Research on Modifications to the Method of Preliminary Estimates of Interprovincial Migration

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1 Catalogue no. 91F0015MIE No. 007 ISSN: X ISBN: Research Paper Demographc documents Research on Modfcatons to the Method of Prelmnary Estmates of Interprovncal Mgraton by Jaosheng He and Margaret Mchalowsk Demography Dvson 1710 Man Buldng, Ottawa, K1A 0T6 Telephone:

2 Research on Modfcatons to the Method of Prelmnary Estmates of Interprovncal Mgraton by Jaosheng He and Margaret Mchalowsk 91F0015MIE ISSN: X ISBN : Demography Dvson 1710 Man Buldng, Ottawa, K1A 0T6 Statstcs Canada How to obtan more nformaton : Natonal nqures lne: E-Mal nqures: nfostats@statcan.ca June 2005 Publshed by authorty of the Mnster responsble for Statstcs Canada Mnster of Industry, 2005 All rghts reserved. The content of ths publcaton may be reproduced, n whole or n part, and by any means, wthout further permsson from Statstcs Canada, subject to the followng condtons: that t s done solely for the purposes of prvate study, research, crtcsm, revew, newspaper summary, and/or for non-commercal purposes; and that Statstcs Canada be fully acknowledged as follows: Source (or Adapted from, f approprate): Statstcs Canada, name of product, catalogue, volume and ssue numbers, reference perod and page(s). Otherwse, no part of ths publcaton may be reproduced, stored n a retreval system or transmtted n any form or by any means, electronc, mechancal, photocopy, for any purposes, wthout the pror wrtten permsson of Lcensng Servces, Marketng Dvson, Statstcs Canada, Ottawa, Ontaro, Canada K1A 0T6. Note of apprecaton Canada owes the success of ts statstcal system to a long-standng partnershp between Statstcs Canada, the ctzens of Canada, ts busnesses, governments and other nsttutons. Accurate and tmely statstcal nformaton could not be produced wthout ther contnued cooperaton and goodwll. Auss dsponble en franças (n o 91F0015MIF au catalogue).

3 Table of contents Page 1. Introducton Dscrepances between the two estmates of mgraton Sx methods of dervng modfed mgraton Dervng modfed out-mgraton and ts flow matrx Dervng modfed n- and out-mgraton separately Dervng modfed net mgraton drectly from regresson Estmatng net mgraton usng the exponentally weghted movng average model Estmatng net mgraton usng the U.S. Bureau of Census approach Estmatng net mgraton usng the frst dfference regresson Further estmates based on method Extenson of method Replacng the fnal estmate of net mgraton wth the new method fnal net mgraton Conclusons References Appendx A. Dervng modfed out- and n-mgraton and mgraton flow matrx Appendx B. Usng the Ewma model to obtan PMN mgraton Appendx C. Net mgraton estmates usng the frst dfference regresson method Appendx D. Statstcal tables D1. Comparson of annual prelmnary (P), fnal (F) estmates of out-, n-, and netmgraton, 1993/94 to 2000/ D2. Prelmnary estmates of nterprovncal mgraton matrx, 1993/94 to 2000/ D3. Fnal estmates of nterprovncal mgraton matrx, 1993/94 to 2000/ D4. Percentage dfference for prelmnary modfed (PM) and prelmnary (P), outmgraton, Canada, provnces and terrtores, 1994/95 to 2000/01 (PM outmgraton based on regresson) D5. Percentage dfference for prelmnary modfed (PM) and prelmnary (P), nmgraton, Canada, provnces and terrtores, 1994/95 to 2000/01 (PM nmgraton based on regresson) Statstcs Canada Catalogue no. 91F0015MIE No Demography Dvson

4 Table of contents - concluded Page Apppendx D. Statstcal tables - concluded D6. Prelmnary modfed (PM) nterprovncal mgraton matrx, 1993/94 to 2000/01, based on orgn-constraned model to redstrbute PM out-mgraton 56 D7. Prelmnary modfed n-mgraton derved from PM out-mgraton and PM net mgraton (n = out + net), 1994/95 to 2000/ D8. Percentage dfference for prelmnary modfed (PM) and prelmnary (P), nmgraton (n = out + net), Canada, provnces and terrtores, 1994/95 to 2000/ Lst of Tables Table 1. Coverage rates (%) of CTB fle (populaton 0 to 17 years and ncome tax fle) (total populaton), Canada, provnces and terrtores, 1998/ Table 2. Average and standard devaton (sd) for the relatve percentage dfference (%) between prelmnary and fnal annual estmates of out-, n- and net mgraton, 1993/94 to 2000/01 8 Table 3. Index of dssmlarty (ID) between fnal and prelmnary mgraton matrces, and between two consecutve fnal mgraton matrces, 1993/94 to 2000/ Table 4. Average and standard (sd) of absolute percentage dfference (%) for prelmnary modfed (PM) estmates and for prelmnary (P) estmates, out-mgraton, 1994/95 to 2000/ Table 5. Percentage of mgrants msallocated (PMIS) between mgraton matrces, usng orgnconstraned model, 1994/95 to 2000/ Table 6. Average and standard devaton (sd) of absolute percentage dfference (%) for prelmnary modfed (PM) estmates and for prelmnary (P) estmates, n-mgraton, 1994/95 to 2000/ Table 7. Classfcaton of provnces and terrtores based on results of the sx methods for modfed net mgraton, 1994/95 to 2000/ Table 8. Average and standard devaton (sd) of absolute percentage dfference (%) for prelmnary modfed (PM) estmates and for prelmnary (P) estmates, n-mgraton, 1994/95 to 2000/ Table 9. Percentage of mgrants msallocated (PMIS) between mgraton matrces, usng the orgndestnaton constraned model, 1994/95 to 2000/ Table 10. The net nterprovncal mgraton based on dfferent sources by provnce and terrtory, 1996 to Statstcs Canada Catalogue no. 91F0015MIE No Demography Dvson

