Planologisch Studiecentrum TNO
Schoemakerstraat 97/Postbus 45
2600 AA Delft
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AN INTER-REGIONAL MIGRATION MODEL:
a disaggregated approach
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Bi b l iothe ek TU Del ft
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C 1995964- - - ~
-PAPER PRESENTED ATTHE 20TH EUROPEAN CONGRESS OF THE REGIONAL SCI ENCE ASSOCIATION.
t1UNICH, IIEST GERMANY, AUGtiST 1980.
ACKNOWLEDGEMENTS
This paper is partly the result of a project on inter-regional mi gr at i on, financed by the Dutch Nat i onal Physical Planning Agency (R.P.D .). The author gratefully acknowledges the useful remarks of the steering committee of this pro- -ject, viz. Prof. Dr. Paul Drewe, Drs. Leo Eichperger, Drs. Rob van der Erf, Dr. Bert van der Knaap and Drs. Jan Scheurwat er. Specia l thanks are due to Hans Blokland, Peter Nijkamp and Leen Hor di j k for their useful discussions andcritic ism. The author is very indebt ed to Hans Heida for his very fruitfu l discussi ons and inspirat ion . The manuscript was careful ly prepared by Kitty van Geest and Er na Valstar .
CONTENTS
4.1. The utility of goods and services 10
4.2. The disaggreoate model 12
4.3 . The aggregation probl ems with
disaggre-gated mode 1s 17 3 8 10 1. I ntroducti on 2. Interaction model s 3. The entropy model 4. A disaggregate model
5. Evaluation
6. Literature references
22 24
Summary
Application of distribution models for the destination of inter-regi-onal migration flows
The research, supported by the Dutch National Physical Planning Agen-cy, is meant to deepen the insights in factors influencing the inten-sity, composition and direction of inter-regional migration flows. Several models describing the migration processes have been developed. They can be characterized as aggregate models, which are derived by a
'top down' approach like the entropy models.
It is the intention of this study to develop a more disaggregated mo-del, which is derived by a 'bottom up' approach. From a theoretical point of view a disaggregate model, like the logit model, can suit the behavourial background. With a proper aggregation procedure, a priori to calibration, an aggregated logit model with the same structure as the entropy model can be derived.
In this paper both types of models wiJl be describedand compared. The entropy model has a drawback in lacking specification errors. But it is very easy to extend that model with an error term and then the method of least squares, can be applied for calibrating the parameters. In logit models, using the Weibull distribution, the specification term has been integrated into the constant term. The logit model as an individual description of the behaviour can be aggregated to groups of households by adding a perception error term for the deviation of average group behaviour. It is this perception of the error term which makes it possible to apply the method of least squares.
It can be concluded that in the end the entropy and aggregated logit models, as fas as they are applied to groups of households, have the same formulation, but are different in interpretation. The interpreta-tion is in favour of the aggregated logit models.
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Samenvat t i ng
In opdracht van de Rijksplanologische Dienst wordt een onderzoek uit-gevoerd om het inzicht te vergroten in de faktoren die van invloed zijn op de omvang, richting en samenstelling van migratiestromen. Ver sc hi l l ende (ty pen) modellen voor de beschrijving van het migratie-proces zijn ontwikkeld. Zij zijn allen te kenmerken als geaggregeerde model l en, welke door een 'top down' benadering worden afgeleid. Een voorbeeld is het entropy model.
Het is de bedoeling van deze studie om vanuit de 'bottom up' benade -ring een meer gedisaggregeerd model te ontwikkelen. Daarmee kan tege -moet worden gekomen aan een theoretisch verantwoorde gedragsbeschrij-ving. Met het logi t model als uitgangspunt kan middels een geschikte aggregatieprocedure een geaggregeerd logit model worden afgeleid. Dit model heeft dezelfde st r ukt uur als het entropy model.
In deze bijdrage worden be i de typen modellen beschreven en vergeleken. Het entropy model kent geen specifikatie fouten, maar deze zijn
gemak-kel i j k te introduceren door het toevoegen van een storinqsterm. Het is tevens mogelijk de methode van de kleinste kwadr at en voor de schnt -ting van de parameter toe te passen.
Bij logit modellen wordt de specifikatie fout gepres ent eerd door de Wei bul l funktie. Deze wordt geïntegreerd in de konstante termvan de nutsfunktie. Het logit model, als individuele"beschr i j vi ng van het gedrag, kan geaggregeerd worden naar groepen van huishoudens door het toevoegen van een perceptie fout als afwijking van het gemiddel-de groepsgedrag. Het is deze term welke ook hier toepassing van de methode van de kleinste kwadraten mogelijk maakt.
Er kan gekonkludeerd worden dat entropy modellen en geaggregeerde logit modellen, voor zover op groepen van huishoudens toegepast, dezelfde formulering kennen, maar verschillend in interpretatie zijn. Deze in-terpretatie doet de voorkeur naar geaggregeerde logi t modellen uitgaan.
