n t h e s e s . Chow t e s t F ( 1 2 , 3 1 6 ) = 5.5301 Mean and S t a n d a r d D e v i a t i o n o f t h e M o d e l I I A 1967 Group I Group I I Whole sample V a r i a b l e ( y I z = 1 ) ( y I z = 0 ) y u n c o n d i t i o n 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 wg 2.0312 (0 . 6498) 2.08863 (0.6742) 2.0567 (0.6604) (Wife w g ) 2 4.5457 (3.0821) 4.8139 (3.2770) 4.6648 (3.1682) Head i n c 7812.65 (4311 .02) 75 2 2 . 16 (5073.70) 7683.64 (4660.26) (Head inc) 2 0.7952x108 (0.1608x10*) 0.8215x108 (0. 1333x109) 0.8069x108 (0.1490x109) Unemploy 3.2090 (10.6461) 4.9735 (25.3447) 3.9927 (18.6509) (Unemploy) 2 123.038 (630.650) 662.838 (5019.12) 362.773 (3382.22) inc/need 242.873 (116.078) 265.384 (146.320) 252.871 (130.657) (inc/need) 2 0.7239x105 (0.8825x105) 0.9170x105 (0.1078x106) 0.8096x105 (0.9776x105) Fecundit 16.1481 (6.0053) 15.0530 (7.5189) 15.6618 (6.7313) (Fecundit) 2 296.635 (190.053) 282.748 (238.273) 290.468 (212.603) Observations 189 151 340 Standard d e v i a t i o n s are i n parentheses. l l ^ l e V B Mean and S t a n d a r d D e v i a t i o n o f t h e M o d e l I I X 1968 Group I Group I I Whole s a m p l e V a r i a b l e ( y | z = 1 ) ( y J z = 0 ) y u n c o n d i t i o n HourWork 749.293 (756.947) 1132.99 (763.656) 928.726 (782.811) B i r t h G a p 0.3812 (0.6178) 0.7862 ( 1 . 5482) 0.5706 (1.1665) l i f e wg 1.9666 (0.5927) 2.0144 (0.6335) 1.9889 (0.6117) ( W i f e wg) 2 4.2169 (2.6983) 4.4564 (2.9939) 4.3289 (2.8387) Head i n c 8552.08 (5424.92) 8499.53 (4702.69) 8527.51 (5092.60) (Head i n c ) 2 0.1024x10* ( 0 . 2 9 6 2 x 1 0 9 ) 0.9422x108 (0. 1099x10*) 0.9858x10« (0.2285x109) Unemploy 2.8488 (12.6523) 2.6376 (11.3457) 2.7500 (12.0418) (Unemploy) 2 167.313 (1072.79) 134.872 (697.642) 152.142 (915.526) i n c / n e e d 264.033 (121.435) 303.673 (149.635) 282.571 (136.594) ( i n c / n e e d ) 2 0.8438x105 (0 . 1080x10*) 0.1145x106 (0. 1231x106) 0.9845x105 ( 0 . 1161x10*) F e c u n d i t 16.5580 (5.8900) 12.5031 (6.9917) 14.6618 (6.7313) ( F e c u n d i t ) 2 308.669 (183.222) 204.906 (2 0 3 . 140) 260.144 (199.365) O b s e r v a t i o n s 181 159 340 S t a n d a r d d e v i a t i o n s a r e i n p a r e n t h e s e s . Mean and S t a n d a r d D e v i a t i o n c f t h e M o d e l I I X 1969 G r o u p I Group I I Whole s a m p l e V a r i a b l e ( y I z = 1 ) ( y J z = 0 ) y u n c o n d i t i o n HourWork 756.927 (781.734) 1195.58 (745.969) 965.932 (794.667) B i r t h G a p 0.3483 (0.6656) 0,5988 (1.4764) 0.4677 (1.1323) W i f e wg 1.8301 (0.5237) 1.9638 (0.6173) 1.8938 (0.5733) ( W i f e wg) 2 3.6218 (2.2029) 4.2351 (2.8226) 3.9140 (2.5321) Head i n c 8609.85 (5525.41) 9097.13 (4544.49) 8842.02 (5080.19) (Head i n c ) 2 0.1045x109 (0.3004x109) 0. 1033x109 (0.1135x109) 0.1039x109 (0.2307x109) Unemploy 2.6938 (8.6431) 2.7284 (15.9771) 2.7103 (12.6585) (Unemploy) 2 81.5407 (377.597) 261.136 (2441.43) 167.112 (1706.86) i n c / n e e d 284.146 (123.874) 328.210 (144.391) 305. 141 (135.644) ( i n c / n e e d ) 2 0.9600x10s (0. 1017x10*) 0.1284x106 (0.1273x106) 0. 1115x106 (0.1156x106) F e c u n d i t 16.0955 (5.9296) 10.9877 (6.5579) 13.6618 (6.7313) ( F e c u n d i t ) 2 294.028 (176.428) 163.469 (172.537) 231.821 (1 8 6 . 158) O b s e r v a t i o n s 178 162 340 S t a n d a r d d e v i a t i o n s a r e i n p a r e n t h e s e s . T a b l e V D Mean and S t a n d a r d D e v i a t i o n o f t h e M o d e l I I X 1970 Group I Group I I Whole s a m p l e V a r i a b l e ( y | z = 1 ) ( y I z = 0 ) y u n c o n d i t i o n HourWork 781.601 (807.945) 1143.93 (758. 429) 959.571 (803.603) B i r t h G a p 0.3757 (0.9357) 0.3713 (1.1745) 0.3735 (1.0581) W i f e wg 1.7079 (0.46924) 1.9147 (0 . 6349) 1.8095 (0.56554) ( W i f e wg) z 3.1358 (1.8479) 4.0669 (2.8769) 3.5931 (2.4500) Head i n c 8253.68 (3700.80) 9687.73 (5921.69) 8958.05 (4963.42) (Head i n c ) 2 0.8174x108 (0.9086x108) 0.1287x109 (0.2492x109) 0.1048x109 (0.1 8 7 5 x 1 0 9 ) Unemploy 9.2269 (32.2653) 3.3802 (15.3756) 6.3552 (25.5447) (Unemploy) 2 1120.17 (5777.52) 246.421 (1535.59) 691.002 (4275.75) i n c / n e e d 312.04 0 (140.308) 364.401 (188.303) 337.759 (167.447) ( i n c / n e e d ) 2 0 . 1 1 6 9 x 1 0 * (0.1181x10*) 0.1680x106 (0.2356x106) 0. 