**Domestic Migration Estimation in Sub-County Areas**

A Case Study in Massachusetts

By Zongli Tang

Massachusetts Institute for Social & Economic Research (MISER)

University of Massachusetts

Amherst, MA 01003

Tel: 413-577-0090

Email: zongli@miser.umass.edu

**Abstract**

A principal challenge in small area population estimation is to find
adequate data and appropriate methods for estimating domestic migration.
Tax return data collected by the Internal Revenue Service (IRS) are
employed by the US Bureau of the Census to conduct population estimates
at the county level. Since the IRS data are not available for
cities/towns, the Bureau has developed the housing-unit method to
estimate population at the sub-county level. However, this method
requires a vast amount of information that is not always easy to
acquire. Data provided by various agencies are often found to be
incomplete and inaccurate. This method is especially problematic when
applied to areas with a large population of minorities, immigrants, and
college students. Moreover, the method is too complex to manage and too
costly in time and financial resources when collecting and analyzing
data on an annual basis. In preparing the 1991-97 population estimates
for Massachusetts at the state and the sub-county levels, I have
developed the city/town ratio method, attempting to address the issues
mentioned above and to propose a new strategy for producing local
population estimates.

This method consists of two major steps. Firstly, converting IRS data by
counties into migrants by cities/towns. Migration can occur at three
levels: 1) state, i.e., migration among states; 2) county, including
migration among states and migration among counties within a state; and
3) subcounty (city/town), including levels 1) , 2), and migration among
cities/towns within a county. In other words, the subcounty level
migration is the total migration for cities/towns. The data of the three
levels of migration can be found in the long form questionnaire of the
1990 census. Using this information, we can first calculate the
city/town proportion of the county migration in Massachusetts, then
calculate the ratio of the migration at the subcounty level to the
migration at the county level for each city/town. These proportions and
ratios are assumed to be constant during the entire period of estimation
(1991-1997) and are calculated separately for in-migration and
out-migration. Once calculated, the proportions are first applied to the
annual IRS data between 1991 and 1997 to generate migration estimates at
the county level for all Massachusetts cities/towns. Then, we convert
migration at the county level into the subcounty level for each
city/town using the ratios.

Secondly, adjusting the migration counts calculated on the IRS data. The
IRS data are only available for population under 65 and may undercount
certain sub-populations. We employ school enrollment data (including
elementary/high school and college enrollment data) and Medicare data to
adjust the domestic migration numbers derived above.

The limitation of this model is the assumption that the proportions and
ratios calculated from the census hold constant during the estimating
period. Recently, I have developed an approach to improve this issue and
apply annual changes to these proportions and ratios. The revised model
will be used in the 1998 estimates.

In addition, migration pattern of the college age population (aged
18-24) is different from that of the other age groups. This would affect
age distribution in population estimates. However, the traditional
cohort-component method, which is used by most institutes of conducting
estimates including the Census Bureau, does not provide any means to
solve this issue. As a result, age distribution in population estimates,
especially for areas with substantial college age population, is often
distorted. In developing the local migration estimation model, I have
found a national-cohort method, attempting to overcome this problem.

This model as mentioned above enables us to produce timely population
estimates by race, sex, and age at the sub-county level in an efficient
and economical manner. We can also generate estimates at state and
county levels using the bottom-up method. A comparison between our
output and that produced by the US Bureau of the Census indicates that
the historical population underestimation for Massachusetts has been
significantly improved. This method could be employed in population
estimates in all areas, especially in the New England region.

-   **Too costly in time, financial and human resources**

-   **Too much data required**

-   **Inaccurate and Incomplete Information**

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-   **Index of persons per house unit: Single houses**

> **Apartments**

-   **Reports of new construction**

-   **Index of occupancy rate: Single houses\-\--housing market**

> **Apartments**

-   **Group Quarters: College Students**

#####  Nursing houses

> **especially problematic when this method is applied to areas**
>
> **with a large amount of population of minorities, immigrants,**
>
> **and college students**
>
> $\begin{array}{r}
> M_{i} = M_{i}^{s} \\
> M_{i}:\text{  total net migration for state;} \\
> M_{i}^{s}:\text{ total net interstate migration for state i}\text{.}
> \end{array}$

