population forecasting with nonstationary multiregional growth matrices

, Stuart H . Sweeney and Kevin J . Konty

Population Forecasting with Nonstationary Muhiregional Growth Matrices

Though the mathematics of multiregional population projections were dejined over twenty years ago, and the methodology has seen some adoption internationally, most practitioners in the United States still use rudimenta y cohort component projections techniques. Both the stationarity assumption and the implicit jive-year retrospective time scale imposed by the census migration data have probably contributed to the limited use of multiregional projections methods. This paper reviews previous attempts to overcome the stationarity assumption and proposes a decompositional approach using log linear models estimated via the ECM algorithm. The paper discusses the advantages of the decompositional approach and implements the model for intra- state migration in Calijornia.

Population forecasts are one of the primary inputs to a wide range of important planning functions carried out by states and localities in the United States. As has been argued eloquently by Isserman (1984) accurate population forecasts are fundamental to good planning. In long-range planning, large forecast errors translate into costs related to over- or underprovision of physical infrastructure. In rapidly growing states, such as California, there is added urgency for reliable short-term forecasts. Transportation, human resources, and environmental planning all rely on near-term forecasts of changes in the spatial population distribution.

In practice, the methods used to construct both long- and short-range population projections have remained rudimentq. Though multiregional population projection methodology was developed by Rogers in the 1960s, simple cohort-component methods are still the most widely taught and applied methodology in practice. This is true despite attempts to write gentle introductions to the methodology for a practitioner audience (Rogers 1985, Isserman 1984) and despite clear demonstrations of the bias introduced through the use of net migration in the cohort-component projection model (Rogers 1990).

An earl version of this aper was resented at the 17th Pacific Conference of the Re ional Science As- sociation Lternational ancfbenefittez from the comments of the discussant, Jacques Le%ent. The authors also thank the anonymous reviewer for very insightful comments and suggestions.

Stuart H. Sweeney is an assistant professor and Kevin]. Konty is a graduate student, both in geography at the University of Calfornia, Santa Barbara. E-mail: [email protected] and [email protected]

Geographicd Analysis, Vol. 34, No. 4 (October 2002) The Ohio State University Submitted: 11/7/01. Revised version accepted: 1/18/02

290 / Geographical Analysis

Why is it that multiregional projection methods have not found wider use to date? There are several reasons. Like the large-scale models in Lee’s requiem (1973), multiregional models are data hungry. Whereas simple cohort-component models can use either residually measured net migration or ignore migration altogether, multiregional models require a matrix of origin-destination flows. Demographically disaggregate flow data have been only available once per decade and the computational burden needed to access the data containing county-to-county flows, and to a lesser extent state-to-state flows, has been prohibitive for some.

The data landscape is becoming increasingly complex, presenting new challenges and new opportunities. In the United States, the decennial census long form, sam- pling approximately one-sixth of the population, has long been the primary source of internal migration data capable of supporting detailed socioeconomic and spatial disaggregation.’ Given mounting political pressure related to privacy concerns and the high cost of the long form, the US. Census Bureau has proposed to supercede the long form with the American Community Survey (ACS) by the 2010 Census.2 The ACS would have a monthly sample of 250,000 compared to approximately 17 million sampled by the 2000 Census long form. Though the monthly ACS will include a 1- year retrospective migration question, providing more timely data, the smaller sample size necessarily means some degradation of the socioeconomic and spatial detail traditionally available in the long form. The ACSAong form trade-offs reflect a general shift in the social science data collection and dissemination.

Though there is an apparent loss in resolution and statistical power if the long form is indeed eclipsed by the ACS, it is likely that social science data resolution will advance on all fronts: spatial scales, temporal scales, and demographic characteristics. The major shift is that those resolutions will be scattered over multiple data sources and will be derived from both sample and administrative sources. For example, administrative records, such as the IRS migration data used in this paper, are increasingly being harnessed as a supplemental source of social science data. In the IRS migration data, high-spatial and -temporal resolution is traded off against information about the characteristics of the migrants. Model-based estimates that stitch together information from disparate data sources are ideally suited to the emerging data environment and will become increasingly common in social science research (Citro 1998).

The stationarity (Markov) assumption and the intrinsic time scale of the multiregional model are also problematic. Given the five-year retrospective flows data, projections made with the model are based on the assumption that the observed pattern of interregional interaction will prevail over the range of the projection period. In ad- dition, the five-year scale of the data imposes five-year increments on the projection period. Both issues are binding for short-term forecasts. A five-year scale is often longer than the time scale used for planning and the stationarity assumption imposes an unlikely future in which no period effects alter the spatial structure of interaction. It was in recognition of the intractability of the latter point that Plane (1993) composed his “Requiem for the Fixed Transition Probability Migrant” shortly after Rogers (1990) composed his “Requiem for the Net Migrant.” Both critiques are valid, and both of the implausible migrant constructs should rest in peace.

1. Since 1940 the census long form has included a retrospective migration question. In all years except 1950 it was a five-year retrospective, but because of the war effort the 1950 census used a one-year retrospective question.

2. Of course, the roposed replacement of the long form de ends on sufficient fundin for the ACS and the playing out orinterest group I;olitics related to the long-Erm. For example, the Feleral Highwa Administration is highly opposed to e iminating the long-form since it rovides detailed journey-to-worz data at the intrametropolitan scale. It is far too early to say whether the &I0 Census will or will not include a long form.

Stuart H. Sweeney and KevinJ. Konty / 291

Though the methodology in this paper is applicable to intercensal estimates of migration and builds on the model-based migration estimates literature (Willekens 1994), the primary goal is to address a practical constraint in multiregional projections. In the most recent round of state projections carried out by the Census Bureau, multiregional projections were used but the methodology report notes the following quandary:

The matched IRS tax returns data set contains nineteen annual observations on each of the 2,550 state-to-state migration flows. The size and detail of this data set offer so many different options for projection models as to create a special type of problem. Because reliable, comprehensive data on migration have been so scarce in the past, professional researchers in this area have not yet developed any consensus as to the best method for projecting migration. Consequently, we were confronted with an overwhelming array of possibilities and have little guidance from the professional literature on making a selection (Campbell 1996).

Ultimately, they chose an approach by Frees (1992) in which individual out-migration rates are modeled using methods for pooled cross-sectiodtime-series methods. Our contribution provides an alternative to modeling time series of transaction matrices. Specifically, we propose that the nonstationarity in transaction matrices can be described by a small set of time-dependent parameters, and that those parameters can be modeled as a function of regional economic variables.

