DURATION DEPENDENCE IN MIGRATION ...

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from participants in these meetings, particularly to Ian McKendrick (NUTIS), and ..... clearly pointed to an initial sharp rise in the chances of considering moving ...
DURATION DEPENDENCE IN MIGRATION BEHAVIOUR: CUMULATIVE INERTIA VERSUS STOCHASTIC CHANGE

Ian Gordon * and

Ian Molho **

* Department of Geography, University of Reading. ** Department of Economics, University of Newcastle-upon-Tyne.

revised draft, July 1994 Final version published in Environment and Planning A, 27, 1961-75, 1995

Acknowledgement Earlier versions of this paper have been presented to a British-Israeli Regional Science meeting and to seminars at Reading University (Economics Department, and NERC Unit for Thematic Information Systems/Applied Statistics Department). We are grateful for helpful comments from participants in these meetings, particularly to Ian McKendrick (NUTIS), and from two anonymous referees.

1. Introduction Duration dependence arises in migration behaviour when people's present migration decisions depend on their history of previous location decisions. A particular hypothesis, familiar to migration scholars, is that of cumulative inertia, based on a perception that over time people build up personal, social and economic attachments to an abode, area and/or job, which discourage further movement. A number of stylised facts lend support to this hypothesis, notably the evidence from many sources that while movement is usually confined to a minority of the population, there are a disproportionate number of repeat movers.

The pioneering work of Myers et al. (1967) stimulated a number of empirical studies, particularly in the 1970s, but these have not provided any very rigorous confirmation of the hypothesis against the alternative explanations which may be offered for these facts (Willis, 1974; Pickles, 1981).

Of these alternative explanations the two most important relate

mobility either to situational factors (such as age, life cycle, and occupation) which change only slowly over time, or to an unobserved (permanent) pre-disposition of individuals to `mover' or `stayer' behaviour (Blumen et al., 1955). Several studies attempting to control for these variables or for other sources of unobserved heterogeneity have actually failed to confirm the existence of a tendency to cumulative inertia (Courgeau, 1973; Plessis-Fraisard, 1975). On the other hand, both Clark and Huff (1977)i and Murphy (1986) have reported clear evidence of duration dependence, but of a form inconsistent with a simple cumulative inertia modelii.

In this paper we provide an alternative theoretical model of duration effects, with evidence to corroborate it, which indicates why they display this more complex pattern, and why earlier studies have provided ambiguous results about the existence of a duration effect. Our analysis indicates that any tendency to cumulative inertia combines with stochastic change in individuals' situation and preferences in ways which produce a non-monotonic relationship between duration and mobility, and confound conventional tests for cumulative inertia. In contrast to the model of cumulative stress/inertia developed by Clark et al. (1979), which provided for a variety of forms of non-monotonicity, our theoretical model (outlined in section

1

2) implies a general tendency for probabilities of movement to first rise and then fall as durations of occupancy increase.

Our empirical work uses information from the 1983 General Household Survey (GHS) on uncompleted durations of residence, and models the answers of a sample of working age male employees to a question on whether they were currently thinking of moving. The approach parallels that of Hughes and McCormick (1985), with 1974 GHS data, in estimating a logit equation including a battery of individual and areal control variables as well as the duration factors in which we are primarily interested. In this study, however, we specify a non-monotonic relationship with duration, and also examine the possible influence of job (as well as residence) duration on movement decisions. The data sources and methodology are discussed in section 3, and the main results are presented in section 4. Our analysis of these data suggests that whilst residential movement displays an apparent tendency toward cumulative inertia, an explicit model of the process leads to clear rejection of that hypothesis.

2. Theory Consider an individual who has just moved into a new place of residence. That person is initially in a chosen position, and hence would appear most unlikely to move again straightaway, under ordinary circumstances. As time passes, however, a number of processes come into play which may alter that situation. These include, on the one hand, the possibility of cumulative inertia as a general behavioural tendency - although, taken together with an initial disposition to stay put, this would seem to rule out mobility altogether. Alternatively, as Speare et al. (1975) and Clark et al. (1979) have suggested there could be a tendency - which they term `cumulative stress' (the converse of cumulative inertia) - for residents to become more dissatisfied with their situation as time passes, perhaps reflecting their progression through the life cycle. This latter hypothesis has been much less widely adopted in the literature on duration effects, perhaps because it is less obviously a function of duration per se.

