10. Spatial Eco No Metric Models- The Empirical Experience

8/7/2019 10. Spatial Eco No Metric Models- The Empirical Experience

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Statistics:the mathematical theory

of ignorance.Morris Kline

CHAPTER NINE

SPATIAL ECONOM ETRIC M ODELS:THE EMPIRICAL EXPERIENCE

Previous discussions on activity-allocation-and-derivation models have concen-

trated on gravity-interaction and econom ic-base concepts. Largely, these are w e ll -structured modelling approaches founded on generally accepted theories of spatialdevelopment. It was observed that more interaction takes place morein closerproximity and the economy is fueled by the multiplier-effect of each dollar ofinvestment. There is a parallel school of thought about the modelling process;however, in which empirical relationships are established between observedeconom ic activities suchas population and employm ent. Econom etric or correlativeequations are then constructed to explain the interaction between these activities.The EMPIRIC model is one of the most widely acclaimed urban models in the

1970s and 1980s. This chapter will review its development and application, andthrough such discussions, we can reexam ine the different approach to comp rehen-sive spatial mode lling ingeneral. It complement sthe more causal mod els presentedin previous ch apters.The chapter starts out with a review of spatial econometrics. This special branch

of statistics has been evo lving rapidly in recent years , fueled ma inly by applicationsranging from image processing to renewed interests in geo-statistics.Thediscussion here can be viewed as part one of two parts. Part one concentrates on"spatial dependence" or the empirical relationship between adjacent geographic

units; such analyses are based mainly oncross-sectionaldata. Part two delves intospatial-temporal models constructed upontime-series information. The idea ofspatial dependence is not new, since we have seen many examplesof spatialrelationship already ranging from remote-sensing applicationsto gravitationalinteraction. The focus here is simply to formalize the information presented thusfar, casting an economic framework within which diverse applications can findahome.

9.1 Spatial econom etrics

In general, one can think of spatial econometrics as dealing withspatial effects,ofwhich there are two kinds:spatial dependenceand spatial heterogeneity.Spatialdependence is best manifested by Tobler's "first law of geography ."This law says


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Location, Transport & Land-use 559

that "everything is related to everything else, but near things are more related thandistant things." Here the notion o f'nearne ss' g oes well beyond Euclidean space, aswe have discussed previously in the "Spatial separation" chapter. Spatialheterogeneity, on the other hand, takes exception to homogeneous dependence

among all points in space. While it does not violate Tobler's "first law," itintroduces additional complexity in modelling in that spatial relationships differfrom one point to another (Anselin 1988).

9.1.1 Spatial dependence vs. heterogeneity.Perhaps the best know n statisticto measure spatial dependence isspatial autocorrelation,defined as the relation-ship or correlation1 between data points in specified orientation in space. It isanalog ous to classic Pearson co rrelation and particularly autocorrelation over time.But the multi-directional nature of spatial dependenceas contrasted with say a

one-d irectional situation in timenecessitates the use of a different me thod olog icalframework.The second type of spatial effect, spatial heterogeneity, suggests the lack of

stability over space in the behavioral or other relationships under study. Moreprecisely, this implies that functional forms and parameters vary with location andare not hom ogen eous throu ghou t the data set. This is likely to occur in econo metricmodels estimated on a cross-sectional data seti.e., data taken at one point in time(a 'snapshot')of dissimilar spatial units. For example, traffic zones are typicallydrastically different in size and shape at the city center compared to suburbs (w ith

those at the center smaller than the suburban ones). Suppose a m odel is calibratedfor a base-year. A separate model may have to be calibrated in the central city vicethe subu rbs, since thehomogeneityassumption may be violated. In contrast to thespatial dependence case, the problems caused by spatial heterogeneity can for themost part be solved by means of standard econometric techniques. An example isthe use of dichotomou s 0 -1 variables to 'switch' betweena central city and suburbanmodel. Specifically, methods that pertain to varying parameters, randomcoefficients and structural instability can easily be adopted to take into accou nt suchvariation over space. W e will illustrate how this can b e don e in sequel.

9.1.2 Connec tivity in space.The very notion of spatial dependence impliesthe need to decide which other units in space have an influence on the p articularunit under consideration. Form ally, this is expressed in the topological notions ofneighborhood and nearest neighbor. This concept has been illustrated in the"Geo graphic Information System s" chapter (Chan 2000). In raster files of digitizedimages, we pointed out that in such a grid system, a pixel has either four imm ediateneighbors each sharing a common side with the subject pixel, or eight adjoiningneighbors, including the ones at the four corners. (Refer to the "Raster datastructure example" figure in the chapter.) The four sharing a side each are closer

*For a formal definition of correlation, see the "Statistical Tools" appe ndix in Chan (20 00).Autocorrelation simply extends the concept from the relationship between data pointsi and j to datapoints between periodt and M-l, as will be explained m ore formally in sequel.


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560 9. Spatial Econom etric Mod els

in Euclidean distance than the other four at the corners. W e referred to the formeras first-orderneighbors and the latter assecond-orderneighbo rs. In analogy to thegame of chess, we also call this therook and bishop case respectively.

In general, consider a system ofri spatial units, labelled / = 1, 2, ...,n \ and a

vector measure x with a valuext observed for each of these spatial units. Thevariable x may represent the grey values in satellite images for example. In suchapplications as image restoration in raster-files, a set of neigh bors ofa spatial uniti can be defined as the collection of those unitsj for which Xj is contained in thefunctional form of the probability ofxi9 conditional upon x at all other locations.Formally, this would yield the set of neighbors fori as Nj, for which P(Xj\x) =P(Xj | xNf) wh ere V is the vector ofXj observations, for a ll / in JVJ, and x is the vector

containing allxk (ki) values in the entire system. Alternatively, and less strictly,the set of neighbo rs^ fori can be taken as {/1P(x,) *P(xt \ xj)}. These are the locationsfor which the conditional marginal probability forxt is not equal to the uncondi-tional marginal probab ility. A s m entioned, application of this conceptis graphicallyillustrated in the Bayesian method of image classification. This is discussed in the"Geographic Information Systems" chapter of Chan (200 0), where the rth pixel isallocated based on an updated probability depending on the grey value of theneighboring pixels j . Note that, in general, neither of these neighborhooddefinitions includes information about the relative location o f the two spa tial un its,but only the neighbor's influence via conditional probabilities.

In order to introduce a spatial aspect to these definitions of neighbors, which

also m akes the link w ith the notionsof spatial stochastic processes,this workingdefinition is suggested: {y|P(x:)*/>(*,|xy) and dt]


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Location^ Transport & Land-use 561

spatial units. The larger the separation, the less the interaction is expected andhence the smaller the weight. This is expressed in a spatial weight matrix W =[w^]for all pairs of geographic units in the study area. The determination of the properspecification for the elements of this matrixwfj is a more difficult and controversial

methodological issue in spatial econometrics. For example, one can use acombination of distance measures and therelative length of the common borderbetween two spatial units. The resulting weights will be asymmetric, unless bothspatial units i and j have the same relative common boundary length, and thedistance between the two units is the same both way s(dtj = d^). Specifically,

where s^ is the proportion of the interior boun dary of uniti that is in contact withunity, and a an d b are calibration param eters. To the extent thats^s^ or dg*d Jt in

general, one cannot expectw tJ = w JU In gravitational interaction, such a relationsh ipis amply illustrated, with say population at /(Nt) and employment at /(Ej) as proxyfor the geometric quantitystj. W e can think ofstj = {Nfijf in this case.

In a similar way, weights can also take into account the relative area of thespatial units:

with Sjj as a binary contiguity factor(Stj = 1 if units / andj arc neighbors and 0otherwise,) and 0t is the share of unit / in the total area of all spatial units in the

system. A competitor of gravity model, the Intervening Opportunity Model,illustrates this weight definition. Here "annular rings" (or zones of influence) aresequenced in terms of distance away from the subject zone, each characterized bythe number of'opportunities' which are in effect proxies for 'area'. Each unit in aring shares a fraction of the residential or employm ent 'opportunities' depending onthe 'size9 of the ring. The exam ple in the "Descriptive Tools" chapter (Chan 200 0)has an employment origin at downtown, and there are five destinations in theneighboring first ring, and the probability of residential location at any destinationis 1/2. The probability of pop ulation living in any first-ring destination is simply9 = e-v2(i)_ e-v2(6) _ Q 555 jjer e the probability of wo rking ati and living in ringy canbe interpreted aswip Sg = 1 between downtown and the first ring, andstJ = 1 sincethe ring com pletely surrounds the downtown origin.

Both weight definitions, in their original design, are closely linked to thephysical features of spatial units on a map, or its digitized image. As with thebinary contiguity measures, they are less useful when the spatial units consist ofdiscrete points or a lattice, since then the notions of boundary length and area arelargely artificial. They are also less meaningful when the spatial-interactionphen om enon un der consideration is determined by purely econom ic factors, w hichmay have little to do w ith the spatial pattern of bound aries on a physical map . T hisis again illustrated by the gravitational-interaction example ab ove, wh ere econom icactivities such as population and employment have to be used as proxies for theboundary variable.