Thus, the methodology developed and applied in this article extends the multiregional modeling framework in three ways: (1) it is consistent with emergent realities of how social scientists will access disaggregate migration data, ( 2 ) it relaxes the stationarity constraint, and ( 3 ) it is a demoeconomic model and allows for period effects to enter through changes in economic variables. The next section of the paper reviews some of the existing methods that have been used to relax the stationarity assumption. Following that a methodology based on Poisson regression via the ECM algorithm is proposed. The paper concludes with an application of the methodology to small area population projections in California.

MULTIREGIONAL POPULATION FORECASTING, WHITHER STATIONARITY?

Multiregional demography is statistically founded on Markov processes. When only a single matrix of origin-destination flows is available, the model implicitly in- vokes the first-order, stationary Markov assumption, that is, that the future is condi- tionally independent of the past given a single period of observation. That assumption is the matrix equivalent of a one-period lag in time series. Conditional independence can also be specified with n significant lag periods leading to an nth-order Markov process. In a projections context, the first-order stationarity assumption is the basis for projecting the stream of spatial population distributions based on the observed dynamics from a single period.

Yet, the system dynamics are undoubtably characterized by structural shifts that are influenced by demographic, economic, and social disturbances. Grounded in hypotheses posed by Vining (1972) and Easterlin (1968), researchers have explored how the baby-boom cohort and economic conditions have interacted through shifts in the interstate migration system (Plane and Rogerson 1986). Plane (1999) hypothesizes that system dynamics are characterized by interregums; that is, periods of prevailing beaten paths give way to new patterns of spatial interaction following significant disturbances on the system. For example, the 1985-90 five-year migration window measured in the 1990 Census is coincident with a major energy disturbance in the Mountain West. Time paths of in- and out-migration for Billings, Montana, and Den- ver, Colorado (Figure l), capture the region-wide effects of that disturbance. Tobler (1995) provides evidence of interregums over decade1 time scales based on the analysis of six decades of the U.S. interstate migration system.


0.12 -

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1984 1986 1988 1990 1992 1994 1996

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FIG. 1. Two-Year Moving Average In- and Out-Migration rates for Selected Labor Markets. Authors’ tabulations based on annual matched IRS tax returns.

Given the existence of those shifts, researchers have searched for methods to characterize the nature of the nonstationarity and to incorporate nonstationarity into projections. That research can be broadly partitioned into demoeconomic approaches and structural approaches.

Nonstationarity through Demoeconomic Interactions One objection to purely demographic models is that social science theories in&-

cate that migration rates are a function of, even endogenous to, other prevailing economic and social processes. In fact, more than a decade ago demoeconomic modeling was heralded as the second major research stage of multiregional demography (Isser- man 1985). Though social science theories of migration clearly offer guidance to build demoeconomic models, data limitations, particularly in the United States, have resulted in few recent attempts to pursue demoeconomic modeling since that initial surge of activity.

Demoeconomic models with endogenously determined migration rates can be di- vided into three classes: time-series models, cross-section models, and transition matrix models. Time-series models attempt to relate a series of explanatory economic variables to a time series of migration flows or rates. Early attempts by Ledent (1978),

Stuart H . Sweeney and Kevin]. Konty / 293

Milne (1981), and Plaut (1981) were limited to time series of net migration and the models generally performed poorly (Isserman 1985). Moreover, as stated above, Rogers (1990) has shown that there are theoretical problems with using net migration as an indicator of system dynamics. Those problems are further complicated by the fact that U.S. net migration series are measured as a residual component of population change (Isserman 1985). Despite those problems regional forecasting models such as the REMI model (Treyz 1993) still use time series of net migration. More re- cently the Census Bureau has modeled individual out-migration rates as a function of origin and destination employment rates where both dependent and independent variables are expressed as the first-differences of the log rate (Campbell 1996).

To adequately incorporate the hypotheses of social science theory requires using data with a broad array of individual characteristics. The decennial census long-form data is the only source with a large enough sample to make valid inferences at the state, or substate, spatial scale^.^ This cross-section data can be used to forecast models if a time series of the explanatory variables is available. The first step is to estimate the parameters of the cross-section model, then to predict a time series of flows using the time series of explanatory variables as input to the model’s parameter estimates. There are two problems with this framework: (1) many relevant explanatory variables are only available in the decennial census data, and (2) the parameter estimates from the single cross-section implicitly assume that the relationship between the migration flows and explanatory variables is constant across time. This assumption is referred to as functional stationarity (Rogerson and Plane 1984).

One recent model using the cross-section method is the Norwegian REGARD model (Stambd et al. 1998). This is one of the few forecasting models in the literature that successfully links the multiregional migration model with changes in the external economy. The REGARD model links production and labor demand to a multiregional population submodel. The submodel uses a simple set of labor market indicator explanatory variables to calibrate a multinomial logit model of migration. Changes in the labor market indicators then induce changes in the interregional migration flows. The models validity is dependent on the time stationarity of the cross- sectional relationships measured at the time of the census.

The third approach to demoeconomic modeling is to focus on the interaction between the economy and a time series of transition probability matrices. Rather than using a single cross- section calibration of spatial interaction, the attempt in these models is to relate changes in spatial interaction patterns over time to an external economic time series or other theoretically relevant variables. Isserman et al. (1985) proposed a model that relates an index of attractivity to changes in individual transition probabilities across two periods. The attractivity index is basically a regional versus national share measure calibrated between two adjacent transition matrices.

Nonstationarity through Structural Relationships

Structural methods use a temporal sequence of transition probability matrices to either infer a series of future transition probability matrices or simply characterize the nature of change in the system dynamics over time. Two approaches have been used in this work: (1) those characterizing temporal relationships between individual transition probabilities, and (2) matrix approaches based on overall spatial and temporal relationships (Rogerson and Plane 1984).

3. The most appropriate way to test social science theories of migration is to use Ion tudinal data and there is an enormous body of good literature using methods appropriate to that data (Wifekens 1999b). In the projections context, however, the small sample sizes for the longitudinal data mean that the results are not generalizable to detailed, or even state, spatial scales.


in the paper is how to parsimoniously capture the complexity of the spatial-temporal correlation structure in the data; the resulting models are evaluated using eleven periods to calibrate the model and two periods for out-of-sample validation.