But there are also likely to be stochastic processes operating, which interact with one or other of these deterministic trends. First, the individual's preferences and situation may change over

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time in unforeseen ways (e.g. they may marry, divorce, get a new job, or ageing itself may alter their values). Secondly the location in which they presently live, or other potential locations, may be subject to change (actual or perceived), and these changes may affect people in different ways, depending on their specific circumstances and values. Both sets of stochastic factors may, in any particular instance, have positive or negative effects on individuals' evaluation of their location relative to others, and hence make them more or less likely to migrate.

We model these processes using a standard human capital framework. For expositional purposes we shall first consider the changing probabilities of movement over time for a stable population, ignoring the compositional changes induced by actual migration. We then extend the model to examine the movement probabilities of the surviving population in each period of time.

The expected utility for an individual a of living in an area i during period t depends on their evaluation of a set of area characteristics k and the probabilities which they attach to each of the possible levels of attainment l for these characteristics:-

where EU(Ri) = the expected utility of the returns from living in i pklT = the subjective probability of characteristic k obtaining at level l in time period T U = a utility function X = 1 when characteristic k obtains at level l (or = 0 otherwise); and the superscripts at denote an evaluation by individual a at time t.

Time point zero we take to be either the time of arrival in the area (in the case of adult in-migrants) or immediately after the time of reaching `adulthood', i.e. freedom to migrate

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independently (in the case of all others). For those living in area i the expected value of staying there over a future lifetime L would then be given by:

where V(Sio) = the present value of the expected utility of staying in i evaluated at time 0, and r = the discount rate. The expected value of moving to another area j may be similarly definediii, with the additional consideration of the various moving costs entailed, both monetary and psychic - including those of replacing place-specific human capital:-

where Cij = movement costs from i to j. Individuals would move from i to j at any time t if:-

But at time point zero this should not be the case (for any j) since residents have just chosen to move to i (in the case of migrants), or not to move away (in the case of others)iv, i.e.:-

As time passes, costs and returns are re-assessed, for the reasons outlined above. Cumulative inertia is likely to induce a negative drift over time in (Vj - Vi), making the current location more attractive - while cumulative stress would have the opposite effect. Other factors described above are essentially random in nature, and hence their combination with inertia/stress yields a random walk with drift:

where αij  0 F(t) is homogeneous, εijt = εijt-1 + eijt ,and e = a random variable.

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With cumulative inertia present (f'(t) < 0), then as time passes the average person becomes more reluctant to move. But the random walk component (ε) means that in every area there are likely to be some people who change their minds about their optimum location as time goes by. This random component is set to zero at time zero, and its mean value remains at zero, but its variance will grow proportionately with the passage of time:Var(εijt) = tσ2

(7)

The proportion of the population who would choose to live elsewhere thus depends on the probability of the differences in equation (6) exceeding zero, i.e. the probability of an individual coming to prefer a move to one of the possible alternative regions over continued residence in i. For this we need to make some assumption about the probability distribution of ε. Given the possibility of independent disturbances occurring in each of a series of very short periods, the normal distribution is a natural assumption, although it does not yield simple analytical solutions for the model.

We can, however, derive qualitative conclusions about its behaviour from consideration of the course of its mean and standard deviation over time. From equations (6) and (7) we may derive the ratio of its mean to its standard deviation as:

This ratio is equivalent to Student's t statistic, with values monotonically related to the probability of movement. Differentiation with respect to time allows us to identify a maximum probability of movement at:αij+f(t) t* = 2 f¢(t)

(2)

which has a positive finite solution, provided that there is a tendency to cumulative inertiav.

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Given this tendency, and a time trend representable as a power function: f(t) = β(t)θ

(3)

where β,θ are parameters β 0, then the turning point is given by: αij 1 t* = (β(2θ-1))θ

(4)

This has real roots, given that β is negative, and in the linear case reduces to: αij t* = β

(5)

At time zero the probability of choosing another location would be zero; after climbing to the maximum value it should decline asymptotically toward zero again. In general the turning point will occur later where the initial relative advantage of the present area (α) was very high, and earlier when relocation costs grow rapidlyvi.

