Modelling and Forecasting Interregional Migration for Multiregional Population Projections

Abstract

In this paper we focus on modelling and forecasting gross interregional migration in a way that can be embedded within multiregional population projections. We revisit, and apply, a family of spatial interaction models first formulated during the 1970s. The classic gravity model—in which migration is positively related to the populations of sending and receiving areas, but inversely related to various types of spatial friction associated with migrating between them—is a special case that is nested within this family of models. We investigate which member of the family of models gives the best fit when modelling five-year migration flows between the 66 Territorial Authorities (TAs) of Aotearoa New Zealand, using 2013 and 2018 census data. We find that predicting migration between two TAs can be improved by taking into account, firstly, an index of the ‘draw’ from all other TAs when modelling out-migration of any TA and, secondly, an index of the ‘competitiveness’ of a TA vis-à-vis all other TAs when modelling in-migration of any TA. We highlight the properties of the statistically-preferred model by simulating the impact on internal migration of an exogenous increase in Auckland’s population. In this model, such a population change affects not only migration flows from and to Auckland, but also migration between other TAs. The usefulness of this approach for population projections is assessed by forecasting the 2013–18 migration matrix by means of 2013 census data only. In this specific case, the model outperforms the classic gravity model in terms of forecasting gross migration, but not net migration.

Introduction

In this paper we focus on modelling and forecasting gross interregional migration in a way that can be embedded within multiregional population projections. Forecasting future inward and outward migration of regions remains challenging. Levels of immigration and emigration can be particularly volatile, but even internal migration may be difficult to forecast.¹ The standard approach for incorporating migration in regional population projections is to apply the cohort component method, in which a certain level of net migration is assumed in future years for each region. Net migration is then allocated across age-sex groups by means of a pre-determined schedule.² Historical patterns of a region’s net migration can be easily estimated as the difference between the region’s total population change over a certain period (usually the time between two population censuses) and observed natural increase, calculated from vital statistics. However, the problem with this net migration approach in population projections is that the four components of each region’s net migration (immigration, domestic inward migration, emigration, domestic outward migration) may have quite different determinants and trends. Consequently, it is useful to model and project these four components separately, but jointly (see e.g. Alimi et al., 2019). Additionally, knowing how a region’s future net migration is made up of these four components is more insightful for end-users of population and migration statistics than is only knowing the projected level of net migration. As Rogers (1990) pointed out, a ‘net migrant’ does not exist, but immigrants, emigrants, people arriving from other regions, and people moving to other regions are readily identifiable population groups.

A model that has turned out to be remarkably successful for analysing the number of migrants between regions and/or countries is the gravity model (Poot et al., 2016). Similar to Newton’s gravity law in physics, the migrant flow (the equivalent of the attraction force) is positively related to the size of the population in the origin and in the destination (the equivalent of two masses), while inversely related to the distance between the origin and destination. This model has been extensively fitted to both internal and international migration flows around the world. However, when trying to model migrant inflows and outflows of sub-national regions, applying the gravity model to the international flows is going to be more complex that applying the model to the interregional flows. This is because cross-border flows may be more closely related to historical patterns, networks, contemporary policies and economic conditions than to population at the origin and destination, and the distance between them (e.g., Poot, 1995). Beyer et al. (2022) find that, even though gravity models can fit bilateral international migration flows quite well at the global level, these models are unable to capture the dynamics of international migration and are, consequently, not suitable for forecasting. Additionally, data on emigration from regions and immigration into regions, by age, sex and country of destination and origin respectively, are often of relatively low quality, or have to be imputed.

In this paper we therefore revisit the question of how to account for migration in multiregional population projections. We focus on the case of Aotearoa New Zealand, a country with a population of 5.3 million of whom close to 30 percent were born abroad.³ International migration has been incorporated in previous and current work on multiregional population projections in New Zealand in a rather simple way (e.g., Cameron, 2018)—essentially through distributing projected national immigration and emigration across the regions in line with regional population shares, as observed in the census, and by applying estimated regional age-sex schedules.⁴ It is difficult to improve on that because of the volatility of New Zealand’s international migration (e.g. Bedford & Poot, 2010). However, for internal migration we can do better than adopting the conventional approach by taking observable determinants of out-migration and in-migration of regions explicitly into account. In this paper, we introduce a family of gravity models that can provide considerable insight into how interregional migration is likely to change when regional populations change. This knowledge can be used when developing multiregional population projections.

There are many mathematical models that can incorporate the gravity principle, which is—as noted above—that migration flows between two locations are positively related to origin and destination population sizes (which are course just two of the many ‘push’ and ‘pull’ variables that one could consider, but population tends to have the greatest predictive power) and inversely related to some measure of the cost associated with migrating from one location to the other. All models that incorporate this gravity principle can be referred to as a family of spatial interaction models. Using internal migration data from the 2013 and 2018 censuses in New Zealand, we identify in this paper the member of this family of spatial interaction models that performs best in terms of within-sample and out-of-sample forecasting of interregional migration. We argue that this spatial interaction model should therefore be considered for incorporating into population projections. Motivated by the results of this paper, we invite readers who wish to incorporate gross internal migration in population projections to consider the family of spatial interaction models rather than to assume a priori that the classic gravity model will yield the best projections. However, generating a set of multiregional population projections for New Zealand with an embedded model of gross internal migration is beyond the scope of the present paper and will be the focus of a follow-up paper.

The model we use originates from a theory of movements that was developed by William Alonso from the beginning of the 1970s and published in its final form in Alonso (1978).⁵ The idea of applying the model to interregional migration and embedding this migration into a system of multiregional demographic accounts—with the potential of projecting the regional populations—was already present in Alonso (1973). However, independently from Alonso, Sir Alan Wilson formulated a family of spatial interaction models in the late 1960s, summarised in Wilson (1971), that is mathematically very similar to Alonso’s theory. Because the formulation by Alonso (1978) provides the most general mathematical description that can be applied to movements between areas (as well as to any other type of paired data), and includes Wilson’s family of spatial interaction models, we shall refer to the model formulated in this paper as the Alonso model of migration.⁶

There have already been several applications of Alonso’s theory to modelling gross interregional migration flows (see de Vries et al., 2001 and Hua, 2001 for reviews), but the only previous application using New Zealand data is Poot (1986). More recently, Anderson and van Wincoop (2003) developed a gravity model of international trade which has a mathematical structure that resembles the Alonso theory of movements, including the presence of two systemic variables which are referred to in the literature on modelling trade flows as outward and inward multilateral resistance respectively. These are equivalent to Alonso’s draw and competitiveness variables (to be defined below). Anderson (2011) formulated a theoretical model of migration that includes multilateral resistance terms, thereby providing a formal justification for the Alonso model of migration in terms of economic theory.

The remainder of the paper is structured as follows. The next section describes the Alonso model of migration and its mathematical properties. A discussion of how the parameters of this model can be estimated is given in a technical appendix. The third section provides an application of the model to the migration between the 66 Territorial Authorities (TAs) of New Zealand. For this purpose, we use the 2008–2013 and 2013–2018 matrices of gross inter-TA migration obtained from the New Zealand Census of Population and Dwellings of 2013 and 2018 respectively. Having estimated the parameters of the model, we demonstrate the properties of the model, particularly vis-à-vis the classic gravity model, by simulating the impact on interregional migration of an exogenous increase in Auckland’s population of 10 percent at the beginning of an intercensal five-year period. The penultimate section assesses how the Alonso model of migration compares with the classic gravity model in terms of forecasting migration flows within sample and out of sample. The final section provides concluding comments.