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562 9. Spatial Econom etric Models

Con sequently, som e have argued for weights with a m ore direct relation to th eparticular phenomenon under study. For example, a general accessibility weight,calibrated between 0 and 1, combines in a logistic function the influence of one ormore modal linkages between two regionsincluding roads, railways, and other

me ans of transport. Formally,

in which kj shows the relative impo rtance of the transportation m o d e/ . Th e sum isover all the m ode s, each of w hich separates the spatial units by a d istance ofd tj ata different unit costCj. a an d p are calibration constants. Similar to Equation (9-2),it should be noted that a smaller weight results from a longer 'distance' separation,

wh ich is really time separation in transportation m odes.An important problem results from the incorporation of calibration parametersin the weights. Typically, these weights are taken to be exogenous and theparameter values are determined prior to, or in steps separate from, the rest of thespatial analysis. This creates problems for estimating and interpreting the results.In particular, it could potentially lead to the inference ofspurious relationships.Remember the validity of estimates is pre-conditioned by the extent to which thespatial structure is correctly reflected in the weights. More impo rtant, it could resultin a circular reasoning, in that the spatial structure, which the analyst may wish to

discover in the data, has to be assumed known before the data analysis is carriedout.When the weight matrixdetermined say by Equations (9-1), (9-2) and (9-

3)is used in a h ypothesis test, the null hy pothesis2 is one of spatial indep endence.Ideally, the w eight m atrix should be related to the relevant, alternative hypo thesisof spatial depend ence, in order to maxim ize the power of the test. How ever, ev enwith an improperly specified weight matrix, a conservative interpretation of arejection of the null will only imply a lack of independence, and not a particulartype of dependence. This practice has its redeeming value however. Although thepower of the test may be compromised, the potential for spurious conclusions isminimized. Unfortunately, application of such a hypothesis test is not simple inpractice.

9.1.4 Spatial lag operators. The ultimate objective of a spatial weight matrixis to relate a variable at one point in space to the observations for that variable inother spatial units. In the more familiar time-series context, one relates theobservation in time periodt to the observation to periodt-k. This is achieved byusing a lag operator, wh ich shifts the variable by one or more periods in time. F orexample,xt_k = L(lc)xt shows the variablesxt shifted k periods back from timet, as aresult of the M i pow er of the lag operator X. No w the time-seriesxt can be related

2For an explanation of a null hypothesis, see the "Statistical Tools" appendix to Chan (2000).


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(a) Shift to first-order neighbors

4(1+1,./)

(b) Shift to( / - I , 7+1)

0"-1 ,7-

Figure 9-1 Illustrating spatial lags

to tim e-series L(2)xt by regression if desired, for such purposes as forecasting. (Thecorrelation coefficient b etween the two time series is the autocorrelation coeffi-cient.)

W hile it is similar in concep t, matters are no t this straightforward in space du eto the many directions in which the shift can take place. As an illustration, co nsiderthe discrete lattice structure in F ig ur e9-1. The v ariablex, observed at location (//),can be shifted in separate ways using two simple contiguity criteria. If the first-order neigh bors are the positions considered for the shift, the alternate positions arexi-ij> xu-i9 xi+\,p xu+i- If the second-order neighbo rs are the shift p ositions, o n theother hand, the spatial-lag p ositions arex^j.^ xmj.l9 xi+lJ+l, x ^ ^ . For an eight-connected-neighbors2 shift, as defined in the "Geographic Information Systems"chapter (Chan 2 000 ), we are talking about the com bination of the four positions inFigure 9-l(a) and the four positions in Figure 9-l(b). Thus three different direc-tions of spatial lag can be easily defined even for a very simple example such as

3The eight-connected neighbors are analogous to the eight more positions of a queen on a chess board.


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564 9. Spatial Econometric Models

this. Corresponding to each of these directions, correlation parameters such asautocorrelation coefficient need to be estimated respectively.

In most applied situations, there are no strong a priori motivations to g uide thechoice of the relevant form of spatial dependence. This problem is compounded

when the spatial arrangement of observations is irregular, since then an infinitenum ber o f directional shifts beco mes possible. Clearly in any statistical analysis, thenum ber of parameters associated w ith all shifted positions qu ickly w ould becom eunw ieldy and preclude any meaningful mo del to be constructed. Moreov er, unlessthe data set is very large and structured in a regular way, the remainingdegrees-of-freedom w ould be insufficient to allow an efficient estimation of these param eters.In the extreme case, this would be analogous to calibrating a two-variable linearregression line through tw o d ata points, where no degree of freedom is left.4

This problem is resolved by considering a weighted sum ofall values belonging

to a given contiguity class, rather than taking each of them individually. Th e termsof the sum are obtained by multiplying the observations in question by theassociated weigh t from the spatial-weight m atrix. Formally,

z ( 0 * / = ^ / w / y * / (9-4)

where Lil) is the spatial-lag op erator associated with contiguity classIJ is the indexof the observations belonging to the contiguity class forj ,N\. In matrix form, thiswould be IPXJ = w (/ )x. Thus in the lattice ofFigure 9-1, either the first-orderneighbors are treated as a group, or the second-order neighbors, or the eight-conn ected-ne ighbors. Then a set of four weigh ts wo uld be defined for both the firstand second order neighbors and a set of eight weights for the eight-connectedneighb ors. The spatial lag of observationxi9 L(l)xf (/ = 1, 2 or 3), would simply b ethe weighted sum of the four or eight neighbors.

W e hav e seen exam ples of the spatial-lag operator in spatial-filter app lications.In the "Rem ote Sensing" chapter (Chan 200 0), we have seen that irrespective of thetype o f image filters used, the basic approach is to sum prod ucts between th e m askcoefficients and the intensities of the pixels under the mask. In a 3x3 mask, forexamp le, there are n ine grey levels of pixels under the mask counting the subjectpixel and the 8-connected neighbors, namelyxl9 xl9 ..., x9. The filter simply usesEquation (9-4) to produce a weighted sum for the subject pixel aswxxx + WJX 2 + ...+ w^c9. The g rey value of the pixel at the center of the mask is then replaced by thisweighted average. The mask is then m oved to the next pixel location in the im ageand the process is repeated. This continues until all pixel locations have beencovered . A lo w -pass spatial-filter' for exam ple, will simply average the pixe ls in theneighborhood, or set all mask coefficientsw to 1, resulting inxl9 x2, ... , x9. Notsurprisingly, the effect is to blur the image, thus removing 'specks' of noisescattered around the image if desired. Clearly, the resulting notion of a spatiallylagged variable is not the same as in time-series analysis. Instead, it is similar to a

4For more discussion of this problem, please consult the "Statistical Tools" appendix to Chan (2000).


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distributed lag such as exponential smoothing in forecasting, where recentobservations are weighted mo re heavily than those in the distant past according toa pre-determined set of weights. It is important to note that the weights used in theconstruction of the lagged variables are taken as given, jus t asa particular time path

can be imposed in estimating a distributed lag. The joint determination of theweights and measures of statistical association, such as correlation or regressioncoefficients, b ecom es a non linearproblem.How ever, by fixing the weightsa priori,this is reduced to a more manageable linear problemat the risk of imposing apotentially wrong structure as explained in the previous section.

Besides the pre-determined weights used in filter m asks, we have seen severalgeneral examples of this practice previously. Recall that in the "Generation,Com petition and Distribution" chapter, a case study of Long Island State Parks wasperformed in Ne w Y ork, where an additive utility function comb ining distance and

travel time n eed to be calibrated a priori:w^distance) + w2(time)+ constant. Herethe weighted sum of distance fromi and the time from i is computed, combin-ingnot unlike a filter maskthe features of both spatially correlated attributescentered at location z. No tice that this was performed exogenous to the mo del byvaluating time using previous studies. Similarly car-operating-expenses per mile(km) were also found exogenously using established cost figures. In the chapter on"Chaos, Catastrophe, Bifurcation," a trip-distribution function showing thepercen tage of trips of 5-minute, 10-minute, 15-minute duration s,etc., was obtained.It was needed as part of an accessibility function for allocating population and

employment in a disaggregate Lowry-Garin Model. This trip-distributionfunctionan exam ple of wh ich appears asconstant + c^duratiori) + c2(duratiori)2-was again calibrated outside the mod el. Here, the trip-distribution functiongenerates a set of weigh tswf for the (/) = 10-minute, (/) = 15-minuteetc. contiguityclasses in the weight matrix w(/ ), as shown in the "Activity Allocation andDerivation" chapter. Additional examples can be cited. But these two, one fromfacility location and the other from land use, represent an adequate illustration ofgeneralized spatially-lagged variables and how they are determined independent ofmeasures o f statistical association in a m o de l

Following this latter example, less-restrictive spatially-lagged-variable can beconstructed from the notion of potential or accessibility. As one recalls, accessibil-ity is the weighted trip-distribution function where the weights are population,employment or other activity variablesxy x. = Y,jMj> b)xj wh ere / is a trip-distribution function of distance and a vecto r of coefficients b . One can think of aweight matrix to the extent that b is a function of x =(


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A related concept to spatial-lag operators iscellular automata.It is tradition-ally used to represent a dynamical system in terms of discrete economic-variables.In this context, it describes the values assumed by a large number of identical cellswith local interactions among them. In this book, they can consist of a lattice of

sites, each w ith a finite set of possiblevalues. A set of discrete rules is now definedfor the evolution of a spatial economy. The values assumed at each site evolvesynchronously in discrete time-steps according to identical rules. The value of aparticular site is decided by the previous values ofa neighborhood ofsites aroundit. For example, if the system is in state / and a specific criterion is met, then thestate changes to j . Cellular automata may be divided into basic classes withdifferent behavior. It is throug h this taxonomy that it appeals to mo delers. Th eseautomata can occasionally generate chaotic trajectories over time. A majordrawback of these models, however, is a lack of rigorous connection between

physical laws in the continuum one is trying to model and the symbol rule thatgenerates the iterated coupled-cells. But it holds promise to explain the type ofspatial-temporal process discussed in this book, as demonstrated in examples ofspatial interaction and image processing above.