A final statement on attempts to introduce nonstationarity is that most of the effort has been on characterizing change rather than forecasting future matrices. The work by Isserman et al. (1985) explicitly poses a projection framework whereas others (no- tably Dorigo and Tobler 1983; Rogers and Wilson 1996) only allude to the projections problem by suggesting that regularities in their respective frameworks might be ex- ploited toward that end.

NONSTATIONARITY THROUGH DECOMPOSITION

The method proposed here builds on log-linear models while incorporating some elements of the literature discussed above. Rather than using the terms causation or transformation, we feel that a more apt description is decomposition, seeing as the multiplicative models decompose the data as a product of parameters. The primary advantage of the decomposition is that the data is partitioned into time-dependent and time-independent parameters. Finally, recall from the introduction, that the primary goals of the framework are, first, to handle incomplete data by accepting marginal distributions from different data sources; second, to provide short-term forecasts of the transition matrix allowing for disaggregation (for example, sex and age); and third, to provide a mechanism to make the migration rates endogenous to regional economic indicators. The work here is heavily indebted to Willekens’ (for example, 1983,1994,1999a) substantial body of literature on log-linear models and incomplete demographic accounts. The primary extension here is the focus on modeling and forecasting time dependence in interaction matrices.

In this paper we ignore the age and sex decomposition and focus on the three-dimensional data account formed by the origin x destination x time cross-classification. Following earlier work on modeling migration flows as Poisson counts (Flowerdew and Aitkin 1982; Willekens 1999a; Lin 1999), each cell count, nF in the three-dimensional data account can be decomposed into a product of parameters,

The data model in (6) is equivalent to a saturated log-linear model and the estimated parameters (equal to the number of observations) will exactly reproduce the observed data counts, ngt. As noted by Lin (1999) it is possible to use the traditional log- linear modeling approach to search for parsimonious descriptions of the data by examining the model characteristics of subsets of the parameters in (6). In almost all Markov transition matrix applications, the dominance of stayers (resulting in under- prediction of the main diagonal) leads to the use of hybrid log-linear models that con- trol for that dominance through either the introduction of factor variables or the use of mixture distributions (Lindsey 1992,1995).

An important element of (6), alluded to above, is that nqt is a product of time- dependent elements, TFT$’T~?~T$”~, and time-independent elements, TT?T~”T*”. Though the latter group are still dependent on the overall configuration of the Jata and are, therefore, somewhat time dependent, it is not unreasonable to assume that they will be stationary over the short run; an assumption equivalent to the use of the estimated covariance structure in the pooled models of Frees (1992). The substantial cross-sectional variation, noted b Frees (1992), is accounted for partially by ~ f ” and

strength of contribution by eaci of those factors may depend on the particular data partially by TY. The terms, T ~ T . B , further increase the explained variation in nqt. The

Stuart H . Sweeney and KevinJ. Konty / 297

set. But, by focusing on forecasting only the time-dependent parameters, we have reduced the amount of variation in the data, and therefore the degree of difficulty, over predicting the individual time series as attempted by Frees (1992). Stated another way, the multiplicative model explicitly reduces the total amount of variation in the data that is time dependent.

Moreover, if the zgDT contributes relative1 little to the explanatory power of (6), those parameters can be dropped, leaving Z , ? T $ ~ Z , ~ as the only important time-dependent set of parameters. Such a reduction would greatly simplify the forecasting problem. In a N region system there are only N + N + 1 series of length T formed by the reduced set of parameters, as opposed to the NxN series of length T as attempted by Frees (1992).

Nonstationarity in the multiregional growth matrices will be introduced through forecasts of the time-dependent parameters in (6) after a test of statistical signifi- cance. Note that this approach is consistent with Rogers' and Wilson's suggestion to search for regularities in parameters. That suggestion is operationalized here using time-series methods to project the time-dependent parameters. By combining the predicted time-dependent parameters with the time-independent parameters, the predicted origin-destination flows are recovered as

where myt* is the model prediction for the future time period, t*, and the hat symbol indicates projected values from the time-series models. Demoeconomic effects can be introduced by incorporating economic series into the time-series models as explanatory variables.

A second important attribute of the decomposition in (6) is that the form of the model is consistent with parameter estimation methods used for incomplete data problems. Specifically, as shown in work by Willekens (1983, 1999a) and Sweeney (1999), the fact that the marginal distributions are the sufficient statistics used to estimated the parameters of (6) allows one to construct model-based predictions of complete data using marginal distributions from different data sources. One way to do this is through the use of an offset (Sweeney 1999). If we write a new model,

where mqt is the expected value, or model predicted count, and is a prior distribution, then the three-dimensional array of mvt will have the lower-order marginal distributions for 0, D, T , OD, OT, and DT equal to the original array nqt but the odds ratio of the interaction for ODT will be borrowed from the prior distribution nit.

This property, can be confirmed by examining the derivatives of the maximum likelihood estimation. Assuming each cell count is an independent Poisson variate gives the likelihood,

with log-likelihood,


Letting mgt = np.tzzf)~DT~T$'%~TzDT, the log-likelihood is Z ( z , z o , ~ ~ ~ , ~ F , z ~ , z ~ ~ , z ~ ~ l n&) with partial derivatives of the f$m

In these equations (ll), for example, the sufficient statistics are n+++, ni++, and ng+; the others are apparent. Setting the partials equal to zero, the parameter estimates are of the form

A n+++ z =

0 D T OD OT DT ' C C C ' & T i T j T t ~ i j Tit T j t i j t

n,,,

In most statistical software, the parameters are estimated using the complete data ngt and iteratively reweighted least squares following McCullagh and Nelder (1989). The software can be tricked to incorporate marginals from different data sources as shown in Sweeney (1999). A more direct route, however, is the expectation-conditional maximization algorithm (ECM) described by McLachlan and Krishnan (1997) and Willekens (1999a).

The ECM algorithm operates as follows. First, a column vector of the parameter estimates, T = (;,if),. . .,?F)', is initialized as a column of ones. Each iteration over a subset of the parameter vector uses the newly estimated parameters to update the next subset. Computationally, this is accomplished by defining a design matrix D and subsets of the design matrix, D - T ~ and the parameter vector T+, that are missing the columns or elements for the specified subset; in this case the origin-destination effect. For example, the origin-destination parameters are estimated by

At the conclusion of a complete maximization step (M-step) the entire parameter vector has been updated thus defining a new expectation (E-step) of the data, mi, = exp[ln(no) + D * ln(T)], where the superscript n indicates the nth iteration. Also, note that the maximization is conditional on the prior data no.

