Chapter 1
Postcensal and intercensal population estimates, Canada, provinces and territories

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• 1.1 Postcensal population estimates, Canada, provinces and territories
  • 1.1.1 Definition and calculation of provincial and territorial postcensal estimates of total population
  • 1.1.2 Provincial and territorial postcensal population estimates by age and sex
  • 1.1.3 Levels of estimates
• 1.2 Intercensal population estimates, Canada, provinces and territories
  • 1.2.1 Provincial and territorial intercensal estimates of total population
  • 1.2.2 Provincial and territorial intercensal population estimates by age and sex

Beginning of text

This chapter describes the methods used by Statistics Canada to calculate postcensal and intercensal estimates for the total population; and for the population by age and sex, at the provincial and territorial levels. The sources of data used to produce these estimates are also given.

1.1 Postcensal population estimates, Canada, provinces and territories

1.1.1 Definition and calculation of provincial and territorial postcensal estimates of total population

Postcensal population estimates are produced using data from the most recent census (adjusted for census net undercoverage (CNU)^{Note 1}) and estimates of the components of demographic growth since that census. The data is corrected from the Census Day to July 1 by taking into account the components of demographic growth between Census Day and June 30 of the census year. The component method used to produce postcensal estimates is a population accounting system, where modifications are made to the current census population adjusted for CNU or the most recent estimate by adding and subtracting the components of demographic growth that occur between July 1 and the reference date of the estimate. The factors of demographic growth and their components are:

Natural increase

• births
• deaths

International migration

• immigrants
• emigrants
• returning emigrants
• net temporary emigration
• net non-permanent residents

Interprovincial migration

• in-migrants
• out-migrants

These components can also be divided into two groups, according to the type of data used: those components for which data are readily available, such as births, deaths, and immigration, and those that have to be estimated, such as interprovincial migration, emigrants, returning emigrants, net temporary emigration, and net non-permanent residents (NPRs).

The two components of natural increase, i.e. births and deaths, have similar methodological approach when it comes to estimation. Provincial and territorial Vital Statistics Acts (or equivalent legislation) render compulsory the registration of all live births and deaths within the province or territory. Vital statistics universe include births and deaths of all Canadians, immigrants and non-permanent residents (NPR) and exclude foreign residents.

International migration represents the movement of population (a change in the usual place of residence) between Canada and a foreign country.

Figure 1.1
International migration flows for Canada

Description for Figure 1.1

In the Demographic Estimates Program, international migration consists of five components: immigration, emigration, returning emigrants, net temporary emigration and net non-permanent residents. International migration flows can be categorized as either permanent or temporary. Permanent flows refer to persons arriving in Canada for permanent residence (immigrants), Canadian citizens or immigrants returning to Canada after previously emigrated from Canada (returning emigrants), and Canadian citizens or immigrants leaving Canada to establish a permanent residence in another country (emigrants). Temporary flows refer to foreigners arriving for temporary stay in Canada and leaving after their stay ends (non-permanent residents), as well as Canadian citizens and immigrants living temporarily abroad who have not maintained a usual place of residence in Canada (temporary emigration).

Net non-permanent residents represent the variation in the number of non-permanent residents between two dates, and net temporary emigration represents the variation in the number of temporary emigrants between two dates. Different methodological approaches are used; one for the immigration component, another one for non-permanent residents and a model based approach for the remaining components of international migration (emigration, returning emigrants, and net temporary emigration).

The last factor of demographic growth that is discussed is the interprovincial migration. While this factor does not affect the total population of Canada, it does affect the provincial and territorial population counts and is a significant challenge for the Demographic Estimates Program.

Table 1.1 shows the sources and references of component data used to generate the postcensal population.

Table 1.1
Sources and references of postcensal population estimates – Component data

Table summary
This table displays the results of Sources and references of postcensal population estimates – Component data. The information is grouped by Components (appearing as row headers) and Sources (appearing as column headers).