The shortcoming of this model is that it does not remove from the population at risk of movement in any period those who may already have moved away. In effect it assumes that constraints are such that nobody ever succeeds in realising their preference for an alternative location. In general this is likely to be less realistic than the alternative limiting assumption, namely that everybody is free to migrate as soon as they would choose to do so, i.e. at the moment when the value of equation (6) first becomes positive.

In this latter case the probability of movement in any period depends on the distribution of first passage times from the original point of excess satisfaction with the current location (α) to a zero level at which alternative locations become equally attractive. Using Fürth's formula (Feller, 1968, XIV.9 problem 14), the probability density for the waiting time to absorption (i.e. migration) for a simple linear drift is given by:

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Ft P0 =

-α 2 2 e-(α+βt) /(2σ t) 2 3 2π σ t

(6)

where P0 = the original population of a cohort at its time of arrival in the current residence Ft = the flow of members of that cohort into serious consideration of moving in the tth quarter after arrival.

The sum of these probabilities to infinity, representing the long term probability of migration, equals unity if β is non-negative, implying that if there were no tendency to cumulative inertia (and infinite lifetimes) everybody would migrate eventually; given a tendency to cumulative inertia, however,(i.e. a negative value for β) the sum is less than one, implying that even in the very long run the probability of movement is less than unity.

The maximum probability, and hence the highest rate of movement is at: t* =

-3 σ2 + 9 σ4 + 4β2α2 2β2

for β¹0

(7)

or at : t*

α2 = 3σ2

for β = 0

(8)

For a given initial advantage (α) the peak will occur earlier where the stochastic disturbances (σ) are strong and/or where there is a strong positive or negative drift (β) in relative evaluations of the area. Since it is the absolute value of β which is important the peak rate of movement can occur after the same duration of residence given either a tendency to cumulative inertia or its converse, cumulative stress. The difference is that in the former case actual rates of movement will be significantly higher (since these vary monotonically, positively with β).

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The probabilities represented in equation 13 relate to the original population rather than to the survivors at time t. They do not therefore directly indicate the risk (or hazard) of migration for the current population, which cannot be represented by any simple expression. Numerical evaluation with a range of parameter values shows, however, that the turning point for the true hazard function (i.e. the point at which chances of migration are maximised) occurs sometime after the turning point for the overall volume of movement (represented by equation 14), being close to it when there is a clear tendency to cumulative inertia. The other difference is that after this peak, the hazard rate falls away rather less sharply than the level of flows. For estimation purposes it is very inconvenient to make direct use of the hazard function as derived here and a series of approximations have been investigated, with the common characteristic that they rise from zero flows at time zero to a peak before falling back again asymptotically toward zero over longer durations.

3. Data and Methodology These models have been tested empirically using data from the General Household Survey (GHS), which is unique among recent British surveys in providing data on duration of residence for a large sample of the population (covering around 11,000 households, and 30,000 individuals in the case of the 1983 GHS used here). Even with a sample restricted (in the interests of homogeneity and independence) to males aged between 16 and 64 who were in employment at the time of the interview, 4,347 cases were available for analysisvii. A limitation of this data set, however, is that it can only provide data on migration intentions, without a follow-up to check on whether these were realised in practice over some succeeding period. The relevant questions in the survey identified individuals who were "at the moment.... seriously thinking of moving" from the present address. 660 respondents fell into this category of potential migrants, representing 15% of the sample. No time interval was specified for the move to be achieved but data from other sources indicate that the majority would be likely to move within a year, and that almost all will move eventuallyviii. Compared with the eventual flow of movers the stock of potential movers will be somewhat biased towards types of people who, through choice or constraint, take longer to successfully complete their search for an alternative location. The latter group possibly includes more long term residents, in

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which case cumulative inertia effects may be somewhat understated in this analysis. In other respects, however, those stating themselves to be `seriously considering' a move are expected to be representative of actual movers, although the time at which they are recorded as considering it inevitably precedes the actual date of moving.

This migration variable was modelled in terms of a battery of explanatory variables available from the GHS. Most of these factors (discussed below) served as control variables in an analysis aimed at identifying the influence of two key variables, the length of residence in the present dwelling and the length of time in the current job. The latter variable was included in the analysis, despite the fact that most of the potential movers identified in the GHS will not be moving their workplace as well as their residence (in contrast to the longer distance movers considered by Hughes and McCormick, 1981), as the only available indicator of possible attachments to the local area, in which the respondent may have occupied other houses before their present dwelling)ix.