The ECM algorithm directly uses the marginal distributions rather than the complete data. This means that if the desired multidimensional array is only available as

Stuart H. Sweeney and Kevin]. Konty / 299

marginal distributions dispersed over several different data sets, model-based estimates of the complete data can still be recovered. Of course, the validity of these model-based estimates need to be assessed very carefully (Sweeney 1999). In the case of the interregional population forecasting this means that non-stationarity can be introduced through modeling the origin x destination x time array from matched IRS records, region x age x immigration effects can be introduced from INS records, and age x sex x origin x destination effects can be introduced as stationary distributions from census data. Any interactions that are not available, such as the age x origin x time interaction, will be represented by parameter values equal to 1. There are several other models within this general framework that will be pursued in subsequent work. One alternative is to use a stochastic ECM and embed future periods as un- known parameters as is common in the MCMC literature.

APPLICATION: SMALL AREA FORECASTS FOR CALIFORNIA

The methodology described above is applied in this section to a six-region system in California. The regionalization of California, shown in Figure 2, is motivated by work in biogeography evaluating the population impacts on biologically diverse and sensitive regions in the Sierra Nevada, in particular the fast-growing Tahoe region. To simplify the application, only the gray-shaded subset of the Californian regions is used in the present analysis. The study region contains a fast-growing region in the Sierra Nevada (Tahoe), the two major population centers in California, the greater Los Angeles-Southern California region (South) and San Francisco region (Bay Area), as well as an important growing population center in the central valley (South Sierra). The application here focuses on modeling the time dependence in the origin-

FIG. 2. California Regions and Study Area (shaded gray). Key: 1 = Bay Area, 2 = Sacramento, 3 = Tahoe, 4 = Mother Lode, 5 = South Sierra, 6 = South


destination matrix. The larger problem including age dependence and immigration is left to future work.

The data used in this section come from matched IRS tax returns compiled by the Census Bureau’s Administrative Records Branch. Migration is inferred when the state or county of residence for a particular social security number differs for succes- sive years. The matching results in a state-to-state-flows series and a county-to- county-flows series. The data report the number of returns and the number of dependents; the former serves as a proxy for the number of households and the latter is a proxy for the number of individuals. The IRS migration data series is reviewed elsewhere by Isserman et al. (1985). The data have been criticized because of poten- tial biases due to the nature of income tax reporting and coverage. Yet it should be kept in mind the sample size (though not a probability sample) is enormous, upward of 90 percent of the total population, compared to the census long-form (17 percent) or its subset, the public use microdata sample (5 percent), used in most cross-sectional migration work. Moreover, the emphasis here is not on whether the exact count of population matches some other series, but whether the patterns of spatial and temporal interaction are reflective of reality, a much weaker threshold for validity. The six-region and thirteen-period data array used in this analysis contains 468 cells (6x6~13).

The first step in the modeling framework outlined above is to fit multiplicative models, of form (5) or (6), to the data where the main diagonal has been constrained to equal zero. Table 1 contains fit characteristics for several model specifications. The standard measure of fit for these models is the deviance (a.k.a. the likelihood ratio); lower values indicate a larger proportion of explained variation. The Akaike information criterion (=deviance+e*number of parameter) is also reported. In the top panel of Table 1, both the deviance and AIC drop as additional terms are added to the model. The bottom panel isolates absolute and percentage contribution that model effects make to the reduction in deviance. There are three important conclusions one can draw from the table. First, the contribution of the origin-destination interaction (OD) is profound; 96 percent of the reduction in deviance is due to that term alone. That finding supports the statement by Frees (1992) that cross-sectional variation dominates time series of migration flows. We should be careful to note, however, that the dominance of the origin-destination interaction may be much less in regions that experience substantial economic shocks, say, the Mountain West.

Next, consider the relative contribution of the time-dependent and time-independent components of the model. Approximately 3.3 percent of the deviance reduction is due to time-dependent parameters. This is both good and bad news. The relatively small contribution means that we have greatly reduced the amount of total variation in the origin-destination flows matrices that need to be explained through time. But does that small contribution also mean that the assumption of stationarity is sufficient? No. First, from a statistical modeling perspective, a likelihood ratio test of the time-independent model versus a model including the time-dependent elements suggests that the latter terms are statistically significant. Second, taking a broader view, it is exactly those marginal changes in the interaction matrix through time that are fundamental to forecasting changing population dynamics.

As noted above, the tractability of using the time-dependent parameters for forecasting hinges on the contribution of the three-way interaction, ODT. In fact, it only adds about 0.44 percent to the reduction in deviance. Dropping the ODT term to simplify the model, given the number of time-dependent parameters it would introduce into the forecasting problem, certainly seems justified.6 Moreover, it may also

6. We should also note that linear and quadratic priors were tested as offsets as proxies for the ODT term. In both cases they contributed little to the overall reduction in deviance as shown in Table 2.

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be the case that focusing on the one-way and two-way time-dependent parameters will eliminate some of the noise characteristic of annual migration flows. This finding is certainly tentative given the small scale of the application. Future work should in- vestigate whether the same property is characteristic of the annual series of state-to- state flows.

The second stage of modeling focuses on time-series methods applied to forecasting the time-dependent parameters, z:, zZT, and zF. The main conceptual leap here is that the time- dependent parameters can be regarded as time-series data and, therefore, forecasted using standard time-series methods. As a demonstration, we test eight different time-series model specifications and include a simple linear trend for comparison. One class of models is autoregressive, AR( 1) and AR(2), with either current or lagged unemployment rates as explanatory variables. The choice of unemployment rates as the explanatory variable is somewhat arbitrary; either employment or income series could have also been used. The main point is that by modeling the small set of time-dependent parameters allows for the inclusion of economic covariates and more sophisticated time-series models. The main constraint with the current data is the brevity of the series, only thirteen periods.

In fact, the ideal time-series approach would be vector autoregressions (VAR) since they are capable of simultaneously estimating the interaction between the time paths of the time-dependent parameters, T:, and zF, and the time paths of economic series. Given the short length of our series we can only implement this approach in a limited way using unemployment as the sole economic variable again. For example, the VAR (1J) models are estimated for one model of the form,

and six each of the form,

and,

(14.1)

(14.2)

(14.3)

Tables 2, 3, and 4 contain model fit characteristics and some of the parameter estimates for individuals models are contained in Appendix 1. Following Frees (1992) we report averages of the R2 and adjusted R2 to assist in the model selection. The AR and VAR models easily beat the linear trends. The two best models are the AR(2) with current and lagged unemployment and the VAR(2,2). Ultimately, because of the poor performance of the autoregressive model in the Mother Lode and South Sierra regions we chose the VAR(2,2). A VAR(2,2,2) would be preferable conceptual1 since there would be simultaneous interaction among unemployment, 7tT, and z IT , but there are simply too few observations to attempt that model. It should be noted that although the average adjusted R2 are not spectacular for any of the models, they com- pare very favorably with the 0.15-0.20 range in Frees (1992) paper.