The Alonso Model of Migration

We shall use the same notation as Alonso (1978). Assume that there are r regions. ${M}_{ij}$ is the flow of migrants from region i to region j over a given period. Alonso assumes that total out-migration from region i is given by

$$M_{ix} = v_{i} D_{i}^{{\alpha_{i} }}$$

(1)

In this equation, ${M}_{ix}$ is total out-migration from region i. Hence the subscript x indicates that migration is summed over all possible j’s, i.e. $M_{ix} = \sum\nolimits_{j} {M_{ij} }$. Total out-migration from region i is the product of two terms. The first term in Eq. (1) is an index ${v}_{i}$, which measures the combined effect of all factors that ‘push’ people out of i. Hence, we can think of ${v}_{i}$ as being a function of various observable determinants of out-migration, such as the size of the region’s population, income per capita, unemployment, house prices, the climate, etc. (for New Zealand, see Alimi et al., 2019). We do not take these factors, other than population, into account in this paper for two reasons: first, it would be hard to project such variables forward and, second, the gravity model with only population of the sending and receiving regions as origin and destination variables already tends to generate a relatively good fit to the data (Poot et al., 2016). Hence we simply account for the fact that larger populations will generate more out-migrants. Mathematically, we write

$$v_{i} = c_{o} P_{i}^{{\upmu }}$$

(2)

in which ${P}_{i}$ refers to the population of i at the beginning of the period over which migration is measured. ${c}_{o}$ and $\mu$ are parameters to be estimated.

The variable ${D}_{i}$ of the second term in Eq. (1), ${D}_{i}^{{\alpha }_{i}}$, is an index that can be interpreted as the joint ‘draw’ of all potential destinations for migrants who leave i. The index is not directly observed by the available data, but can be calculated as shown below. We can think of ${D}_{i}$ as the ‘proximity-weighted’ relative pull of all other destinations. ${\alpha }_{i}$ is a parameter. Alonso (1978) allows this parameter to vary across origins, but it is clear that such origin-specific parameters cannot be statistically identified unless a long time-series of out-migration is observed. In applications of this model to date, researchers have therefore simply assumed that ${\alpha }_{i}\equiv \alpha$, i.e. the parameter is the same in all regions.

In-migration of region j can be described analogously by

$$M_{xj} = w_{j} C_{j}^{{\beta_{j} }}$$

(3)

in which ${M}_{xj}=\sum_{i}{M}_{ij}$; ${w}_{j}$ is an index which measures the combined effect of all factors that ‘pull’ people to j, and ${C}_{j}$ is the ‘competitiveness’ of destination j in the presence of other potential destinations, which is the ‘proximity-weighted’ relative push from those other destinations. ${\beta }_{j}$ is a parameter that varies with j in the most general case, but in the absence of long time series of ${M}_{xj}$, we shall here also assume that this parameter does not vary across regions, i.e. ${\beta }_{j}\equiv \beta$.

For the index ${w}_{j}$ we assume something similar to Eq. (2), namely that

$$w_{j} = c_{d} P_{j}^{\nu }$$

(4)

in which ${P}_{j}$ refers to the population of j at the beginning of the period over which migration is measured and ${c}_{d}$ and $\nu$ are parameters to be estimated. Together, ${D}_{i}$ and ${C}_{j}$ can be referred to as ‘balancing factors’ (e.g. Wilson, 1971) or ‘multilateral resistances’ (e.g. Anderson, 2011).

To incorporate the gravity property of migration, Alonso (1978) assumes that any specific migration flow ${M}_{ij}$ is, all else remaining the same, proportional to an index ${t}_{ij}$, which represents the ‘ease of migration between i and j’ (and which is the reciprocal of barriers or resistance to migration between i and j). Specifically, ${M}_{ij}$ is assumed to satisfy the following equation:

$$\frac{{M_{ij} }}{{M_{ix} M_{xj} }} = \frac{{t_{ij} }}{{D_{i} C_{j} }}$$

(5)

To measure the index ${t}_{ij}$ it is assumed that

$$t_{ij} = c_{od} F_{ij}^{ - \delta }$$

(6)

where ${F}_{ij}$ refers to a measure of the ‘friction’ of migrating between i and j, such as the straight-line distance between the geographic centroids of i and j (in our empirical analysis, we shall actually consider the effects of several geographical barriers simultaneously). ${c}_{od}$ and $\delta$ are parameters to be estimated.

To complete the model, we must assume that the sum of the number of migrants who leave any location according to Eq. (1) is equal to the sum of the number of migrants who arrive at any location according to Eq. 3, i.e. there is no ‘adding up’ problem. Mathematically, this means that

$$\sum\nolimits_{i} {M_{ix} = } \sum\nolimits_{j} {M_{xj} = } M_{xx}$$

(7)

in which ${M}_{xx}$ refers to the total number of migrants across all regions. Importantly, it can be shown that the model described by Eqs. (1 – 7) is consistent with any pre-specified level of ${M}_{xx}$. The ‘scale’ of migration is a macro-parameter that can be set exogenously. In our application we set ${M}_{xx}$ equal to the total number of migrants that is observed in the country in the period used to estimate the model parameters. In projections we set ${M}_{xx}$ exogenously, for example by assuming that the average propensity to migrate does not change over time, or by assuming that it changes according to a pre-determined function of time. This highlights that the Alonso model of migration can allocate any given number of migrants across origins and destinations, but it does not uniquely determine the aggregate number of migrants ${M}_{xx}$.

If we substitute (1) and (3) into (5) and use that ${\alpha }_{i}\equiv \alpha$ and ${\beta }_{j}\equiv \beta$, we obtain an equation for ${M}_{ij}$ that resembles a single-equation gravity model:

$$M_{ij} = v_{i} D_{i}^{\alpha - 1} w_{j} C_{j}^{\beta - 1} t_{ij}$$

(8)

As noted above, the values of ${D}_{i}$ and ${C}_{j}$ are not directly observed but must be calculated. If we first calculate ${M}_{ix}$ by using (8), we get

$$M_{ix} = \sum\nolimits_{i} {M_{ij} = } \sum\nolimits_{i} {v_{i} } D_{i}^{\alpha - 1} w_{j} C_{j}^{\beta - 1} t_{ij} = M_{ix} \sum\nolimits_{j} {D_{i}^{ - 1} } w_{j} C_{j}^{\beta - 1} t_{ij}$$

(9)

Dividing both sides by ${M}_{ix}$, and rearranging, we get

$$D_{i} = \sum\nolimits_{j} {w_{j} } C_{j}^{\beta - 1} t_{ij} = \sum\nolimits_{j} {M_{xj} } C_{j}^{ - 1} t_{ij}$$

(10)

Similarly, we can derive that

$$C_{j} = \sum\nolimits_{i} {v_{i} } D_{i}^{\alpha - 1} t_{ij} = \sum\nolimits_{i} {M_{ix} } D_{i}^{ - 1} t_{ij}$$

(11)

Equations (10) and (11) together constitute a system of 2r nonlinear equations that express ${D}_{i}$ and ${C}_{j}$ as functions of the indexes ${v}_{i}$, ${w}_{j}$, ${t}_{ij}$ and the parameters $\beta$ and $\alpha$. Alternatively, ${D}_{i}$ and ${C}_{j}$ are simply functions of ${M}_{ix}$, ${M}_{xj}$ and ${t}_{ij}$. Unfortunately, this system of equations does not have a closed form solution. Hence, it must be solved by some iterative approximation algorithm. Additionally, a normalisation is needed because it is easy to see that if $\left\{{D}_{i}^{*},{C}_{j}^{*}\right\}$ is a solution, $\left\{{\lambda D}_{i}^{*},{\frac{1}{\lambda }C}_{j}^{*}\right\}$ is also a solution for any constant $\lambda$.⁷

These somewhat cumbersome properties of the Alonso model of migration have undoubtedly reduced the appeal of this model among applied researchers (see also Poot, 2024). However, mathematical and statistical software, such as MATLAB, R or Stata, make solving Eqs. (10 and 11) straightforward and computationally fast. Moreover, we shall show that the classic gravity model of migration, and Wilson’s spatial interaction models, are special cases of the Alonso model. It is this flexibility and, as we shall see in subsequent sections, the potential to incorporate the best fitting model in a population projections methodology, that gives the Alonso model its appeal.