5 1. Introducton Mgraton s an mportant ssue for several reasons. Frst, t affects the sze and composton of the populaton of the areas of orgn and of destnaton; thus, t comprses a fundamental element of the demographc, socal, and cultural structure of a naton and ts regons. Second, a large porton of mgraton s labour force flow that represents the transfer of dfferent sklls. As a result, t has sgnfcant repercussons for the economc performance of both sendng and recevng areas. Fnally, mgraton s a major factor n determnng government programs, such as revenue transfers and costsharng programs among varous levels of government. In response to the mportance of nternal mgraton ssues, the Populaton Estmates Program at Statstcs Canada has been producng mgraton estmates usng admnstratve sources of data. As such, there are two versons of mgraton estmates currently avalable at Statstcs Canada: prelmnary (P) and fnal (F). The P and F estmates are based on the monthly Chld Tax Beneft (CTB) fle and the annual ncome tax return fle, respectvely; these two fles are provded by Canada Customs and Revenue Agency (CCRA). The CTB s an ncome supplement provded by the federal government to those famles wth chldren of less than 18 years of age, and under a certan ncome threshold. The man advantage of the CTB and the ncome tax return data s the reasonably good qualty of coverage and accuracy. They are also a tmely source of data: the CTB fles are after the reference date by about four months whle the ncome tax return fle lags the CTB fles by about 12 months. Hence, these data sources represent a tmely and contnuous data base for populaton estmates. The P estmates of nternal mgraton are produced by Demography Dvson. The CTB fle s adjusted to account for the coverage of chldren aged 0 to 17 years, and the dfferental propensty to mgrate between famles recevng and not recevng ths beneft (Statstcs Canada, 2003). The monthly CTBbased mgraton estmates, of reasonably good qualty, have been avalable snce the month of July Pror to ths date, famly allowance fles were used for the purpose of estmates (Bédard, 1994; Bédard and Mchalowsk, 1994). The coverage rate of the CTB fle, defned as the rato of the number of chldren actually recevng CTB to the total number of chldren under 17 years of age, vares more by provnce and terrtory than t does over a tme perod. Table 1 presents the coverage rates of the CTB fle for the provnces and terrtores n 1998/99. They vary from a low of 69.2% n the Yukon Terrtory to a hgh of 86.2% n Prnce Edward Island, wth an average and standard devaton of 78.4% and 6.3%, respectvely. The F estmates are produced by Small Area and Admnstratve Data Dvson (SAADD). The annual data of ncome tax returns are used to develop these estmates. They are subsequently dsaggregated nto monthly estmates n accordance wth monthly schedules of the P estmates. Development of the F Statstcs Canada Catalogue no. 91F0015MIE No Demography Dvson

6 estmates of mgraton nvolves four man steps (SAADD, 2002): (1) Geocodng of ncome tax records; (2) estmatng non-flng dependents of tax flers, by age group and sex; (3) dentfyng the number, age and gender of mgrant tax flers; and (4) adjustng for the populaton not covered by the CCRA taxaton system. Coverage of tax flers, defned as the rato of SAADD s populaton at-rsk-to-mgrate to the total number of populaton estmated by Demography Dvson, also vares among provnces and terrtores, and over tme. Table 1 ncludes the coverage rates of tax flers for 1998/99. It shows the rate changes from 81.9% n the Yukon Terrtory to 92.7% n Prnce Edward Island, wth an average and standard devaton of 88.0% and 3.6%, respectvely. When compared to the CTB coverage rates, the rate of tax flers s hgher and ts varance s smaller. Table 1. Coverage rates (%) of CTB fle (populaton 0 to 17 years and ncome tax fle (total populaton), Canada, provnces and terrtores, 1998/99 Provnce/terrtory CTB fle Income tax fle (1) (2) (3) Canada N.L P.E.I N.S N.B Que Ont Man Sask Alta B.C Y.T N.W.T Note: The coverage rates of ncome tax fle ndcate the at-rsk-to-mgrate populaton defned as those who fled tax returns for two consecutve years and ther dependents. Source: Statstcs Canada The P estmates of mgraton dffer from the F estmates of mgraton n that a dfferent tme frame s used to defne a move. A person can have more than one nterprovncal move accordng to the CTBbased data, whereas for the same person, n an extreme case, no move s recorded based on the annual ncome tax data. Consequently, the two estmates have shown consderable dscrepances, (although other causes may also contrbute to ths dsagreement). In general, the number of mgrants from P estmates exceeds those from F estmates (Statstcs Canada, 2003). For the perod from 1993/94 to 2000/01, the proporton of F estmates s about 76% to 86% of the P estmates. A long-standng ssue Statstcs Canada Catalogue no. 91F0015MIE No Demography Dvson

7 facng the Populaton Estmates Program s how to reduce the dscrepances between the P and the F estmates of mgraton (Wlknson, memo, 2000). The objectves of the present report are twofold. The frst s to descrbe the dscrepances between F and P estmates. The second objectve s to ntroduce and descrbe sx approaches to decrease the dscrepances usng the regresson method, smple tme seres analyss, and the U.S. Bureau of Census method. The organsaton of the report s as follows: Secton 2 provdes an overvew of the dscrepances between the two versons of nterprovncal mgraton estmates based manly on the ndex of dssmlarty. Secton 3 descrbes the approaches that are used to obtan the modfed prelmnary mgraton estmates. Secton 4 descrbes further estmates based on one of the methods (Method 3) specfed n Secton 3. Conclusons are presented n secton Dscrepances between the two estmates of mgraton Before dealng wth the ssue of reducng the dscrepancy between the two estmates, t s worthwhle to dentfy how these two estmates of nterprovncal mgraton dffer. We frst look at the dscrepances between the two estmates for out- and for n-mgraton. The dscrepancy s measured as a percentage dfference (PD) and s defned as PD = Prelmnar y Fnal Fnal x 100 Thus, a postve PD ndcates that the P estmate s greater than the F estmate, and a negatve value reflects the opposte. Table 2 presents the summary statstcs (average and standard devaton) of ths measure for the provnces and terrtores usng the data between 1993/94 and 2000/01 (96 cases). It shows that the eght-year average of PDs for out-mgraton ranges from a hgh of 26.0% n Newfoundland and Labrador to a low of 14.8% n the Northwest Terrtores. Ths suggests that P estmates of out-mgraton are about 15% to 26% hgher than F estmates n the provnces and terrtores. Based on the standard devatons (column 2), the varaton of PDs s more pronounced for small provnces and terrtores than for large provnces. For n-mgraton the average of PDs vares between 47.8% n Newfoundland and Labrador and 17.3% n Alberta. Overall, wth the excepton of Ontaro, Alberta and Brtsh Columba, the average of PDs for n-mgraton s hgher than that for out-mgraton n the rest provnces and terrtores. Ths mples that more dscrepances are observed between the two versons of estmates for n-mgraton than for out-mgraton. Statstcs Canada Catalogue no. 91F0015MIE No Demography Dvson