1. Introduction
Changes in the composition of the population and its distribution over the country are the results of births, deaths, and migration both internal and external. Dwing to the sharp fall in the number of births and the increase in migration figures over recent decades, the effects of migration on the extent andcomposition of regional populations is becoming significant.
Of the large number of movements of population in Holland, some 57%took place within the same munici pality, while 75%remained within the same region . The balance, 25%, moved to other regions. Particularly the lat t er, the inte r-regi onal migrants, can exert a considerab le infl uence on the distri but i on of the population. Among all the housing movements, the inter-regional migrants form a spe-ci al group; they do not mere ly change thi s municipality of residence in which they live, but they also change their labour market and housi ng market areas . For the other inter - regi onal migration can
65 55 60 50 45 40
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197 5 1977 1970 Per 1000 of the average population 1965 1960Trend of the level of domestic migration af ter 1945 Source : CBS, 1976, 1979 Fi gure
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1946 1950 700 6~ Domestic migration in absolute figures ---lI: 1000 7~ 550 600 500 400 450- 2 ~
accordingly be defined as:
A change of housing district, where the district of establishment forms part of a different spatial entity, distinguished by virtue, inter alia, of its labour market, housing market and services struc -tures, from the district of departure .
Dwing to the difference in reorientation, or the (potential) dif-ference in factors and weights, separate treatment of intra-regional and inter-regional migration is preferable. This study will concen-trate on inter-regional migration.
The particular consequences of inter-regional migration makes it des-irable, if not essential, in the interests of spatial and environmen-tal policy, to obtain sufficient reliable information regarding the nature and effect of factors and variables, influencing the inten-sity, direction and composition of inter-regional migration flows. The phenomenon ofinter-regional migration in Holland is the
empi-rical subject of this paper. The questions asked of this empiempi-rical subject, or identification subject, relate to the general question:
'who migrates where and under the influence of what factors?'. Since it is a household which is migrating, migration takes place as a consequence of a decision by that household.
In order to reduce the uncertainty regarding the characteristics of the anticipated migration flow, and to measure the extent of the influence of a number of variables, an analysis'of the migration behaviour will be required. A formalised reproduction of the assump-tions relating thereto may lead to the application of an existing or a newly developed migration model.
Many models for describing migration flows have been developed.
Good summaries of these are provided by Ter Heide (1965), Drewe (1972), Shaw (1975) and Alonso (1977). For Holland, regional migration models have been developed, among others, by Drewe (1976), Janknegt (1976) and Somermeyer (1971). A general feature of these applications is
that these models include an aggregate estiw.ate of the migrants and th at the provinces are selecte~ as regions. The material available did not permit of any more precise partition (apart form the sub-division of the migrants into heads of families. A migration research
project however considering the intensity, com posi t i on and directi on of the mi gr at i on f~ows ought to hold the decision of the household central. Mobility and the consequences thereof can only be explained within a theoretical framework based on the mot i vat i on and choi ce of the smallest decision unit. Never t hel ess , among the cons i derabl e amount of research into migration, only a very small proportion is really devoted to the forming and testing of theories. Only recent ly has the need for disaggregrate models begun to become apparant. When use was made of a model based on a theory, the models concerned were aggregate models which are capable of supplying a description of the volume of the migration flow, but never any motivation as to why the household migrated. In other words, attention has been directed above all to the distribution of the migration for the flows between the various regions.
For some time now the need for disaggregate models has become manifes t . The starting point for the development of such models is the behavoria l background. To get information on the distributi on of the mi gr at io n flows involved an appropriate aggregation procedure is neces sa ry. Thi s subject gets very much attention in literature. For thi s study an a priori procedure has been applied and the result is a mode l const ruc-tion which is very similar to the entropy model. In thi s paper bot h types of models will be compared: the entropy model as an aggregate model and the aggregated logit model as a disaggregate model .
In chapter two some general features of interaction models will be discussed. The entropy model will be dealt with in chapter 3. In chap-ter 4, the disaggregate model will be derived. Combining the specifi c-ation and perception errors the variant for this study will be presen -ted. The paper wil l end with a comparison between both the entropy and the aggregated logi t model.
2. Interaction models
Inter-regional migration models describe the migration movement as an interaction between geographical areas. Consequently, they are to be included with the so-cal l ed interaction models (Wilson, 1974). The interactions have a point of origin in one area, a point of
destina 4 destina
-tion in another and an ac-tion between the two areas. These aspects give rise to the migration movements to:
origins of migration from a region i, i =1, ...., N.
_ flows of migration between regions and j; i, j =1, .... , N;
i
j. _ arrivals of migration in a region j, j=
1, .... , N.These three aspects can be expressed in various ways in a model, as aresult ofwhich different types of interaction models arise. The most important difference between interaction models is probably the various ways in which origins (or arrivals) and flows are descri-bed in a model (Alonso, 1977). This may occur both directly and indi-rectly. In the case of the direct model, origins and fl ows are descri-bed at the same time: without any additivitv conditions. In the case of the indirect model the origins and flows are described in separate sub-models, mainly sequentially and sometimes with feedback effects. In the latter case, the origins are to be considered as input data, a point of departure for the distribution over the regions: the flows are distributed over the areas based on origins, and hence with addi-tivity conditions.