In calculating the balancing factors ${D}_{i}$ and ${C}_{j}$ iteratively, it is convenient to start with ${C}_{j}^{0}=1$ (${C}_{j}^{0}$ is the starting value of ${C}_{j}$) for all j and substitute these values into the right hand side of Eq. (10) to obtain ${D}_{i}^{1}$, the value of ${D}_{i}$ in the first iteration. We can then obtain ${C}_{j}^{1}$ by substituting ${D}_{i}^{1}$ into Eq. (11). Next, ${D}_{i}^{2}$ is found by substituting ${C}_{j}^{1}$ into Eq. (10) and so on until a solution is found. In practice, the resulting values of ${D}_{i}$ and ${C}_{j}$ usually converge quite quickly (see also Anderson & Yotov, 2010). Poot (1986, footnote 9) found using New Zealand inter-urban migration data that successive approximations to the solution were within the bounds of a very small tolerance in less than 10 iterations. Once a stable solution is found after n iterations, normalisation is achieved by dividing the values of ${C}_{j}^{n}$ by their mean $\overline{{C }^{n}}=\frac{1}{r}\sum_{j}{C}_{j}^{n}$, so that they end up having a mean of 1 (and consequently by multiplying the values of ${D}_{i}^{n}$ by $\overline{{C }^{n}}$).

The Classic Gravity Model

There are several special cases of interest. Consider first $\alpha =1$ and $\beta =1$. Substituting this into Eq. (8), using Eqs. (2), (4) and (6), and defining $\gamma ={c}_{o}{c}_{d}{c}_{od},$ yields

$$M_{ij} = v_{i} w_{j} t_{ij} = \gamma P_{i}^{\mu } P_{j}^{\nu } F_{ij}^{ - \delta }$$

(12)

This is equivalent to the ‘classic’ or ‘unconstrained’ gravity model, in which we can calculate ${M}_{ij}$ without having to know the values of ${M}_{ix}$ and ${M}_{xj}$. The unconstrained gravity model has strong implications for prediction. For example, a change in population ${P}_{i}$ would affect in-migration from all regions j to region i and out-migration from region i to all regions j, but would have no impact on any other migration flows. The strength of the Alonso model is that it allows the possibility that, for example, more in-migration into region i leads to less in-migration into regions that may be considered to be ‘competing’ with i for migrants. Similarly, more migration out of region i may ‘crowd out’ out-migration from other regions. In the standard gravity model, these competition and crowding-out effects are all equal to zero. Using the Alonso model, we can test whether such competition and crowding-out effects are actually present in the data.

The Doubly-Constrained Gravity Model

In the second special case, we assume that $\alpha =0$ and $\beta =0$. This yields what Wilson (1971) refers to as the ‘doubly constrained’ gravity model (referred to as the ‘inelastic’ gravity model by Alonso, 1978), in which ${M}_{ij}$ is calculated by

$$M_{ij} = v_{i} D_{i}^{ - 1} w_{j} C_{j}^{ - 1} t_{ij} = \gamma P_{i}^{\mu } D_{i}^{ - 1} P_{j}^{\nu } C_{j}^{ - 1} F_{ij}^{ - \delta }$$

(13)

after ${D}_{i}$ and ${C}_{j}$ are calculated with (10) and (11). Here, the row sums (total out-migration) ${M}_{ix}$ are now fully determined by ${v}_{i}$ (see Eq. (1) with $\alpha =0$) while the column sums (total in-migration) ${M}_{xj}$ are given by ${w}_{j}$ (Eq. (3) with $\beta =0$). Calculating ${M}_{ij}$ then according to Eq. (13) is essentially a bi-proportional adjustment to ‘fill in’ the migration matrix when row sums and column sums are given and the individual flows ${M}_{ij}$ are positively related to ${t}_{ij}$ (and hence inversely related to ${F}_{ij}$).

Attraction-Constrained Spatial Interaction Models

In the case when $\alpha =1$ and $\beta =0$, the inflow into any given destination is predetermined by ${w}_{j}$ and there are no competition effects from other destinations. Wilson (1971) refers to these models as ‘attraction-constrained’ spatial interaction models, while Alonso (1978) refers to them as ‘pull’ models.

Production-Constrained Spatial Interaction Models

The case when $\alpha =0$ and $\beta =1$ can be referred to as ‘production-constrained’ or ‘push’ models. This special case and the three others defined above are shown in Fig. 1, reproduced from Alonso (1978).

Fig. 1

Special cases of the theory of movements. Source: modified from Alonso (1978, Fig. 9.1). This figure shows that all members of the most general family of spatial interaction models, as formulated by Alonso’s (1978) theory of movements, can be represented by a point in the parameter space ($\alpha$, $\beta$) corresponding to draw and competitiveness effects respectively. The figure highlights the special cases and the names assigned by Alonso to these cases. Alternative names that were introduced by Wilson (1971) are discussed in the text. Alonso (1978) argues that all points in the square are feasible and that a point such as A is empirically plausible. The point labelled ‘1971–76 population model’ represents the estimated parameter values that were obtained when applying the Alonso model to 1971–76 inter-urban migration data in New Zealand and considering sizes of urban populations as the only push and pull variables. Re-estimating the Alonso model of 1971–76 migration with several demographic, economic and climate variables yielded the parameter estimates represented by the point labelled ‘1971–76 all vars model’. The point labelled ‘2013–18 population model’ corresponds to the estimates of the Alonso model of 2013–18 migration reported in Table 1, column (8), using again population as the only push and pull variables

The General Alonso Model of Movements

In the most general case, the values of $\alpha$ and $\beta$ are likely to be somewhere between 0 and 1. In this case, competition and crowding-out effects both matter. Point A in Fig. 1 is a theoretical example suggested by Alonso (1978), without actually considering any migration or other movements data. The point labelled ‘1971–76 population model’ represents the estimated parameter values that were obtained when applying the Alonso model to 1971–76 interurban migration data in New Zealand and considering sizes of urban populations as the only push and pull variables (see Poot, 1986). Re-estimating the Alonso model of 1971–76 migration with several demographic, economic and climate variables yielded the parameter estimates represented by the point labelled ‘1971–76 all vars model’ (see also Poot, 1986). Finally, the point labelled ‘2013–18’ corresponds to the estimates of the Alonso model of 2013–18 migration reported in Table 1, using again population as the only push and pull variables. Comparing the points, it is clear that competition and crowding-out effects are important in the New Zealand context, but internal migration has become more ‘attraction constrained’ (i.e. more like a ‘pull’ model) over time.

Table 1 Parameter estimates of models of inter-TA migration

Substituting Eq. (2) into (1), and Eq. (4) into (3), and recalling Eqs. (8), (10) and (11), and $\gamma ={c}_{o}{c}_{d}{c}_{od}$, the following Alonso model of interregional migration results:

$$D_{i} = \sum\nolimits_{j} {c_{d} } c_{od} P_{j}^{\nu } C_{j}^{\beta - 1} F_{ij}^{ - \delta }$$

(14)

$$C_{j} = \sum\nolimits_{i} {c_{o} } c_{od} P_{i}^{\mu } D_{i}^{\alpha - 1} F_{ij}^{ - \delta }$$

(15)

$$M_{ij} = \gamma P_{i}^{\mu } D_{i}^{\alpha - 1} P_{j}^{\nu } C_{j}^{\beta - 1} F_{ij}^{ - \delta }$$

(16)

Several methods have been suggested in the literature for estimating the parameters of this model with observed gross migration flows ${M}_{ij}$, populations ${P}_{1}$,…, ${P}_{n}$, and observed ‘frictions’ to migration, such as geographical distance, ${F}_{ij}$ (see Hua, 2001).⁸ The estimation method used in this paper is described in an appendix. Once the parameters are known, gross migration can be calculated for any given set of regional populations.⁹

The proportionality constant $\gamma$ has a special role in this. The Alonso model is a cross-sectional migration-allocation model that equates the aggregate supply of migrants ${\sum }_{i}{M}_{ix}$ to the aggregate demand for migrants ${\sum }_{j}{M}_{xj}$ irrespective of what the total number of migrants ${M}_{xx}$ actually is. We can assign a specific value to $\gamma$ that is consistent with a given level of overall mobility ${M}_{xx}$.¹⁰ In estimation, it is reasonable to assume that the estimated aggregate number of migrants equals the observed aggregate number of migrants. When populations change, we can again pre-assign any realistic value to ${M}_{xx}$ and change $\gamma$ accordingly. This allows us to generate more plausible migration projections than would be the case with the standard gravity model. For example, if estimated $\mu =\nu =1$, which is commonly observed in the literature, a doubling of all populations would lead to aggregate migration increasing fourfold in the classic gravity model (Eq. (12) with $\alpha =\beta =1$, i.e. ${M}_{ij}=\gamma {P}_{i}{P}_{j}{{F}_{ij}}^{-\delta }$), whereas a roughly doubling of aggregate migration would be more realistic (because, without further information, it would be reasonable to assume that an individual’s propensity to migrate will not change when the population has only been ‘scaled up’).