TAB

LE 1

Sp

ecifi

catio

ns of

the

Mul

tiplic

ativ

e M

odel

Log-

likel

ihoo

d M

odel

Satu

rate

d M

odel

D

evia

nce

Obs

Pa

rms

DF

P(ch

i>D

) A

IC

0,D

0.D.T

0D:O

D

O,D

,T,O

D

0.D

.T.O

D.O

T O

;D;T

;OD

;DT

O,D

,T,O

D,O

T,D

T

O,D

,T,O

D,O

T,D

T w

ith p

rior

l**

0.D

.T.O

D.O

T.D

T w

ith ~

rior

2

22.0

97.0

38

20.9

76.9

72

22;0

97:0

38

20:9

87:9

14

22,0

97,0

38

22,0

56,8

34

22,0

97,0

38

22,0

67,7

77

22,0

97,0

38

22,0

78,0

60

22,0

97,0

38

22,0

83,6

26

22,0

97,0

38

22,0

92,1

31

22,0

97,0

38

22,0

93,3

74

22,0

97.0

38

22.0

93.9

90

2.24

0.13

3 2;

218;

248

80,4

08

58,5

23

37.9

57

26,8

24

9,81

5

7,32

8 6.

097

468

13

468

26

468

49

468

62

468

140

468

140

468

218

468

218

468

218

455

442

419

406

328

328

250

250

250

<.00

01

2,24

0,00

0 <.

0001

2,

220,

000

<.OOO

1 80

,506

<.

0001

58

,647

<.

OOO1

38

,237

<.O

OO1

27,1

04

<.00

01

10,2

51

<.OOO

1 7,

764

<.OOO

1 6,

533

Effe

ct

CR

D*

PRD

*

OfD

O

D

T OT

DT

prio

r1

prio

r2

Tim

e-in

depe

nden

t (O

,D,O

D)

Tim

e-de

pend

ent (

T,O

T,D

T)

Une

xpla

ined

(O

DT)

130

0.00

6%

2,15

9,72

5 96

.252

%

21,8

85

0.97

5%

20,5

66

0.91

7%

31,6

99

1.41

3%

2,48

7 0.

111%

3,

718

0.16

6%

2,15

9,85

5 96

.258

%

74,1

50

3.30

5%

9,81

5 0.

437%

NOTE: A

utho

rs c

alcu

latio

ns.

* CR

D=c

ontr

ibut

ion t

o re

duct

ion in

devi

ance

; PR

D=p

erce

nt re

duct

ion i

n de

vian

ce.

** Pr

ior 1

IS a

sim

ple l

inea

r tre

nd of

the

OD

T se

ries

; pri

or 2

is th

e eith

er a linear o

r qua

drat

ic fit

dep

endi

ng on

whi

ch h

as th

e hig

her a

djus

ted-

R2.


be the case that focusing on the one-way and two-way time-dependent parameters will eliminate some of the noise characteristic of annual migration flows. This finding is certainly tentative given the small scale of the application. Future work should in- vestigate whether the same property is characteristic of the annual series of state-to- state flows.

The second stage of modeling focuses on time-series methods applied to forecasting the time-dependent parameters, T:, zgT, and ZY. The main conceptual leap here is that the time- dependent parameters can be regarded as time-series data and, therefore, forecasted using standard time-series methods. As a demonstration, we test eight different time-series model specifications and include a simple linear trend for comparison. One class of models is autoregressive, AR( 1) and AR(2), with either current or lagged unemployment rates as explanatory variables. The choice of unemployment rates as the explanatory variable is somewhat arbitrary; either employment or income series could have also been used. The main point is that by modeling the small set of time-dependent parameters allows for the inclusion of economic covariates and more sophisticated time-series models. The main constraint with the current data is the brevity of the series, only thirteen periods.

In fact, the ideal time-series approach would be vector autoregressions (VAR) since they are capable of simultaneously estimating the interaction between the time paths of the time-dependent parameters, z:, zgT, and z y , and the time paths of economic series. Given the short length of our series we can only implement this approach in a limited way using unemployment as the sole economic variable again. For example, the VAR (1 , l ) models are estimated for one model of the form,

and six each of the form,

and,

(14.1)

(14.2)

(14.3)

Tables 2, 3, and 4 contain model fit characteristics and some of the parameter estimates for individuals models are contained in Appendix 1. Following Frees (1992) we report averages of the R2 and adjusted R2 to assist in the model selection. The AR and VAR models easily beat the linear trends. The two best models are the AR(2) with current and lagged unemployment and the VAR(2,2). Ultimately, because of the poor performance of the autoregressive model in the Mother Lode and South Sierra regions we chose the VAR(2,2). A VAR(2,2,2) would be preferable conceptud since there would be simultaneous interaction among unemployment, zZT, and T&, but there are simply too few observations to attempt that model. It should be noted that although the average adjusted R2 are not spectacular for any of the models, they com- pare very favorably with the 0.15-0.20 range in Frees (1992) paper.

TABLE 2 Time Series Model Fits

AR(l), UNE(0) AR(l) , UNE(1) AR(1). UNE(O), UNE(1) VAR( 1,l) AR(2). UNE(0) AR(2): UNE(1) AR(2), UNE(O), UNE(1) VAR(2,2) Linear trend only

0.355 0.431 0.595 0.554 0.409 0.535 0.728 0.735 0.005

0.226 0.305 0.422 0.465 0.211 0.360 0.547 0.603

-0.086

Overall average 0.435 0.297

NOTE: Al l ( . ) indicates autore ressive with ( ) lags and UNE (.) indicates use of unemplo inent as a regressor usin either the current period (0). the prior period (1) or 110%. VAN., . ) is ;vector autoregression with either one prrioYlags on each variable (I,?) or two lags (22).