#### 

#### 

#### 

####  

$\begin{array}{r}
M_{i,j} = M_{i,j}^{s} + M_{i,j}^{c} \\
M_{i,j}:\text{  total net migration for county j  in state i;} \\
M_{i,j}^{s}:\text{  net interstate migration for the given county;} \\
M_{i,j}^{c}:\text{ net intercounty migration }(\text{within state i})\text{ for the given county}\text{.}
\end{array}$

#### 

#### 

#### 

#### 

#### 

#### 

$\begin{array}{r}
M_{i,j,k} = M_{i,j,k}^{s} + M_{i,j,k}^{c} + M_{i,j,k}^{t} \\
M_{i,j,k}:\text{  total net migration for k city/town in county j and state i;} \\
M_{i,j,k}^{s}:\text{  net interstate migration for the given city/town;} \\
M_{i,j,k}^{c}:\text{ net intercounty migration }(\text{within state i})\text{ for the given city/town;} \\
M_{i,j,k}^{t}:\text{ net inter-MCD migration }(\text{within county j and state i})\text{ for the given city/town}\text{.}
\end{array}$

#### 

####  

####  

#### 

#### 

####  

####   {#section-16 .unnumbered}

#### 

1.  **Calculate ratios for each city/town, using the data from Long Form
    Questionnaire in the 1990 Census.**

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i.  $\frac{M_{i,j,k}^{s} + M_{i,j,k}^{c}}{\sum_{k = 1}^{n}\left( M_{i,j,k}^{s} + M_{i,j,k}^{c} \right)}$Calculate
    proportion of county migration (in migration and out migration
    separately) for Cities/Towns:

> **Terms defined previously.**

ii. $\frac{M_{i,j,k}^{s} + M_{i,j,k}^{c} + M_{i,j,k}^{t}}{M_{i,j,k}^{s} + M_{i,j,k}^{c}}$Calculate
    the ratio of city/town migration (in migration and out migration
    separately) to county migration:

> $\begin{array}{r}
> 2\text{.}\text{  Apply the above proportions to annual IRS data to get internal migration for each city/town at } \\
> \text{county level }(M_{\text{i,j,k}}^{s} + M_{i,j,k}^{c}),\text{  on the assumption that the proportions remain constant during the } \\
> \text{period 1991-1995}\text{.}
> \end{array}$**Terms defined previously.**
>
> $\begin{array}{r}
> 3\text{.}\text{  Apply the ratios to the migration calculated above to get internal migration for each city/town } \\
> \text{at city/town level }(M_{i,j,k}^{s} + M_{i,j,k}^{c} + M_{i,j,k}^{t})\text{ on the assumption that the ratios remain constant during} \\
> \text{ the period 1991-1995}\text{.}
> \end{array}$
>
> $$

-   $\begin{array}{r}
    \frac{p_{i,j,k}^{\text{90}}}{S_{i,j,k}^{\text{90}}} \\
    p_{i,j,k}^{\text{90}}\text{:  population aged 5-17 in the 1990 census for a given city/town;} \\
    S_{i,j,k}^{\text{90}}\text{:  students grade 0-12 in the 1990 SAC data for a given city/town}\text{.}
    \end{array}$**Calculate coverage rate of school enrollment data in
    1990:**

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-   **Apply the coverage rate to annual SAC data to get expected
    population aged 5-17;**

-   $\begin{array}{r}
    M_{i,j,k}^{a} = p_{i,j,k}^{E} - p_{i,j,k}^{e} + M_{i,j,k} \\
    M_{x}^{a}\text{:     adjusted net migration for a given city/town} \\
    p_{i,j,k}^{E}\text{:  expected population for a given city/town} \\
    p_{i,j,k}^{e}\text{:  estimated population before adjustment for a given city/town} \\
    M_{i,j,k}\text{:  net migration before adjustment for a given city/town}
    \end{array}$**Adjust migration aged 5-17 using the expected
    population:**