Calculating migration over a period by means of inserting the population at the beginning of a period into the Alonso model allows us to project the population of any region i forward, period after period, using the projected levels of births, deaths, immigration and emigration:

$$P_{i} \left( {projected} \right) = P_{i} + M_{xi} - M_{ix} + natural increase + immigration - emigration$$

(17)

A New Zealand Application

In this application, we model the migration between TAs in New Zealand. The data are obtained from the 2013 and 2018 population censuses and yield two five-year inter-TA gross migration matrices: 2008–2013 and 2013–2018.¹¹ There are 66 TAs. Intra-TA residential mobility is ignored. Hence the total number of observed flows in each migration matrix is 66 × 65 = 4,290. The results for the spatial interaction models are reported in Table 1. Because estimation requires taking the natural logarithm of the migration flows, the common problem arises of how to account for cases of zero migration. The use of counts models, such as the Poisson or negative binomial models (e.g. Verbeek, 2004, Sect. 7.3), has become increasingly popular in modelling movements matrices (e.g. Ramos, 2016). The results reported in Table 1 were generated by simply replacing cases of zero migration with migration of one person before estimation, but negative binomial counts models were also estimated for comparison. The latter yielded very similar results in terms of parameter estimates and goodness of fit, and are therefore not included in Table 1. Goodness of fit statistics are included in Table 1. The top half of the table refers to the 2008–2013 period and the bottom half to 2013–2018.

One goodness of fit statistic that is specific to flow matrices is the ‘relative error’ measure RE (Plane, 1982):

$${\text{RE}} = 100\left( {\sum\nolimits_{i = 1}^{r} {\sum\nolimits_{j = 1}^{r} {\frac{{M_{ij} }}{{M_{xx} }}\frac{{\left| {\hat{M}_{ij} - M_{ij} } \right|}}{{M_{ij} }}} } } \right) = \frac{100}{{M_{xx} }}\left( {\sum\nolimits_{i = 1}^{r} {\sum\nolimits_{j = 1}^{r} {\left| {\hat{M}_{ij} - M_{ij} } \right|} } } \right)$$

(18)

This statistic represents the weighted average of the absolute percentage errors made in predicting the individual origin-to-destination migration flows, with the weights equal to the relative sizes of the flows. When predicted total migration ${\widehat{M}}_{xx}$ is equal to observed total migration ${M}_{xx}$, RE can also be interpreted as measuring twice the percentage of migrants who are ‘mis-allocated’ by the model.

Column (1) of Table 1 reports estimates of the classic gravity model, estimated with loglinear OLS and the 2008–2013 data. The coefficients of the origin and destination population are about 0.9 and 1.0 respectively. The spatial friction variable ${F}_{ij}$ is here represented by three variables, each with their own $\delta$. The first is distance with a coefficient of about -0.8. The second is a contiguity dummy variable which takes the value one when origin TA and destination TA are contiguous. Its coefficient is about 1.2. Hence, for a given distance, migration flows are ${e}^{1.2}$=3.3 times as large when the TAs are contiguous than when they are not, holding other variables constant. The third spatial friction variable is a Cook Strait dummy which takes the value one when migration between the origin and destination TA involves crossing Cook Strait between the North Island and South Island of New Zealand. Migration flows crossing Cook Strait are about ${e}^{-0.5}$=0.6, or 60 percent, of intra-island migration flows, holding other variables constant.

As described in the previous section, the classic gravity model of column (1) corresponds with $\alpha =1$ and $\beta =1.$ The adjusted ${R}^{2}$ of this model is about 0.8, which demonstrates the excellent fit of the classic model. The fit can be improved upon further by introducing a range of push and pull factors that drive migration, but the objective of the present study is to only use past origin and destination populations as the input into projecting future gross migration. The correlation coefficient of observed and predicted ${M}_{ij}$ is about 0.75. However, the model predicts about 10 percent more migration than actually occurred (556,794 instead of 506,331).

RE in column (1) suggests an average weighted percentage error in predicting the individual origin-to-destination flows of about 64 percent before scaling, which appears rather high and inconsistent with the high R squared. However, the challenge of accurately predicting each cell in a migration matrix of 4,290 cells – with individual flows varying between 1 and more than 10,000 – is considerable.¹² The prediction errors can be reduced by, firstly, scaling aggregate migration down to make ${\widehat{M}}_{xx}$ equal to ${M}_{xx}$, secondly by introducing fixed effects (FEs) and, thirdly, by replacing the classic gravity model with the Alonso model. Column (8) shows that fitting the Alonso model reduces RE to about 56 percent with the 2013–2018 data, indicating that the model ‘misallocates’ only a little over one quarter of all migrants in the rather large migration matrix with 66 origins and 66 destinations. This error could be undoubtedly reduced further by including demographic, economic, social and other determinants of the migration flows, but it would be challenging to find plausible values for many of these additional determinants in any population projections.

Column (5) shows the identical model as in column (1), but now estimated with 2013–2018 data. The fit of the model is better than with 2008–2013 data. Total predicted migration in column (5) is only 6 percent more than observed migration. The other goodness of fit statistics in column (5) are also notably better than in column (1), with greater adjusted ${R}^{2}$ and correlation, and smaller RE.

To assess the extent to which additional origin- and destination-specific variables could have improved the fit, and as the first step in estimating the Alonso model (as discussed in the appendix), we replace origin and destination population with a full set of origin and destination fixed effects (FEs) in columns (2) and (6).¹³ Given estimation with one cross-section only, the origin and destination population now have to be dropped as variables to avoid perfect collinearity.¹⁴

The introduction of origin and destination FEs leads to greater adjusted ${R}^{2}$, as would be expected given that the FEs soak up a lot of the variation. For 2013–2018, the correlation between predicted and observed ${M}_{ij}$ also improves slightly with introduction of FEs, but declines slightly with the 2008–2013 data. More problematic is the fact that the FEs model increases the relative prediction error RE. Finally, the FEs model predicts about 20 percent more aggregate migration in both periods than actually occurred.

However, the FEs models are helpful in order to calculate unbiased coefficients of the spatial friction variables. These coefficients change compared with the conventional gravity model: the coefficient of distance increases to about -1, while the coefficient of the contiguity dummy halves. The Cook Strait effect remains roughly 0.5. These coefficients are also quite stable over time.

Columns (3) and (7) take the coefficients of the spatial friction variables as given by columns (2) and (6) and then re-estimates the coefficients of the origin and destination population with the 2008–2013 and 2013–2018 data respectively. This is done by transforming the dependent variable from the natural logarithm of migration from TA i to TA j (log M_ij) to y_ij, which is the natural logarithm of ${M}_{ij}{{F}_{ij}}^{\widehat{\delta }}$, as elaborated in the appendix ($\widehat{\delta }$ is represented in Table 1 by 3 coefficients). Finally, columns (4) and (8) show the results of estimating the Alonso model.

In terms of goodness of fit, it is impossible to improve on a cross-sectional gravity model with a full set of origin and/or destination FEs. However, when using the models of columns (2) and (6) for projecting gross migration, information on changes in origin and destination population is redundant. The models of columns (3) and (7) re-introduce origin and destination populations but, by imposing the classic gravity model structure (in which the coefficients of the ‘draw’ effect on out-migration ${D}_{i}$ and the ‘competition’ effect on in-migration ${C}_{j}$ are set to 1), the fit has deteriorated notably. After introducing Alonso’s general model (in which ${D}_{i}$ and ${C}_{j}$ are estimated, as shown in columns (4) and (8), the fit improves again greatly.

Hence the advantage of the Alonso model is that it replaces a black box mechanical method to ‘fill in’ the gross migration matrix through fixed effects with one in which there is an explicit role for change in the size of the TA populations and also for calculation of the theoretically-plausible substitution and crowding-out effects of regional population change by means of the estimated balancing factors. Additionally, we shall see that, in terms of out-of-sample forecasting of gross migration flows, the Alonso model generally outperforms the classic gravity model.