Estimation was carried out in two stages. In the first stage the probability that an individual was seriously thinking of moving (p(M)) was modelled using a standard logit formulation: K

exp(α0+ å αkXk+f(R)+g(J)) p(M) =

k=1

K

(9)

1 + exp(α0+ å αkXk+f(R)+g(J)) k=1

where X represents a vector of control variables, R = residence length up to the date of the interview, and J = length of time in current job.

The duration of residence function implied by the first passage model of equation (13) had to be approximated at this stage by a linear function, the empirically preferred form being: f(R) = η1(R) + η2log(R)

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(10)

In the case of the job duration function, for which there were no clear a priori expectations, an appropriate form was also established by experiment during the empirical workx.

The control variables included a wide range of personal characteristics (age, race, family situation, state of health and educational attainment, social group, type of employment and attitudes to their job) and background factors of relevance to migration decisions (housing quality and tenure, usual earnings, region of residence and neighbourhood type), for which data were available from the GHS. In an attempt to control for `unobserved heterogeneity' in terms of varying attitudes to mobility, information on numbers of moves in the recent past was also included among the independent variablesxi.

Initial investigation of this wide set of variables was undertaken using Ordinary Least Squares (OLS) techniques. Logit regression was then carried out with a more restricted model, and further insignificant effects were deleted, testing down sequentially until a final parsimonious representation was reached. This model was used as the basis for all subsequent analysis. It is notable, however, firstly that OLS and logit models did not differ substantially in terms of signs and significance of coefficients, and secondly that the duration variables of interest in this study remained significant throughout the model selection procedure.

A second stage of the empirical analysis involved fitting a version of the theoretically derived `first passage' model to the duration function calibrated in these logit regressions. For this purpose the stock of individuals `seriously considering' movement at a particular point in time (defined in terms of quarters) was related to the flow of those first considering such a move in each preceding quarter. The number of potential movers at any point in time is represented as a (positive) function of past decisions to consider a move and a (negative) function of actual moves:

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where St = the stock of those seriously considering migration t quarters after arrival, Mq = migration in the qth quarter after arrival, and Fqis as defined for Equation (13).

The rate of actual movement is assumed proportionate to the stock of potential movers one quarter earlier: Mt = μSt-1

(11)

where μ is a parameter, 0 < μ  1.

The population at risk of movement in any period is defined as all those who have not yet moved:

where Pt = the population of the cohort remaining at their residence t quarters after arrival.

The hazard function is then given by: St p(c | t) = P t where p(c|t) = the probability of seriously considering a move, conditional on t quarters of residence.

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(12)

4. Results The final estimated logit model is presented as Model 5 in Table 1 (with coefficients for the set of control variables being reported separately in the Appendixxii). The first point to note from Table 1 is that we do find a highly significant effect of residence duration on mobility, even when all the other influences have been controlled. And, as the theoretical analysis in section 2 predicted, it is far from monotonic in form. Experiments with residence duration bands clearly pointed to an initial sharp rise in the chances of considering moving with increasing durations of residence, followed by a subsequent decline. This pattern was so markedly skewed that a simple quadratic formulation was unable to capture the observed shape. This explains why Hughes and McCormick (1985) were unable to detect any significant non-monotonicity in the residence duration effect in their experiments using a quadratic function. Similarly, the peaked form of the duration function explains why Plessis-Fraisard (1975) was unable to discern a consistent duration of residence effect across a number of duration bands. The best fitting representation of the functional form proved to be the mixed exponential and power function obtained when equation 17 is logged. This involved entering the residence duration variable into the logit model both linearly and in logged form. The fitted relationship has a turning point at (0.6152/0.0894)  7 years (uncannily close to the `seven year itch' which has been reported for some other forms of mobility!). The form of this relationship is consistent with the random walk hypothesis of stochastic variation in areal preferences but no direct conclusions can be drawn at this stage as to any tendency to cumulative inertia.