-   $\begin{array}{r}
    M_{i,j,k}^{a} = \left( p_{i,j,k}^{E} - p_{i,j,k}^{e} \right) \div \frac{p_{n}^{\text{90}}}{p_{n,m}^{\text{90}}} + M_{i,j,k} \\
    p_{n}^{\text{90}}\text{:  national population aged 5-17 in the 1990 census,} \\
    p_{n,m}^{\text{90}}\text{:  the married national population aged 25-49 in the 1990 census;} \\
    M_{i,j,k}^{a}\text{:  adjusted migration aged 25-49 for a given city/town;} \\
    M_{i,j,k}\text{:  net migration aged 25-49 for a given city/town before adjustment}\text{.}
    \end{array}$**Adjust migration aged 25-49:**

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-   $\begin{array}{r}
    M_{j}^{a} = M_{j}^{s} + M_{j} \\
    M_{j}^{a}:\text{  adjusted net migration aged 18-24 for a given county} \\
    M_{j}^{s}\text{:  net student migration for county j calculated from IPEDS data on the assumption} \\
    \text{           that these students are not covered by IRS data} \\
    M_{j}\text{:   net migration aged 18-24 for country j before adjustment}
    \end{array}$**Adjust migration aged 18-24:**

![](media/image1.wmf)

![](media/image2.wmf)

  --------------- ----------- ------------ ----------- ------------ ----------- ------------ ----------- ------------ ----------- ------------
                                                                                                                                  

  **Variation**   **19**      **91**       **19**      **92**       **19**      **93**       **19**      **94**       **19**      **95**

                  **Count**   **%**        **Count**   **%**        **Count**   **%**        **Count**   **%**        **Count**   **%**

  **(+, -) 5%**   **331**     **94.3%**    **311**     **88.6%**    **284**     **80.9%**    **274**     **78.1%**    **248**     **70.7%**

  **(+, -)10%**   **349**     **99.4%**    **344**     **98.0%**    **335**     **95.4%**    **326**     **92.9%**    **312**     **88.9%**

  **(+, -)15%**   **351**     **100.0%**   **349**     **99.4%**    **343**     **97.7%**    **344**     **98.0%**    **340**     **96.9%**

  **(+, -)20%**   **351**     **100.0%**   **351**     **100.0%**   **348**     **99.1%**    **348**     **99.1%**    **345**     **98.3%**

  **(+, -)25%**   **351**     **100.0%**   **351**     **100.0%**   **350**     **99.7%**    **349**     **99.4%**    **348**     **99.1%**

  **(+,-)30%**    **351**     **100.0%**   **351**     **100.0%**   **351**     **100.0%**   **349**     **99.4%**    **349**     **99.4%**

  **(+, -)35%**   **351**     **100.0%**   **351**     **100.0%**   **351**     **100.0%**   **350**     **99.7%**    **349**     **99.4%**

  **(+, -)40%**   **351**     **100.0%**   **351**     **100.0%**   **351**     **100.0%**   **350**     **99.7%**    **349**     **99.4%**

  **(+,-)40%+**   **351**     **100.0%**   **351**     **100.0%**   **351**     **100.0%**   **351**     **100.0%**   **351**     **100.0%**

                                                                                                                                  
  --------------- ----------- ------------ ----------- ------------ ----------- ------------ ----------- ------------ ----------- ------------

$\begin{array}{r}
p_{t}^{\text{18} - \text{24}} = p_{t}^{\text{18}} + p_{t}^{\text{19}} + \text{.}\text{.}\text{.}\text{.}\text{.}\text{.}p_{t}^{\text{24}} \\
p_{t + 1}^{\text{18} - \text{24}} = {\left( p_{t}^{\text{17}} - D_{t\text{.}t + 1}^{\text{17}} + {\text{IM}_{t,t + 1}^{\text{17}}}_{}^{} \right) + \left( p_{t}^{\text{18}} - D_{t\text{.}t + 1}^{\text{18}} + {\text{IM}_{t,t + 1}^{\text{18}}}_{}^{} \right)}_{}^{} + \text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\left( p_{t}^{\text{23}} - D_{t\text{.}t + 1}^{\text{23}} + \text{IM}_{t,t + 1}^{\text{23}} \right) \\
\text{          } = p_{t}^{\text{17}} + p_{t}^{\text{18}} + \text{.}\text{.}\text{.}\text{.}\text{.}\text{.}p_{t}^{\text{23}} - D_{t\text{.}t + 1}^{\text{17} - \text{23}} + {\text{IM}_{t,t + 1}^{\text{17} - \text{23}}}_{}^{} \\
p_{t + 1}^{\text{18} - \text{24}} - p_{t}^{\text{18} - \text{24}} = p_{t}^{\text{17}} - p_{t}^{\text{24}} - D_{t\text{.}t + 1}^{\text{17} - \text{23}} + {\text{IM}_{t,t + 1}^{\text{17} - \text{23}}}_{}^{}
\end{array}$