1420x106 (0.1868x106) F e c u n d i t 15.8150 (5.8220) 9.3952 (6. 0260) 12.6618 (6.7313) ( F e c u n d i t ) 2 i 283.815 (164.463) 124.365 (141.679) 205.497 (172.990) O b s e r v a t i o n s 173 167 340 S t a n d a r d d e v i a t i o n s a r e i n p a r e n t h e s e s . T a b l e V E Mean and S t a n d a r d D e v i a t i o n o f t h e M o d e l I I X 2211 Group I Group I I Whole s a m p l e V a r i a b l e ( y I z = 1 ) { y | z = 0 ) y u n c o n d i t i o n HourWork 797.360 (789.206) 1175.35 (734.960) 993.026 (783.650) B i r t h G a p 0.3659 (0.9530) 0.2898 ( 1 . 1064) 0.3265 (1.0345) l i f e wg 1.6824 (0.5206) 1.8392 (0.5805) 1.7636 (0.5572) ( W i f e wg) 2 3.09972 (2.0359) 3.7178 (2. 4179) 3.4197 (2.2598) Head i n c 8283.83 (3356.62) 9668.88 (5530.66) 9000.80 (4657.06) (Head i n c ) 2 0.7982x108 (0.6195x108) 0.1239x109 (0. 2085x109) 0.1026x109 (0.1574x109) Unemploy 8.2317 (31.4318) 7.0483 (26.3531) 7.6191 (28.8773) (Unemploy) 2 1049.70 (5537.04) 740.217 (3553.56) 889.495 (4498.73) i n c / n e e d 333.982 (140.231) 387.778 (199.672) 361.829 (175.389) ( i n c / n e e d ) 2 0.1311x10* (0.1130x10*) 0. 1 9 0 0 x 1 0 * (0.2454x10*) 0 . 1616x10* (0.1 9 5 2 x 1 0 * ) F e c u n d i t 15.0366 (5.7939) 8.5171 (5.9904) 11.6618 (6.7313) ( F e c u n d i t ) 2 259.463 (154.686) 108.222 (126.997) 181.174 (159.872) O b s e r v a t i o n s 164 176 340 S t a n d a r d d e v i a t i o n s a r e i n p a r e n t h e s e s . 76 T a b l e VI A The C o m p a r i s o n o f t h e Two Stacje A i t k e n and 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 X 1967 V a r i a b l e ( y 1 Two S t a g e 2 = 1 ) S i n g l e Eqn ( y l Two S t a q e z = 0 ) S i n g l e Eqn c o n s t a n t - 24.8795 (536.694) 1558.7693 (602.6486) 1704.61 (555.908) 1267.707 (599.956) B r i t h G a p 206.277 (79.5456) 111.2579 (95.1705) 6.7483 (40.4629) -3.7038 (42.2918) W i f e wg 360.029 (402.916) -704.1122 (446.8012) - 7 89.138 (413.982) -184.1179 (445.6689) ( H i f e w g ) 2 - 5 0 . 9 9 4 8 (83.6112) 176.6363 (92.1762) 100.529 (84.9228) -18.0680 (90.5690) Head i n c -0.2742 (0.03637) -0.2652 (0.0420) - 0 . 2 6 4 3 (0.0351) -0.2698 (0.0390) (Head i n c ) 2 0 . 6 2 1 x 1 0 - s ( 0 . 1 x 1 0 - s ) 0.567x10-s ( 0 . 1 x 1 0 - 5 ) 0 . 4 3 3 x 1 0 - 5 ( 0 . 1 x 1 0 - 5 ) 0 . 4 2 0 x 1 0 - 5 ( 0 . 1 x 1 0 - 5 ) Unemploy 37.2640 (9. 8779) 42.6361 (12.7363) - 1 . 1740 (6.1164) - 1.5681 (6.9694) (Unemploy) 2 - 0 . 5 9 6 3 (0. 1677) -0.7200 (0.2139) - 0.0102 (0.0309) -0.0193 (0.0352) i n c / n e e d 10.5357 (1.7873) 8.9754 (2.1005) 10.3676 (1.6288) 10.1893 (1.8183) ( i n c / n e e d ) 2 -0.0095 (0.0025) -0.0072 (0.0030) -0.0067 (0.0022) -0.0065 (0.0024) F e c u n d i t 20.4197 ( 4 4 . 1752) -4.2753 (49.6485) 29.1138 (40.8465) 12.6560 (43.4798) ( F e c u n d i t ) 2 -1.5124 (1.4128) -0.7383 (1.5 827) -1.6923 (1.3143) - 1 . 3 2 6 1 (1.4035) S t a n d a r d d e v i a t i o n s a r e i n p a r e n t h e s e s . Chow T e s t : Two S t a g e , F ( 1 2 , 3 1 6 ) = 3.4416 S i n g l e Egn, F ( 1 2 , 3 1 6 ) = 2.2980 J 77 T a b l e V I B The 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 and 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 X 1968 V a r i a b l e ( y Two S t a g e 2 = 1 ) S i n g l e Egn ( y Two S t a g e 2 = 0 ) S i n g l e Egn c o n s t a n t B r i t h G a p H i f e wg ( H i f e wg) 2 Head i n c (Head i n c ) 2 Unemploy (Unemploy) 2 i n c / n e e d ( i n c / n e e d ) 2 F e c u n d i t ( F e c u n d i t ) 2 719.489 (497.420) 58.6397 (67.7885) 54.4460 (374.372) 8.3943 (81.0106) -0.1846 (0.0229) 0 . 2 4 2 x 1 0 - 5 (0.6x10-*) 6.7462 (8.6091) -0.0856 (0.0990) 6.6379 (1.5732) -0.0031 (0.0023) 4.6712 (38.6399) -1.2779 (1.2737) 2402.071 (617.6222) -77.0932 (90.6618) -1036.6126 (465.0873) 251.1645 (100.2469) -0.1902 (0.0301) 0. 1 8 0 x 1 0 - 5 ( 0 . 8 x 1 0 - 6 ) 1.8356 (12.4788) - 0 . 0 2 3 1 ( 0 . 1458) 4.0753 (2.1409) 0.0009 (0.0032) -3.7857 (46.3501) -0.9524 (1.5159) 1207.37 (506.332) - 1 1.9464 (39.5871) - 5 2 4 . 5 4 5 (413.052) 73.1776 (86.7298) -0.2294 (0.0358) 0 . 3 6 3 x 1 0 - 5 ( 0 . 1 x 1 0 - 5 ) -0.2577 (13.3125) 0.0541 (0.2178) 11.0835 (1.2870) -0.0077 (0.0014) - 2 3 . 4 3 1 0 (34.3664) 0.2166 (1.2025) 1344.832 (580.5073) -30.6823 (43. 1462) -542.6887 (475.2111) 66.3974 (98.8818) -0.2265 (0.0429) 0 . 