8 There s no clear pattern wth respect to PDs of net mgraton. A postve PD s observed for about 51% of the 96 cases n the perod 1993/94 to 2000/01. Ths suggests that, for about half of the cases, P estmates of net mgraton are greater than F estmates; the other half show the opposte. The average of PDs for net mgraton n ths eght-year perod ranges from a low of -66.5% n Ontaro to a hgh of 115.6% n Prnce Edward Island, wth a much larger standard devaton, n comparson to that for outand n- mgraton. Ths ndcates that the dstrbuton of net mgraton among the provnces and terrtores tends to be dspersed more from the average than that of out- and n- mgraton. One of the mplcatons, therefore, arsng from ths table s that modellng net mgraton wll be more challengng than modellng out- and n- mgraton. Table 2. Average and standard devaton (sd) for the relatve percentage dfference (%) between prelmnary and fnal annual estmates of out-, n- and net mgraton, 1993/94 to 2000/01 Provnce/terrtory Out-mgraton In-mgraton Net mgraton average sd average sd average sd N.L P.E.I N.S N.B Que Ont Man Sask Alta B.C Y.T N.W.T Source: Statstcs Canada, Demography Dvson. To look at how a specfc mgraton flow, wth orgn and destnaton, dffers between the two estmates, we use the ndex of dssmlarty (ID) to examne the overall dscrepancy n dstrbuton between the prelmnary and the fnal mgraton flow matrces n the same year, and, between two consecutve annual mgraton matrces of the F estmates. Specfcally, two steps are taken n the calculaton of ths ndex. 1. Calculate the percentage dstrbuton of mgraton n mgraton matrces a and b, respectvely. Matrces a and b are the prelmnary and the fnal mgraton matrx, respectvely, or the matrx n year t and n year (t-1) for the fnal mgraton seres. Ths calculaton s performed by dvdng Statstcs Canada Catalogue no. 91F0015MIE No Demography Dvson

9 the number of mgrants of each cell by the total number of mgrants of the matrx and then multplyng by Calculate the ndex. It s based on the absolute dfferences between the percentages for each cell of the mgraton matrces a and b. The ID between the two mgraton matrces s then one-half of the sum of the absolute dfferences between the respectve cells of the matrces (Shryock and Swanson, 2004). The formula may be wrtten as follows: where a m j and ID = 1 2, j a m j m b j b m j represent the percentage of mgrants of a cell for nterprovncal mgraton matrces a and b, respectvely. For example, f the ID s equal to 10%, t means that 10% of the mgrants n matrx a wll have to be transferred to dfferent cells (here provnce or terrtory) n order to make ther dstrbuton the same as that of mgraton matrx b (Duncan and Duncan, 1955). (1) Table 3 shows the ndex between the prelmnary and the fnal orgn-destnaton mgraton flow matrx (column 2), and between two consecutve fnal mgraton matrces (column 3). For the prelmnary vs. fnal mgraton flows, the ndex ranges from 3.4% to 6.1%, wth an average of 4.2%. Wth the excepton of the 6.1% n 1999/2000, t vares from 3.5% to 4.7%. Ths suggests that, although there are relatvely large dscrepances between the two versons of out- or n-mgraton, as dscussed above, the dscrepances n dstrbuton as a percentage between the two estmates s relatvely small. Table 3. Index of dssmlarty (ID) between fnal and prelmnary mgraton matrces, and between two consecutve fnal mgraton matrces, 1993/94 to 2000/01 Year ID between F and P ID between Fs (1-year lag) (1) (2) (3) 1993/ / / / / / / / average sd Source: Statstcs Canada, Demography Dvson. Statstcs Canada Catalogue no. 91F0015MIE No Demography Dvson

10 Column 3 of Table 3 shows the ndex calculated usng two consecutve annual mgraton matrces of F estmates. For example, the 1993/94 ndex of 2.7% s calculated usng the 1993/94 and the 1992/93 fnal mgraton matrces. The ndex ranges from 2.7% to 5.9%, wth an average of 4.1%. Agan, the ndces for the eght-year perod demonstrate that the orgn-destnaton mgraton flows are relatvely stable, and the dstrbuton as a percentage s close. Over tme, ths relatve stablty of fnal orgndestnaton flows suggests that a prevous year matrx of F estmates could be used for dervaton of a modfed mgraton matrx. 3. Sx methods of dervng modfed mgraton Our purpose s, not only to reduce dscrepances between P and F estmates of out- and n-mgraton, but also to reduce dscrepances between the two estmates of net mgraton. Sx methods have been tested for ths purpose. 3.1 Dervng modfed out-mgraton and ts flow matrx The frst method nvolves two steps to derve of prelmnary modfed (PM) estmates of out-mgraton. In the frst step, a smple regresson between F and P estmates of provnce and terrtory outflows s used to derve PM estmates of out-mgraton. In the second step, PM estmates of out-mgraton by provnce and terrtory are converted nto an orgn-destnaton flow matrx by usng an out-mgraton (orgn) constraned modelng method (see Appendx A for detals). Table 4. Average and standard (sd) of absolute percentage dfference (%) for prelmnary modfed (PM) estmates and for prelmnary (P) estmates, out-mgraton, 1994/95 to 2000/01 PM-F /Fx100 P-F /Fx100 Year average sd average sd (1) (2) (3) (4) (5) 1994/ / / / / / / Source: Statstcs Canada, Demography Dvson. Statstcs Canada Catalogue no. 91F0015MIE No Demography Dvson