TABLE 3 Time Series Model Fits, by Series

Variable

Average adl-R2 Modrl D*T O*T Average

0.163 0.288 0.226 AR(1), UNE(0) AR( l ) , UNE(1) AR(l), UNE(O), UNE( VAR(1,l) AR(Z), UNE(0) AR(2). UNE(1) AR(2); UNE(Oj, UNE( VAR(2,Z) Linear trend only Total

I

0.277 0.333 0.389

0.333 0.511 0.541

0.305 0.422 0.465

0.152 0.270 0.211 0.384 0.337 0.360 0.515 0.578 0.547 0.594 0.612 0.603

-0.083 -0.089 -0.086 0.264 0.329 0.297

Variable

Average R2 Model D*T O*T Average ~

AR(1), UNE(0) ARf1). UNE(1) AR(1); UNE(Oj, UNE(1) VAR( 1,l) AR(2), UNE(0) AR(Z), UNE(1) AR(2). UNE(O), UNE(1) VAR(2.2)

0.303 0.408 0.533 0.491 0.364 0.552 0.709 0.729

0.407 0.454 0.657 0.617 0.453 0.518 0.747 0.741

~~~

0.355 0.431 0.595 0.554 0.409 0.535 0.728 0.735

Linear trend only 0.008 0.002 0.005 Total 0.410 0.460 0.435


TABLE 4 Time Series Model Fits, by Region

Average adj-Re Model’ Bay Sac Tahoe Mother S.Sierra South Grand Total

AR(1). UNE(0) ~ ~ _ _ _ _ _ _ _ _ _ _

0.228 0.377 0.389 0.049 0.142 0.169 0.226 ARilj : UNEilj 0.392 0.605 0.419 0.150 -0.001 0.263 0.305 ~ ~~~ ~~~ ~ ~~~ ~ ~~~ ~ ~~~ ~ ~ ~ , . ~~ ~

AR(l), UNEiOj, UNE(1) 0.738 0.699 0.532 0.081 0.068 0.413 0.422 VAR( 1,l) 0.745 0.800 0.567 0.279 0.038 0.360 0.465 AR(2), UNE(0) 0.159 0.292 0.331 0.071 0.303 0.112 0.211 AR(2), UNE(1) 0.355 0.648 0.371 0.267 0.294 0.228 0.360 AR(2). UNE(O), UNE(1) 0.780 0.707 0.663 0.262 0.321 0.547 0.547 VAR(2,2) 0.729 0.786 0.713 0.578 0.369 0.441 0.603 Linear trend only -0.088 -0.087 -0.087 -0.077 -0.090 -0.084 -0.086

Total 0.395 0.474 0.381 0.158 0.135 0.236 0.297

Average RL Model’ Bay Sac Tahoe Mother SSierra South Grand Total

AR(1), UNE(0) 0.356 0.481 0.491 0.207 0.285 0.307 0.355

AR(l), UNE(O), UNE(1) 0.816 0.790 0.672 0.356 0.347 0.589 0.595 VAR(1,l) 0.788 0.833 0.639 0.399 0.198 0.467 0.554 AR(2), UNE(0) 0.369 0.469 0.498 0.303 0.477 0.334 0.409 AR(2), UNE(1) 0.531 0.744 0.543 0.467 0.487 0.438 0.535 AR(2), UNE(O), UNE(1) 0.868 0.824 0.798 0.557 0.593 0.728 0.728 VAR(2,2) 0.819 0.857 0.809 0.719 0.579 0.628 0.735 Linear trend only 0.003 0.004 0.004 0.012 0.001 0.006 0.005

Total 0.506 0.568 0.498 0.334 0.315 0.390 0.435

AR(l), UNE(1) 0.503 0.677 0.525 0.305 0.181 0.397 0.431

*See note on Table 2.

Given the parameters of the VAR(2,2) model it is possible to forecast the T:, zZT, and T$” parameters and to simulate effects of changes in unemployment. Figures 3, 4, anh 5 contain line graphs of the observed and forecasted values of the parameters. Three regions are highlighted with bold lines: a solid bold line for Tahoe, a bold line with an open box symbol for Bay Area, and bold line with a solid circle for the South. In Figure 3 unemployment and the parameter values are projected simultaneously. Notice that the parameters for the two major population centers track quite closely to each other. Another interesting feature of all the models is that the parameter values track back toward the value of 1 over the projection interval. This is intuitively ap- pealing since it implies that as our information declines we revert back to the average stationary matrix represented in the time-independent parameters.

Figure 4 contains a baseline scenario where unemployment is held constant at the 1998 level and Figure 5 introduces a two-period increase in the unemployment rate during 2000-01. The origin and destination parameters react strongly to the shock with the largest impact in the less populous Tahoe region (solid black line). For Tahoe, both arrivals and departures increase with unemployment, whereas in the Bay Area and South regions departures decrease and arrivals increase. Characteristic of the tangled relationship between unemployment and migration, the six regions react differently to the shock in both direction and magnitude. But, again, the main point of this demonstration is to show the feasability of the approach. Given good forecasts of the time-dependent parameters using economic covariates, the shifts in the entire migration system can be forecast into the future. Extensions to this work are under- way that test the generalizability of our results, examining larger systems (the state- to-state flows) and a more complete set of economic variables.

FIG. 3. VAR(2) Forecasts of Parameters. Key: 1 = Bay Area, 2 = Sacramento, 3 = Tahoe, 4 = Mother Lode, 5 = South Sierra, 6 = South

1 2 1

4 D T - 1 4 -4- DT-2 -DT-3 --A- DT-4 -8- DT-5 + DT-6

1.1 i 0 8 O 9 i

1

I I -

I

0.9

0 8

-0-OT-1 I- OT-2 - OT-3 -4- OT-4 -4- OT-5 + OT-6 I

FIG. 4. VAR(2) Forecasts of Parameters, Constant Unemployment. Key: 1 = Bay Area, 2 = Sacra- mento, 3 = Tahoe, 4 = Mother Lode, 5 = South Sierra, 6 = South

Desiinat&n*Time

1 0.9 ' 0.8 1

I 1 0.7

1 0.6 I Sanple observations ~ I Predictions ~ ~

OrigLn*T@e

Unemploy_ment( Dest) I

1 0.14 I I 0.12 1 1 0.1 I

I

' 0.08 1 ' 0.06 1 1 0.04 , I 0.02 l o

1 -0- DT-1 -4- DT-2 -DT-3 1 +DT-4 1 -+- DT-5 +DT_6

-C-OT-I I l -OT-2 -0T-3 1 + OT-5 +OT-6

-+ or-4

FIG. 5. VAR(2) Forecasts of Parameters, Systemic Shock to All Regions. Key: 1 = Bay Area, 2 = Sacra- mento, 3 = Tahoe, 4 = Mother Lode, 5 = South Sierra, 6 = South


CONCLUSION

There is a long-standing interest among spatially minded demographers and econ- omists to understand how migration systems change over time. There is already a ma- ture literature on this topic, though much of the literature is oriented toward characterizing that change rather than forecasting change. This paper outlines a new approach, building on the earlier work, by blending the structural approach of multiplicative models with a time-series approach to characterize regularities in the time path of change. The framework is promising: first, because it completely character- izes the substantial cross-sectional variation of migration systems and isolates the time dependence in a reduced set of parameters; second, it is suited to the frag- mented data delivery system in the United States; and third, it models migration change as a function of regional economic series. The last attribute is particularly important, since the temporal behavior of economic series is relatively well understood compared to the behavior of migration systems.

The paper concludes with a limited empirical application of the concepts. Though the framework looks promising in the limited application, the validity and usefulness of the approach will depend on larger applications with longer time series.

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~~

~ ~