The subdivision of indirect models into two submodels fits very well with the subdivision into phases of decision-making in migration
pro-cesses (Speare et.al., 1974). The subdivision into phases is important since various studies have shown that, when deciding to migrate, dif
-ferent factors may play a part compared with the case of the choice of a destination for migration (e.g. Morrison, 1977; Lowry, 1964).
For analytical purposes, it is possible to bring the decision-making process down to two phases:
(1) the decision to migrate (generation of migration); and
(2) the decision regarding the destination of the migration (distri
-bution of migration).
The following is an example of a direct interaction model:
where:
Tij the number of interaction between and j;
Qi a variable for i ;
Qj a variable for j;
D.. a variable for ij;
1J
ao al a2 a3: reaction coefficients.
Dependingon thefield of application,it is ooss ib Ie to give a more preci se content to the system characteris ti cs , whil e the reacti on coefficients can be determined on the basis of empirical material . The fol lowing is an example of an indirect int er acti on model:
where: T. 1. ( 2.2) T. 1. x. J a and:
the number of origins from area i; an attributes vector of area i; an attributes vector of area j;
an attri butes vector of relatio nships between areas and j; a vector of parameters. T? Q. Ai QC:2 D~ ~ lJ 1 J 1J where : Q.
L
T? 1 j 1J and ( 2.3) (2.4) (2.5)By substituting (2.4) and (2.5) in (2.3), result is the so-called
"sin gl y constrained" int er act i onmodel \~ '1i c h is shown here as a dis-tribution model since a number of origins, known a ~riori, and from
6
-:lodel s (2.1) and (2.6) are different in principl e, this being caused by the additivity condition:
(a) The rroportionality factor, or the constant term of (2.1), has disappeared from (2.6); this is now a zone-dependent "balancing " factor, Ai' The function of this factor is the regulation of the com-petition element between the interactions (~ilson, 1974).
(b) The migration elasticity of a region of origin al, has now dis-appeared, or in other words, since the model in (2.6) is homogeneous
fr om the zero degree, the elasticity is equal to 1. All features of the area of origin have now been integrated away from the model into the production variable of the interactions, 0i' The characteristics of the region of ori gin have been replaced by the number of origins.
In order to determine the number of departures from a region, a separate generation model for (2.2) is now required.
(c) By stating the additivity conditions, the direct model of (2.1 )
is unravelled into two sub-model s , the generation model and the dis-tribution model. In the case of the disdis-tribution model, the totalof the parts to be explained, the number of destinations, is equal to the whole, the number of origins. This avoids the possibility of the model being able to explode, which can lead, particularly in the case of forecasts, to disastrous developments.
(d) 5ince the distribut ion model of (2.6) is based on a given number of orig ins, it is of considera ble impor t ance to construct the genera-tion of the interactions, presented in model (2.2) carefully, and to make feedback effects possible. In this manner, a si mult aneous system is obtained for the descrintion of the interaction process with which different effe cts can be analysed which cannot be achieved with the direct model.
With a formulat ion in shares or probabilities, the probability of a (2.6)
0
0:
2 D<: ~ J 1JT~.lJ
'direct' interaction from region to region j is mainly defined (Drewe, 1976) as:
(2.7)
where:
Bi : the population of area i.
There are certain objections attaching to this formulation. In the first place, it may be observed that the probabilities, generally speaking, do not add up to 1, and in the second place such a model does not satisfy the additivity conditions. This can also signify that, despite any good results with a calibration, the model is like-ly to explode as a forecasting model.
A possible solution to the above ~roblem is provided by splitting the model into a generation model and J distribution model:
Pij = Pi . . Pj/i (2.8)
The first part of the right-hand component of equation (2.8) refers to a generation model of migration from i, when for instance use can be made of a Poisson process formulation. The second part relates to the distribution character of the migration. By this formulation,
it is possible to satisfy the additivity conditions, so that the
problems of (2.7) are also solved at the same time.
Summarising the foregoing, it may be stated that a migration model is concerned with the generation of the migration and the distribu-tion of this migradistribu-tion over the areas, so that the additivity condi-tions require to be imposed on the origins as logical demands made of the migration model. Only then all system conditions of a logical nature can be satisfied (Somermeyer, 1967). The direct interaction model is then also less suitable, so that a simultaneous generation and distribution model results for the description of the migration phenomenon.
8
-mul t i pl i cat i ve direct migration models (2.1) there are also additive
direct migrati on models. Examples may be found in Fabricant (1970),
Feeney (1973) and Hawrylshym (1977). With this type of model, the volume of migration is usually linked to popul at ion , emp l oyment, i
n-come and the li ke. By the structure of these model5 (the additive
character ) all logical demands are in fact violated. No further
attention can therefore be devoted to this type of model.