11 The smple regresson method has the followng procedure: annual F estmates of out-mgraton are regressed on annual P estmates of out-mgraton usng three-year poolng data; also, the regresson s forced to have a zero constant. PM estmates are calculated assumng a one-year lag n data avalablty (the usual SAADD producton schedule). Suppose we would lke to derve PM estmates of outmgraton for 1995/96. We use the prevous three-year data (1994/95, 1993/94 and 1992/93) to obtan the annual regresson parameter. The annual regresson parameter s then used to obtan PM estmates of out-mgraton for 1995/96. Table 4 shows that the summary statstcs for PM out-mgraton estmates are far better than P estmates of out-mgraton. In the second step, we use a one-year prevous nterprovncal mgraton matrx of F estmates to redstrbute annual PM out-mgraton. The redstrbuton results n a modfed orgn-destnaton mgraton flow matrx. Ths mgraton matrx s evaluated aganst the matrx of the F estmates usng the percentage of mgrants msallocated (PMIS). PMIS s a goodness-of-ft statstc, ndcatng the percentage of mgrants who would have to be dsplaced to other provnces and terrtores n order for the modfed orgn-destnaton matrx to match the flow matrx of F estmates. PMIS shows that the modfed mgraton matrces are much closer to the matrx of F estmates than to the mgraton matrx of the P estmates (see Table 5). Table 5. Percentage of mgrants msallocated (PMIS) between mgraton matrces, usng orgnconstraned model, 1994/95 to 2000/01 PMIS Year Matrces PM and F Matrces P and F (1) (2) (3) 1994/ / / / / / / average sd Source: Statstcs Canada, Demography Dvson. Results are further analysed by lookng at the numbers for net mgraton. Table 7 shows that when PM and P estmates are compared to F, n 66% of the cases, PM compares better than or close to the P estmates (a case here means net mgraton for a partcular provnce or terrtory n a partcular year: the Statstcs Canada Catalogue no. 91F0015MIE No Demography Dvson

12 total number of cases s 84,.e., 12 7). However, the actual mprovement s acheved n only 51% of cases. 3.2 Dervng modfed n- and out-mgraton separately In the second method, the regresson method s used to derve prelmnary modfed n-mgraton (PMI) and prelmnary modfed out-mgraton (PMO) separately. Ther dfference serves as an estmate of prelmnary modfed net mgraton (PMN). Specfcally, PMI and PMO are derved based on the annual regresson parameter, whch s estmated usng three-year poolng and the zero-constant-regresson procedure. To derve PMI, the F estmates of n-mgraton are regressed on the P estmates of nmgraton; to derve PMO, the F estmates of out-mgraton are regressed on the P estmates of outmgraton. Table 6. Average and standard devaton (sd) of absolute percentage dfference (%) for prelmnary modfed (PM) estmates and for prelmnary (P) estmates, n-mgraton, 1994/95 to 2000/01 PM-F /Fx100 P-F /Fx100 Year average sd average sd (1) (2) (3) (4) (5) 1994/ / / / / / / Source: Statstcs Canada, Demography Dvson. For each year beng estmated, total PMI and total PMO, across all provnces and terrtores, are not necessarly dentcal. In such stuatons, totals are constraned to be the same numbers by allocatng one-half of the dfference to PMI and the other half to PMO. These dfferences are then allocated to ndvdual provnces and terrtores on a proportonal bass. Smlar to the degree of mprovement n PMO, PMI s far better than P estmates of n-mgraton (see Table 6). The results of PMN usng ths method are evaluated aganst the net mgraton of F and P estmates. As ndcated n Table 7, ths method of estmatng the modfed net mgraton results n 46% of cases beng better than the P estmates of net mgraton, but 54% of cases beng worse than the P estmates of net mgraton. Statstcs Canada Catalogue no. 91F0015MIE No Demography Dvson

13 Table 7. Classfcaton of provnces and terrtores based on results of the sx methods for modfed net mgraton, 1994/95 to 2000/01 Year Redstrbuton of Regresson of Regresson of EWMA U.S.Census Regresson of PMO "n" and "out" "net" Bureau method "frst dfference" Method 1 Method 2 Method 3 Method 4 Method 5 Method 6 B 1 C 2 W 3 B 1 C 2 W 3 B 1 C 2 W 3 B 1 C 2 W 3 B 1 C 2 W 3 B 1 C 2 W / / / / / / / total % Notes: 1. B better, PMN mgraton s better than P estmates of net mgraton when F n -PM n / F n s less than F n -P n / F n, F n, P n, and PM n are fnal, prelmnary and prelmnary modfed net mgraton, respectvely. 2. C close, dfference between the two terms, F n - P n / F n and F n - PM n / F n s not greater than W worse, the dfference between the two terms, F n - P n / F n and F n - PM n / F n s greater than Source: Statstcs Canada, Demography Dvson. 3.3 Dervng modfed net mgraton drectly from regresson In ths method, modfed net mgraton s obtaned drectly from regressng the fnal net mgraton on prelmnary net mgraton. As n Method 1 and Method 2, the regresson s based on the 3-year poolng data approach. Due to the nature of the data on the number of net mgraton, the constant parameter of the regresson equaton always equals to zero. In Table 7, when compared to F estmates (wth the excepton of two cases), PMN mgraton s ether better than or close to the prelmnary seres of net mgraton. Specfcally, n 50% of the cases, PMN mgraton s better than the P estmates of net mgraton; n 48% of the cases, PMN mgraton s close to the P estmates of net mgraton. Further analyss shows that n most cases the dfference (cases of the F category close ) between the two terms, P F PM n n and les n a narrow range between F F 0.01 and Therefore, dscrepances arsng from ths method, compared wth those from the prevous two methods are much smaller. Overall, judgng from the PMN estmates, ths method s better than Method 1 and Method 2. Statstcs Canada Catalogue no. 91F0015MIE No Demography Dvson