9.1.5 Problem s with modelling spatial effects. A substantial part of spatialeconometric analysis is based on data collected for spatial units with irregular andarbitrary boundaries, such as political subdivisions and traffic zones. Nonetheless,the interpretation of the various models and their policy implications are often

made spatially in general. This implies that there is some unique and identifiablespatial structure, with clear statistical properties, independent from the way inwhich the data are organized in spatial units. Unfortunately, matters are not thisstraightforwardthe way subareal boundaries are drawn has a direct bearing uponstructure and the interpretation ofa mo del. We have already alluded to this earlierin this chapter, but it bears repeating.

The modifiable-areal-unitproblem addresses the fact that statistical measuresfor cross-sectional data are sensitive to the way in which the spatial units areorganized. Specifically, the level-of-aggregation and the way contiguous units arecombined affects the various measures of association, such as spatial-autocorrela-tion coefficients and parameters in a regression mo del. This problem is an old on e,and has been referred to as the micro-macro aggregation problem in econo metricsand the ecolog ical fallacy prob lem in sociology. The modifiable-areal-unit prob lemcan best be explained as a combination of two familiar prob lems in econom etrics:aggregation an d identification.

9.1.5.1 SPATIAL AGGRE GATION. AS is well known, aggregation is onlymeaningful if the underlying phenomenon is homogeneous across the units ofobservation. If this is not true the inherent heterogeneity and structural instabilityshould be accounted for in the various aggregation schemes (Chan 1991). In otherwords, unless there is a homogeneous spatial process underlying the data, anyaggregation will tend to neutralize variation in the data and lead to misleadingresults. Because a model's coefficients are found by explaining variations in


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observed data, the less the variation to be explained in aggregated data, the lessreliable the model will be. The reduced variability in aggregate data also results ina high level of collinearity between variables at the aggregate level, which does no texist in the disaggregate level. It would impart bias to estimated coefficients. For

examp le, a coefficient of -0 .4 may b e estimated using disaggregate data, whereasa value of -0 .3 m ay be obtained for the same parameter using aggregate data.

The concept of spatial aggregation goes well beyond the straightforwarddefinition of areal units such as census tracts or traffic zones. When one employsEq uation (9 -4) to 'pre-process' the data, one also has performed spatial aggregation .For exam ple, the m emb ership in the contiguity class for areal uniti, N }9 can varydepending on whether a 4-connected neighbor is defined, or an 8-connectedneighbo r; there is no reason one cannot go beyond a 3x3 'mask' to a 4x4 , 5x5 andso on. Depend ing on the definition adopted, the resulting data base would look very

different. In short, spatial aggregation introduces a bias on the model-buildingprocess. To account for this bias, this aspect of the modifiable-areal-unit problemshould be considered a specification issue, related to the form of spatial hetero gen e-ity. It is no t solely an issue determined b y the spatial organization o f the data. A lsoan integral part of model specification ismodel identification,which can be thoughtof as the ability to discern the und erlying m odel structure.

9.1.5.2 IDENTIFICATION DEFINED. Consider a simultaneous-equation m odelas illustrated in the "Descriptive Tools" chapter (Chan 2000 ) under the "Econom et-

rics approach" section. The identification problem arises when a system of morethan o ne equation con tains two-way causality5, so that there is a sort of imbalancebetween the dependent{endogenous)and the independent{exogenous)variables.Since the exo genous v ariables are the source of information for estimating m odelparameters, too few of them might preclude parameter estimation of all theendogenous variables. Equations that do not contain the sufficient number ofexogenou s variables, and wh ose parameters cannot be estimated, are calledunder-identified equations.

The identification problem can be illustrated by a simple example (Kanafani

1983). Suppose a demand and supply model system is given by

(9-5)

where V is the inter-regional trade,p" is the unit price, a0, a1? po an d $Y arestructural parameters to be estimated, andaD and a s are error or disturbance termsin each of these two equations. Both equations are said to beunder-identifiedsinceit is not possible to estimate their parameter values from empirical observations.

5For a discussion of the confusion between causality and correlation, see the sub-section on "Arrowdiagrams" in the "Descriptive Methods" chapter (Chan 2000). It is explained in the appendix that thedirection of the arrow shows causality, while correlation between two factors exists irrespective of thearrow direction.


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568 9. SpatialEconometric Models

TradeV

(a) Unidentified

Model K

Pnce p

TradeV

(b) IdentifiedDemandFunction

Price p

TradeV

(c) IdentifiedSupplyFunction

Price p"

Figure 9-2 Under-identified and exactly-identified systems (Kanafani 1983)

There are simply not enough exogenous variable in both equations. Referring toFigure 9-2(a), it can be seen that the two equ ations are to be calibrated by a clusterof data at around only "one point"(Ve,pfJ). Hen ce, these observations on trade andprice cannot be used to estimate parameters of either equation, since they willprovide only "one point" on each cu rve. Observations thus obtained w ill result inan average of the location of the equilibrium point, and a rather poor average atthat, as suggested by the possible scatter in the data illustrated inFigure 9-2(a).This scatter is reflected by the error termsaD and a s. In order to identify either thedemand equation or the supply equation, it is necessary for this model to identifyan additional exo genou s variable. For the sake of fixing ideas, we will assume th eerror terms aD an d a s to be negligible (or set at zero value) from th is po int forwarduntil further notice.

As illustrated in Figure 9-2(b), an additional exogenous v ariable, sayxs thatcontributes toward locating the supply function b ut not that of the dem and function,will allow making observations of trade and price at different levels ofxs. Theseobservations made on different supply curves will permit estimating the demand


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Location^ Transport & Lund-use 569

curve. This is analogous to adding an exogenous variable in the mod el, which w illchange it to

When xs does not influence the demand, i.e., wh en a2 = 0, it becomes possible tomake observations of V and p' at different values ofxs. This way one obtainsobservations of the equilibrium points at different supply curves. Such observationswill allow the identification of the dem and function, as illustrated inFigure 9-2(b).In other wo rds, a0 and c^ can now be estimated.

Similarly, to estimate the supply function, it would be necessary to have anadditional exo genou s variable, sayxD, which affects the demand without affecting

the supply. The m odel then becom esD

l }

With both p2 = 0 and p3 = 0 or when exo genous variablesxs an d xD do not influencesupply, it is no w possible to obtain observations of the equilibrium p oint at differentvalues of xD. This will allow estimating po and $x of the supply function, asillustrated in Figure 9-2(c).

This example illustrates the fact that for an equation to be identifiable, it isnecessary that a certain relation exists between the endo genous and the ex ogeno usvariab les. In general, it is sufficient to have the vector of param eters for the m od elequations set all independent and orthogonal. In practice, it often suffices that thefollowing relationship between the num ber of variables in an equation be m et: LetEf be the numb er of exogeno us variables in the whole model,E" be the num ber ofendogenous variables, ef be number of exogenous variables left in any modelequation after estimation, and e" b e the number of endogenous v ariables left: thenfor any model equation to be identifiable, the following relationship must hold:

(E '-ef) + (E "- e")>E " -1 . (9-8)W hen it is strict equality, the equation is said to beexactly identified, and when

a strict inequality holds, the equation is said to beover-identified.It is und erstoodthat an under-identified equation is one for which the inequality is reversed. It ispossib le to estimate param eters for exactly identified and over-identified equations,but not for eq uations that are under-identified. In other words, the total numbe r ofexogenou s and end ogenou s estimated parameters, including the intercept, must beequal to or exceed the total numb er of exogeno us variables. This is analogous to thedegree-of-freedom in a regular regression equ ation, wh ich is the numb er of'useful'pieces of information after estimation, or [ (total number of data) - (number ofestimated parameters)] . A regression equation can be estimated only if the nu mb erof useful pieces of information equal to or exceed the number of estimatedparameters, taking into account the 'intercept1 parameter.


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In the examp le used earlier in this section, the following relationsh ips hold . ForEquation (9-5), ' = 0,e'= 0, E"= 2, e" = 2 for both equations. Applying Equ ation(9-8) yields (0 - 0) + (2 - 2) < (2 - 1) which is the same for both equations andsuggests that neither is identifiable. For Equation (9-7), however,Er = 29e'=l9E"

= 2 , e" = 2 for both equations. Again, applying Equation (9-8) yields (2 - 1) +(2 - 2) = (2 - 1), wh ich show s that both equations are exactly identified. The samegoes for Equation (9-6), wh ere (1 - 0) + (2 - 2) = (2 - 1).