The Alonso model (columns (4) and (8)) yields a correlation with the actual migration flows that is larger than the correlation with the predictions of the ‘black box’ FEs model of columns (2) and (6) (0.80 vs. 0.75 and 0.82 vs. 0.80 respectively). Additionally, after scaling, the weighted relative prediction error RE of the Alonso model is smaller than in the classic gravity model but a little larger than in the FE model (59.7 vs. 57.4 and 55.5 vs. 53.0 respectively). It can be also seen from Table 1 that the Alonso model requires more downscaling of aggregate migration than the FEs model to bring predicted aggregate migration down to observed aggregate migration. Finally, the adjusted R² of the Alonso models are of course also smaller than those of the FEs model, because no observed origin and/or destination variables can improve over a full set of FEs. Nonetheless, the slightly inferior within-sample fit is a price worth paying for a migration model that provides an understanding of how regional populations, and the ‘draw’ and ‘competitiveness’ effects, actually drive the migration process, as well as providing the potential to generate better out-of-sample forecasts.

Columns (4) and (8) in Table 1 show that models in which the coefficients of the balancing factors are much smaller than one fit the data better in many ways that the classic model. These coefficients are also greater than zero, which implies that migration is neither fully ‘supply constrained’ nor fully ‘demand constrained’. The maximum likelihood estimates suggest values for the coefficients of ${D}_{i}$ and ${C}_{j}$ of about 0.6 and 0.5 respectively in column (4) and in column (8).¹⁵ This compares with 0.6 and 1.0 in the corresponding regression of Poot (1986), who modelled the 1971–76 migration of male workers between 24 Main Urban Areas (MUAs) of New Zealand.¹⁶ However, including a range of other variables (such as age structure, income and climate) into the 1971–76 migration model reduced the coefficient of ${C}_{j}$ to 0.7. These point estimates are all shown in Fig. 1. We find that the ‘draw’ effect of the system of urban areas on out-migration is still the same as it was during the 1970s, but the smaller value for the coefficient of ${C}_{j}$ implies that the competition effect on in-migration has become less important, as was already noted in the previous section. Hence internal migration appears to have become more ‘demand-constrained’ in the 2010s than in the 1970s. The likely reason for this is that the New Zealand housing market has become much more constrained by a scarcity of available land for new development in the 2010s as compared with the 1970s, leading to housing shortages and an increased cost of housing in the larger TAs (e.g. Nunns, 2021), which constrains the migration flows to these areas.

Simulating the Impact of Exogenous Population Growth

The way in which migration between the TAs is affected by the draw effect and competition effect is best illustrated by means of a simulation with the best, and most recent, estimates of the Alonso model parameters, as given in column (8) of Table 1. In the simulation we assume an exogenous 10 percent increase in the population of the Auckland TA, for example due to a policy that encourages greater immigration. The populations of the other TAs are assumed to remain unchanged.

There are two stages in doing a simulation with the Alonso model. In the first stage, we account only for the change in the population of Auckland through the estimated coefficients of the origin and destination populations. In the second stage, we account for the change in the ‘draw’ effect on out-migration of all TAs, and the ‘competition’ effect on in-migration of all TAs, that arise even though only Auckland’s population has changed. The technical details are given in the Appendix.

The first stage effect of Auckland’s population change can be simply gauged from the coefficients on the population of the origin TA and the destination TA in column (8). Given that the model is loglinear, the 10 percent population growth will initially increase migration out of Auckland by e^0.952x0.1 = 1.0998, i.e. 9.98 percent, irrespective of the destination TA. Similarly, the migration to Auckland from each of the TAs will increase by e^0.965x0.1 = 1.1013, i.e. 10.13 percent, irrespective of the origin TA. None of the other migration flows change. This is illustrated in the ‘heat map’ of the first stage percentage changes in the bilateral migration flows in Fig. 2.

Fig. 2

Heat map of change in internal migration resulting from 10 percent population growth in Auckland—classic gravity model. The figure shows how each cell in the 66 × 66 inter-TA migration matrix (New Zealand, 2013–2018) changes when the population of Auckland is exogenously increased by 10 percent, using the Alonso model but assuming that the draw and competition variables of all TAs have not yet changed (which is equivalent to simulating with the classic gravity model). In this case, only migration flows to Auckland and from Auckland change, all by about 10 percent

Aggregating the changes in migration in the Auckland row and column of the migration matrix in Fig. 2, and recalling that in the first stage, which is equivalent to the classic gravity model, migration between the other TAs does not change when only Auckland’s population grows, the total number of internal migrants in New Zealand increases from the observed 657,575 in the 2013–2018 period to 685,011. This is an increase of 4.2 percent. We can check the plausibility of this outcome by considering the number of migrants that can be expected when it is assumed that an individual’s propensity to migrate remains unchanged after the population growth of 10 percent in Auckland. Given that the population of the Auckland TA is about one third of that of New Zealand, the expected increase in the total number of migrants under this assumption is 1/3 × 10 percent plus 2/3 × 0 percent, which is 3.3 percent. The total number of migrants would then be 677,302. This is of a similar magnitude as was calculated above with the regression coefficients. Because we have to set the total number of internal migrants in the second stage of the Alonso model exogenously and because it is reasonable to assume that the propensity to migrate may increase somewhat after the population growth in Auckland (particularly if it makes the population more youthful), we assume that the final number of migrants in the second stage is 685,011.

In the Alonso model, the growth in Auckland’s population changes its position in New Zealand’s migration system. This impacts on the ‘draw’ away from each TA and the ‘competitiveness’ of each TA, referred to as the ‘second stage’ of the simulation. The re-calculated values of the balancing factors are reported in Table 2. The draw and competitiveness indexes are highly correlated. They basically both reflect the spatial distribution of New Zealand’s population. In migration models in which ‘pull’ variables and ‘push’ variables measure very different things, the draw and competition indexes could convey very different information and be relatively uncorrelated.

Table 2 Effects of 10 percent population growth in Auckland on NZ internal migration—Alonso model

The competitiveness index has been normalised to have a mean of one. By Auckland being the largest TA in the country, the distance-from-Auckland-weighted out-migration from the other TAs—which defines the competitiveness index value of Auckland (see Eq. (11))—is relatively small (0.41 in Table 2). Similarly, the distance-from-Auckland-weighted in-migration into the other TAs, which determines the draw index (see Eq. (10)) is also relatively small (0.73). The draw and competitiveness effects of Auckland are obviously important for migration from and to the other TAs in the northern half of the North Island, yielding much larger values. Note that a similar explanation can be given for the relatively small value of the draw and competitiveness index values for Christchurch and Wellington, vis-à-vis their adjacent TAs (although the index values are above the means in the latter city). Finally, it is clear that most of the draw and competitiveness effects with small values are found in the South Island, which acts as the population periphery of New Zealand in the internal migration system.

Nijkamp and Poot (1987) calculated—theoretically by means of calculus (p. 381) and by means of simulation (p. 385)—what may ultimately be expected to happen to the migration matrix when the parameters are as we estimated before our simulation. In the case of a gravity model in which the coefficient of the destination population is greater than that of the origin population (which is the case here: 0.965 vs. 0.952, see column (8) of Table 1) and in which the coefficient of the draw effect is greater than of the competitiveness effect (also the case here: 0.622 vs. 0.511, see column (8) of Table 1), population redistribution through migration favours the largest regional population. Auckland, the largest TA in New Zealand, has net outward migration (at a net migration rate (NMR) of -33.5 per 1000 population, see Table 2). Hence, we theoretically predict that its net outward migration would decline following the population shock.

This is indeed confirmed by the simulation. Taking the indirect (second stage) effects of the 10 percent increase in Auckland’s population size into account, Table 2 shows that the final change in Auckland’s out-migration is 10.36 percent (up from 9.98 percent initially) while the final change in in-migration is 12.85 percent (up from 10.13 percent initially). Other TAs, particularly those relatively close to Auckland, send more migrants to Auckland, which implies a substitution away from other potential destinations. This can be clearly seen in Fig. 3, which shows the heat map of the final percentage changes. Shades of red signal a decline in migration and shades of blue an increase, with a large change corresponding to a darker colour. Comparing Figs. 2 and 3, we see that the migration to and from Auckland has increased (and in most cases ultimately more than directly), but migration between the other TAs has often decreased, particularly in the TAs that are geographically relatively close to Auckland. However, the light blue cells, particularly in the southern half of the North Island, show that there are some ‘non-Auckland’ migration flows that increase following the increase in Auckland’s population.