14 3.4 Estmatng net mgraton usng the exponentally weghted movng average model Another class of models, whch s based on tme seres data and s often used for estmaton, s called the movng average model. One such model s the exponentally weghted movng average (EWMA) model (see Appendx B for detals). In usng the EWMA approach, we assume that a reasonably good estmaton of net mgraton reles more on the most recent values than on earler ones. We use only F estmates of net mgraton to obtan PMN mgraton seres. The value of the weghtng parameter α s gven 0.7. Table 7 ndcates that, relatve to F estmates, only 37% of the cases are better than P estmates of net mgraton. The performance of the method vares among the provnces and terrtores and over the seven-year perod. The best results are found n the year 1999/2000 n whch there are eght provnces and terrtores showng better PMN estmates than the prelmnary net mgraton. In contrast, the year 1998/99 shows the worst performance (ten provnces have worse PMN estmates relatve to P data of net mgraton). We also tested the use of other values of α, but the results are unsatsfactory. For each year under estmaton, the total number of the modfed net mgraton across all provnces and two terrtores s not necessarly zero. In such crcumstances, totals are constraned to zero by allocatng the dscrepances to ndvdual provnces and terrtores on a proportonal bass. Two features of ths method prevent us from mprovng the estmates. Frst, obtanng the values of α s a tral-and-error process and, therefore, n general, the EWMA approach s a heurstc procedure. Second, the approach reles completely on past values, and often s unable to predct net mgraton wth reasonable accuracy when the data seres comes to a turnng pont. In other words, only when the tme seres data on net mgraton demonstrates a reasonable stablty can ths approach be approprate n estmatng net mgraton. 3.5 Estmatng net mgraton usng the U.S. Bureau of Census approach For populaton under the age of 65 years, the U.S. Census Bureau estmates nternal mgraton through the address matchng of federal tax returns for successve years. For those aged 65 years and over, the Bureau uses county-level nformaton on the number of medcare enrollees from the Health Care Fnancng Admnstraton (U.S. Census Bureau, 2000). Based on the U.S Census Bureau approach, a smlar method can be establshed for Canada s estmates of nternal mgraton. Statstcs Canada Catalogue no. 91F0015MIE No Demography Dvson

15 (1) Calculate the net mgraton rate. For a partcular geographcal area, usng the number of nmgrants and out-mgrants, as well as the number of at-rsk-to-mgrate persons provded by SAADD, the net mgraton rates for provnces and terrtores were calculated. (2) Derve the net number of mgrants. The net mgraton rates for provnces and terrtores are then appled to the prelmnary populaton estmates (avalable from Demography Dvson) at the begnnng of the mgraton perod. As n Method 4, the total number of modfed net mgraton for each year under estmaton across all provnces and two terrtores s not necessarly zero. In such crcumstances, totals are constraned to zero by allocatng the dscrepances to ndvdual provnces and terrtores on a proportonal bass. Table 7 ndcates that, overall, 43% of the cases (the modfed net mgraton) are better than the P estmates of net mgraton. The performance of ths approach vares over the seven-year perod. It depends on the consstency of the prevous year s net mgraton rates (derved from ncome tax fles) to the mgraton patterns of the reference year. 3.6 Estmatng net mgraton usng the frst dfference regresson Ths proposed method s based on regressng the frst dfference of fnal net mgraton aganst that of prelmnary net mgraton. In other words, the dfference of fnal net mgraton between two consecutve years, year t and year t-1, s regressed aganst the dfference of prelmnary net mgraton between year t and year t-1. Ths s the frst dfference regresson method. The modfed fnal net mgraton s obtaned usng the regresson parameter (for detals, see Appendx C). The results ndcate that when the regresson s based on the 24 observatons, the modfed net mgraton n 45% of the 84 cases s better than the P estmates, and the rest s ether close (13%) or much worse (42%). On the other hand, when the regresson s based on removng the outlers, slghtly better results are shown. Specfcally, 48% of cases are better than the P net mgraton, 14% are close, and 38% are worse (or 52% are worse than the P estmates of net mgraton). Overall, the modfed estmates of net mgraton usng ths method are not satsfactory snce ts performance s not as good as that of Method 3 Statstcs Canada Catalogue no. 91F0015MIE No Demography Dvson

16 4. Further estmates based on Method Extenson of Method 3 In Method 3, PMN s obtaned usng the regresson method. In ths secton we would lke to demonstrate that we can have the estmates of PMI and the orgn-destnaton mgraton matrx. We obtaned PMO n the frst method by regressng the fnal out-mgraton aganst the prelmnary outmgraton. Based on PMN and PMO estmates, we obtaned PMI by summng PMN and PMO. The summary statstcs are ncluded n Table 8 - when comparng PMI wth F estmates of n-mgraton. Comparng the summary statstcs of PMI by ths extenson (Table 8) wth that usng the regresson method (Table 6), except for 1997/98, the average absolute dfference for PMI mgraton usng ths approach, s smaller than those usng the regresson method; whle the summary statstcs are approxmately smlar for the year 1999/2000. Therefore, we have derved the best possble modfed estmates for net-, out-, and n-mgraton through Method 3 and ts extenson. Table 8. Average and standard devaton (sd) of absolute percentage dfference (%) for prelmnary modfed (PM) estmates and for prelmnary (P) estmates, n-mgraton, 1994/95 to 2000/01 year PM-F /Fx100 P-F /Fx100 average sd average sd (1) (2) (3) (4) (5) 1994/ / / / / / / Note: PM n-mgraton s the sum of PMO and PMN. Source: Statstcs Canada, Demography Dvson. Usng an orgn-destnaton (also called doubly) constraned model, we redstrbute PMO and PMI derved from ths further estmate to obtan the orgn-destnaton mgraton flow matrx. These PM matrces are evaluated aganst the F mgraton matrx. Table 9 shows the evaluaton results based on the percentage of mgraton msallocated. It ndcates that, for seven years, PMIS between the PM and the F mgraton matrces s consderably lower than that between the P and the F mgraton matrces. The PMIS between PM and F mgraton matrces n Table 9 ranges from a low of 2.8% to a hgh of 5.6%, wth an average of 4.2%; whereas, PMIS between the P and F matrces vares between 8.6% and 15.1%, Statstcs Canada Catalogue no. 91F0015MIE No Demography Dvson