Components	Sources
Base Population	May 10, 2011 Census of Population adjusted for census net undercoverage (including the adjustment for incompletely enumerated Indian reserves and demographic adjustment if needed). 2011 Census: Statistics Canada, Census of Canada, 2011, Catalogue no. 98-310-X. Census net undercoverage: See The Daily, September 26, 2013. Incompletely enumerated Indian reserves: See The Daily, September 26, 2013.
Births and deaths	Statistics Canada, Health Statistics Division. Statistics Canada, Demography Division, catalogue no. no. 91-215-X, annual, catalogue no. 91-002-X, quarterly.
Immigrants	Data based on the: immigrant files provided by Citizenship and Immigration Canada (CIC).
Emigrants	Data produced by Demography Division using: data from Canada Revenue Agency (CRA) Canada Child Tax Benefit files (CCTB) program; tax data calculated using T1FF file provided by the Income Statistics Division of Statistics Canada; data provided by the U.S. Department of Homeland Security, Office of Immigration Statistics; data on the number of adult and children emigrants from T1FF file used for the provincial distribution of adults.
Returning emigrants	Data produced by Demography Division using: data from Canada Revenue Agency (CRA) Canada Child Tax Benefit files (CCTB) program; 2011 National Household Survey – question on the place of residence one year ago.
Net temporary emigration	Data produced by Demography Division using: data from the Reverse Record Check (RRC) of the 2011 Census; 2011 National Household Survey – question on the place of residence 5 years ago; estimates of returning emigrants for 2006 to 2011 intercensal period.
Net non-permanent residents	Data produced by Demography Division using: Field Operational Support System files (FOSS) from CIC. These files document the number of persons holding permits, authorizations or claiming refugee status. CIC is currently transitioning towards a new system: the Global case management system (GCMS). Demography division should be using this system starting with the March 2016 release.
Interprovincial migration	Data produced by Demography Division using: CCTB-based adjusted migration data for children; factors corresponding to the ratio of the migration rate of all children to the migration rate of eligible children calculated using CRA tax file data; factors used to calculate adult migration and corresponding to the ratio of the adult to child migration rates, calculated on a three-year basis using CRA tax file data.

Estimates of population are first produced for each province and territory, and then summed to obtain an estimate of the population of Canada.

The component method used in estimating total provincial and territorial populations is expressed as follows:

Equation 1.1:

$${\text{P}}_{(\text{t}+\text{i})}={\text{P}}_{\text{t}}+{\text{B}}_{(\text{t},\text{t}+\text{i})}-{\text{D}}_{(\text{t},\text{t}+\text{i})}+{\text{I}}_{(\text{t},\text{t}+\text{i})}-({\text{E}}_{(\text{t},\text{t}+\text{i})}+\text{Δ}\text{T}{\text{E}}_{(\text{t},\text{t}+\text{i})})+\text{R}{\text{E}}_{(\text{t},\text{t}+\text{i})}+\text{Δ}{\text{NPR}}_{(\text{t},\text{t}+\text{i})}+\text{Δ}{\text{N}}_{(\text{t},\text{t}+\text{i})}$$

where for each province and territory:

1.1.2 Provincial and territorial postcensal population estimates by age and sex

Postcensal estimates of the population by age and sex are produced using the cohort component approach, where the population is aged from year to year and the components are organized according to age and sex cohorts. A cohort is a group of persons who experience a certain event in a specified period of time. For the calculation of age and sex estimates, birth cohorts (those persons born during the same year) by sex are used. Therefore the data required for the cohort component method include demographic events, such as deaths, immigration and emigration, that can be directly linked to persons belonging to the same birth cohorts by sex.

Chapter 9 describes the application of the cohort component approach in greater detail. The chapters on the separate components will detail the manner in which the components are organized by age and sex.