W hat should one do if a system of equations is under-identified? The n aturalcou rse of action is to devise a reasonab ly realistic set of identifying assum ptions(Kane 1968). B asically, there are only three ways by wh ich under-identification canbe rem oved. Either (a) we constrain the values of certain structural parameters, (b)w e find enoug h exo genous variables that affect one equation but no t the others, or(c) we constrain the relative behavior of the random -error termsa(a s) or c(aD). This

last alternative effectively removes the error terms from the equations since theremaining constant terms can no w b e incorporated into the respective intercepts a0andp0 .

W e have illustrated methods (a) and (b) adequately in our discussions ab ove.Let us now give equ al time to the error terms. Assum ing the variance of the errorterms c(aD) = 0, this would render the d emand curve identifiable in Equation (9-6)wh en the 'error' term a ^ can now the 'absorbed' as a constant in a0. Similarly,setting o(as) = 0 in Equ ation (9-7) would allow the supply curve to be identified.These assumptions w ould adequ ately portray the data having been g enerated by one

curve shifting along a second roughly stationary one, as explained above. Becauseerror terms arise as a result of a comp lex of unknow n forces, however, assumptionsregarding the properties of a are peculiarly hard to defend. At best, one's choice ofadditional exogenous variables is apt to be only slightly less arbitrary. Nor is itpossible to guarantee in adv ance that the cho sen exog enous variables will, in fact,have the hoped-for properties. Consequently, most econometricians prefer toconstrain the structure of their model. Here at least, a priori reasoning can play alegitimate supporting role. Of the many restrictions one might impose on thestructural coefficients, these represent the mo st frequently encou ntered varieties: (a)set a parameter to zero; (b) two coefficients are set equal; (c) coefficients areestimated outside the m odel.

9.1.5.3 SPATIAL IDENTIFICATION. For proper identification of the structureof spatial dependence,an analysis o f spatial association is typically carried ou t byrelating a variable to its spatially lagged counterpart. The latter is constructed as aweighted sum (typically a linear combination) of the observations in the system, asalluded to above. The association is indicated by a correlation or regressioncoefficient. For example, a variable x would be related to W(/)x, where Q> is aspatial autoregressive coefficient matrix and W(/ ) is the spatial weig ht matrix for the/th contiguity class. Clearly a different choice of W(/ ) will result in a different >,and therefore the measure of spatial association is indeterminate. Drawing ourexperience from image processing , however, the model specification should looklike


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LocationjTmnsport & Land-use 571

x2(l)9/^2(7+1)

0 .

0 .

9 1

0

0

9 2

9 1

0

q>2 .

0 .

9 / 0 .

9/ 0

9/

0 .

0 .

0 .

V ( 0

Here the (/+l)th entries in each row include the subject spatial-unititself. Noticethat there are only / calibrated au toregressive-coefficients, with the coefficient for

the subject unit being unity by definition. Conceptually, we are expanding, the /-entry ve ctorin g =ZJ, qw-,-*,-We have seen examples of this in the spatial filters discussed in Section 9.1.4.Clearly, a choice of different w eightsw tJ is likely to result in a different estim ate forq>, as mentioned previously. Interestingly enou gh, Anselin (1988 ) conjectured there

may be some relationship between spatial weights and technical coefficients of aninput-output model. Measures of interconnectedness that are based on suchtechnical coefficients would be equally applicable to spatial weights. I share this


15/53


stipulation in light of how we use gravitational interaction to estimate technicalcoefficients in the "Spatial Equilibrium" chapter.

The seeming indeterminacy ofcp is mostly a problem in exploratory dataanalysis, since there is insufficient structure in the data as such to derive the prop er

spatial mod el. In the mod el-driven approach that is takenhere, a priori (theoretical)reason s dictate the particular form for the identification constraints. This is similarto the approach taken for systems of simultaneous econ ometric equations,as we w illshow. Com peting specifications can subsequently be comp ared by means of mo delspecification tests and model selection procedures,

9.2 A taxonom y of spatial econom etric models

In classifying spatial econometric models, a fundamental distinction is made

between a simultaneousand a conditionalspatial process. They have a significantimpact on estimation and testing strategies, as we will see later. The conditionalmodel, as the nam e implies, is based on a conditional probab ility specification,whereas the simultaneous m odel is expressed as a joint pro bability. Both mod els aremost often presented in anautoregressiveform, i.e., where the value of a variableat one point is related to its values in the rest of the spatial system (A nselin 1988).

9,2.1 Simultaneous vs. conditional model.The simultaneous model isconstructed based on a stochastic process (or "random fields") on a regular lattice

in the form of a stochastic difference equation. In the following equationsj9 k- 1 , 0, 1, ... are spatial indices, Z is the variable under consideration, a areparameters to be estimated, anda is the independent, no rmally-distributed randomvariables:

m n

E E *m-ltn-uZj+lMu= V V /'* ( 9 " 1( ) )1=0 u=0

For m =1, n =\J =-1 ,0 ; A; = -1 ,0 ; for example, we can write:

a i , i Z - i , - i + a i ,o Z -i,o + a o, i Z o,-i + a o,o Z o,o = ^-i,-i (7 = - l , * = - l )a U Z - i , o + a i , o Z -u + a o , i Z o ,o + a o ,o Z o , i = ^- i , o O ' = - l , * = 0 )

a i 5 iZ o,-i + a i,o Z o,o + a o,i Z i , - i + a o,o Z i ,o = a o,-i (7 = 0 , * = - l ) ( y " '

a i ,i Z o,o + a i 5 oZ o, i + a o, i Z i 5 o

+ a o , o Z U= ^ o , o O ' = 0 , * = 0 ) .

Notice that Z is used from this point on as a more general notation thanx toaccentuate spatial data. Refer to Figure 9-3(a), where the equations are illustrated.Let us now re-label the four grid points as / = 1,2, 3 ,4 as shown inFigure 9-3(b)where7= - 1 1 k= - 1 is equivalencedto i = \J = - 1 /k= Oto / = 2J = 0 /k= - 1to / = 3, an dj = 0 /k = 0 to / = 4. In matrix notation, this is equivalent to a first-order spatial autoregressive mode l: Z =


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Location}Tmnsport & Land-use 573

(a) Spatial indexy, k

(-1,1) (0.1)-

y=- l , =0

0,1)

y=0, k= 0

(-1,0) (d ,b ):r :z :z (1 JO)

(-1 -1)-

i

i !i !

-(0-1) (1,-1)

(a) Spatial index i

1 3

- 1

Figure 9-3 Simultaneous spatial econometric model on a lattice

wh ere equivalence has been established betweena tj and the {< pi9wv}. One can nowwrite Equation (9-12) out in more detail:

Z2=


17/53


(l-(plwll)Zl-q> lwnZ2-(plwl3Z3-(plwHZ4=a l

C 1"" ^ 2 ^ 2 2 )Z 2 " < P 2W2 3 Z3-


18/53


autoregressive model. Suffice to say here that when only cross-sectional data areavailable, an autoregressive mod el is the methodology to follow. W hen bo th cross-sectional and time-series information are available, however, spatial time-seriesmay represent an attractive alternative to analyzing and forecasting land-use

patterns.

9.2.2 Spatial linear regression models for cross-section data. The modelspecification here pertains to the situation where observations are available for across-section of spatial units at one point in time. Consonant with the thrust in thischapter, the general modelling framework is provided first, from which specificmo dels can b e obtained and illustrated. Th is is done by imposing certain constraintson the param eters of the general formulation. The general mod el consists of a setof simultaneous equations, including an equation on the error term:

,z+ZB + aa = T W 2 a + e .

In this specification, p is&Kx 1 vector of parameters associated with exogen ous(i.e., not lagged-dependent) variables Z, which is represented in ann*Kmatrix.This is to be differentiated from the spatially-lagged dependent variable W ^ .Q>is the coefficient matrix of the spatially-lagged dependent variable, and y is thecoefficient matrix in a spatial autoregressive structure for the disturbance a. The

error vector 8 is taken to be normally distributed withE~/V(0 , ) , and the diagonalelements of the error-covariance matrix E = [covfoey)] asEu = hj(aTy) (h{ > 0). Itcomes with a general diagonal covariance matrix as mentioned. The diagonalelements allow for heteroscedasticity6 as a function ofm'+l exogenous variables y ,including a constant term. Them -parameter a are associated with the non-constantterms such that fora = 0, it follows thath = a 1 the classic hom oscedastic situationwith a constant variance.

The two n*n matrices Wj and W2 can be standardized or unstandardizedspatial-weight matrices, i.e., they can be normalized as fractions that sum to one or

otherwise. They are associated respectively with a spatial autoregressive-process inthe dependent variable and in the disturbance term. This allows for the twoprocesses to be driven by a different spatial structure. For exam ple, the reg ularvariable m ay hav e spatial dependence on a spatial order defined for the8-connectedneighbors while the disturbance term for the 4-connected neighbors. In all, themodel has 3n+K+m' unknown parameters, which are in vector form:

J , y y , a r (9-17)

6As mentioned in the "Statistical Tools" appendix to Chan (2000), residuals to a regression ishomoscedastic if the error is normally distributed with a constant variance around the regression line.Heteroscedasticity is defined as the opposite phenomenon when this property is not found.