3 3 9 x 1 0 - 5 ( 0 . 2 x 1 0 - 5 ) 21.6627 (17 . 0604) -0.2880 (0.2769) 11.2963 (1.5380) -0.0082 (0.0017) -41.8059 (38.0776) 1.1036 (1.3420) S t a n d a r d d e v i a t i o n s a r e i n p a r e n t h e s e s . Chow T e s t : Two S t a g e , F ( 1 2 , 3 1 6 ) = 9.2483 S i n g l e E g n , ? (12,316) = 7.3582 78 T a b l e V I C The 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 and 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 ~ ~ J 9 6 9 ~ V a r i a b l e ( y 1 Two S t a g e 2 = 1 ) S i n g l e Eqn ( y I Two S t a g e z = 0 ) S i n g l e Eqn c o n s t a n t 1074.15 (584. 199) 1910.3313 (685.601) - 6 1 . 8 5 6 5 (515.343) 24.0713 (606.944) B r i t h G a p 44.8480 (69.4932) -51.3668 ( 8 3 . 1572) - 5 3 . 8 0 7 5 (43.6123) -67.4059 (49.6914) H i f e wg -240.4760 (501.982) -828.3737 (601.6902) 479.664 (446.460) 635.7993 (537.3614) ( W i f e w g ) 2 81.9762 (118.436) 224.4707 (142.5031) - 1 3 5 . 4 5 3 (95.4104) -182.5847 (114.5202) Head i n c -0.2054 (0.0246) - 0 . 1907 (0.0299) - 0 . 1750 (0.0387) - 0.1901 (0.0473) (Head i n c ) 2 0 . 2 67x10-s ( 0 . 5 x 1 0 - 6 ) 0.228x10-s ( 0 . 6 x 1 0 - 6 ) 0.257x10~s ( 0 . 1 x 1 0 - s ) 0 . 3 0 8 x 1 0 - 5 ( 0 . 2 x 1 0 - 5 ) Unemploy -5.1539 (12.7613) -11.4448 (16.3863) 6.1288 (7.0696) 2.2814 (8.8619) (Unemploy) 2 0.1846 (0.3000) 0.2350 (0.3862) -0.0385 (0.0462) -0.0215 (0.059) i n c / n e e d 9.6 972 (1.8008) 7.3298 (2.2416) 7.8545 (1.2175) 7.5424 (1.4784) ( i n c / n e e d ) 2 - 0 . 0 0 7 3 (0.002) -0.0049 (0.0031) -0.0050 . (0.0013) -0.0050 (0.0015) F e c u n d i t -52.7559 (40.7958) -40.0683 (45. 8751) 75.8987 (35.6987) 70.3725 (40.4924) ( F e c u n d i t ) 2 0.3216 (1.4042) 0.0498 (1.5686) -3.2084 (1.4490) -2.7817 (1.6690) S t a n d a r d d e v i a t i o n s a r e i n p a r e n t h e s e s . Chow T e s t : Two S t a g e , F ( 1 2 , 3 1 6 ) = 7.7696 S i n g l e E g n , F (12,316) = 3.6634 79 T a b l e V I D The 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 and 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 X J.970 V a r i a b l e ( Y 1 Two S t a g e z = 1 ) S i n g l e Egn ( y l Two S t a g e z = 0 ) S i n g l e Egn c o n s t a n t 1412.61 (559.445) 1680.997 (656.8402) 4 7.0384 (441.321) 52.8269 (517.0958) B r i t h G a p - 9 2 . 9 1 8 9 (52.7555) - 1 13.2762 (59.9558) - 3 4 . 0 4 7 9 (51.4447) - 3 3 . 8650 (57.1928) W i f e wg -652.953 (523.368) -736.9992 (627.373) 5 9 0 . 1 7 3 (38 7 . 140) 704.6772 (468.432) ( W i f e w g ) 2 157.065 (132.528) 167.3082 (158.5716) - 1 4 6 . 9 1 1 (84.1822) -178.8685 (101.6526) Head i n c -0.2830 (0.0422) -0.2874 (0.0511) - 0 . 1548 (0.0214) -0.1588 (0.0254) (Head i n c ) 2 0.470x10-s ( 0 . 2 x 1 0 - 5 ) 0 . 4 5 3 x 1 0 - 5 ( 0 . 2 x 1 0 - 5 ) 0. 1 9 9 x 1 0 - 5 ( 0 . 5 x 1 0 - 6 ) 0 . 2 1 1 x 1 0 - 5 ( 0 . 5 x 1 0 - 6 ) Onemploy 4.5144 (3.5196) 5.7146 (4.5743) - 3 . 1778 ( 6 . 8682) 0.1341 (9.1211) (Onemploy) 2 -0.0290 (0.0199) -0.0382 (0.0257) -0.0035 (0.0687) -0.0872 (0.0915) i n c / n e e d 10.6796 (1.3660) 9.3285 (1.6508) 5.5279 (0.6995) 5.3371 (0.8330) ( i n c / n e e d ) 2 -0.0074 (0.0015) -0.0061 (0.0018) - 0 . 0 0 2 5 (0.0005) -0.0027 (0.0006) F e c u n d i t -15.1719 (41.0756) 4.4495 (45.8121) 74.9506 (32.0058) 75.9236 (35.1332) ( F e c u n d i t ) 2 -0.8335 (1.487 8) -1.3858 (1.6470) -3.7018 ( 1 . 4362) -3.5510 (1.5960) S t a n d a r d d e v i a t i o n s a r e i n p a r e n t h e s e s . Chow T e s t : Two S t a g e , F (12,316) = 26.339 S i n g l e E g n , F ( 1 2 f 3 1 6 ) = 2.7094 T a b l e VI E The 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 and 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 I l j t 197J V a r i a b l e ( y 1 Two S t a g e 2 = 1 ) S i n g l e Egn ( y I Two S t a g e z = 0 ) S i n g l e Egn c o n s t a n t 1330.05 (586. 100) 1496.4514 (664.1467) 839.732 (493.843) 587.5898 (566.7747) B r i t h G a p -58.3862 (56.9849) -62.948 (61.6013) -112.064 (54.4209) -89.8988 (59.1419) W i f e wg -473.837 (531.652) 896.2939 (609.2837) -104.060 (491.017) 205.7050 (575.294) ( W i f e w g ) 2 130.288 (135.215) 224.4905 (155.0258) 8.5510 (116.298) -75.6943 (135.7505) Head i n c -0.2869 (0.0619) -0.2370 (0.0728) -0.1368 (0.0241) -0.1402 (0.0274) (Head i n c ) 2 0 . 