Fig. 3

Heat map of change in internal migration resulting from 10 percent population growth in Auckland—Alonso model. The figure shows how each cell in the 66 × 66 inter-TA migration matrix (New Zealand, 2013–2018) changes when the population of Auckland is exogenously increased by 10 percent, using the Alonso model with the parameters of column (8) in Table 1. A blue cell signals an increase in migration and a red cell a decrease. A darker colour implies a larger change. All cells in the migration matrix are affected by the change in Auckland’s population, because this changes the relative draw out of all TAs and the relative competitiveness of all TAs in the migration system. Out-migration from Auckland and in-migration into Auckland now change more than in Fig. 2, by 10.36 percent and 12.85 percent respectively (see Table 2)

The heat map of Fig. 3 suggests an interesting departure—at least in percentage change terms—from Tobler’s first law of geography, in which he states that spatially everything is related to everything else, but nearby locations are more related to each other than locations that are further apart (e.g. Tobler, 1970). On the one hand, the greatest changes (positive or negative) in migration occur in the TAs near Auckland, where the population shock took place. On the other hand, relatively large increases (blue) are also observed in migration between TAs in the lower half of the North Island while relatively large decreases (red) occur in migration between TAs in the South Island. Hence, because Auckland is so important in the overall migration system, even TAs that are relatively far away may be considerably impacted by population change in Auckland.

The final two columns of Table 2 show that the simulated initial change in a TA’s NMR following the increase in Auckland’s population is much smaller than the simulated final change. In most cases, the NMR is becoming more negative as Auckland absorbs more migrants. However, Auckland’s NMR (which is -33.5 per 1000 population between 2013 and 2018) initially becomes less negative by 1.26 per 1000 population, and ultimately by 3.57 per 1000 population. In most TAs, except in the far south, the NMR declines because the percentage change in out-migration is larger than the percentage change in in-migration.

Many other simulations could be considered. As a general rule, the smaller the population that changes, the smaller the effect on the gross migration matrix (see also Poot, 1986, Table 3, which compares the impact of strong economic growth in Gisborne, a small regional city, with that of a public sector cutback in Wellington, the capital city). In the case of a population projection, the most recent TA populations can be inserted in Eq. (16) to project the internal migration in the following five-year period. An adjustment is then made in which all flows are scaled to an aggregate level of mobility ${\widehat{M}}_{xx}$ that is considered appropriate for that period. Next, the balancing factors are recalculated with Eqs. (10) and (11). The projected migration matrix that incorporates the indirect effects (i.e. the change in the balancing factors) is then recalculated with Eq. (16). Finally, the flows are scaled once more so that they add again to ${\widehat{M}}_{xx}$. This process can be repeated with projected TA populations in subsequent periods, but the assumption that the estimated parameters remain constant over time becomes less tenable the further forward the projections go. However, this caveat does not only apply to the Alonso model, but also holds for the simple gravity model.

Table 3 Comparing forecast errors of the classic gravity model and the Alonso model of migration

The Performance of the Alonso Model When Generating Interregional Migration Forecasts

The ultimate test of whether the Alonso model can be usefully embedded in a multiregional population projections methodology is to assess how the model performs in out-of-sample forecasting. For this, we assume that we only have the 2013 census data available and we forecast the 2013–2018 gross migration matrix. We compare two different forecasting models:

• 1. The classic gravity model without FEs. The parameter estimates can be found in Table 1, column (1).

• 2. The Alonso model in which the coefficient of the draw effect and of the competitiveness effect are set at their optimal values, as determined by the iterative maximum likelihood method described in the appendix. The parameter estimates can be found in Table 1, column (4).

No model with FEs is included in this comparison, even though Table 1 shows that these perform better within sample. FEs models are less desirable when they are estimated cross-sectionally, as we have done in this paper, because cross-sectional FEs models do not allow for information about population change in each of the TAs to be incorporated into the forecasts.

The results are reported in Table 3. For comparison, we provide the within-sample forecasts in the top half of the Table 3. Hence the RE values reported in the top half of Table 3 are those of the corresponding columns of Table 1. However, we now also include two additional forecasting error measures, namely the mean algebraic percentage error (MALPE) and the mean absolute percentage error (MAPE). These are calculated as follows:

$${\text{MALPE }} M_{ij} = \frac{100}{{r\left( {r - 1} \right)}}\left( {\sum\nolimits_{i = 1}^{r} {\sum\nolimits_{j = 1}^{r} {\frac{{\hat{M}_{ij} - M_{ij} }}{{M_{ij} }}} } } \right)$$

(19)

$${\text{MAPE }} M_{ij} = \frac{100}{{r\left( {r - 1} \right)}}\left( {\sum\nolimits_{i = 1}^{r} {\sum\nolimits_{j = 1}^{r} {\frac{{\left| {\hat{M}_{ij} - M_{ij} } \right|}}{{M_{ij} }}} } } \right)$$

(20)

Additionally, we calculate the MALPE and MAPE of total out-migration of TA i, ${M}_{ix}$, and of total in-migration of TA j, ${M}_{xj}$, in the same way as in Eqs. (19 and 20) above. We also recalculate these measures by weighting each percentage error with the population share ${P}_{i}/{P}_{x}$. Hence for ${M}_{ix}$ we calculate

$${\text{WMALPE }} M_{ix} = \frac{100}{r}\left( {\sum\nolimits_{i = 1}^{r} {\frac{{P_{i} }}{{P_{x} }}\frac{{\hat{M}_{ix} - M_{ix} }}{{M_{ix} }}} } \right)$$

(21)

$${\text{WMAPE }} M_{ix} = \frac{100}{r}\left( {\sum\nolimits_{i = 1}^{r} {\frac{{P_{i} }}{{P_{x} }}\frac{{\left| {\hat{M}_{ix} - M_{ix} } \right|}}{{M_{ix} }}} } \right)$$

(22)

and analogously for in-migration ${M}_{xj}$. Finally, we calculate the average difference (and absolute difference) between the forecast of the net migration rate $\left({\widehat{M}}_{xi}-{\widehat{M}}_{ix}\right)/{P}_{i}$ and the actual net migration rate $\left({M}_{xi}-{M}_{ix}\right)/{P}_{i}$, i.e.:

$${\text{MALDE }} \left( {M_{xi} - M_{ix} } \right)/P_{i} = \frac{100}{r}\left( {\sum\nolimits_{i = 1}^{r} {\frac{{\hat{M}_{xi} - \hat{M}_{ix} }}{{P_{i} }} - \frac{{M_{xi} - M_{ix} }}{{P_{i} }}} } \right)$$

(23)

$${\text{MADE }} \left( {M_{xi} - M_{ix} } \right)/P_{i} = \frac{100}{r}\left( {\sum\nolimits_{i = 1}^{r} {\left| {\frac{{\hat{M}_{xi} - \hat{M}_{ix} }}{{P_{i} }} - \frac{{M_{xi} - M_{ix} }}{{P_{i} }}} \right|} } \right)$$

(24)

and the population-weighted equivalents corresponding to Eqs. (23) and (24). We note that $\text{WMALDE } \left({M}_{xi}-{M}_{ix}\right)/{P}_{i}$ = 0, because $\sum_{i=1}^{r}\left({M}_{xi}-{M}_{ix}\right)=\sum_{i=1}^{r}\left({\widehat{M}}_{xi}-{\widehat{M}}_{ix}\right)=0$ (total internal out-migration in a country equals total internal in-migration).

For both within-sample forecasting and out-of-sample forecasting we set the level of aggregate migration exogenously. For within-sample forecasting, we set predicted total migration 2008–2013 to be equal to the observed total migration: 506,331. For out-of-sample forecasting, we have to make an assumption about how much internal migration might have been expected between 2013 and 2018 based on the information available in 2013. Without any further information, the simplest assumption would be to assume that the propensity to migrate remained the same over the 2013–2018 period as over the 2008–2013 period. Hence the percentage increase in aggregate migration is then equal to the expected percentage increase in population between 2013 and 2018. The population increased between 2008 and 2013 by 4.3 percent. Hence, assuming that this rate of growth would continue over the following five years, the forecast of total migration between 2013 and 2018 is 528,050.

