Chapter 7
Interprovincial migration

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• 7.1 Relevant concepts and data sources
• 7.2 Preliminary interprovincial migration estimates, Canada, provinces and territories
  • 7.2.1 Estimation methodology
  • 7.2.2 Monthly, quarterly and annual estimates
• 7.3 Final interprovincial migration estimates, Canada, provinces and territories
  • 7.3.1 Estimation methodology
  • 7.3.2 Monthly, quarterly and annual estimates
• 7.4 Interprovincial migration by age and sex

Beginning of text

Interprovincial migration represents movement between provinces or territories involving a change in the usual place residence. Intraprovincial migration (or subprovincial migration) also involves a change in usual place of residence, but these movements occur within the same province or territory. The term internal migration may refer to either interprovincial migration or intraprovincial migration, or both. This chapter focuses on the interprovincial migration component. For information on intraprovincial migration, see Chapter 8 on subprovincial population estimates.

As is the case for emigration, there is no provision for recording interprovincial migration in Canada. Consequently, this component of demographic growth has to be estimated using administrative data. The methods used to prepare estimates of interprovincial migration will be discussed in this chapter.^{Note 1}

7.1 Relevant concepts and data sources

Interprovincial migration is the movement of persons from one province or territory to another, involving a change to their usual place of residence. These movements are measured by comparing the place of residence at the beginning and end of the measurement period.

Canada Revenue Agency (CRA) provides Statistics Canada with data from the Canada Child Tax Benefit (CCTB) program and personal income tax returns, which are used to estimate interprovincial migration. Monthly, quarterly and annual interprovincial migration estimates are produced using these sources.

Since 1976, personal income tax records have become the official data source for final interprovincial migration estimates. The population covered by tax data is more comprehensive than that of the CCTB; in general, interprovincial migration estimates based on tax data are considered to be of higher quality than those produced using data from the CCTB. However, tax data are not timely enough for use in preliminary estimates. Statistics Canada receives tax data only after annual processing, compared to CCTB data which are available on a monthly basis. Therefore, final estimates are calculated annually using the T1 Family File (T1FF),^{Note 2} while preliminary estimates^{Note 3} are calculated using CCTB data.

Interprovincial migration is a unique demographic phenomenon, in that it is closely tied to the measurement period. In fact, as a result of return migration and multiple migrations, the number of migrants may differ substantially depending on the duration of the measurement period. The following diagram illustrates this situation using a case of multiple migration.

Figure 7.1 Mock migration flow of a migrant starting the year in province A and ending it in province C, by measurement period

Description for Figure 7.1

Even though the migrant moves twice in the same year, only one move is recorded at the annual level (A to C). As well, since both moves occur in the second quarter, only one move is recorded at the quarterly level (A to C). Lastly, since the moves occur in the fourth and sixth months, two moves are recorded at the monthly level (A to B and B to C). The result is different migration flows depending on the measurement period.

Since interprovincial migration estimates are calculated for the month, quarter and year, the number of migrants must be calculated for each measurement period. Although the number of migrants may differ, net interprovincial migration^{Note 4} must be identical for each measurement period. For example, annual net migration in the figure above is -1 for A, 0 for B and +1 for C regardless of whether it is calculated using monthly periods or quarterly periods. However, given the nature of the administrative data and the fact that some individuals are absent at the beginning and end of each period, an adjustment must be made to ensure that the net migration is identical.

7.2 Preliminary interprovincial migration estimates, Canada, provinces and territories

7.2.1 Estimation methodology

Preliminary estimates of migration between provinces and territories are produced using CCTB data, along with data from personal income tax records, processed by Income Statistics Division. Because the CCTB program is not universal and does not provide direct information on the number of adult migrants, this estimation requires three adjustment factors to derive the complete estimate of interprovincial migration:

1. the program’s partial coverage (e.g., those who have not applied for the CCTB);
2. the differential propensity to migrate between children who are registered to the CCTB program and all children;
3. the differential propensity to migrate of adults versus children.

The three adjustment factors use CCTB and T1FF data. The methods for estimating the interprovincial migration rates of children (aged 0 to 17 years) and adults (aged 18 years and older) are described in the next section.