6 4 8 x 1 0 - 5 ( 0 , 3 x 1 0 - 5 ) 0.397x10-5 ( 0 . 4 x 1 0 - 5 ) 0. 176x10-5 (0.5x10-*) 0. 182x10-5 ( 0 . 6 x 1 0 - 6 ) Onemploy 4.8257 (4. 1312) 4.1735 (4.9665) 3.2804 (5. 1358) -3.0460 (6.4734) (Onemploy) 2 -0.0239 (0.0240) -0.0218 (0.0291) -0.0163 (0.0380) 0.0274 (0.0481) i n c / n e e d 8.7383 (1.5652) 8.4099 (1.7937) 4.8126 (0.8318) 4.8945 (0.9606) ( i n c / n e e d ) 2 -0.0062 (0.0018) -0.0061 (0.0021) -0.0021 (0.0006) -0.0023 (0.0007) F e c u n d i t -19.4112 (42.7355) -8. 1950 (46.0473) 29.3953 (30.9769) 36.3512 (33.4791) ( F e c u n d i t ) 2 -0.2104 (1.6331) -0.4393 (1.7509), -1.0301 (1.5451) -1.1797 (1.6928) S t a n d a r d d e v i a t i o n s a r e i n p a r e n t h e s e s . Chow T e s t : Two S t a g e , F (12,316) = 48.711 S i n g l e E g n , F(12,316) = 5.5301 T a b l e V I I A P r o b a b i l i t y 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 J 9 6 7 V a r i a b l e C o e f f i c i e n t Asym s t d v Asym t - : c o n s t a n t - 2 . 4 2 7 9 7 * 0.74520 3.25817 B i r t h G a p - 0 . 4 1 6 7 1 * 0.09670 4.30938 W i f e wg 0.354 86 0.54061 0.65641 ( W i f e wg) 2 -0.09620 0.11071 0.86890 Head i n c 0.06513 0.04625 1.40828 (Head i n c ) 2 0.488x10-* 0.00135 0.03629 Unemploy 0.02830 0.01896 1.49258 (Unemploy) 2 -0.00046. 0.00031 1.48280 i n c / n e e d -0.02214 0.22531 0.09826 ( i n c / n e e d ) 2 -0.02807 0.03065 0.91577 F e c u n d i t 0.28355* 0.05580 5.08141 ( F e c u n d i t ) 2 - 0 . 0 0 7 3 5 * 0.00177 4. 14357 Lo g o f l i k e l i h o o d f u n c t i o n = -191.969 a f t e r 10 i t e r a t i o n s . * s i g n i f i c a n t i n 95% u n d e r H: p a r a m e t e r e s t i m a t e s = 0 . 0 n o t e : 1 Head i n c = 1 i n c / n e e d = $1,000 100 1 (Head i n c ) 2 =$1,000,000 1 ( i n c / n e e d ) 2 = 10,000 T a b l e V I I B P r o b a b i l i t y F u n c t i o n E s t i m a t e s o f t h e M o d e l I I X 1968 V a r i a b l e C o e f f i c i e n t fisym s t d v Asym t - : c o n s t a n t -3,05 5 36* 0.80555 3.79287 B i r t h G a p -0.44 8 6 1 * 0.10617 4.22535 W i f e wg 1.00167 0.60746 1.64895 ( W i f e w g ) 2 -0.22005 0.12970 1.69655 Head i n c 0.11504* 0.03902 2.94809 (Head i n c ) 2 -0.00055 0.00090 0.61233 Unemploy -0.01095 0.01661 0.65922 (Unemploy) 2 0.00015 0.00022 0.66373 i n c / n e e d - 0.33192 0.24119 1.37618 ( i n c / n e e d ) 2 -0.00278 0.03091 0.08988 F e c u n d i t 0.24864* 0.05810 4.27937 ( F e c u n d i t ) 2 - 0 . 0 0 4 8 9 * 0.00196 2.49738 Log o f l i k e l i h o o d f u n c t i o n = - 176.713 a f t e r 13 i t e r a t i o n s . * s i g n i f i c a n t i n 95$ u n d e r H: p a r a m e t e r e s t i m a t e s = 0 . 0 n o t e : 1 Head i n c = 1 i n c / n e e d = $1,000 100 1 (Head i n c ) 2 =$1,000,000 1 ( i n c / n e e d ) 2 = 10,000 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 1969 Variable Coefficient Asym stdv Asym t-: constant -2.39040* 0.77684 3. 07709 BirthGap -0.46689* 0.10790 4. 32698 Hife wg 1.67102* 0.69282 2. 41191 (Hife wg) 2 -0.39129* 0.15509 2. 52292 Head inc 0.07168 0.03768 1. 90244 (Head inc) 2 -0.00032 0.00088 0. 36407 Unemploy 0.05337* 0.02657 2. 00867 (Unemploy) 2 -0.00109* 0.00054 2. 03169 inc/need -0.48319* 0.20561 2. 35006 (inc/need) 2 0.02887 0.0 2326 1. 24127 Fecundit 0.11981* 0.05352 2. 23850 (Fecundit) 2 -0.00025 0.00204 0. 12416 Log of li k e l i h o o d function = -174.969 after 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 estimates = 0.0 n o t e : 1 Head inc = $1,000 1 (Head i n c ) 2 =$1,000,000 1 inc/need = 100 1 (inc/need) 2 = 10,000 T a b l e V I I D P r o b a b i l i t y 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 1970 V a r i a b l e C o e f f i c i e n t Asym s t d v Asym t - i c o n s t a n t - 2 . 0 3 5 2 9 * 0.76525 2.65963 B i r t h G a p - 0 . 2 9 6 8 2 * 0.09165 3.23849 W i f e wg 1.25770 0.73180 1.71864 ( W i f e wg) 2 - 0 . 3 6 7 6 5 * 0.17337 2.12059 Head i n c 0.08875 0.05830 1.52238 (Head i n c ) 2 -0.00170 0.00210 0.81031 Unemploy 0.00164 0.00909 0.18042 (Unemploy) 2 0 . 4 1 1 x 1 0 - * 0.834x10-* 0.49300 i n c / n e e d - 0 .27551 / 0.15846 1.73869 ( i n c / n e e d ) 2 0.01443 0.01493 0.96648 F e c u n d i t 0.07757 0.05195 1.49326 ( F e c u n d i t ) 2 0.00157 0.00212 0.73948 Log o f l i k e l i h o o d f u n c t i o n = -172.197 a f t e r 15 i t e r a t i o n s . * s i g n i f i c a n t i n 95% u n d e r H: p a r a m e t e r e s t i m a t e s = 0 . 