17 wth an average of 11.3%. It, therefore, suggests that the PM mgraton matrx based on the orgndestnaton constraned model s much closer to the fnal matrx than the prelmnary s to the fnal matrx. Compared to PMIS n Table 5, the summary statstcs of PMIS usng the orgn-destnaton constraned model are slghtly better. Table 9. Percentage of mgrants msallocated (PMIS) between mgraton matrces, usng the orgn-destnaton constraned model, 1994/95 to 2000/01 PMIS Year Matrces PM and F Matrces P and F (1) (2) (3) 1994/ / / / / / / average sd Note: The PM mgraton matrx s estmated usng 1-year prevous fnal mgraton matrx to redstrbute annual PM out- and PM n-mgraton. Source: Statstcs Canada, Demography Dvson. 4.2 Replacng the fnal estmate of net mgraton wth the new method fnal net mgraton Method 3 also was tested by usng the new method F estmates of net mgraton data. These data are based on the calculaton of the current method nflaton factor ; however, the extreme nflaton factors are modfed wth the natonal average, rather than the ndvdual provncal averages (see Wlknson, 2003). 1 Evaluaton ndcates that 48% of cases are better than the P estmates of net mgraton, and 52% of cases are close to the P. Ths result s qute close to that of usng the F estmates of net mgraton wth Method 3. Smlar results are obtaned when Method 1 and Method 2 are tested usng the new method estmates of mgraton as a reference source (He and Mchalowsk, 2003a). 1. Ths method was adopted by SAADD for the producton of estmates of nternal mgraton startng wth the 2001/02 perod. Statstcs Canada Catalogue no. 91F0015MIE No Demography Dvson

18 5. Conclusons The applcaton of the smple regresson method to derve PMO (Method 1) resulted n data much converged to the fnal seres of estmates of out-mgraton rather than to the ones based on the CTB-data. When these estmates were converted nto an orgn-destnaton mgraton matrx, flows were not n our range of expectaton. In other words, the regresson-based PMO and ts subsequent redstrbuton provde us wth reasonably good estmates of out-mgraton, but unsatsfactory net mgraton estmates. As our ultmate goal s to obtan better convergence of the prelmnary and the fnal net mgraton data, we have tested fve other feasble approaches to drectly estmate net mgraton (PMN). The underlyng dea s, that once we have good estmates of net mgraton, the mgraton flow matrx can be modelled. In Method 2, modfed net mgraton s the result of the dfference between PMI and PMO. In Method 3, modfed net mgraton s obtaned drectly by regressng fnal net mgraton on prelmnary net mgraton. In Method 4, modfed net mgraton s obtaned usng the EWMA model based on F estmates of net mgraton data. In Method 5, net mgraton rates are derved from one-year lagged ncome tax return data and appled to P estmates of populaton. In the last method, Method 6, modfed net mgraton s derved by regressng the frst dfference of net fnal data aganst the frst dfference of net prelmnary data. We have concluded that the best results for PMN mgraton are obtaned from the drect regresson of F estmates of net mgraton on P estmates of net mgraton. The worst results are from the EWMA approach. All other methods produce results between these two extremes. A common methodologcal feature for the frst three methods s the use of the smple regresson model based on prelmnary and fnal mgraton data. These three methods perform better relatve to the other three methods. One of the reasons may be the smultaneous relance on fnal and prelmnary data, whether t s only out-mgraton data, out-mgraton and n-mgraton data, or the balance of the two n the form of net mgraton data. Method 3 works best among the three regresson methods. Not only do 50% of the cases show better PMN mgraton estmates; as well, 47% of the cases have PMN close to prelmnary net mgraton. These 47%, although beng worse, dsplay a small margn of deteroraton when compared to the other methods. Method 2, dervng PMI and PMO separately, does not work well. In usng Method 2, we notced that the degree of the dscrepances between prelmnary and fnal seres of data for n-mgraton and for out-mgraton s dfferent. Usng the regresson model to derve modfed n-mgraton and modfed out-mgraton separately, and then combnng the two to obtan the modfed net mgraton, Statstcs Canada Catalogue no. 91F0015MIE No Demography Dvson

19 does not work for most of the cases - better modfed n-mgraton and out-mgraton cannot guarantee better modfed net mgraton. Method 1 delvers modfed net mgraton estmates whch are more consstent wth F estmates than those obtaned through Method 2 but stll wth a lower degree of consstency than the results of Method 3. The relatvely worst results, obtaned when applyng Method 4, suggest that relyng exclusvely on fnal data tme seres s not approprate for dervng modfed net mgraton. It seems that the tme seres of the fnal mgraton data mght not be the best predctor of the mgraton pattern for the reference perod. Ths s especally true when the mgraton pattern undergoes abrupt change. The performance of Method 5 ndcates that the mgraton patterns arsng from only the taxflers and ther dependents ( unadjusted ncome tax data) do not fully reflect the patterns of total populaton n the reference perod. Ths concluson s relatve to the performance of the other fve methods. Moreover, wth the excepton of Ontaro and Quebec, net mgraton rates for most provnces and terrtores are too fluctuatng to be approprate for dervng modfed net mgraton estmates. The performance of Method 6 cannot match that of Method 3 because the percentage of cases belongng to the worse category s much hgher. Based on the above evdence from the 1994/95 to 2000/01 perod, t seems that the methods n whch fnal and prelmnary mgraton data are combned to estmate the prelmnary net mgraton (Method 3) are the best approach to acheve mproved convergence between the P and the F estmates of nternal mgraton for the Populaton Estmaton Program. Ths approach allows for the smoothng of some erratc patterns dsplayed by CTB-based estmates whle preservng ths source s ablty to capture current shfts n mgraton patterns. Fnally, the estmates of nternal mgraton dscussed here may be compared wth data from census and census coverage studes (Reverse Record Check). Table 10 presents net nterprovncal/terrtoral mgraton from these sources from the 1996 to 2001 perod. Notwthstandng annual dscrepances between tax-returns-based and CTB-based estmates of mgraton, t seems that ther 5-year perod data are very comparable to each other and to the census and Reverse Record Check numbers. Statstcs Canada Catalogue no. 91F0015MIE No Demography Dvson