Interprovincial migration of children

Given that income tax returns are produced annually and are not available when preliminary estimates are being produced, preliminary estimates of interprovincial migration are based on monthly microdata from CCTB administrative files, from which the number of child migrants can be extracted. The data can be used to derive information on the usual province or territory of residence of children registered in the CCTB program. Therefore, monthly, quarterly and annual migration can be estimated by comparing two microdata files.

The estimation of interprovincial migration of children involves two adjustment factors. The first factor reflects the CCTB program's coverage: the ratio of the number of children as estimated by Demography Division, to the number of children who are registered to the CCTB program, for each month and province and territory. The second factor, which corrects for any biases resulting from the differential propensity to migrate between children who are registered to the CCTB program and all children, is obtained by comparing the interprovincial out-migration rates of children who are registered to the CCTB program with the out-migration rates for all children (aged 0 to 17). The adjustment factor is calculated for each province and territory and is based on the two last available years of T1FF.

A matrix is used to represent interprovincial migration between 13 provinces and territories of origin (denoted by j) and 13 destination provinces and territories (denoted by k). Each item (j,k) of the matrix represents both the number of out-migrants from province or territory of origin j and the number of in-migrants to destination province or territory k.

The formula used to estimate each item (j,k) of the matrix for the interprovincial migration of children is as follows:

Equation 7.1:

${}_{\text{j},\text{k}}\text{M}{}^{0-17}={}_{\text{j},\text{k}}{}^{\text{C}\text{C}\text{T}\text{B}}\text{M}{}^{0-17}\times \frac{1}{{}_{\text{j}}{}^{\text{C}\text{C}\text{T}\text{B}}\text{ℜ}}\times {}_{\text{j}}\text{G}$

where

The equation above includes the adjustments for incomplete coverage of CCTB-children who are registered to the CCTB program (Equation 7.2) and for the differences in the propensities of this group to migrate as compared to all children (Equation 7.3).

The coverage rates are calculated as follows:

For each province or territory,

Equation 7.2:

${}_{\text{j}}{}^{\text{C}\text{C}\text{T}\text{B}}\text{ℜ}=\left[\frac{{}_{\text{j}}{}^{\text{C}\text{C}\text{T}\text{B}}\text{P}{}^{0-17}}{{}_{\text{j}}{}^{\text{D}\text{e}\text{m}}\text{P}{}^{0-17}}\right]$

where

The differential migration propensity is based on a set of ${}_{\text{j}}\text{G}$ factors. Using tax data, the ${}_{\text{j}}\text{G}$ factors are obtained by dividing the interprovincial out-migration rates for all children by the out-migration rates of children who are registered to the CCTB program, as follows:

For each province and territory:

Equation 7.3:

${}_{\text{j}}\text{G}=\left[\frac{{}_{\text{j}}{}^{\text{T}\text{A}\text{X}}\text{R}\text{a}\text{t}\text{e}\_{\text{M}}^{0-17}}{{}_{\text{j}}{}^{\text{C}\text{C}\text{T}\text{B}}\text{R}\text{a}\text{t}\text{e}\_{\text{M}}^{0-17}}\right]=\frac{\left[\frac{{}_{\text{j}}{}^{\text{T}\text{A}\text{X}}\text{M}{}^{0-17}}{{}_{\text{j}}{}^{\text{T}\text{A}\text{X}}\text{P}{}^{0-17}}\right]}{\left[\frac{{}_{\text{j}}{}^{\text{C}\text{C}\text{T}\text{B}}\text{M}{}^{0-17}}{{}_{\text{j}}{}^{\text{C}\text{C}\text{T}\text{B}}\text{P}{}^{0-17}}\right]}$

where

The ${}_{\text{j}}\text{G}$ factor will equal one if the migration rate of children who are registered to the CCTB program and the migration rate of all children are identical. A value of ${}_{\text{j}}\text{G}$ that is greater (or less) than one indicates that the propensity to migrate of children who are registered to the CCTB program is less (or greater) than that of all children.