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Several familiar spatial model structures result when subvectors of the parametervector (9-17) are setto zero. Specifically, the following situations correspond to thefour traditional spatial autoregressive-models discussed in the literature:

a. W hen = 0, a = 0, or fixing m'+l parameters, results in the linear-regression m odel w ith a spatial au toregressive-disturbance:

z = Z p + ( I - y W 2 r1 e. (9-20)

d. For a = 0 or fixingm 1 parameters, the mixed-regressive-spatial-autoregressivemodel with a spatial-autoregressive disturbance is obtained:

Z=O>W1z+Zp+(/-yW2) ~xz (9-21)

which is the full model specification in Equation (9-15).

Four more specifications are obtained by allowing heteroscedasticity of a

specific form (i.e., a specific h (ar

y ) in the models(9-17) through (9-21).) H owever,our taxonom y above focuses primarily on the specification of spatial d ependenceand cross-sectional data. The introduction of the time dimension considerablyincreases the comp lexity of issues that canbe taken into accoun t in the specificationof spatial econometric-models. We will defer to the "Spatial Time Series" chapterfor handling these complexities. For the time being, we will simply illustratethrough a case study of applying these cross-sectional mod els in real-wo rld lan d-useforecasting. We will concentrate on the case of the mixed-regressive-spatial-autoregressive mod el of Equation (9-19), wh ich has been im plemented in a widely

disseminated model: EMPIRIC. We will also show how one gets around thelimitation of a cross-sectional model by a finite difference equation set, whichintroduces a small dose of time-series information when the difference is takenbetween two cross-sectional data sets.


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93 The EMPIRIC mode!

EMPIRIC is a linear and simplified backup version of the original PolymetricM ode l that consists of a set of non linear finite-difference equation s. The Polym etric

M odel, developed in the early 1960's by the Traffic Research Corporation (now partof Kates, Peat, Marwick & Co.), was discarded because it placed too muchrequirement on the usernot only in terms of resources but also his/her apprecia-tion for technical intricacies. EM PIRIC b ecame a widely used m odel in transporta-tion planning partly because of its simplicity and partly because of its continualdevelopm ent and maintenance by K ates, Peat, Marw ick &Co. (KPM & Co.). It wassubsequently incorporated in the U. S. Federal Highway Administration's UrbanTransportation Planning Package during the early 1970's and is therefore in thepublic domain.

The mo del is discussed for several reasons. First, it is a demonstrable exam pleof an empirical, econometric model (rather than a structured, 'theory-based' modelsuch as Lowry), using statistical correlations established among variables.7 In itsbasic version, EMPIRIC is designed as a straightforward projection of futureactivities and landuse, without any pretense of underlying theoretical construct thatexplains the developm ent process. Second, it represents one of the mo st widely u sedland-use models, and as such, serves as an example of a 'successful' modellingeffort. It is possibly the only model that had been applied across at least sevenmetropolitan areas in the United States and Canada. The question is: does its

success have an ything to d o with its simple, empirically-based m odel structure orthere other intangibles involved? Third, there is a pedagogic reason for discussingthe EM PIRIC m odel. It illustrates how spatial econom etrics play a role in the realwo rld. Th e discussion here presents the mod el no t only in terms of spatial statisticsbut also the general econo metric literature.

9.3.1 A n example of EM PIRIC model. Let us describe the EMPIRIC modelin greater details through an example. Following the arrow diagram techniquepresented in the chapter on "Descriptive Tools" (Chan 2000), the structural

relationship between the identified variablespopulation, white- vs. blue-collaremploym ent, and accessibility isshownin Figure 9-4. It is portrayed, for exam ple,that the population in the forecast period is not only affected by the base-yearpopulation, but also employment in the forecast year. We call population thedependent (endogenous) variable and the rest independent variables, and anequation can be written to realize this relationship:

(forecast pop) =a (base-yr pop) +b (forecast white-collar emp) +c (forecast-yr access)

7 See chapter on "Descriptive Techniques" in Chan (2000) for a comparison between theory-based andempirical models.


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Forecast Time-Period Base Time-Period

...T Population < Population

/ """White-Collar Em ployment White-Collar Em ploym entI\ Blue-Collar Employment Blue-Collar Em ployment

Accessibility

Legend

Primary Order

Secondary Order

Tertiary Order

Figure 9-4 Arrow diagram for EMPIRIC

where a, b, and c are calibration coefficients. As noted in the chapter on "Descrip-tive Too ls," the relationship and hence the equation is merely a postulation by themodel builder. The validity of the postulation can only be established by statisticalverification and validation using available data.

After the arrow diagram is completed in full, a final set of equations can beconstructed as shown in Equation (9-22) below . No tice the intertwining relationshipbetween the dependent and independent variables. A dependent variable such asforecast population (that appears in the left-hand-side in the first equation) servesas an independent v ariable in the second equation (on the right-hand-side), w hileaccessibility remains an independent variable throughout. It is through such asimultaneous equation formulation that the interdependence of the various factorsis mo delled. T his is an example of a simultaneous econom etric mod el (rather thana conditional mo del.) W e may also like to contrast this with our experience withthe Low ry M odel, where the interrelationship between the population (household)sector and the basic and service employment sectors is modelled in a sequential(rather than a simultaneous) fashion.

W e show below the same three equations corresponding to the three depend entvariables respectively: forecast population, white-collar, and blue-collar employ-ment. Such equations remain the same for every subareal unit (such as a trafficzone) and each forecast period:

AN = 0. 3234 A^ - 0.0064 iV+1.9258A^A*F=0.4166AN-0.0061FF+0.9640Aw (9-22)AB "= 0.1562 A N - 0.01 30B "+ 0.9993 Au


22/53

Location^Tmnsport & Land-use 579

wh ere AN is the change-in-subareal-share-of-total-population,AW isthe change insubareal-share-of-white-collar-employment, andAB" is the change-in-subareal-share-of-blue-collar-employment. N is the base-year-subareal-share-of-total-population, Wis the base-year-subareal-share-of-white-collar-employm ent, an dB "

is the base-year-subareal-share-of-blue-collar-employment.At' is the change-in-subareal-share-of-transportation-accessibility-to-employment, andAu is the chan ge-in-subareal-share-of-accessibility-to-population. No tice this is in the form of amixed-regressive-spatial-autoregressive mo del of Eq uation(9-15), or Equation (9 -19).

To see this, one needs only to recognize these variable definitions:zT = (AN,

and p, = (-0.006 4,-0.00 61,-0.0 130 f where zp =

0.3234 0 0 |

o 04166 o |. A s all ud ed to alr ea dy , a co m m o n0 0 0.1562J

AW, AB"),

z , p , + z2p2.

z, =

w,=

N0

0

0

I1

0 0W 0

0 B

1 0

0 0 ,0 0

factor is specified o n top of other explanatory variab les, with z2 =tat'

0 0 AM

and p2

= (1.9258 , 0.9640, 0.9993)7. We will specify in more detail the weight matrixWland the observation matrix Z in sequel. Meanwhile, let us define the concept of*change-in-subareal-shares. Alternatively, one can view E MP IRIC as a special caseof Equation (9-21) when p is a single vector of (-0.0 06 4,-0.0061, -0 .0130)r an d

1.9258 0 0

where [I- yW2] = o o.964o o and e = (At Au Aiif.

9.3.2 The concept of shares.Notice the variables in the equations do notrepresent the absolute values of the subareal population, employment andaccessibility. Instead, they are shares of regional totals, where the 'shares' of eachactivity for each sub area is com putedas the ratio of the activity-level for the subareato the regional total. Thus if the calibration-year(t - At)population in subareai is22,000 out of the regional total of 2,050,000, for exam ple, the share of that subareais 22,000 /2,050,000 or 0.107. T he difference betw een the shares so comp uted for

two p oints in time (say between calibration yeart-At and base year t) is then takenas the change-in-share for the subarea of interest. Thus if subarea-f s share inpopu lation is 0.140 in the base year, the change over the two time-periods is 0.140- 0.107 = 0 .033 , wh ich indicates that a high er percent of regional population islocated in the subarea under discussion over the time-period A.

The policy measure such as accessibility is defined similarly as changes inshare in the equations. Let us say the accessibility to em ployment from subareai istt(b ) in the base year and tt(c ) in the calibration year and the total regionalaccessibility is t(b) an d t(c) respectively. Subarea-fs shares of accessibility are

tj(b)/t(b) an d t t(c)l t(c) correspondingly. Therefore,

At'=^t(b) t(c)

(9-23)


23/53

580 9. Spatial Econom etvic Mod els

There are several reasons wh y subareal shares, rather than absolute nu m bers,are used. First, they reflect the 'competitive' subareal component relative to theregion , show ing the relative concentrations ofactivities. Second, the numbers arebounded between zero and one, thus easing the calibration process, in that the

coefficients are prevented from being too large or too small because of the valuesof the variab les. In this dimensionless form, the calibrated coefficients can also becompared to the calibration experiences in another city more readily since anorm alized form is used in both cities.