20 Table 10. The net nterprovncal mgraton based on dfferent sources by provnce and terrtory, 1996 to 2001 Tax Provnce/terrtory CTB based 2001 Census 2001 RRC Method 3 old method new method (1) (2) (3) (4) (5) (6) (7) N.L. -32,075-37,577-32,395-31,035-38,157-29,350 P.E.I , N.S. -6,363-5,832-5,083-1,290 5,037-4,593 N.B. -8,412-11,593-5,455-8,430-17,268-4,916 Que. -69,047-69,128-77,886-57,310-48,685-70,908 Ont. 68,906 70,939 56,499 51,885 62,559 52,381 Man. -21,041-23,892-14,880-18,585-25,418-13,488 Sask. -25,424-28,553-24,264-24,925-32,017-22,304 Alta. 137, , , , , ,934 B.C. -37,529-27,656-30,888-23,620-36,547-29,507 Y.T. -3,088-2,770-3,821-2, ,525 N.W.T. -3,378-3, , ,016 Nvt average 31,812 32,430 35,599 26,377 40,506 30,101 sd 51,747 52,536 56,366 43,605 57,055 49,186 Notes: The average s calculated usng the absolute values of the net mgraton... not avalable for specfc reference perod 1. For the CTB-based and Method 3, estmates for Nunavut are ncluded wth the Northwest Terrtores. Statstcs Canada Catalogue no. 91F0015MIE No Demography Dvson

21 References Allen, B Estmatng nterstate mgraton, Demography Workng Paper 99/2, Canberra: Australan Bureau of Statstcs. Bédard, M Methodology for the Adjustment of the Chld Tax Beneft Data for Use n Estmatng Prelmnary Interprovncal Mgraton. Ottawa: Demography Dvson, Statstcs Canada. Bédard, M. and M. Mchalowsk Methodology for the Estmaton of Interprovncal Mgraton by Age, Sex and Martal Status for the Post-1991 Perod. Ottawa, Demography Dvson, Statstcs Canada. Draper, N.R. and H. Smth Appled Regresson Analyss (2 nd edton). New York: John Wley & Sons, Inc. Duncan, O.D. and B. Duncan Resdental dstrbuton and occupatonal stratfcaton. Amercan Journal of Socology 60 (5): de Vres, J Interprovncal mgrants and ther language characterstcs, Research Paper No.18, Ottawa: Socal and Economc Studes Dvson, Statstcs Canada. Fnne, R Inter-provncal mgraton n Canada: a longtudnal analyss of movers and stayers and the assocated ncome dynamcs. Canadan Journal of Regonal Scence 22(3): He, J. and M. Mchalowsk, 2003a. Modfcatons to the current method for the prelmnary estmates of nternal mgraton usng the proposed fnal data. Demography Dvson, Statstcs Canada b. How to make the best use of ncomplete nformaton: modellng the nternal mgraton of the Canadan populaton. Demography Dvson, Statstcs Canada. Ln, Zhengx Foregn-born vs natve-born Canadans: A comparson of ther nter-provncal labour moblty. Cat. No. 11F0019MIE Ottawa: Busness and Labour Market Analyss Dvson, Statstcs Canada. Nckson, M Geographc moblty n Canada, October 1964 October Specal Labour Force Studes no.4. Ottawa: Labour Dvson, Domnon Bureau of Statstcs. Pndyck, R.S. and D.L.Rubnfeld Econometrc Models and Economc Forecasts (second edton). New York: McGraw-Hll Book Company. Pooler, J Modellng nterprovncal mgraton usng entropy-maxmzng methods. The Canadan Geographer 31(1): Shryock, H.S. and D.A. Swanson (eds.) The Methods and Materals of Demography (second edton). Amsterdam: Elsever Academc Press. Small Ara and Admnstratve Data Dvson (SAAD) Mgraton Estmates. Ottawa: Statstcs Canada, Cat. no. 91C0025. Statstcs Canada Catalogue no. 91F0015MIE No Demography Dvson

22 Statstcs Canada Populaton and Famly Estmaton Methods at Statstcs Canada. Ottawa: Demography Dvson, Statstcs Canada, Catalogue No XIE. U.S. Census Bureau Methodology for Estmates of State and County Total Populaton. Wlknson, P Current Status of Converson Model Project. Memo, Ottawa: Demography Dvson, Statstcs Canada Estmaton of nternal mgraton: alternatve methods for calculatng coverage adjustment factors (report no. 4). Ottawa: Demography Dvson, Statstcs Canada. Wlson, A.G A statstcal theory of spatal dstrbuton models. Transportaton Research 1: Statstcs Canada Catalogue no. 91F0015MIE No Demography Dvson