Interprovincial migration of adults

The second factor for adjusting the estimates produced using CCTB data is used in the calculation of the number of adult migrants. Estimates of adults migrating across provincial or territorial boundaries are obtained by calculating modelled migration ratios of adults to children, called F factors. F factors, which are calculated using the most recent tax data, are the ratio of the adult migration rate to the child migration rate, by province or territory of origin and by destination province or territory, as follows:

For each province and territory of origin and each destination province and territory,

Equation 7.4:

${}_{\text{j},\text{k}}\text{F}{}_{\text{t}-3,\text{t}}=\left[\frac{{}_{\text{j},\text{k}}\text{A}\text{v}\text{g}\_\text{R}\text{a}\text{t}\text{e}\_{\text{M}}^{18+}}{{}_{\text{j},\text{k}}\text{A}\text{v}\text{g}\_\text{R}\text{a}\text{t}\text{e}\_{\text{M}}^{0-17}}\right]=\frac{\left[\frac{{}_{\text{j},\text{k}}{}^{\text{T}\text{A}\text{X}}\text{M}{}_{\text{t}-3,\text{t}-2}^{18+}+{}_{\text{j},\text{k}}{}^{\text{T}\text{A}\text{X}}\text{M}{}_{\text{t}-2,\text{t}-1}^{18+}+{}_{\text{j},\text{k}}{}^{\text{T}\text{A}\text{X}}\text{M}{}_{\text{t}-1,\text{t}}^{18+}}{{}_{\text{j}}{}^{\text{D}\text{e}\text{m}}\text{P}{}_{\text{t}-2}^{18+}}\right]}{\left[\frac{{}_{\text{j},\text{k}}{}^{\text{T}\text{A}\text{X}}\text{M}{}_{\text{t}-3,\text{t}-2}^{0-17}+{}_{\text{j},\text{k}}{}^{\text{T}\text{A}\text{X}}\text{M}{}_{\text{t}-2,\text{t}-1}^{0-17}+{}_{\text{j},\text{k}}{}^{\text{T}\text{A}\text{X}}\text{M}{}_{\text{t}-1,\text{t}}^{0-17}}{{}_{\text{j}}{}^{\text{D}\text{e}\text{m}}\text{P}{}_{\text{t}-2}^{0-17}}\right]}$

where

The ${}_{\text{j},\text{k}}\text{F}$ factors once applied to the child migration rates (based on adjusted CCTB data) produces estimates of adult interprovincial migration rates. These are then multiplied by the provincial or territorial populations to obtain estimates of the number of adults migrating between provincial and territorial borders, as follows:

For each province and territory,

Equation 7.5:

${}_{\text{j},\text{k}}\text{M}{}^{18+}=\frac{{}_{\text{j},\text{k}}\text{M}{}^{0-17}}{{}_{\text{j}}{}^{\text{D}\text{e}\text{m}}\text{P}{}^{0-17}}\times {}_{\text{j}}{}^{\text{D}\text{e}\text{m}}\text{P}{}^{18+}\times {}_{\text{j},\text{k}}\text{F}{}_{\text{t}-3,\text{t}}$

where

Finally, after estimates for the total numbers of interprovincial child and adult migrants have been separately calculated by province or territory of origin j and destination province or territory k, interprovincial migration estimates are obtained by adding the number of interprovincial child and adult migrants as follows:

For each province and territory,

Equation 7.6:

${}_{\text{j},\text{k}}\text{M}={}_{\text{j},\text{k}}\text{M}{}^{0-17}+{}_{\text{j},\text{k}}\text{M}{}^{18+}$

7.2.2 Monthly, quarterly and annual estimates

The monthly, quarterly and annual number of migrants can be calculated using monthly CCTB microdata. However, adjustments must be made to the numbers of migrants to align the net interprovincial migration of the different measurement periods for which estimates are derived.

Aligning monthly net numbers with quarterly net numbers

Each quarter, monthly data must be adjusted to align the monthly net numbers with the quarterly net numbers.

First, the difference between the net values of the sum of the three monthly matrices and those of the quarterly matrix is calculated for each combination of origin and destination:

Equation 7.7:

${}_{\text{j},\text{k}}{}^{\text{P}}\text{Δ}{}_{\text{t},\text{t}+2}=\left({}_{\text{k},\text{j}}{}^{\text{P}}\text{S}{\text{M}}_{\text{t},\text{t}+2}-{}_{\text{j},\text{k}}{}^{\text{P}}\text{S}{\text{M}}_{\text{t},\text{t}+2}\right)-\left({}_{\text{k},\text{j}}{}^{\text{P}}\text{Q}{}_{\text{t},\text{t}+2}-{}_{\text{j},\text{k}}{}^{\text{P}}\text{Q}{}_{\text{t},\text{t}+2}\right)$

where

These differences are used to change the number of migrants to create a new matrix in which the net numbers match those of the quarterly matrix.