3. The entropy model
One of the most used types of a model for spatial int er act i ons is the so-called distribution model. Anumber of variants are to bedi stin
-guished ; the best known of these is the gravity model. This model was
introduced by Carey and Ravenstein (1889) in geogr aphy. It is to be
consi der ed as analogous to Newton 's law. Never has a type of model
been used more of ten and at the same time beenmore of ten reviled .
Despite themany good advantages of the model in a statistical sense, it is judged unfavourably owing to its excessively mechanistic charac-ter and its inadequat e theoretical base. Consequent ly, on many o
cca-si ons an attempthas been made to give the gravity model a better
(behavi our a l ) substructure. Many derivations have been discovered in the meantime. A very well-known derivation is the entropy model of Wilson (1969, 1970, 1974, 1978) with which he der ive s a family of in
-teract i on models. The entropy approach is based on the idea that the
sys t em in question strives for the most probable macro-situation
(maximum entropy) if all the mi cr o- si tu at i ons of the system concerne d
are equally probable and at the same time certain subsi di ary condi
-tions are also satisfied. This assumptionof a Maxwell-Boltzman situa-tion (Feller, 1965) is not justified since the real isat i on of an in
-teraction between two are as may be dependent on the chara ct er is t ic s
of the areas and the comMuni cati on facilities between those areas. If a given probability distribu ti on of realisati on is assumed as between
the areas i andj, say P.., and if there are no 1imitations on the lJ
nUMber of destinations, then the res ul ting migrati on flow between i and j can be described by (van Est, 1976):
where : T.. lJ C. p .. O. J lJ l 'l: C. p.. . J 1J .] (3.1)
Cj the charac te r ist ic vari abl e of regio n or zone j.
The entropy model can be genera lized further by adding a vector of
charac ter istic var ia bles for destination re~ion j, xj, and a vector
for distances variables between regions i and j, 0 .. , and by adding lJ
a srecif ica t io ner r or , ci j . It is also possi bl e to formulate the model in a probabilistic way, sa that then as a "s i ngl y constrained" gravity model resul t s:
where :
exp{a'xj + b'Dij + ci j} l:exp{a'x. + b'D
i J· + ci J' }
j J
a,b vectors of reactio n coef f ic ie nts.
(3.2)
\Jith (3.2), a fai rl y general analogous formulation of the gravity
model has now been derived. This formulation is in essence identical to the formulatio n accordi ng to equation (2.6). The derivation of the entropy models sti ll retain s a mechanis tic character, however. It is sti ll based on the idea that the (aggregate) system concerned str ives for maxim isa t i on of the entropy. Whether the individuals concerned also see this as their motivat ion is doubtful since, in the case of the entropy model, this consideration is in fact con
-sci ously excl uded. Ader i vation accordi ng to (3.2) is then to be
preferred since, by introducing the probability Pij certain struc
-tura l feat ures can be int r oduced. As a res ul t, however, a complete
behavioura l substructure is st i ll not provided. The entropy model, like the gravity model, has found it s application as an analogous
model with in the social sciences. Conseq uent ly for this type of model
toa a search is made for a better, behavi our al , substructure. Still, the crucial quest io n is: "What is the mot ivation of the individual,
4
- 10
-or the household maki ng an intera ction ?". The entro!1Y model can not answer thi s o,ues tio n, because the only behavioural char act er i s t i cs are insert ed int o the sub- condi t i ons (Van Zuylen, 1975). This means that also the entropy model is rat her a mechanistic ty~e of mode l . In addi-ti on impact st udi es are hardly possible, since the b-parameters are Lagrange multipliers andposs es ses therefore only validity in the o p-timum.
Despite the somewhat mechani stic conception of the entropymodel it-self, a more behavioural methad of approximation is also possible. If the entropy model is written in probability notation, it may be looked upon as a geometrical programming model. Nijkamp (1975) pointed out that the objective function of the dual problem can be looked upon as a primary problem of a maximisation of a utility function, namely the minimisation of the total casts of the interactions of the spatial problem concerned. The entropy model is thus to be interpreted in an economiesense, while the equilibrium condition of the geomet r i cal
problemin question is characterised by a gravity formulation. The real question of the entropy model is now as to whether the utility funct i on of the primary problem (a logarithmic perception of the casts of the interaction) is valid? As long as this question is not adequ -ately answered, the point of departure and therefore the value of the entropy model remains a matter for discussion; it is at most a special case within the theory of cons umer behaviour.
4. A disaggregate model
4.1. The utility of goods and services
For same time now, use ha5 increasingly been made of so-called dis -aggregate models. The need for this type of model arose from the dis -satisfact io nwith aggregate models, si nce there was for the most part a lack of any behavioural substructure. The disaggregate models have been developed above all in biology (Fi nney, 1947). In the sixties, the first applications were m~de in transport planni ng. Great pioneers in this field were Stopher (1968), Domencich and McFadden (1975) and Ben Akiva (1973). In Holland, a number of applications were made in the seventies. Mentionmay be made of Donnea (1971), Goudappel and Coffeng (1974) , Ruygro k (1979) and Baanders et al. (1979).