In retrospect, we now know that the NZ resident population increased by 10.8 percent between 2013 and 2018 and that total intercensal migration increased by even more, 29.9 percent. The 2008–2013 migration matrix was strongly affected by the 2007–2009 Global Financial Crisis and the resulting recession, which lowered mobility. In contrast, the 2013–2018 period was characterised by high levels of immigration leading to strong population growth and buoyant economic conditions (NZPC, 2021).

Despite the inability to incorporate these unforeseen out-of-sample trends in the migration forecasts, the out-of-sample forecasting errors are not hugely larger than the corresponding ones within sample. Interestingly, for the individual flows ${M}_{ij}$ the out-of-sample average forecasting errors are actually smaller than the within-sample forecasting errors, which is the opposite of what we would have expected. The usual expectation is confirmed for the forecasting errors of out-migration, in-migration and net migration, where the out-of-sample errors are larger than within sample (except for WMALPE ${M}_{xj}$ in the classic gravity model, and WMAPE ${M}_{ix}$ and WMALPE ${M}_{xj}$ in the Alonso model).

Table 3 shows that, as expected, it is harder to predict the individual gross migration flows ${M}_{ij}$ than total out-migration ${M}_{ix}$ or total in-migration ${M}_{xj}$. The MAPEs are somewhat more informative about this than the MALPEs, given that the latter can be strongly influenced by a small number of large under- or over-predictions. The out-of-sample MALPE of ${M}_{ij}$, -9 percent, is impressively small with the Alonso model, but the MAPE of ${M}_{ij}$ is 72.2 percent with this model, showing that many individual flows are quite badly predicted. However, the classic gravity model is even worse in this respect (81.2 percent). Out of sample, RE of the Alonso model is slightly larger than RE of the classic gravity model (57.4 percent versus 56.3 percent respectively), but MALPE ${M}_{ij}$ and MAPE ${M}_{ij}$ are both smaller with the Alonso model.

Population-weighted out-of-sample forecast errors are in all but one case smaller than unweighted forecast errors (the exception is WMAPE ${M}_{ij}$). In terms of forecasting gross migration, the Alonso model outperforms the classic gravity model in 7 out the 11 possible comparisons. This is the case both within-sample and out-of-sample. However, when considering net migration, the classic model outperforms the Alonso model in all three valid comparisons – both within-sample and out-of-sample (as noted above, $\text{WMALDE } \left({M}_{xi}-{M}_{ix}\right)/{P}_{i}$ equals zero in all possible models). It remains possible of course that more accurate forecasting of gross migration by means of the Alonso model in other case studies (for example by including additional predictors) may contribute to this model also generating better forecasts of net migration. This is yet to be investigated with different data and model specifications.

Concluding Comments

In this paper, we focused on modelling and forecasting gross interregional migration in a way that can be embedded within multiregional population projections. We find—as in the classic gravity model of migration—that there is a positive relationship between migration and the populations of sending and receiving Territorial Authorities (TAs) in New Zealand, while inter-TA migration is inversely related to various types of spatial friction between the TAs. The estimated parameters are quite stable over the two considered periods: 2008–2013 and 2013–2018. Additionally, we find that modelling, and forecasting, the migration between two TAs can be improved by taking into account, firstly, an index of the ‘draw’ from the other TAs when modelling out-migration and, secondly, an index of the ‘competitiveness’ of a TA vis-à-vis the other TAs when modelling in-migration. The introduction of these two variables yields a migration system in which all flows are interdependent and are sensitive to population change in any of the TAs, and particularly the larger ones. In a specific simulation of the impact on 2013–2018 internal migration of growth of Auckland’s population, we find that such a population shock affects not only the migration flows from and to Auckland, but also the migration flows between other TAs. When forecasting the 2013–2018 migration matrix by means of only 2013 census data, we find that our preferred model outperforms the classic gravity model in terms of the majority of measures of forecast errors of gross migration. However, in terms of forecasting net migration, the classic model performs better.

Clearly, accurately forecasting a large internal migration matrix remains a challenge. Even using our best model more than one quarter of the migrants are ‘misallocated’ and the mean absolute percentage errors (MAPEs) of forecasting a TA’s out-migration and in-migration out of sample are about 28 percent and 42 percent respectively. However, there are several ways in which this modelling and forecasting of interregional migration can be improved upon in future work, hopefully leading to smaller forecast errors. The first of these is to pool additional five-year inter-TA migration matrices in the estimation of the Alonso model, such that it becomes possible to estimate and incorporate stable cross-sectional fixed effects (FEs), jointly with the impact of changing populations over time.

Sadly, as was the case in the 2018 census, the 2023 New Zealand census includes only a question on address one year ago and not five years ago. Consequently, the 2018–2023 migration matrix may again have to be imputed by linking a person’s usual address at the two successive censuses. The quality of the data will also depend on census response rates, which have been declining in New Zealand (e.g. Kukutai & Cormack, 2018). Nonetheless, the methodology described in this paper can be applied to any source of migration data, including administrative data—which are becoming increasingly common. Using pooled data will best suit a migration system where the push and pull factors between areas are stable over time. Where large structural changes are occurring, this approach may lead to more biased forecasts.

Besides adding origin TA and destination TA FEs to the forecasting model, it is possible that the forecasts will also improve by taking account of the changing age structure of each of the TAs. Poot (1986) found that the out-migration from an urban area ${M}_{ix}$ is negatively related to the percentage of the population aged 40 and over. Such age-composition variables are of course readily available in population projections. Additionally, it is relatively straightforward to formulate and apply a probabilistic/stochastic extension of the internal migration modelling described in this paper. This could take the form of successive draws of values of the key parameters from plausible intervals. Such stochastic projections could even incorporate deterministic trends such as an increase in the extent to which inward migration into a TA is demand-constrained, as shown by the declining value of the estimated parameter $\beta$ seen in Fig. 1.

We have argued that there is considerable benefit in producing modelled forecasts of gross internal (and international) migration as part of a population projection methodology, rather than simply assuming certain levels of net migration up to the forecasting horizon. However, whether our endogenously-derived forecasts of migration by means of spatial interaction models may yield better population projections than the conventional ones remains an open question. A useful exercise would be to project the 2023 TA population with conventional assumptions about net migration and compare the results with those derived with the Alonso model of gross migration, in both cases using only information up to, and including, the 2018 census.¹⁷ Finally, similar to comparing the effect of spatial frictions on internal migration across countries around the world (as in Stillwell et al., 2016), it would also be interesting and worthwhile to estimate—and compare—Alonso’s draw and competition variables and their coefficients in modelling gross internal migration in countries other than New Zealand.

Appendix

Estimating the Parameters of the Alonso Model

Several steps are needed to estimate the parameters of the model. There are several variations in these steps found in the literature (reviewed in Hua, 2001, and de Vries, 2015). Here we adopt a relatively straightforward procedure that yielded plausible estimates in Poot (1986), as well as in the current context.

Step (1) is the estimation of the parameter δ (or the set of parameters that correspond to all the friction variables, such as ${\delta }_{1}$, ${\delta }_{2}$ and ${\delta }_{3}$ in Table 1). The early literature suggested several ways to remove from Eq. (16) variables that vary only by i or by j, but not by both, through a transformation of ${M}_{ij}$ before estimation.¹⁸ However, a simpler way – which is adopted here – is to initially treat ${{P}_{i}}^{\mu }{D}_{i}^{\alpha -1}$ and ${{P}_{j}}^{\nu }{C}_{j}^{\beta -1}$ as origin and destination fixed effects (FEs) respectively. It is then straightforward to estimate $\delta$ in Eq. (16) by Ordinary Least Squares (OLS), after taking the natural logarithm of both sides (as shown in columns (2) and (6) of Table 1).