Each item (j,k) of the adjusted matrix is obtained as follows:

Equation 7.8:

${}_{\text{j},\text{k}}{}^{\text{P}}\text{S}{\text{M}}_{\text{t},\text{t}+2}^{*}={}_{\text{j}.\text{k}}{}^{\text{P}}\text{S}{\text{M}}_{\text{t},\text{t}+2}+{}_{\text{j}.\text{k}}{}^{\text{P}}\text{Δ}{}_{\text{t},\text{t}+2}\times \frac{{}_{\text{j}.\text{k}}{}^{\text{P}}\text{S}{\text{M}}_{\text{t},\text{t}+2}}{{}_{\text{j}.\text{k}}{}^{\text{P}}\text{S}{\text{M}}_{\text{t},\text{t}+2}+{}_{\text{k}.\text{j}}{}^{\text{P}}\text{S}{\text{M}}_{\text{t},\text{t}+2}}$

where

Various adjustments are made to matrix ${}^{\text{P}}\text{S}{\text{M}}_{\text{t},\text{t}+2}^{*}$ to avoid negative values or division by zero.

Lastly, for each of the three months in the quarter, the adjusted monthly interprovincial migration matrix is derived using the matrix resulting from the equation above. Each matrix item (j,k) is adjusted as follows:

Equation 7.9:

${}_{\text{j},\text{k}}{}^{\text{P}}\text{M}{}_{\text{t}}^{*}={}_{\text{j},\text{k}}{}^{\text{P}}\text{M}{}_{\text{t}}\times \frac{{}_{\text{j},\text{k}}{}^{\text{P}}\text{S}{\text{M}}_{\text{t},\text{t}+2}^{*}}{{}_{\text{j},\text{k}}{}^{\text{P}}\text{S}{\text{M}}_{\text{t},\text{t}+2}}$

where

This matrix contains the preliminary monthly numbers of migrants adjusted so that net numbers match the quarterly net interprovincial numbers.

Aligning monthly net numbers with annual net numbers

Each year, the monthly interprovincial migration estimates must be adjusted to align the monthly net numbers with the net numbers of the annual matrix.

The adjustment involves calculating the difference between the net numbers of the sum of the 12 monthly matrices ${}^{\text{P}}\text{M}{}_{\text{t}}^{*}$ and the net numbers of the annual matrix for each combination of origin and destination:

Equation 7.10:

${}_{\text{j},\text{k}}{}^{\text{P}}\text{Δ}{}_{\text{t},\text{t}+11}=\left({}_{\text{k},\text{j}}{}^{\text{P}}\text{S}{\text{M}}_{\text{t},\text{t}+11}^{*}-{}_{\text{j},\text{k}}{}^{\text{P}}\text{S}{\text{M}}_{\text{t},\text{t}+11}^{*}\right)-\left({}_{\text{k},\text{j}}{}^{\text{P}}\text{A}{}_{\text{t},\text{t}+11}-{}_{\text{j},\text{k}}{}^{\text{P}}\text{A}{}_{\text{t},\text{t}+11}\right)$

where

These differences are then used to change the number of migrants to create a new matrix in which the monthly net numbers match the annual net numbers.

Each item (j,k) of this matrix is calculated as follows:

Equation 7.11:

${}_{\text{j},\text{k}}{}^{\text{P}}\text{S}{\text{M}}_{\text{t},\text{t}+11}^{**}={}_{\text{j},\text{k}}{}^{\text{P}}\text{S}{\text{M}}_{\text{t},\text{t}+11}^{*}+{}_{\text{j},\text{k}}{}^{\text{P}}\text{Δ}{}_{\text{t},\text{t}+11}\times \frac{{}_{\text{j},\text{k}}{}^{\text{P}}\text{S}{\text{M}}_{\text{t},\text{t}+11}^{*}}{{}_{\text{j},\text{k}}{}^{\text{P}}\text{S}{\text{M}}_{\text{t},\text{t}+11}^{*}+{}_{\text{k},\text{j}}{}^{\text{P}}\text{S}{\text{M}}_{\text{t},\text{t}+11}^{*}}$

where

Adjustments are made to matrix ${}^{\text{P}}\text{S}{\text{M}}_{\text{t},\text{t}+11}^{**}$ to avoid negative values or division by zero.

