An Alternative to Fixed Transition Probabilities for the Projection of Interprovincial Migration in Canada

Abstract

Internal migration plays a critical role in subnational population projections. The multiregional model is often seen as a gold standard, for its capacity to project several interconnected regions simultaneously and coherently. However, undesirable effects may occur when assumptions of constant transition probabilities are used. This paper investigates these limits, explores a few solutions provided in the literature and describes the alternative methodology used by Statistics Canada in its most recent edition of population projections for the Canadian provinces and territories. Among other things, the new method is shown to improve the consistency between internal migration assumptions and results and to facilitate the projection of the uncertainty associated with this component.

Appendices

Annex 1

Tracking Changes in Net Migration Rates in the Context of Fixed Migration Intensities

We can track simply the evolution of the NMR in the multiregional model, knowing that it is equal to the in-migration rate minus the out-migration rate, and assuming stable populations to remove effects of the age structure on these aggregated rates. In this context, the sum of all age- and sex-specific out-migration rates from a region i, $\sum\nolimits_{asj} {m_{askj}^{t,t + 1} } = \sum\nolimits_{j} {m_{kj}^{t,t + 1} } .$

Holding origin- and destination-specific out-migration rates constant through a projection implies that the total out-migration rate from a region k towards all other regions of the system, $\sum\nolimits_{j} {m_{kj}^{t,t + 1} } ,$ is also time invariant, and therefore equals $m_{kj}^{\text{ref}} .$ In contrast, region k’s in-migration rate for flows originating from a region i, $imr_{ik}^{t,t + 1} ,$ is not a parameter of the projection. It is rather the result of the out-migration rates of the other regions of the system and their population sizes, and so is the total in-migration rate for region k, $\sum\nolimits_{i} {imr_{ik}^{t,t + 1} } ,$, as Eq. 1 shows:

$$\mathop \sum \limits_{i} imr_{ik}^{t,t + 1} = \mathop \sum \limits_{i \ne k} \left( {P_{i}^{t} m_{ik}^{\text{ref}} } \right)/P_{k}^{t} .$$

(12)

Two observations can be made at this point: (1) when population sizes evolve during the projections, the in-migration rates evolve too; and (2) since the out-migration rates remain constant, the variations in k’s NMRs will be solely the consequences of changes in the projected in-migration rates.

Equations 13 and 14 track how a change in the population size of one of the region of the system affect k’s projected NMR (nmr _k ), that is, in regards to variations in sizes of one region other than k, here region x (Eq. 13), or in regards to variations in k’s own population size (Eq. 14):

$$\frac{\partial }{{\partial P_{x} }}{\text{nmr}}_{k} = \frac{\partial }{{\partial P_{x} }}\left(\mathop \sum \limits_{i \ne x,k} \left( {P_{i}^{ref} m_{ik}} \right)+\left({P_{x} m_{xk}}\right)\right)/P_{k}^{ref} - \frac{\partial }{{\partial P_{x} }}\mathop \sum \limits_{j \ne k} m_{kj} = \frac{{m_{xk} }}{{P_{k}^{ref} }} - 0 = \frac{{m_{xk} }}{{P_{k}^{ref} }},$$

(13)

$$\frac{\partial }{{\partial P_{k} }}{\text{nmr}}_{k} = \frac{\partial }{{\partial P_{k} }}\mathop \sum \limits_{i \ne k} \left( {P_{i}^{ref} m_{ik} } \right)/P_{k} - \frac{\partial }{{\partial P_{k} }}\mathop \sum \limits_{j \ne k} m_{kj} ={ -} \mathop \sum \limits_{i \ne k} \left( {P_{i}^{ref} m_{ik} } \right)/\left( {P_{k}} \right)^{2} - 0 = {-} \mathop \sum \limits_{i \ne k} \left( {P_{i}^{ref} m_{ik} } \right)/\left( {P_{k} } \right)^{2} .$$

(14)

The mechanism leading to variations in NMRs become obvious: the NMR of a region k will rise with population increases in the other regions of the system and fall with increases in its own population (ceteris paribus).

