Immigration / Emigration Model¶

Data¶

Statistics Canada does not contain immigration/emigration data broken down by the necessary groups (age, sex, etc), so we do not have exact data for this category. Instead, we use the data from the birth and death models.

Population Data¶

We use the Statistics Canada population data that was generated and saved as: processed_data/{time_delta_tag}/birth/initial_pop_distribution_prop.csv.

This table contains the number of people in a given age, sex, province, and projection scenario, along with the number of births for that timepoint. This data is the net number of people, factoring in death, immigration, and emigration.

Column	Type	Description
`timepoint`	`int`	the starting date / time of the time interval that the data applies to
`age`	`int`	the age of the person in years
`province`	`str`	the province of the person (e.g., `AB` = Alberta, `BC` = British Columbia, etc.)
`n_age`	`int`	the number of people in a given age group, time interval, province, and projection scenario
`n_birth`	`int`	the number of births in that time interval, province, and projection scenario
`prop`	`float`	the proportion of the population in that age group, time interval, province, and projection scenario relative to the number of births in that time interval, province, and projection scenario
`prop_male`	`float`	the proportion of the population in a given age group, time interval, province, and projection scenario who are male
`projection_scenario`	`str`	the projection scenario used to generate the data

Mortality Data¶

We use the Statistics Canada population data that was generated and saved as: processed_data/{time_delta_tag}/life_table.csv.

Column	Type	Description
`timepoint`	`int`	the starting date / time of the time interval that the data applies to
`age`	`int`	the age of the person in years
`province`	`str`	the province of the person (e.g., `AB` = Alberta, `BC` = British Columbia, etc.)
`sex`	`str`	`F` = female, `M` = male
`prob_death`	`float`	the probability that a person of the given age and sex, living in the given province, will die during the given time interval.
`se`	`float`	the standard error on the probability of death

Model¶

To obtain the net migration, for anyone aged > 0, we compute the number of people in each age group where \(\Delta n\) is the net migration, \(n_{(\text{age},\ \text{timepoint})}\) is the number of people at the current age and timepoint, \(n_{(\text{age}-1,\ \text{timepoint}-1)}\) is the number of people one year younger at the previous year, and \(q_{x_{(\text{age}-1,\ \text{year}-1)}}\) is the sex-specific probability of death for that younger cohort. This is computed separately for each combination of age, sex, province, and projection scenario. Age 0 is excluded because newborns are handled separately by the birth model.

\[\Delta n_{(a,\ s,\ t)} = n_{(a,\ s,\ t)} - n_{(a-1,\ s,\ t-1)} \cdot \left(1 - q_{x_{(a-1,\ s,\ t-1)}}\right)\]

where \(a\) = age, \(s\) = sex, \(t\) = year.

If \(\Delta n > 0\), the surplus is attributed to immigration. If \(\Delta n < 0\), the deficit is attributed to emigration.

Processed Data¶

The migration model produces a single processed data file generated by leap/data_generation/migration_data.py, covering the provinces CA (all of Canada) and BC (British Columbia).

Migration Table¶

Saved as: leap/processed_data/{time_delta_tab}/migration/migration_table.csv.

Each row corresponds to a unique combination of year, province, age, sex, and projection_scenario. The table records the signed net migration (\(\Delta n\)) for each group, along with derived columns used by the simulation at runtime for both immigration and emigration.

\(\Delta n\) is computed independently for each sex. Because males and females are computed separately, it is possible for one sex to have net immigration while the other has net emigration for the same age and year.

Column	Type	Description
`year`	`int`	the calendar year
`province`	`str`	the province abbreviation (`BC` or `CA`)
`age`	`int`	the age in years
`sex`	`str`	`M` = male, `F` = female
`projection_scenario`	`str`	the StatCan population projection scenario
`delta_n`	`float`	the signed net migration for this age, sex, year, province, and projection scenario; positive values indicate net immigration, negative values indicate net emigration
`prop_migrants_birth`	`float`	`delta_n` divided by the number of births that year; signed — positive for net immigration cells, negative for net emigration cells
`prop_immigrants_year`	`float`	for cells where `delta_n > 0`, each age and sex group’s share of all immigrants arriving in a given year (denominator is the sum of positive `delta_n` values only); zero for emigration cells
`prop_emigrants_year`	`float`	for cells where `delta_n < 0`, each age and sex group’s share of all emigrants leaving in a given year (denominator is the sum of negative `delta_n` values only); zero for immigration cells
`prob_emigration`	`float`	for cells where `delta_n < 0`, the per-person annual probability of emigrating, computed as \(\|\Delta n\| / N\); zero for immigration cells

prop_migrants_birth is computed as:

\[\text{prop migrants birth}_{(a,\ s,\ t)} = \dfrac{\Delta n_{(a,\ s,\ t)}}{n^{\text{birth}}_{(t)}}\]

where \(a\) = age, \(s\) = sex, \(t\) = year.

The total number of immigrant agents created in a given year is:

\[\begin{split}i_{(t)} = \left\lceil n_{(t)} \cdot \sum_{\substack{a,\ s \\ \Delta n > 0}}\ \text{prop migrants birth}_{(a,\ s,\ t)} \right\rceil\end{split}\]

where \(n_{(t)}\) is the number of simulated births in that year.

prop_immigrants_year is computed as:

\[\begin{split}\text{prop immigrants year}_{(a,\ s,\ t)} = \dfrac{\Delta n_{(a,\ s,\ t)}}{\sum_{\substack{a,\ s \\ \Delta n > 0}}\ \Delta n_{(a,\ s,\ t)}} \quad \text{if } \Delta n > 0, \text{ else } 0\end{split}\]

prob_emigration is computed as:

\[\text{prob emigration}_{(a,\ s,\ t)} = \dfrac{|\Delta n_{(a,\ s,\ t)}|}{N_{(a,\ s,\ t)}} \quad \text{if } \Delta n < 0, \text{ else } 0\]

At runtime, the simulation uses prob_emigration directly in a Bernoulli trial each year to determine whether an agent emigrates:

\[\text{emigrates} \sim \text{Bernoulli}(p_{\text{emigrate}}(\text{sex}, \text{age}, \text{year}))\]

Agents aged 0 are excluded — newborns never emigrate.

Both immigration and emigration are rooted in the same StatCan population-level counts (\(\Delta n\)), and are converted into agent-level operations to fit LEAP’s microsimulation framework:

For immigration, rows where delta_n > 0 are used. prop_migrants_birth (positive) determines how many immigrant agents to create at each timepoint, and prop_immigrants_timepoint determines the age and sex of each agent.
For emigration, rows where delta_n < 0 are used. prob_emigration is applied to each existing agent individually via a Bernoulli trial for each timepoint.