0 n o t e : 1 Head i n c = 1 i n c / n e e d = $1,000 100 1 (Head i n c ) 2 =$1,000,000 1 ( i n c / n e e d ) 2 = 10,000 T a b l e V I I E P r o b a b i l i t y F u n c t i o n E s t i m a t e s o f t h e J o d e l I I X 1971. V a r i a b l e C o e f f i c i e n t Asym s t d v Asym t - : c o n s t a n t -1.29702 0.72149 1.79768 B i r t h G a p - 0 . 1 9 2 2 5 * 0.07349 2.61615 W i f e wg 0.21367 0.71663 0.29816 ( W i f e w g ) 2 -0.10158 0.17430 0.58281 Head i n c 0. 1 5 9 3 1 * 0.07094 2.24571 (Head i n c ) 2 -0.00592 0.00315 1.87641 Onemploy -0.00493 0.00689 0.71503 (Onemploy) 2 0.340x10-* 0.459x10-* 0.74210 i n c / n e e d -0.18304 0.15392 1.18917 ( i n c / n e e d ) 2 0.00748 0.01437 0.52037 F e c u n d i t 0.04998 0.04688 1.06614 ( F e c u n d i t ) 2 0.00228 0.00206 1.10551 Log o f l i k e l i h o o d f u n c t i o n = -180.441 a f t e r 11 i t e r a t i o n s . * s i g n i f i c a n t i n 95% u n d e r H: p a r a m e t e r e s t i m a t e s = 0.0 n o t e : 1 Head i n c = 1 i n c / n e e d = $1,000 100 1 (Head i n c ) 2 =$1,000,000 1 ( i n c / n e e d ) 2 = 10,000 86 C h a p t e r V I I I C o n c l u s i o n The p r o p o s e d b a s i c m o d e l , w h i c h i n v o l v e s d i s c r e t e and c o n t i n u o u s d e p e n d e n t v a r i a b l e s , i s e s t i m a t e d by s e p a r a t i n g t h e mo d e l i n t o a s i m p l e r e g r e s s i o n model and a p r o b a b i l i t y e g u a t i o n m o d e l . The r e g r e s s i o n model c a n be e s t i m a t e d by o r d i n a r y l e a s t - s q u a r e s . I t i s s u g g e s t e d t h a t t h e p r o b a b i l i t y e g u a t i o n model be f o r m u l a t e d a s a l o g i s t i c f u n c t i o n and e s t i m a t e d by u s i n g t h e maximum . l i k e l i h o o d method. W h i l e t h e b a s i c m o d e l i s e x t e n d e d i n t o a s y s t e m o f e q u a t i o n s , i t c a n be s e p a r a t e d i n t o a s y s t e m o f r e g r e s s i o n e g u a t i o n s and a s y s t e m o f l o g i s t i c e g u a t i o n s . The s y s t e m o f r e g r e s s i o n e q u a t i o n s c a n be e s t i m a t e d by Z e l l n e r ' s two s t a g e method i n o r d e r t o g a i n e f f i c i e n c y . The p r o b a b i l i t y model c a n be e s t i m a t e d by t h e method o f N e r l o v e and P r e s s . I t i s i n t e r e s t i n g t h a t , t h i s b a s i c model c a n be e x t e n d e d by c o n s i d e r i n g t h e c o r r e l a t i o n o f d i s t u r b a n c e s i n t h e r e g r e s s i o n e g u a t i o n s , t o be a model w i t h c o n s t r a i n t s , t o be a model w i t h j o i n t l y d e p e n d e n t d i s c r e t e v a r i a b l e s , o r t o be a s i m u l t a n o u s - e g u a t i o n m o d e l . Those e x t e n d e d m o d e l s a r e s o l v a b l e . One e x t e n s i o n o f t h i s m o d e l , w h i c h h a s n o t been s o l v e d w i t h v e r i f i c a t i o n , i s t h e r e c u r s i v e m o d e l . The b a s i c m o d el i s more common i n s o c i a l s c i e n c e , a l t h o u g h t h e r e i s n o t much l i t e r a t u r e . The e c o n o m i c model i n t h i s t h e s i s i s a s t u d y on t h e l a b o u r s u p p l y o f A m e r i c a n m a r r i e d women f r o m 1967 t o 1971. We f i n d t h a t t h e number o f h o u r s worked by m a r r i e d women i s a f f e c t e d by t h e age o f t h e i r y o u n g e s t c h i l d v e r y much, and s l i g h t l y by t h e i r 87 head's income. There i s some eff 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 incomes over needs, yet the s i g n i f i c a n c e of these factors varies from year to year. The b i r t h gap has a s i g n i f i c a n t e f f e c t on the probability of the wife having a c h i l d . not older than 6 years of age. Therefore, the r e s u l t s t e l l us that the married woman's r o l e i n the labour market i s guite dependent upon her family planning. 