Having obtained the estimate $\widehat{\delta }$, step (2) consists of finding the ‘first stage’ solution values of the system variables ${D}_{i}$ and ${C}_{j}$ by solving (25) and (26) below, using the iterative procedure discussed earlier in the paper.

$$\hat{D}_{i} \left( 1 \right) = \sum\nolimits_{j} {M_{xj} } \left( {\hat{C}_{j} \left( 1 \right)} \right)^{ - 1} F_{ij}^{{ - \hat{\delta }}}$$

(25)

$$\hat{C}_{j} \left( 1 \right) = \sum\nolimits_{i} {M_{xj} } \left( {\hat{D}_{i} \left( 1 \right)} \right)^{ - 1} F_{ij}^{{ - \hat{\delta }}}$$

(26)

Note that we apply a normalisation in which the average value of ${\widehat{C}}_{j}$ is equal to 1.

In step (3), we substitute ${\widehat{D}}_{i}(1)$ and ${\widehat{C}}_{j}(1)$ into (16), and re-arrange the terms to yield:

$$M_{ij} F_{ij}^{{\hat{\delta }}} = \gamma P_{i}^{\mu } \left( {\hat{D}_{i} \left( 1 \right)} \right)^{\alpha - 1} P_{j}^{\nu } \left( {\hat{C}_{j} \left( 1 \right)} \right)^{\beta - 1}$$

(27)

We find estimates of $\gamma,\mu$,$\alpha$, $\nu$ and $\beta$ in Eq. (27) by OLS regression, after taking the natural logarithm of both sides.

In step (4), we calculate the first-stage predicted values of migration from the estimated Eq. (27):

$$\hat{M}_{ij} \left( 1 \right) = \hat{\gamma }P_{i}^{{\widehat{{}}\left( 1 \right)}} \left( {\hat{D}_{i} \left( 1 \right)} \right)^{{\hat{\alpha }\left( 1 \right) - 1}} P_{j}^{{\widehat{{}}\left( 1 \right)}} \left( {\hat{C}_{j} \left( 1 \right)} \right)^{{\hat{\beta }\left( 1 \right) - 1}} F_{ij}^{{ - \hat{\delta }}}$$

(28)

but we rescale aggregate migration by setting $\widehat{\gamma }$ such that ${\widehat{M}}_{xx}\left(1\right)={M}_{xx}$.

We then repeat step (2) and re-calculate the balancing factors by replacing the actual migration flows with the predicted migration flows from Eq. (28):

$$\hat{D}_{i} \left( 2 \right) = \sum\nolimits_{j} {\hat{M}_{xj} } \left( 1 \right)\left( {\hat{C}_{j} \left( 2 \right)} \right)^{ - 1} F_{ij}^{{ - \hat{\delta }}}$$

(29)

$$\hat{C}_{j} \left( 2 \right) = \sum\nolimits_{i} {\hat{M}_{ix} } \left( 1 \right)\left( {\hat{D}_{i} \left( 2 \right)} \right)^{ - 1} F_{ij}^{{ - \hat{\delta }}}$$

(30)

We then repeat step (3) by re-estimating the regression model (27), now with the new values ${\widehat{D}}_{i}(2)$ and ${\widehat{C}}_{j}(2)$ replacing ${\widehat{D}}_{i}(1)$ and ${\widehat{C}}_{j}(1)$. This yields the second stage parameter estimates and predictions of the migration flows:

$$\hat{M}_{ij} \left( 2 \right) = \hat{\gamma }P_{i}^{{\widehat{\mu }\left( 2 \right)}} \left( {\hat{D}_{i} \left( 2 \right)} \right)^{{\hat{\alpha }\left( 2 \right) - 1}} P_{j}^{{\widehat{\nu }\left( 2 \right)}} \left( {\hat{C}_{j} \left( 2 \right)} \right)^{{\hat{\beta }\left( 2 \right) - 1}} F_{ij}^{{ - \hat{\delta }}}$$

(31)

Here we again modify $\widehat{\gamma }$, such that after scaling ${\widehat{M}}_{xx}\left(2\right)={M}_{xx}$.

While we can repeat steps (3) and (4) as many times as needed to reach the best fit (e.g. minimum Root Mean Squared Error, RMSE), we find that in practice two iterations are sufficient to reach this goal.¹⁹

Having obtained the best estimates of all parameters, it is straightforward to test whether the migration system can be described by one of the special cases discussed in this paper and shown in Fig. 1. This involves statistical tests of whether the values of $\alpha$ and $\beta$ corresponding to these special cases are within the 95 percent confidence intervals of the parameter estimates.

Simulation with the Alonso Model

To calculate the effect of a change in population, for simulation or for population projection, involves again several steps. Replacing the old population vector ${P}_{1}$,…, ${P}_{r}$ by a new vector ${P}_{1}^{*}$,…, ${P}_{r}^{*}$, the first step is to substitute these populations in Eq. (16) to yield new ‘first stage’ migration levels:

$$M_{ij}^{*} \left( 1 \right) = \hat{\gamma }P_{i}^{{*}{\widehat{\mu }\left( 2 \right)}} \left( {\hat{D}_{i} \left( 2 \right)} \right)^{{\hat{\alpha }\left( 2 \right) - 1}} P_{j}^{{*}{\widehat{\nu }\left( 2 \right)}} \left( {\hat{C}_{j} \left( 2 \right)} \right)^{{\hat{\beta }\left( 2 \right) - 1}} F_{ij}^{{ - \hat{\delta }}}$$

(32)

Given the loglinear structure of the model, it is clear that the predicted percentage change in ${M}_{ij}$ will be in this first step simply $\hat{\mu }$ $(2)$ times the percentage change in ${P}_{i}$ plus $\hat{v}$ $(2)$ times the percentage change in ${P}_{j}$, identical to what it would be in the standard gravity model. In the second step we adjust $\widehat{\gamma }$, if necessary, to ensure that the aggregate level of internal migration is realistic (in terms of overall population mobility) at the new population levels. In the third step we recalculate the balancing factors, using Eqs. (10 and 11):

$$D_{i}^{*} \left( 1 \right) = \sum\nolimits_{j} {M_{xj}^{*} } \left( 1 \right)\left( {C_{j}^{*} \left( 1 \right)} \right)^{ - 1} F_{ij}^{{ - \hat{\delta }}}$$

(33)

$$C_{j}^{*} \left( 1 \right) = \sum\nolimits_{i} {M_{ix}^{*} } \left( 1 \right)\left( {D_{i}^{*} \left( 1 \right)} \right)^{ - 1} F_{ij}^{{ - \hat{\delta }}}$$

(34)

As before, the balancing factors are normalised such that the mean of ${C}_{j}^{*}(1)$ is equal to one. Finally, in the fourth step, we recalculate the migration flows that result when we take into account that the ‘draw’ of migrants and the ‘competitiveness’ for migrants has changed in each region:

$$M_{ij}^{*} \left( 2 \right) = \hat{\gamma }P_{i}^{{*}{\widehat{\mu }\left( 2 \right)}} \left( {D_{i}^{*} \left( 1 \right)} \right)^{{\hat{\alpha }\left( 2 \right) - 1}} P_{j}^{{*}{\widehat{\nu }\left( 2 \right)}} \left( {C_{j}^{*} \left( 1 \right)} \right)^{{\hat{\beta }\left( 2 \right) - 1}} F_{ij}^{{ - \hat{\delta }}}$$

(35)

We adjust $\widehat{\gamma }$ such that aggregate migration remains the same as before, i.e. ${M}_{xx}^{*}\left(2\right)={M}_{xx}^{*}(1)$. While we could in principle iterate with Eqs. (33) to (35) further to generate ${D}_{i}^{*}\left(k\right),$ ${C}_{j}^{*}(k)$, ${M}_{xx}^{*}\left(k+1\right)$, the resulting changes in migration are likely to be very small compared to the difference between migration as calculated in Eq. (32) (where the balancing factors have not changed yet) and migration as calculated in Eq. (35) (where the balancing factors have been recalculated). Hence, we assess the total impact of simulated population change as being equal to the sum of the direct effect $\left({M}_{ij}^{*}\left(1\right)-{\widehat{M}}_{ij}(2)\right)$, which does not take the change in the balancing factors into account, and the indirect effect due to the changed balancing factors $\left({M}_{ij}^{*}\left(2\right)-{M}_{ij}^{*}\left(1\right)\right)$. Note that ${\widehat{M}}_{ij}(2)$ represents here the pre-population change modelled/benchmark level of migration, as calculated in Eq. (31), and not the observed migration ${M}_{ij}$.