$\begin{array}{r}
p_{t}^{\text{17}} - p_{t}^{\text{24}} = C_{t,t + 1} \\

\end{array}$

$\begin{array}{r}
\text{If both sides divided by }p_{t}^{\text{18} - \text{24}}\text{, then,} \\
 \\
 \\
\frac{p_{t + 1}^{\text{18} - \text{24}} - p_{t}^{\text{18} - \text{24}}}{p_{t}^{\text{18} - \text{24}}} = \frac{C_{t,t + 1} - D_{t\text{.}t + 1}^{\text{17} - \text{23}} + {\text{IM}_{t,t + 1}^{\text{17} - \text{23}}}_{}^{}}{p_{t}^{\text{18} - \text{24}}} \\
\text{                               } = \frac{C_{t,t + 1}}{p_{t}^{\text{18} - \text{24}}} - \frac{D_{t\text{.}t + 1}^{\text{17} - \text{23}}}{p_{t}^{\text{18} - \text{24}}} + \frac{\text{IM}_{t,t + 1}^{\text{17} - \text{23}}}{p_{t}^{\text{18} - \text{24}}} \\
\text{                                     } \downarrow \text{                   } \downarrow \text{                   } \downarrow \\
\text{                        cohort change    death       net immigration} \\
\text{                        rate                    rate          rate} \\

\end{array}$

Assume: the national cohort change rate is the same as the local, then,

$\begin{array}{r}
\frac{p_{t + 1}^{\text{18} - \text{24}} - p_{t}^{\text{18} - \text{24}}}{p_{t}^{\text{18} - \text{24}}} = \frac{C_{t,t + 1}}{p_{t}^{\text{18} - \text{24}}} - \frac{D_{t\text{.}t + 1}^{\text{17} - \text{23}}}{p_{t}^{\text{18} - \text{24}}} + \frac{\text{IM}_{t,t + 1}^{\text{17} - \text{23}}}{p_{t}^{\text{18} - \text{24}}} \\
\text{                                           } \downarrow \text{                   } \downarrow \text{                   } \downarrow \\
\text{                        national cohort         local death        local immigration} \\
\text{                        change   rate             rate                    rate} \\
 \\
\text{as, } \\
\frac{C_{t,t + 1}}{p_{t}^{\text{18} - \text{24}}} = \frac{p_{t + 1}^{\text{18} - \text{24}} - p_{t}^{\text{18} - \text{24}}}{p_{t}^{\text{18} - \text{24}}} + \frac{D_{t\text{.}t + 1}^{\text{17} - \text{23}}}{p_{t}^{\text{18} - \text{24}}} - \frac{\text{IM}_{t,t + 1}^{\text{17} - \text{23}}}{p_{t}^{\text{18} - \text{24}}} \\
 \\
\text{therefore,} \\
 \\
\frac{p_{t + 1}^{\text{18} - \text{24}} - p_{t}^{\text{18} - \text{24}}}{p_{t}^{\text{18} - \text{24}}} = \frac{p_{t + 1}^{\text{18} - \text{24}} - p_{t}^{\text{18} - \text{24}}}{p_{t}^{\text{18} - \text{24}}} + \frac{D_{t\text{.}t + 1}^{\text{17} - \text{23}}}{p_{t}^{\text{18} - \text{24}}} - \frac{\text{IM}_{t,t + 1}^{\text{17} - \text{23}}}{p_{t}^{\text{18} - \text{24}}} - \frac{D_{t\text{.}t + 1}^{\text{17} - \text{23}}}{p_{t}^{\text{18} - \text{24}}} + \frac{\text{IM}_{t,t + 1}^{\text{17} - \text{23}}}{p_{t}^{\text{18} - \text{24}}} \\
 \\
\frac{p_{t + 1}^{\text{18} - \text{24}} - p_{t}^{\text{18} - \text{24}}}{p_{t}^{\text{18} - \text{24}}} - \left( \frac{D_{t\text{.}t + 1}^{\text{17} - \text{23}}}{p_{t}^{\text{18} - \text{24}}} - \frac{D_{t\text{.}t + 1}^{\text{17} - \text{23}}}{p_{t}^{\text{18} - \text{24}}} \right) + \left( \frac{\text{IM}_{t,t + 1}^{\text{17} - \text{23}}}{p_{t}^{\text{18} - \text{24}}} - \frac{\text{IM}_{t,t + 1}^{\text{17} - \text{23}}}{p_{t}^{\text{18} - \text{24}}} \right)
\end{array}$