88 Appendix A Least-Sguares Estimation Here we follow a l l the notations defined in chapter 2. Therefore the conditional regression w i l l be written as following: r y i , Y z ] = [ X » , X * J rA 0-, + £u,v] «-0 B-» w i l l c a l l Y = £Y», Y 2 ] , X = £ X » # X * ] , and E = generalized multivariate regression 1, we know Re £u,v]. From so Hence, Since, then, A+ = ( X f X ) ~ » X « Y (X»X) = r (Xi) S f X 1 X 2] = r (X1) ' (X1) L ( X 2 ) »J *- (X»X)" 1 = r ( X 1 1 X 1 ) - 1 »- 0 0 0 ~i (X 2) • ( X 2 ) J 0 T ( X 2 ' X 2 ) - u r A + 0 T = r ( x a » X i ) - i 0 -, rXi'Y 1 0 T I 0 -B+-» «• 0 ( X 2 « X 2 ) - 1 J >- 0 X 2 » Y 2 J A+ = £ X 1 • X 1 J - 1 (X1) ' Y 1 B+ = £ X 2 «X 2 ]-i ( X 2 ) « Y 2 Cov+(u,v) = rVar+(u) 0 ^ =E+E + «/n L 0 V a r + ( v ) J E+ = £Y>—X * A + Y2-X 2B+] Cov+(U,V) = 1 R (Y » - X i A + ) ' (Y* - X * A * ) (Y1 -X1 A+) • (Y2-X2B+) nMY 2 -X2B+) • (Y*-X*A+) (Y2-X2B+) • (Y2-X2B+) J = r (Y 1—X*A+) ' (Y»-X»A+)/n 0 T i- 0 ( Y 2 - X 2 B + ) M Y 2 - X 2 B + ) / n J For unbiased estimators of Var +(U) and Var +(V), we have 89 Var+(u) = ( Y 1 - X 1 A + ) * v a r + ( v ) = (Y2-X2B+) » where n 1 i s t h e t o t a l number o f (Yi-X^A+J / (n J-k). (Y2-X2B+)/ (n2-k) o b s e r v a t i o n s when z = 1 ; n 2 = n- F o o t n o t e 1 P r e s s , J . , A p p l i e d M u l t i v a r i a t e A n a l y s i s , 1 9 7 2 , p p . 2 2 0 . 90 Appendix B L i k e l i h o o d R a t i o T e s t f o r M i c r o R e g r e s s i o n C o e f f i c i e n t V e c t o r E g u a l i t y 1 d Under t h e h y p o t h e s i s o f c h a p t e r 3, H: S =.,.= S , The s y s t e m o f e g u a t i o n s c a n be w r i t t e n as r X S 8 + I • i I . . I I • I I. I I* I I. I I • i I. i I. I I d| I d| I dj L y J L X J L U J or, Y* = X*W + U* (B1) We d e f i n e a t r a n s f o r m a t i o n T, s u c h t h a t E ( T U*U*»T«) = v a r ( U * ) I . L e t TY* = Y ° , TX* = X ° , and TU* = U ° . Then t h e l i k e l i h o o d f u n c t i o n , L ( U ° ) , under t h e h y p o t h e s i s i s -dk/2 L(U*) = (2 v a r ( U O ) ) exp (-U° • U V ( 2 v a r (u°)) ) (B2) The maximum l i k e l i h o o d e s t i m a t o r s f o r e q u a t i o n (B1) a r e Var+(U*) = UO + ,UO +/dk = (YO - x o w * ) ' (Y0-XQW+)/dk and W+ = { X O « X O ) - i X 0 1 YO Hence i f we r e w r i t e e g u a t i o n (B2) i n t e r m s o f t h e s e e s t i m a t o r s , t h e n -dk/2 I (U*+) = (2 var+(U * ) ) exp (-dk/2) 91 L i k e w i s e , we t r a n s f o r m t h e v a r i a b l e s i n e g u a t i o n (1) o f c h a p t e r 3 by T and e x p r e s s t h e maximum l i k e l i h o o d f u n c t i o n i n t e r m s o f 0+ t h e n -dk/2 L(U+) = {2 V a r ( U + ) ) e x p {-dk/2) So, t h e e s t i m a t e d l i k e l i h o o d r a t i o , r i s r = L (D*+) /I (U+) -dk/2 = [ Var+ ( u * ) / V a r + (u) ] o r , - 2 1 o g ( r ) = d k l o g [ V a r + { u * ) / V a r * {u) ] H h i c h i s a s y m p t o t i c a l l y d i s t r i b u t e d as C h i s g u a r e s w i t h ( d - 1 ) n d e g r e e o f f r e e d o m . 92 A p p e n d i x C P a r a m e t e r E s t i m a t e s f o r L a b o u r E q u a t i o n 1967 - 71 w i t h o u t ( O n e m p l o y ) 2 When unemployment o f t h e h e a d was c o n s i d e r e d a s a l i n e a r r e l a t i o n s h i p w i t h t h e a n n u a l h o u r s w orked by t h e w i f e i n t h e r e g r e s s i o n m o d e l , we f o u n d t h a t t h e y a r e n e g a t i v e l y c o r r e l a t e d . We t a b u l a t e d t h e r e s u l t s o f m o d e l I a s f o l l o w i n g : Group I G r o u p I I V a r i a b l e ( y | z = 1 ) ( y j z = 0 c o n s t a n t 1728.3230 478.4102 (4.0485) (1.2482) B i r t h G a p - 4 2 .8763 -36.5538 (1.7460) (2.8869) W i f e wg -714.5180 360.3229 (11.0354) (3.0861) ( W i f e w g ) 2 192.8221 -113.3932 (15.6143) (6.6019) Head i n c -0.2069 -0.1740 (217.8773) (182.9149) (Head i n c ) 2 0 . 2 8 9 2 x 1 0 - 5 0 . 2 3 3 7 x 1 0 - 5 (97.1418) (52.2196) Unemploy -0.2405 -2.5008 (0.0486) (4.7875) i n c / n e e d 6.5660 6.1298 (78.8038) (207.5199) ( i n c / n e e d ) 2 -0.0040 -0.0032 (19.0210) (77.3457) F e c u n d i t -13.2484 43.1432 (0.4368) (9.7703) ( F e c u n d i t ) 2 -0.5110 -1.7794 (0.5569) (10.9088) o b s e r v a t i o n 885 815 R 2 0.2521 0.3197 A s y m t o t i c t v a l u e s a r e i n p a r e n t h e s e s . 