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Table 1: Location History Effects With and Without Control Variables ________________________________________________________________________ Model 1

Model 5 (Including Controls) ________________________________________________________________________ Constant

- 1.304 (-20.77)

Model 2

Model 3

Model 4

- 1.452 - 1.257 -1.626 -1.516 (-18.86) (-14.92) (-12.46) (- 4.48)

Residence Length

-0.0458 -0.0859 (-7.97) (-6.96)

-0.0803 (-6.45)

-0.0881 (-6.93)

-0.0894 (-6.49)

Residence Length (logged)

0.2960 (3.79)

0.3147 (4.00)

0.4906 (5.24)

0.6152 (6.08)

-0.0286 (-5.08)

-0.0179 (-2.67)

Job Length

-0.0305 (-5.43)

Number of moves in Last Five Years

Likelihood Ratio Test n

0.1746 (3.94)

73.14 4347

88.37 4347

0.0836 (1.71)

121.10

136.20

388.68

4347

4347

4347

(t statistics in parentheses). Notes:1. Models 1 to 4 exclude all other explanatory variables. The results in column 5 are taken from the full model including control variables (see Appendix). 2. The likelihood ratio test compares the relevant model with one which just includes a constant term.

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In the case of the job length variable, no such complex functional relationship was in evidence, but rather a straightforward monotonic relationship, involving the expected negative and significant coefficient on the job length variable. It should be noted, however, that most of the moves planned by respondents to the GHS would not have involved a change of job. Hence neither cumulative inertia nor random walks with respect to their job satisfaction can have much relevance to their movement plans. Rather the effect of job duration can only be in reflecting another aspect of attachment to an area - or else in proxying a general `stayer' rather than `mover' disposition in particular respondents. In fact the overall effect of job length is estimated to be about half that of residence length (evaluated at the mean), and is also clearly of less significance in statistical terms, as one might expect given the more indirect nature of the causal connection. Some tests for interaction effects were performed between the residence and job duration variables, but no significant effects were detected. Also some tests were performed by splitting the sample into four groups (short job/short residence; short job/long residence; long job/short residence; and long job/long residence), but no significant differences between the sub-samples were detected.

The large number of control variables included in the model is intended to capture heterogeneity in the sample that might lead to a spurious correlation between the duration variable and movement intentions. And, as previous studies have suggested, we do find that the simple (monotonic) duration of residence pattern evident in bi-variate analyses is much weakened when variables such as age are controlled for. A further attempt has been made to control for unobserved `mover/stayer' variations within the sample by including in the regressions a variable measuring the number of actual moves in the last 5 years. The estimated coefficient on this variable has the anticipated positive sign, but is only on the margins of significance (t ratio = 1.71, compared to a critical value of 1.65 at the 5% level), suggesting that the control variables have largely done their work.

As we have already indicated, this rather full battery of control variables by no means eliminates the duration of residence effect on mobility - but it does significantly alter its

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apparent form. In Table 1, results based on models which exclude the control variables are compared with the duration effects estimated in their presence. Model 1 includes only a (monotonic) residence duration effect which is negative and highly significant. Model 2 adds in the log (residence duration) variable which attracts the expected positive sign, the coefficients on both variables being significant. Model 3 adds in the job length variable, and Model 4 the `number of moves in the last 5 years' variable. Both these effects are significant. Interestingly, the latter effect is much stronger in the absence of the other control variables than in their presence, as one would expect. In general, the residence duration effects retain their significance in all the versions of the model, but the job duration variable loses some of its significance when the control variables are added in.

Figure 1 plots the relationship between the probability of considering moving and the length of residence. The broken line shows the relationship estimated in the absence of all control variables (i.e., Model 2 from Table 1), whilst the solid line shows the relationship estimated in their presence. The latter exhibits a more gentle increase in the first few years, and a flatter peak after a much longer period of time, before the negative portion of the function takes over. The difference between the two is consistent with the view that excluding control variables leads to a (spuriously) stronger negative relationship between residence durations and movement intentions than in fact exists, as previous authors have argued. However, we have shown that when these controls are included there is actually much clearer evidence of the peaked form of duration function which our theory predicts, and the substantive significance of this model is enhanced, since the initial period of rising mobility is extended well beyond the first year or so.