Lastly, for each of the 12 months, the adjusted monthly interprovincial migration matrix is derived by obtaining each item (j,k) as follows:

Equation 7.12:

${}_{\text{j},\text{k}}{}^{\text{P}}\text{M}{}_{\text{t}}^{**}={}_{\text{j},\text{k}}{}^{\text{P}}\text{M}{}_{\text{t}}^{*}\times \frac{{}_{\text{j},\text{k}}{}^{\text{P}}\text{S}{\text{M}}_{\text{t},\text{t}+11}^{**}}{{}_{\text{j},\text{k}}{}^{\text{P}}\text{S}{\text{M}}_{\text{t},\text{t}+11}^{*}}$

where

This matrix contains the preliminary monthly numbers of migrants adjusted so that the net numbers match the annual net interprovincial numbers.

Aligning quarterly net numbers with annual net numbers

Each year, when the annual matrix is created, the quarterly net numbers must also be aligned with the annual net numbers.

The adjustment involves calculating the difference between the net numbers of the quarterly matrix and those of the sum of the matrices for the three months of the quarter. Note that net interprovincial migration in the preliminary monthly matrices has already been aligned with annual net numbers.

Equation 7.13:

${}_{\text{j},\text{k}}{}^{\text{P}}\text{Δ}{}_{\text{t},\text{t}+2}^{*}=\left({}_{\text{k},\text{j}}{}^{\text{P}}\text{Q}{}_{\text{t},\text{t}+2}-{}_{\text{j},\text{k}}{}^{\text{P}}\text{Q}{}_{\text{t},\text{t}+2}\right)-\left({}_{\text{k},\text{j}}{}^{\text{P}}\text{S}{\text{M}}_{\text{t},\text{t}+2}^{**}-{}_{\text{j},\text{k}}{}^{\text{P}}\text{S}{\text{M}}_{\text{t},\text{t}+2}^{**}\right)$

where

The differences are then used to change the quarterly number of migrants so that the net numbers of the quarterly matrix match those of the sum of the net numbers of the monthly matrices for the three corresponding months.

Each item (j,k) of the adjusted matrix is obtained as follows:

Equation 7.14:

${}_{\text{j},\text{k}}{}^{\text{P}}\text{Q}{}_{\text{t},\text{t}+2}^{*}={}_{\text{j},\text{k}}{}^{\text{P}}\text{Q}{}_{\text{t},\text{t}+2}+{}_{\text{j},\text{k}}{}^{\text{P}}\text{Δ}{}_{\text{t},\text{t}+2}^{*}\times \frac{{}_{\text{j},\text{k}}{}^{\text{P}}\text{Q}{}_{\text{t},\text{t}+2}}{{}_{\text{j},\text{k}}{}^{\text{P}}\text{Q}{}_{\text{t},\text{t}+2}+{}_{\text{k},\text{j}}{}^{\text{P}}\text{Q}{}_{\text{t},\text{t}+2}}$

where

Adjustments are made to matrix ${}_{\text{j},\text{k}}{}^{\text{P}}\text{Q}{}_{\text{t},\text{t}+2}^{*}$ to avoid negative values or division by zero.

This matrix contains the preliminary quarterly number of migrants adjusted so that the net numbers match the annual net interprovincial numbers.

7.3 Final interprovincial migration estimates, Canada, provinces and territories

7.3.1 Estimation methodology

Final estimates of interprovincial migration are produced using personal income tax data. By comparing the place of residence at the time of filing, it is possible to identify for those persons who file returns for two consecutive years, those who move, their place of origin and their destination. The development of migration data involves four main steps^{Note 5}:

1. geocoding of tax records;
2. estimation of non-filing dependants of tax filers, by age groups and gender;
3. identification of migrant tax filers by age group and gender;
4. adjustment for the population not covered by the T1FF.

The four steps of the estimation methodology are described below.

Step 1 – Geocoding

The migration data developed from the taxation records are estimates of migration flows between census divisions (CDs) or census metropolitan areas (CMAs). The geographic coding of CDs and CMAs on the tax records is done primarily on the basis of the postal code, which is a part of the mailing address. Tax records also contain a locality code assigned by Revenue Canada based on place name, which is a combination of Standard Geographic Classification (SGC) codes used by the census. About 99% of all tax records are filed with a postal code, or are assigned one, based on the filer's address by matching the address to a file similar to the Postal Code Directory from Canada Post. As the tax return is usually submitted several months after the end of the tax year, the postal codes correspond to those existing in the spring of the year following the tax year.