Changes in shares, instead of absolute numbers, are used again for goodreason. In order to perform projection incrementally as in EMPIRIC, it is morestatistically meaningful to relate variables such as population-growth to theimprovement-in-accessibility, both of which are quantified in terms of changes.Thus if one forecasts from calibration-year(t-At) to base-year (f) to forecast-year

(t+At) to future-year (t+2At\ these time periods are summarized into threeAtincrements in the model: calibration-to-base years, base-to-forecast years, andforecast-to-future years.

In general, the concept of 'share' fits in well within theallocation function ofan activity-allocationmodel,wh ile the concept of incremental change is cong ruousto theprojection function of land-use models. The future allocations are translatedback into absolute population and employment levels by the following simpleprocedu re. The subareal shares are multiplied by the exogenou sly specified controltotals. The forecast increments of subareal growth or decline are then added to theobserve d initial values to obtain the final figure of interest. The EM PIRIC mo delis a good examp le of thisshift-share analysisthat was described previously in thechapter on "Economic Concepts" (Chan 2000).

Example

Suppose the following equations hav e been calibrated for a three-zone city:

where E stands for em ploym ent. Part of the hypothetical data base used for this calibrationis shown in the columns marked by "projection year" and "base year" inTable 9-1.

Using the se given numerical values fromTable 9-1, the two unknow nsAN and AE canbe solved by the two equations:M=ofoAN+ + l'% (-oooso) The results areAW = -0 .00617andAE = -0.00732, which are entered in the 'forecast' columns of the table. (It will be a goodexercise for the readers to fill in the m issing entries of this table.)

To transform these shares into absolute activity levels, we use the following formula:

(forecast activity) = [(change in share) + (base-yr share)](forecast-yr control-total).

Thus(forecast pop) = [-0.00733 + 0.387 2](l 16,230) = 44,148

(forecast emp) = [-0.00617 + 0.3671](42 300) = 15,267.

Both numbers are included under the forecast columns inTable 9-1.


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Projection year Access to pop Access to emp Base year Forecast

c,,u Access to Access to d i n ere ChnveS u b " D OO e m o Base Chnge Base Chnge _

U l m g e U l m g e Zonal Zonalarea PP e m P Pop Emp in inyr in yr ni T , T , pop emp

T , , T , , Level Level pop emp T . T ,Level Share Level ShareL e v e l s h a r e L e v e l s h a r e share share L e v e l L e v e l

1 2 4 , 0 1 6 0 . 3 0 9 6 1 1 , 9 33 0 . 2 31 2 4 , 3 1 1 - 0 . 0 0 2 6 1 2 , 1 3 5 - 0 . 0 0 3 0 3 8 , 6 0 0 8 , 7 0 0 - 0 . 0 7 3 3 - 0 . 0 0 6 1 7 4 4 , 1 4 8 1 5 , 2 6 7

2 2 7 , 7 9 3 1 8 , 1 8 8 2 7 , 7 9 3 1 8 , 2 0 3 4 3 , 2 0 0 9 , 6 0 0

3 2 5 , 7 7 1 2 0 , 2 1 3 2 5 , 7 7 1 2 0 , 2 1 3 1 7 , 9 00 5 , 4 0 0

R e -, 7 7 , 5 8 0 1.00 5 0 , 3 3 4 1 .0 0 7 7 , 8 7 5 0 5 0 , 5 5 1 0 9 9 , 7 0 0 2 3 , 7 0 0 0 0 1 1 6 , 2 3 0 4 2 , 3 0 0

t o t a l

Table 9-1 Partial data base for hypothetical city

9.4 Forecast sequence

We recall that the Lowry Model uses cross-sectional data for its calibration; andforecasting is performed only for a single time period into the future. E M PIR IC, on

the other hand, is one of the few models that operates on cross-sectional dataorganized in two points in time. Consonant with census updates in the U.S., theEMPIRIC forecasting periodAt is in five- or ten-year increments. The model isapp lied typically for several forecasting incrementsfor example from he base-yeart to forecast-year t+At, from forecast-year t+At to future year t+2Atand so on. Theoutput of EMPIRIC'S first period forecast is input for the second etc. For the initialforecast increment, the model is invoked by inputting this information:

a. Calib ration- and base-year activity and land-use by subarea (such as a trafficzone), corresponding to time periodst-At and t;

b . Planning policy that represents changes in the freeway system, sewag efacilities and zoning ordinances and the like during the time incrementAt;

c. Region al-control-totals of population and employm ent for the base-periodt.Notice that the activity- and land-use data specified in (a) is generated by themodel for each of the future time-increments once the information for theoriginal period (between calibration-year and base-year) is assembled. Thepolicy and regional control-total projections, however, have to be providedoutside the model for each forecast incremen t.Figure 9-5 illustrates thecalibration and forecasting steps of EMPIRIC.

Referring to Figure 9-5, subareal activity- and land-use data are the firstcategory of input into the EMPIRIC model. Activity data consist of demographicinformation such as the number of families/households, population for eachsubarea, and similar employment data. Land-use data, on the other hand, is usually


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582 9. Spatial Econom etric Mod els

V Forecast >\ Possible Forecast >

ProjectedPlanning

Policy

Calibration-YearActivity and

Land-Use

t-At

ProjectedRegional

Control-Total

ProjectedPlanning

Policy

(MODEL)*

Base YearActivityand

Land-Use

t

ProjectedRegional

Control-Total

CTalifrrat ion

ForecastActivityand

Land-Use

t + At t + 2At

Fig ure 9-5 Block diagram of forecasting procedure

expressed in terms of acreage by type-of-use, again disaggregatedby subarea.Output o f the EM PIRIC m odel again includes subareal activity and land-use data,wh ich are in turn used as input to the next forecast increment of the m odel.

EMPIRIC recognizes thatthe study area cannot be treated in isolation, theinteraction w ith the neighbo ring areas hasto be modelled. An external subarea isused to reflect the influence of the 'outside' (external world). The data used forinput are labelled external householdand employment. Regional activityis yetanother category of input into EMPIRIC.The activity data consist of regionalcontrol totals of popu lation and emp loyment. Su ch regional-data input is requiredfor each forecastperiod. They are allocated amon g the subareas by the mod el everytime. Finally, policy-planning changesare input to the EMPIRIC model. Directpolicy changes include infrastructure such as transportation and utilities (sewage,water etc). Indirect yet im portant p olicy chang es such as density control and landavailability are also used as inputs into the EMPIRIC model. Again,the policyvariables have to be specified for each forecast period. Table9-2 summarizes thevarious inputs and ou tputs to the model.It is to be noted that activity/land-use dataare needed for two periods o f timethe base-year and calibration-yearfor mod elcalibration.

9.5 M odel calibration

The technique used to calibrate the EM PIRIC m odel involves the determination ofregression coefficientvalues. As m entioned previously, these values are calculatedusing data from two h istoric time-periods, simply because EM PIRICis a difference-equation model. The regression coefficients so obtained are then verified byobserving how well the calibrated mod el can reproduce the forecast-year data. Such


26/53

Location,Tmnsport & Land-use 583

Input Output

Activity/LandUse

Subareal

INTERNAL demographic

N o. of familiesN o. of householdsPopulation

employmentN o. of employee

land-useAcreages by use

Regional

EXTERNAL

total household total employment

demographic employment

v

Policy

DIRECT

transportation utilities

(e.g. water, sewer)

INDIRECT

density control /aJ availability

Table 9-2 EMPIRIC input-output information

a verification is facilitated by the FORC ST block of the program , which 'forecasts'the base-year / activity distribution and land use from the calibration-year -A/. Theforecasted data are subsequently verified b y the "reliability testing" block .

An examination of the regression package used in the model calibrationprocess reveals that it consists basically of five programs. The programs are: (i)Graphing block, (ii) Data correlation block, (iii) Factor analysis block, (iv)Regression block, and (v) Reliability testing block. While the functions of theseprograms may be quite apparent from their names, some discussion on the


27/53

58 4 9. Spatial Econom etric Models

regression b lock is necessary. The regression b lock is the program that estimatesthe mod el coefficients. Included in the estimation procedure are these com ponen ts:

a. Ord inary least squa res: a standard least-squares regression pack age, used in theinitial stages of calibration and in the construction of independent sub-modelsoperating on the output of the simultaneous-equations model. (More will besaid about these sub-models later.)

b. Two-stage least-squares: a standard, 'simultaneous'-equationsregression--package, used to estimate the coefficients in the basic EMP IRIC m odel u singthe sequential, iterative method described in the chapter on "Descriptive Toolsfor Analysis."

co Maximum likelihood: same as for above, but used in unbiased estimators,again as described in the "Descriptive analysis" chapter for an introduction tothis techniqu e.

d. Simultaneous least squares: same as for (b) and (c) above, but based on themethod of indirect least squares for special 'triangular* form of the simulta-neou s equation set. (See "Descriptive T ools" chapter in Chan (2000).)