93 E i b l i o g r a p h y B e r k s o n , J . , ( 1 9 5 1 ) , "Why I P r e f e r L o g i t s t o P r o b i t s " , B i o m e t r i c s ^ December 1951, pp. 327-339. B e r k s o n , J . , (1953), "A S t a t i s t i c a l l y P r e c i s e and R e l a t i v e l y S i m p l e Method o f E s t i m a t i n g t h e B i o a s s a y w i t h Q u a n t a l R e s p o n s e , B a s e d on t h e L o g i s t i c F u n c t i o n " , 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 J o u r n a l ^ , S e p t e m b e r 1953, pp. 565-599. B e r k s o n , J . , ( 1 9 5 5 ) , "Maximum L i k e l i h o o d and Minimum X 2 E s t i m a t e s o f t h e L o g i s t i c F u n c t i o n " , J o u r n a l o f 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 0 , 1955, pp.130-161. B e r n d t E. R. and T. J . H a l e s , ( 1 9 7 4 ) , " D e t e r m i n a n t s o f Wage R a t e s f o r M a r r i e d Women: R e s u l t s f r o m P a n e l D a t a " , D i s c u s s i o n P a p e r No.74-05, D e p a r t m e n t o f E c o n o m i c s , U n i v e r s i t y o f B r i t i s h C o l u m b i a , March 1974. B e r n d t , E. B, and T. J . W a l e s , (1974), " L a b o u r S u p p l y and F e r t i l i t y B e h a v i o u r o f M a r r i e d Women: An E m p i r i c a l A n a l y s i s " , D i s c u s s i o n P a p e r 74-27, U n i v e r s i t y o f B r i t i s h C o l u m b i a , December 1974. B i s h o p , Y. M. M., ( 1 9 6 9 ) , " F u l l C o n t i n g e n c y T a b l e s , L o g i t s , and S p l i t C o n t i n g e n c y T a b l e s " , B i o m e t r i c s ^ V o l . 25, 1969, pp. 383- 400. B l i s s , C. I . , (1 9 3 4 ) , "The Method o f P r o b i t s " , S c i e n c e , V o l . 7 9 , No.2037, J a n u a r y 1934, pp.38-39. B l i s s , C. I . , ( 1 9 3 4 ) , "The Method o f P r o b i t s — A C o r r e c t i o n " , S c i e n c e , V o l . 7 9 , No.2053, May 1934, pp. 409-410. B o x , M. J . , D. D a v i e s , a n d W. H. Swann, ( 1 9 6 9 ) , N o n - l i n e a r O p t i m i z a t i o n T e c h n i q u e s , , O l i v e r and Boy d L t d . , E d i n b u r g h , 1969. B u s e , A. ( 1 9 7 2 ) , " A T e c h n i c a l B e p o r t on B i n a r y D e p e n d e n t 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 S c i e n c e s " , A Commissoned R e s e a r c h P r o j e c t o f t h e A l b e r t a Human B e s o u r c e s R e s e a r c h C o u n c i l , Edmonton, A l b e r t a , 1972. C h a m b e r s , E. A., and D. B. C o x , ( 1 9 6 7 ) , " D i s c r i m i n a t i o n b e t w e e n A l t e r n a t i v e 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 , ( 1 9 6 0 ) , " T e s t s o f E g u a l i t y Between S e t s o f C o e f f i c i e n t s 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 x V o l . 2 8 , No.3, J u l y 1960, pp. 591-605. C o r n f i e l d , J . and N. M a n t e l , (1950), "Some New A s p e c t s o f t h e A p p l i c a t i o n o f Maximum L i k e l i h o o d t o t h e C a l c u l a t i o n o f t h e Dosage B e s p o n s e C u r v e " , 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 94 i s s o c i a t i o j Q x V o l . 45, 1950, pp. 181-210, C o x , D. S. ( 1 9 7 0 ) , The A n a l y s i s o f B i n a r y . J ) a t a x M e t h u e n , L o n d o n , 1970. D e m p e t e r , A. P., ( 1 9 7 1 ) , "An O v e r v i e w o f M u l t i v a r i a t e D a t a A n a l y s i s " , J o u r n a l o f M u l t i v a r i a t e A n a l y s i s I x 1971, pp. 316- 346. D e m p s t e r , A, P., ( 1 9 7 2 ) , " A s p e c t s o f t h e M u l t i n o m i a l L o g i t M o d e l " , M u l t i v a r i a t e A n a l y s i s I I I X E d i t e d by P. B. K r i s h n a i a h , A c a d e m i c ' p r e s s , 1972, pp. 129-142. F i n n e y , D. J . , ( 1 9 4 7 ) , P r o b i t A n a l y s i s ^ C a m b r i d g e U n i v e r s i t y P r e s s , C a m b r i d g e , E n g l a n d , 1947. ( 3 r d e d i t i o n , 1971). G o l d b e r g e r , A. S., ( 1 9 6 4 ) , E c o n o m e t r i c T h e o r y x W i l e y , New Y o r k , 1964. Goodman, L. A., ( 1 9 7 0 ) , "The M u l t i v a r i a t e A n a l y s i s o f 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 C l a s s i f i c a t i o n s " , 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. Goodman, L. A., ( 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 t o t h e A n a l y s i s o f D i c h o t o m o u s V a r i a b l e s " , A m e r i c a n S o c i o l o g i c a l R e v i e w , V o l . 