In order to test the hypothesis of cumulative inertia, however, we have to translate the empirical residence duration function into implied coefficients on the theoretical `first passage' model. This was accomplished in a second stage of the empirical analysis by using a non-linear least squares regression to fit this model to values for the proportion of the population at different duration lengths who would be seriously considering a move, as predicted by the residence duration function estimated in the first stage logit regressions. As compared with the

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raw data used for those regressions this measure of movement probability at specific durations has been averaged, smoothed and controlled for the influence of factors other than duration. The fitted model is represented by equations (13) and (18) to (21). In effect what is estimated is an equation representing the substitution of the first 4 of these equations into (21), with an additive disturbance term; because of the recursive nature of the model it is not very helpful, however, to present this in reduced form. This block of equations contains 4 parameters, one of which (α) may be set arbitrarily to scale the expected utility function; this was set to a value of -100. A second parameter (μ), representing the proportion of the potential migrant stock who actually move in the subsequent period, was set on the basis of independent evidence as to the success rate of intending migrants and the current rates of household movement8; translated into quarterly terms the value adopted for this parameter was 0.167. The other two parameters (β, representing the trend of inertia or stress; and σ2, representing the variance of steps in the random walk) were estimated directly from the non-linear least squares regression.

A good fit was achieved with quarterly observations over a range of durations up to 25 years (with an R2 value of 0.985). Estimated values for the two fitted parameters were: σ2 = 486.9

β = 3.404

Since the dependent variable in this regression is based on smoothed migration probabilities rather than raw data, for an arbitrary number of observations, significance tests are meaningless. However, not only are the t statistics for both parameters very strong (56 and 25 respectively) but if β is constrained to be negative the R2 value falls to 0.582xiii. The positive coefficient for this parameter, indicating a tendency to cumulative stress rather than cumulative inertia as durations of residence increase, thus appears to be substantively significant. In other words there appears to be a systematic tendency, over and above the stochastic variation, for the `fit' between individuals and their homes to deteriorate over time.

Of course, many of those experiencing this growing `mismatch' may resolve it by very local moves, and cumulative inertia may still hold if we consider only those seeking longer distance

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moves out of their current neighbourhood. A possible explanation for the tendency to cumulative stress among those making local, essentially housing-related moves could be the efficiency of the housing market in generating a range of housing within any locality allowing individuals to satisfy their housing needs quite closely on most dimensions of concern (given their budget constraints). If that is the case, then the great majority of conceivable changes in an individual's circumstances - whether increases or decreases in family size, or in income, changes in taste or workplace location - are likely to reduce satisfaction with the current residence, rather than displaying the symmetry assumed by the random walkxiv. For other sorts of locational shift, involving longer distances and motivated by environmental or employment-related factors, this is much less likely to be the case, since few will have the opportunity to match all of their preferences so perfectly in relation either to job or neighbourhood. Longer distance migration may well thus reflect a tendency to cumulative inertia. However, at the level of all residential moves, despite superficial evidence to the contrary, cumulative inertia does not appear to operate, and observed duration effects are essentially attributable to stochastic variation in the relative evaluations of current and alternative residences.

5. Conclusions This paper presents a new theoretical framework and supporting empirical evidence on the relationship between movement probabilities and length of stay. The novelty of the theoretical model consists in recognising that, while longer residence in an area may generally increase the `costs' of any subsequent move, the passage of time will always lead a minority (at least) of the population to re-evaluate their original preferences in favour of some other area, job or house. The argument is that initially the migrant is in a chosen position, and is therefore extremely unlikely to move. It is only after the passage of time, bringing about changes in the individual, the area of residence and/or other areas, that agents are likely to seriously consider moving. Despite the accumulation of local ties which inhibit movement, the `itch' induced for some members of the population by stochastic changes in these variables will then lead to some migration from any area. The overall duration profile of movement depends on the way in which these two tendencies interact at different durations of residence. The specific implication of our model, which incorporates both duration-related processes, is that movement probabilities will initially rise before falling off - and that this pattern will be

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observed irrespective of whether there is, or is not, a tendency to cumulative inertia effects. The only qualification to be made is that in some cases the turning point could be delayed beyond the expected life-time of residents.

The empirical section of the paper tested this model against data on one type of mobility, using survey data on migration intentions, defined so as to include all residential moves. For this data set most of the relevant hypotheses were confirmed. Duration of residence effects were indeed significant, even after controlling for a battery of other influences, but they followed the non-monotonic pattern predicted by the theoretical analysis, rather than the traditional assumption of continuous declines in the disposition to move. However, in this particular case there was no evidence that cumulative inertia played any role in the duration pattern of mobility.