Step 2 – Estimation of dependants

Since the tax records source file has no direct information on the number and characteristics of non-filing dependants, this information must be imputed. Following the 1988 tax reforms, the estimation of tax filers' dependants is obtained from the T1FF. The family system creates families by linking all filing family members together, using spousal social insurance numbers, marital status, and matched addresses. This system imputes a non-filing spouse whenever a filer has declared himself or herself married but was not linked to a filing spouse. Children are imputed based on CCTB data, which carry the social insurance number of the parent receiving the benefit. Newborns are added to tax filer records from Statistics Canada's vital statistics birth database, and in addition a historical file of imputed children is created from the previous year's tax file^{Note 6}.

The family data from the tax file are then used to create a file of individual tax filers for the migration system. The migration file contains information for each tax filer and assigns dependants to each tax filer based on the number of filers in the family, and the number of dependants. If only one parent in the family has filed a tax return, then all dependants, including the spouse, if one exists, are added to the tax filer's record. If both parents filed tax returns, and their family record contained dependant children, then each tax filer would receive half the count of dependants.

If not already present, the age of the dependant spouse is estimated in the family system. The age of imputed children is provided (date of birth) on CCTB records, birth files from vital statistics, and from the Historical Family File. The family system does not impute gender for non-filing children. The gender for each dependant child is assigned a male to female ratio based on information from the most recent census.

Step 3 – Identifying migrant tax filers

After assigning the geographic codes and non-filing dependants to each tax filer, records for two consecutive years are matched by social insurance number. Only records of tax filers who are present in both years are retained. Migrant tax filers are identified by comparing current and previous geography codes (CDs or CMAs). Thus, it is only possible to determine migrant status for those who file in two consecutive years. The identification of migrants is based on their address at the time of filing, and thus the migration period is not precisely one year, but it is assumed to be close. The assumed reference period is around April to April, of the year following each tax year. However, in the Demographic Estimates Program the given estimates cover the period from July to June. Tax filers' non-filing dependants are assumed to have the same migration behaviour as that of the filers to whom they are assigned.

Step 4 – Coverage adjustment factor

The final step in the estimation process is an adjustment for coverage, done at the CD to CMA level for gender and specific age groups. This adjustment is required to estimate those migrants who did not file two consecutive personal income tax returns, which involves the calculation of coverage adjustment factors (or inflation factors). Population estimates by CD to CMA ratios are used as the denominator to the same population for whom two consecutive income tax returns could be linked to create coverage ratios.

For migration estimates prior and up to 2000/2001, provincial adjustment ratios were substituted in place of the CD to CMA ratio in the few cases where coverage was abnormally high or low (outlier values in the coverage ratios). Beginning with 2001/2002 migration data, high and low coverage cases were identified with a different methodology and a Canadian adjustment ratio was used in place of the CD to CMA ratio for these outlier coverage ratios. After a review and analysis of these methods, it was determined that cases where the coverage ratios were deemed outlier, represented small domains having little impact on the measure of migration, moreover, it was known that replacing the coverage ratios with a national ratio had the effect of reducing the true measure of migration flows. Consequently, beginning in 2006/2007, the identification and treatment of high and low coverage cases was discontinued and the coverage was simply adjusted to the Demography Division data at the CD level by age and sex group. The same adjustment is also done for CMAs.

The adjustment ratios are applied to the counts of interprovincial out-migrants derived in Step 3 to obtain an estimate of total migration. The basic assumption is that the population not covered by the taxation system has the same rate of migration as those for whom it can be measured.

7.3.2 Monthly, quarterly and annual estimates

Final annual estimates are obtained directly from T1FF data. However, since these data are annual, models must be used to calculate monthly and quarterly estimates. As well, these estimates must be adjusted to align the net interprovincial migration of the different measurement periods for which estimates are derived. These adjustments are similar to those for the preliminary estimates. However, for clarity, they are reproduced here.

As is the case with preliminary estimates, matrix representations are used for final estimates. A matrix is used to represent interprovincial migration between 13 provinces and territories of origin (denoted by j) and 13 destination provinces and territories (denoted by k). Each item (j,k) of the matrix represents both the number of out-migrants from province or territory of origin j and the number of in-migrants to destination province or territory k.