Item (a) is used in variable definition, as inpu t to calibration and in sub-m odeldevelopmen t. O ne of the other sub-programs (b) through (d) is used to calibrate thecoefficients of the simultaneous-equation model. The output of each of theseprog ram s is a set of coefficients, together w ith associated goodness-of-fit statisticssuch as the co efficient-of-multiple-determinationR2, th e t and F statistics. Together

all the five programs (i) through (v) formalize the variables, estimate thecoefficients, and re-estimate coefficients using input from the factor-analysis-program block and the initial stages ofcalibration. In the calibration proc ess, two-stage least squares is the main tool employed, rather than indirect least squares, asthe reader can imagine.

As an exam ple, take the three-equation m odel presented earlier:

AN = a lAW+blN+clAt/

AW = a 2AN+ b 2 W+ c 2Au (9-24)

AB//

=a 3AN+b3B//

+c3Au.Using the calibration and base-year data sets, the coefficients of a,b, an d c areestimated through regression calibration. It follows the procedure outlined inthe"Descriptive Tools" chapter8 and the "Statistical Tools" append ix in Chan(2000). The estimated coefficients (shown above in Equation (9-22)) are thenstatistically evaluated and tested for their significance. Th ese coefficients can sho weither positive or neg ative relationships among the variables, wh ich either agree ordisagree with intuition. Further evaluation of the coefficients is done by forecastingthe distribution of the three dependent variables for the base-yeart based on the

8For a numerical example showing how two-stage least squares is performed step-by-step, see the"Calibration" section of the "Descriptive Tools" chapter (Chan 2000).


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calibration-year datat-At. The com mitted transportation system for the b ase yearis used as policy inpu t. The results o f the 'forecast' are then statistically com paredwith the observed data for the base-yeart. Through an iterative process suchas two-stage least-squares, the coefficients are fine tuned un til they replicate base-year data

satisfactorily.It should also be no ted that, strictly speaking, the calibration of the m ode l has

to be performed for each study area that is being applied. However, there are newresearch efforts to transfer calibration results from one city to another. Thecalibration process is a major part of the modelling effort and requires a largequan tity of data to be used. B eing able to transfer calibration results from a similarcity will save a great deal of time and money. Putman (1979) tabulated theregression coefficients of EMPIRIC studies performed in six cities in the Uniteddisagree with intuition. Further evaluation of the coefficients is done by forecasting

the distribution of the three dependent variables for the base-yeart based on thecalibration-year datat-At. The com mitted transportation system for the base yearis used as policy inpu t. The results of the 'forecast' are then statistically com paredwith the observed data for the base-yeart. Through an iterative process suchas two-stage least-squares, the coefficients are fine tuned u ntil they replicate base-year datasatisfactorily.

It should also be noted that, strictly speaking, the calibration of the model hasto be performed for each study area that is being app lied. How ever, there are newresearch efforts to transfer calibration results from one city to another. The

calibration process is a major part of the modelling effort and requires a largequan tity of data to be used. Being able to transfer calibration results from a similarcity will save a great deal of time and money. Putman (1979) tabulated theregression coefficients of EMPIRIC studies performed in six cities in the UnitedStates. A fair amount of consistency is found among the population equations inTable 9-3. Althou gh there are occasional exceptions both in signs and magnitud e,some interesting patterns emerge. The change-in-share of a region's totalpopu lation found in each subarea mo ves with the change-in-share of the adjacentpopu lation class. For exam ple, lower income mo ves with change-in-share o f lowerincome, and lower middle income moves with change-in-shares of lower incomeand upper midd le income and so on. On the other hand, change-in-share mov es inopposition to its own concentration in the base year and m oves w ith concentrationsof the next higher incom egroup. In other words, changes in subareal shares of eachincome group move with changes in shares of the next higher and next lowerincome group, and away from concentrations of their own income group towardconcentrations of the next higher incom e group .

Pu tman pointed o ut that the patterns found in these coefficients are quiteconsistent with hypo theses regarding peop les' desires for increased socioeconom icstatus. They are also consistent with hypo theses regarding peo ples' unw illingnessto live among groups very different from their current economic status. Thecoefficients of other variables in the population equations as well as those of theemployment equations do not exhibit any where the same degree of uniformity.They point to the site-specific nature of the calibration process. Counterintuitive


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586 P. Spatial Econom etric Mod els

Dependent

variable

Change

lowincome

population

Changein share

low-middleincome

population

Changein share

upper-middle

incomepopulation

Changein ^linrc

upperincome

population

Low

-0.119(I )2

+0.194to

+0.53(5)

-0.125to

-0.14(2)

-0.42

to-0.507(2)

Population by income (independent variable)

Change in share

Lowermiddle

+0.129to

+0.637(6)

N.A3-

+0.434to

+0.658(6)

-0.282

to+0.685(3)

Uppermiddle

-0.295to

-0.367(2)

+0.307to

+0.781(6)

N.A.

+0.504

to+0.83(4)

Upper

-0.281to

-0.39(2)

N.A.

-0 .16 (1 )

N.A.

Low

-0.199to

+0.133(5)

N.A.

-0.16(1)

N.A.

Base year share

Lowermiddle

+0.294to

+0.36(3)

-0.054to

-0.353(4)

N.A.

N.A.

Uppermiddle

-0.109to

+0.258(2)

-0.334to

+0.10(3)

-0.219to

-0.27(3)

+0.219(1)

Upper

XT AJN.A.

N . A .

- 0 . 1 5 5to

+ 0 . 11 3(2)

- 0 . 2 7 8

to- 0 . 4 8 1

(4 )

1 The six calibrations include Atlanta, Denver, Washington DC, Twin Cities Puget Sound and Boston.2 The num ber of data points are included in parenthesis. For example, (3) means three coefficients are included in therange of numbers citied.3 N.A . stands for not applicable, meaning n o coefficients were specified in the structural equation to be calibrated.

Table 9-3 Popu lation coefficients in six EM PIRIC calibrations1

signs of the coefficients may even suggest spurious correlations that may work inshort-term forecasts, but are not suitable for long-run applications.

9.6 Definition of accessibility and developab ility

A really key item in each land-use mo delis the way accessibility and developab ilitymeasures are quantified, since they really make up the pivotal concepts in activityallocation and forecasting. T hroug hou t this boo k, accessibility has been recogn izedas one of the determinants of facility location and land development. It lies at theheart of spatial price theory. Developability, on the other hand, reflects thedevelopm ent potential of a subarea from such considerations as zoning ordinancesand the economic and non-economic opportunities that exist. Much of regional-


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Location,Transport & Land-use 587

science research has b een focusing on the capacity of a subarea in taking o n newactivities.

9.6.1 Accessibility. Instead of presenting the equation for accessibility

directly, let us try to derive it. First, we recall that accessibility is a meaningfulm easure only w hen applied toward a particular type of activities and with respectto a particular location in the study area. For example,, the accessibility toemploym ent opportunities is defined for a specific residential zone, and likewise theaccessibility to shopping can also be defined for the same zone. This reflects thatlocations accessible to work may be different from those to shops, and theaccessibility to shops may be different from home versus the office.

Accessibility is related to travel time. In general shorter travel times meanbetter accessibility or proximity, and vice versa. Finally, accessibility for a suba rea

has to reflect travel not o nly to one w ork or shop location, but also other possiblelocations. In this way it summ arizes the proximityto job and shop opportunities ofthe region as an aggregate. This is normally taken into account by taking aweighted average of all possible proximities to these opportunities. Emp loymen t o rshopping activities in the target subareas are often used as weights. Sinceemploym ent or shopp ing activities are often m anifested in terms of work or shoptrips, trip vo lumes are often used as weights in the place of actual num ber o f job sor retail floor space. Such a concept of accessibility has been introduced earlier inthis book including in the "Generation, Competition and Distribution" chapter,where state parks are located vis-a-vis accessibility to the local population.

An equation that takes all the abo ve factors into account is given below for astudy area consisting ofri subareas, tracts, zones or districts:

(9-25)uk=}2ng=iNgQx^-auTgky

Here T is the trav el time, and the interaction propensity a is corresponding to theX-value of the accessibility-opportunity model, representing the probability oflocation (see chapters on "Economic Methods" and "Descriptive Tools" in Chan(2000)). The exponential-function formulation considers both the attractiveness ofdestination and its proximity, since it discounts the interaction with subareas faraway from h It should be noted that the above accessibility definitions aresomewhat different from that used in the Lowry model. Instead of distributingpopulation and employment, the econometric model here relates the location ofpopulation and employmentto a num ber of factors, including proximity to job s andprox imity to the labor market respectively. The difference relates to the fundamen-tal conceptual distinction between the two m odelling approach es. We highlight thisdifference by the indexg-k instead ofk-g, although the two are equivalent so longas travel time and cost betweeng and k are symmetrical.

For ex amp le, in the hypo thetical case study to be discussed later in this chapter,the accessibility to employm ent for the calibration period is computed for all the 10districts. The numerical value ranges from 154 807 for district 3 to 31 039 for


31/53


district 6, showing that the proximity to work locations in the Boston area is bestif one lives in district 3 and worst in district 8. Notice the numerical value foraccessibility is purely an index and best thought of as simply a scale. (For thisreason, w e norm alize accessibility eventually between0 and 1 as described earlier.)