Dutch applications, too, were directed to communications and transport proble ms. For a be~ter founded behavioural substructure of the migra-tion movement, disaggregate models can also be useful.
Disaggregate models are based on the smallest decision unit, which is formed in this case by the family household. A certain rationality in the behaviour of the households is taken as the basis. This means that use can be made of the socio-economie theory of consumer beha -vi our . Here the rule is that a househol d in a given choice situation makes the choice whereby, given all the trouble, costs and benefits, it maximises its utility (difference between benefits and charges) un-der certain condition.
In the classical theory of consumer behaviour, the basis taken is a rationally acting individual who endeavours to distribute his income over the goods and ser vices in such a manner th at the marginal utility of the expenditure on each good, divided by the price for the goods concerned, is equal to the delimiting utility of the cash income. The conventio nal theor ies of consumer behaviour have given an insight into the choice behaviour of individuals within the economie system. But there are various objections to the foregoing formulations. Kl aasse n (1972) ment i ons four obj ect i ons , namel y the statie character of the theory , the consumption of goods without regard for time or trouble, the lack of any time budget restrietion, and the infinite divisib i lity of the goods .
In addition to these objections, there is also an other important pro-blemconnect ed with the utility theory. The goods admittedly posses a subjec ti ve util i satio n value , but this is intrinsic inthe good. This construction is abandoned in modern theory consumer behaviour. A theory has been evolved by Lancaster (1966) on the attributes of the goods in -stead of the int rin si c val ue of the goods themselves. The utility func-tion is th en also expressed in terms of the attributes concerned. This implies th at the choice process is now formulated on a collection of attributes and not on a col lect io n of the goods themselves. The new approach possess es many advantages. Not only is the indivisibility problemof Klaassen solved thereby, but the theory can also be rende-red opera t io nal much more simply (discrete choice problems now require
12
-under certain conditions with respect to the alternatives concerned. The above formulation of the theory of consumer behaviour is the point of departure for the deri vati on of a di saggregate choi ce model. to be solved). In the case of a migration process, it is a question of the description of the attributes of the relevant choice possibi-lities. In the case of inter-regional migration, this relates to the characteristics of the housing and labour market areas. The modern theory of consumer behaviour is based on a rational decision-maker who is able to place his preferences in order and makes the best choice
for himself from a set of alternatives on the basis of his personal socio-economic circumstances:
If the collection B comprises all the relevant choice possibili-ties, where each alternative is described by the attributes vec-tor, and the personal circumstances of the consumer are described by vector s, then the choice process of the consumer is to be descri bed by:
(4.1 ) MAX U(x,s)
Xe:B
4.2. The disaggregate model
In modern utility theory a consumer with a vector of characteristics s evaluates each alternative j out of a particular set with a vector of attributes x. Then for every alternative there exist a measurable relationship between the benefits and evaluation of that alternative:
V(Xj,5). l t is a non- stochastic component, repre'senting the average preferences of the population concerned. This is the "strict utility" component, which is quite similar to the classical theory.
Next to the "strict util ity", there is added a stochastic component,
e:(Xj,sj, for the deviating behaviour from average, and the
non-obser-vable variables in the choice process. This means that not all house-holds within the same (homogeneous) group in question consider the same attributes and/or evaluate then equally; i.e. a specification
error or "random utilities" component. Putting both components toget-her, the utility function which a household experiences may be written as:
(4.2)
(4
.4)
On the basis of a certain rationality, the household willapt for that alternat ive which provides the greatest utility, or the household will opt for alternative j in set B, sothat:P [jEB] = P [U(xi,s) > U(xi,s); \ti ; i ij] =
=P [E(Xj,S) - E(Xi,S) > V(xi,s) - V(xj,s); \ti; i ij] (4.3) In further process i ng, the stochastic term E plays a very important part. With different assumptions regarding the farm of the distribu
-tion of the stochastic component, various types of model can be deri
-ved. The bes t known types her e are the fol lowing:
(a) a normal distribution; (b) a Cauchydistribut ion ;
(c) a Weibull distribution.
If a joint normal distribution is selected for the stochastic compo
-nent, a so-ca lled "probit " model is found to result. From a statisti
-cal point of view (Finney, 1947), this model can conveniently be used
in a linear choice situation. For appl ications of the probit model, reference may be made, among others, to Donnea (1971) and Hausman and Wise (1978). In a multiple choice situation, there are the necessary arithmet ica l difficul t ies . But these do not offset the robustness of the assumptions with regard to the independence of the stochastic com
-ponents. If the stochastic components are distributed independently Cauchy-wise, a so-ca l led Ar ct an model results . The derivations with this type of model are simple, but still prohibitive by numeri cal in
-tegratio n methods. It has now been demons t r at ed by McFadden (1975) that, if the stochasticcomponents are independently Weibull distribu
-ted, the foll owi nq "logit" model will result: . eX[J {V(xj .s ) -
a
}
P [jEB] =
~ exp {V(xk, s ) - ak}
k
The () term is the locati on parameter of the Weibull distribution and is to be interpreted as a specific effect of the various alternatives.