Further work is required to establish the prevalence of such effects in relation to length of residence in a locality rather than just that in a single house, and also for other, or more specific, types of movement. This work should include specific analyses of job-related moves, but also work on job-changers who do not change their place of residence, since the prediction of a non-monotonic duration effect applies to all kinds of movement. If marriages, as well as residential histories, are subject to the hypothesised `7 year itch', these may also appropriately be modelled in terms of the interaction of cumulative inertia and stochastic variation.

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Plessis-Fraisard, M. (1975) `Age, length of residence and the probability of migration', Working Paper 107, Department of Geography, University of Leeds. Speare, A., Goldstein, S. and Frey, W. (1975) Residential Mobility, Migration and Metropolitan Change, Cambridge, Mass: Ballinger. Willis, K. (1974) Problems in Migration Analysis, Farnborough, Hants: Saxon House.

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Notes i. Pickles et al. (1982), who introduce more sophisticated controls on heterogeneity and non-stationarity, have however, questioned the significance of the duration of stay effects in Clark and Huff's data. ii. Morrison's (1967) analysis of Dutch data, controlled only by age, suggested that migration probabilities were more or less constant for the first 2 years of residence, falling thereafter; his fitted model is formally non-monotonic, but the fitted values imply maximum mobility within 1 or 2 months of residence. iii. For simplicity, and since we are only seeking to model the likelihood at particular points in time of moving rather than staying (not the direction of movement), we refer only to a single prospective destination j, with attribute values reflecting the currently most attractive alternative location. iv. This is one key difference from Clark et al's (1979) model, in which the probability of moving (again) remains high immediately after a move has been accomplished; the authors acknowledge this to be the least realistic aspect of their model. v. Otherwise, with a tendency to cumulative stress, probabilities of movement would show continuous increases with longer durations of residence. vi. It may be noted that the former condition is more likely to affect previous migrants than the indigenous population, since migrants demonstrably preferred this to other locations even when they had to incur movement costs to get there; some natives, however, would prefer to be elsewhere were it not for the movement costs (which are included in α). vii. The study was pitched at the individual rather than the household level because household structure is endogenous to migration decisions. The influence of household structure on migration decisions of individuals is captured in this analysis by a series of explanatory variables describing individuals' family circumstances. viii. Data from a 1980 survey of actual and potential migrants indicates that at least 90% of potential movers actually make a move within 5 years, half of them within a year (Gordon et al., 1983). By way of comparison 1981 Census data shows that about 10% of working age males move house each year - a figure which is compatible with the estimated proportion of potential movers accomplishing their move during the following year, given that almost a third of movers only start to consider their move during the year in which it actually occurs. ix. Birthplace, of the individual and/or their parents, in the same house/area, and duration of residence in the area could also have been relevant variables, but were not available from the GHS, while country of birth and ethnic origin, which were available, proved empirically insignificant. x. The job length variable was constructed from survey responses relating to 13 uneven time bands ranging from `less than 4 weeks' to `40 years or more,' using the mid-points of the bands to approximate the modal job length within each band, since this yielded similar but marginally preferable results to alternative approximations. xi. Non-stationarity, the other major problem highlighted by Pickles et al. (1982), is not directly controlled for in this analysis. Since the dependent variable (expressed migration intentions) relates to a single period of time, this is less of an issue than when individuals' migration histories are being analysed; nevertheless, non-stationarity in the parameter (α) representing comparative evaluations of the area at `time zero' (the point of in-migration or adulthood) cannot be ruled out - or tested with this data set. xii. In the interests of space we have omitted any discussion of the results for the control variables, though this is available in an earlier working paper obtainable on request. xiii. Estimates of this parameter are sensitive to assumed values for μ, since the accumulation over time of a stock

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of potential movers depends in part (inversely) on the success of such potential movers in translating intentions into action. Negative estimates for the β parameter (implying a tendency to cumulative inertia) only start to emerge, however, when μ falls below .05, which would imply an annual success rate for potential movers of less than 20%, and an mobility rate less than a third of that actually observed. xiv. This is the implicit assumption of Clark et al's (1979) analysis of intra-urban mobility which suggests that unexpected events and life cycle transitions are likely to cause increasing stress and mismatches with accommodation.

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