23 Appendx A. Dervng modfed out- and n-mgraton and mgraton flow matrx Let F and P represent the respectve annual fnal and prelmnary out-mgraton orgnated n provnce or terrtory (note also F = F, and P = P, where F and P represent total number of outmgratons of F and P estmates for any partcular year). Then a smple regresson model can be establshed as follows: F = α + β P + e (A1) where α and β are parameters to be estmated, and e ndcates the presence of error. It must be noted that our man concern s wth statstcal relatonshp or statstcal explanaton, not wth any causal process nvolved n these two estmates. Fttng Eq. (A1) to every observed annual out-mgraton data, from 1993/94 to 2000/01 would provde the basc parameters for subsequent estmaton. The second step concerns the dervaton of the prelmnary modfed estmates of out-mgrants. Let PM represent the PM estmates of out-mgratons for provnce or terrtory. The followng relaton can be wrtten for annual prelmnary modfed estmates of out-mgratons, PM = α + ˆ β (A2) P It should be noted that the parameters n Eq. (A2) are obtaned usng the regresson equaton as shown n Eq. (A1). That s, the annual fnal s regressed on the annual prelmnary estmates. Ths regresson procedure has the followng features: (1) there s an assumpton of a one-year lag between the data used for fttng the regresson and the reference perod of estmatng the PM verson of out-mgraton; (2) the prevous three-year data on both P and F are pooled; and (3) the regresson s forced to be wth zero-constant. Suppose we would lke to estmate PM out-mgraton for 1995/96, gven the data on P of the same year. The prevous three-year (.e., 1994/95, 1993/94 and 1992/93) data on P and on F estmates are pooled to ncrease the number of observatons and use Eq. (A1) to obtan the annual regresson parameters. These annual regresson parameters are used to obtan the PM estmates of out-mgratons for 1995/96. The regresson procedure s based on three consderatons. (1) There s an approxmate one-year lag between P and F estmates of mgraton. In other words, the avalablty of F estmates of nternal mgraton lags behnd P estmates by one year. (2) We would lke to have a reasonable number of observatons for the regresson. (3) The zero-constant approach can avod the PM wth negatve numbers. For estmatng PM n-mgraton, follow the same procedure and ft Eq. (A2) to the observed annual n-mgraton data to obtan the regresson parameters, and use the parameters to obtan modfed n-mgraton estmates. Statstcs Canada Catalogue no. 91F0015MIE No Demography Dvson

24 For dervng the orgn-destnaton mgraton matrx, we use a one-year lagged mgraton matrx of F estmates to redstrbute PM out-mgraton. Ths derved mgraton matrx s evaluated aganst the matrx of F estmates usng the percentage of mgrants msallocated (PMIS). An orgn or outmgraton-constraned model s used for dervng the flow matrx usng PM out-mgraton and, ths model can be specfed as PM j = A PMO F j (A3) subject to constrant where j PM = j PMO (A4) 1 A = F j (A5) j where PM j s the redstrbuted or PM mgraton from provnce to provnce j, PMO s the total PM estmates of outmgrants from provnce, and F j s the one-year prevous mgraton matrx of the F estmates. Notce that Eq. (A4) ensures that the total number of outmgrants from the redstrbuted matrx wll match the observed number of outmgrants of the PM estmates, and Eq. (A5) ensures that ths condton be satsfed (A s called the orgn balancng factor). Based on Eqs. (A3) (A5), the redstrbuton procedure for PM estmates of out-mgraton usng a oneyear lagged matrx can be expressed by the followng statement: Prelmnary modfed estmates of matrx n year t = annual PM out-mgraton n year t lagged annual matrx of the F estmates n year t-1 the balancng factor A (A6) Followng the conventon, the redstrbuton procedure employs a goodness-of-ft statstc, the percentage of mgrants msallocated (PMIS), to evaluate the redstrbuted or prelmnary modfed mgraton matrx aganst the matrx of the F estmates. The PMIS between PM and the F estmates of mgraton matrces can be calculated usng the followng equaton: 50 PMIS = PM j F j, (A7) F, j Where PMIS represents the percentage of mgrants msallocated n the mgraton matrx, F s total number of mgrants of fnal estmates, and PM F s the absolute dfference between the prelmnary modfed estmates of mgrants and the F estmates of mgrants. Any calculated value for ths measure ndcates the percentage of mgrants that would have to be dsplaced to other provnces or terrtores n order for PM mgraton matrx to match the matrx of the F estmates. In other words, j j Statstcs Canada Catalogue no. 91F0015MIE No Demography Dvson

25 PMIS ndcates what percentage of mgrants would have to be dsplaced to other provnces and terrtores n order for the modfed orgn-destnaton matrx to match the flow matrx of F estmates. The PMIS for P and for prelmnary modfed estmates can be compared to dentfy whether the prelmnary modfed estmates of mgraton matrx are close to the F estmates. Suppose further we would lke to derve the orgn-destnaton flow matrx usng both modfed out- and modfed n-mgraton. How can such a matrx be derved? Gven the vectors of PM out- and PM nmgraton as dscussed above, we use an orgn-destnaton mgraton (or out-mgraton-n-mgraton) constraned model to derve the flow matrx. In theory, such a model can be derved by analogue wth statstcal mechancs as n Wlson (1971). Past applcaton of the model to Canadan data can be found, for example, n Pooler (1987). The model s used for redstrbutng PMO and PMI, and can be specfed as PM j = A PMO B PMI F j j j (A8) subject to constrants PM = PMO ( j ) n j= 1 j (A9) where n = 1 PM j = PMI ( j ) j (A10) 1 n A = B j PMI j Fj j= 1 (A11) n 1 B j = A PMO Fj = 1 (A12) where PMI j s the total prelmnary modfed n-mgraton to provnce j, and all other terms are defned above. Notce that Eqs. (A9) and (A10) ensure that the total number of out-mgraton and n-mgraton from the redstrbuted flow matrx wll match the observed number of PMO and PMI mgraton. Eqs. (A11) and (A12) ensure that these two condtons be satsfed (A s called the orgn balancng factor and B j destnaton balancng factor). Statstcs Canada Catalogue no. 91F0015MIE No Demography Dvson

26 Based on Eqs. (A8) (A12), the procedure for redstrbutng prelmnary modfed estmates of outmgraton and n-mgraton usng a one-year prevous fnal matrx may be expressed by the followng statement: Prelmnary modfed estmates of flow matrx n year t = PMO n year t PMI n year t matrx of the fnal data n year t-1 the balancng factor A the balancng factor B j. (A13) The goodness-of-ft statstc, the percentage of mgrants msallocated (PMIS), ntroduced n Eq. (A7) s stll used for the evaluaton of the newly derved flow matrx. Statstcs Canada Catalogue no. 91F0015MIE No Demography Dvson