By int egr atin g thi s ef f ect int o the dete rmi ni s ti c component, that is by mak i ng the c parameter part of the V functi on, (4.6) can be writ
-The derivation of the logit model can also be effected in a different manner. This different method makes use of the normal distribution, which constitutes an advantage compared with the double exponential
distribution of Weibull. A method is presented by Sen (1978) with a
statistical basis instead of the behavio ura l logi t model s . The dif-ference compared wit h the logit formulation is that now the x varia-bles are now assumed to be stochastica l ly distributed and no dis tur-bance term is int roduced. No randum ut ili t y is then assumed either. This is quite logical since it is a purely statistical derivation. A choice has to be made fromthe three modelsdiscussed. This choice is in fact fairly simple. The logit model possesses good statistical characteristics, is convenient to handle froma mathematical point
of vi ew and fits into the specification regarding the st ochas tic
com-ponent of the utility function. The other two models have their li mi-tations as regards the characteristics mentioned to a greater or les-ser degree. The choice is then also in favour of the logit model. In addition to various disadvantages, the disaggregate model shows many advantages and possibilities . The most.i mpor t ant advantage is that this type of model connects with the individual choi ce process: the introduction of variability in choice behaviour is the most impor-tant advantage (McFadden , 1975; Ruijgrok, 1975).
Another advantage of the disaggregate models is that, for estimating and testing parameters, many fewer observations are necessary than with aggregate.models . This advantage is quite important for the cali
-br at i on of a model , but freq uent ly subordinated to the object of the
investigatio n. As in the present inves ti gati on, the int ensi t y , compo
-sition and direction of the migrat ion flows are al l-important. The
disaggregate approximation supplies inf or mati on regarding individua l behaviour, which is important for motivation, but in itself does not supply any information regarding the flows. In the last resort, the aim is to provide information regarding the flows. Thi s implies that ten more simply as follows:
exp {V(x. ,s)} P
[j
EB] = JL
exp {V(xk, s)} k - 14-(4
.5
)
the individual's behaviour also requires to be tested against the ag-gregation of individual interactions which give rise to the flow. As aresult, knowledge is required regarding the flows themselves. A li-mited random sample, in itself suffic ient for calibration of a model, is generally much too small for reasonable reliable statements regar-di ng the flows.
The formulat ion of the logit model is very similar to the orginal for
-mulat ion.of the gravity model . Thi s notwithstanding, the logit model
can only be incl uded in the category of gravity models provided extra
condit i ons are added in respect of independence and separability. It
is a st rong point that logi t models do not require the conditions of separabil ity of the vari abl es and independence of destinat ions in or
-der to be derived by means of the theory of consumer behaviour. If, for instance , potentia l formulations are not strictly necessary (i.e. if there is no spatial auto-cor re lation), the axiom of Luce (1959) regar
-ding the.independence of irrelevant alternatives is self-evident. For
the formulation of the logit model, this means that:
V(xi,s) - V(x j .s )
(4.6)
The ratio between the two probabi lities is entirely determined by the "strict utility" function of the two alternatives i and J. The other
alternatives have no infl uence on this . Through this restriction, the possibility is opened up of the introduction of a new alternative.
With the int r oducti on of a new and independent alternative, it is only necessary for the denominator of (4.5) to be expanded with the attri
-butes of the new alternative ; a recalcu lat ion is not necessary since the ralative ratio remains unaltered. This estimate is only possible
if the (new ) alternatives are comple te ly distinct and independent.
In the fore goin g obse rvatio n, it has been assumed that Luce's axiom
would imply that the rela t i ve probabili ti es are only dependent on the
areas involved. The str uct ure of the other regions would not exert any
influence at all. In other wor ds , by V(xj , s ), only those variables are
descri bed which rela t e to the regio n j, where these are not dependent
on other regi ons. For region j, variab les now require to be introduced
as a potentia l , whereby elements of the tot al spatial structure can be introduced. 5uppose that:
- 16
-It is now important to show what influence is exerted on the probabi-lity P
j/ i if in area 9 the attraction variable changes. For this, the marginal probability dAg is investigated. It then follows that:
(4.10)
(4.9)
(4.8
)
(4.7)
dP j/ i = P{
d~l
_ \' P .' d-1 } dAg ct j/i Jg ~ m/l mgFrom (4.10), it is seen that the requirement of independence is no longer met, while the potential is only a function of j itself. It is not possible to make any pronouncement regarding the positive or the negative character of the marginal change. It is above all the dis-tance or possible function of the disdis-tance which exerts an influence. The effect of a marginal change of the potentials in region 9 upon the probability of a migration to region j can be divided into an "i nt r a-regional" and an "inter-regional" effect:
V(Xj,5) ctX1j + eX~ J +(l and that:
X~J
Akdj~
whereby the variable xl is defined as the potentialof A and the dis-tance from the region, and x2 is another arbitrary set of variables. Then the following will apply for Pj/ i:
exp {ctX} +
e
X~
+ 'ó} PJ'/iI
exp {ctX1+s
x
2+ 'ó}m m m
(4.11 )
If ct is positive and the distance between the two regions concerned is relatively small, than is the "int r a- r egi onal " effect great compared with the "inter-regional" effect. This may impl y that for instance the creation of recreational areas in region j can effect a positive in-fluence, while the creation of such areas in other regions far away from region j can influence the migration to region j in a ne9ative respect.