~~~~

~~

APP

END

IX T

ABL

E 1

Tim

e Se

ries P

aram

eter

s for

AR

(2) a

nd V

AR

(2) S

peci

ficat

ions

Bay

Are

a (1

):

Des

tinat

ion*

Tim

e 0 ri g

i n * T

i m e

Ti

me

AR

VA

R

AR

VA

R

AR

VAR

Mod

el

d d

unem

p 0

0

unem

p t

t un

emp

Con

stan

t 1.

0002

(s

tand

ard e

rror

) (0

.013

6)

-0.0

355

(0.3

632)

(s

.e. la

g 1)

La

g 1

Lag

2 -

(s.e

. lag

2)

Une

mp

Lag1

(s

e)

-0.0

599

(0.3

647)

-

Une

mp

Lag2

-

(se)

1.07

91

0.09

85

(0.0

252)

(0

.011

1)

-0.2

253

-0.0

354

(0.3

654)

(0

.162

3)

-0.0

919

0.03

40

(0.3

244)

(0

.136

1)

3.44

09

1.44

36

(1.1

204)

(0

.353

4)

-0.6

282

-0.4

706

(1.6

286)

(0

.713

3)

0.99

94

(0.0

156)

-0

.012

7 (0

.330

6)

-0.0084

(0.3

326)

0.89

19

(0.0

15)

-0.4

718

(0.3

224)

-3

.978

5 (0

.524

6)

-0.1

601

(0.1

96)

-0.3

205

(1.4

271)

0.10

32

(0.0

0677

) -0

.136

9 (0

.313

5)

1.40

55

(0.3

513)

-0

.081

2 (0

.187

) - 1.

0089

(1

.384

)

1.00

33

0.99

48

(0.0

134)

(0

.006

86)

0.89

40

0.37

94

(0.2

485)

(0

.232

8)

-0.7

942

- 1.

2073

(0

.247

9)

(0.3

204)

-

4.90

39

(2.3

576)

-

-7.2

457

(2.3

453)

0.06

72

(0.0

0308

) 0.

0934

(0

.034

7)

-0.0

540

(0.0

459)

1.

4792

(0

.336

8)

-0.6

207

(0,3

544)

Sacr

amen

to (2

):

Des

tinat

ione

Tim

e O

rigin

*Tim

e Ti

me

AR

VAR

AR

VA

R A

R

VAR

Mod

el

d d

unem

p 0

0

unem

p t

t un

emp

Con

stan

t (s

tand

ard e

rror

) La

g 1

(s.e

. lag

1)

Lag

2 (s

.e. la

g 2)

Une

mp

Lag2

(s

e)

0.99

36

(0.0

293)

0.

4030

(0

.331

5)

-0.1

743

(0.3

32)

0.85

67

(0.0

338)

-0

.189

1 (0

.442

) -0

.202

3 (0

.245

7)

-6.0

316

(1.4

683)

-0

.010

2 (2

.529

3)

0.10

29

(0.0

0924

) 0.

0571

(0

.148

5)

0.03

78

(0.0

974)

1.

3288

(0

.382

) 0.

0333

(0

.892

4)

1.00

35

(0.0

158)

0.

2052

(0

.318

9)

(0.3

199)

-0

.224

5

1.07

99

(0.0

206)

(0.3

538)

(0.2

734)

3.

9715

(0

.925

3)

1.28

35

(2.0

894)

-0.5

152

-0.2

585

0.09

63

(0.0

0738

) -0

.165

0 (0

.186

4)

-0.0

149

(0.1

36)

1.34

77

(0.3

809)

0.

1341

( 1

.092

9)

1.00

33

0.99

83

(0.0

134)

(0

.005

47)

0.89

40

0.59

99

(0.2

485)

(0

.258

6)

-0.7

942

-1.3

774

(0.2

479)

(0

.425

3)

-

4.89

70

(2.7

407)

-

-6.0

083

(2.5

139)

0.07

20

(0.0

0342

) 0.

1250

(0

.035

7)

0.97

76

(0.3

807)

-0

.006

8 (0

.059

3)

-0.1

658

(0.3

469)

Tah

oe (3):

~ ~~

~

Des

bnah

on*E

me

Ong

n*T

ime

Tim

e

AR

VAR

AR

VAR

AR

VAR

d d

unem

p 0

0

unem

p t

t un

emp

Mod

el

Con

stan

t (s

tand

ard e

rror

) 0.

9973

(0

.042

7)

0.62

55

(0.0

928)

-0

.074

9 1.

0064

(0

.045

6)

(0.0

437)

1.

2159

(0

.043

6)

(0.4

703)

(0.3

005)

-0.4

055

-0.0

886

~ ~~

0.99

86

(0.0

0609

) 0.

6866

(0

.262

2)

-1.4

333

(0.4

506)

4.

4270

(2

.495

1)

-5.2

439

(2.1

95)

1.00

33

0.89

40

(0.2

485)

(0.2

479)

(0,0

134)

- 0.

7942

0.06

55

(0.0

0412

) 0.

1461

(0

.040

8)

-0.0

134

(0.0

714)

0.09

84

(0.0

0936

) -0

.030

8

-0.0

052

(0.0

928)

(0.1

234)

La

g 1

(s.e

. lag

1)

%2

(s

.e. la

g 2)

U

nem

p L

agl

(se)

U

nem

p La

g2

(se)

-0.0

201

(0.3

236)

0.

7708

(0

.252

2)

0.02

63

(0.1

296)

0.