1.1.3 Levels of estimates

The production of population estimates between censuses entails the use of data from administrative files or surveys. The quality of population estimates therefore depends on the availability of a number of administrative data files that are provided to Statistics Canada by federal, provincial and foreign government departments. Since some components are not available until several months after the reference date, three kinds of postcensal estimates are produced: preliminary postcensal (PP), updated postcensal (PR)^{Note 2} and final postcensal (PD). When all the components are preliminary, the estimate is described as preliminary postcensal. When they are all final, the estimate is referred to as final postcensal. Any other combination of levels is referred as updated postcensal estimates. The delay between the reference date and the release date is three months for preliminary estimates and two to three years for final estimates.

1.2 Intercensal population estimates, Canada, provinces and territories

Intercensal estimates are estimates of population for reference dates found between two censuses. They are produced following each census in order to reconcile previous postcensal estimates with the new census counts adjusted for CNU, thus assuring the internal consistency of the estimation system.

The production of intercensal estimates involves two main steps:

1. the calculation of the error of closure;
2. the linear distribution of the error of closure according to the number of days between the two censuses.

The error of closure is defined as the difference between the postcensal population estimates on Census Day and the population enumerated in that census (after adjustment for CNU). Assuming that the coverage studies that follow each enumeration are unbiased, the adjusted intercensal figures are considered exact.

More specifically, the error of closure is calculated as:

Equation 1.2:      $\text{ε}=\text{P}-\text{P}$

where

The error of closure comes from two sources: measurement errors in any of the components of demographic growth over the intercensal period and errors from the measurement of census coverage for the current and previous censuses.

The error of closure can be calculated for any disaggregated group, or for any summation of such disaggregation up to and including the total population. The disaggregation of the CNU is modeled as the sample size is not sufficient enough to give reliable disaggregated estimates.

1.2.1 Provincial and territorial intercensal estimates of total population

For the production of intercensal estimates it is assumed that the error of closure is a linear function of the time elapsed since the previous census. The production of intercensal estimates of total population involves two steps: the calculation of the error of closure ($\text{ε}$) as in Equation 1.2, and the distribution of this error uniformly over the intercensal period by an arithmetic function.

Once we have calculated the error of closure we are able to produce the intercensal population estimates for the five years between the two censuses. The intercensal estimates and the residuals are calculated for each month in the intercensal period.

To produce an intercensal estimate of the population at time t we need the following information:

1. The Census dates (α and β).
2. The date of the intercensal estimate to be produced (t).
3. The error of closure at the end of the period $({\text{ε}}_{\text{β}}).$
4. The postcensal estimate of the population at date $\text{t}$ $({\text{P}}_{\text{t}}).$

Intercensal estimates at time t are obtained using the following formula:

Equation 1.3:      $\text{I}{\text{P}}_{\text{t}}={\text{P}}_{\text{t}}-(\frac{\text{t}-\text{α}}{\text{β}-\text{α}}){\text{ε}}_{\text{β}}$

Intercensal estimates are then rounded to the nearest integer.

The residual is calculated for each month in the intercensal period. This residual is an added component that is used to balance the adjustments made to the population for the error of closure. It is calculated as follows:

For the month containing the date of the previous census of the intercensal period under consideration (m(α), for example, May 2006):

Equation 1.4:      $\text{Re}\text{s}\text{i}{\text{d}}_{\text{m}(\text{α})}={\text{P}}_{\text{m}(\text{α})+1}-\text{I}{\text{P}}_{\text{m}(\text{α})+1}$

For the months m(t) between the two censuses m(α) and m(β) (for example, June 2006 to April 2011):

Equation 1.5:      $\text{Re}\text{s}\text{i}{\text{d}}_{\text{m}(\text{t})}={\text{P}}_{\text{m}(\text{t})+1}-\text{I}{\text{P}}_{\text{m}(\text{t})+1}-{\displaystyle \sum _{\text{k}=\text{m}(\text{α})}^{\text{m}(\text{t})-1}\text{Re}\text{s}\text{i}{\text{d}}_{\text{k}}}$

For the month containing the date of the recent census of the intercensal period under consideration (m(β), for example, May 2011):

Equation 1.6:      $\text{Re}\text{s}\text{i}{\text{d}}_{\text{m}(\text{β})}={\text{P}}_{\text{β}}-\text{E}{\text{C}}_{\text{β}}-{\displaystyle \sum _{\text{k}=\text{m}(\text{α})}^{\text{m}(\text{β})-1}\text{Re}\text{s}\text{i}{\text{d}}_{\text{k}}}$

where

The sum of all these residuals equals the error of closure.