From the formulation of the potential in (4.11) , it may be deduced that, if the area 9 in which the change takes place is near to j, the influence will be predominant ly posit ive , and that in the case of re -mot e areas a negative influ ence wi l l generally be the result. In such a case, the spatia l structure wi l l need to be analysed carefully. 4.3. Theaggrega t i on pr oblems with disa ggre gated models
Disaggregate models are based on an indi vi dual pattern, but the final result whic h is desi red is the descrip t io n of the exte nt and composi
-tion of the various flows. This invol ves an aggregation procedure bei ng necessary in orde r to move from the indi vi dual behaviour to a collective description. The problemwit h disaggregate models is that sucha model is not mer el y direc t ed to the descript io n of the beha
-viour of the smal lest uni t of decis ion , but that also the explanatory
variab les are specific to that unit (Ru ijgrok and Wieleman (1975).
These variab les may rel at e to the attr ibu tes of the alternatives (the regio ns) as well as to the characteristics of the decisionmaking unit (t he households) . A model with a disaggreg ated endogenous variable and aggregated exogenous variab les (see Ric hards and Mars 1975) is accor
-dingly not auto matica l ly to be desig nat ed as a disaggr egat e model .
The assumption at the out set that the households (of a given group) react in like measure to the same val ues of varia bles (e.g . zonal mag
-nitudes, income catego ries, and sa on) does not of itself make amodel
disaggregated. This is a hypot hesi s whic h fi r st has to be proved. This
notwith standin g, it woul d be desirabl e to exchange the indiv idua l ob
-servations for zonal and group equal ivalents respective ly . There are several reasans for such an approac h. In the first place , in general ,
an aggregatio n over indiv idua l observations is generally not the same as a subst i t ut io n of the mean val ues of the independent variables .
That is in general:
(4.12)
where
P
m the share of the popul at i on choosing alternat ive m; Pm the probabi l ity of choosi ngalterna t ive in according to18
-The logit model is a non-linea r model 50 that equation (4.11) will
only be an equality if identical values of the independent variables can be assigned for every household in the population. The shape of the distributi onof those var i abl es has an influence on the magnitude of the error. The application of zonal and gr oup values as mean va-lues of the variables may then cause difficulties.
A second problem lies in the availabil ity of indidual data as far as applications are concerned. Especially making forecasts is hardly pos-sible because individual data of the population are then not at dis-posal. In this case a simplified aggregation procedure is necessary. The objecti ve thereby is to minimize the aggregation error to an
ag-greable level.
Several methods of aggregation have been developed. The procedure is that af ter the calibration of the parameters of the disaggregate model, the aggregation will be performed. In Koppelman (1976) a number of methods is described. In this paper the following methods can be men
-tioned:
_ the naive method, with which use will be made of:
P P (x,s) (4.13)
m m
Because of the inequality of (4.12) Talvitie (1973) added a correc t io n term, k , 50 that:
(4. 14) The correction term can be interpreted as a measure of the homo gen-eity of the population conce r ned. For to big a value of this term it
is better to divide the populatio n into homogeneous subgroups. With a weighing summation of the subgroups the probabil ity of the whole po -pulation can be calcu lated (le Clercq et al, 1976):
P L P . Nw (4•15)
m i~ m\~ "N -where
P
mw the aggregated probabil ity for alternative m of households
in group w; N
w the number of households on group w; N the total number of households.
enumeration, with which a summation is proposed for all the indivi
-dual functions:
P =
L
P (xh,sh)m h m (4.16)
The enumeration procedure is exact if all values of the ingependent variables for all members of the populatio n concerned are known. Mostly this condition will not be met. Approximations have then to be made, by usin g, for ins t ance, a sampl i ngof particular distri
-butions of the variables.
- groupenumeration or class ification, with which mean values of the independent variables per group will be used:
(4.17)
where the shares for each group are presented by fg.
- util ity class ifica tion , with whi ch the grouping will not be done
over the seperate and independent variables, but over the utility
funct io n it sel f:
-P
=
L
f Pm(V )m q q q (4.18)
The first th ree me thods are di smi ssed by Lierop and Nijkamp (1979) as too arbitrary andeven desig nated as marginal in comparison with the aggregate gravity and entropy models . They advocate an aggregation method developed byRei d (1978) which is based on a utility
classifi-cation. Here the procedure sugges t ed by Theil (1967) is passed over,
whereby a disturbance term is added to the explanatory variables of
a give n alter native wit hi n a groupof house hol ds. Themethod of Theil
is an a priori meth od, whic h means that theaggregation procedure can be applied before the calibration of the parameters. This method can
here be appl ied on an analo geous way. Whils t Theil makes an adaption to mean group values, in this case it is proposed to transform the
independent variab les to zonal and group values with a correction
term for the indiv i dual devat io ns. The deter minis t i c part of the uti
(4.22) (4.21) (4.20) (4.19) 20 -ng =~
I I
8p.'!'ph (4.23) hg p h hg the number of households.