1857

(0

.331

7)

-0.0

327

(0.3

236)

-0

.090

2 (0

.175

2)

0.02

52

0.03

54

(0.0

914)

(0

.331

6)

0.95

71

-

(0.3

538)

-0

.105

3 -

(0.5

295)

3.84

75

(1.4

202)

9.

7530

(2

.041

6)

1.36

61

(0.3

922)

1.

0177

(0

.394

3)

-2.9

645

(1.4

378)

-1

.195

6 (3

.929

6)

-0.1

954

(1.0

943)

- 0.

1782

(0

.347

8)

Mot

her

Lod

e (4

):

Des

tinat

ion*

T,m

e O

rigi

n*Ti

me

Th

e

AR

VA

R AR

VA

R AR

VA

R M

odel

d

d un

emp

0

0

unem

p t

t un

emp

Con

stan

t 0.

9938

(s

tand

ard e

rror

) (0

.016

3)

Lag

1

-0.1

862

(s.e

. lag

1)

(0.3

162)

La

g 2

-0.3

898

(s.e

. lag

2)

(0.3

162)

U

nem

p L

agl

-

(se)

U

nem

p La

g2

-

be)

0.91

36

(0.0

447)

0.

1071

(0

.245

1)

-0.5

765

(0.2

415)

4.

1020

(1

.486

5)

-2.5

540

(1.2

449)

0.03

87

0.06

42

(0.0

892)

-0

.020

0 (0

.087

1)

0.94

78

(0.3

579)

-0

.002

0 (0

.379

8)

(0.0

202)

1.

0035

1.

1204

(0

.025

5)

(0.0

377)

0.

1609

(0

.018

) 1.

0033

(0

.013

4)

0.89

40

(0.2

485)

-0

.794

2 (0

.247

9)

0.99

87

(0.0

078)

0.

1069

(0

.006

28)

0.10

16

(0.0

565)

0.

0088

(0

.068

8)

1.05

49

(0.3

464)

-0

.255

9 (0

.344

8)

0.11

66

0.05

85

(0.3

38)

(0.3

975)

0.

0125

(0

.195

) 0.

7532

(0

.244

3)

-0.0

401

-0.0

012

(0.3

399)

(0

.275

9)

-

3.95

71

(1.0

929)

-

-1.8

400

(1.4

808)

0.06

03

(0.1

507)

1.

2846

(0

.354

9)

-0.4

926

(0.7

889)

-1.1

076

(0.3

012)

2.

4871

(1

.538

3)

-3.1

354

( 1.5

238)

APP

END

IX T

AB

LE 1

(con

tinue

d)

Sou

th S

ierr

a (5

):

Des

tinat

ion*

Tim

e O

rigi

n*T

ime

lim

e

AR

VAR

AR

VA

R AR

VA

R d

d un

emp

n

n

nnem

p t

t un

emp

Mod

el

Con

stan

t (s

tand

ard e

rror

) 1.

0080

1.

0586

(0

.018

9)

(0.0

312)

0.

7584

0.

7796

(0

.260

7)

(0.2

162)

0.18

17

0.08

59

(0.1

051)

-0

.090

2 (0

.102

3)

1.25

91

(0.3

266)

-0

.347

5 (0

.348

4)

(0,0

199)

1.o

O01

(0

.008

45)

0.28

78

(0.2

925)

-0

.390

3 (0

.293

6)

1.00

43

(0.0

0687

) 0.

0731

(0

.335

6)

(0.3

415)

0.

6205

(0

.848

6)

0.26

69

(0.8

663)

-0.5

525

0.13

74

(0.0

0753

) -0

.094

0 (0

.103

9)

-0.1

107

(0.1

049)

1.

4154

(0

.262

6)

1.00

33

0.89

40

(0.2

485)

(0.2

479)

(0.0

134)

-0.7

942

0.99

75

(0.0

066)

~~

0.12

85

(0.0

0634

) 0.

1180

(0

.068

) -0

.068

0 (0

.069

8)

1.28

06

(0.3

57)

-0.4

615

(0.3

635)

0.48

61

(0.2

516)

L

ag2 -

(s.e

. lag

2)

Une

mp

Lag1

(s

e)

-0.7

234

-0.8

364

(0.2

607)

(0

.206

2)

- 1.

0605

(0

.259

3)

2.89

42

(1.3

525)

-

1.88

71

(0.7

885)

(0.7

95)

-

-0.7

467

Une

mp

Lag2

(s

e)

-0.5

753

(0.2

709)

-3

.594

0 (1

.368

7)

Sou

th (6

):

Des

tinat

inn*

Tir

ne

Ori

gin8

Tim

e T

ie

AR

VAR

AR

VAR

AR

VAR

Model

d d

unem

p 0

0

iem

p

t t

unem

p

Con

stan

t (s

tand

ard e

rror

) 1.

0038

1.

0450

(0

.012

9)

(0.0

249)

0.

0821

(0

.007

4)

1.00

06

1.00

23

(0.0

0455

) (0

.002

83)

-0.1

551

-0.5

381

(0.3

247)

(0

.278

6)

-0.0

440

-0.3

261

(0.3

267)

(0

.285

5)

-

- 1.

5666

(0

.453

2)

-

0.85

37

(0.4

473)

~_

__

_~

0.06

27

(0.0

0456

) 0.

0166

(0

.140

7)

0.12

62

1.50

39

(0.2

287)

-0

.734

8

(0.1

44)

(0.2

24)

1.oo

oO

(0.0

0742

) 0.

5204

(0

.284

9)

-1.0

501

(0.3

327)

0.06

76

(0.0

0401

) 0.

0792

(0

.042

6)

(0.0

496)

-0

.051

6

1.00

33

0.89

40

(0.2

485)

(0.2

479)

(0,0

134)

-0.7

942

Lag

1

(s.e

. lag

1)

0.45

49

0.39

04

(0.3

201)

(0

.333

) 0.

0205

(0

.130

3)

-0.2

336

0.31

32

(0.3

201)

(0

.312

7)

-

2.64

62

(1.0

509)

(1.3

691)

-

- 1.

4339

0.04

79

(0.1

031)

1.

5478

(0

.306

7)

(0.4

361)

-0

.887

4

3.12

09

(2.1

476)

1.

4522

(0

.321

3)

Une

mp

Lag2

(s

e)

-4.0

385

(2.0

972)

-0

.687

5 (0

.314

2)

population forecasting with nonstationary multiregional growth matrices

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