i.  calculate total annual cohort change for national population aged
    18-24

> $\begin{array}{r}
> \frac{P_{t + 1}^{\text{18} - \text{24}}}{P_{t}^{\text{18} - \text{24}}} \\
> P_{t + 1}^{\text{18} - \text{24}}\text{:   national population aged 18-24 in the current year;} \\
> P_{t}^{\text{18} - \text{24}}\text{:  national population aged 18-24 in the previous year}\text{.}
> \end{array}$

ii. calculate the differences between national and local death rates and
    differences between national and local immigration rates.

iii. calculate single age cohort change for national population aged
     18-24

$\begin{array}{r}
\frac{p_{t + 1}^{a}}{p_{t}^{a}} \\
p_{t + 1}^{a} = \frac{P_{t + 1}^{a}}{P_{t + 1}^{\text{18} - \text{24}}} \\
p_{t}^{a} = \frac{P_{t}^{a}}{P_{t}^{\text{18} - \text{24}}}
\end{array}$

iv. Apply the national cohort changes and the differences of deaths and
    immigrants to Massachusetts population aged 18-24 in t year to get
    the preliminary population estimates in t+1 year.

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i.  Apply IRS migration rates to the preliminary population estimates
    calculated above to get migration estimates for this age group;

ii. Deduct migration from the preliminary population estimates to get
    the estimate population.

  ----------- ------------ ------------ ------------ ------------ -------------
                                                                  

  **Age       **4/1/80**   **4/1/90**   **7/1/90**   **1995       **1995
  Group**                                            Census**     MISER**

  **0-4**     **337215**   **421349**   **420569**   **412133**   **414831**

  **5-9**     **374734**   **376621**   **374056**   **416765**   **404930**

  **10-14**   **461332**   **347017**   **344776**   **382215**   **372836**

  **15-19**   **548239**   **416030**   **406268**   **354713**   **398667**

  **20-24**   **552902**   **517194**   **516820**   **401063**   **488565**

  **25-29**   **489175**   **556320**   **556755**   **500988**   **511605**

  **30-34**   **446395**   **542330**   **545582**   **547276**   **529536**

  **35-39**   **348552**   **481173**   **484699**   **525054**   **522426**

  **40-44**   **278748**   **433373**   **438248**   **466436**   **467025**

  **45-49**   **278261**   **333333**   **336147**   **410249**   **424455**

  **50-54**   **306574**   **262601**   **263980**   **316669**   **317010**

  **55-59**   **310995**   **252174**   **251423**   **244230**   **242300**

  **60-64**   **277384**   **261905**   **261468**   **226381**   **224741**

  **65-69**   **235574**   **251383**   **250872**   **238632**   **229188**

  **70-74**   **187041**   **207331**   **207991**   **219142**   **211253**

  **75-79**   **137828**   **159653**   **160608**   **175108**   **163020**

  **80-84**   **92180**    **106299**   **106742**   **119251**   **112934**

  **85+**     **73908**    **90339**    **91119**    **104261**   **108997**

                                                                  
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**[Advantages]{.underline} [Disadvantages]{.underline}**