Annex 2

Tracking Changes in Net Migration Rates in the NMRP model

It is possible to track the evolution of the NMRs in the NMRP model, as was done in Annex 1 in the context of fixed transition probabilities. From the point of view of region k, the adjusted out-migration rate at time t is

$$m_{k}^{t,t + 1} = \mathop \sum \limits_{j \ne k} \left( {m_{kj}^{\text{ref}} \;*\;\frac{{P_{j}^{t} /\left(\mathop \sum \nolimits_{j} P_{j}^{t}+P_{k}^{\text{t}}\right) }}{{P_{j}^{\text{ref}} /\left(\mathop \sum \nolimits_{j} P_{j}^{\text{ref}}+P_{k}^{\text{ref}}\right) }}} \right).$$

(15)

The (resultant) in-migration rate is

$${\text{imr}}_{k}^{t,t + 1} = \mathop \sum \limits_{i \ne k} \frac{{( P_{i}^{t} m_{ik}^{\text{ref}} )}}{{P_{k}^{t} }}\;*\; \frac{{P_{k}^{t} /\mathop \sum \nolimits_{i} P_{i}^{t} }}{{P_{k}^{\text{ref}} /\mathop \sum \nolimits_{i} P_{i}^{\text{ref}} }}.$$

(16)

The partial derivatives of the NMR of a region k in respect to a change (increase) in its own population or in the population of another region (x) in the system are shown by Eqs. 17 and 18, respectively:

$$\frac{\partial }{\partial P_{k}}{\text{nmr}}_{k} = \frac{{\mathop \sum \nolimits_{i} (P_{i}^{\text{ref}} )\left( {P_{k}^{\text{ref}} \mathop \sum \nolimits_{j \ne k} m_{kj}^{\text{ref}} - \mathop \sum \nolimits_{i \ne k} P_{i} m_{ik}^{\text{ref}} } \right)}}{{P_{k}^{\text{ref}} \left( {\mathop \sum \nolimits_{j \ne k} P_{j}^{\text{ref}}+P_{k} } \right)^{2} }},$$

(17)

$$\begin{aligned} \frac{\partial }{\partial P_{x}}{\text{nmr}}_{k} \\ \quad ={ -} \frac{{\mathop \sum \nolimits_{i} (P_{i}^{\text{ref}} )\left( {\left( {\mathop \sum \nolimits_{j \ne x} P_{j}^{\text{ref}} P_{k}^{\text{ref}} m_{kx}^{\text{ref}} } \right) - \left( {\mathop \sum \nolimits_{i \ne x} P_{i}^{\text{ref}} P_{x}^{\text{ref}} m_{xk}^{\text{ref}} } \right) - \left( {\mathop \sum \nolimits_{j \ne x,k} P_{k}^{\text{ref}} P_{x}^{\text{ref}} m_{kj}^{\text{ref}} } \right) + \left( {\mathop \sum \nolimits_{i \ne x,k} P_{i}^{\text{ref}} P_{x}^{\text{ref}} m_{ik}^{\text{ref}} } \right)} \right)}}{{P_{k}^{\text{ref}} P_{x}^{\text{ref}} \left( {\mathop \sum \nolimits_{j \ne x} P_{j}^{\text{ref}}+P_{x} } \right)^{2} }}. \\ \end{aligned}$$

(18)

In the NMRP model, the origin–destination-specific out-migrations rates are not fixed, but subject to change following a system of first-order difference equations. The consequence is that the NMRP model loses the Markovian properties displayed by the multiregional model and its long-run behaviour remains dependant of the initial conditions. This can be seen in Eqs. 17 and 18. Indeed, the signs of the partial derivatives are defined in the numerator, which contains constant terms, the initial out-migration rates and population sizes. Thus, the direction in which the NMR evolves for a given change in the population sizes is specified by the initial conditions. At the limit (when the change in population size reaches towards infinity), the partial derivative will converge towards zero but will keep the same sign. The changing term, that is the changing population size, is found in the denominators only, and has a weighting effect on the numerator. Now in practice, when population sizes of more than one region are allowed to change, the NMRs result from interactions with all regions in the system, including constant effects (numerators) and changing effects (denominators). As a result, the way the NMR of a region will evolve during the projection is not restricted to a single direction.