When the relative frequency of the observed shar e for the dest i nat i on
choi ce is presented by f g, then the error term equals:
n =
.L
I
{l n f - ln Ph} (4.24)9 h h 9
9
X
ph: the variable p for household h.
where:
For all households, h, within a particular group, 5g, the utility
function can be deseribed as:
then follows:
If the proposed transformation is:
X =X;I!; + '!'
ph P ph
where
After manipulating with Taylor series on the logarithmic terms the resul t is:
=
~ (4.25)n
g f
g
From (4.25) it appears that the error term is approximately equal to
the difference between the observed relative frequency of group 5g
and the average probability.of the group (obtained byenumeration),
divided by the relative frequency. In other words the expectation of
(4.26)
The variance is inversely proportionate to a factor of the number and
share of the alternatives involved. From this then follovis an agg
re-gation to the whole population by means of weighted summatio n of the
groups with their means as arguments and the inverse of the variance
as weights. However, this method can only be applied where the gr
oup-ing method is such that the variation in the probability of an
alter-native within the group is sufficiently small. As a consequence , the
method by means of grouping does not become arbitrary, but in fact
places the emphasis on the need for a good classifi cation of the
po-pulation into groups of households.
In addition to the aggregation procedures, there are already other,
frequently fai rly involved and complex, methods which have been
de-veloped. Examples may be found, for instance, in McFadden (1971), Reid
(1978) and Van Lierop and Nijkamp (1979). When applying intra-regi onal
studies, these may perhaps be of interest, but for an inter-regional
study such refinements count for too little compared with the dif
fi-.cul t y and associated cost. For the present study, mos t regional
vari-ables are to be considered as zonal averages. For the individual h
ouse-holds within a given group, these zonal variables, conforming with Theil, are looked upon as average values so that a correction is
de-sirable by means of a disturbance term. In that case , the probability
of a migration of households of group Sg is to be defined as:
p _ exp {V(Xgj i ,Sg) + ngj}
gj/i-
2
exp {V(Xgmi,Sg) + ngm} m
The formulation of the logit model in (4.27) shows a considerable
mea-sure of agreement with the logit model of (4.5) . The difference lies
only in the addition of the disturbance term. This addition makes it
now possible to estimate the parameters in the model by the method of
least squares (van Est and van Setten, 1977 en 1979). Bij relating to
the 9rouP of households g, a considerable measure of agreement also
arises with the original gravity model and the entropy model of (3.2).
22
-observed. In the following chapter, this will be considered more clo -sely.
5. Evaluation
In the previous section, various types of both aggregate as well as disaggregate models have been discussed. Most of the attention has been paid to the entropy and the logi t model. The first one , see equ -ation (3.2) , is an example of an aggregate model andcan becharacterized as a 'top down' approach: a description of macro-states . The entropy model lacks the capacity to take into account indiv i dual behaviour as well as indiv idual circ ums ta nces . The model is des i gnat ed as a gro up-specific multi -nominal distribut ion model, acceptable in the class of aggregate models on methodological grounds. It is also a co-called 'strict' choice model since no stochastic components are included in the choice behaviour. The deviations, as a consequence of specifica-tion errors are incor por at ed into the disturbance term.
The logit model is a model which is directed to indi vi dual observations and individ ual circumstances . By means of a suitab le a priori agg re-gation procedure, it is possible to derive an aggregated model for groups of households showing considerable agreement with the model of (3.2.). In this aggregated logi t model (4.3), the indiv idua l observa -tions of expla natory variab les are exchanged for zonal and group val ues for these factors, when the disturbance term takes care of the correc-tion. The disturbance term in the model of (4.24) then stands for the percept io n error of the exogenous variables . Themodel (4.24) is de-rived from indi vi dual behaviour and aggregated to groups of households, as a result of which it is to be looked upon"as a 'bot t om up' approxi -mation. Since stochastic components are admitted to the choice beha -viour of the households, this model may al so be des igned as a 'ra ndom ' model .
Both models have the same structure, relating to groups of households with response to the same exogenous vari ab1es. However, the int er pre-tation is clearly dif ferent. The difference is expressed in the co n-stant term and the disturbance term. In the case of entropy model, the specification error is accommodated in the disturbance term, E, while
term, a, (cf. 4.4) . The model (3.2) lacks the perception error through the 'strict' characte: of the model , while the perception error is spe-cified, in the case of model (4.24) , in the disturbance term, Q.
The advantages of the aggregated logi t model are evident. In particu-lar, the behavioura l substructu re fits in better with the theoretical conception of migratio n behaviviour so that, from this point of view, choic e of that model is justified .
- 24
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