Monthly estimates

Monthly interprovincial migration estimates are derived as follows.

First, a matrix containing the sum of the monthly number of migrants in the 12 months of the year is created using the annual matrix of the T1FF and the preliminary estimates derived from CCTB data.

For each province or territory of origin j and destination province or territory k, item (j,k) of the matrix is calculated as follows:

Equation 7.15:

${}_{\text{j},\text{k}}{}^{\text{D}}\text{S}{\text{M}}_{\text{t},\text{t}+11}={}_{\text{j},\text{k}}{}^{\text{P}}\text{S}{\text{M}}_{\text{t},\text{t}+11}^{**}\times \frac{{}_{\text{j},\text{k}}{}^{\text{D}}\text{A}{}_{\text{t},\text{t}+11}}{{}_{\text{j},\text{k}}{}^{\text{P}}\text{A}{}_{\text{t},\text{t}+11}}$

where

An adjustment is then applied to the matrix so that the sum of monthly net interprovincial migration matches that of the annual matrix.

The adjustment involves calculating the difference between the net numbers of the two matrices for each combination of origin and destination:

Equation 7.16:

${}_{\text{j},\text{k}}{}^{\text{D}}\text{Δ}{}_{\text{t},\text{t}+11}=\left({}_{\text{k},\text{j}}{}^{\text{D}}\text{S}{\text{M}}_{\text{t},\text{t}+11}-{}_{\text{j},\text{k}}{}^{\text{D}}\text{S}{\text{M}}_{\text{t},\text{t}+11}\right)-\left({}_{\text{k},\text{j}}{}^{\text{D}}\text{A}{}_{\text{t},\text{t}+11}-{}_{\text{j},\text{k}}{}^{\text{D}}\text{A}{}_{\text{t},\text{t}+11}\right)$

where

The differences calculated above are then used to change the number of migrants to create a new matrix in which the net numbers match those of the annual migration matrix.

Each item (j,k) of the adjusted matrix is obtained as follows:

Equation 7.17:

${}_{\text{j},\text{k}}{}^{\text{D}}\text{S}{\text{M}}_{\text{t},\text{t}+11}^{*}={}_{\text{j},\text{k}}{}^{\text{D}}\text{S}{\text{M}}_{\text{t},\text{t}+11}+{}_{\text{j},\text{k}}{}^{\text{D}}\text{Δ}{}_{\text{t},\text{t}+11}\times \frac{{}_{\text{j},\text{k}}{}^{\text{D}}\text{S}{\text{M}}_{\text{t},\text{t}+11}}{{}_{\text{j},\text{k}}{}^{\text{D}}\text{S}{\text{M}}_{\text{t},\text{t}+11}+{}_{\text{k},\text{j}}{}^{\text{D}}\text{S}{\text{M}}_{\text{t},\text{t}+11}}$

where

Various adjustments are made to matrix ${}^{\text{D}}\text{S}{\text{M}}_{\text{t},\text{t}+11}^{*}$ to avoid negative values or division by zero.

Lastly, for each of the 12 months, the adjusted monthly interprovincial migration matrix is derived using the matrix resulting from Equation 7.17 and preliminary monthly estimates derived from the CCTB.

Equation 7.18:

${}_{\text{j},\text{k}}{}^{\text{D}}\text{M}{}_{\text{t}}^{*}={}_{\text{j},\text{k}}{}^{\text{P}}\text{M}{}_{\text{t}}^{**}\times \frac{{}_{\text{j},\text{k}}{}^{\text{D}}\text{S}{\text{M}}_{\text{t},\text{t}+11}^{*}}{{}_{\text{j},\text{k}}{}^{\text{P}}\text{S}{\text{M}}_{\text{t},\text{t}+11}^{**}}$

where

This matrix contains the final monthly numbers of migrants adjusted so that the sum of the net numbers matches that of the annual matrix.

Quarterly estimates

When monthly interprovincial migration estimates are produced, quarterly migration estimates can be derived.

First, final quarterly matrices must be derived using information from preliminary matrices and final monthly matrices for the three months in each quarter produced above.