3 7 , 1972, pp. 28-46. G u n d e r s o n , M., ( 1 9 7 4 ) , " B e t e n t i o n o f T r a i n e e s — A S t u d y W i t h D i c h o t o m o u s D e p e n d e n t V a r i a b l e s " , J o u r n a l o f E c o n o m e t r i c s 2 X 1974, pp. 79-93. J o h n s o n , J . , ( 1 9 7 2 ) , E c o n o m e t r i c M e t h o d s , 2nd E d i t , Mcgraw B i l l , 1972. N e r l o v e , M. and S. J . P r e s s , ( 1 9 7 3 ) , U n i v a r i a t e and M u l t i v a r i a t e L c g z L i n e a r and L o g i s t i c M o d e l s , S a n t a M o n i c a , C a l i f . : Rand C o r p o r a t i o n R e p o r t R-1306, 1973. N e t e r , J . , and 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 o f t h e C o r r e l a t i o n C o e f f i c i e n t w i t h 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 . , ( 1 9 7 2 ) , A p p l i e d M u l t i v a r i a t e A n a l y s i s , H o l t , R i n e h a r t and W i n s t o n , New Y o r k , 1972. S c h m i d t , P. and P. S t r a u s s , ( 1 9 7 4 ) , " E s t i m a t i o n o f M o d e l s w i t h J o i n t l y D e p e n d e n t Q u a l i t a t i v e V a r i a b l e s : A S i m u l t a n e o u s L o g i t A p p r o a c h " , U n i v e r s i t y o f C a r o l i n e , 1974. S c h m i d t , P. And P. S t r a u s s , ( 1 9 7 5 ) , " E s t i m a t i o n o f M o d e l s w i t h J o i n t l y D e p e n d e n t Q u a l i t a t i v e V a r i a b l e s : A S i m u l t a n e o u s L o g i t A p p r o a c h " , E c o n o m e t r i c a x V o l . 4 3 , No.4, J u l y 1975, pp. 745-755. S u r v e y R e s e a r c h C e n t r e , U n i v e r s i t y o f M i c h i g a n ( 1 9 7 2 ) , A P a n e l 95 S t u d y o f Income D y n a m i c s , Ann A r b o r , 1972. 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 L i n e a r L o g i t M o d e l " , I n t e r n a t i o n a l E c o n o m i c R e v i e w , V o l . 1 0 , No.3, O c t o b e r 1969, pp.~251-259T 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 " , A m e r i c a n J o u r n a l o f S o c i o l o g y ^ V o l . 7 6 , 1970, pp. 103-154. T h e i l , H. ( 1 9 7 0 ) , P r i n c i p l e s o f E c o n o m e t r i c s ^ New Y o r k : J o h n W i l e y and S o n s , 1970. T c b i n , J . , ( 1 9 5 5 ) , "The A p p l i c a t i o n o f M u l t i v a r i a t e P r o b i t A n a l y s i s t o E c o n o m i c S u r v e y D a t a " , C o w l e s F o u n d a t i o n D i s c u s s i o n P a p e r No. 1, December 1, 1955. W o n n a c o t t , H. J , And 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. Z e l l n e r , A. and H. T h e i l , ( 1 9 6 2 ) , " T h r e e - S t a g e S g u a r e s : S i m u l t a n e o u s E s t i m a t i o n o f S i m u l t a n e o u s E g u a t i o n s " , E c o n o m e t r i c a , V o l . 3 0, No. 1, J a n u a r y 1962, pp. 54-78. Z e l l n e r , , A., ( 1 9 6 2 ) , "An E f f i c i e n t Method o f E s t i m a t i n g S e e m i n g l y 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 o f 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 7 , 1962, pp. 348- 3 6 8 . " Z e l l n e r , A. and D. S. Huang, ( 1 9 6 2 ) , " F u r t h e r P r o p e r t i e s o f E f f i c i e n t E s t i m a t o r s f o r S e e m i n g l y U n r e l a t e d R e g r e s s i o n E g u a t i o n s " , I n t e r n a t i o n a l E c o n o m i c R e v i e w , V o l . 3 , No.3, S e p t e m b e r 1962, pp. 3 0 0 - 3 1 3 . Z e l l n e r , A., ( 1 9 6 3 ) , " E s t i m a t o r s f o r S e e m i n g l y U n r e l a t e d 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 mple R e s u l t s " , J o u r n a l o f 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. 977-9 9 2 . Z e l l n e r , A. and T. H. L e e , ( 1 9 6 5 ) , " J o i n t 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 D i s c r e t e Random V a r i a b l e s " , E c o m e t r i c a x V o l . 3 3 , No.2, A p r i l 1 965, pp. 382-394.

Cite

Citation Scheme:

    

Usage Statistics

Country Views Downloads
United Kingdom 4 0
United States 4 0
France 3 0
China 2 1
City Views Downloads
Unknown 7 3
Beijing 2 1
Ashburn 2 0
Mountain View 1 0
Sunnyvale 1 0

{[{ mDataHeader[type] }]} {[{ month[type] }]} {[{ tData[type] }]}

Share

Share to:

Comment

Related Items