1.2.2 Provincial and territorial intercensal population estimates by age and sex

The error of closure for each sex and single year of age is the difference between the population estimates and the census counts (after adjustment for CNU^{Note 3}). The method is the same as for the total population. The production of the intercensal estimates by age and sex involves three steps:

1. the calculation of the error of closure by age and sex;
2. the distribution of this error;
3. a final adjustment to ensure consistency with total population estimates calculated independently.

With the exception of ages between 0 and 4 years, and 100 years and over, the error of closure associated with each sex and single year of age is distributed linearly, as a function of the time elapsed since the previous census. Distributing the error of closure between censuses following specific cohorts generates intercensal estimates. Figure 1.2 shows the method for distributing the error of closure.

To calculate an intercensal estimate at time t for a given province (or territory) p, a particular age a and sex s, we must first define the following:

1. The dates of the two censuses (α and β).
2. The date of the estimate (t).
3. The error of closure by province, age and sex $({\text{ε}}_{\text{p},\text{a},\text{s}}).$
4. The postcensal estimates of the population at time t for province p, age a and sex s $({\text{P}}_{\text{t}}(\text{p},\text{a},\text{s})).$
5. The variable n which denotes the number of whole years that separates t and β. For example, if t = 1st of July 2008 and β = 10th May 2011, then n = 2.

The following Lexis diagram (Figure 1.2) is used to illustrate a general example of the intercensal estimate by age and sex.

Figure 1.2 Lexis diagram showing intercensal estimation

Description for Figure 1.2

The intercensal estimate at time t for province or territory p, for age a and sex s is calculated differently based on the date and age:

Equation 1.7:

$$\text{I}\text{P}{\text{'}}_{\text{t}}(\text{p},\text{a},\text{s})={\text{P}}_{\text{t}}(\text{p},\text{a},\text{s})-\left[\left(\frac{\text{t}-{\text{β}}_{-(\text{n}+1)}}{{\text{β}}_{-\text{n}}-{\text{β}}_{-(\text{n}+1)}}\right){\text{f}}_{\text{t},\text{a},\text{a}+\text{n}}\text{ε}(\text{p},\text{a}+\text{n},\text{s})+\left(\frac{{\text{β}}_{-\text{n}}-\text{t}}{{\text{β}}_{-\text{n}}-{\text{β}}_{-(\text{n}+1)}}\right){\text{f}}_{\text{t},\text{a},\text{a}+\text{n}+1}\text{ε}(\text{p},\text{a}+\text{n}+1,\text{s})\right]$$

where

To calculate this fraction we use:

These are assigned as follows:

If  $\text{x}=\text{a}+\text{n},$ then  $\text{p}={\text{β}}_{-(\text{a}+\text{n}+1)}$ and  $\text{q}={\text{t}}_{-\text{a}};$
If  $\text{x}=\text{a}+\text{n}+1,$ then  $\text{p}={\text{t}}_{-(\text{a}+1)}$ and  $\text{q}={\text{β}}_{-(\text{a}+\text{n}+1)}.$

Once we have p and q, we calculate  ${\text{f}}_{\text{t},\text{a},\text{x}}$ as follows:

Equation 1.8:      ${\text{f}}_{\text{t},\text{a},\text{x}}=\frac{{\text{A}}_{\text{p},\text{q}}(\text{α},\text{t})}{{\text{A}}_{\text{p},\text{q}}(\text{α},\text{β})}$

where

It is noteworthy that this area is relative to the size of the cohort (q-p). These results remain valid given that the size of the cohort cancels out in the calculation of f_t,a,x.