The accessibility definition adopted in Equation (9-25) can be used to co mp utethe spatial weights w tj in the mixed-regressive-spatial-autoregressive modelofEquation (9-15).One can readily see the connection between accessibilityand thespatially-lagged variable aliasfor spatial weightsin Equation (9-4).In brief, w fj= fitjj.a) = exp(-ar^), and the spatially-lagged variable is Lxt = EjW^- = Xi, =

Sj x7exp (-ar^). T he change-in-accessibility matrixAtf 0 0

A H 0

0 AM

as defined by Equ ation

(9-23) is simply the common-factor specification Z2. Notice only a single class ofspatial contiguity (/ = 1) is involved here, and the spatially-lagged variable isexactly the definition of accessibility, with the activity variablex being employmentE or popu lation TV, depend ing on whether accessibility-to-workit*) or accessibility-to-shop (u) is considered. Remember these accessibilitiesare used to locateresidence and wh ite/blue collar emp loyment respectivelyin the EMPIRIC Modelof Equation (9-22).To be noted also is that the spatial-lag is normalized to be 0-1ranged, as alluded to already.In sum, the norm alized commo n-factor-specificationterm Z2p2 is simply

K b) Kc)

0

0

u(b) u{c)

u,(b) Uj {c)

u(b) u{c)

1.9258

0.9640

0.9993

(9-26)

where Z is now a matrix consisting of the com bination of three weights correspond-ing to activity variablesE, N, N.

9.6.2 Developability.The other key conceptthe developm ent opp ortunityfora particular activity typeis defined in a rather ad-hoc fashion. It essentiallyassumes that a subarea's development potential (called developability)is related tothe amount of developable landand the fraction of the total gross area devotedtothe activity o f interest .Thisis accomplished in the following manner:

(9-27)

where A", A* represents thenet area for housing and service-industry developm entin subareay respectively,Aj is the gross area, and A.= A.-A^-Af-Af-Aj1 is thedevelopable area after accountingfor the undevelopable and used areas. Here theused area includesall the functional purposes suchas residential (H), manufactur-


32/53


ing (B) and service(R). EM PIRIC normally assumes that undevelop able areas havebeen accounted for exogenously by redefining gross area asAj9 which alreadyexcludes the undevelopable area(i.e.,A- = Aj-Aj).

In the nex t section, the reader will see how the capacity or land developability

index for population is computed and accounted for by the first of the two inEquation (9-27), while the index for service industry by the second of the two.Notice there is one serious inadequacy of the land developability definition ingeneralthat an area is not developable unless prior developments have beenlocated there.

9.7 Supportive analysis

The EM PIRIC m odel consists of16 subroutines divided into four groups. The first

group, Data Assembly Programs, prepares basic data and develops historicalrecords. The second g roup, Calibration Program s, computes the various coefficientsin the regression p rogram s, and calibrates the simultaneous eq uations in general (asdescribed p reviously). T he third grou p, Forecasting, substitutes the base-year andpolicy data into the calibrated equations for projecting subregional or subarealactivities. The fourth group, Land Consumption, is an accounting routine used todetermine the am ount of acreage by use from the projected activity levels.

The Data Assembly and Land Consumption programs can be thought of assupportive program s. Under the D ata Assembly Program, one has a variety of utility

programs:a. Data stack block -expand data column -wise for inclusion of new variables,b. Data modification block- for insertion ofa new v alue in a table,c. Subregion stacking block -expand data row-wise for inclusion o f a new

subregion or subarea,d. Town aggregation block- used to combine data row-wise from two or more

subareas into a single area,e. Data combination block- comb ine or transform two or more of the variables

column-w ise into a new variable(s),f. Data difference block - computes changes in subregional orsub-areal

activities,g. Percent and normalization block -converts variable columns to subregional

or subareal shares,h. Accessibility block -computes subregional or subareal accessibilities to an

activity,L Report generation block- prints selected information from input or output

data.

As expected, the Calibration and Forecasting programs form the core of themodel. They were discussed in some detail in sections 9.5 and 9.4 respectively andwill not be elaborated on further here. The initial forecasts of subregional orsubareal population and employment are translated into equivalent changes inclassified small-area land use by means of the fourth, Lan d Consum ption, prog ram.


33/53


Figure 9-6 - Base-year network

This is again carried out using data assembled for the same two points in time. Itaccepts as input the output of the simultaneous equation module, together withdevelopability in each small area, including the range of permissible developm entdensities for each activity. Its output is a detailed updated accounting of land use,broken down by type, within each small area for each forecast year. Where asubregion or subarea is capacitated, the acreage assigned for new development isscaled dow n. This way, the total acreage would b e reduced to exactly the n um berat the base year of the forecast interval plus that becoming available during theinterval due to declines of activities. It can be seen, therefore, that the EMPIRICmo del, besides the basic set of simultaneous equations, is supported by a num berof routines. It can be though t of as a collection o f programs and is really a packag eof econometric and special purpose software adapted for land-use forecasting.Ag ain, the mo del is available from theU.S. Federal Highw ay Adm inistration un dertheir urban transportation plann ing com puter package.

9.8 A hypothetical study

A hypo thetical city is divided into a lattice structure of10 subregions as illustratedin Figure 9-7 (Thompson 1972, Kates, Peat, Marwick & Co. 1971). The area istraversed by a river in the middle. Sub regions 3 and 4 represent dow ntown, wh ilesubregions 2, 7 and 10 represent relatively new and rapidly growing suburbanareas. The assumed base-year highway network at timet shown in Figure 9-6consists of a predominantly radial arterial highway system, supp lemented by onecross-town link and a partially completed inner beltway at the northern part oftown. Th e network includes only two river bridges. Also shown isFigure 9-7, th ecalibration-year network at timet-At. The only difference between the calibration-and base-year networks is that the partially completed innerbelt is built during theelapsed time incrementAt. The study area is reminiscent of Boston, M assachusetts.


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Location }Tmnsport & Lund-use 591

Figure 9-7 - Calibration year network

To simplify the exam ple, the following assumptions are made:

a. No transit system exists or is proposed for the future;b. Urb an grow th d uring th e period between calibration- and base-year wa s

influenced by concurrent improvement in the highway system (i.e., thepartially com pleted beltway);

c. The comm itted network in the forecast year con sists of an inner beltwayaround the city, and including two additional river bridges at where theinnerbelt crosses the river.

A simple three-equation model is to be developed involving three dependentand five independent variables:

Dependent endogenons variablesCalibration-to-base-year-change in subregional-share-of-total-populationCalibration-to-base-year-change in subregional-share-of-white-collar-employment

Calibration-to-base-year-change in subregional-share-of-blue-collar-employmentIndependent exogennons variablesCalibration-year subregional-share-of-total-populationCalibration-year subregional-share-of-white-collar-employmentCalibration-year subregional-share-of-blue-collar-employmentCalibration-to-base-year-change in subregional-share-of-accessibility-to-total-employmentCalibration-to-base-year-change in subregional-share-of-accessibility-to-total-population.

Table 9-4 summarizes the raw data for the ten subregions, with separateaccessibilities for the calibration and base years, where accessibility is defined by


35/53


Sub-region

1

2

3

4

5

67

g

9

10

Total

Population

Calibra-tionyear

220,000

415,000

205,000

190,000

140,000

290,000230,000

g o , o o o

lg5,000

95,000

2,050,000

Baseyear

3g6,000

432,000

179,000

186,000

228,000

390,000

321,000

196,000

256,000

176,000

2,750,000

White CollarEmployment

Calibra-tionyear

22,000

55,000

184,000

91,000

9,000

830,000

22,000

5,000

35,000

4,000

510,000

Baseyear

47,000

56,000

240,000

115,000

30,000

133,000

28,000

26,000

51,000

21,000

747,000

Blue CollarEmployment

Calibra-tionyear

21,000

110,000

40,000

24,000

38,000

7,000

36,000

14,000

18,000

5,000

313,000

Baseyear

35,000

128,000

44,000

22,000

58,000

14,00043,000

25,000

24,000

10,000

403,000

Accessibilityto Population

Calibra-tionyear

139,963

228,669

214,657

196,850

121,945

172,577184,564

75,996

165,163

88,379

1,588,763

Baseyear

171,661

228,669

214,657

196,850

166,945

201,490196,260

111,321

179,105

131,966

1,798,924

Accessibilityto Employment

Calibra-tionyear

69,683

130,501

154,807

137,335

61,288

97,898

110,007

31,039

87,012

36,010

923,760

Baseyear

82,748

138,542

154,807

137,371

71,892

103,817

118,818

42,627

94,967

52,297

997,887

Table 9-4 Raw data for calibration

Equ ation (9-2 5). No tice it is throug h this separation that non -linearity is taken o utof the model. As discussed in Section 9.1.4. this represents a piecewise linearapproximation of the accessibility termXt (or the spatially-lagged variable X(/)x7).The shares and change-in-shares of each activity and policy variable for eachsubregion is computed using a similar formula as shown for accessibility inEquation (9-23). Table 9-5 summarizes the changes-in-shares together with thecalibration-year activity shares.

The three-eq uation m ode l relat

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10. Spatial Eco No Metric Models- The Empirical Experience

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