For each province or territory of origin j and destination province or territory k, each item (j,k) of the quarterly migration matrix is calculated as follows:

Equation 7.19:

${}_{\text{j},\text{k}}{}^{\text{D}}\text{Q}{}_{\text{t},\text{t}+2}=={}_{\text{j},\text{k}}{}^{\text{P}}\text{Q}{}_{\text{t},\text{t}+2}^{*}\times \frac{{}_{\text{j},\text{k}}{}^{\text{D}}\text{S}{\text{M}}_{\text{t},\text{t}+2}^{*}}{{}_{\text{j},\text{k}}{}^{\text{P}}\text{S}{\text{M}}_{\text{t},\text{t}+2}^{**}}$

where

An adjustment is then applied to quarterly matrix ${}^{\text{D}}\text{Q}{}_{\text{t},\text{t}+2}$ to align net numbers with those of the sum of the three corresponding monthly matrices.

The adjustment involves calculating the difference between the net numbers of the quarterly matrix and those of the sum of the monthly matrices for the three months of the quarter. Note that net interprovincial migration in the final monthly matrices has already been aligned with the annual net numbers.

Equation 7.20:

${}_{\text{j},\text{k}}{}^{\text{D}}\text{Δ}{}_{\text{t},\text{t}+2}=({}_{\text{k},\text{j}}{}^{\text{D}}\text{Q}{}_{\text{t},\text{t}+2}-{}_{\text{j},\text{k}}{}^{\text{D}}\text{Q}{}_{\text{t},\text{t}+2})-({}_{\text{k},\text{j}}{}^{\text{D}}\text{S}{\text{M}}_{\text{t},\text{t}+2}^{*}-{}_{\text{j},\text{k}}{}^{\text{D}}\text{S}{\text{M}}_{\text{t},\text{t}+2}^{*})$

The differences are then used to change the number of migrants in the quarterly matrix to align the quarterly net numbers with those of the sum of the three monthly matrices for the three months of the quarter.

Each item (j,k) of the adjusted matrix is obtained as follows:

Equation 7.21:

${}_{\text{j},\text{k}}{}^{\text{D}}\text{Q}{}_{\text{t},\text{t}+2}^{*}={}_{\text{j},\text{k}}{}^{\text{D}}\text{Q}{}_{\text{t},\text{t}+2}+{}_{\text{j},\text{k}}{}^{\text{D}}\text{Δ}{}_{\text{t},\text{t}+2}\times \frac{{}_{\text{j},\text{k}}{}^{\text{D}}\text{Q}{}_{\text{t},\text{t}+2}}{{}_{\text{j},\text{k}}{}^{\text{D}}\text{Q}{}_{\text{t},\text{t}+2}+{}_{\text{k},\text{j}}{}^{\text{D}}\text{Q}{}_{\text{t},\text{t}+2}}$

where

Various adjustments are made to matrix ${}_{\text{j},\text{k}}{}^{\text{D}}\text{Q}{}_{\text{t},\text{t}+2}^{*}$ to avoid negative values or division by zero.

This matrix contains the final quarterly numbers of migrants adjusted so that the net numbers match those of the sum of the three corresponding monthly matrices.

7.4 Interprovincial migration by age and sex

Final interprovincial migration estimates are derived from the T1FF file consisting solely of five broad age groups (under 18, 18 to 24, 25 to 44, 45 to 64, and 65 years and over) by sex, as well as the origin and destination province or territory.

To estimate the number of interprovincial migrants by age and sex, the following steps are followed:

1. calculate migration by sex and five-year age cohorts using the number of migrants by province of origin and destination obtained from the one-year ago mobility question from the most recent census;
2. split the migration cohort into single-year of age using Sprague's multipliers;
3. calculate a percentage distribution by single year of age for each broad age-sex group (under 18, 18 to 24, 25 to 44, 45 to 64, and 65 years and over), using the data obtained in step (ii);
4. apply the distribution obtained in step (iii) (which remains constant for the five-year postcensal estimates) to the interprovincial migration estimates by broad age group and sex, in order to disaggregate them into a distribution by single-year of age;
5. finally, to eliminate inconsistencies from one age to another, the number of migrants by single-year of age, is aggregated by five-year age groups, to which Sprague's multipliers are again applied.

To produce the preliminary interprovincial migration estimates by age and sex at the provincial and territorial level, the most recent age and sex distribution of final estimates is applied.

Notes