To calculate  ${\text{A}}_{\text{p},\text{q}}(\text{i},\text{j}),$ we need to derive the following variables:

We then calculate the area using the following formula:

Equation 1.9:      ${\text{A}}_{(\text{p},\text{q})}(\text{i},\text{j})=\left(\frac{(\text{j}\text{'}-\text{i}\text{'})(\text{i}\text{'}-\text{p})+\frac{{(\text{j}\text{'}-\text{i}\text{'})}^2}{2}}{\text{q}-\text{p}}\right)+(\text{j}"-\text{i}")$

In the case where p=q, we set  ${\text{f}}_{\text{t},\text{a},\text{x}}=1.$ This is arbitrary and will not affect the outcome since the condition to have p=q implies that this term of the equation is nil.

In this case the general formula described previously can still be used. In fact we can show that if (t_-a <= α) and (a <= agemax-n-2) the formula reduces to the following expression:

Equation 1.10:

$$\text{I}\text{P}{\text{'}}_{\text{t}}(\text{p},\text{a},\text{s})={\text{P}}_{\text{t}}(\text{p},\text{a},\text{s})-\left[\frac{\text{t}-\text{α}}{\text{β}-\text{α}}\right]\left[\left(\frac{\text{t}-{\text{β}}_{-(\text{n}+1)}}{{\text{β}}_{-\text{n}}-{\text{β}}_{-(\text{n}+1)}}\right)\text{ε}(\text{p},\text{a}+\text{n},\text{s})+\left(\frac{{\text{β}}_{-\text{n}}-\text{t}}{{\text{β}}_{-\text{n}}-{\text{β}}_{-(\text{n}+1)}}\right)\text{ε}(\text{p},\text{a}+\text{n}+1,\text{s})\right]$$

Once the age cohort reaches agemax-n-1, we have to take into account cohorts that are as old or older than the maximum age that is released in the estimates program (agemax) at the time of the recent census (β). At agemax-n-1, we use the error of closure for agemax-1 and agemax.

Equation 1.11:

$$\begin{array}{l}\text{I}\text{P}{\text{'}}_{\text{t}}(\text{p},\text{a},\text{s})={\text{P}}_{\text{t}}(\text{p},\text{a},\text{s})-\\ \left[\frac{\text{t}-\text{α}}{\text{β}-\text{α}}\right]\left[\left(\frac{\text{t}-{\text{β}}_{-(\text{n}+1)}}{{\text{β}}_{-\text{n}}-{\text{β}}_{-(\text{n}+1)}}\right)\text{ε}(\text{p},\text{a}+\text{n},\text{s})+\left(\frac{\left(\frac{{\text{β}}_{-\text{n}}-\text{t}}{{\text{β}}_{-\text{n}}-{\text{β}}_{-(\text{n}+1)}}\right){\text{P}}_{\text{t}}(\text{p},\text{a},\text{s})}{\left(\frac{{\text{β}}_{-\text{n}}-\text{t}}{{\text{β}}_{-\text{n}}-{\text{β}}_{-(\text{n}+1)}}\right){\text{P}}_{\text{t}}(\text{p},\text{a},\text{s})+{\displaystyle \sum _{\text{i}=\text{a}+1}^{\text{a}\text{g}\text{e}\text{max}}{\text{P}}_{\text{t}}(\text{p},\text{i},\text{s})}}\right)\text{ε}(\text{p},\text{a}\text{g}\text{e}\text{max},\text{s})\right]\end{array}$$

In the case where  ${\displaystyle \sum _{\text{i}=\text{a}}^{\text{agemax}}{\text{P}}_{\text{t}}(\text{p},\text{i},\text{s})=0}$, we suppose a uniform distribution and the equation reduces to:

Equation 1.12:

$$\text{I}\text{P}{\text{'}}_{\text{t}}(\text{p},\text{a},\text{s})={\text{P}}_{\text{t}}(\text{p},\text{a},\text{s})-\left[\frac{\text{t}-\text{α}}{\text{β}-\text{α}}\right]\left[\left(\frac{\text{t}-{\text{β}}_{-(\text{n}+1)}}{{\text{β}}_{-\text{n}}-{\text{β}}_{-(\text{n}+1)}}\right)\text{ε}(\text{p},\text{a}+\text{n},\text{s})+\left(\frac{\left(\frac{{\text{β}}_{-\text{n}}-\text{t}}{{\text{β}}_{-\text{n}}-{\text{β}}_{-(\text{n}+1)}}\right)}{\left(\frac{{\text{β}}_{-\text{n}}-\text{t}}{{\text{β}}_{-\text{n}}-{\text{β}}_{-(\text{n}+1)}}\right)+\text{n}+1}\right)\text{ε}(\text{p},\text{a}\text{g}\text{e}\text{max},\text{s})\right]$$

In this last case, we are looking at the age cohorts that are at agemax or older at time β.

Equation 1.13:

$${\text{IP'}}_{\text{t}}{\text{(p,a,s)=P}}_{\text{t}}\text{(p,a,s)-}\left[\frac{\text{t-α}}{\text{β-α}}\right]\left(\frac{{\text{P}}_{\text{t}}\text{(p,a,s)}}{\left(\frac{{\text{β}}_{\text{-n}}\text{-t}}{{\text{β}}_{\text{-n}}{\text{-β}}_{\text{-(n+1)}}}\right){\text{P}}_{\text{t}}\text{(p,agemax-n-1,s)+}{\displaystyle \sum _{\text{i=agemax-n}}^{\text{agemax}}{\text{P}}_{\text{t}}\text{(p,i,s)}}}\right)\text{ε(p,agemax,s)}$$

In the case where,  ${\displaystyle \sum _{\text{i=agemax-n-1}}^{\text{agemax}}{\text{P}}_{\text{t}}\text{(p,i,s)=0}}$, we suppose a uniform distribution and the equation reduces to:

Equation 1.14:

$$\text{I}\text{P}{\text{'}}_{\text{t}}(\text{p},\text{a},\text{s})={\text{P}}_{\text{t}}(\text{p},\text{a},\text{s})-\left[\frac{\text{t}-\text{α}}{\text{β}-\text{α}}\right]\left(\frac{1}{\left(\frac{{\text{β}}_{-\text{n}}-\text{t}}{{\text{β}}_{-\text{n}}-{\text{β}}_{-(\text{n}+1)}}\right)+\text{n}+1}\right)\text{ε}(\text{p},\text{a}\text{g}\text{e}\text{max},\text{s})$$

Adjustment of the intercensal estimate to maintain coherence with the intercensal population estimated by province

Since the error of closure is estimated by cohorts, the intercensal estimates by age and sex will not exactly match the total by province as measured in the first part of this chapter. A final adjustment is done to ensure that both estimates are consistent.

Equation 1.15:      $\text{I}{\text{P}}_{\text{t}}(\text{p},\text{a},\text{s})=\left(\frac{\text{I}{\text{P}}_{\text{t}}^{\text{t}\text{o}\text{t}\text{a}\text{l}}(\text{p})}{{\displaystyle \sum _{\text{i},\text{j}}\text{I}\text{P}{\text{'}}_{\text{t}}(\text{p},\text{i},\text{j})}}\right)\text{I}\text{P}{\text{'}}_{\text{t}}(\text{p},\text{a},\text{s})$

where  $\text{I}{\text{P}}_{\text{t}}^{\text{t}\text{o}\text{t}\text{a}\text{l}}(\text{p})$ is the intercensal estimate measured for province p using Equation 1.3 to 1.6 at time t.

The special case where the intercensal population estimates become negative

It can happen, although rarely, that for certain age and sex cohorts for certain provinces or territories that have very low counts, negative population count are assigned with the above mentioned methodology. In these cases, the counts will be set to zero and the difference will be redistributed proportionately in all the other cohorts. The estimates are then rounded to the nearest integer.

Notes